Reality Properties of Conjugacy Classes in algebraic Groups

Abstract: Let $G$ be an algebraic group defined over a field $k$. We call $g\in G$ {\bf real} if $g$ is conjugate to $g^{-1}$ and $g\in G(k)$ as {\bf $k$-real} if $g$ is real in $G(k)$. An element $g\in G$ is {\bf strongly real} if $\exists h\in G$, $h^{2}=1$ (i.e. $h$ is an {\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two involutions in $G(k)$). For classical groups, with some mild exceptions, over an arbitrary field $k$ of characteristic not 2, we prove that $k$-real semisimple elements are strongly $k$-real. We compute an obstruction to reality and prove some results on reality specific to fields $k$ with $cd(k)\leq 1$. Finally, we prove that in a group $G$ of type $G_2$ over $k$, characteristic of $k$ different from 2 and 3, any real element in $G(k)$ is strongly $k$-real. This extends our results in \cite{st}, on reality for semisimple and unipotent real elements in groups of type $G_2$.


Groebner-Shirshov Bases for Lie Algebras: after A. I. Shirshov

Abstract: In this paper, we review Shirshov's method for free Lie algebras invented by him in 1962 which is now called the Groebner-Shirshov bases theory.


Cohomology of Oriented Tree Diagram Lie Algebras

Abstract: Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this paper, we use Hodge Laplacian to study the cohomology of these Lie algebras. The "total rank conjecture" and "$b_2$-conjecture" for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler-Poincare principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.


Chaining Techniques and their Application to Stochastic Flows

Abstract: We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow.


Abelian Ideals and Cohomology of Symplectic Type

Abstract: For symplectic Lie algebras $\mathfrak{sp}(2n,\mathbb{C})$, denote by $\mathfrak{b}$ and $\mathfrak{n}$ its Borel subalgebra and maximal nilpotent subalgebra, respectively. We construct a relationship between the abelian ideals of $\mathfrak{b}$ and the cohomology of $\mathfrak{n}$ with trivial coefficients. By this relationship, we can enumerate the number of abelian ideals of $\mathfrak{b}$ with certain dimension via the Poincare polynomials of Weyl groups of type $A_{n-1}$ and $C_n$.


Singular cobordism categories

Abstract: Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the main theorem of [7] to the case of cobordism categories of embedded d-dimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov h-principle.


k-Wise Independent Random Graphs

Abstract: We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling phenomena where only k-wise independence is assumed to hold, and is also useful when the relevant graphs are so huge that handling G(N,p) graphs becomes infeasible, and cheaper random-looking distributions (such as k-wise independent ones) must be used instead. Unfortunately, many well-known properties of random graphs in G(N,p) are global, and it is thus not clear if they are guaranteed to hold in the k-wise independent case. We explore the properties of k-wise independent graphs by providing upper-bounds and lower-bounds on the amount of independence, k, required for maintaining the main properties of G(N,p) graphs: connectivity, Hamiltonicity, the connectivity-number, clique-number and chromatic-number and the appearance of fixed subgraphs. Most of these properties are shown to be captured by either constant k or by some k= poly(log(N)) for a wide range of values of p, implying that random looking graphs on N vertices can be generated by a seed of size poly(log(N)). The proofs combine combinatorial, probabilistic and spectral techniques.


A note on Chern character, loop spaces and derived algebraic geometry

Abstract: In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.


Geometric optics and boundary layers for Nonlinear Schrodinger equations

Abstract: We justify supercritical geometric optics in small time for the defocusing semiclassical Nonlinear Schrodinger Equation for a large class of non-necessarily homogeneous nonlinearities. The case of a half-space with Neumann boundary condition is also studied.


Notes on link homology

Abstract: This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.


Maximal integral point sets over $\mathbb{Z}^2$

Abstract: Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and every additional point $p_{n+1}$ destroys this property. Here we consider such sets for a given cardinality and with minimum possible diameter. We determine some exact values via exhaustive search and give several constructions for arbitrary cardinalities. Since we cannot guarantee the maximality in these cases we describe an algorithm to prove or disprove the maximality of a given integral point set. We additionally consider restrictions as no three points on a line and no four points on a circle.


Maximal integral point sets in affine planes over finite fields

Abstract: Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane $\mathbb{F}_q^2$ over a finite field $\mathbb{F}_q$, where the formally defined squared Euclidean distance of every pair of points is a square in $\mathbb{F}_q$. It turns out that integral point sets over $\mathbb{F}_q$ can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case integral point sets can be restated as cliques in Paley graphs of square order. In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over $\mathbb{F}_q$ for $q\le 47$. Furthermore, we give two series of maximal integral point sets and prove their maximality.


Integral point sets over finite fields

Abstract: We consider point sets in the affine plane $\mathbb{F}_q^2$ where each Euclidean distance of two points is an element of $\mathbb{F}_q$. These sets are called integral point sets and were originally defined in $m$-dimensional Euclidean spaces $\mathbb{E}^m$. We determine their maximal cardinality $\mathcal{I}(\mathbb{F}_q,2)$. For arbitrary commutative rings $\mathcal{R}$ instead of $\mathbb{F}_q$ or for further restrictions as no three points on a line or no four points on a circle we give partial results. Additionally we study the geometric structure of the examples with maximum cardinality.


Pointwise Trichotomy for Skew-Evolution Semiflows on Banach Spaces

Abstract: The paper introduces the notion of skew-evolution semiflows and presents the concept of pointwise trichotomy in the case of skew-evolution semiflows on a Banach space. The connection with the classic notion of trichotomy presented by us in a previous paper in 2006 for evolution operators, is also emphasized, as well as some characterizations. The approach of the theory is from uniform point of view. The study can also be extended to systems with control whose state evolution can be described by skew-evolution semiflows.


Khovanov homology and star-like isotopies

Abstract: A star-like isotopy for oriented links in 3-space is an isotopy which uses only Reidemeister II moves with opposite orientations and Reidemeister III moves with alternating orientations when checking the strands clockwise (or anticlockwise). We define a link polynomial derived from the Jones polynomial which is, in general, only invariant under star-like isotopies and we categorify it.


Bounds for the minimum oriented diameter

Abstract: We consider the problem of finding an orientation with minimum diameter of a connected bridgeless graph. Fomin et. al. discovered a relation between the minimum oriented diameter an the size of a minimal dominating set. We improve their upper bound.


Bounds for the minimum diameter of integral point sets

Abstract: Geometrical objects with integral sides have attracted mathematicians for ages. For example, the problem to prove or to disprove the existence of a perfect box, that is, a rectangular parallelepiped with all edges, face diagonals and space diagonals of integer lengths, remains open. More generally an integral point set $\mathcal{P}$ is a set of $n$ points in the $m$-dimensional Euclidean space $\mathbb{E}^m$ with pairwise integral distances where the largest occurring distance is called its diameter. From the combinatorial point of view there is a natural interest in the determination of the smallest possible diameter $d(m,n)$ for given parameters $m$ and $n$. We give some new upper bounds for the minimum diameter $d(m,n)$ and some exact values.


Integral point sets over $\mathbb{Z}_n^m$

Abstract: There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of $\mathbb{Z} / \mathbb{Z}n$, and study the properties of the resulting combinatorial structures.


Universal star products

Abstract: One defines the notion of universal deformation quantization: given any manifold $M$, any Poisson structure $\P$ on $M$ and any torsionfree linear connection $\nabla$ on $M$, a universal deformation quantization associates to this data a star product on $(M,\P)$ given by a series of bidifferential operators whose corresponding tensors are given by universal polynomial expressions in the Poisson tensor $\P$, the curvature tensor $R$ and their covariant iterated derivatives. Such universal deformation quantization exist. We study their unicity at order 3 in the deformation parameter, computing the appropriate universal Poisson cohomology.


There are integral heptagons, no three points on a line, no four on a circle

Abstract: We give two configurations of seven points in the plane, no three points in a line, no four points on a circle with pairwise integral distances. This answers a famous question of Paul Erd\H{o}s.


Weak approximation of stochastic partial differential equations: the non linear case

Abstract: We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise.


LERF and the Lubotzky-Sarnak conjecture

Abstract: We prove that every closed hyperbolic 3-manifold has a family of (possibly infinite sheeted) coverings with the property that the Cheeger constants in the family tend to zero. This is used to show that, if in addition the fundamental group of the manifold is LERF, then it satisfies the Lubotzky-Sarnak conjecture.


On the minimum diameter of plane integral point sets

Abstract: Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets $\mathcal{P}$, which are sets of $n$ points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter $d(2,n)$ of a plane integral point set consisting of $n$ points arises. We give some new exact values and describe state-of-the-art algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for $d(2,n)$ achieving the known upper bound $n^{c_2\log\log n}$ up to a constant in the exponent. A famous question of Erd\H{o}s asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by $\dot{d}(2,n)$. Recently $\dot{d}(2,7)=22 270$ could be determined via an exhaustive search.


Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's

Abstract: We present a general result of transverse nonlinear instability of 1-d solitary waves for Hamiltonian PDE's for both periodic or localized transverse perturbations. Our main structural assumption is that the linear part of the 1d model and the transverse perturbation "have the same sign". Our result applies to the generalized KP-I equation, the Nonlinear Schr\"odinger equation, the generalized Boussinesq system and the Zakharov-Kuznetsov equation and we hope that it may be useful in other contexts.


Surface subgroups of Kleinian groups with torsion

Abstract: We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.


Enumeration of integral tetrahedra

Abstract: We determine the numbers of integral tetrahedra with diameter $d$ up to isomorphism for all $d\le 1000$ via computer enumeration. Therefore we give an algorithm that enumerates the integral tetrahedra with diameter at most $d$ in $O(d^5)$ time and an algorithm that can check the canonicity of a given integral tetrahedron with at most 6 integer comparisons. For the number of isomorphism classes of integral $4\times 4$ matrices with diameter $d$ fulfilling the triangle inequalities we derive an exact formula.


Tilting modules arising from ring epimorphisms

Abstract: Given a ring R, we investigate tilting modules of the form S \oplus S/R for some injective ring epimorphism R \to S. In particular, we are interested in tilting modules arising from Schofield's universal localization. For some rings, in this way one obtains a classification of all tilting modules.


Plurisubharmonicity in a General Geometric Context

Abstract: Recently the authors have explored new concepts of plurisubharmonicity and pseudoconvexity, with much of the attendant analysis, in the context of calibrated manifolds. Here a much broader extension is made. This development covers a wide variety of geometric situations, including, for example, Lagrangian plurisubhamonicity and convexity. It also applies in a number of non-geometric situations. Results include: fundamental properties of $P^+$-plurisubharmonic functions, plurisubharmonic distributions and regularity, $P^+$-convex domains and $P^+$-convex boundaries, topological restrictions on and construction of such domains, continuity of upper envelopes, and solutions of the Dirichlet problem for related Monge-Ampere-type equations. Many results in this paper have been generalized in recent work of the authors. However, this article covers many cases of geometric interest, and certain convexity assumptions here allow the use of classical analytic methods, making the exposition more accessible.


On the underestimation of model uncertainty by Bayesian K-nearest neighbors

Abstract: When using the K-nearest neighbors method, one often ignores uncertainty in the choice of K. To account for such uncertainty, Holmes and Adams (2002) proposed a Bayesian framework for K-nearest neighbors (KNN). Their Bayesian KNN (BKNN) approach uses a pseudo-likelihood function, and standard Markov chain Monte Carlo (MCMC) techniques to draw posterior samples. Holmes and Adams (2002) focused on the performance of BKNN in terms of misclassification error but did not assess its ability to quantify uncertainty. We present some evidence to show that BKNN still significantly underestimates model uncertainty.


Packing index of subsets in Polish groups

Abstract: For a subset $A$ of a Polish group $G$, we study the (almost) packing index $\ind_P(A)$ (resp. $\Ind_P(A)$) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is (almost) disjoint (in the sense that $|xA\cap yA|<|A|$ for any distinct points $x,y\in S$). Subsets $A\subset G$ with small (almost) packing index are small in a geometric sense. We show that $\ind_P(A)\in \IN\cup\{\aleph_0,\cc\}$ for any $\sigma$-compact subset $A$ of a Polish group. If $A\subset G$ is Borel, then the packing indices $\ind_P(A)$ and $\Ind_P(A)$ cannot take values in the half-interval $[\sq(\Pi^1_1),\cc)$ where $\sq(\Pi^1_1)$ is a certain uncountable cardinal that is smaller than $\cc$ in some models of ZFC. In each non-discrete Polish Abelian group $G$ we construct two closed subsets $A,B\subset G$ with $\ind_P(A)=\ind_P(B)=\cc$ and $\Ind_P(A\cup B)=1$ and then apply this result to show that $G$ contains a nowhere dense Haar null subset $C\subset G$ with $\ind_P(C)=\Ind_P(C)=\kappa$ for any given cardinal number $\kappa\in[4,\cc]$.


Symmetric monochromatic subsets in colorings of the Lobachevsky plane

Abstract: We prove that for each partition of the Lobachevsky plane into finitely many Borel pieces one of the cells of the partition contains an unbounded centrally symmetric subset.


The expansion for the overlap function

Abstract: In this work, it is proved the complete expansion for the second moment of the overlap function for the Sherrington-Kirkpatrick model. It is a technical result which takes advantage of the cavity method and other induction arguments.


A little noticed right triangle

Abstract: Given a right triangle ABC, with the ninety degree angle at A; consider the triangle O1OO2.Where the point O is the midpoint of the hypotenuseBC(and so the center of the triangle ABC's circumcircle), the point O1 being the triangle AOB's circumcenter, and the point O2 being the triangle AOC circumcenter. The triangle O1OO2 is actually a right triangle similar to ABC. In the early part of this work, some geometrical formulas are derived, including some lengths of segments from a nearby trapezoid. After that, the focus changes. We examine the case when ABC is a Pythagorean triangle, and find the precise conditions under which the triangle O1OO2 is also Pythagorean. A numerical table of values is also offered at the end of the paper.


Groebner-Shirshov Bases: Some New Results

Abstract: In this survey article, we report some new results of Groebner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.


Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic--parabolic systems

Abstract: Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a class of hyperbolic-parabolic systems including the Navier--Stokes equations of compressible gas- and magnetohydrodynamics, establishing that linear and nonlinear stability are both equivalent to an Evans function, or generalized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in particular, it has been shown by Costanzino, Humpherys, Nguyen, and Zumbrun to hold for sufficiently large-amplitude layers for isentropic ideal gas dynamics, with general adiabiatic index $\gamma \ge 1$. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for compressive (``shock-type'') layers in other than the nearly-constant case. The analysis, as in the strictly parabolic case, proceeds by derivation of detailed pointwise Green function bounds, with substantial new technical difficulties associated with the more singular, hyperbolic behavior in the high-frequency/short time regime.


Splitting the spectral flow and the SU(3) Casson invariant for spliced sums

Abstract: We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.


Metabelian representations, twisted Alexander polynomials, knot slicing, and mutation

Abstract: Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional linear representations of the fundamental group of X. This provides a simpler and faster algorithm to compute these twisted polynomials, allowing us to prove that 16 (of 18 previously unknown) algebraically slice knots of 12 or fewer crossings are not slice. We also use this improved algorithm to prove that the 24 mutants of the pretzel knot P(3,7,9,11,15), corresponding to permutations of (7,9,11,15), represent distinct concordance classes.


Carath\'eodory, Helly and the others in the max-plus world

Abstract: Carath\'eodory's, Helly's and Radon's theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also their max-plus counterparts: namely, the colorful Carath\'eodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem -- Sierksma's conjecture --, although still open for the usual convexity, is shown to be true in the max-plus settings.


Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3

Abstract: Let M be a closed, connected, orientable 3-manifold. The purpose of this paper is to study the Seiberg-Witten Floer homology of M given that S^1 X M admits a symplectic form. In particular, we prove that M fibers over the circle if M has first Betti number 1 and S^1 X M admits a symplectic form with non-torsion canonical class.


A bijection on core partitions and a parabolic quotient of the affine symmetric group

Abstract: Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\ell$-cores index minimal length coset representatives for $\widetilde{S_{\ell}} / S_{\ell}$ where $\widetilde{S_{\ell}}$ denotes the affine symmetric group and $S_{\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type $A_{\ell-1}$. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse.


A Note on Approximate Liftings

Abstract: In this paper, we prove approximate lifting results in the C$^{\ast}$-algebra and von Neumann algebra settings. In the C$^{\ast}$-algebra setting, we show that two (weakly) semiprojective unital C*-algebras, each generated by $n$ projections, can be glued together with partial isometries to define a larger (weakly) semiprojective algebra. In the von Neumann algebra setting, we prove lifting theorems for trace-preserving *-homomorphisms from abelian von Neumann algebras or hyperfinite von Neumann algebras into ultraproducts. We also extend a classical result of S. Sakai \cite{sakai} by showing that a tracial ultraproduct of C*-algebras is a von Neumann algebra, which yields a generalization of Lin's theorem \cite{Lin} on almost commuting selfadjoint operators with respect to $\Vert\cdot\Vert_{p}$ on any unital C*-algebra with trace.


Topological Stable Rank of Nest Algebras

Abstract: We establish a general result about extending a right invertible row over a Banach algebra to an invertible matrix. This is applied to the computation of right topological stable rank of a split exact sequence. We also introduce a quantitative measure of stable rank. These results are applied to compute the right (left) topological stable rank for all nest algebras. This value is either 2 or infinity, and rtsr(T(N)) = 2 occurs only when N is of ordinal type less than omega^2 and the dimensions of the atoms grows sufficiently quickly. We introduce general results on `partial matrix algebras' over a Banach algebra. This is used to obtain an inequality akin to Rieffel's formula for matrix algebras over a Banach algebra. This is used to give further insight into the nest case.


A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups

Abstract: We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>\epsilon $ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $\epsilon =\epsilon (n)>0$ is an absolute constant.


Coverage Probability of Wald Interval for Binomial Parameters

Abstract: In this paper, we develop an exact method for computing the minimum coverage probability of Wald interval for estimation of binomial parameters. Similar approach can be used for other type of confidence intervals.


Optimal Explicit Binomial Confidence Interval with Guaranteed Coverage Probability

Abstract: In this paper, we develop an approach for optimizing the explicit binomial confidence interval recently derived by Chen et al. The optimization reduces conservativeness while guaranteeing prescribed coverage probability.


A Strong Tits Alternative

Abstract: We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite subgroups of $GL_d(\Bbb{F}_q)$ as well as other diophantine properties of non-discrete subgroups of Lie groups.


Presentations of finite simple groups: a computational approach

Abstract: All nonabelian finite simple groups of rank $n$ over a field of size $q$, with the possible exception of the Ree groups $^2G_2(3^{2e+1})$, have presentations with at most $80 $ relations and bit-length $O(\log n +\log q)$. Moreover, $A_n$ and $S_n$ have presentations with 3 generators$,$ 7 relations and bit-length $O(\log n)$, while $\SL(n,q)$ has a presentation with 7 generators, $2 5$ relations and bit-length $O(\log n +\log q)$.


On Estimation and Optimization of Mean Values of Bounded Variables

Abstract: In this paper, we develop a general approach for probabilistic estimation and optimization. An explicit formula and a computational approach are established for controlling the reliability of probabilistic estimation based on a mixed criterion of absolute and relative errors. By employing the Chernoff-Hoeffding bound and the concept of sampling, the minimization of a probabilistic function is transformed into an optimization problem amenable for gradient descendent algorithms.


The generalized Lefschetz number of homeomorphisms on punctured disks

Abstract: We compute the generalized Lefschetz number of orientation-preserving self-homeomorphisms of a compact punctured disk, using the fact that homotopy classes of these homeomorphisms can be identified with braids. This result is applied to study Nielsen-Thurston canonical homeomorphisms on a punctured disk. We determine, for a certain class of braids, the rotation number of the corresponding canonical homeomorphisms on the outer boundary circle. Also,it is shown that the canonical homeomorphisms corresponding to some braids are pseudo-Anosov.


Relative support varieties

Abstract: We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.


Lotsize optimization leading to a $p$-median problem with cardinalities

Abstract: We consider the problem of approximating the branch and size dependent demand of a fashion discounter with many branches by a distributing process being based on the branch delivery restricted to integral multiples of lots from a small set of available lot-types. We propose a formalized model which arises from a practical cooperation with an industry partner. Besides an integer linear programming formulation and a primal heuristic for this problem we also consider a more abstract version which we relate to several other classical optimization problems like the p-median problem, the facility location problem or the matching problem.


The Top-Dog Index: A New Measurement for the Demand Consistency of the Size Distribution in Pre-Pack Orders for a Fashion Discounter with Many Small Branches

Abstract: We propose the new Top-Dog-Index, a measure for the branch-dependent historic deviation of the supply data of apparel sizes from the sales data of a fashion discounter. A common approach is to estimate demand for sizes directly from the sales data. This approach may yield information for the demand for sizes if aggregated over all branches and products. However, as we will show in a real-world business case, this direct approach is in general not capable to provide information about each branch's individual demand for sizes: the supply per branch is so small that either the number of sales is statistically too small for a good estimate (early measurement) or there will be too much unsatisfied demand neglected in the sales data (late measurement). Moreover, in our real-world data we could not verify any of the demand distribution assumptions suggested in the literature. Our approach cannot estimate the demand for sizes directly. It can, however, individually measure for each branch the scarcest and the amplest sizes, aggregated over all products. This measurement can iteratively be used to adapt the size distributions in the pre-pack orders for the future. A real-world blind study shows the potential of this distribution free heuristic optimization approach: The gross yield measured in percent of gross value was almost one percentage point higher in the test-group branches than in the control-group branches.


A lower bound for the principal eigenvalue of the Stokes operator in a random domain

Abstract: This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\"{u}trich for localization of the PE of the Schr\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE.


The C1 generic diffeomorphism has trivial centralizer

Abstract: Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.


Smooth Yamabe invariant and surgery

Abstract: We prove a surgery formula for the smooth Yamabe invariant $\sigma(M)$ of a compact manifold $M$. Assume that $N$ is obtained from $M$ by surgery of codimension at least 3. We prove the existence of a positive number $\Lambda_n$, depending only on the dimension $n$ of $M$, such that $$ \sigma(N) \geq \min{\sigma(M),\Lambda_n}. $$


Sur la g\'eom\'etrie systolique des vari\'et\'es de Bieberbach

Abstract: The systole of a compact non simply connected Riemannian manifold is the smallest length of a non-contractible closed curve ; the systolic ratio is the quotient $(\mathrm{systole})^n/\mathrm{volume}$. Its supremum on the set of all the riemannian metrics, is known to be finite for a large class of manifolds, including the $K(\pi,1)$. We study the optimal systolic ratio of compact, 3-dimensional non orientable Bieberbach manifolds, and prove that it cannot be realized by a flat metric.


Probabilistic computation of wind farm power generation based on wind turbine dynamic modeling

Abstract: This paper addresses the problem of predicting a wind farm's power generation when no or few statistical data is available. The study is based on a time-series wind speed model and on a simple dynamic model of a DFIG wind turbine including cut-off and cut-in behaviours. The wind turbine is modeled as a stochastic hybrid system with three operation modes. Numerical results, obtained using Monte-Carlo simulations, provide the annual distribution of a wind farm's active power generation. For different numbers of wind turbines, we compare the numerical results obtained using the dynamic model with those obtained considering the wind turbine's steady-state power curve. Simulations show that the wind turbine's dynamics do not need to be considered for analyzing the annual distribution of a wind farm generation.


Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

Abstract: We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).


Torsion points on elliptic curves over function fields and a theorem of Igusa

Abstract: If F is a global function field of characteristic p>3, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.


Coherent structures and isolated spectrum for Perron-Frobenius cocycles

Abstract: We present an analysis of one-dimensional models of dynamical systems that possess 'coherent structures'; global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron-Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron-Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalise the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.


Representations of quivers via reflection functors

Abstract: These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear algebra is sufficient for proving Gabriel's theorem. Thus we classify the quivers of finite representation type and study their representations using reflection functors. Further material has been added after giving this course in Bielefeld in summer 2010. This includes a discussion of regular representations and wild phenomena. In particular, two classical examples are covered: representations of the Kronecker quiver and representations of the Klein four group.


Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

Abstract: We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times\CR^d$ for some unbounded interval $I\subset\CR$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\CR^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\{G(t,s)\}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.


An asymptotic result for Brownian polymers

Abstract: We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_t$ (corresponding to the location of the end of the polymer at time $t$) for a particular type of repelling interaction function without compact support.


Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Abstract: Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1<p<\infty. Our main result states that the following assertions are equivalent: (1) dom((sqrt(D_V* B D_V)) = dom(D_V) and Meyer's inequalities hold for D_V* B D_V; (2) D_V D_V* B admits a bounded H-infinity calculus on the closure of the range of D_V; (3) dom(sqrt(V*BV)) = dom(V) and Meyer's inequalities hold for V*BV; (4) VV*B admits a bounded H-infinity calculus on the closure of the range of V. Moreover, if these conditions are satisfied, then dom(L) = dom(D_V^2) \cap dom(D_A). The equivalence of (1)-(4) is a non-symmetric generalisation of the classical Meyer inequalities (which correspond to the case G=H, V=I, B=I). A one-sided version of the main result, giving L^p-boundedness of the associated Riesz transforms in terms of a square function estimate, is also obtained. As an application let -A generate an analytic C_0-contraction semigroup on a Hilbert space H and let -L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A.


Extreme flatness of normed modules and Arveson-Wittstock type theorems

Abstract: We show in this paper that a certain class of normed modules over the algebra of all bounded operators on a Hilbert space possesses a homological property which is a kind of a functional-analytic version of the standard algebraic property of flatness. We mean the preservation, under projective tensor multiplication of modules, of the property of a given morphism to be isometric. As an application, we obtain several extension theorems for different types of modules, called Arveson-Wittstock type theorems. These, in their turn, have, as a straight corollary, the `genuine' Arveson-Wittstock Theorem in its non-matricial presentation. We recall that the latter theorem plays the role of a `quantum' version of the classical Hahn-Banach theorem on the extension of bounded linear functionals. It was originally proved by Wittstock (1981), and a crucial preparatory step was done by Arveson (1969).


On countably compact 0-simple topological inverse semigroups

Abstract: We describe the structure of 0-simple countably compact topological inverse semigroups and the structure of congruence-free countably compact topological inverse semigroups.


On linearly ordered $H$-closed topological semilattices

Abstract: We give a criterium when a linearly ordered topological semilattice is $H$-closed. We also prove that any linearly ordered $H$-closed topological semilattice is absolutely $H$-closed and we show that every linearly ordered semilattice is a dense subsemilattice of an $H$-closed topological semilattice.


Semigroup Closures of Finite Rank Symmetric Inverse Semigroups

Abstract: We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations $\mathscr{I}_\lambda^n$ of the rank $\leqslant n$ is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.


Quenched large deviations for multidimensional random walk in random environment: a variational formula

Abstract: We take the point of view of the particle in a multidimensional nearest neighbor random walk in random environment (RWRE). We prove a quenched large deviation principle and derive a variational formula for the quenched rate function. Most of the previous results in this area rely on the subadditive ergodic theorem. We employ a different technique which is based on a minimax theorem. Large deviation principles for RWRE have been proven for i.i.d. nestling environments subject to a moment condition and for ergodic uniformly elliptic environments. We assume only that the environment is ergodic and the transition probabilities satisfy a moment condition.


Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. II

Abstract: We find many conditions equivalent to the model-theoretical property $\lambda \stackrel{\kappa}{\Rightarrow} \mu$ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices.


On chains in $H$-closed topological pospaces

Abstract: We study chains in an $H$-closed topological partially ordered space. We give sufficient conditions for a maximal chain $L$ in an $H$-closed topological partially ordered space such that $L$ contains a maximal (minimal) element. Also we give sufficient conditions for a linearly ordered topological partially ordered space to be $H$-closed. We prove that any $H$-closed topological semilattice contains a zero. We show that a linearly ordered $H$-closed topological semilattice is an $H$-closed topological pospace and show that in the general case this is not true. We construct an example an $H$-closed topological pospace with a non-$H$-closed maximal chain and give sufficient conditions that a maximal chain of an $H$-closed topological pospace is an $H$-closed topological pospace.


Cluster algebras and semipositive symmetrizable matrices

Abstract: In this paper, we give a description of the skew-symmetrizable matrices and their mutation classes which are determined by the generalized Cartan matrices of affine type.


Monads and comonads in module categories

Abstract: Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\Hom_A(\C,-)$-modules (called $\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\C$-comodules and $\Hom_A(\C,-)$-modules are equivalent provided $\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom_R(H-)$ is a Hopf bimonad on $\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\Hom_R(H,-)$-bimodules are both equivalent to $\M_R$.


Stabilit\'{e} du comportement des marches al\'{e}atoires sur un groupe localement compact

Abstract: Dans cet article nous d\'{e}montrons un th\'{e}or\`{e}me de stabilit\'{e} des probabilit\'{e}s de retour sur un groupe localement compact unimodulaire, s\'{e}parable et compactement engendr\'{e}. Nous d\'{e}montrons que le comportement asymptotique de $F^{*(2n)}(e)$ ne d\'{e}pend pas de la densit\'{e} $F$ sous des hypoth\`{e}ses naturelles. A titre d'exemple nous \'{e}tablissons que la probabilit\'{e} de retour sur une large classe de groupes r\'{e}solubles se comporte comme $\exp(-n^{1/3})$.


On Semi-Modular Subalgebras of Lie Algebras Over Fields of Arbitrary Characteristic

Abstract: This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. It is shown that, in certain circumstances, including for all solvable algebras, for all Lie algebras over algebraically closed fields of characteristic p > 0 that have absolute toral rank less than or equal to 1 or are restricted, and for all Lie algebras having the one-and-a-half generation property, the conditions of modularity and semi-modularity are equivalent, but that the same is not true for all Lie algebras over a perfect field of characteristic three. Semi-modular subalgebras of dimensions one and two are characterised over (perfect, in the case of two-dimensional subalgebras) fields of characteristic different from 2, 3.


Homotopy exponents for large H-spaces

Abstract: We show that H-spaces with finitely generated cohomology, as an algebra or as an algebra over the Steenrod algebra, have homotopy exponents at all primes. This provides a positive answer to a question of Stanley.


On Exponential Stability for Skew-Evolution Semiflows on Banach Spaces

Abstract: The paper emphasizes the property of stability for skew-evolution semiflows on Banach spaces, defined by means of evolution semiflows and evolution cocycles and which generalize the concept introduced by us in a previous paper. There are presented several general characterizations of this asymptotic property out of which can be deduced well known results of the stability theory. A unified treatment in the uniform and in the nonuniform setting is given. The main results are also formulated in discrete time.


A Littelmann path model for crystals of Generalized Kac-Moody algebras revisited

Abstract: A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an isomorphism theorem and a character formula of Borcherds-Kac-Weyl type for the crystals. In the symmetrizable case, the isomorphism theorem implies that the crystals constructed by this path model coincide with those of Jeong, Kang, Kashiwara and Shin obtained by taking the limit at q=0 in the quantized enveloping algebra.


On certain enumeration problems in two-dimensional topology

Abstract: We announce a solution to several enumeration problems in topology of surfaces. This includes an enumeration of homotopy classes of sections of locally trivial fiber bundles over surfaces and a computation of non-abelian 1-cohomology of surfaces.


Remarks on a normal subgroup of GA_n

Abstract: We show that the subgroup generated by locally finite polynomial automorphisms of k^n is normal in GA_n. Also, some properties of normal subgroups of GA_n containing all diagonal automorphisms are given.


Differential modules on p-adic polyannuli

Abstract: We consider variational properties of some numerical invariants, measuring convergence of local horizontal sections, associated to differential modules on polyannuli over a nonarchimedean field of characteristic zero. This extends prior work in the one-dimensional case of Christol, Dwork, Robba, Young, et al. Our results do not require positive residue characteristic; thus besides their relevance to the study of Swan conductors for isocrystals, they are germane to the formal classification of flat meromorphic connections on complex manifolds.


Rationality of the moduli spaces of plane curves of sufficiently large degree

Abstract: We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.


On an identity of Ky Fan

Abstract: We give several applications of an identity for sums of weakly stationary sequences due to Ky Fan.


The rationality of the moduli space of curves of genus 3 after P. Katsylo

Abstract: This article is a survey of P. Katsylo's proof that the moduli space of smooth projective complex curves of genus 3 is rational. We hope to make the argument more comprehensible and transparent by emphasizing the underlying geometry in the proof and its key structural features.


Geometry of whips and chains

Abstract: We study the geometry of the inextensible string (the whip) and its discrete approximation (the chain). In the absence of gravity, both motions represent geodesic motions on certain manifolds. We show how the motion of the chain converges to that of a whip, and how the curvature of the chain's configuration space converges to that of the whip's configuration space. Finally we speculate on the analogous approximation of an incompressible fluid by a discrete system.


The supersingular locus of the Shimura variety of GU(1,n-1) II

Abstract: We complete the study of the supersingular locus in the fiber at $p$ of a Shimura variety attached to a unitary similitude group $\GU(1,n-1)$ over $\QQ$ in the case that $p$ is inert. This was started by the first author in \cite{Vo_Uni} where complete results were obtained for $n =2,3$. The supersingular locus is uniformized by a formal scheme $\Ncal$ which is a moduli space of so-called unitary $p$-divisible groups. It depends on the choice of a unitary isocrystal $\Nbf$. We define a stratification of $\Ncal$ indexed by vertices of the Bruhat-Tits building attached to the reductive group of automorphisms of $\Nbf$. We show that the combinatorial behaviour of this stratification is given by the simplicial structure of the building. The closures of the strata (and in particular the reduced irreducible components of $\Ncal$) are identified with (generalized) Deligne-Lusztig varieties. We show that the Bruhat-Tits stratification is a refinement of the Ekedahl-Oort stratification and also relate the Ekedahl-Oort strata to Deligne-Lusztig varieties. We deduce that the supersingular locus is locally a complete intersection, that its irreducible components and each Ekedahl-Oort stratum in every irreducible component is isomorphic to a Deligne-Lusztig variety, and give formulas for the number of irreducible components of every Ekedahl-Oort stratum of the supersingular locus.


Rooted induced trees in triangle-free graphs

Abstract: For a graph $G$, let $t(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree. Further, for a vertex $v\in V(G)$, let $t^v(G)$ denote the maximum number of vertices in an induced subgraph of $G$ that is a tree, with the extra condition that the tree must contain $v$. The minimum of $t(G)$ ($t^v(G)$, respectively) over all connected triangle-free graphs $G$ (and vertices $v\in V(G)$) on $n$ vertices is denoted by $t_3(n)$ ($t_3^v(n)$). Clearly, $t^v(G)\le t(G)$ for all $v\in V(G)$. In this note, we solve the extremal problem of maximizing $|G|$ for given $t^v(G)$, given that $G$ is connected and triangle-free. We show that $|G|\le 1+\frac{(t_v(G)-1)t_v(G)}{2}$ and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^v(n)=\lceil {1/2}(1+\sqrt{8n-7})\rceil$, improving a recent result by Fox, Loh and Sudakov.


Symmetric links and Conway sums: volume and Jones polynomial

Abstract: We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial.


Suppression of unbounded gradients in a SDE associated with the Burgers equation

Abstract: We consider the Langevin equation describing a stochastically perturbed by uniform noise non-viscous Burgers fluid and introduce a deterministic function that corresponds to the mean of the velocity when we keep the value of position fixed. We study interrelations between this function and the solution of the non-perturbed Burgers equation. Especially we are interested in the property of the solution of the latter equation to develop unbounded gradients within a finite time. We study the question how the initial distribution of particles for the Langevin equation influences this blowup phenomenon. It is shown that for a wide class of initial data and initial distributions of particles the unbounded gradients are eliminated. The case of a linear initial velocity is particular. We show that if the initial distribution of particles is uniform, then the mean of the velocity for a given position coincides with the solution of the Burgers equation and in particular does not depend on the constant variance of the stochastic perturbation. Further, for a one space space variable we get the following result: if the decay rate of the power-behaved initial particles distribution at infinity is greater or equal $|x|^{-2},$ then the blowup is suppressed, otherwise, the blowup takes place at the same moment of time as in the case of the non-perturbed Burgers equation.


Inseparable local uniformization

Abstract: It is known since the works of Zariski in early 40ies that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, i.e. any valuation can be uniformized by a purely inseparable alteration.


K3 surfaces with Picard rank 20

Abstract: We determine all complex K3 surfaces with Picard rank 20 over Q. Here the Neron-Severi group has rank 20 and is generated by divisors which are defined over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We then apply our methods to general singular K3 surfaces, i.e. with Neron-Severi group of rank 20, but not necessarily generated by divisors over Q.


Generalized Navier Boundary Condition and Geometric Conservation Law for surface tension

Abstract: We consider two-fluid flow problems in an Arbitrary Lagrangian Eulerian (ALE) framework. The purpose of this work is twofold. First, we address the problem of the moving contact line, namely the line common to the two fluids and the wall. Second, we perform a stability analysis in the energy norm for various numerical schemes, taking into account the gravity and surface tension effects. The problem of the moving contact line is treated with the so-called Generalized Navier Boundary Conditions. Owing to these boundary conditions, it is possible to circumvent the incompatibility between the classical no-slip boundary condition and the fact that the contact line of the interface on the wall is actually moving. The energy stability analysis is based in particular on an extension of the Geometry Conservation Law (GCL) concept to the case of moving surfaces. This extension is useful to study the contribution of the surface tension. The theoretical and computational results presented in this paper allow us to propose a strategy which offers a good compromise between efficiency, stability and artificial diffusion.


Modular polynomials for genus 2

Abstract: Modular polynomials are an important tool in many algorithms involving elliptic curves. In this article we investigate their generalization to the genus 2 case following pioneering work by Gaudry and Dupont. We prove various properties of these genus 2 modular polynomials and give an improved way to explicitly compute them.


Giroux torsion and twisted coefficients

Abstract: We explain the effect of applying a full Lutz twist along a pre-Lagrangian torus in a contact 3-manifold, on the contact invariant in Heegaard Floer homology with twisted coefficients.


The Weyl group of type $A_1$ root systems extended by an abelian group

Abstract: We investigate the class of root systems $R$ obtained by extending an $A_1$-type irreducible root system by a free abelian group $G$. In this context there is a Weyl group $W$ and a group $U$ with the presentation by conjugation. Both groups are reflection groups with respect to a discrete symmetric space $T$ associated to $R$. We show that the natural homomorphism $U\to W$ is an isomorphism if and only if an associated subset $T^{ab}\setminus\{0\}$ of $G_2=G/2G$ is 2-independent, i.e. its image under the map $G_2\to G_2\otimes G_2, g\mapsto g\otimes g$ is linearly independent over the Galois field $F_2$.


Coarse differentiation and multi-flows in planar graphs

Abstract: We show that the multi-commodity max-flow/min-cut gap for series-parallel graphs can be as bad as 2, matching a recent upper bound Chakrabarti, Jaffe, Lee, and Vincent for this class, and resolving one side of a conjecture of Gupta, Newman, Rabinovich, and Sinclair. This also improves the largest known gap for planar graphs from 3/2 to 2, yielding the first lower bound that doesn't follow from elementary calculations. Our approach uses the {\em coarse differentiation} method of Eskin, Fisher, and Whyte in order to lower bound the distortion for embedding a particular family of shortest-path metrics into $L_1$.


Oscillatory Integral Decay, Sublevel Set Growth, and the Newton Polyhedron

Abstract: Using some resolution of singularities methods of the author, a generalization of a well-known theorem of Varchenko relating decay of oscillatory integrals to the Newton polyhedron is proven. They are derived from analogous results for sublevel integrals, proven here. Varchenko's theorem requires a certain nondegeneracy condition on the faces of the Newton polyhedron on the phase. In this paper, it is shown that the estimates of Varchenko's theorem also hold for a significant class of phase functions for which this nondegeneracy condition does not hold. Thus in problems where one wants to switch coordinates to a coordinate system where Varchenko's estimates are valid, one has greater flexibility. Some additional estimates are also proven for more degenerate situations, including some too degenerate for the Newton polyhedron to give the optimal decay in the sense of Varchenko.


Fixed point-free isometric actions of topological groups on Banach spaces

Abstract: We show that every non-precompact topological group admits a fixed point-free continuous action by affine isometries on a suitable Banach space. Thus, precompact groups are defined by the fixed point property for affine isometric actions on Banach spaces. For separable topological groups, in the above statements it is enough to consider affine actions on one particular Banach space: the unique Banach space envelope of the universal Urysohn metric space, known as the Holmes space. At the same time, we show that Polish groups need not admit topologically proper (in particular, free) affine isometric actions on Banach spaces (nor even on complete metric spaces): this is the case for the unitary group of the separable infinite dimensional Hilbert space with strong operator topology, the infinite symmetric group, etc.


Adaptive nonparametric estimation in heteroscedastic regression models. Part 2: Asymptotic efficiency

Abstract: The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper (2007) for estimation of unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk. It means that the asymptotic quadratic risk for this procedure coincides with a sharp lower bound.


A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs

Abstract: For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.


Comparison of secondary invariants of algebraic K-theory

Abstract: In this paper we prove that the multiplicative character of A. Connes and M. Karoubi and the determinant invariant of L. G. Brown, J. W. Helton and R. E. Howe agree up to a canonical homomorphism.


Structural Ramsey theory of metric spaces and topological dynamics of isometry groups

Abstract: In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces - called ultrahomogeneous - is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore the different aspects of this connection.


Holomorphic Representation of Constant Mean Curvature Surfaces in Minkowski Space: Consequences of Non-Compactness in Loop Group Methods

Abstract: We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an open dense subset, after doubling the size of the real form $SU_{1,1}$, and prove several results concerning the behavior of the surface as the boundary of this open set is encountered. We then use the generalized Weierstrass representation to create and classify new examples of spacelike CMC surfaces in $\real^{2,1}$. In particular, we classify surfaces of revolution and surfaces with screw motion symmetry, as well as studying another class of surfaces for which the metric is rotationally invariant.


Invariance of a Shift-Invariant Space

Abstract: A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.


Automorphisms of curves fixing the order two points of the Jacobian

Abstract: Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism \sigma that fixes pointwise all the order two points of Pic}^0(X), then we prove that X is hyperelliptic with \sigma being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms \sigma' of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with \sigma' being its hyperelliptic involution.


Linear Optimization over a Polymatroid with Side Constraints -- Scheduling Queues and Minimizing Submodular Functions

Abstract: Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a {\it grouping} algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of $O(n^3LP(n))$, $n$ being the number of job classes, and LP(n) being the computational efforts of solving a linear program with no more than $n$ variables and $n$ constraints. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. %Each time the algorithm is applied, it identifies a subset; and We show the minimizing subset can be identified by applying the grouping algorithm $n$ times. Hence, this results in a algorithm that minimizes a submodular function with an effort of $O(n^4LP(n))$.


Automorphisms and Verma modules for Generalized Schr\"{o}dinger-Virasoro algebras

Abstract: Let $\mathbb{F}$ be a field of characteristic 0, $G$ an additive subgroup of $\mathbb{F}$, $\alpha\in \mathbb{F}$ satisfying $\alpha\notin G, 2\alpha\in G$. We define a class of infinite-dimensional Lie algebras which are called generalized Schr\"{o}dinger-Virasoro algebras and use $\mathfrak{gsv}[G,\alpha]$ to denote the one corresponding to $G$ and $\alpha$. In this paper the automorphism group and irreducibility of Verma modules for $\mathfrak{gsv}[G,\alpha]$ are completely determined.


Hilbert domains quasi-isometric to normed vector spaces

Abstract: We prove that a Hilbert domain which is quasi-isometric to a normed vector space is actually a convex polytope.


Hilbert geometry for convex polygonal domains

Abstract: We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.


On the spectrum of asymptotic slopes

Abstract: The slopes of maximal subbundles of rank $s$ divided by the degree of the map under various pull backs form a bounded collection of numbers called the $s$-spectrum of the bundle. We study the supremum of the $s$-spectrum and determine it in terms of the Harder Narasimhan filtration of the bundle.


Counting conics in complete intersections

Abstract: We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.


Existence and boundedness of solutions for a singular phase field system

Abstract: This paper is devoted to the mathematical analysis of a thermomechanical model describing phase transitions in terms of the entropy and order structure balance law. We consider a macroscopic description of the phenomenon and make a presentation of the model. Then, the initial and boundary value problem is addressed for the related PDE system, which contains some nonlinear and singular terms with respect to the temperature variable. Existence of the solution is shown along with the boundedness of both phase variable $\chi$ and absolute temperature $\theta$. Finally, uniqueness is proved in the framework of a source term depending Lipschitz continuously on $\theta$.


Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process

Abstract: For a bivariate L\'evy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised Ornstein-Uhlenbeck (GOU) process is defined as V_t:=e^{\xi_t}(z+\int_0^t e^{-\xi_{s-}}d\eta_s), t\ge0, where $z\in\mathbb{R}.$ We define necessary and sufficient conditions under which the infinite horizon ruin probability for the process is zero. These conditions are stated in terms of the canonical characteristics of the L\'evy process and reveal the effect of the dependence relationship between $\xi$ and $\eta.$ We also present technical results which explain the structure of the lower bound of the GOU.


Finite Dimensional Intuitionistic Fuzzy Normed Linear Space

Abstract: Following the definition of intuitionistic fuzzy n-norm [ 3 ], we have introduced the definition of intuitionistic fuzzy norm (in short IFN) over a linear space and there after a few results on intuitionistic fuzzy normed linear space and finite dimensional intuitionistic fuzzy normed linear space. Lastly, we have introduced the definitions of intuitionistic fuzzy continuity and sequentially intuitionistic fuzzy continuity and proved that they are equivalen


Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

Abstract: We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.


The Terwilliger Algebra of a Distance-Regular Graph of Negative Type

Abstract: Let $\Gamma$ denote a distance-regular graph with diameter $D \ge 3$. Assume $\Gamma$ has classical parameters $(D,b,\alpha,\beta)$ with $b < -1$. Let $X$ denote the vertex set of $\Gamma$ and let $A \in MX$ denote the adjacency matrix of $\Gamma$. Fix $x \in X$ and let $A^* \in MX$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $MX$ generated by $A, A^*$. We call $T$ the {\em Terwilliger algebra} of $\Gamma$ with respect to $x$. We show that up to isomorphism there exist exactly two irreducible $T$-modules with endpoint 1; their dimensions are $D$ and $2D-2$. For these $T$-modules we display a basis consisting of eigenvectors for $A^*$, and for each basis we give the action of $A$


Introducing Ramanujan's Class Polynomials in the Generation of Prime Order Elliptic Curves

Abstract: In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.


Dedekind Zeta motives for totally real fields

Abstract: Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.


On percolation in random graphs with given vertex degrees

Abstract: We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.


On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

Abstract: If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional eigenspace relative to a fixed complex ring class character provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is nonzero. This represents the analogue in the function field setting of a theorem for rational elliptic curves due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.


An inequality between multipoint Seshadri constants

Abstract: Let X be a projective variety of dimension n and L be a nef divisor on X. Denote by e_d(r;X,L) the d-dimensional Seshadri constant of r very general points in X. We prove that e_d(rs;X,L) >= e_d(r;X,L)e_d(s;P^n,O_{P^n}(1)).


Poles of regular quaternionic functions

Abstract: This paper studies the singularities of Cullen-regular functions of one quaternionic variable. The quaternionic Laurent series prove to be Cullen-regular. The singularities of Cullen-regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of Cullen-regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution.


Geometry of plane sections of the infinite regular skew polyhedron {4,6|4}

Abstract: The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In this paper we present in detail the results for the skew polyhedron {4,6|4} and in particular we show that in this case a fractal arises and that such a fractal can be generated through an elementary algorithm, which in turn allows us to verify for this case a conjecture of S.P.Novikov that such fractals have zero measure.


Optimal systems of subalgebras for a nonlinear Black-Scholes equation

Abstract: This paper has been withdraw by the auhor due to a mistake in classification of the algebra


Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups

Abstract: We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. Moreover we determine the principal isotropy algebras for all isometric actions on G2.


Non-algebraic Hyperkaehler manifolds

Abstract: We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0,2 and 4. The paper gives structure results for "generalised hyperkaehler" manifolds and studies nef lines bundles.


Approximating optimization problems over convex functions

Abstract: Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.


Sub-Riemannian geodesics on the 3-D sphere

Abstract: The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.


An Artin-Rees Theorem and applications to zero cycles

Abstract: We prove an Artin-Rees type theorem for algebraic cycles and give an application to zero cycles.


A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

Abstract: We consider a transmission wave equation in two embedded domains in $R^2$, where the speed is $a1 > 0$ in the inner domain and $a2 > 0$ in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strictly convex and $a1 > a2$ . As a consequence of this inequality, uniqueness and Lip- schitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.


Vortex-like finite-energy asymptotic profiles for isentropic compressible flows

Abstract: Bidimensional incompressible viscous flows with well-localised vorticity are well-known to develop vortex structures. The purpose of the present paper is to recover the asymptotic profiles describing these phenomena for homogeneous finite-energy flows as asymptotic profiles for near-equilibrium isentropic compressible flows. This task is performed by extending the sharp description of the asymptotic behaviour of near-equilibrium compressible flows obtained by David Hoff and Kevin Zumbrun to the case of finite-energy vortex-like solutions.


Constructive solution of a bilinear optimal control problem for a Schr\"odinger equation

Abstract: Often considered in numerical simulations related to the control of quantum systems, the so-called monotonic schemes have not been so far much studied from the functional analysis point of view. Yet, these procedures provide an efficient constructive method for solving a certain class of optimal control problems. This paper aims both at extending the results already available about these algorithms in the finite dimensional case (i.e., the time-discretized case) and at completing those of the continuous case. This paper starts with some results about the regularity of a functional related to a wide class of model in quantum chemistry. Those enable us to extend an inequality due to


An inverse problem for Schr\"odinger equations with discontinuous main coefficient

Abstract: This paper concerns the inverse problem of retrieving a stationary potential for the Schr\"odinger evolution equation in a bounded domain of RN with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schr\"odinger equation with discontinuous coefficients, result also interesting by itself.


Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 2: Asymptotic efficiency

Abstract: Asymptotic efficiency is proved for the constructed in part 1 procedure, i.e. Pinsker's constant is found in the asymptotic lower bound for the minimax quadratic risk. It is shown that the asymptotic minimax quadratic risk of the constructed procedure coincides with this constant.


Adaptive nonparametric estimation in heteroscedastic regression models. Part 1: Sharp non-asymptotic Oracle inequalities

Abstract: An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed.


The Yamabe problem with singularities

Abstract: Let $(M,g)$ be a compact Riemannian manifold of dimension $n\geq 3$. Under some assumptions, we prove that there exists a positive function $\varphi$ solution of the following Yamabe type equation \Delta \varphi+ h\varphi= \tilde h \varphi^{\frac{n+2}{n-2}} where $h\in L^p(M)$, $p>n/2$ and $\tilde h\in \mathbb R$. We give the regularity of $\varphi$ with respect to the value of $p$. Finally, we consider the results in geometry when $g$ is a singular Riemannian metric and $h=\frac{n-2}{4(n-1)}R_g$, where $R_g$ is the scalar curvature of $g$.


Robust control of a bimorph mirror for adaptive optics system

Abstract: We apply robust control technics to an adaptive optics system including a dynamic model of the deformable mirror. The dynamic model of the mirror is a modification of the usual plate equation. We propose also a state-space approach to model the turbulent phase. A continuous time control of our model is suggested taking into account the frequential behavior of the turbulent phase. An H_\infty controller is designed in an infinite dimensional setting. Due to the multivariable nature of the control problem involved in adaptive optics systems, a significant improvement is obtained with respect to traditional single input single output methods.


Amenability of algebras of approximable operators

Abstract: We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.


Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I

Abstract: Let $\omega ,\omega_0$ be appropriate weight functions and $q\in [1,\infty ]$. We introduce the wave-front set, $\WF_{\mathscr FL^q_{(\omega)}}(f)$ of $f\in \mathscr S'$ with respect to weighted Fourier Lebesgue space $\mathscr FL^q_{(\omega)}$. We prove that usual mapping properties for pseudo-differential operators $\op (a)$ with symbols $a$ in $S^{(\omega _0)}_{\rho, 0}$ hold for such wave-front sets. Especially we prove \WF_{\mathscr FL^q_{(\omega /\omega_0)}}(\op (a)f)\subseteq \WF_{\mathscr FL^q_{(\omega)}}(f) \subseteq \WF_{\mathscr FL^q_{(\omega /\omega_0)}}(\op (a)f)\ttbigcup \Char (a). %% Here $\Char (a)$ is the set of characteristic points of $a$.


On Parallel Sections of a Vector Bundle

Abstract: We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the Frobenius Theorem.


Push-outs of derivations

Abstract: Let A be a Banach algebra and let X be a Banach A -bimodule. In studying the bounded Hochschild cohomology groups H^1(A,X) it is often useful to extend a given derivation D: A-> X to a Banach algebra B containing A as an ideal, thereby exploiting (or establishing) hereditary properties. This is usually done using (bounded/unbounded) approximate identities to obtain the extension as a limit of operators b->D(ba)-b.D(a), (a in A) in an appropriate operator topology, the main point in the proof being to show that the limit map is in fact a derivation. In this paper we make clear which part of this approach is analytic and which algebraic by presenting an algebraic scheme that gives derivations in all situations at the cost of enlarging the module. We use our construction to give improvements and shorter proofs of some results from the literature and to give a necessary and sufficient condition that biprojectivity and biflatness are inherited to ideals.


Biharmonic maps and morphisms from conformal mappings

Abstract: Inspired by the all-important conformal invariance of harmonic maps on two-dimensional domains, this article studies the relationship between biharmonicity and conformality. We first give a characterization of biharmonic morphisms, analogues of harmonic morphisms investigated by Fuglede and Ishihara, which, in particular, explicits the conditions required for a conformal map in dimension four to preserve biharmonicity and helps producing the first example of a biharmonic morphism which is not a special type of harmonic morphism. Then, we compute the bitension field of horizontally weakly conformal maps, which include conformal mappings. This leads to several examples of proper (i.e. non-harmonic) biharmonic conformal maps, in which dimension four plays a pivotal role. We also construct a family of Riemannian submersions which are proper biharmonic maps.


Lower bounds of martingale measure densities in the Dalang-Morton-Willinger theorem

Abstract: For a $d$-dimensional stochastic process $(S_n)_{n=0}^N$ we obtain criteria for the existence of an equivalent martingale measure, whose density $z$, up to a normalizing constant, is bounded from below by a given random variable $f$. We consider the case of one-period model (N=1) under the assumptions $S\in L^p$; $f,z\in L^q$, $1/p+1/q=1$, where $p\in [1,\infty]$, and the case of $N$-period model for $p=\infty$. The mentioned criteria are expressed in terms of the conditional distributions of the increments of $S$, as well as in terms of the boundedness from above of an utility function related to some optimal investment problem under the loss constraints. Several examples are presented.


Collision probabilities in the rarefaction fan of asymmetric exclusion processes

Abstract: We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate $p\in(1/2,1]$ and to the left at rate $1-p$, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the initial state has first-class particles to the left of the origin, second-class particles at sites 0 and 1, and holes to the right of site 1. We show that the probability that the two second-class particles eventually collide is $(1+p)/3p$, where a_collision_ occurs when one of the particles attempts to jump over the other. This also corresponds to the probability that two ASEP processes, started from appropriate initial states and coupled using the so-called "basic coupling", eventually reach the same state. We give various other results about the behaviour of second-class particles in the ASEP. In the totally asymmetric case ($p=1$) we explain a further representation in terms of a multi-type particle system, and also use the collision result to derive the probability of coexistence of both clusters in a two-type version of the corner growth model.


On convex functions and the finite element method

Abstract: Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.


Large semilattices of breadth three

Abstract: A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\kappa$ elements.


Solitary waves for linearly coupled nonlinear Schrodinger equations with inhomogeneous coefficients

Abstract: Motivated by the study of matter waves in Bose-Einstein condensates and coupled nonlinear optical systems, we study a system of two coupled nonlinear Schrodinger equations with inhomogeneous parameters, including a linear coupling. For that system we prove the existence of two different kinds of homoclinic solutions to the origin describing solitary waves of physical relevance. We use a Krasnoselskii fixed point theorem together with a suitable compactness criterion.


Multidimensional Version of Lagrange's Problem on Mean Motion

Abstract: The famous mean motion problem which goes back to Lagrange as follows: to prove that any exponential polynomial with exponents on the imaginary axis has an average speed for the amplitude, whenever the variable moves along a horizontal line. It was completely proved by B. Jessen and H. Tornehave in Acta Math.77, 1945. Here we give its multidimensional version.


Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

Abstract: We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$.


Parabolicity of maximal surfaces in Lorentzian product spaces

Abstract: In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and $M^2\times\mathbb{R}_1$ is endowed with the Lorentzian product metric $<,>=<,>_M-dt^2$. In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain $\Omega\subseteq M$ is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in $M^2\times\mathbb{R}_1$.


On Type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations

Abstract: Local regularity of axially symmetric solutions to the Navier-Stokes equations is studied. It is shown that under certain natural assumptions there are no singularities of Type I.