Cohomologies of harmonic bundles on quasi-compact Kaehler manifolds

Abstract: In this note, we survey our recent work concerning cohomologies of harmonic bundles on quasi-compact Kaehler manifolds.


Standard representation of multivariate functions on a general probability space

Abstract: It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an extension of this to multivariate functions. This is motivated by some recent constructions of random graphs.


Lawvere completion and separation via closure

Abstract: For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.


There are 1,132,835,421,602,062,347 nonisomorphic one-factorizations of $K_{14}$

Abstract: We establish by means of a computer search that a complete graph on 14 vertices has 98,758,655,816,833,727,741,338,583,040 distinct and 1,132,835,421,602,062,347 nonisomorphic one-factorizations. The enumeration is constructive for the 10,305,262,573 isomorphism classes that admit a nontrivial automorphism.


Symplectic Homogenization

Abstract: Let $H(q,p)$ be a Hamiltonian on $T^*T^n$. We show that the sequence $H_{k}(q,p)=H(kq,p)$ converges for the $\gamma$ topology defined by the author, to $\bar{H}(p)$. This is extended to the case where only some of the variables are homogenized, that is the sequence $H(kx,y,q,p)$ where the limit is of the type ${\bar H}(y,q,p)$ and thus yields an "effective Hamiltonian". We give here the proof of the convergence, and the first properties of the homogenization operator, and give some immediate consequences for solutions of Hamilton-Jacobi equations, construction of quasi-states, etc. We also prove that the function $\bar H$ coincides with Mather's $\alpha$ function which gives a new proof of its symplectic invariance proved by P. Bernard. A previous version of this paper relied on the former "On the capacity of Lagrangians in $T^*T^n$ which has been withdrawn. The present version of Symplectic Homogenization does not rely on it anymore.


Risk management for analytical methods: conciliating objectives of methods, validation phase and routine decision rules

Abstract: In the industries that involved either chemistry or biology, such as pharmaceutical industries, chemical industries or food industry, the analytical methods are the necessary eyes and hear of all the material produced or used. If the quality of an analytical method is doubtful, then the whole set of decision that will be based on those measures is questionable. For those reasons, being able to assess the quality of an analytical method is far more than a statistical challenge; it's a matter of ethic and good business practices. Many regulatory documents have been releases, primarily ICH and FDA documents in the pharmaceutical industry (FDA, 1995, 1997, 2001) to address that issue.


On the capacity of Lagrangians in the cotangent disc bundle of the torus

Abstract: The paper is wihdrawn due to a critical error in the argument using the spectral sequence


Support of Non-separable Multivariate Scaling Function

Abstract: We make an estimation of the support of a multivariable scaling function for an arbitrary dilation matrix. We give a method of calculating the values of the scaling function on a tight set using the knowledge of the size of the support.


The Sasaki Cone and Extremal Sasakian Metrics

Abstract: We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five dimensional case.


Characterizations of algebras of rapidly decreasing generalized functions

Abstract: The well-known characterizations of Schwartz space $\mathcal{S}$ of rapidly decreasing functions is extended to the algebra $\mathcal{G}_{\mathcal{S}}$ of rapidly decreasing generalized functions and to the algebra $\mathcal{G}_{% \mathcal{S}}^{\infty}$ of regular rapidly decreasing generalized functions.


Products of Factorial Schur Functions

Abstract: The product of any finite number of factorial Schur functions can be expanded as a $Z[y]$-linear combination of Schur functions. We give a rule for computing the coefficients in such an expansion which generalizes a specialization of the Molev-Sagan rule, which in turn generalizes the classical Littlewood-Richardson rule.


An Abelian Category of Motivic Sheaves

Abstract: The goal of this paper is to construct a category of motivic "sheaves" on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the categoriesof constructible sheaves for the classical and etale topologies. Moreover, there is a tannakian subcategory of motivic local systems with a realization functor into the category of variations of mixed Hodge structures. Conversely, all basic geometric examples of the latter come from this motivic category.


Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree condition

Abstract: The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of $t_2(K(n_1,n_2,...,n_k))$. In this paper, we prove that if $n\geq 3$, and K(n,n) is 3-edge-colored such that every vertex has color degree 3, then $t_3(K(n,n))=3.$


On The Isoperimetric Spectrum of Graphs and Its Approximations

Abstract: In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set of $n$ disjoint subsets of the vertex set of the graph. We show that the second mean isoperimetric constant in this general setting, coincides with (the mean version of) the classical Cheeger constant of the graph, while for the rest of the spectrum we show that there is a fundamental difference between the $n$th isoperimetric constant and the number obtained by taking the minimum over all $n$-partitions. In this direction, we show that our definition is the correct one in the sense that it satisfies a Federer-Fleming-type theorem, and we also define and present examples for the concept of a supergeometric graph as a graph whose mean isoperimetric constants are attained on partitions at all levels. Moreover, considering the ${\bf NP}$-completeness of the isoperimetric problem on graphs, we address ourselves to the approximation problem where we prove general spectral inequalities that give rise to a general Cheeger-type inequality as well. On the other hand, we also consider some algorithmic aspects of the problem where we show connections to orthogonal representations of graphs and following J.~Malik and J.~Shi ($2000$) we study the close relationships to the well-known $k$-means algorithm and normalized cuts method.


On Certain Hypotheses in Optimal Control Theory and the Relationship of the Maximum Principle with the Dynamic Programming Method Proposed by L. I. Rozonoer

Abstract: In this paper we will study three hypotheses proposed by L. I. Rozonoer (Automation and Remote Control, 2003, vol.64, no.8, pp.1237--1240) in optimal control theory in order to derive conditions for the existence of an optimal control under all initial conditions, and the relationships between Pontryagin maximum principle and the dynamic programming method.


Cayley 4-form, comass, and triality isomorphisms

Abstract: Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds.


Complex product manifolds cannot be negatively curved

Abstract: We show that if $M = X \times Y$ is the product of two complex manifolds (of positive dimensions), then $M$ does not admit any complete K\"ahler metric with bisectional curvature bounded between two negative constants. More generally, a locally-trivial holomorphic fibre-bundle does not admit such a metric.


An elementary approach to some rigidity theorems

Abstract: Using elementary comparison geometry, we prove: Let $(M,g)$ be a simply-connected complete Riemannian manifold of dimension $\ge 3$. Suppose that the sectional curvature $K$ satisfies $ -1-s(r) \le K \le -1$, where $r$ denotes distance to a fixed point in $M$. If $\lim_{r \rt \infty} e^{2r}s(r) =0$, then $(M,g)$ has to be isometric to ${\mathbb H}^n$. The same proof also yields that if $K$ satisfies $-s(r) \le K \le 0$ where $\lim_{r \rt \infty} r^2s(r)=0$, then $(M,g)$ is isometric to $\R^n$, a result due to Greene and Wu. Our second result is a local one: Let $(M,g)$ be any Riemannian manifold. For $a \in \R$, if $K \le a$ on a geodesic ball $B_p(R)$ in $M$ and $K = a$ on $\partial B_p(R)$, then $K= a $ on $B_p(R)$.


Walk versus Wait: The Lazy Mathematician Wins

Abstract: In this recreational mathematics note, we address a simple, yet instructive question: Justin has to travel a distance of d miles along a bus route. Along this route, there are n bus stops i, each spaced at a distance of d_i from the starting point. At each bus stop, Justin is faced with a choice: to walk or to wait. If he walks on, he can still catch a bus at the next bus stop--but if a bus passes him while he walks, he is almost assured a longer wait. We model Justin's decision constraint and completely solve the model in a special case. The answer is intuitive: the optimal strategy is the laziest.


Cellularity of Cyclotomic Birman--Wenzl--Murakami algebras

Abstract: We show that the cyclotomic Birman-Wenzl-Murakami algebras are cellular by producing a cellular basis of affine tangle diagrams.


The kernel and continuity ideals of homomorphisms from C_0(\Omega)

Abstract: We give a description of the continuity ideals and the kernels of homomorphisms from the algebras of continuous functions on locally compact spaces into Banach algebras.


Uncountable families of prime z-ideals in C_0(R)

Abstract: Denote by $\continuum=2^{\aleph_0}$ the cardinal of continuum. We construct an intriguing family $(P_\alpha: \alpha\in\continuum)$ of prime $z$-ideals in $\C_0(\reals)$ with the following properties: If $f\in P_{i_0}$ for some $i_0\in\continuum$, then $f\in P_i$ for all but finitely many $i\in \continuum$; $\bigcap_{i\neq i_0} P_i \nsubset P_{i_0}$ for each $\i_0\in \continuum$. We also construct a well-ordered increasing chain, as well as a well-ordered decreasing chain, of order type $\kappa$ of prime $z$-ideals in $\C_0(\reals)$ for any ordinal $\kappa$ of cardinality $\continuum$.


Nonsplitting in Kirchberg's ideal-related KK-theory

Abstract: A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.


Nonparametric sequential prediction of time series

Abstract: Time series prediction covers a vast field of every-day statistical applications in medical, environmental and economic domains. In this paper we develop nonparametric prediction strategies based on the combination of a set of 'experts' and show the universal consistency of these strategies under a minimum of conditions. We perform an in-depth analysis of real-world data sets and show that these nonparametric strategies are more flexible, faster and generally outperform ARMA methods in terms of normalized cumulative prediction error.


Euler Numbers and polynomials associated with zeta functions

Abstract: In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.


Properties of Expectations of Functions of Martingale Diffusions

Abstract: Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitz continuous then f(t,x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t,x) will be convex in x and decreasing in t. We also define the marginal support of a process and show that it almost surely contains the paths of the process. Although f need not be jointly continuous, we show that it will be continuous on the marginal support of X. We prove these results for a generalization of diffusion processes that we call `almost-continuous diffusions', and includes all continuous and strong Markov processes.


Explicit fibrant replacement for discrete G-spectra

Abstract: If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also difficult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete G-spectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.


Hyperelliptic plane curves of type (d,d-2)

Abstract: In a previous paper, we classified and constructed all rational plane curves of type (d,d-2). In this paper, we generalize these results to irreducible plane curves of type (d,d-2) with positive genus.


Near-ideal model selection by $\ell_1$ minimization

Abstract: We consider the fundamental problem of estimating the mean of a vector $y=X\beta+z$, where $X$ is an $n\times p$ design matrix in which one can have far more variables than observations, and $z$ is a stochastic error term--the so-called "$p>n$" setup. When $\beta$ is sparse, or, more generally, when there is a sparse subset of covariates providing a close approximation to the unknown mean vector, we ask whether or not it is possible to accurately estimate $X\beta$ using a computationally tractable algorithm. We show that, in a surprisingly wide range of situations, the lasso happens to nearly select the best subset of variables. Quantitatively speaking, we prove that solving a simple quadratic program achieves a squared error within a logarithmic factor of the ideal mean squared error that one would achieve with an oracle supplying perfect information about which variables should and should not be included in the model. Interestingly, our results describe the average performance of the lasso; that is, the performance one can expect in an vast majority of cases where $X\beta$ is a sparse or nearly sparse superposition of variables, but not in all cases. Our results are nonasymptotic and widely applicable, since they simply require that pairs of predictor variables are not too collinear.


Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems

Abstract: We prove the global classical solvability of initial-boundary problems for semilinear first-order hyperbolic systems subjected to local and nonlocal nonlinear boundary conditions. We also establish lower bounds for the order of nonlinearity demarkating a frontier between regular cases (classical solvability) and singular cases (blow-up of solutions).


Convolution-Dominated Operators on Discrete Groups

Abstract: We study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for $x\in G$ and some $a\in \ell ^1(G)$. This class of "convolution-dominated" matrices forms a Banach-*-algebra contained in the algebra of bounded operators on $\ell ^2(G)$. Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that $G$ is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized $L^1$-algebras and the symmetry of group algebras.


The Lex-Plus-Powers Conjecture holds for pure powers

Abstract: We prove Evans' Lex-Plus-Powers Conjecture for ideals containing a monomial regular sequence.


Old and new examples of scale functions for spectrally negative Levy processes

Abstract: We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations.


An exact minimum degree condition for Hamilton cycles in oriented graphs

Abstract: We show that every sufficiently large oriented graph with minimum in- and outdegree at least (3n-4)/8 contains a Hamilton cycle. This is best possible and solves a problem of Thomassen from 1979.


On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

Abstract: Let $X$ be a finite sequence of length $m\geq 1$ in $\mathbb{Z}/n\mathbb{Z}$. The \textit{derived sequence} $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length 1 is reached, determines a triangle, the \textit{Steinhaus triangle $\Delta X$ generated by the sequence $X$}. We say that $X$ is \textit{balanced} if its Steinhaus triangle $\Delta X$ contains each element of $\mathbb{Z}/n\mathbb{Z}$ with the same multiplicity. An obvious necessary condition for $m$ to be the length of a balanced sequence in $\mathbb{Z}/n\mathbb{Z}$ is that $n$ divides the binomial coefficient $\binom{m+1}{2}$. It is an open problem to determine whether this condition on $m$ is also sufficient. This problem was posed by Hugo Steinhaus in 1963 for $n=2$ and generalized by John C. Molluzzo in 1976 for $n\geq3$. So far, only the case $n=2$ has been solved, by Heiko Harborth in 1972. In this paper, we answer positively Molluzzo's problem in the case $n=3^k$ for all $k\geq1$. Moreover, for every odd integer $n\geq3$, we construct infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. This is achieved by analysing the Steinhaus triangles generated by arithmetic progressions. In contrast, for any $n$ even with $n\geq4$, it is not known whether there exist infinitely many balanced sequences in $\mathbb{Z}/n\mathbb{Z}$. As for arithmetic progressions, still for $n$ even, we show that they are never balanced, except for exactly 8 cases occurring at $n=2$ and $n=6$.


Calculating conjugacy classes in Sylow $p$-subgroups of finite Chevalley groups

Abstract: In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial in $q$ with integer coefficients.


Algebras that satisfy Auslander's condition on vanishing of cohomology

Abstract: Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture - by a 2003 counterexample due to Jorgensen and Sega - motivates the consideration of the class of rings that do satisfy Auslander's condition. We call them AC rings and show that an AC Artin algebra that is left-Gorenstein is also right-Gorenstein. Furthermore, the Auslander-Reiten Conjecture is proved for AC rings, and Auslander's G-dimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.


Extended Hyperbolicity

Abstract: Given a complex space $X$, we cosidered the problem of finding a {\it hyperbolic model} of $X$. This is an object $\ip(X)$ with a morphism $i:X\to \ip(X)$ in such a way that $\ip(X)$ is ``hyperbolic'' in a suitable sense and $i$ is as close as possible to be an isomorphism. Using the theory of model categories, we found a definition of hyperbolic simplicial sheaf (for the strong topology) that extends the classical one of Brody for complex spaces. We prove the existence of hyperbolic models for any simplicial sheaf. Furthermore, the morphism $i$ can be taken to be a cofibration and an affine weak equivalence (in an algebraic setting, Morel and Voevodsky called it an $\aff$ weak equivalence). Imitating one possible definition of homotopy groups for a topological space, we defined the {\it holotopy} groups for a simplicial sheaf and showed that their vanishing in ``positive'' degrees is a necessary condition for a sheaf to be hyperbolic. We deduce that if $X$ is a complex space with a non zero holotopy group in positive degree, then its hyperbolic model (that in general will only be a simplicial sheaf) cannot be weakly equivalent to a hyperbolic complex space (in particular is not itself hyperbolic). We finish the manuscript by applying these results and a {\it topological realization functor}, constructed in the previous section, to prove that the hyperbolic models of the complex projective spaces cannot be weakly equivalent to hyperbolic complex spaces.


Invariant tensors and graphs

Abstract: We describe a correspondence between GL_n-invariant tensors and graphs, and show how this correspondence accomodates various types of symmetries and orientations.


Operator-valued Herglotz kernels and functions of positive real part on the ball

Abstract: We describe several classes of holomorphic functions of positive real part on the unit ball; each is characterized by an operator-valued Herglotz formula. Motivated by results of J.E. McCarthy and M. Putinar, we define a family of weighted Cauchy-Fantappi\`e pairings on the ball and establish duality relations between certain pairs of classes, and in particular we identify the dual of the positive Schur class. We also establish the existence of self-dual classes with respect to this pairing, and identify some extreme points of the positive Schur class.


Le lemme fondamental pour les algebres de Lie

Abstract: We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.


Semigroups of valuations dominating local domains

Abstract: A new criterion is given for a semigroup to be the semigroup of a valuation dominating an equicharacteristic local domain. The criterion is used to construct examples of well ordered subsemigroups of the positive rational numbers which are of ordinal type omega, but are not the value semigroup of a valuation on an equicharacteristic noetherian local domain. This shows that the necessary conditions on value semigroups given in Appendix 3 to Zariski and Samuel's book ``Commutative Algebra'' are not sufficient.


On the wonderful compactification

Abstract: These lecture notes explain the construction and basic properties of the wonderful compactification of a complex semisimple group of adjoint type. An appendix discusses the more general case of a semisimple symmetric space.


An Alternative Prior Process for Nonparametric Bayesian Clustering

Abstract: Prior distributions play a crucial role in Bayesian approaches to clustering. Two commonly-used prior distributions are the Dirichlet and Pitman-Yor processes. In this paper, we investigate the predictive probabilities that underlie these processes, and the implicit "rich-get-richer" characteristic of the resulting partitions. We explore an alternative prior for nonparametric Bayesian clustering -- the uniform process -- for applications where the "rich-get-richer" property is undesirable. We also explore the cost of this process: partitions are no longer exchangeable with respect to the ordering of variables. We present new asymptotic and simulation-based results for the clustering characteristics of the uniform process and compare these with known results for the Dirichlet and Pitman-Yor processes. We compare performance on a real document clustering task, demonstrating the practical advantage of the uniform process despite its lack of exchangeability over orderings.


Analytic continuation from a family of lines

Abstract: Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.


The representations of cyclotomic BMW algebras

Abstract: In this paper, we prove that the cyclotomic BMW algebras B2p+1,n are cellular in the sense of [16]. We also classify the irreducible B2p+1,nmodules over a field.


Minimality and nonergodicity on a family of flat surfaces in genus 3

Abstract: We prove that a certain family of flat surfaces in genus 3 does not fulfill Veech's Dichotomy. These flat surfaces provide uncountably many minimal but nonergodic directions. The conditions on this family are a combinatorical one and an irrationality condition. The Arnoux-Yoccoz surface fulfills this conditions.


Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II

Abstract: An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by rational numbers field. This proves a weak form of a conjecture of Ginzburg proposed in 1987.


Analytic Continuation of q-Euler numbers and polynomials

Abstract: In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $\zeta_{E,q}(s)$ in terms of nested series of $\zeta_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials.


On Universal Binary Hermitian Forms

Abstract: Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's list of universal quadratic forms and on the Bhargava-Hanke 290-Theorem. Our methods bypass nearly all of the ad hoc universality arguments required in the original classification.


Global Gorenstein dimensions of polynomial rings and of direct products of rings

Abstract: In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.


Toric Ideals of Flow Polytopes

Abstract: A referee found an error in the proof of the Main Theorem ("toric ideals of flow polytopes are generated in degree 3") that we could not fix. More precisely, the proof of Lemma 4.2.(ii) is incorrect. The results on Gr\"obner bases are untouched by this. ----- We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gr\"obner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m\times n)$-transportation polytopes, a similar result holds: they have Gr\"obner bases of at most degree $\lfloor mn/2\rfloor$. We construct a family of examples, where this bound is sharp.


Some examples of absolute continuity of measures in stochastic fluid dynamics

Abstract: A non linear Ito equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto-Sivashinsky equation and in the Navier-Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for large time is characterized.


Adjusted Bayesian inference for selected parameters

Abstract: We address the problem of providing inference from a Bayesian perspective for parameters selected after viewing the data. We present a Bayesian framework for providing inference for selected parameters, based on the observation that providing Bayesian inference for selected parameters is a truncated data problem. We show that if the prior for the parameter is non-informative, or if the parameter is a "fixed" unknown constant, then it is necessary to adjust the Bayesian inference for selection. Our second contribution is the introduction of Bayesian False Discovery Rate controlling methodology,which generalizes existing Bayesian FDR methods that are only defined in the two-group mixture model.We illustrate our results by applying them to simulated data and data froma microarray experiment.


Parametrizing recollement data

Abstract: We give a general parametrization of all the recollement data for a triangulated category with a set of generators. From this we deduce a characterization of when a perfectly generated (or aisled) triangulated category is a recollement of triangulated categories generated by a single compact object. Also, we use homological epimorphisms of dg categories to give a complete and explicit description of all the recollement data for (or smashing subcategories of) the derived category of a k-flat dg category. In the final part we give a bijection between smashing subcategories of compactly generated triangulated categories and certain ideals of the subcategory of compact objects, in the spirit of Henning Krause's work. This bijection implies the following weak version of the Generalized Smashing Conjecture: in a compactly generated triangulated category every smashing subcategory is generated by a set of Milnor colimits of compact objects.


Quantum triads: an algebraic approach

Abstract: A concept of quantum triad and its solution is introduced. It represents a common framework for several situations where we have a quantale with a right module and a left module, provided with a bilinear inner product. Examples include Van den Bossche quantaloids, quantum frames, simple and Galois quantales, operator algebras, or orthomodular lattices.


Sharp estimates of the Kobayashi metric and Gromov hyperbolicity

Abstract: Let D be a smooth relatively compact and strictly J-pseudoconvex domain in a four dimensional almost complex manifold (M,J). We give sharp estimates of the Kobayashi metric. Our approach is based on an asymptotic quantitative description of both the domain D and the almost complex structure J near a boundary point. Following Z.M.Balogh and M.Bonk, these sharp estimates provide the Gromov hyperbolicity of the domain D.


On torsion torsionfree triples

Abstract: We study torsion torsionfree(=TTF) triples in abelian and triangulated categories. (Notice that TTF triples in a triangulated category are essentially in bijection with recollement data for this triangulated category.) In particular, we complete Jans' characterization of split TTF triples on a category of modules, prove a weak version of the Generalized Smashing Conjecture, use homological epimorphisms of differential graded(=dg) categories to give an explicite description of all the TTF triples in the derived category of a k-flat dg category and develope an unbounded approach to Koenig's theorem on recollements of right bounded derived categories of ordinary algebras.


The quotient of a complete symmetric variety

Abstract: We study the quotient of a completion of a symmetric variety G/H under the action of H. We prove that this is isomorphic to the closure of the image of an isotropic torus under the action of the restricted Weyl group. In the case the completion is smooth and toroidal we describe the set of semistable points.


Twisted Yangians and Mickelsson Algebras II

Abstract: We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.


A remark on amoebas in higher codimensions

Abstract: It is shown that tube sets over amoebas of algebraic varieties (and, more generally, of almost periodic holomorphic chains) of dimension q are q-pseudoconcave in the sense of Rothstein. This is a direct consequence of a representation of such sets as supports of positive closed currents.


A New Approach on Constant Angle Surfaces in E^3

Abstract: In this paper we study constant angle surfaces in Euclidean 3-space. Even that the result is a consequence of some classical results involving the Gauss map (of the surface), we give another approach to classify all surfaces for which the unit normal makes a constant angle with a fixed direction.


On the Neron-Severi group of surfaces with many lines

Abstract: For a binary quartic form $\phi$ without multiple factors, we classify the quartic K3 surfaces $\phi(x,y)=\phi(z,t)$ whose Neron-Severi group is (rationally) generated by lines. For generic binary forms $\phi$, $\psi$ of prime degree without multiple factors, we prove that the Neron-Severi group of the surface $\phi(x,y)=\psi(z,t)$ is rationally generated by lines.


Dimension vs. Genus: A surface realization of the little k-cubes and an E_{\infty}-operad

Abstract: We define a new $E_{\infty}$ operad based on surfaces with foliations which contains $E_k$ sub-operads. We construct CW models for these operads and provide applications of these models by giving actions on Hochschild complexes -thus making contact with string topology-, by giving explicit cell representatives for the Dyer-Lashof-Cohen operations for the 2-cubes and by constructing new $\Omega$ spectra. The underlying novel principle is that we can trade genus in the surface representation vs. the dimension $k$ of the little $k$-cubes.


Polyhedral hyperbolic metrics on surfaces

Abstract: In the last section of \cite{CompHyp} it is proved that the map $\mathcal{I}$ is a finite-sheeted covering map between $\mathcal{P}$ and $\mathcal{M}$. As $\mathcal{M}$ is simply connected it is deduced that $\mathcal{I}$ is a homeomorphism. The fact that $\mathcal{P}$ is connected is missing. Here we provide a proof.


Maximum and entropic repulsion for a Gaussian membrane model in the critical dimension

Abstract: We consider the real-valued centered Gaussian field on the four-dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field.


Sur les ensembles d'entiers reconnaissables

Abstract: Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and V recognizable, then E is a finite union of arithmetical progressions.


Hamiltonian handleslides for Heegaard Floer homology

Abstract: A $g$-tuple of disjoint, linearly independent circles in a Riemann surface of genus $g$ determines a `Heegaard torus' in its $g$-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic Lagrangian submanifolds. This provides an alternative route to the handleslide-invariance of Ozsvath-Szabo's Heegaard Floer homology.


Counting One-Vertex Maps

Abstract: The number of distinct maps (pre-maps) with a single vertex and valence $d$ is computed for any value of $d$. The types of maps (pre-maps) that we consider depend on whether the underlaying graph (pre-graph) is signed or unsigned and directed or undirected.


Sampling-Based Resolution-Complete Algorithms for Safety Falsification of Linear Systems

Abstract: In this paper, we describe a novel approach for checking safety specifications of a dynamical system with exogenous inputs over infinite time horizon that is guaranteed to terminate in finite time with a conclusive answer. We introduce the notion of resolution completeness for analysis of safety falsification algorithms and propose sampling-based resolution-complete algorithms for safety falsification of linear time-invariant discrete time systems over infinite time horizon. The algorithms are based on deterministic incremental search procedures, exploring the reachable set for feasible counter examples to safety at increasing resolution levels of the input. Given a target resolution of inputs, the algorithms are guaranteed to terminate either with a reachable state that violates the safety specification, or prove that no input exists at the given resolution that violates the specification.


Polar orthogonal representations of real reductive algebraic groups

Abstract: We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras.


Discrete bidding games

Abstract: We study variations on combinatorial games in which, instead of alternating moves, the players bid with discrete bidding chips for the right to determine who moves next. We consider both symmetric and partisan games, and explore differences between discrete bidding games and Richman games, which allow real-valued bidding. Unlike Richman games, discrete bidding game variations of many familiar games, such as chess, Connect Four, and even Tic-Tac-Toe, are suitable for recreational play. We also present an analysis of Tic-Tac-Toe for both discrete and real-valued bidding.


No Finite Invariant Density for Misiurewicz Exponential Maps

Abstract: For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane.


Minimal polynomial of an exponential automorphism of C^n

Abstract: We show that the minimal polynomial of a polynomial exponential automorphism F of C^n (i.e. F=exp(D), where D is a locally nilpotent derivation) is of the form \mu_F(T)=(T-1)^d, d=min{m \in N: D^m(X_i)=0 for i=1,...,n}.


Towards a classification of the tridiagonal pairs

Abstract: Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \in End(V)$ that satisfy (i)--(iv) below: (i) Each of $A,A^*$ is diagonalizable. (ii) There exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. (iii) There exists an ordering $\{V^*_i\}_{i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq \delta$, where $V^*_{-1}=0$ and $V^*_{\delta+1}=0$. (iv) There is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\em tridiagonal pair} on $V$. Let $E^*_0$ denote the element of $End(V)$ such that $(E^*_0-I)V^*_0=0$ and $E^*_0V^*_i=0$ for $1 \leq i \leq d$. Let $D$ (resp. $D^*$) denote the $K$-subalgebra of $End(V)$ generated by $A$ (resp. $A^*$). In this paper we prove that the span of $E^*_0 D D^*DE^*_0$ equals the span of $E^*_0D E^*_0DE^*_0$, and that the elements of $E^*_0 D E^*_0$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.


From Bombieri's Mean Value Theorem to the Riemann Hypothesis

Abstract: From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.


A proof of the DDVV conjecture and its equality case

Abstract: In this paper, we give a proof of the DDVV conjecture which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. Furthermore we solved the problem of its equality case.


Auslander-Reiten theory for simply connected differential graded algebras

Abstract: Peter Jorgensen introduced the Auslander-Reiten quiver of a simply connected Poincare duality space. He showed that its components are of the form ZA_infty and that the Auslander-Reiten quiver of a d-dimensional sphere consists of d-1 such components. In this thesis we show that this is the only case where finitely many components appear. More precisely, we construct families of modules, where for each family, each module lies in a different component. Depending on the cohomology dimensions of the differential graded algebras which appear, this is either a discrete family or an n-parameter family for all n.


Covering theorems for Artinian rings

Abstract: The covering properties of Artinian rings which depend on their additive structure only, are investigated.


On Potentially 3-regular graph graphic Sequences

Abstract: For given a graph $H$, a graphic sequence $\pi=(d_1,d_2,...,d_n)$ is said to be potentially $H$-graphic if there exists a realization of $\pi$ containing $H$ as a subgraph. In this paper, we characterize the potentially $H$-graphic sequences where $H$ denotes 3-regular graph with 6 vertices. In other words, we characterize the potentially $K_{3,3}$ and $K_6-C_6$-graphic sequences where $K_{r,r}$ is an $r\times r$ complete bipartite graph. One of these characterizations implies a theorem due to Yin [25].


Structure of normal twisted group rings

Abstract: In this paper we describe those groups G and commutative rings K for which the twisted group rings K*G are f-normal.


Invariant boundary distributions for finite graphs

Abstract: Let $\Gamma$ be the fundamental group of a finite connected graph $\mathcal G$. Let $\mathfrak M$ be an abelian group. A {\it distribution} on the boundary $\partial\Delta$ of the universal covering tree $\Delta$ is an $\mathfrak M$-valued measure defined on clopen sets. If $\mathfrak M$ has no $\chi(\mathcal G)$-torsion then the group of $\Gamma$-invariant distributions on $\partial\Delta$ is isomorphic to $H_1(\mathcal G,\mathfrak M)$.


A stronger model for peg solitaire, II

Abstract: The main problem addressed here is to decide whether it is possible or not to go from a given position on a peg-solitaire board to another one. No non-trivial sufficient conditions are known, but tests have been devised to show impossibility. We expose the way these tests work in a unified formalism and provide a new test which is strictly stronger than all previous ones.


On Appell Sets and the Fueter-Sce Mapping

Abstract: It is proved, that the recently discussed Appell polynomials in Clifford algebras are the Fueter-Sce extension of the complex monomials z^k. Furthermore, it is shown, for which complex functions the Fueter-Sce extension and the extension method using Appell polynomials coincide.


The Goto numbers of parameter ideals

Abstract: Let Q be a parameter ideal of a Noetherian local ring (R,m). The Goto number g(Q) of Q is the largest integer g such that Q:m^g is integral over Q. We examine the values of g(Q) as Q varies over the parameter ideals of R. We concentrate mainly on the case where dim R = 1, and many of our results concern parameter ideals of a numerical semigroup ring.


Meromorphic functions of one complex variable. A survey

Abstract: This is an appendix to the English translation of the book by A. A. Goldberg and I. V. Ostrovskii, Distribution of values of meromorphic functions, Moscow, Nauka, 1970. An English translation of this book is to be published soon by the AMS. In this appendix we survey the results obtained on the topics of the book after 1970.


Some remarks on tangent martingale difference sequences in $L^1$-spaces

Abstract: Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X-valued martingales f and g with tangent martingale difference sequences one has \[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L^1. We state several problems related to (*) and other decoupling inequalities.


Combinatorics in affine flag varieties

Abstract: The Littelmann path moel gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann and others has demonstrated a connection between this model and the geometry of the loop Grassmannian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the "cells" in generalized Mirkovic-Vilonen intersections.


Koppelman formulas and the $\dbar$-equation on an analytic space

Abstract: Let $X$ be an analytic space of pure dimension. We introduce a formalism to generate intrinsic weighted Koppelman formulas on $X$ that provide solutions to the $\dbar$-equation. We prove that if $\phi$ is a smooth $(0,q+1)$-form on a Stein space $X$ with $\dbar\phi=0$, then there is a smooth $(0,q)$-form $\psi$ on $X_{reg}$ with at most polynomial growth at $X_{sing}$ such that $\dbar\psi=\phi$. The integral formulas also give other new existence results for the $\dbar$-equation and Hartogs theorems, as well as new proofs of various known results.


Hermitian integral geometry

Abstract: We give in explicit form the principal kinematic formula for the action of the affine unitary group on $\C^n$, together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.


Nonparametric estimation of a convex bathtub-shaped hazard function

Abstract: In this paper, we study the nonparametric maximum likelihood estimator (MLE) of a convex hazard function. We show that the MLE is consistent and converges at a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function is positive and strictly convex. Moreover, we establish the pointwise asymptotic distribution theory of our estimator under these same assumptions. One notable feature of the nonparametric MLE studied here is that no arbitrary choice of tuning parameter (or complicated data-adaptive selection of the tuning parameter) is required.


Convergence of multi-dimensional quantized $SDE$'s

Abstract: We quantize a multidimensional $SDE$ (in the Stratonovich sense) by solving the related system of $ODE$'s in which the $d$-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the $ODE$ converge toward the solution of the $SDE$. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for $\frac 1q$-H\" older distance, $q>2$, in $L^p(\P)$.


Generalized incidence theorems, homogeneous forms, and sum-product estimates in finite fields

Abstract: In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in \cite{HI07} and \cite{HIKR07}. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions. In particular, our machinery enables us to prove that if $E \subset {\Bbb F}_q^d$, $d \ge 4$, the $d$-dimensional vector space over a finite field ${\Bbb F}_q$, of size much greater than $q^{\frac{d}{2}}$, and if $E$ is a product set, then the set of volumes of $d$-dimensional parallelepipeds determined by $E$ covers ${\Bbb F}_q$. This result is sharp as can be seen by taking $E$ to equal to $A \times A \times ... \times A$, where $A$ is a sub-field of ${\Bbb F}_q$ of size $\sqrt{q}$. In three dimensions we establish the same result if $|E| \gtrsim q^{{15/8}}$. We prove in three dimensions that the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^{{3/2}}$. Finally we show that in three dimensions the set of volumes covers a positive proportion of ${\Bbb F}_q$ if $|E| \ge Cq^2$, without any further assumptions on $E$, which is again sharp as taking $E$ to be a 2-plane through the origin shows.


Biflatness and Pseudo-Amenability of Segal algebras

Abstract: We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, L1(G), and the Fourier algebra, A(G), of a locally compact group, G.


Jumping Numbers on Algebraic Surfaces with Rational Singularities

Abstract: In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.


Global fixed points for centralizers and Morita's Theorem

Abstract: We prove a global fixed point theorem for the centralizer of a homeomorphism of the two dimensional disk $D$ that has attractor-repeller dynamics on the boundary with at least two attractors and two repellers. As one application, we show that there is a finite index subgroup of the centralizer of a pseudo-Anosov homeomorphism with infinitely many global fixed points. As another application we give an elementary proof of Morita's Theorem, that the mapping class group of a closed surface $S$ of genus $g$ does not lift to the group of diffeormorphisms of $S$ and we improve the lower bound for $g$ from 5 to 3.


On Existence and Uniqueness of Universal Enveloping Locally C*-Algebra for a Locally JB-Algebra

Abstract: A theorem is presented on existence and uniqueness up to the topological *-isomorphism of universal locally C*-algebra for an arbitrary locally JB-algebra.


Quantum invariants can provide sharp Heegaard genus bounds

Abstract: Using Seifert fibered three-manifold examples of Boileau and Zieschang, we demonstrate that the Reshetikhin-Turaev quantum invariants may be used to provide a sharp lower bound on the Heegaard genus which is strictly larger than the rank of the fundamental group.


On lattices and their ideal lattices, and posets and their ideal posets

Abstract: For P a poset or lattice, let Id(P) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that Id(P) is always, and id(P) often, "essentially larger" than P. In the first vein, we find that a poset P admits no "<"-respecting map (and so in particular, no one-to-one isotone map) from Id(P) into P, and, going the other way, that an upper semilattice S admits no semilattice homomorphism from any subsemilattice of itself onto Id(S). The slightly smaller object id(P) is known to be isomorphic to P if and only if P has ascending chain condition. This result is strengthened to say that the only posets P_0 such that for every natural number n there exists a poset P_n with id^n(P_n)\cong P_0 are those having ascending chain condition. On the other hand, a wide class of cases is noted here where id(P) is embeddable in P. Counterexamples are given to many variants of the results proved.


Simple Jordan conformal superalgebras

Abstract: We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.


Lifting Group Actions and Nonnegative Curvature

Abstract: We show that all vector bundles over CP^2 which are not spin admit a complete metric with nonnegative sectional curvature. In the proof we construct a nonnegatively curved metric on the corresponding principle bundle by showing that it admits a cohomogeneity one action with singular orbits of codimension 2. This is closely related to the problem of when an action of G on the base of an L principle bundle lifts to the total space, such that the lift commutes with L. We solve this lifting problem for all SO(k) principle bundles over a 4-dimensional simply connected base B with G a cohomogeneity one action on B.


A Note on Banach Principle for JW-algebras

Abstract: In the sequel we establish the Banach Principle for semifinite JW-algebras without direct summand of type I sub 2, which extends the recent results of Chilin and Litvinov on the Banach Principle for semifinite von Neumann algebras to the case of JW-algebras.


Sifting Function Partition for the Goldbach Problem

Abstract: All sieve methods for the Goldbach problem sift out all the composite numbers; even though, strictly speaking, it is not necessary to do so and which is, in general, very difficult. Some new methods introduced in this paper show that the Goldbach problem can be solved under sifting out only some composite numbers. In fact, in order to prove the Goldbach conjecture, it is only necessary to show that there are prime numbers left in the residual integers after the initial sifting! This idea can be implemented by using one of the three methods called sifting function partition by integer sort, sifting function partition by intervals and comparative sieve method, respectively. These are feasible methods for solving both the Goldbach problem and the problem of twin primes. An added bonus of the above methods is the elimination of the indeterminacy of the sifting functions brought about by their upper and lower bounds.


On the monochromatic Schur Triples type problem

Abstract: We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \geq 2$. We also find a new upper bound for the minimum number, over all $r$-colorings of $[1,n]$, of monochromatic Schur triples, for $r \geq 3$.


Equivariant Ricci flow with surgery and applications to finite group actions on geometric 3-manifolds

Abstract: We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This answers a question posed by Thurston in [32].


One curious proof of Fermat's little theorem

Abstract: We give a proof of Fermat's little theorem which does not use nor arithmetic(Euclidean algorithm) neither algebra (group theory), but it rather employs the field of the formal power series Q((x)). The note is an example of a mathematical joke, though it contains a rigorous proof. (The paper will appear in print exactly as in the version v3).


Symmetric subgroups in modular group algebras

Abstract: Let V(KG) be a normalised unit group of the modular group algebra of a finite p-group G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=<G,S*>, where S* is a set of symmetric units of V(KG).


Analysis of the physical Laplacian and the heat flow on a locally finite graph

Abstract: We study the physical Laplacian and the corresponding heat flow on an infinite, locally finite graph with possibly unbounded valence.


Classification problems for system of forms and linear mappings

Abstract: We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric, skew-symmetric, or Hermitian forms, (iii) isometric or selfadjoint operators on a space with nonsingular symmetric, or skew-symmetric, or Hermitian form are obtained over any field of characteristic not 2 up to classification of Hermitian forms over its finite extensions.


A Modified Version of Free Orbit-Dimension of von Neumann Algebras

Abstract: Based on the notion of free orbit-dimension introduced by D. Hadwin and J. Shen [4], we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent on the generating set, and we extend some results in [4] to von Neumann algebras that are not finitely generated.


Semiclassical second microlocal propagation of regularity and integrable systems

Abstract: We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold of $T^*X.$ The construction of the calculus, closely analogous to one performed by Bony in the setting of homogeneous Lagrangians, proceeds via the consideration of a model case, that of the zero section of $T^*\mathbb{R}^n,$ and conjugation by appropriate Fourier integral operators. We prove a propagation theorem for the associated wavefront set analogous to H\"ormander's theorem for operators of real principal type. As an application, we consider the propagation of Lagrangian regularity on invariant tori for quasimodes (e.g. eigenfunctions) of an operator with completely integrable classical hamiltonian. We prove a secondary propagation result for second wavefront set which implies that even in the (extreme) case of Lagrangian tori with all frequencies rational, provided a nondegeneracy assumption holds, Lagrangian regularity either spreads to fill out a whole torus or holds nowhere locally on it.


Rigidity of Conformally Compact Manifolds with the Round Sphere as the Conformal Infinity

Abstract: In this paper we prove that under a lower bound on the Ricci curvature and an asymptotic assumption on the scalar curvature, a complete conformally compact manifold $(M^{n+1},g)$, with a pole $p$ and with the conformal infinity in the conformal class of the round sphere, has to be the hyperbolic space.


Introduction to the Prisoners Versus Guards Game

Abstract: We introduce a two-player game in which one and his/her opponent attempt to pack as many ``prisoners'' as possible on the squares of an n-by-n checkerboard; each prisoner has to be ``protected'' by at least as many guards as the number of the other prisoners adjacent. Initially, the board is covered entirely with guards. The players take turns adjusting the board configuration using one of the following rules in each turn: I. Replace one guard with a prisoner of the player's color. II. Replace one prisoner of either color with a guard and replace two other guards with prisoners of the player's color. We analyze winning strategies for small n (n<5) and the maximum number of prisoners in general. We show that this maximum is less than (7n^2+4n)/11 and conjecture it is more likely 3n^2/5+O(n).


Classification of 5-dimensional MD-algebras having commutive derived ideals

Abstract: In this paper, we study a subclass of the class of MD-algebras, i.e., the class of solvable real Lie algebras such that the K-orbits of its corresponding connected and simply connected Lie groups are either orbits of dimension zero or orbits with maximal dimensions. Our main result is to classify, up to isomorphism, all the 5-dimensional MD-algebras having commutative derived ideals.


Dynamics of Automorphisms of Compact Complex Manifolds

Abstract: We give an algebro-geometric approach towards the dynamics of automorphisms/endomorphisms of projective varieties or compact K\"ahler manifolds, try to determine the building blocks of automorphisms /endomorphisms, and show the relation between the dynamics of automorphisms/endomorphisms and the geometry of the underlying manifolds.


Batch kernel SOM and related Laplacian methods for social network analysis

Abstract: Large graphs are natural mathematical models for describing the structure of the data in a wide variety of fields, such as web mining, social networks, information retrieval, biological networks, etc. For all these applications, automatic tools are required to get a synthetic view of the graph and to reach a good understanding of the underlying problem. In particular, discovering groups of tightly connected vertices and understanding the relations between those groups is very important in practice. This paper shows how a kernel version of the batch Self Organizing Map can be used to achieve these goals via kernels derived from the Laplacian matrix of the graph, especially when it is used in conjunction with more classical methods based on the spectral analysis of the graph. The proposed method is used to explore the structure of a medieval social network modeled through a weighted graph that has been directly built from a large corpus of agrarian contracts.


A Note on Testing of Hypothesis

Abstract: In this paper, a problem of testing is discussed when the samples have been drawn from the normal distribution. The study of hypothesis testing is also extended to Baye's set up.


Inequalities and Ehrhart $\delta$-Vectors

Abstract: For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$-vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope.


Cyclotomic Solomon Algebras

Abstract: This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type $G(r,1,n)$. As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for certain `reflection subgroups'. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in $r$. This allows us to define a deformation, or $q$-analogue, of these algebras which depends on a parameter $q$. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.


Groups not acting on manifolds

Abstract: In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.


Instant Evaluation and Demystification of $\zeta(n),L(n,\chi)$ that Euler,Ramanujan Missed - I

Abstract: For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive integral argument in first variable.On using functionalequations this results in instant evaluation of Riemann Zeta function at positive even integral values of its argument and of Dirichlet L series at positive integral values of its argument,when the argument and the corresponding Dirichlet character are both even or both odd.We also obtain finite sum expression for any Dirichlet L series,when its argument is one.We also deal with Lerch's Zeta function on similar lines.


A more accurate treatment of the problem of drawing the shortest line on a surface

Abstract: E727 in the Enestrom index. This is a translation from the Latin original "Accuratior evolutio problematis de linea brevissima in superficie quacunque ducenda" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop equations that give the geodesics on this surface. I am new to the calculus of variations, so it is not clear to me what steps follow from results that are previously known (like the Euler-Lagrange equation in the calculations) and what steps follow from earlier in this paper. I would appreciate comments from any readers who are familiar with calculus of variations.


Gelfand-Kirillov Conjecture and Harish-Chandra Modules for Finite W-Algebras

Abstract: We address two problems regarding the structure and representation theory of finite W-algebras associated with the general linear Lie algebras. Finite W-algebras can be defined either via the Whittaker model of Kostant or, equivalently, by the quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of the finite W-algebras. The second main result is a parametrization of finite families of irreducible Harish-Chandra modules by the characters of the Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Harish-Chandra modules for the finite W-algebras.


Characteristic classes of $\ai$-algebras

Abstract: Standard combinatorial construction, due to Kontsevich, associates to any $\ai$-algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We propose an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite dimensional Lie algebras. An application to topological conformal field theories is given.


Diophantine inequality for equicharacteristic excellent Henselian local domains

Abstract: G. Rond has proved a Diophantine type inequality for the field of quotients of the convergent or formal power series ring in multivariables. We generalize his theorem to the field of the quotients of an excellent Henselian local domain in equicharacteristic case.


A BGG-type resolution for tensor modules over general linear superalgebra

Abstract: We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.


Computation of 2-groups of narrow logarithmic divisor classes of number fields

Abstract: We present an algorithm for computing the 2-group of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK2(F).


Sur le sous-groupe des \'el\'ements de hauteur infinie du K2 d'un corps de nombres

Abstract: By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for any number field F.


Approche logarithmique des noyaux \'etales sauvages des corps de nombres

Abstract: We study the l-part of the the wild \'etale kernels WK2i(F) of an arbitary number field F for a given prime l in connection with the logarithmic l-class groups. From the logarithmic arithmetic we deduce rank formulas, periodicity and reflection theorems, triviality characterizations and various consequences.


G\'en\'eralisation d'un Th\'eor\`eme d'Iwasawa

Abstract: We extend to convenient finite quotients of a noetherian Lambda-module the classical result of K. Iwasawa giving the asymptotic expression of the l-part of the number of ideal class groups in Zl-extensions of number fields. Then, in the arithmetic context, we compute the three characters associated by this way to the l-groups of T-infinitesimal S-classes in the cyclotomic tower and relate them to the classical invariants and the decomposition characters associated to the finite sets of places S and T. A main tool in this study is the so-called Spiegelungssatz of Georges Gras, which exchanges (wild or tame) ramification and decomposition. The main results of this arithmetical part extend those we obtained with Christian Maire in a previous article. The most intricate study of the wild contribution of the sets S and T involves a generalization of a classical result of R. Greenberg on the genus theory of cyclotomic towers.


A new Algorithm for the Computation of logarithmic l-Class Groups of Number Fields

Abstract: We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.


Estimating of $P(Y<X)$ in the Exponential case Based on Censored Samples

Abstract: In this article, the estimation of reliability of a system is discussed $p(y<x)$ when strength, $X$, and stress, $Y$, are two independent exponential distribution with different scale parameters when the available data are type II Censored sample. Different methods for estimating the reliability are applied. The point estimators obtained are maximum likelihood estimator, uniformly minimum variance unbiased estimator, and Bayesian estimators based on conjugate and non informative prior distributions. A comparison of the estimates obtained is performed. Interval estimators of the reliability are also discussed.


Groebner bases of nested configurations

Abstract: In this paper we introduce a new and large family of configurations whose toric ideals possess quadratic Groebner bases. As an application, a generalization of algebras of Segre-Veronese type will be studied.


Relative Hyperbolic Extensions of Groups and Cannon-Thurston Maps

Abstract: Let $1\to (K,K_1)\to (G,N_G(K_1))\to(Q,Q_1)\to 1$ be a short exact sequence of pairs of finitely generated groups with $K$ strongly hyperbolic relative to proper subgroup $K_1$. Assuming that for all $g\in G$ there exists $k\in K$ such that $gK_1g^{-1}=kK_1k^{-1}$, we prove that there exists a quasi-isometric section $s\colon Q \to G$. Further we prove that if $G$ is strongly hyperbolic relative to the normalizer subgroup $N_G(K_1)$ and weakly hyperbolic relative to $K_1$, then there exists a Cannon-Thurston map for the inclusion $i\colon\Gamma_K\to \Gamma_G$.


On the graph-connectivity of skeleta of convex polytopes

Abstract: Given a $d$-dimensional convex polytope $P$ and nonnegative integer $k$ not exceeding $d-1$, let $G_k (P)$ denote the simple graph on the node set of $k$-dimensional faces of $P$ in which two such faces are adjacent if there exists a $(k+1)$-dimensional face of $P$ which contains them both. The graph $G_k (P)$ is isomorphic to the dual graph of the $(d-k)$-dimensional skeleton of the normal fan of $P$. For fixed values of $k$ and $d$, the largest integer $m$ such that $G_k (P)$ is $m$-vertex-connected for all $d$-dimensional polytopes $P$ is determined. This result generalizes Balinski's theorem on the one-dimensional skeleton of a $d$-dimensional convex polytope.


On the extremal rays of the cone of positive, positive definite functions

Abstract: The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.


The Riemannian manifolds with boundary and large symmetry

Abstract: Sixty years ago, S. B. Myers and N. E. Steenrod ({\it Ann. of Math.} {\bf 40} (1939), 400-416) showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. Recently A. V. Bagaev and N. I. Zhukova ({\it Siberian Math. J.} {\bf 48} (2007), 579-592) proved the same result for a Riemannian orbifold. In this paper, we firstly show that the isometry group of a Riemannian manifold $M$ with boundary has dimension at most ${1/2} \dim M (\dim M-1)$. Then we completely classify such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension.


A Survey of Simple Permutations

Abstract: We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.


Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

Abstract: We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $\tau$ between the lower half and the upper half of the box, and the maximal flow $\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\tau$ in rational directions. Kesten and Zhang have proved the law of large numbers for $\tau$ and $\phi$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\tau /s$ and $\phi /s$ are abnormally small. For $\tau$, the box can have any orientation, whereas for $\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\tau /s$ and $\phi /s$, and we improve the conditions required to obtain the law of large numbers for these variables.


Detecting change-points in a discrete distribution via model selection

Abstract: This paper is concerned with the detection of multiple change-points in the joint distribution of independent categorical variables. The procedures introduced rely on model selection and are based on a penalized least-squares criterion. Their performance is assessed from a nonasymptotic point of view. Using a special collection of models, a preliminary estimator is built. According to an existing model selection theorem, it satisfies an oracle-type inequality. Moreover, thanks to an approximation result demonstrated in this paper, it is also proved to be adaptive in the minimax sense. In order to eliminate some irrelevant change-points selected by that first estimator, a two-stage procedure is proposed, that also enjoys some adaptivity property. Besides, the first estimator can be computed with a complexity only linear in the size of the data. A heuristic method allows to implement the second procedure quite satisfactorily with the same computational complexity.


On Tsfasman--Vl\u{a}du\c{t} Invariants of Infinite Global Fields

Abstract: In this article we study certain asymptotic properties of global fields. We consider the set of Tsfasman-Vladuts invariants of infinite global fields and answer some natural questions arising from their work. In particular, we prove the existence of infinite global fields having finitely many strictly positive invariants at given places, and the existence of infinite number fields with certain prescribed invariants being zero. We also give precisions on the deficiency of infinite global fields and on the primes decomposition in those fields.


Laws of Large Numbers for Continuous Belief Measures on Compact Spaces

Abstract: We prove for outer continuous belief measures defined on compact spaces strong and weak laws of large numbers as Kolmogorov's one for measures. These results contribute to M. Marinacci's (Journal of Economic Theory 84 (1999) 145-195) though with different methods.


On the multiple q-Genocchi and Euler numbers

Abstract: The purpose of this paper is to present a systemic study of some families of multiple q-Genocchi and euler numbers by using multivariate q-Volkenborn integral. From the studies of those q-Genocchi numbers and polynomials of higher order we derive some interesting identities related to q-Genocchi numbers and polynomials of higher order.


Imprecise Markov chains and their limit behaviour

Abstract: When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n goes to infinity: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.


Law of large numbers for non-additive measures

Abstract: Our aim is to give for some classes non-additive measures some limit theorems. For balanced games we obtain a weak and strong law of large numbers for bounded random variables, a sharper conclusion is obtain with exact games. We provide an extension to upper enveloppe measures.


Hamilton-Pontryagin Integrators on Lie Groups: Introduction and Structure-Preserving Properties

Abstract: In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned Runge-Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and St\"{o}rmer-Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.


Paires de structures de contact sur les vari\'et\'es de dimension trois

Abstract: We introduce a notion of positive pair of contact structures on a 3-manifold which generalizes a previous definition of Eliashberg-Thurston and Mitsumatsu. Such a pair gives rise to a locally integrable plane field $\lambda$. We prove that if $\lambda$ is uniquely integrable and if both structures of the pair are tight, then the integral foliation of $\lambda$ doesn't contain any Reeb component whose core curve is homologous to zero. Moreover, the ambient manifold carries a Reebless foliation. We also show a stability theorem "\`a la Reeb" for positive pairs of tight contact structures.


Equilibrium States for Partially Hyperbolic Horseshoes

Abstract: In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \cite{DHRS07}. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of $C^\infty$ potentials with phase transition.


New results on lower bounds for the number of (at most k)-facets

Abstract: In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq k)$-edges for a fixed $0\leq k\leq \lfloor n/3 \rfloor -1$; 2. We give a simple construction showing that the lower bound $3\binom{k+2}{2}+3\binom{k-\lfloor \frac{n}{3} \rfloor+2}{2}$ for the number of $(\leq k)$-edges of a planar point set appeared in [Aichholzer et al. New lower bounds for the number of ($\leq k$)-edges and the rectilinear crossing number of $K_n$. {\em Disc. Comput. Geom.} 38:1 (2007), 1--14] is optimal in the range $\lfloor n/3 \rfloor \leq k \leq \lfloor 5n/12 \rfloor -1$; 3. We show that for $k < \lfloor n/(d+1) \rfloor$ the number of $(\leq k)$-facets of a set of $n$ points in general position in $\mathbb{R}^d$ is at least $(d+1)\binom{k+d}{d}$, and that this bound is tight in the given range of $k$.


Moderate deviations for random fields and random complex zeroes

Abstract: Moderate deviations for random complex zeroes are deduced from a new theorem on moderate deviations for random fields.


Conjectural estimates on the Mordell-Weil and Tate-Shafarevich groups of an abelian variety

Abstract: We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the N\'eron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators of the free part of the Mordell-Weil group. The method is an extension of the algorithm proposed by Yu. Manin for finding a basis for the non-torsion rational points of an elliptic curve defined over the rationals.


Lower bounds for measurable chromatic numbers

Abstract: The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10,..., 24, and we give a new proof that it grows exponentially with the dimension.


On Wittgenstein's philosophy of mathematics

Abstract: Mathematics cannot anymore be assimilated to a linguistic game, where formal proofs are strongly differentiated with conjectural thinking, without building any category of knowledge to understand the passage (Wittgenstein's gist). Nowadays, philosophy has to face with the growing, exponential ramified tree of speculative mathematical thinking. Our main (problematical) theses are: 1. In mathematics, there is no empirical automatism, and no separate, physical-like motricity. 2: The irreversible-synthetical must force to complexify the exegetical game of philosophy; numerical experiments in algebra and in number theory are a kind of letting blow up all possible problems; 4. The nature of mathematical questioning still remains in question.


On the robustness of power-law random graphs in the finite mean, infinite variance region

Abstract: We consider a conditionally Poissonian random graph model where the mean degrees, `capacities', follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component which is super-small in the sense that the typical distance between vertices is of the order of $\log\log N$. The shortest paths travel through a core consisting of nodes with high mean degrees. In this paper we derive upper bounds of the typical distance when an upper part of the core is removed, including the case that the whole core is removed.


Stability in the Stefan problem with surface tension (I)

Abstract: We develop a high-order energy method to prove asymptotic stability of flat steady surfaces for the Stefan problem with surface tension - also known as the Stefan problem with Gibbs-Thomson correction.


Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

Abstract: We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology.


Heat equation approach to index theorems on odd dimensional manifolds

Abstract: D.Freed has formulated and proved an index theorem on odd dimensional spin manifolds with boundary. The proof is based on analysis by Calderon and Seeley. In this note we are going to give a proof of this theorem using the heat kernels methods for boundary conditions of Dirichlet and Von Neumann type. Moreover we consider also the Atiyah-Patodi-Singer spectral boundary condition which is not considered in Freed's paper. As a direct consequence of the method, we will obtain some information about isospectral invariants of the boundary conditions. This proof does not uses the cobordism invariance of index and are easily generalized to family case.


Simultaneous analysis of Lasso and Dantzig selector

Abstract: We exhibit an approximate equivalence between the Lasso estimator and Dantzig selector. For both methods we derive parallel oracle inequalities for the prediction risk in the general nonparametric regression model, as well as bounds on the $\ell_p$ estimation loss for $1\le p\le 2$ in the linear model when the number of variables can be much larger than the sample size.


A Combinatorial Interpretation for Certain Relatives of the Conolly Sequence

Abstract: For any integer s >= 0, we derive a combinatorial interpretation for the family of sequences generated by the recursion (parameterized by s) h_s(n) = h_s(n - s - h_s(n - 1)) + h_s(n - 2 - s - h_s(n - 3)), n > s + 3, with the initial conditions h_s(1) = h_s(2) = ... = h_s(s+2) = 1 and h_s(s+3) = 2. We show how these sequences count the number of leaves of a certain infinite tree structure. Using this interpretation we prove that h_s sequences are "slowly growing", that is, h_s sequences are monotone nondecreasing, with successive terms increasing by 0 or 1, so each sequence hits every positive integer. Further, for fixed s the sequence h_s(n) hits every positive integer twice except for powers of 2, all of which are hit s+2 times. Our combinatorial interpretation provides a simple approach for deriving the ordinary generating functions for these sequences.


Some (big) irreducible components of the moduli space of minimal surfaces of general type with $p_g=q=1$ and $K^2=4$

Abstract: In this paper we study the minimal surfaces of general type with $p_g=q=1$ and $K^2=4$ whose Albanese general fibre has genus 2, classifying those such that the direct image (under the Albanese morphism) of the bicanonical sheaf is sum of line bundles. We find 8 unirational families, all of dimension strictly bigger than the expected one. These families are pairwise disjoint irreducible components of the moduli space of minimal surfaces of general type.


$G$-Parking Functions, Acyclic Orientations and Spanning Trees

Abstract: Given an undirected graph $G=(V,E)$, and a designated vertex $q\in V$, the notion of a $G$-parking function (with respect to $q$) was independently developed and studied by various authors, and has recently gained renewed attention. This notion generalizes the classical notion of a parking function associated with the complete graph. In this work, we study properties of {\em maximum} $G$-parking functions and provide a new bijection between them and the set of spanning trees of $G$ with no broken circuit. As a case study, we specialize some of our results to the graph corresponding to the discrete $n$-cube $Q_n$. We present the article in an expository self-contained form, since we found the combinatorial aspects of $G$-parking functions somewhat scattered in the literature, typically treated in conjunction with sandpile models and closely related chip-firing games.


Decay and Continuity of Boltzmann Equation in Bounded Domains

Abstract: Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{\infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $% L^{\infty}$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.


Hessian and gradient estimates for three dimensional special Lagrangian Equations with large phase

Abstract: We derive a priori interior Hessian and gradient estimates for special Lagrangian equation of phase at least a critical value in dimension three.


Monoidal categories of comodules for coquasi Hopf algebras and Radford's formula

Abstract: We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford's $S^4$ formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a monoidal isomorphism between certain double dual functors.


Hierarchical selection of variables in sparse high-dimensional regression

Abstract: We study a regression model with a huge number of interacting variables. We consider a specific approximation of the regression function under two ssumptions: (i) there exists a sparse representation of the regression function in a suggested basis, (ii) there are no interactions outside of the set of the corresponding main effects. We suggest an hierarchical randomized search procedure for selection of variables and of their interactions. We show that given an initial estimator, an estimator with a similar prediction loss but with a smaller number of non-zero coordinates can be found.


Projective normality of quotient varieties modulo finite groups

Abstract: In this note, we prove that for any finite dimensional vector space $V$ over an algebraically closed field $k$, and for any finite subgroup $G$ of $GL(V)$ which is either solvable or is generated by pseudo reflections such that the $|G|$ is a unit in $k$, the projective variety $\mathbb P(V)/G$ is projectively normal with respect to the descent of $\mathcal O(1)^{\otimes |G|}$.