Substochastic semigroups and densities of piecewise deterministic Markov processes

Abstract: Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.


Square-Difference-Free Sets of Size Omega(n^{0.7334...})

Abstract: A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = \Omega(n^{0.5(1+ \log_{65} 7)}) = \Omega(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = \Omega(n^{0.5(1+ \log_{205} 12)})= \Omega(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.


An analog of the Furstenberg-Katznelson-Weiss theorem on triangles in sets of positive density in finite field geometries

Abstract: We prove that if the cardinality of a subset of the 2-dimensional vector space over a finite field with $q$ elements is $\ge \rho q^2$, with $\frac{1}{\sqrt{q}}<<\rho \leq 1$, then it contains an isometric copy of $\ge c\rho q^3$ triangles.


Automorphisms of the truth-table degrees are fixed on some cone

Abstract: Let Dtt denote the set of truth-table degrees. A bijection p from Dtt to Dtt is an automorphism if for all truth-table degrees x and y we have x <=tt y if and only if p(x) <=tt p(y). We say an automorphism p is fixed on some cone if there is a degree b such that for all x >=tt b we have p(x) = x. We first prove that for every 2-generic real X we have X' is not tt below X + 0'. We next prove that for every real X >=tt 0' there is a real Y such that Y + 0' =tt Y' =tt X. Finally, we use this to demonstrate that every automorphism of the truth-table degrees is fixed on some cone.


Campedelli surfaces with fundamental group of order 8

Abstract: We prove that an etale cover Y of degree 8 of a Campedelli surface S is a complete intersection of four quadrics in P^6, obtaining as a consequence that Y is the universal cover of S, the covering group G=Gal(Y/S)is the topological fundamental group of S and that G cannot be the dihedral group of order 8. This paper patches up an incomplete manuscript of the third author.


3--symmetric and 3--decomposable drawings of $K_n$ (extended version)

Abstract: Even the most superficial glance at the vast majority of crossing-minimal geometric drawings of $K_n$ reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply {\em 3-symmetric}) . And second, they all are {\em 3-decomposable}, that is, there is a triangle $T$ enclosing the drawing, and a balanced partition $A, B, C$ of the underlying set of points $P$, such that the orthogonal projections of $P$ onto the sides of $T$ show $A$ between $B$ and $C$ on one side, $B$ between $A$ and $C$ on another side, and $C$ between $A$ and $B$ on the third side. In fact, we conjecture that all optimal drawings are 3-decomposable, and that there are 3-symmetric optimal constructions for all $n$ multiple of 3. In this paper, we show that any 3-decomposable geometric drawing of $K_n$ has at least $0.380029\binom{n}{4}+\Theta(n^3)$ crossings. On the other hand, we produce 3-symmetric and 3-decomposable drawings that improve the {\em general} upper bound for the rectilinear crossing number of $K_n$ to $0.380488\binom{n}{4}+\Theta(n^3)$. We also give explicit 3-symmetric and 3-decomposable constructions for $n<100$ that are at least as good as those previously known.


Optimized Schwarz preconditioning for SEM based magnetohydrodynamics

Abstract: A recent theoretical result on optimized Schwarz algorithms demonstrated at the algebraic level enables the modification of an existing Schwarz procedure to its optimized counterpart. In this work, it is shown how to modify a bilinear FEM based Schwarz preconditioning strategy originally presented in [Fischer, JCP 133:84 1997] to its optimized version. The latter is employed to precondition the pseudo--Laplacian operator arising from the spectral element discretization of the magnetohydrodynamic equations in Elsasser form.


A new series of compact minitwistor spaces and Moishezon twistor spaces over them

Abstract: In recent papers math.DG/0701278 and arXiv:0705.0060, we gave explicit description of some new Moishezon twistor spaces. In this paper, developing the method in the papers much further, we explicitly give projective models of a number of new Moishezon twistor spaces, as conic bundles over some rational surfaces (called minitwistor spaces). These include the twistor spaces studied in the papers as very special cases. Our source of the result is a series of self-dual metrics with torus action constructed by D. Joyce. Actually, for arbitrary Joyce metrics and U(1)-subgroups of the torus which fixes a torus-invariant 2-sphere, we first determine the associated minitwistor spaces in explicit forms. Next by analyzing the meromorphic maps from the twistor spaces to the minitwistor spaces, we realize projective models of the twistor spaces of all Joyce metrics, as conic bundles over the minitwistor spaces. Then we prove that for any one of these minitwistor spaces, there exist Moishezon twistor spaces with only C*-action whose quotient space is the given minitwistor space. This result generates numerous Moishezon twistor spaces which cannot be found in the literature (including the author's papers), in quite explicit form.


Affine Deligne-Lusztig varieties in affine flag varieties

Abstract: This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.


Projective models of the twistor spaces of Joyce metrics

Abstract: We provide a simple algebraic construction of the twistor spaces of arbitrary Joyce's self-dual metrics on the 4-manifold H^2 x T^2 that extend smoothly to nCP^2, the connected sum of complex projective planes. Indeed, we explicitly realize projective models of the twistor spaces of arbitrary Joyce metrics on nCP^2 in a CP^4-bundle over CP^1, and show that they contain the twistor spaces of H^2 x T^2 as dense non-Zariski open subsets. In particular, we see that the last non-compact twistor spaces can be realized in rank-4 vector bundles over CP^1 by quite simple defining equations.


A Discrete Construction for Gaussian Markov Processes

Abstract: In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$.


On a variance for twins of $k-$free numbers in arithmetic progressions

Abstract: In this paper, we give a new upper bound of Barban-Davenport-Halberstam type for twins of $k-$free numbers in arithmetic progressions.


Quantile tomography: using quantiles with multivariate data

Abstract: The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development.


A bijective enumeration of labeled trees with given indegree sequence

Abstract: For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence $\lambda = 1^{e_1}2^{e_2} \ldots$ of a tree on the vertex set $\set{1,2,\ldots,n}$ is a partition of $n-1$. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Pr\"ufer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a $q$-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.


Non-commutative A-G mean inequality

Abstract: In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality $$a^{r}b^{1-r}+(r-1)b\geq ra$$ for two positive numbers $a,b$ and $r> 1$. We show that under some assumptions the non-commutative analogue for $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to $r$-mean. The case $0<r<1$ is also considered. In particular, we give a new characterization of the geometric mean.


A nonparametric estimator of the spectral density of a continuous-time Gaussian process observed at random times

Abstract: In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a nonparametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and application to biological data are also provided.


Existence results for quasilinear elliptic boundary value problems via topological methods

Abstract: In this paper, existence and localization results of $C^1$-solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder.


Graph braid groups and right-angled Artin groups

Abstract: We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free and leave the corresponding statement for $n$-braid groups as a conjecture along with few other conjectures about graphs whose braid groups of index $\le 4$ are right-angled Artin groups.


Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries

Abstract: The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail.


The volume conjecture for augmented knotted trivalent graphs

Abstract: We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.


Resonances for a diffusion with small noise

Abstract: We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.


Bilinear Hilbert transforms along curves I. The monomial case

Abstract: We establish an L^2 \times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.


Generalizations of Han's Hook Length Identities

Abstract: Han recently discovered new hook length identities for binary trees. In this paper, we extend Han's identities to binomial families of trees. Moreover, we present a bijective proof of one of the identities for the family of ordered trees.


Kummer surfaces associated to (1,2)-polarized abelian surfaces

Abstract: The aim of this paper is to describe the geometry of the generic Kummer surface associated to a $(1,2)$-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an elliptic fibration with twelve singular fibers of type $I_2.$


Del Pezzo Surfaces of degree 6 over an arbitrary field

Abstract: We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field $F$. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen $K$-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface.


A generalised Joyce construction for a family of nonlinear partial differential equations

Abstract: We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation.


Constant scalar curvature metrics on toric surfaces

Abstract: This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/


Uniform uniform exponential growth of subgroups of the mapping class group

Abstract: Let Mod(S) denote the mapping class group of a compact, orientable surface S. We prove that finitely generated subgroups of Mod(S) which are not virtually abelian have uniform exponential growth with minimal growth rate bounded below by a constant depending only, and necessarily, on S. For the proof, we find in any such subgroup explicit free group generators which are "short" in any word metric. Besides bounding growth, this allows a bound on the return probability of simple random walks.


Cullen-regular quaternionic functions in a Fueter operator framework

Abstract: We show characterizations of the class of Cullen-regular functions in the sense of Gentili-Struppa for any domain $\Omega$ in terms of the Fueter operator. We then state a Integral Theorem and discuss how it can be used to define a more general version of Cullen-regularity, that does not require the function to be of class $C^{1}$.


On the calculation of the cohomology of the third finite subset space of spheres

Abstract: In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.


Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry

Abstract: We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their oo-categories QC(X) of quasicoherent sheaves. We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on a fiber product with functors between the categories of sheaves on the factors (thus realizing functors as integral transforms, generalizing a theorem of Toen for ordinary schemes). As a first application, for a perfect stack X, consider QC(X) with its usual monoidal tensor product. Then our main results imply the equivalence of the Drinfeld center (or Hochschild cohomology category) of QC(X), the trace (or Hochschild homology category) of QC(X) and the category of sheaves on the loop space of X. More generally, we show that the E_n-center and the E_n-trace (or E_n-Hochschild cohomology and homology categories respectively) of QC(X) are equivalent to the category of sheaves on the space of maps from the n-sphere into X. This directly verifies geometric instances of the categorified Deligne and Kontsevich conjectures on the structure of Hochschild cohomology. As a second application, we use our main results to calculate the Drinfeld center of categories of linear endofunctors of categories of sheaves. This provides concrete applications to the structure of Hecke algebras in geometric representation theory. Finally, we explain how all of the above results can be interpreted in the context of topological field theory.


Operator-valued dyadic BMO spaces

Abstract: We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\|\pi_B\|$ in terms of the norm of the "sweep" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities.


Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Abstract: Let $\alpha$ be a quadratic Poisson bivector on a vector space $V$. Then one can also consider $\alpha$ as a quadratic Poisson bivector on the vector space $V^*[1]$. Fixed a universal deformation quantization (prediction some weights to all Kontsevich graphs [K97]), we have deformation quantization of the both algebras $S(V^*)$ and $\Lambda(V)$. These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on $\alpha$, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [T1].


Spectral parameterization for the power sums of quantum supermatrix

Abstract: A parameterization for the power sums of GL(m|n) type quantum (super)matrix is obtained in terms of it's spectral values.


Normal forms for real quadratic forms

Abstract: We investigate the non-diagonal normal forms of a quadratic form on R^n, in particular for n=3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of \R^5.


On normal contact pairs

Abstract: We consider manifolds endowed with a contact pair structure. To such a structure are naturally associated two almost complex structures. If they are both integrable, we call the structure a normal contact pair. We generalize the Morimoto's Theorem on product of almost contact manifolds to flat bundles. We construct some examples on Boothby--Wang fibrations over contact-symplectic manifolds. In particular, these results give new methods to construct complex manifolds.


On the equipartition of energy for critical NLW

Abstract: We study some qualitative properties of global solutions to the following focusing and defocusing critical $NLW$: \begin{equation*} \Box u+ \lambda u|u|^{2^*-2}=0, \hbox{} \lambda\in {\mathbf R} \end{equation*} $$\hspace{2cm} u(0)=f\in \dot H^1({\mathbf R}^n), \partial_t u(0)=g\in L^2({\mathbf R}^n)$$ on ${\mathbf R}\times {\mathbf R}^n$ for $n\geq 3$, where $2^*\equiv \frac{2n}{n-2}$. We will consider the global solutions of the defocusing $NLW$ whose existence and scattering property is shown in \cite{shst}, \cite{sb} and \cite{bg}, without any restriction on the initial data $(f,g)\in \dot H^1({\mathbf R}^n) \times L^2({\mathbf R}^n)$. As well as the solutions constructed in \cite{pecher} to the focusing $NLW$ for small initial data and to the ones obtained in \cite{mk}, where a sharp condition on the smallness of the initial data is given. We prove that the solution $u(t, x)$ satisfies a family of identities, that turn out to be a precised version of the classical Morawetz estimates (see \cite{mor1}). As a by--product we deduce that any global solution to critical $NLW$ belonging to a natural functional space satisfies: $$\lim_{R\to \infty}\frac 1R \int_{\mathbf R} \int_{|x|<R} |\nabla_{x} u(t,x)|^2 \hbox{} dxdt $$ $$=\lim_{R\to \infty} \frac 1{2R} \int_{\mathbf R} \int_{|x|<R} (|\nabla_{t,x} u(t,x)|^2 + \frac{2 \lambda}{2^*} |u(t,x)|^{2^*}) \hbox{} dxdt$$ $$=\int_{{\mathbf R}^n} (|\nabla_{t, x} u(0, x)|^2+ \frac{2 \lambda}{2^*} |u(0, x)|^{2^*}) \hbox{} dx.$$


Application of topological radicals to calculation of joint spectral radii

Abstract: It is shown that the joint spectral radius $\rho(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $\rho_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{\rho(a):a\in M^{n}\right\} ^{1/n}) . \] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras.


The Auslander-Reiten conjecture for Gorenstein rings

Abstract: The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this note is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$.


Evaluation for moments of a ratio with application to regression estimation

Abstract: Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977) 289--292], we propose sharper bounds for the moments of randomly weighted sums and for the $L^p$-deviations from the asymptotic normal law when the central limit theorem holds. We indicate suitable applications in finance and censored data analysis and focus on the applications in the field of functional estimation.


Topological chaos: what may this mean ?

Abstract: We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others.


Properties and refinements of the fused lasso

Abstract: We consider estimating an unknown signal, both blocky and sparse, which is corrupted by additive noise. We study three interrelated least squares procedures and their asymptotic properties. The first procedure is the fused lasso, put forward by Friedman et al. [Ann. Appl. Statist. 1 (2007) 302--332], which we modify into a different estimator, called the fused adaptive lasso, with better properties. The other two estimators we discuss solve least squares problems on sieves; one constrains the maximal $\ell_1$ norm and the maximal total variation seminorm, and the other restricts the number of blocks and the number of nonzero coordinates of the signal. We derive conditions for the recovery of the true block partition and the true sparsity patterns by the fused lasso and the fused adaptive lasso, and we derive convergence rates for the sieve estimators, explicitly in terms of the constraining parameters.


Diamond-alpha Integral Inequalities on Time Scales

Abstract: The theory of the calculus of variations was recently extended to the more general time scales setting, both for delta and nabla integrals. The primary purpose of this paper is to further extend the theory on time scales, by establishing some basic diamond-alpha dynamic integral inequalities. We prove generalized versions of H\"{o}lder, Cauchy-Schwarz, Minkowski, and Jensen's inequalities. For the particular case when alpha is equal to one or alpha is equal to zero one gets, respectively, correspondent delta and nabla inequalities. If we further restrict ourselves by fixing the time scale to the real or integer numbers, then we obtain the classical inequalities, whose role in optimal control is well known. By analogy, we trust that the diamond-alpha integral inequalities we prove here will be important in the study of control systems on times scales.


Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term

Abstract: In this paper we consider the Cauchy boundary value problem for the abstract Kirchhoff equation with a continuous nonlinearity m : [0,+\infty) --> [0,+\infty). It is well known that a local solution exists provided that the initial data are regular enough. The required regularity depends on the continuity modulus of m. In this paper we present some counterexamples in order to show that the regularity required in the existence results is sharp, at least if we want solutions with the same space regularity of initial data. In these examples we construct indeed local solutions which are regular at t = 0, but exhibit an instantaneous (often infinite) derivative loss in the space variables.


Logarithms and Square Roots of Real Matrices

Abstract: In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n x n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is, there is a real matrix, X, such that e^X = A. Furthermore, if the eigenvalues, xi, of X satisfy the property -pi < Im(xi) < pi, then X is unique. It is also known that under the same condition every real n x n matrix, A, has a real square root, that is, there is a real matrix, X, such that X^2 = A. Moreover, if the eigenvalues, rho e^{i theta}, of X satisfy the condition -pi/2 < theta < pi/2, then X is unique. These theorems are the theoretical basis for various numerical methods for exponentiating a matrix or for computing its logarithm using a method known as scaling and squaring (resp. inverse scaling and squaring). Such methods play an important role in the log-Euclidean framework due to Arsigny, Fillard, Pennec and Ayache and its applications to medical imaging. Actually, there is a necessary and sufficient condition for a real matrix to have a real logarithm (or a real square root) but it is fairly subtle as it involves the parity of the number of Jordan blocks associated with negative eigenvalues. As far as I know, with the exception of Higham's recent book, proofs of these results are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in the topics of this paper may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham. In these notes, I present a unified exposition of these results and give more direct proofs of some of them using the Real Jordan Form.


Arithmetic Laplacians

Abstract: We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, and analyze their spaces of solutions.


On multiplicative conditionally free convolution

Abstract: Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The result is applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.


Diamond-alpha Polynomial Series on Time Scales

Abstract: The objective of this paper is twofold: (i) to survey existing results of generalized polynomials on time scales, covering definitions and properties for both delta and nabla derivatives; (ii) to extend previous results by using the more general notion of diamond-alpha derivative on time scales. We introduce a new notion of combined-polynomial series on a time scale, as a convex linear combination of delta and nabla generalized series. Main results are formulated for homogenous time scales. As an example, we compute diamond-alpha derivatives on time scales for delta and nabla exponential functions.


Isoperimetric Problems of the Calculus of Variations on Time Scales

Abstract: We prove a necessary optimality condition for isoperimetric problems on time scales in the space of delta-differentiable functions with rd-continuous derivatives. The results are then applied to Sturm-Liouville eigenvalue problems on time scales.


An example of a weighted algebra $L_p^w(G)$ on uncountable group

Abstract: We construct examples of weighted algebras $L_p^w(G)$ with $1<p\le 2$ on uncountable free groups. For $p>2$ no weighted algebras exist on these groups. From the other side, we prove that an amenable group on which exist weighted algebras with $p>1$ must be sigma-compact.


Convolution and Cross-Correlation of Ramanujan-Fourier Series

Abstract: This paper uses the machinery of almost periodic functions to prove that even without uniform convergence the connection between a pair of almost periodic functions and the constants of the associated Fourier series exists for both the convolution and cross-correlation. The general results for two almost periodic functions are narrowed and applied to Ramanujan sums and finally applied to support the specific relation of the Wiener-Khinchin formula for arithemic functions with a Ramanujan-Fourier Series.


The Bar-Natan skein module of the solid torus and the homology of (n,n) Springer varieties

Abstract: This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n,n) Springer variety. The results build on Khovanov's work with crossingless matchings and the cohomology of the (n,n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-manifold and boundary curve system.


Rouquier blocks of the cyclotomic Hecke algebras of G(de,e,r)

Abstract: The "Rouquier blocks" of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the "families of characters", defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series, G(de,e,r), thus completing their calculation for all complex reflection groups.


Notes on Convex Sets, Polytopes, Polyhedra, Combinatorial Topology, Voronoi Diagrams and Delaunay Triangulations

Abstract: Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new.


A Khasminskii type averaging principle for stochastic reaction-diffusion equations

Abstract: We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems.


Averaging principle for a class of stochastic reaction-diffusion equations

Abstract: We consider the averaging principle for stochastic reaction-diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE's.


Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions

Abstract: One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the action of the unit quaternions, SU(2), on reals^3 (i.e., the action is defined in terms of multiplication in a larger algebra containing both the group Spin(n) and reals^n). The group Spin(n), called a spinor group, is defined as a certain subgroup of units of an algebra, Cl_n, the Clifford algebra associated with reals^n. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p, q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful "8-periodicity theorem" of Elie Cartan and Raoul Bott (with proofs).


Sums of norm spheres are norm shells and lower triangle inequalities are sharp

Abstract: The statements in the title are explained and proved, as a little exercise in elementary normed vector space theory at the level of Chapter 5 of Dieudonn\'e's "Foundations of Mathematical Analysis". A connection to recent moment bounds for submartingales is sketched.


On the norm convergence of nonconventional ergodic averages

Abstract: We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to an arbitrary number of commuting probability-preserving transformations always converge to some limit in L^2. We prove the corresponding result for a collection of commuting actions of a larger discrete Abelian group, and gives convergence that is uniform in the start-point of the averages. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.


On submanifolds with tamed second fundamental form

Abstract: We show that a complete submanifold $M$ with tamed second fundamental form in a complete Riemannian manifold $N$ with sectional curvature $K_{N}\leq \kappa \leq 0$ are proper, (compact if $N$ is compact). In addition, if $N$ is Hadamard then $M$ has finite topology. We also show that the fundamental tone is an obstruction for a Riemannian manifold to be realized as submanifold with tamed second fundamental form of a Hadamard manifold with sectional curvature bounded below.


Width of homoclinic zone for quadratic maps

Abstract: We study several families of planar quadratic diffeomorphisms near a Bogdanov-Takens bifurcation. For each family, the associate bifurcation diagram can be deduced from the interpolating flow. However, a zone of chaos confined between two lines of homoclinic bifurcation that are exponentially close to one-another is observed. The goal of this paper is to test numerically an accurate asymptotic expansion for the width of this chaotic zone for different families.


Nathanson's Heights and the CSS Conjecture for Cayley Graphs

Abstract: Let $G$ be a finite directed graph, $\beta(G)$ the minimum size of a subset $X$ of edges such that the graph $G' = (V,E \smallsetminus X)$ is directed acyclic and $\gamma(G)$ the number of pairs of nonadjacent vertices in the undirected graph obtained from $G$ by replacing each directed edge with an undirected edge. Chudnovsky, Seymour and Sullivan \cite{CSS07} proved that if $G$ is triangle-free, then $\beta(G) \leq \gamma(G)$. They conjectured a sharper bound (so called the "CSS conjecture") that $\beta(G) \leq \dfrac{\gamma(G)}{2}$. Nathanson and Sullivan verified this conjecture for the directed Cayley graph $\Cay(\bbZ/N\bbZ, E_A)$ whose vertex set is the additive group $\bbZ/N\bbZ$ and whose edge set $E_A$ is determined by $E_A = {(x,x+a) : x \in \bbZ/N\bbZ, a \in A}$ when $N$ is prime in \cite{NS07} by introducing "height". In this work, we extend the definition of height and the proof of CSS conjecture for $\Cay(\bbZ/N\bbZ, E_A)$ to any positive integer $N$.


Universal spaces for manifolds equipped with a closed integral k-form

Abstract: In this note we prove that any integral closed k-form $\phi ^k$, $k\ge 3$, on a m-dimensional manifold $M^m$, $m \ge k$, is the restriction of a universal closed k-form $h^k$ on a universal manifold $U^{d(m,k)}$ as a result of an embedding of $M^m$ to $U^{d(m,k)}$.


Abstract: In this note we introduce a Yang-Mills bar equation on complex vector bundles over compact Hermitian manifolds as the Euler-Lagrange equation for a Yang-Mills bar functional. We show the existence of a non-trivial solution of this equation over compact K\"ahler manifolds as well as a short time existence of the negative Yang-Mills bar gradient flow. We also show a rigidity of holomorphic connections among a class of Yang-Mills bar connections over compact K\"ahler manifolds of positive Ricci curvature.


Periods and elementary real numbers

Abstract: The periods, introduced by Kontsevich and Zagier, form a class of complex numbers which contains all algebraic numbers and several transcendental quantities. Little has been known about qualitative properties of periods. In this paper, we compare the periods with hierarchy of real numbers induced from computational complexities. In particular we prove that periods can be effectively approximated by elementary rational Cauchy sequences. As an application, we exhibit a computable real number which is not a period.


Asymptotic enumeration of constellations and related families of maps on orientable surfaces

Abstract: We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps and m-constellations -- for example, large maps of genus g with even face degrees are bipartite with probability tending to 1/2^{2g}. A special case of our results implies former conjectures of Gao.


M\"obius transformations and the Poincar\'e distance in the quaternionic setting

Abstract: In the space $\hh$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\Delta_{\hh}$ and of the open half-space $\hh^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of $\Delta_{\hh}$ and $\hh^{+}$ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M\"obius transformations, and use it to define the analogous of the Poincar\'e distances on $\Delta_{\hh}$ and $\hh^{+}$. We easily deduce that there exists no isometry between the quaternionic Poincar\'e distance of $\Delta_{\hh}$ and the Kobayashi distance inherited by $\Delta_{\hh}$ as a domain of $\mathbb{C}^{2}$, in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.


Strong cleanness of the $2\times 2$ matrix ring over a general local ring

Abstract: A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings $R$ for which ${\mathbb M}_n(R)$ is strongly clean. For a general local ring $R$ and $n>1$, however, it is unknown when the matrix ring ${\mathbb M}_n(R)$ is strongly clean. Here we completely determine the local rings $R$ for which ${\mathbb M}_2(R)$ is strongly clean.


On Smarandache Bryant Schneider group of a Smarandache loop

Abstract: The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some properties of the Bryant Schneider Group of a loop are shown to be true for the Smarandache Bryant Schneider Group of a Smarandache loop. Some interesting and useful cardinality formulas are also established for a type of finite Smarandache loop.


Topological methods in analysis of periodic and chaotic canard-type trajectories

Abstract: We investigate the role of topological methods in the analysis of canard-type periodic and chaotic trajectories. In the first part of the paper, we apply topological degree to the analysis of multi-dimensional canards. The second part is devoted to an application of a special corollary of the Poincare-Bendixson theorem to the existence of periodic two-dimensional canards.


A Limit Theorem for Products of Free Unitary Operators

Abstract: This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.


Alternatives to Pearson's and Spearman's Correlation Coefficients

Abstract: This article presents several alternatives to Pearson's correlation coefficient and many examples. In the samples where the rank in a discrete variable counts more than the variable values, the mixtures that we propose of Pearson's and Spearman's correlation coefficients give better results.


Freiman-Ruzsa-type theory for small doubling constant

Abstract: In this paper, we study the linear structure of sets $A \subset \mathbb{F}_2^n$ with doubling constant $\sigma(A)<2$, where $\sigma(A):=\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in a small affine subspace. We also show that $A$ can be covered by at most four shifts of some subspace $V$ with $|V|\leq |A|$. Finally, we classify all binary sets with small doubling constant.


Gr\"unbaum Colorings of Toroidal Triangulations

Abstract: We prove that if G is a triangulation of the torus and \chi(G) \neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.


Representation Growth of Linear Groups

Abstract: Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When $\Gamma$ is an arithmetic group satisfying the congruence subgroup property then $\calz_\Gamma(s)$ has an ``Euler factorization". The "factor at infinity" is sometimes called the "Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representations of suitable open subgroups $U$ of the associated simple group $G$ over the associated local field $K$. Here we show a surprising dichotomy: if $G(K)$ is compact (i.e. $G$ anisotropic over $K$) the abscissa of convergence goes to 0 when $\dim G$ goes to infinity, but for isotropic groups it is bounded away from 0. As a consequence, there is an unconditional positive lower bound for the abscissa for arbitrary finitely generated linear groups. We end with some observations and conjectures regarding the global abscissa.


The Player's Effect

Abstract: In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a "small" set that, when eliminated, leaves every set with little effect. Finally, we ask whether there always exists a player with positive effect. We answer this question differently in various scenarios, depending on the properties of the function and the distribution of players' inputs. More specifically, we show that if the function is non-monotone or the distribution is only known to be pairwise independent, then it is possible that all players have 0 effect. If the distribution is pairwise independent with minimal support, on the other hand, then there must exist a player with "large" effect.


Collinear triples in permutations

Abstract: Let $\alpha:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples in the graph of $\alpha$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once proved $\Psi(\alpha)\geq(q-1)/4$ and conjectured the sharp bound should be $\Psi(\alpha)\geq(q-1)/2$. In this note we indicate that the Cooper-Solymosi conjecture is true.


Some conjectures about q-Fibonacci polynomials

Abstract: In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.


Small volume expansions for elliptic equations

Abstract: This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form $\nabla \cdot D^\eps \nabla u^\eps=0$ or $(-\Delta + q^\eps) u^\eps=0$ with prescribed Neumann conditions. The theory is well-known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order $\eps^{2d}$, where $d$ is dimension and $\eps$ is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most $\eps^{d+1}$ rather than the expected $\eps^d$. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.


New Proof on some sharp double integral Inequalities of the Hermite-Hadamard Type

Abstract: In this paper, we derive a new proof on some sharp double integral inequalities of the Hermite-Hadamard type. Our approach is mainly based on well-known Taylor's theorem with the integral remainder.


Connected components of strata of quadratic differentials over Teichmuller space

Abstract: In this paper, we study connected components of strata of the space of quadratic differentials lying over $\T_g$. We use certain general properties of sections of line bundles to put a upper bound on the number of connected components, and a generalized version of the Gauss map as an invariant to put a lower bound on the number of such components. For strata with sufficiently many zeroes of the same order we can state precisely the number of components.


Motivic Integration on Toric Stacks

Abstract: We present a decomposition of the space of twisted arcs of a toric stack. As a consequence, we give a combinatorial description of the motivic integral associated to a torus-invariant divisor of a toric stack.


Denominators in cluster algebras of affine type

Abstract: The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.


Uniform Bounds on Pre-Images under Quadratic Dynamical Systems

Abstract: For any elements b,c of a number field K, let G(b,c) denote the backwards orbit of b under the map f_c: C-->C given by f_c(x)=x^2+c. We prove an upper bound on the number of elements of G(b,c) whose degree over K is at most some constant B. This bound depends only on b, [K:Q], and B, and is valid for all b outside an explicit finite set. We also show that, for any N>3 and any b in K outside a finite set, there are only finitely many pairs of complex numbers (y,c) for which [K(y,c):K]<2^(N-3) and the value of the N-th iterate of f_c(x) at x=y is b. Moreover, the bound 2^(N-3) in this result is optimal.


The mixing advantage is less than 2

Abstract: Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results.


The Elliptic Hypergeometric Functions Associated to the Configuration Space of Points on an Elliptic Curve I : Twisted Cycles

Abstract: We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate the {\it twisted homology group}, with coefficients in the local system associated to a power function $g^{\alpha}$ of an elliptic function $g$, and the intersection form. Applying these calculations, we describe the {\it connection matrices} representing the linear isomorphisms induced from analytic continuations of the functions defined by the integrations of $g^{\alpha}$ over twisted cycles.


Distance-based clustering of sparsely observed stochastic processes, with applications to online auctions

Abstract: We propose a distance between two realizations of a random process where for each realization only sparse and irregularly spaced measurements with additional measurement errors are available. Such data occur commonly in longitudinal studies and online trading data. A distance measure then makes it possible to apply distance-based analysis such as classification, clustering and multidimensional scaling for irregularly sampled longitudinal data. Once a suitable distance measure for sparsely sampled longitudinal trajectories has been found, we apply distance-based clustering methods to eBay online auction data. We identify six distinct clusters of bidding patterns. Each of these bidding patterns is found to be associated with a specific chance to obtain the auctioned item at a reasonable price.


Twisted Poincar\'e Lemma and Twisted \v{C}ech-de Rham Isomorphism in case of Projective Line

Abstract: In this paper, we give a direct proof of the twisted Poincar\'{e} lemma by using the integrations over regularized paths. This method tells us a concrete description of the \v{C}ech-de Rham isomorphism.


Consistency of restricted maximum likelihood estimators of principal components

Abstract: In this paper we consider two closely related problems : estimation of eigenvalues and eigenfunctions of the covariance kernel of functional data based on (possibly) irregular measurements, and the problem of estimating the eigenvalues and eigenvectors of the covariance matrix for high-dimensional Gaussian vectors. In Peng and Paul (2007), a restricted maximum likelihood (REML) approach has been developed to deal with the first problem. In this paper, we establish consistency and derive rate of convergence of the REML estimator for the functional data case, under appropriate smoothness conditions. Moreover, we prove that when the number of measurements per sample curve is bounded, under squared-error loss, the rate of convergence of the REML estimators of eigenfunctions is near-optimal. In the case of Gaussian vectors, asymptotic consistency and an efficient score representation of the estimators are obtained under the assumption that the effective dimension grows at a rate slower than the sample size. These results are derived through an explicit utilization of the intrinsic geometry of the parameter space, which is non-Euclidean. Moreover, the results derived in this paper suggest an asymptotic equivalence between the inference on functional data with dense measurements and that of the high dimensional Gaussian vectors.


Stanley decompositions and localization

Abstract: We study the behavior of Stanley depth under the operation of localization with respect to a variable.


Spectral gap for the interchange process in a box

Abstract: We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree.


Quotient categories, stability conditions, and birational geometry

Abstract: This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.


Degrees of maps between Grassmann manifolds

Abstract: Let $f:G_{n,k}\longrightarrow G_{m,l}$ be any continuous map between any two distinct complex Grassmann manifolds of the same dimension where the target is not the complex projective space. We show that, for any given $k,l$, the degree of $f$ is zero provided that $m,n$ are sufficiently large. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^*$ of elements of a set of algebra generators of $H^*(G_{m,l};\mathbb{Q})$ is determined upto a sign, $\pm$, if the degree of $f$ is non-zero. Our proofs cover the case of quaternionic Grassmann manifolds as well.


Notes on algebra and geometry of polynomial representations

Abstract: Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following template. Assume that in a neighborhood of a boundary point the semi-algebraic set A can be described by an irreducible polynomial f. Then f is a factor of a certain multiplicity of some of the polynomials p_1,...,p_m. Special cases when A is elementary closed, elementary open, a polygon, or a polytope are considered separately.


On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

Abstract: We consider a family of slightly extended version of the Raynaud's surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H^1(X, Z^{-1})\ne 0. The surfaces are at least normal but smooth under a special condition. We compute the cohomologies H^i(X, Z^n), for intergers i and n, and study their (non-)vanishing. Finally, we give a fairly large family of non Mumford-Szpiro type polarizations Z_{a,b} with Kodaira non-vanishing.


Zeros of p-adic forms

Abstract: A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.


Steady periodic water waves under nonlinear elastic membranes

Abstract: This is a study of two-dimensional steady periodic travelling waves on the surface of an infinitely deep irrotational ocean, when the top streamline is in contact with a membrane which has a nonlinear response to stretching and bending, and the pressure in the air above is constant. It is not supposed that the waves have small amplitude. The problem of existence of such waves is addressed using methods from the calculus of variations. The analysis involves the Hilbert transform and a Riemann-Hilbert formulation.


Ideals generated by submaximal minors

Abstract: The goal of this paper is to study irreducible families W(b;a) of codimension 4, arithmetically Gorenstein schemes X of P^n defined by the submaximal minors of a t x t matrix A with entries homogeneous forms of degree a_j-b_i. Under some numerical assumption on a_j and b_i we prove that the closure of W(b;a) is an irreducible component of Hilb^{p(x)}(P^n), we show that Hilb^{p(x)}(P^n) is generically smooth along W(b;a) and we compute the dimension of W(b;a) in terms of a_j and b_i. To achieve these results we first prove that X is determined by a regular section of the twisted conormal sheaf I_Y/I^2_Y(s) where s=deg(det(A)) and Y is a codimension 2, arithmetically Cohen-Macaulay scheme of P^n defined by the maximal minors of the matrix obtained deleting a suitable row of A.


Local time steps for a finite volume scheme

Abstract: We present a strategy for solving time-dependent problems on grids with local refinements in time using different time steps in different regions of space. We discuss and analyze two conservative approximations based on finite volume with piecewise constant projections and domain decomposition techniques. Next we present an iterative method for solving the composite-grid system that reduces to solution of standard problems with standard time stepping on the coarse and fine grids. At every step of the algorithm, conservativity is ensured. Finally, numerical results illustrate the accuracy of the proposed methods.


Ideal clones: Solution to a problem of Czedli and Heindorf

Abstract: Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In a 2001 article, G. Czedli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.


A martingale approach to minimal surfaces

Abstract: We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.


Intermittence and nonlinear parabolic stochastic partial differential equations

Abstract: We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\sL$ in dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics \cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the two exactly-solvable cases where $u_0=\delta_0$ and $u_0\equiv 1$.


The Classification Theorem for Compact Surfaces And A Detour On Fractals

Abstract: The purpose of these notes is to present a fairly complete proof of the classification Theorem for compact surfaces. Other presentations are often quite informal (see the references in Chapter V) and we have tried to be more rigorous. Our main source of inspiration is the beautiful book on Riemann Surfaces by Ahlfors and Sario. However, Ahlfors and Sario's presentation is very formal and quite compact. As a result, uninitiated readers will probably have a hard time reading this book. Our goal is to help the reader reach the top of the mountain and help him not to get lost or discouraged too early. This is not an easy task! We provide quite a bit of topological background material and the basic facts of algebraic topology needed for understanding how the proof goes, with more than an impressionistic feeling. We hope that these notes will be helpful to readers interested in geometry, and who still believe in the rewards of serious hiking!


New congruences for central binomial coefficients

Abstract: Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.


Hulls and Husks

Abstract: The aim of this note is to prove an analog of the flattening decomposition theorem for reflexive hulls. The main applications are: the construction of the moduli space of varieties of general type, improved flatness conditions and criteria for simultaneous normalizations.


Cyclic homology of crossed products

Abstract: We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homology of E relative to K. The first one in the general setting and the second one (which generalizes those previously found by several authors) when f takes its values in K.


Homotopy of unitaries in simple C*-algebras with tracial rank one

Abstract: Let $\epsilon>0$ be a positive number. Is there a number $\delta>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<\epsilon \forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that $\phi: C\to A$ is a unital monomorphism and suppose that $v\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $\phi.$ It is shown that there is a continuous path of unitaries $\{v(t): t\in [0,1]\}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $\phi,$ provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented.


Homological and homotopical higher-order filling functions

Abstract: We construct groups in which FV^3(n) != \delta^2(n). This construction also leads to groups G_k, k >= 3 for which \delta^{k}(n) is not subrecursive.


Rational and algebraic series in combinatorial enumeration

Abstract: Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers a(n), but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions.


A characterization of Dirac morphisms

Abstract: Relating the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion, we characterize Dirac morphisms, i.e. maps which pull back (local) harmonic spinor fields onto (local) harmonic spinor fields.


Descendent bounds for effective divisors on the moduli space of curves

Abstract: The slope of the moduli space of genus g curves is bounded from below by 60/(g+4) via a descendent calculation.


Limit theorems for additive c-free convolution

Abstract: In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods.


Hardy inequalities for general elliptic operators with improvements

Abstract: We obtain optimal generalized versions of Hardy inequalities, which as special cases contain Hardy's inequality and Hardy's inequality involving the distance function to the boundary of $ \Omega$. In addition we obtain neccesary and sufficient conditions to add improvements in the form of non negative potentials.


Functional moderate deviations for triangular arrays and applications

Abstract: Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables.


Embeddings between operator-valued dyadic BMO spaces

Abstract: We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterisations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces.


Logarithmic decay of hyperbolic equations with arbitrary boundary damping

Abstract: In this paper, we study the logarithmic stability for the hyperbolic equations by arbitrary boundary observation. Based on Carleman estimate, we first prove an estimate of the resolvent operator of such equation. Then we prove the logarithmic stability estimate for the hyperbolic equations without any assumption on an observation subboundary.


Curvature extrema and four-vertex theorems for polygons and polyhedra

Abstract: Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their evolutes is obtained. Also are considered higher-dimensional analogs of the four-vertex theorem for regular and shellable triangulations.


Classifications of Linear Controlled Systems

Abstract: This paper is devoted to a study of linear, differential and topological classifications for linear controlled systems governed by ordinary differential equations. The necessary and sufficient conditions for the linear and topological equivalence are given. It is also shown that the differential equivalence is the same as the linear equivalence for the linear controlled systems.


Computational Geometric Optimal Control of Rigid Bodies

Abstract: This paper formulates optimal control problems for rigid bodies in a geometric manner and it presents computational procedures based on this geometric formulation for numerically solving these optimal control problems. The dynamics of each rigid body is viewed as evolving on a configuration manifold that is a Lie group. Discrete-time dynamics of each rigid body are developed that evolve on the configuration manifold according to a discrete version of Hamilton's principle so that the computations preserve geometric features of the dynamics and guarantee evolution on the configuration manifold; these discrete-time dynamics are referred to as Lie group variational integrators. Rigid body optimal control problems are formulated as discrete-time optimization problems for discrete Lagrangian/Hamiltonian dynamics, to which standard numerical optimization algorithms can be applied. This general approach is illustrated by presenting results for several different optimal control problems for a single rigid body and for multiple interacting rigid bodies. The computational advantages of the approach, that arise from correctly modeling the geometry, are discussed.


Groebner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras

Abstract: In this paper, we establish the Composition-Diamond lemma for associative algebras with multiple linear operators. As applications, we obtain Groebner-Shirshov bases of free Rota-Baxter algebra, $\lambda$-differential algebra and $\lambda$-differential Rota-Baxter algebra, respectively. In particular, linear bases of these three free algebras are respectively obtained, which are essentially the same or similar to those obtained by Ebrahimi-Fard and Guo, and Guo and Keigher recently by using other methods.


Einstein solvmanifolds attached to two-step nilradicals

Abstract: A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in one of the rare cases when the complete set of affine invariants is known: for the two-step nilpotent Lie algebras with the two-dimensional center. Informally speaking, we prove that such a Lie algebra is an Einstein nilradical, if it is defined by a matrix pencil having no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also discuss the connection between the property of a two-step nilpotent Lie algebra and its dual to be an Einstein nilradical.


On the coordinate ring of spherical conjugacy classes

Abstract: Let G be a simple algebraic group over an algebraically closed field of characteristic zero and X be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring of X into simple G-modules.


On the Holonomy of Kaluza-Klein metrics

Abstract: We investigate Kaluza-Klein metrics with a recurrent light-like vector field over a pseudo-Riemannian manifold.


Tensor products of type III factor representations of Cuntz-Krieger algebras

Abstract: We introduced a non-symmetric tensor product of any two states or any two representations of Cuntz-Krieger algebras associated with a certain non-cocommutative comultiplication in previous our work. In this paper, we show that a certain set of KMS states is closed with respect to the tensor product. From this, we obtain formulae of tensor product of type {\rm III} factor representations of Cuntz-Krieger algebras which is different from results of the tensor product of factors of type {\rm III}.


Multidimensional Hahn polynomials, intertwining functions on the symmetric group and Clebsch-Gordan coefficients

Abstract: We generalize a construction of Dunkl, obtaining a wide class intertwining functions on the symmetric group and a related family of multidimensional Hahn polynomials. Following a suggestion of Vilenkin and Klymik, we develop a tree-method approach for those intertwining functions. We also give a group theoretic proof of the relation between Hahn polynomials and Clebesh-Gordan coefficients, given analytically by Koornwinder and by Nikiforov, Smorodinski\u{i} and Suslov. Such relation is also extended to the multidimensional case.


A Note on Fuzzy Real and Complex Field

Abstract: Using the concept of fuzzy field, we have considered the fuzzy field of real and complex numbers and thereafter we have established a few standard results of real and complex numbers with respect to a membership function.


The Euler characteristic as a polynomial in the Chern classes

Abstract: In this paper we obtain some explicit expressions for the Euler characteristic of a rank n coherent sheaf F on P^N and of its twists F(t) as polynomials in the Chern classes c_i(F), also giving algorithms for the computation. The employed methods use techniques of umbral calculus involving symmetric functions and Stirling numbers.


On the Calculation of gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\widetilde{A}$ by Using Gr\"obner Bases

Abstract: Let $A=K< X_1,...,X_n> /< {\cal G}>$ be a $K$-algebra defined by a finite Gr\"obner basis ${\cal G}$. It is shown how to use the Ufnarovski graph $\Gamma ({\bf LM}({\cal G}))$ and the graph of $n$-chains $\Gamma_{\rm C}({\bf LM}({\cal G}))$ to calculate gl.dim$G^{\mathbb{N}}(A)$ and gl.dim$\widetilde{A}$, where $G^{\mathbb{N}}(A)$, respectively $\widetilde{A}$, is the associated $\mathbb{N}$-graded algebra of $A$, respectively the Rees algebra of $A$ with respect to the $\mathbb{N}$-filtration $FA$ of $A$ induced by a weight $\mathbb{N}$-grading filtration of $K< X_1,...,X_n>$.


Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces

Abstract: We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove this, we base ourselves on the recent progress in the problem of the validity of Hardy inequalities in variable exponent Lebesgue spaces.


Abstract: Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of Borel's conjecture is stated, asserting that the cohomology can be computed using automorphic forms.


Principal eigenvalue for random walk among random traps on Z^d

Abstract: Let $(\tau_x)_{x \in \Z^d}$ be i.i.d. random variables with heavy (polynomial) tails. Given $a \in [0,1]$, we consider the Markov process defined by the jump rates $\omega_{x \to y} = {\tau_x}^{-(1-a)} {\tau_y}^a$ between two neighbours $x$ and $y$ in $\Z^d$. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension.


A necessary condition for dynamic equivalence

Abstract: If two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-B\"acklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be "trivial" (or linear controllable).


Free subalgebras of Lie algebras close to nilpotent

Abstract: We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations.


Covariance of centered distributions on manifold

Abstract: We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. However, most of the available solutions use coordinate specific definition that makes them less versatile. %We propose improvements in two directions. We define the distributions of interest in coordinate independent way by utilizing co-variant 2-tensors. Then we study the relation of these distributions to their Euclidean counterparts. In particular, we are interested in relating the covariance to the tensor that controls distribution concentration. We find approximating expression for this relation in general and give more precise formulas in case of manifolds of constant curvature, positive or negative. Results are confirmed by simulation studies of the standard normal distribution on the unit-sphere and hyperbolic plane.


Lecture by Michael Hopkins: the string orientation of tmf

Abstract: These are the notes of a lecture held by Michael Hopkins in march 2007, at the Talbot workshop.


Maximum likelihood estimation in a partially observed stratified regression model with censored data

Abstract: The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also obtained.


A Microscopic Convexity Principle for Nonlinear Partial Differential Equations

Abstract: We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.


Conformal metrics on $\R^{2m}$ with constant Q-curvature

Abstract: We study the conformal metrics on $\R^{2m}$ with constant Q-curvature $Q$ having finite volume, particularly in the case $Q\leq 0$. We show that when $Q<0$ such metrics exist in $\R^{2m}$ if and only if $m>1$. Moreover we study their asymptotic behavior at infinity, in analogy with the case $Q>0$, which we treated in a recent paper. When Q=0, we show that such metrics have the form $e^{2p}g_{\R^{2m}}$, where $p$ is a polynomial such that $2\leq \deg p\leq 2m-2$ and $\sup_{\R^{2m}}p<+\infty$. In dimension 4, such metrics are exactly the polynomials $p$ of degree 2 with $\lim_{|x|\to+\infty}p(x)=-\infty$.


Construction of eternal solutions for a semilinear parabolic equation

Abstract: Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution -- the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singularities in both backward and forward time. However, semilinear parabolic equations can also support nontrivial eternal solutions. This article shows how nontrivial eternal solutions can be constructed for a semilinear equation that has at least two distinct equilibrium solutions. The resulting eternal solution is a heteroclinic orbit which connects the two given equilibria.


Which powers of holomorphic functions are integrable?

Abstract: We show that every limit of log canonical thresholds of n-variable functions is also a log canonical threshold of an (n-1)-variable function.


Splitting Polytopes

Abstract: A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].


Square-free discriminants of Frobenius rings

Abstract: Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the Frobenius element $\pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $\pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $\pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$.


Finding and investigating exact spherical codes

Abstract: In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the 20-point code in $\mathbb{R}^6$ and the 24-point code in $\mathbb{R}^7$, which are both the union of two cross polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.


Asymptotic normality of wavelet estimators of the memory parameter for linear processes

Abstract: We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$.


Central Limit Theorems for arrays of decimated linear processes

Abstract: Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then establish central limit theorems for arrays of squares of such decimated processes. These theorems are used to obtain the asymptotic behavior of estimators of the spectral density at specific frequencies. Another application, treated elsewhere, concerns the estimation of the long-memory parameter in time-series, using wavelets.


Notes on character sheaves

Abstract: In the first section we study a functor of Bezrukavnikov, Finkelberg and Ostrik defined on character sheaves; we compute it in a Grothendieck group taking weights into account. In the second section we enlarge the class of character sheaves to a larger class of simple perverse sheaves which behaves well under tensor product (unlike the character sheaves themselves).


On Certain Vanishing Identities For Gromov-Witten Invariants

Abstract: In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we will also prove their conjectures in low genus cases.


Constructing Big Indecomposable modules

Abstract: Let $R$ be local Noetherian ring of depth at least two. We prove that there are indecomposable $R$-modules which are free on the punctured spectrum of constant, arbitrarily large, rank.


An exotic sphere with positive sectional curvature

Abstract: We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.


Extension Theorems for Paraboloids in the Finite Field Setting

Abstract: In this paper we study the $L^p-L^r$ boundedness of the extension operators associated with paraboloids in vector spaces over finite fields.In higher even dimensions, we estimate the number of additive quadruples in the subset $E$ of the paraboloids, that is the number of quadruples $(x,y,z,w) \in E^4$ with $x+y=z+w.$ As a result, in higher even dimensions, we improve upon the standard Tomas-Stein exponents which Mockenhaupt and Tao obtained for the boundedness of extension operators for paraboloids by estimating the decay of the Fourier transform of measures on paraboloids. In particular, we obtain the sharp $L^p-L^4$ bound up to endpoints in higher even dimensions. Moreover, we also study the $L^2-L^r$ estimates.In the case when -1 is not a square number in the underlying finite field, we also study the $L^p-L^r$ bound in higher odd dimensions.The discrete Fourier analytic machinery and Gauss sum estimates make an important role in the proof.


Probabilistic proofs of hook length formulas involving trees

Abstract: Recently, Han discovered two formulas involving binary trees which have the interestig property that hooklengths appear as exponents. The purpose of this note is to give a probabilistic proof of one of Han's formulas. Yang has generalized Han's results to ordered trees. We show how the probabilistic approach can also be used in Yang's setting, as well as for a generalization of Han's formula in terms of certain infinite trees.


Some Remarks on the Algebra of Bounded Dirichlet Series

Abstract: We examine the algebra of all Dirichlet Series bounded on the right half plane. We consider the analogue of the Corona theorem in this setting, and show that it is false, i.e. the right half-plane is not dense in the maximal ideal space. We also prove some refinements of the Hille-Bohnenblust theorem, where both probabilistic and deterministic devices are used, and we show how the proof is carried out for each.


Gradient-like observers for invariant dynamics on a Lie group

Abstract: This paper proposes a design methodology for non-linear state observers for invariant kinematic systems posed on finite dimensional connected Lie groups, and studies the associated fundamental system structure. The concept of synchrony of two dynamical systems is specialised to systems on Lie groups. For invariant systems this leads to a general factorisation theorem of a nonlinear observer into a synchronous (internal model) term and an innovation term. The synchronous term is fully specified by the system model. We propose a design methodology for the innovation term based on gradient-like terms derived from invariant or non-invariant cost functions. The resulting nonlinear observers have strong (almost) global convergence properties and examples are used to demonstrate the relevance of the proposed approach.


The Analogue of Dedekind Eta Functions for Calabi-Yau Manifilolds II. (Algebraic, Analytic Discriminants and the Analogue of Baily-Borel Compactification of the Moduli Space of CY Manifolds.)

Abstract: In this paper we construct the analogue of Dedekind eta-function on the moduli space of polarized CY manifolds. We prove that the L-two norm of eta is the regularized determinants of the Laplacians of the CY metric on (0,1) forms. We construct the analogue of the Baily-Borel Compactification of the moduli space of polarized CY and prove that it has the same properties as the Baily-Borel compactification of the locally symmetric Hermitian spaces. We proved that the compactification constructed in the paper is the minimal.


A note on non-reduced Picard schemes

Abstract: The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension $\kappa\leq0$, that this phenomenon can be bounded for $\kappa=2$ (general type) and that it is as bad as can be in $\kappa=1$.


On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs

Abstract: We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete orthonormal basis of finitely supported eigenfunctions.


Cut Points and Diffusions in Random Environment

Abstract: In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.


Concentration of measure via approximated Brunn--Minkowski inequalities

Abstract: We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.


Lyapunov control of a quantum particle in a decaying potential

Abstract: A Lyapunov-based approach for the trajectory generation of an $N$-dimensional Schr{\"o}dinger equation in whole $\RR^N$ is proposed. For the case of a quantum particle in an $N$-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.


A uniformly spread measure criterion

Abstract: We prove that if all shifts of a measure in the Euclidean space are close in a sense to each other, then this measure is close to the Lebesgue one.