Application of a curvature adjusted method in image segmentation

Abstract: This article deals with flow of plane curves driven by the curvature and external force. We make use of such a geometric flow for the purpose of image segmentation. A parametric model for evolving curves with uniform and curvature adjusted redistribution of grid points will be described and compared.


Mould expansions for the saddle-node and resurgence monomials

Abstract: This article is an introduction to some aspects of \'Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler's vector field $x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}$, and for which the formal normalisation is shown to be resurgent in $1/x$. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.


The Dynamical Mordell-Lang Conjecture

Abstract: We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $\phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $\phi$ are algebraic, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(\bP^1)^g$ has only finite intersection with any curve contained in $(\bP^1)^g$. We also show that our result holds for indecomposable polynomials $\phi$ with coefficients in $\bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $\bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(\phi,\phi)$ on $\bA^2$.


Virtual Bridge Number One Knots

Abstract: We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\in \N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i.$


Robustly estimating the flow direction of information in complex physical systems

Abstract: We propose a new measure to estimate the direction of information flux in multivariate time series from complex systems. This measure, based on the slope of the phase spectrum (Phase Slope Index) has invariance properties that are important for applications in real physical or biological systems: (a) it is strictly insensitive to mixtures of arbitrary independent sources, (b) it gives meaningful results even if the phase spectrum is not linear, and (c) it properly weights contributions from different frequencies. Simulations of a class of coupled multivariate random data show that for truly unidirectional information flow without additional noise contamination our measure detects the correct direction as good as the standard Granger causality. For random mixtures of independent sources Granger Causality erroneously yields highly significant results whereas our measure correctly becomes non-significant. An application of our novel method to EEG data (88 subjects in eyes-closed condition) reveals a strikingly clear front-to-back information flow in the vast majority of subjects and thus contributes to a better understanding of information processing in the brain.


Singularities of the Secant Variety

Abstract: We give positivity conditions on the embedding of a smooth variety which guarantee the normality of the secant variety, generalizing earlier results of the author and others. We also give classes of secant varieties satisfying the Hodge conjecture as well as a result on the singular locus of degenerate secant varieties.


Ternary cyclotomic polynomials having a large coefficient

Abstract: Let $\Phi_n(x)$ denote the $n$th cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that $a_n(k)$, the coefficient of $x^k$ in $\Phi_n(x)$, satisfies $|a_n(k)|\le (p+1)/2$ in case $n=pqr$ with $p<q<r$ primes (in this case $\Phi_n(x)$ is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example $|a_n(k)|\le 3p/4$). Here we show that, nevertheless, Beiter's conjecture is false for every $p\ge 11$. We also prove that given any $\epsilon>0$ there exist infinitely many triples $(p_j,q_j,r_j)$ with $p_1<p_2<... $ consecutive primes such that $|a_{p_jq_jr_j}(n_j)|>(2/3-\epsilon)p_j$ for $j\ge 1$.


Petri map for rank two bundles with canonical determinant

Abstract: We prove the Bertram-Feinberg-Mukai conjecture for a generic curve $C$ of genus $g$ and a semistable vector bundle $E$ of rank two and determinant $K$ on $C$, namely we prove the injectivity of the Petri-canonical map $S^2(H^0(E))\to H^0(S^2(E))$.


Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories

Abstract: We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.


The Wickstead Problem

Abstract: In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.


A proof of Goldbach's conjecture that all even numbers greater than four are the sum of two primes

Abstract: In this paper I introduce a model which allows one to prove Goldbachs hypothesis. The model is produced by studying Goldbach partitions as displayed by an inverted mirror image of all the primes up to some even number equal to the last prime plus three. The bottom half of the model is then moved to the right in steps of two which exhibit the Goldbach partitions for the next even number. As long as the model contains all the primes up to the resulting even number minus three, then Goldbachs hypothesis can be proven if it can be shown that each move must produce a Goldbach partition until one reaches the next prime plus one. I show that this must be the case.


Particle Approximation of the Wasserstein Diffusion

Abstract: We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein Diffusion, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya.


General runner removal and the Mullineux map

Abstract: We prove a new `runner removal theorem' for $q$-decomposition numbers of the level 1 Fock space of type $A^{(1)}_{e-1}$, generalising earlier theorems of James--Mathas and the author. By combining this with another theorem relating to the Mullineux map, we show that the problem of finding all $q$-decomposition numbers indexed by partitions of a given weight is a finite computation.


Local Ramsey theory. An abstract approach

Abstract: It is shown that the known notion of selective coideal can be extended to a family $\mathcal{H}$ of subsets of $\mathcal{R}$, where $(\mathcal{R},\leq,r)$ is a topological Ramsey space in the sense of Todorcevic (see \cite{todo}). Then it is proven that, if $\mathcal{H}$ selective, the $\mathcal{H}$-Ramsey and $\mathcal{H}$-Baire subsets of $\mathcal{R}$ are equivalent. This extends the results of Farah in \cite{farah} for semiselective coideals of $\mathbb{N}$. Also, it is proven that the family of ${\cal H}$--Ramsey subsets of ${\cal R}$ is closed under the Souslin operation.


Large Deviations for Riesz Potentials of Additive Processes

Abstract: We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^{t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p(t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2$. We obtain results about the large deviations and laws of the iterated logarithm for $\zeta_{t}$.


Degree k Linear Recursions Mod(p)

Abstract: Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If $F_{k,n}(t_1,...,t_k) = f_n$, then $$f_n - \sum_{j=1}^k t_j f_{n-j} = 0,$$ and $\{f_n\}$ is a linear recursion of degree $k$. On the one hand, the periodic properties of such sequences modulo a prime $p$ are discussed, and are shown to be rela ted to the prime structure of certain algebraic number fields; for example, the arithmetic properties of the period ar e shown to characterize ramification of primes in an extension field. On the other hand, the structure of the semiloca l rings associated with the number field is shown to be completely determined by Schur-hook polynomials.


Uniqueness of a pre-generator for $C_0$-semigroup on a general locally convex vector space

Abstract: The main purpose is to generalize a theorem of Arendt about uniqueness of $C_0$-semigroups from Banach space setting to the general locally convex vector spaces, more precisely, we show that cores are the only domains of uniqueness for $C_0$-semigroups on locally convex spaces. As an application, we find a necessary and sufficient condition for that the mass transport equation has one unique $L^1(\R^d,dx)$ weak solution.


A polynomial parametrization of torus knots

Abstract: For every odd integer $N$ we give an explicit construction of a polynomial curve $\cC(t) = (x(t), y (t))$, where $\deg x = 3$, $\deg y = N + 1 + 2\pent N4$ that has exactly $N$ crossing points $\cC(t_i)= \cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pad\'e approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.


The general quadruple point formula

Abstract: Maps between manifolds $M^m\to N^{m+\ell}$ ($\ell>0$) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morin-maps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4-secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to $\Sigma^2\Sigma^0$ multisingularities.


Multi-linear multipliers associated to simplexes of arbitrary length

Abstract: In this article we prove that the $n$-linear operator whose symbol is the characteristic function of the simplex $\Delta_n = \xi_1 < ... < \xi_n$ is bounded from $L^2 \times ... \times L^2$ into $L^{2/n}$, generalizing in this way our previous work on the "bi-est" operator (which corresponds to the case $n=3$) as well as Lacey-Thiele theorem on the bi-linear Hilbert transform (which corresponds to the case $n=2$).


Towards BAD conjecture

Abstract: For $\alpha, \beta, \delta \in [0,1], \alpha +\beta = 1 $ we consider sets $$ {\rm BAD}^* (\alpha, \beta ;\delta) = \left\{\xi = (\xi_1,\xi_2) \in [0,1]^2: ,\inf_{p\in \mathbb{N}} \max \{(p\log(p+1))^\alpha ||p\xi_1||, (p\log (p+1))^\beta ||p\xi_2||\} \ge \delta \right\}. $$ We prove that for different $(\alpha_1,\beta_1), (\alpha_2,\beta_2), \alpha_1 +\beta_1 = \alpha_2 +\beta_2 = 1 $ and $\delta $ small enough $$ {\rm BAD}^* (\alpha_1, \beta_1 ;\delta) \bigcap {\rm BAD}^* (\alpha_2, \beta_2 ;\delta) \neq \varnothing . $$ Our result is based on A. Khintchine's construction and an original method due to Y. Peres and W. Schlag.


Positivity results on ribbon Schur function differences

Abstract: There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur-positivity is essentially a product of two chains.


Limits Of One Dimensional Diffusions

Abstract: In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov.


An equivariant index formula for elliptic actions on contact manifolds

Abstract: Given an elliptic action of a compact Lie group $G$ on a co-oriented contact manifold $(M,E)$ one obtains two naturally associated objects: A $G$-transversally elliptic operator $\dirac$, and an equivariant differential form with generalised coefficients $\mathcal{J}(E,X)$ defined in terms of a choice of contact form on $M$. We explain how the form $\mathcal{J}(E,X)$ is natural with respect to the contact structure, and give a formula for the equivariant index of $\dirac$ involving $\mathcal{J}(E,X)$. A key tool is the Chern character with compact support developed by Paradan-Vergne \cite{PV1,PV}.


Morse Inequalities for Orbifold Cohomology

Abstract: This paper begins the study of Morse theory for orbifolds, or more precisely for differentiable Deligne-Mumford stacks. The main result is an analogue of the Morse inequalities that relates the orbifold Betti numbers of an almost-complex orbifold to the critical points of a Morse function on the orbifold. We also show that a generic function on an orbifold is Morse. In obtaining these results we develop for differentiable Deligne-Mumford stacks those tools of differential geometry and topology -- flows of vector fields, the strong topology -- that are essential to the development of Morse theory on manifolds.


Refinements of Milnor's Fibration Theorem for Complex Singularities

Abstract: Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_\epsilon$ be a ball in $U$ of sufficiently small radius $\epsilon>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_\theta$, with axis $V$, which leads to a fibration $\Phi$ of the whole space $(X \cap \mathbb{B}_\epsilon) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_\epsilon) \setminus V$ is the usual Milnor fibration $\phi = \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_\eta) \cap \mathbb{B}_\epsilon$ is the Milnor-L\^e fibration of $f$. Each element of the pencil $X_\theta$ meets transversally the boundary sphere $\mathbb{S}_\epsilon = \partial \B_\epsilon$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $\phi$ over antipodal points in the circle. Furthermore, the space ${\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\mathbb{R} \mathbb{P}^1$ with the $X_\theta$ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.


A priori bounds for some infinitely renormalizable quadratics: III. Molecules

Abstract: In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These {\it a priori bounds} imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.


Gluing of Surfaces with Polygonal Boundaries

Abstract: By pairwise gluing of edges of a polygon, one produces two-dimensional surfaces with handles and boundaries. In this paper, we count the number ${\cal N}_{g,L}(n_1, n_2, ..., n_L)$ of different ways to produce a surface of given genus $g$ with $L$ polygonal boundaries with given numbers of edges $n_1, n_2, >..., n_L$. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursive relations between ${\cal N}_{g,L}$. We show that Harer-Zagier numbers appear as a particular case of ${\cal N}_{g,L}$ and derive a new explicit expression for them.


Tame nonsmooth inverse mapping theorems

Abstract: We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity of some principal minors, contractiblity) of the space of Jacobi's matrices at smooth points.


Takacs' asymptotic theorem and its applications: A survey

Abstract: The book of Lajos Tak\'acs \emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$ P_{k,i}=\mathrm{P}{\sup_{1\leq n\leq\rho(i)}(N_n-n)<k-i},\leqno(*) $$ where $N_n=\nu_1+\nu_2...+\nu_n$ is a sum of mutually independent, nonnegative integer and identically distributed random variables, $\pi_j=\mathrm{P}\{\nu_k=j\}$, $j\geq0$, $\pi_0>0$, and $\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i\geq1$. (If there is no such $n$, then $\rho(i)=\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution type: $Q_0\neq0$ and $Q_k=\sum_{j=0}^k\pi_jQ_{k-j+1}$. Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey concerns only with one of the areas of application associated with asymptotic behavior of $Q_k$ as $k\to\infty$. The theorem on asymptotic behavior of $Q_k$ as $k\to\infty$ and further properties of that limiting sequence are given on pages 22-23 of the aforementioned book by Tak\'acs. In the present survey we discuss applications of Tak\'acs' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Many of the results presented in this survey have appeared recently, and some of them are new. In addition, further applications of Tak\'acs' theorem are discussed.


Lie Group Action and Stability Analysis of Stationary Solutions for a Free Boundary Problem Modelling Tumor Growth

Abstract: In this paper we study asymptotic behavior of solutions for a multidimensional free boundary problem modelling the growth of nonnecrotic tumors. We first establish a general result for differential equations in Banach spaces possessing a local Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either $D_A(\theta)$ or $D_A(\theta,\infty)$ type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient $\gamma$ is larger than a threshold value $\gamma^\ast$ then the unique stationary solution is asymptotically stable modulo translations, provided the constant $c$ representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small, whereas if $\gamma< \gamma^\ast$ then this stationary solution is unstable.


Asymptotic Stability of the Stationary Solution for a Hyperbolic Free Boundary Problem Modeling Tumor Growth

Abstract: In this paper we study asymptotic behavior of solutions for a free boundary problem modeling the growth of tumors containing two species of cells: proliferating cells and quiescent cells. This tumor model was proposed by Pettet et al in {\em Bull. Math. Biol.} (2001). By using a functional approach and the $C_0$ semigroup theory, we prove that the unique stationary solution of this model ensured by the work of Cui and Friedman ({\em Trans. Amer. Math. Soc.}, 2003) is locally asymptotically stable in certain function spaces. Key techniques used in the proof include an improvement of the linear estimate obtained by the work of Chen et al ({\em Trans. Amer. Math. Soc.}, 2005), and a similarity transformation.


Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform

Abstract: We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of $L^p$ spaces, with $p$ close enough to 1. We also prove that the trilinear Hilbert transform is unbounded in a similar range of $L^p$ spaces.


A Duality Result for Moduli Spaces of Semistable Sheaves Supported on Projective Curves

Abstract: We show that the moduli space M(r,c) of semistable sheaves on n-dimensional projective space with support of dimension one, with multiplicity r and with Euler characteristic c is isomorphic to M(r,-c).


Nonsmoothable group actions on elliptic surfaces

Abstract: Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of our previous paper.


Cyclotomic expansion of exceptional spectral measures

Abstract: We find explicit formulae for the circular spectral measures of $E_7,E_8$. This leads to a number of general observations regarding the ADE circular measures: these are linear combinations of measures supported by the roots of unity, with real density given by certain degree 3 polynomials.


Calculating Colimits Compositionally

Abstract: We show how finite limits and colimits can be calculated compositionally using the algebras of spans and cospans, and give as an application a proof of the Kleene Theorem on regular languages.


Variational inference for large-scale models of discrete choice

Abstract: Discrete choice models are commonly used by applied statisticians in numerous fields, such as marketing, economics, finance, and operations research. When agents in discrete choice models are assumed to have differing preferences, exact inference is often intractable. Markov chain Monte Carlo techniques make approximate inference possible, but the computational cost is prohibitive on the large data sets now becoming routinely available. Variational methods provide a deterministic alternative for approximation of the posterior distribution. We derive variational procedures for empirical Bayes and fully Bayesian inference in the mixed multinomial logit model of discrete choice. The algorithms require only that we solve a sequence of unconstrained optimization problems, which are shown to be convex. Extensive simulations demonstrate that variational methods achieve accuracy competitive with Markov chain Monte Carlo, at a small fraction of the computational cost. Thus, variational methods permit inferences on data sets that otherwise could not be analyzed without bias-inducing modifications to the underlying model.


An invariant regarding Waring's problem for cubic polynomials

Abstract: We compute the equation of the 7-secant variety to the Veronese variety (P^4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariantof plane cubics as a pfaffian.


On $p$-harmonic map heat flows for {$1\leq p< \infty$} and their finite element approximations

Abstract: Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a $BV$-solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms $\cA\cdot\nab\bv$ and $\cA\wedge\nab\bv$; which pair a divergence-$L^1$, or a divergence-measure, tensor field $\cA$, and a $BV$-vector field $\bv$. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the $p$-harmonic map heat flow, and the convergence of the proposed numerical method is also established.


The Viterbo Transfer as a Map of Spectra

Abstract: Let $L$ and $N$ be two smooth manifolds of the same dimension. Let $j\colon L\to T^*N$ be an exact Lagrange embedding. We denote the free loop space of $X$ by $\Lambda X$. C. Viterbo constructed a transfer map $(\Lambda j)^! \colon H^*(\Lambda L) \to H^*(\Lambda N)$. This transfer was constructed using finite dimensional approximation of Floer homology. In this paper we define a family of finite dimensional approximations and realize this transfer as a map of Thom spectra: $(\Lambda j)_! \colon (\Lambda N)^{-TN} \to (\Lambda L)^{-TL+\eta}$, where $\eta$ is a virtual vector bundle classified by the tangential information of $j$.


Betti numbers of monomial ideals and shifted skew shapes

Abstract: We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a fixed degree, and combinatorial interpretations for the Betti numbers of other classes of ideals, all of which are independent of the coefficient field.


An Extended Bracket Polynomial for Virtual Knots and Links

Abstract: This paper defines a new invariant of virtual knots and links that we call the extended bracket polynomial, and denote by <<K>> for a virtual knot or link K. This invariant is a state summation over bracket states of the oriented diagram for K. Each state is reduced to a virtual 4-regular graph in the plane and the polynomial takes values in the module generated by these reduced graphs over the ring Q[A,A^{-1}]. The paper is relatively self-contained, with background information about virtual knots and long virtual knots. We give numerous examples applying the extended bracket, including a new proof of the non-triviality of the Kishino diagram and the flat Kishino diagram and non-classicality of single crossing virtualizations. The paper has a section on the estimation of virtual crossing number using the extended bracket state sum. Examples are given of virtual knots with arbitrary minimal embedding genus and arbitrarily high positive difference between the virtual crossing number and the minimal embedding genus. A simplification of <<K>> is introduced and denoted by A[K]. This simplified extended bracket, the arrow polynomial, is a polynomial in an infinite set of variables. It is quite strong (detecting the flat Kishino diagram for example) and easily computable. The paper contains a description of a computer algorithm for A[K] and uses the arrow polynomial, in conjunction with the extended bracket polynomial to determine the minimum genus surfaces on which some virtual knots can be represented.


Double Ore Extensions

Abstract: A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the determinant of the DE-data are studied. Using the double Ore extension, we construct 26 families of Artin-Schelter regular algebras of global dimension four in a sequel paper.


Double Extension Regular Algebras of Type (14641)

Abstract: We construct several families of Artin-Schelter regular algebras of global dimension four using double Ore extension and then prove that all these algebras are strongly noetherian, Auslander regular, Koszul and Cohen-Macaulay domains. Many regular algebras constructed in the paper are new and are not isomorphic to either a normal extension or an Ore extension of an Artin-Schelter regular algebra of global dimension three.


Artin-Schelter Regular Algebras, Subalgebras, and Pushouts

Abstract: Take $A$ to be a regular quadratic algebra of global dimension three. We observe that there are examples of $A$ containing a dimension three regular cubic algebra $C$. If $B$ is another dimension three regular quadratic algebra, also containing $C$ as a subalgebra, then we can form the pushout algebra $D$ of the inclusions $i_1:C\hookrightarrow A$ and $i_2:C\hookrightarrow B$. We show that for a certain class of regular algebras $C\hookrightarrow A,B$, their pushouts $D$ are regular quadratic algebras of global dimension four. Furthermore, some of the point module structures of the dimension three algebras get passed on to the pushout algebra $D$.


Leavitt path algebras and direct limits

Abstract: An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.


Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874)

Abstract: Throughout the whole year of 1874, C. Jordan and L. Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in 1870 a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics, not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be looking into the 1874 quarrel, our purpose will be to show how the complex identities of practices such as Jordan s canonical reduction and Kronecker s invariant computation highlight some cultural issues such as tacit knowledge and perceptions of history peculiar to individuals or communities as well as some local ways of thinking such as disciplinary ideals and internal philosophies of generality and simplicity.


Convex Entropy Decay via the Bochner-Bakry-Emery approach

Abstract: We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli-Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear.


Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties

Abstract: For the root system of type $A$ we introduce and study a certain extension of the quadratic algebra invented by S. Fomin and the first author, to construct a model for the equivariant cohomology ring of the corresponding flag variety. As an application of our construction we describe a generalization of the equivariant Pieri rule for double Schubert polynomials. For a general finite Coxeter system we construct an extension of the corresponding Nichols-Woronowicz algebra. In the case of finite crystallographic Coxeter systems we present a construction of extended Nichols-Woronowicz algebra model for the equivariant cohomology of the corresponding flag variety.


Minima in branching random walks

Abstract: Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\mathbf {E}M_n$ when the branching random walk has bounded branching and step size.


Local Lipschitz geometry of weighted homogeneous surfaces

Abstract: We compute Hoelder Complexes,i.e. the complete bi-Lipschitz invariants, for germs of real weighed homogeneous algebraic or semialgebraic surfaces.


Twisting quasi-alternating links

Abstract: Quasi-alternating links are homologically thin for both Khovanov homology and knot Floer homology. We show that every quasi-alternating link gives rise to an infinite family of quasi-alternating links obtained by replacing a crossing with an alternating rational tangle. Consequently, we show that many pretzel links are quasi-alternating, and we determine the thickness of Khovanov homology for "most" pretzel links with arbitrarily many strands.


A_2-web immanants

Abstract: We describe the rank 3 Temperley-Lieb-Martin algebras in terms of Kuperberg's A_2-webs. We define consistent labelings of webs, and use them to describe the coefficients of decompositions into irreducible webs. We introduce web immanants, inspired by Temperley-Lieb immanants of Rhoades and Skandera. We show that web immanants are positive when evaluated on totally positive matrices, and describe some further properties.


Rate of relaxation for a mean-field zero-range process

Abstract: We study the zero-range process on the complete graph. It is a Markov chain model for a microcanonical ensemble. We prove that the process converges to a fluid limit. The fluid limit rapidly relaxes to the appropriate Gibbs distribution.


New directions in Nielsen-Reidemeister theory

Abstract: The purpose of this expository paper is to present new directions in the classical Nielsen-Reidemeister fixed point theory. We describe twisted Burnside-Frobenius theorem, groups with $R_\infty$ \emph{property} and a connection between Nielsen fixed point theory and symplectic Floer homology.


Even perfect polynomials over $F_2$ with four prime factors

Abstract: A perfect polynomial over the binary field $\F_2$ is a polynomial $A \in \F_2[x]$ that equals the sum of all its divisors. If $\gcd(A,x^2-x) \neq 1$ then we call $A$ even. The list of all even perfect polynomials over $\F_2$ with at most 3 prime factors in known. The object of this paper is to give the list of all even perfect polynomials over $\F_2$ with four prime factors. These are all the known perfect polynomials with four prime factors over $\F_2$.


Full Algebra of Generalized Functions and Non-Standard Asymptotic Analysis

Abstract: We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau's solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn-Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by J.F. Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.


Vector spaces with an order unit

Abstract: We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple extensions to an order (semi)norm on the entire space. We single out three of these (semi)norms for further study and discuss their significance for operator algebras and operator systems. In addition, we introduce a functorial method for taking an ordered space with an order unit and forming an Archimedean ordered space. We then use this process to describe an appropriate notion of quotients in the category of Archimedean ordered spaces.


Characters of unipotent groups over finite fields

Abstract: Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called "admissible pairs" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form G(F_q) (where G is a unipotent algebraic group over F_q), the geometry of G, and the theory of character sheaves.


On Extremal k-Graphs Without Repeated Copies of 2-Intersecting Edges

Abstract: The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptions) when k=4.


Generalized Complex and Dirac Structures on Homogeneous Spaces

Abstract: We partially describe equivariant Dirac and generalized complex structures on a homogeneous space $G/K$ by giving equivalent data involving only the Lie algebra. We consider real semisimple adjoint orbits in any semisimple Lie algebra over $\mathbb R$ and real nilpotent orbits in $sl_n (\mathbb R)$. We give a complete classification for Riemannian symmetric spaces and for a compact group modulo a closed, connected subgroup containing a Cartan subgroup.


A Brunn-Minkowski Inequality for Symplectic Capacities of Convex Domains

Abstract: In this work we prove a Brunn-Minkowski-type inequality in the context of symplectic geometry and discuss some of its applications.


Exponentiating $2\times2$ and $3\times3$ Matrices Done Right

Abstract: We derive explicit formulas for calculating $e^A$, $\cosh{A}$, $\sinh{A}, \cos{A}$ and $\sin{A}$ for a given $2\times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3\times3$ matrix $A$. These formulas are expressed exclusively in terms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients.


Two Categories of Dirac Manifolds

Abstract: We define two categories of Dirac manifolds, i.e. manifolds with complex Dirac structures. The first notion of maps I call \emph{Dirac maps}, and the category of Dirac manifolds is seen to contain the categories of Poisson and complex manifolds as full subcategories. The second notion, \emph{dual-Dirac maps}, defines a \emph{dual-Dirac category} which contains presymplectic and complex manifolds as full subcategories. The dual-Dirac maps are stable under B-transformations. In particular we get two structures of a category on Hitchin'sgeneralized complex manifolds, i.e., two reasonable notions of generalized complex maps. We also generalize further to get categories of Dirac manifolds for which the Dirac structures lie in arbitrary exact Courant algebroids. As an example, we consider the case of a Lie group with a complex Dirac structure and establish conditions for which multiplication is a Dirac map.


On the Limit Law of a Random Walk Conditioned to Reach a High Level

Abstract: We consider a random walk with a negative drift and with a jump distribution which under Cram\'er's change of measure belongs to the domain of attraction of a spectrally positive stable law. If conditioned to reach a high level and suitably scaled, this random walk converges in law to a nondecreasing Markov process which can be interpreted as a spectrally-positive L\'evy %-Khinchin process conditioned not to overshoot level one.


Chern class formulas for $G_2$ Schubert loci

Abstract: We define degeneracy loci for vector bundles with structure group $G_2$, and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein-Gelfand-Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli-Thom-Porteous, Kempf-Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, clarifying a previous computation of Edidin and Graham.


The Veronese Construction for Formal Power Series and Graded Algebras

Abstract: Let $(a_n)_{n \geq 0}$ be a sequence of complex numbers such that its generating series satisfies $\sum_{n \geq 0} a_nt^n = \frac{h(t)}{(1-t)^d}$ for some polynomial $h(t)$. For any $r \geq 1$ we study the transformation of the coefficient series of $h(t)$ to that of $h^{< r >}(t)$ where $\sum_{n \geq 0} a_{nr} t^n = \frac{h^{< r >}(t)}{(1-t)^d}$. We give a precise description of this transformation and show that under some natural mild hypotheses the roots of $h^{< r >}(t)$ converge when $r$ goes to infinity. In particular, this holds if $\sum_{n \geq 0} a_n t^n$ is the Hilbert series of a standard graded $k$-algebra $A$. If in addition $A$ is Cohen-Macaulay then the coefficients of $h^{< r >}(t)$ are monotonely increasing with $r$. If $A$ is the Stanley-Reisner ring of a simplicial complex $\Delta$ then this relates to the $r$th edgewise subdivision of $\Delta$ which in turn allows some corollaries on the behavior of the respective $f$-vectors.


Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations

Abstract: In this paper, we consider a global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations (\textit{ANS}). In order to do so, we first introduce the scaling invariant Besov-Sobolev type spaces, $B^{-1+\frac{2}{p},{1/2}}_{p}$ and $B^{-1+\frac{2}{p},{1/2}}_{p}(T)$, $p\geq2$. Then, we prove the global wellposedness for (\textit{ANS}) provided the initial data are sufficient small compared to the horizontal viscosity in some suitable sense, which is stronger than $B^{-1+\frac{2}{p},{1/2}}_{p}$ norm. In particular, our results imply the global wellposedness of (\textit{ANS}) with high oscillatory initial data.


Analysis of nonlinear modes of variation for functional data

Abstract: A set of curves or images of similar shape is an increasingly common functional data set collected in the sciences. Principal Component Analysis (PCA) is the most widely used technique to decompose variation in functional data. However, the linear modes of variation found by PCA are not always interpretable by the experimenters. In addition, the modes of variation of interest to the experimenter are not always linear. We present in this paper a new analysis of variance for Functional Data. Our method was motivated by decomposing the variation in the data into predetermined and interpretable directions (i.e. modes) of interest. Since some of these modes could be nonlinear, we develop a new defined ratio of sums of squares which takes into account the curvature of the space of variation. We discuss, in the general case, consistency of our estimates of variation, using mathematical tools from differential geometry and shape statistics. We successfully applied our method to a motivating example of biological data. This decomposition allows biologists to compare the prevalence of different genetic tradeoffs in a population and to quantify the effect of selection on evolution.


Uniqueness of real closure * of regular rings

Abstract: In this paper we give a characterisation of real closure * of regular rings, which is quite similar to the characterisation of real closure * of Baer regular rings seen in [4]. We also characterize Baer-ness of regular rings using near-open maps. The last part of this work will concentrate on classifying the real closure * of Baer and non-Baer regular rings (upto isomorphisms) using continuous sections of the support map, we construct a topology on this set for the Baer case. For the case of non-Baer regular rings, it will be shown that almost no information of the ring structure of the Baer hull is necessary in order to study the real and prime spectra of the Baer hull. We shall make use of the absolutes of Hausdorff spaces in order to give a construction of the spectra of the Baer hulls of regular rings. Finally we give example of a Baer regular ring that is not rationally complete.


Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment

Abstract: We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle.


Poisson structures and generalized Kahler structures

Abstract: Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler structures and provide non-trivial examples of generalized Kahler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bi-Hermitian structures constructed from Poisson structures.


Cubicity, Boxicity and Vertex Cover

Abstract: A $k$-dimensional box is the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as $box(G)$, is the minimum integer $k$ such that $G$ is the intersection graph of a collection of $k$-dimensional boxes. A unit cube in $k$-dimensional space or a $k$-cube is defined as the cartesian product $R_1 \times R_2 \times ... \times R_k$ where each $R_i$ is a closed interval on the real line of the form $[a_i, a_{i}+1]$. The {\it cubicity} of $G$, denoted as $cub(G)$, is the minimum $k$ such that $G$ is the intersection graph of a collection of $k$-cubes. In this paper we show that $cub(G) \leq t + \left \lceil \log (n - t)\right\rceil - 1$ and $box(G) \leq \left \lfloor\frac{t}{2}\right\rfloor + 1$, where $t$ is the cardinality of the minimum vertex cover of $G$ and $n$ is the number of vertices of $G$. We also show the tightness of these upper bounds. F. S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph $G$, $box(G) \leq \left \lfloor\frac{n}{2} \right \rfloor$, where $n$ is the number of vertices of $G$, and this bound is tight. We show that if $G$ is a bipartite graph then $box(G) \leq \left \lceil\frac{n}{4} \right\rceil$ and this bound is tight. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to $\frac{n}{4}$. Interestingly, if boxicity is very close to $\frac{n}{2}$, then chromatic number also has to be very high. In particular, we show that if $box(G) = \frac{n}{2} - s$, $s \geq 0$, then $\chi(G) \geq \frac{n}{2s+2}$, where $\chi(G)$ is the chromatic number of $G$.


Q-systems as cluster algebras

Abstract: Q-systems first appeared in the analysis of the Bethe equations for the XXX-model and generalized Heisenberg spin chains. Such systems are known to exist for any simple Lie algebra and many other Kac-Moody algebras. We formulate the Q-system associated with any simple, simply-laced Lie algebras g in the language of cluster algebras, and discuss the relation of the polynomiality property of the solutions of the $Q$-system in the initial variables, which follows from the representation-theoretical interpretation, to the Laurent phenomenon in cluster algebras.


The uniform primality conjecture for elliptic curves

Abstract: An elliptic divisibility sequence, generated by a point in the image of a rational isogeny, is shown to possess a uniformly bounded number of prime terms. This result applies over the rational numbers, assuming Lang's conjecture, and over the rational function field, unconditionally. In the latter case, a uniform bound is obtained on the index of a prime term. Sharpened versions of these techniques are shown to lead to explicit results where all the irreducible terms can be computed.


Square root voting in the Council of the European Union: Rounding effects and the Jagiellonian Compromise

Abstract: In recent years, enlargement of the European Union has brought with it renewed discussion of voting arrangements in the Council of the EU. During these negotiations, the Polish government proposed a voting scheme that gives each country a voting weight proportional to the square root of its population, and sets a quota according to an optimality condition ("Jagiellonian Compromise"). In this paper, the optimal quota is found exactly for the current population data from the 27 EU member states, and it is found that rounding of the voting weights can be used to improve the voting scheme.


The Banff Challenge: Statistical Detection of a Noisy Signal

Abstract: Particle physics experiments such as those run in the Large Hadron Collider result in huge quantities of data, which are boiled down to a few numbers from which it is hoped that a signal will be detected. We discuss a simple probability model for this and derive frequentist and noninformative Bayesian procedures for inference about the signal. Both are highly accurate in realistic cases, with the frequentist procedure having the edge for interval estimation, and the Bayesian procedure yielding slightly better point estimates. We also argue that the significance, or $p$-value, function based on the modified likelihood root provides a comprehensive presentation of the information in the data and should be used for inference.


On effaceability of certain $\delta$-functors

Abstract: We prove a conjecture of the first author for $GL_2(F)$, where $F$ is a finite extension of $Q_p$.


Some unbounded functions of intermittent maps for which the central limit theorem holds

Abstract: We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map $T$ of $[0, 1]$ with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of $f\circ T^i$, when $f$ belongs to a large class of unbounded functions from $[0, 1]$ to ${\mathbb R}$. We also prove other limit theorems and moment inequalities.


Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

Abstract: Let there be given a probability measure $\mu$ on the unit circle $\TT$ of the complex plane and consider the inner product induced by $\mu$. In this paper we consider the problem of orthogonalizing a sequence of monomials $\{z^{r_k}\}_k$, for a certain order of the $r_k\in\mathbb{Z}$, by means of the Gram-Schmidt orthogonalization process. This leads to a basis of orthonormal Laurent polynomials $\{\psi_k\}_k$. We show that the matrix representation with respect to the basis $\{\psi_k\}_k$ of the operator of multiplication by $z$ is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of Delvaux and Van Barel. We show that the shape of the snake is determined by the order in which the monomials $\{z^{r_k}\}_k$ are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for $\mu$. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.


Free pluriharmonic majorants and noncommutative interpolation

Abstract: In this paper, we initiate the study of sub-pluriharmonic curves in Cuntz-Toeplitz algebras and free pluriharmonic majorants on noncommutative balls. We are lead to a characterization of the noncommutative Hardy space $H^2_{\bf ball}$ in terms of free pluriharmonic majorants, and to a Schur type description of the unit ball of $H^2_{\bf ball}$. These results are used to solve a multivariable commutant lifting problem and provide a description of all solutions.


Graph limits and exchangeable random graphs

Abstract: We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability.


Solution of the Pompeiu problem (II)

Abstract: This paper has been withdrawn by the author.


Positive divisors on quotients of $\bar{M}_{0,n}$ and the Mori cone of $\bar{M}_{g,n}$

Abstract: We prove that if $m \ge n-3$ then every $S_m$-invariant F-nef divisor on the moduli space of stable $n$-pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points.


Regularity of radial minimizers of reaction equations involving the p-Laplacian

Abstract: We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.


The copies of any permutation pattern are asymptotically normal

Abstract: We prove that the number of copies of any given permutation pattern $q$ has an asymptotically normal distribution in random permutations.


Some Progress in Conformal Geometry

Abstract: This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.


Central cross-sections make surfaces of revolution quadric

Abstract: We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revolution in higher dimensions.


A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case

Abstract: In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state $\rho=p$, based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove {\em a priori} estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.


Heteroclinic Travelling Waves of Gradient Diffusion Systems

Abstract: We establish existence of travelling waves to the gradient system $u_t = u_{zz} - \nabla W(u)$ connecting two minima of $W$ when $u : \R \times (0,\infty) \larrow \R^N$, that is, we establish existence of a pair $(U,c) \in [C^2(\R)]^N \by (0,\infty)$, satisfying \[ \{{array}{l} U_{xx} - \nabla W (U) = - c U_x U(\pm \infty) = a^{\pm}, {array}. \] where $a^{\pm}$ are local minima of the potential $W \in C_{\textrm{loc}}^2(\R^N)$ with $W(a^-)< W(a^+)=0$ and $N \geq 1$. Our method is variational and based on the minimization of the functional $E_c (U) = \int_{\R}\Big\{{1/2}|U_x|^2 + W(U) \Big\}e^{cx} dx$ in the appropriate space setup. Following Alikakos-Fusco \cite{A-F}, we introduce an artificial constraint to restore compactness and force the desired asymptotic behavior, which we later remove. We provide variational characterizations of the travelling wave and the speed. In particular, we show that $E_c(U)=0$.


Transversely non simple knots

Abstract: By proving a connected sum formula for the Legendrian invariant $\lambda_+$ in knot Floer homology we exhibit infinitely many transversely non simple knots.


Nestings of Matchings and Permutations and North Steps in PDSAWs

Abstract: We present a simple bijective proof of the fact that matchings of [2n] with N nestings are equinumerous to partially directed self avoiding walks confined to the symmetric wedge defined by y=+-x, with n east steps and N north steps. A very similar construction connects permutations with N nestings and PDSAWs remaining below the x-axis, again with N north steps. Furthermore, both bijections transport several combinatorially meaningful parameters.


Prescribing valuations of the order of a point in the reductions of abelian varieties and tori

Abstract: Let G be the product of an abelian variety and a torus defined over a number field K. Let R be a K-rational point on G of infinite order. Call n_R the number of connected components of the smallest algebraic K-subgroup of G to which R belongs. We prove that n_R is the greatest positive integer which divides the order of (R mod p) for all but finitely many primes p of K. Furthermore, let m>0 be a multiple of n_R and let S be a finite set of rational primes. Then there exists a positive Dirichlet density of primes p of K such that for every l in S the l-adic valuation of the order of (R mod p) equals v_l(m).


On pairs of commuting nilpotent matrices

Abstract: Let $B$ be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition $\lambda$. Then it is known that its nilpotent commutator $N_B$ is an irreducible variety and that there is a unique partition $\mu$ such that the intersection of the orbit of nilpotent matrices corresponding to $\mu$ with $N_B$ is dense in $N_B$. We prove that map $D$ given by $D(\lambda)=\mu$ is an idempotent map. This answers a question of Basili and Iarrobino and gives a partial answer to a question of Panyushev. In the proof, we use the fact that for a generic matrix $A \in N_B$ the algebra generated by $A$ and $B$ is a Gorenstein algebra. Thus, a generic pair of commuting nilpotent matrices generates a Gorenstein algebra. We also describe $D(\lambda)$ in terms of $\lambda$ if $D(\lambda)$ has at most two parts.


Two variants of the support problem for products of abelian varieties and tori

Abstract: Let G be the product of an abelian variety and a torus defined over a number field K. Let P and Q be K-rational points on G. Suppose that for all but finitely many primes p of K the order of (Q mod p) divides the order of (P mod p). Then there exist a K-endomorphism f of G and a non-zero integer c such that f(P)=cQ. Furthermore, we are able to prove the above result with weaker assumptions: instead of comparing the order of the points we only compare the radical of the order (radical support problem) or the l-adic valuation of the order for some fixed rational prime l (l-adic support problem).


Coverage processes on spheres and condition numbers for linear programming

Abstract: This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\alpha)$ be the probability that $n$ spherical caps of angular radius $\alpha$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\alpha)$ in the case $\alpha\in[\pi/2,\pi]$ and an upper bound for $p(n,m,\alpha)$ in the case $\alpha\in [0,\pi/2]$ which tends to $p(n,m,\pi/2)$ when $\alpha\to\pi/2$. In the case $\alpha\in[0,\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius $\alpha$ that are needed to cover $S^m$. Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem $\exists x\in\mathbb{R}^{m+1}Ax\le0,x\ne0$ where $A\in\mathbb{R}^{n\times(m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.


On the motivic spectra representing algebraic cobordism and algebraic K-theory

Abstract: We show that the motivic spectrum representing algebraic $K$-theory is a localization of the suspension spectrum of $\mathbb{P}^\infty$, and similarly that the motivic spectrum representing periodic algebraic cobordism is a localization of the suspension spectrum of $BGL$. In particular, working over $\mathbb{C}$ and passing to spaces of $\mathbb{C}$-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic $K$-theory and periodic algebraic cobordism are $E_\infty$ motivic spectra.


Voting, the symmetric group, and representation theory

Abstract: We show how voting may be viewed naturally from an algebraic perspective by viewing voting profiles as elements of certain well-studied $\mathbb{Q}S_n$-modules. By using only a handful of simple combinatorial objects (e.g., tabloids) and some basic ideas from representation theory (e.g., Schur's Lemma), this allows us to recast and extend some well-known results in the field of voting theory.


Transfinite diameter notions in C^N and integrals of Vandermonde determinants

Abstract: We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in C^N. An ingredient is a formula of Rumely which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in C^N. Our results extend to certain weights and measures defined on cones in R^N.


On the Lego-Teichmuller game for finite $G$ cover

Abstract: Given a smooth, oriented, closed surface $\Sigma$ of genus zero, possibly with boundary, let $\tilde{\Sigma} \longrightarrow \Sigma$ be a given $G$-cover of $\Sigma$, where $G$ is a given finite group. Let $S_{n}$ denote the standard sphere with $n$ holes. There are many ways of gluing together several $G$-cover of $S_{n}$ to construct the $G$-cover $\ts \longrightarrow \Sigma$, of $\Sigma$. We let $M(\tilde{\Sigma} ,\Sigma)$ be the set of all ways to construct the given $G$-cover, $\tilde{\Sigma} \longrightarrow \Sigma$, of $\Sigma$ from gluing of several $G$-covers of $S_{n}$, here $n$ may vary. In this paper, we define some simple moves and relation which will turn $M(\tilde{\Sigma} ,\Sigma)$ into a connected and simply-connected complex. This will be used in the future paper to construct $G$-equivariant Modular Functor. This $G$-equivariant Modular Functor will be an extension of the usual Modular Functor.


The Skorokhod problem in a time-dependent interval

Abstract: We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.


On Ext-indices of ring extensions

Abstract: In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index and we show that the Auslander-Reiten conjecture is true for this type of rings.


Smooth and palindromic Schubert varieties in affine Grassmannians

Abstract: Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical Grassmannian manifold. It has a Schubert cell decomposition indexed by the coroot lattice or equivalently by the minimal length coset representatives for the affine Weyl group modulo the Weyl group for G. The closure of an affine Schubert cell is a finite dimensional projective variety that we call an affine Schubert variety. In this paper we completely determine the smooth and palindromic (rationally smooth) affine Schubert varieties.


Approximation of the joint spectral radius using sum of squares

Abstract: We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.


On homotopy of volterrian quadratic stochastic operators

Abstract: In the present paper we introduce a notion of homotopy of two Volterra operators which is related to fixed points of such operators. It is establish a criterion when two Volterra operators are homotopic, as a consequence we obtain that the corresponding tournaments of that operators are the same. This, due to \cite{Ga1}, gives us a possibility to know some information about the trajectory of homotopic Volterra operators. Moreover, it is shown that any Volterra q.s.o. given on a face has at least two homotopic extension to the whole simplex.


Hilbert schemes of finite abelian group orbits and Grobner fans

Abstract: Let $G$ be a finite abelian subgroup of $PGL(r-1,K)=\mathrm{Aut}(\P^{r-1}_K)$. In this paper, we prove that the normalization of the $G$-orbit Hilbert scheme $\Hilb^G(\P^{r-1})$ is described as a toric variety, which corresponds to the Gr\"obner fan for some homogeneous ideal $I$ of $K[x_1, ..., x_r]$.


Random matrices, free probability, planar algebras and subfactors

Abstract: Using a family of graded algebra structures on a planar algebra and a family of traces coming from random matrix theory, we obtain a tower of non-commutative probability spaces, naturally associated to a given planar algebra. The associated von Neumann algebras are II$_{1}$ factors whose inclusions realize the given planar algebra as a system of higher relative commutants. We thus give an alternative proof to a result of Popa that every planar algebra can be realized by a subfactor.


The Cops & Robber game on series-parallel graphs

Abstract: The Cops and Robber game is played on undirected finite graphs. $k$ cops and one robber are positioned on vertices and take turn in moving along edges. The cops win if, after a move, a cop and the robber are on the same vertex. A graph is called $k$-copwin, if the cops have a winning strategy. It is known that planar graphs are 3-copwin (Aigner & Fromme, 1984) and that outerplanar graphs are 2-copwin (Clarke, 2002). In this short note, we prove that series-parallel (i.e., graphs with no $K_4$ minor) graphs are 2-copwin. It is a well-known trick in the literature of cops & robber games to define variants of the game which impose restrictions on the possible strategies of the cops (see Clarke, 2002). For our proof, we define the ``cops & robber game with exits''. Our proof yields a winning strategy for the cops.


Model selection for quantum homodyne tomography

Abstract: This paper deals with a non-parametric problem coming from physics, namely quantum tomography. That consists in determining the quantum state of a mode of light through a homodyne measurement. We apply several model selection procedures: penalized projection estimators, where we may use pattern functions or wavelets, and penalized maximum likelihood estimators. In all these cases, we get oracle inequalities. In the former we also have a polynomial rate of convergence for the non-parametric problem. We finish the paper with applications of similar ideas to the calibration of a photocounter, a measurement apparatus counting the number of photons in a beam. Here the mathematical problem reduces similarly to a non-parametric missing data problem. We again get oracle inequalities, and better speed if the photocounter is good.


C^0-rigidity of Poisson brackets

Abstract: Consider a functional associating to a pair of compactly supported smooth functions on a symplectic manifold the maximum of their Poisson bracket. We show that this functional is lower semi-continuous with respect to the product uniform (C^0) norm on the space of pairs of such functions. This extends previous results of Cardin-Viterbo and Zapolsky. The proof involves theory of geodesics of the Hofer metric on the group of Hamiltonian diffeomorphisms. We also discuss a failure of a similar semi-continuity phenomenon for multiple Poisson brackets of three or more functions.


Attractive nearest-neighbor spin systems on the integers in a randomly evolving environment

Abstract: We consider spin systems on $\Z$ (i.e.\ interacting particle systems on $\Z$ in which each coordinate only has two possible values and only one coordinate changes in each transition) whose rates are determined by another process, called a background process. A canonical example is the so called contact process in randomly evolving environment (CPREE), introduced and analysed by E. Broman and furthermore studied by J. Steif and the author, where the marginals of the background process independently evolve as 2-state Markov chains and determine the recovery rates for a contact process. We prove that under certain conditions on the rates there are at most two extremal stationary distributions. The proof follows closely the ideas of Liggett's proof of a corresponding theorem for spin systems on $\Z$ without a background process.


Stein's method on Wiener chaos

Abstract: We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It\^o integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Ess\'een bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.


On A two-variable p-adic l_q function

Abstract: We prove that a two-variable p-adic l_q-function has the series p-adic expansion which interpolates a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given this note is due to Washington


$\R$-trees, dual laminations, and compact systems of partial isometries

Abstract: Let $\FN$ be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-tree with a very small, minimal action of $\FN$ with dense orbits. For any basis $\CA$ of $\FN$ there exists a {\em heart} $K_{\CA} \subset \bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$, for each $a_{i} \in \CA$, defines a tree $T_{(K_{\CA}, \CA)}$ which contains an isometric copy of $T$ as minimal subtree.


Finite unions of balls in C^n are rationally convex

Abstract: It is shown that the rational convexity of any finite union of disjoint closed balls in C^n follows easily from the results of Duval and Sibony.


The first Hochschild cohomology group of a schurian cluster-tilted algebra

Abstract: Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type \tilde{\mathbb{A}}.


Rigid objects in higher cluster categories

Abstract: We study maximal $m$-rigid objects in the $m$-cluster category $\mathcal C_H^m$ associated with a finite dimensional hereditary algebra $H$ with $n$ nonisomorphic simple modules. We show that all maximal $m$-rigid objects in these categories have exactly $n$ nonisomorphic indecomposable summands, and that any almost complete $m$-rigid object in $\mathcal C_H^m$ has exactly $m+1$ nonisomorphic complements. We also show that the maximal $m$-rigid objects and the $m$-cluster tilting objects in these categories coincide, and that the class of finite dimensional algebras associated with maximal $m$-rigid objects is closed under certain factor algebras.


An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $H^2 \times R$

Abstract: In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $H^2\times R$. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary $C$ is a Jordan curve homologous to zero in the asymptotic boundary of $ H^2\times R,$ say $\partial_\infty H^2\times R$, such that $C$ is contained in a slab between two horizontal circles of $\partial_\infty H^2\times R$ with width equal to $\pi.$ We construct minimal vertical graphs in $H^2\times R$ over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains $\Om$ in $H^2\times \{0\}$ are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.


On the rate of convergence and Berry-Esseen type theorems for a multivariate free central limit theorem

Abstract: We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operator-valued Cauchy transforms of the normalized partial sums in an operator-valued free central limit theorem and the Cauchy transform of the limiting operator-valued semicircular element.


Homogenization of reflected semilinear PDE with nonlinear Neumann boundary condition

Abstract: We study the homogenization problem of semi linear reflected partial differential equations (reflected PDEs for short) with nonlinear Neumann conditions. The non-linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and uses weak convergence of associated reflected generalized backward differential stochastic equations (reflected GBSDEs in short).


Note sur la conjecture de Leopoldt

Abstract: We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units


A general Lagrange Theorem

Abstract: The ordinary continued fractions expansion of a real number is based on the Euclidean division. Variants of the latter yield variants of the former, all encompassed by a more general Dynamical Systems framework. For all these variants the Lagrange Theorem holds: a number has an eventually periodic expansion if and only if it is a quadratic irrational. This fact is surely known for specific expansions, but the only proof for the general case that I could trace in the literature follows as an implicit corollary from much deeper results by Boshernitzan and Carroll on interval exchange transformations. It may then be useful to have at hand a simple and virtually computation-free proof of a general Lagrange Theorem.


Plongements l-adiques et l-nombres de Weil

Abstract: We define l-adic analogs of classical Weil numbers in connexion both with complex or l-adic imbeddings of number fields and real or l-adic absolute values. As an application we give some consequences related to the Iwasawa theory of cyclotomic towers.


Compactification l-adique de R

Abstract: We construct a compact topological group Rl which contains both the real additive group R and the l-adic one Ql (for a given prime number l) as dense subgroups; thus we study some of its properties. This construction gives an arithmetic description of the so-called l-adic solenoid classically defined in terms of foliations.


On tropical and Kapranov ranks of tropical matrices

Abstract: We prove that, for any g greater or equal than 3, a matrix g x 5 with tropical rank 3 has Kapranov rank 3.


A Spectral sequence for polynomially bounded cohomology

Abstract: We construct an analogue of the Lyndon-Hochschild-Serre spectral sequence in the context of polynomial cohomology, for group extensions. If G is an extension of Q by H, then the spectral sequence converges to the polynomial cohomology of G. For the polynomial extensions of Noskov with normal subgroup isocohomological, the E_2 term is the polynomial cohomology of Q with coefficients in the polynomial cohomology of H. When both Q and H are isocohomological G must be as well. By referencing results of Connes-Moscovici and Noskov, if Q and H are both isocohomological and have the Rapid Decay property of Jolissaint, then G satisfies the Novikov Conjecture.


SLE and the free field: Partition functions and couplings

Abstract: Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\SLE$ and the free field with appropriate boundary conditions; this involves $\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, showing that, in a precise sense, chordal $\SLE$ is the solution of a stochastic "differential" equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general $\kappa>0$.


When Do Random Subsets Decompose a Finite Group?

Abstract: Let A,B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group; i.e. (AB union BA) = G. The study of this event gives rise to a group invariant we call \Theta(G). \Theta(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes \sqrt{\Theta(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\sqrt{\Theta(G)|G|\log|G|}, then with high probability (AB union BA) does not equal G. If A and B are larger than (1+c)\sqrt{\Theta(G)|G|\log|G|} then (AB union BA) equals G with high probability.


Modular classes of Lie algebroid morphisms

Abstract: We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.


Renormalisation-induced phase transitions for unimodal maps

Abstract: The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t \log|Df|$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.


Sets of double and triple weights of trees

Abstract: Let T be a weighted tree with n leaves. Let D_{i,j} be the distance between the leaves i and j. Let D_{i,j,k}= (D_{i,j} + D_{j,k} +D_{i,k})/2. We will call such numbers "triple weights" of the tree. In this paper, we give a characterization, different from the previous ones, for sets indexed by 2-subsets of a $n$-set to be double weights of a tree. By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an $n$-set to be the set of the triple weights of a tree with $n$ leaves. Besides we propose a slight modification of Saitou-Nei's Neighbour-Joining algorithm to reconstruct trees from the data D_{i,j}.


Approximation of sets defined by polynomials with holomorphic coefficients

Abstract: Let X be an analytic set defined by polynomials whose coefficients a_1,...,a_s are holomorphic functions. We formulate conditions such that for all sequences {a_(1,n)},...,{a_(s,n)} of holomorphic functions converging locally uniformly to a_1,...,a_s respectively the following holds true. If a_(1,n),...,a_(s,n) satisfy the conditions then the sequence of the sets {X_n} obtained by replacing a_j by a_(j,n) in the polynomials, converge to X.


Stability of homogeneous bundles on P^3

Abstract: We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver representation is a parallelepiped are stable, for instance the bundles E whose minimal free resolution is of the kind 0 --> S^{l_1, l_2, l_3} V (t) --> S^{l_1 +s, l_2, l_3} V (t+s) --> E --> 0 are stable.


Identities and Inequalities for Tree Entropy

Abstract: The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede-Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.


A note on generalized equivariant homotopy groups

Abstract: In this paper, we generalize the equivariant homotopy groups or equivalently the Rhodes groups. We establish a short exact sequence relating the generalized Rhodes groups and the generalized Fox homotopy groups and we introduce $\Gamma$-Rhodes groups, where $\Gamma$ admits a certain co-grouplike structure. Evaluation subgroups of $\Gamma$-Rhodes groups are discussed.


Gibbs Sampling for a Bayesian Hierarchical General Linear Model

Abstract: We consider a Bayesian hierarchical version of the normal theory general linear model which is practically relevant in the sense that it is general enough to have many applications and it is not straightforward to sample directly from the corresponding posterior distribution. Thus we study a block Gibbs sampler that has the posterior as its invariant distribution. In particular, we establish that the Gibbs sampler converges at a geometric rate. This allows us to establish conditions for a central limit theorem for the ergodic averages used to estimate features of the posterior. Geometric ergodicity is also a key component for using batch means methods to consistently estimate the variance of the asymptotic normal distribution. Together, our results give practitioners the tools to be as confident in inferences based on the observations from the Gibbs sampler as they would be with inferences based on random samples from the posterior. Our theoretical results are illustrated with an application to data on the cost of health plans issued by health maintenance organizations.


Almost all integer matrices have no integer eigenvalues

Abstract: For a fixed $n\ge2$, consider an $n\times n$ matrix $M$ whose entries are random integers bounded by $k$ in absolute value. In this paper, we examine the probability that $M$ is singular (hence has eigenvalue 0), and the probability that $M$ has at least one rational eigenvalue. We show that both of these probabilities tend to 0 as $k$ increases. More precisely, we establish an upper bound of size $k^{-2+\epsilon}$ for the probability that $M$ is singular, and size $k^{-1+\epsilon}$ for the probability that $M$ has a rational eigenvalue. These results generalize earlier work by Kowalsky for the case $n=2$ and answer a question posed by Hetzel, Liew, and Morrison.


Complete semi-conjugacies for psuedo-Anosov homeomorphisms

Abstract: Suppose $S$ is a surface of genus $\ge 2 $, $f: S \to S$ is a surface homeomorphism isotopic to a pseudo-Anosov map $\alpha$ and suppose $\ti S$ is the universal cover of $S$ and $F$ and $A$ are lifts of $f$ and $\alpha$ respectively. We show there is a semiconjugacy $\Theta : \ti S \to \bar \L^s \times \bar \L^u$ from $F$ to $\bar A$, where $\bar \L^s$ ($\bar \L^u$) is the completion of the $R$-tree of leaves of the stable (resp. unstable) foliation for $A$ and $\bar A$ is the map induced by $A$. We also generalize a result of Markovich and show that for any $g \in Homeo(S)$ which commutes with $f$ and has identity lift $G : \ti S \to \ti S$ and for any $(c,w)$ in the image of $\Theta$ each component of $\Theta^{-1}(c,w)$ is $G$-invariant.


Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS

Abstract: The singular parabolic problem $u_t=\Delta u -\frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $R^N$ with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage --represented here by $\lambda$. As a byproduct, we prove that for sufficiently large $\lambda$, finite-time quenching must occur near the maximum point of the varying dielectric permittivity profile $f(x)$.


C*-algebras associated to product systems of Hilbert bimodules

Abstract: Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems X, the universal representation of X in its Cuntz-Nica-Pimsner algebra is isometric.


Continuity, curvature, and the general covariance of optimal transportation

Abstract: Let M and \bar M be n-dimensional manifolds equipped with suitable Borel probability measures \rho and \bar\rho. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee smoothness of the optimal map pushing \rho forward to \bar\rho; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M \times \bar M. H\"older continuity of optimal maps was established for rougher mass distributions by Loeper, still relying on a key result of Trudinger & Wang which required certain structure on the domains and the cost. We go on to develop this theory for mass distributions on differentiable manifolds -- recovering Loeper's Riemannian examples such as the round sphere as particular cases -- give a direct proof of the key result mentioned above, and revise Loeper's H\"older continuity argument to make it logically independent of all earlier works, while extending it to less restricted geometries and cost functions even for subdomains M and \bar M of R^n. We also give new examples of geometries satisfying the hypotheses -- obtained using submersions and tensor products -- and some connections to spacelike Lagrangian submanifolds in symplectic geometry.


Block-diagonal semidefinite programming hierarchies for 0/1 programming

Abstract: Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported.


Orthogonal polynomials and partial differential equations on the unit ball

Abstract: Orthogonal polynomials of degree $n$ with respect to the weight function $W_\mu(x) = (1-\|x\|^2)^\mu$ on the unit ball in $\RR^d$ are known to satisfy the partial differential equation $$ [ \Delta - \la x, \nabla \ra^2 - (2 \mu +d) \la x, \nabla \ra \right ] P = -n(n+2 \mu+d) P $$ for $\mu > -1$. The singular case of $\mu = -1,-2, ...$ is studied in this paper. Explicit polynomial solutions are constructed and the equation for $\nu = -2,-3,...$ is shown to have complete polynomial solutions if the dimension $d$ is odd. The orthogonality of the solution is also discussed.


On splitting polynomials with noncommutative coefficients

Abstract: It is shown that for every splitting of a polynomial with noncommutative coefficients into linear factors $(X-a_{k})$ with $a_{k}$'s commuting with coefficients, any cyclic permutation of linear factors gives the same result and all $a_{k}$ are roots of that polynomial. Examples are given and analyzed from Galois theory point of view.


Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Abstract: Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.