Equivariant orbifold structures on the projective line and integrable hierarchies

Abstract: Let $\CP^1_{k,m}$ be the orbifold structure on $\CP^1$ obtained via uniformizing the neighborhoods of 0 and $\infty$ respectively by $z\mapsto z^k$ and $w\mapsto w^m.$ The diagonal action of the torus on the projective line induces naturally an orbifold action on $\CP^1_{k,m}.$ In this paper we prove that if k and m are co-prime then Givental's prediction of the equivariant total descendent Gromov-Witten potential of $\CP^1_{k,m}$ satisfies certain Hirota Quadratic Equations (HQE for short). We also show that after an appropriate change of the variables, similar to Getzler's change in the equivariant Gromov-Witten theory of $\CP^1$, the HQE turn into the HQE of the 2-Toda hierarchy, i.e., the Gromov-Witten potential of $\CP^1_{k,m}$ is a tau-function of the 2-Toda hierarchy. More precisely, we obtain a sequence of tau-functions of the 2-Toda hierarchy from the descendent potential via some translations. The later condition, that all tau-functions in the sequence are obtained from a single one via translations, imposes a serious constraint on the solution of the 2-Toda hierarchy. Our theorem leads to the discovery of a new integrable hierarchy (we suggest to be called the Equivariant Bi-graded Toda Hierarchy). We conjecture that this new hierarchy governs, i.e., uniquely determines, the equivariant Gromov-Witten invariants of $\CP^1_{k,m}.$


Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport

Abstract: We state an improved version of the conjecture of Langlands and Rapoport, and we prove the conjecture for a large class of Shimura varieties. In particular, we obtain the first proof of the (original) conjecture for Shimura varieties of PEL-type.


A new characterization for the m-quasiinvariants of S_n and explicit basis for two row hook shapes

Abstract: In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. While many properties of those spaces were proven from this definition, an explicit computation of a basis was only done in certain cases. In particular, Feigin and Veselov computed bases for the m-quasiinvariants of dihedral groups, including S_3, and Felder and Veselov computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a basis for the isotypic component indexed by the partition [n-1,1]. This builds on a previous paper in which we computed a basis for S_3 via combinatorial methods.


Towards a proof of the conjecture of Langlands and Rapoport

Abstract: A conference talk discussing the conjecture of Langlands and Rapoport concerning the structure of the points on a Shimura variety modulo a prime of good reduction.


Vanishing theorems for toric polyhedra

Abstract: A toric polyhedron is a reduced closed subscheme of a toric variety that are partial unions of the orbits of the torus action. We prove vanishing theorems for toric polyhedra. We also give a proof of the $E_1$-degeneration of Hodge to de Rham type spectral sequence for toric polyhedra in any characteristic. Finally, we give a very powerful extension theorem for ample line bundles.


Combinatorial Aspects of Elliptic Curves

Abstract: Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor in addition to the usual number theoretical interpretations. In particular we prove that $N_k = - W_k(q, - N_1)$ where W_k(q,t) is a (q,t)-analogue of the number of spanning trees of the wheel graph. Additionally we develop a determinantal formula for N_k where the eigenvalues can be explicitly written in terms of q, N_1, and roots of unity. We also discuss here a new sequence of bivariate polynomials related to the factorization of N_k, which we refer to as elliptic cyclotomic polynomials because of their various properties.


Random generation of finitely generated subgroups of a free group

Abstract: We give an efficient algorithm to randomly generate finitely generated subgroups of a given size, in a finite rank free group. Here, the size of a subgroup is the number of vertices of its representation by a reduced graph such as can be obtained by the method of Stallings foldings. Our algorithm randomly generates a subgroup of a given size n, according to the uniform distribution over size n subgroups. In the process, we give estimates of the number of size n subgroups, of the average rank of size n subgroups, and of the proportion of such subgroups that have finite index. Our algorithm has average case complexity $\O(n)$ in the RAM model and $\O(n^2\log^2n)$ in the bitcost model.


The barnes G function and its relations with sums and products of generalized Gamma convolution variables

Abstract: We give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables, and that the reciprocal of the Barnes G-function has a L\'{e}vy-Khintchin type representation. These results lead us to introduce the so called generalized gamma convolution variables.


The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

Abstract: We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.


Galilean geometry of motions

Abstract: In this paper we show that Galilean group is a matrix Lie group and find its structure. Then provide the invariants of special Galilean geometry of motions, by Olver's method of moving coframes, we also find the corresponding $\{e\}-$structure.


Index theory for actions of compact Lie groups on C*-algebras

Abstract: We study the index theory for actions of compact Lie groups on C*-algebras with an emphasis on principal actions. Given an invariant semifinite trace on the C*-algebra we obtain semifinite spectral triples. For circle actions we consider the relation to the dual Pimsner-Voiculescu sequence. On the way we show that the notions ``saturated'' and ``principal'' are equivalent for actions by compact Lie groups.


Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology

Abstract: We consider the change-of-rings spectral sequence as it applies to Hochschild cohomology, obtaining a description of the differentials on the first page which relates it to the multiplicative stucture on cohomology. Using this information, we are able to completely describe the cohomology structure of monogenic algebras as well as some information on the structure of the cohomology in more general situations. We also show how to use the spectral sequence to reprove and generalize results of M. Auslander et. al. about homological epimorphisms. We derive from this a rather general version of the long exact sequence due to D. Happel for a one-point (co)-extension of a finite dimensional algebra and show how it can be put to use in concrete examples.


Complex cobordism and algebraic topology

Abstract: This is a historical survey, beginning where Atiyah and Sullivan leave off...


Soft ideals and arithmetic mean ideals

Abstract: This article investigates the soft-interior and the soft-cover of operator ideals. These operations, and especially the first one, have been widely used before, but making their role explicit and analyzing their interplay with the arithmetic mean operations is essential for the study of the multiplicity of traces (see arXiv:0707.3169v1 [math.FA]). Many classical ideals are "soft", i.e., coincide with their soft interior or with their soft cover, and many ideal constructions yield soft ideals. Arithmetic mean (am) operations were proven to be intrinsic to the theory of operator ideals by the work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators and arithmetic mean operations at infinity were studied in arXiv:0707.3169v1 [math.FA]. Here we focus on the commutation relations between these operations and soft operations. In the process we characterize the am-interior and the am-infinity interior of an ideal.


Invariants of Closed 3-Manifolds via Nullhomotopic Filling Dehn Spheres

Abstract: We provide a calculus for the presentation of closed 3-manifolds via nullhomotopic filling Dehn spheres and we use it to define an invariant of closed 3-manifolds by applying the state-sum machinery. As a potential application of this invariant, we show how to get lower bounds for the Matveev complexity of P2-irreducible closed 3-manifolds. We also describe an easy algorithm for constructing a nullhomotopic filling Dehn sphere of each closed 3-manifold from any of its one-vertex triangulations.


Measure-valued equations for Kolmogorov operators with unbounded coefficients

Abstract: Given a real and separable Hilbert space H we consider the measure-valued equation \begin{equation*} \int_H\phi(x)\mu_t(dx)- \int_H\phi(x)\mu(dx)= \int_0^t(\int_HK_0\phi(x)\mu_s(dx))ds, \end{equation*} where K_0 is the Kolmogorov differential operator \[ K_0\phi(x)=\frac12\textrm{Trace}\big[BB^*D^2\phi(x)\big]+< x,A^*D\phi(x)>+< D\phi(x),F(x)>, \] $x\in H$, $\phi:H\to \Rset$ is a suitable smooth function, $A:D(A)\subset H\to H $ is linear, $F:H\to H$ is a globally Lipschitz function and $B:H\to H$ is linear and continuous. In order prove existence and uniqueness of a solution for the above equation, we show that $K_0$ is a core, in a suitable way, of the infinitesimal generator associated to the solution of a certain stochastic differential equation in H. We also extend the above results to a reaction-diffusion operator with polinomial nonlinearities.


Gibbs Rapidly Samples Colorings of G(n,d/n)

Abstract: Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd\H{o}s-R\'enyi random graph G(n,d/n). While the average degree in G(n,d/n) is d(1-o(1)), it contains many nodes of degree of order $\log n / \log \log n$. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring or the Ising model, the mixing time of Gibbs sampling is at least $n^{1 + \Omega(1 / \log \log n)}$. High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including coloring. In this work consider sampling q-colorings and show that for every $d < \infty$ there exists $q(d) < \infty$ such that for all $q \geq q(d)$ the mixing time of Gibbs sampling on G(n,d/n) is polynomial in $n$ with high probability. Our results are the first polynomial time mixing results proven for the coloring model on G(n,d/n) for d > 1 where the number of colors does not depend on n. They extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. The results also generalize to the hard-core model at low fugacity and to general models of soft constraints at high temperatures.


An exotic shuffle relation of $\zeta(\{2\}^m)$ and $\zeta(\{3,1\}^n)$

Abstract: In this short note we will provide a new and shorter proof of the following exotic shuffle relation of multiple zeta values: $$\zeta(\{2\}^m \sha\{3,1\}^n)={2n+m\choose m} \frac{\pi^{4n+2m}}{(2n+1)\cdot (4n+2m+1)!}.$$ This was proved by Zagier when n=0, by Broadhurst when $m=0$, and by Borwein, Bradley, and Broadhurst when m=1. In general this was proved by Bowman and Bradley in \emph{The algebra and combinatorics of shuffles and multiple zeta values}, J. of Combinatorial Theory, Series A, Vol. \textbf{97} (1)(2002), 43--63. Our idea in the general case is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by WZ-method.


Picard-graded Betti numbers and the defining ideals of Cox rings

Abstract: Let X be a smooth projective variety with torsion-free Picard group. We introduce complexes of vector spaces whose homology determines the structure of the minimal free resolution of the Cox ring of X over the polynomial ring and show how the homology of these complexes can be studied by purely geometric methods. As an application of these techniques we give a simple new proof of a characterization of the Cox rings of Del Pezzo surfaces (of degree >1) conjectured by Batyrev and Popov.


Jet Riemann-Lagrange Geometry Applied to Evolution DEs Systems from Economy

Abstract: The aim of this paper is to construct a natural Riemann-Lagrange differential geometry on 1-jet spaces, in the sense of nonlinear connections, generalized Cartan connections, d-torsions, d-curvatures, jet electromagnetic fields and jet Yang-Mills energies, starting from some given non-linear evolution DEs systems modelling economic phenomena, like the Kaldor model of the bussines cycle or the Tobin-Benhabib-Miyao model regarding the role of money on economic growth.


Geometrical Objects on the First Order Jet Space $J^1(T,R^5)$ Produced by the Lorenz Atmospheric DEs System

Abstract: The aim of this paper is to construct natural geometrical objects on the 1-jet space J^1(T,R^5), where $T/subset R$, like a non-linear connection, a generalized Cartan connection, together with its d-torsions and d-curvatures, a jet electromagnetic d-field and a jet Yang-Mills energy, starting from the given Lorenz atmospheric DEs system and the pair of Euclidian metrics $/Delta = (1,/delta_{ij})$ on $T/times R^5$.


Characteristic classes of the Hilbert schemes of points on non-compact simply-connected surfaces

Abstract: We prove a closed formula expressing any multiplicative characteristic class evaluated on the tangent bundle of the Hilbert schemes of points on a non-compact simply-connected surface. As a corollary, we deduce a closed formula for the Chern character of the tangent bundles of these Hilbert schemes. We also give a closed formula for the multiplicative characteristic classes of the tautological bundles associated to a line bundle on the surface. We finally remark which implications the results here have for the Hilbert schemes of points of an arbitrary surface.


A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

Abstract: The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced but a notion ubiquitous in the literature on operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem which extends theorems by A. Neumann, by Arveson and Kadison, and by Antezana, Massey, Ruiz and Stojanoff. We also list ten open questions that we encountered in the development of this material.


Operator valued frames

Abstract: We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends to higher multiplicity (e.g., multiframes) their dilation approach. We prove several results for operator-valued frames concerning their parametrization, duality, disjointeness, complementarity, and composition and the relationship between the two types of similarity (left and right) of such frames. We then apply these notions to prove that the collection of multiframe generators for the action of a discrete group on a Hilbert space is norm pathwise-connected precisely when the von Neumann algebra generated by the right representation of the group has no minimal projections. The proof is obtained by parametrizing this collection by a class of partial isometries in a larger von Neumann algebra. In the multiplicity one case this class reduces to the unitary class which is path-connected in norm, but in the infinite multiplicity case this class is path connected only in the strong operator topology and the proof depends on properties of tensor product slice maps.


Functional Equation for Theta Series

Abstract: In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata Lectures on Theta I (1983), Birkhauser].


Origin of the numerals, Al biruni testimony

Abstract: The origin of the numerals that we inherited from the arabo-Islamic civilization remained one enigma. The hypothesis of the Indian origin remained, with controversies, without serious rival. It was the dominant hypothesis since more of one century. Its partisans found to it and constructed a lot of arguments. The testimonies of the medieval authors have been interpreted to its advantage. The opposite opinions have been dismissed and ignored. An amalgam between the history of our modern numerals and the Indian mathematics history is made. Rational contradictions often passed under silence. A meticulous observation of the numerals permits to affirm that our numerals are in fact more or less modified Arabic letters. The "Ghubari" shape of the numerals shows that the symbol of a numeral corresponds to the Arabic letter whose numerical value is equal to this numeral. The numerals don't have a simple resemblance with some Arabic letters, but every number looks like the Arabic letter whose numerical value is equal to this numeral. The elements of the ''Abjadi'' calculation gives us a theoretical support, independent of the letters and numerals, witch explains our observation. Besides a re-lecture of the testimonies of the medieval authors, particularly the testimony of Al-Biruni, that is probably at the origin of all others testimonies speaking of the Indian origin of the numerals, is in agreement with the fact that our numerals are Arabic letters. We have there a second way concerning the origin of our modern numerals that is only to its beginnings. The deepened researches are necessary to understand the history of our numerals better. A rigorous re-lecture of the medieval testimonies with a new mind imposes itself.


Operator valued frames on C*-modules

Abstract: Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.


Valdivia compact groups are products

Abstract: It is shown that every Valdivia compact group is homeomorphic to a product of metrizable compacta.


Root systems for asymmetric geometric representations of Coxeter groups

Abstract: Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.


Supercuspidal characters of reductive p-adic groups

Abstract: We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu's construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu's construction, Fourier transforms of orbital integrals, and certain signs and cardinalities that are described explicitly in terms of the datum associated to the representation and of the element at which the character is evaluated.


Finite index subgroups of the modular group and their modular forms

Abstract: Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois representation arising from noncongruence cuspforms, and Atkin and Swinnerton-Dyer congruences.


Morita equivalences of cyclotomic Hecke algebras of type G(r,p,n)

Abstract: We prove a Morita reduction theorem for the cyclotomic Hecke algebras H_{r,p,n}({q,Q})$ of type G(r,p,n). As a consequence, we show that computing the decomposition numbers of H_{r,p,n}(Q) reduces to computing the p'-splittable decomposition numbers of the cyclotomic Hecke algebras H_{r',p',n'}(Q'), where $1\le r'\le r$, $1\le n'\le n$, $ p'\mid p$ and where the parameters Q' are contained in a single $(\epsilon,q)$-orbit and $\epsilon$ is a primitive p'th root of unity.


On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

Abstract: We study a variant of a problem considered by Dinaburg and Sinai on the statistics of the minimal solution to a linear Diophantine equation. We show that the signed ratio between the Euclidean norms of the minimal solution and the coefficient vector is uniformly distributed modulo one. We reduce the problem to an equidistribution theorem of Anton Good concerning the orbits of a point in the upper half-plane under the action of a Fuchsian group.


Some connections between results and problems of De Giorgi, Moser and Bangert

Abstract: Using theorems of Bangert, we prove a rigidity result which shows how a question raised by Bangert for elliptic integrands of Moser type is connected, in the case of minimal solutions without self-intersections, to a famous conjecture of De Giorgi for phase transitions.


Non-real zeros of linear differential polynomials

Abstract: Lower bounds are given for the number of non-real zeros of a second order linear differential polynomial with constant coefficients in a real entire function with finitely many non-real zeros.


On the irrelevant disorder regime of pinning models

Abstract: Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that is now rigorous. In this work, we exploit interpolation and replica coupling methods to obtain sharper results on the irrelevant disorder regime of pinning models. In particular, in this regime, we compute the first order term in the expansion of the free energy close to criticality and this term coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and quenched averaged correlation length exponents coincide, while, in general, they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.


On the geometry of cohomogeneity one manifolds with positive curvature

Abstract: This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics.


Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles

Abstract: In this paper we give some examples of almost para-hyperhermitian structures on the tangent bundle of an almost product manifold, on the product manifold $M\times\mathbb{R}$, where $M$ is a manifold endowed with a mixed 3-structure and on the circle bundle over a manifold with a mixed 3-structure.


On the base locus of the linear system of generalized theta functions

Abstract: Let $\cM_r$ denote the moduli space of semi-stable rank-$r$ vector bundles with trivial determinant over a smooth projective curve $C$ of genus $g$. In this paper we study the base locus $\cB_r \subset \cM_r$ of the linear system of the determinant line bundle $\cL$ over $\cM_r$, i.e., the set of semi-stable rank-$r$ vector bundles without theta divisor. We construct base points in $\cB_{g+2}$ over any curve $C$, and base points in $\cB_4$ over any hyperelliptic curve.


Weighted Strichartz estimates for radial Schr\"odinger equation on noncompact manifolds

Abstract: We prove global weighted Strichartz estimates for radial solutions of linear Schr\"odinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity.


Approximation by Several Rationals

Abstract: Following T. H. Chan, we consider the problem of approximation of a given rational fraction a/q by sums of several rational fractions a_1/q_1, ..., a_n/q_n with smaller denominators. We show that in the special cases of n=3 and n=4 and certain admissible ranges for the denominators q_1,..., q_n, one can improve a result of T. H. Chan by using a different approach.


How Much Work Does It Take To Straighten a Plane Graph Out?

Abstract: We prove that if one wants to make a plane graph drawing straight-line then in the worst case one has to move almost all vertices.


Turan's theorem inverted

Abstract: Turan's theorem implies that every graph of order n with more edges than the r-partite Turan graph contains a complete graph of order r+1. We show that the same premise implies the existence of much larger graphs. We also prove corresponding stability theorems. These results complete work started by Erdos in 1963.


Limit laws for boolean convolutions

Abstract: We study the distributional behavior for products, and for sums of boolean independent random variables in an infinitesimal triangular array. We show that the limit laws of boolean convolutions are determined by the limit laws of free convolutions, and vice versa. We further use these results to show several connections between the limiting distributional behavior of classical convolutions and that of boolean convolutions. The proof of our results is based on the analytical apparatus developed for free convolutions.


On the noncommutative spin geometry of the standard Podles sphere and index computations

Abstract: The purpose of the paper is twofold: First, known results of the noncommutative spin geometry of the standard Podles sphere are extended by discussing Poincare duality and orientability. In the discussion of orientability, Hochschild homology is replaced by a twisted version which avoids the dimension drop. The twisted Hochschild cycle representing an orientation is related to the volume form of the distinguished covariant differential calculus. Integration over the volume form defines a twisted cyclic 2-cocycle which computes the q-winding numbers of quantum line bundles. Second, a "twisted" Chern character from equivariant K0-theory to even twisted cyclic homology is introduced which gives rise to a Chern-Connes pairing between equivariant K0-theory and twisted cyclic cohomology. The Chern-Connes pairing between the equivariant K0-group of the standard Podles sphere and the generators of twisted cyclic cohomology relative to the modular automorphism and its inverse are computed. This includes the pairings with the twisted cyclic 2-cocycle associated to the volume form, and the one corresponding to the "no-dimension drop" case. From explicit index computations, it follows that the pairings with these cocycles give the q-indices of the known equivariant 0-summable Dirac operator on the standard Podles sphere.


Kouchnirenko type formulas for local invariants of plane analytic curves

Abstract: Let f(x,y)=0 be an equation of plane analytic curve defined in the neighborhood of the origin and let $\pi:M\to(\Cn^2,0)$ be a local toric modification. We give a formula which connects a number of double points \delta_0(f)$ with a sum $\sum_p \delta_p(\tilde f)$ which runs over all intersection points of the proper preimage of f=0 with the exceptional divisor.


On a Gibbs characterization of normalized generalized Gamma processes

Abstract: We show that a Gibbs characterization of normalized generalized Gamma processes, recently obtained in Lijoi, Pr\"unster and Walker (2007), can alternatively be derived by exploiting a characterization of exponentially tilted Poisson-Kingman models stated in Pitman (2003). We also provide a completion of this result investigating the existence of normalized random measures inducing exchangeable Gibbs partitions of type $\alpha \in (-\infty, 0]$.


Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

Abstract: We prove a general extrinsic rigidity theorem for homogeneous varieties in $\mathbb{CP}^N$. The theorem is used to show that the adjoint variety of a complex simple Lie algebra $\mathfrak{g}$ (the unique minimal G orbit in $\mathbb{P}\mathfrak{g}$) is extrinsically rigid to third order. In contrast, we show that the adjoint variety of $SL_3\mathbb{C}$, and the Segre product $\mathit{Seg}(\mathbb{P}^1\times \mathbb{P}^n)$, both varieties with osculating sequences of length two, are flexible at order two. In the $SL_3\mathbb{C}$ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques.


When almost all sets are difference dominated

Abstract: We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,...,N}, chosen randomly according to a binomial model with parameter p(N), with N^{-1} = o(p(N)). We show that the random subset is almost surely difference dominated, as N --> oo, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference- to the sumset. If p(N) = o(N^{-1/2}) then almost all sums and differences in the random subset are almost surely distinct, and in particular the difference set is almost surely about twice as large as the sumset. If N^{-1/2} = o(p(N)) then both the sum and difference sets almost surely have size (2N+1) - O(p(N)^{-2}), and so the ratio in question is almost surely very close to one. If p(N) = c N^{-1/2} then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_{f,g} N^{-1/2}, for some computable constant c_{f,g}, such that one form almost surely dominates below the threshold, and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p.


An improved Julia-Caratheodory theorem for Schur-Agler mappings of the unit ball

Abstract: We adapt Sarason's proof of the Julia-Caratheodory theorem to the class of Schur-Agler mappings of the unit ball, obtaining a strengthened form of this theorem. In particular those quantities which appear in the classical theorem and depend only on the component of the mapping in the complex normal direction have K-limits (not just restricted K-limits) at the boundary.


Norms and spectral radii of linear fractional composition operators on the ball

Abstract: We give a new proof that every linear fractional map of the unit ball induces a bounded composition operator on the standard scale of Hilbert function spaces on the ball, and obtain norm bounds analogous to the standard one-variable estimates. We also show that Cowen's one-variable spectral radius formula extends to these operators. The key observation underlying these results is that every linear fractional map of the ball belongs to the Schur-Agler class.


Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

Abstract: We prove that the norm of a weighted composition operator on the Hardy space H^2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H^2, and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.


Commensurability of geometric subgroups of mapping class groups

Abstract: Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly disconnected). We characterise the subsurfaces which lead to virtually abelian geometric subgroups. We provide algebraic and geometric conditions under which two geometric subgroups are commensurable. We also describe the commensurator of a geometric subgroup in terms of the stabiliser of the underlying subsurface. Finally, we show some applications of our analysis to the theory of irreducible unitary representations of mapping class groups.


Central limit theorems for multiple Skorohod integrals

Abstract: In this paper, we prove a central limit theorem for a sequence of iterated Shorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed.


Stability and intersection properties of solutions to the nonlinear biharmonic equation

Abstract: We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\Delta^2 \phi = \phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\geq p_c$. Then, we focus on the supercritical case $p\geq p_c$ and we show that the graphs of no two solutions intersect one another.


Convergence of Kaehler-Ricci flow with integral curvature bound

Abstract: Let $g(t)$, $t\in [0, +\infty)$, be a solution of the normalized K\"ahler-Ricci flow on a compact K\"ahler $n$-manifold $M$ with $c_{1}(M)>0$ and initial metric $g (0)\in 2\pi c_{1}(M)$. If there is a constant $C$ independent of $t$ such that $$ \int_{M}|Rm(g(t))|^{n}dv_{t}\leq C,$$ then, for any $t_{k}\to \infty$, a subsequence of $(M, g(t_{k}))$ converges to a compact orbifold $(X, h)$ with only finite many singular points $\{q_{j}\}$ in the Gromov-Hausdorff sense, where $h$ is a K\"ahler metric on $X\backslash \{q_{j}\}$ satisfying the K\"ahler-Ricci soliton equation, i.e. there is a smooth function $f$ such that $$Ric(h)-h=\nabla\bar{\nabla}f, {\rm and}\it \nabla \nabla f=\bar{\nabla} \bar{\nabla} f=0. $$


Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

Abstract: We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.


Sums of products of congruence classes and of arithmetic progressions

Abstract: Consider the congruence class R_m(a)={a+im:i\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b)+R_m(c)R_m(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+\ell m) for some i,j,k,\ell\in Z. It is proved that if gcd(a,b,c,d,m)=1, then R_m(a)R_m(b)+R_m(c)R_m(d) is equal to the congruence class R_m(ab+cd), and that the sum of products set P_m(a)P_m(b)+P_m(c)P_m(d) eventually coincides with the infinite arithmetic progression P_m(ab+cd).


Abstract: Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities and their associated skew group rings. In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit, the numerical version of the McKay correspondence for canonical orders which, relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order.


Loop Products and Closed Geodesics

Abstract: We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.


Derivatives of embedding functors I: the stable case

Abstract: For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula $V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal calculus, in the case when $N$ is a nice open submanifold of a Euclidean space. This leads to a description of the derivatives of this functor when $N$ is a tame stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in $M$, and a certain ``homotopy bundle of spectra'' over this space of trees. The $n$-th derivative is then described as the ``spectrum of restricted sections'' of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor $\Ebar(M, N\times V)$, involving a similar construction with certain spaces of connected graphs (instead of forests) with points marked in $M$.


Decomposition of Cartan Matrix and conjectures on Brauer character degrees

Abstract: Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{\alpha \in F[G]|J\alpha =0\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is related to what kind of blocks in $N$ covered by $B$. We mainly consider a block $B$ of $G$ which covers a block $b$ of $N$ with $l(b)=1$. In two cases, we prove Willems' conjecture holds for these blocks, which covers some true cases by Holm and Willems. Furthermore We give an affirmative answer to a question by Holm and Willems in our cases. Some other results about Cartan invariants are presented in our paper.


Dynamics of symmetric holomorphic maps on projective spaces

Abstract: We consider complex dynamics of a critically finite holomorphic map from P^k to P^k, which has symmetries associated with the symmetric group S_{k+2} acting on P^k, for each k \ge 1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.


Generating mapping class groups of nonorientable surfaces with boundary

Abstract: We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.


A comparison theorem for simplicial resolutions

Abstract: It is well known that Barr and Beck's definition of comonadic homology makes sense also with a functor of coefficients taking values in a semi-abelian category instead of an abelian one. The question arises whether such a homology theory has the same convenient properties as in the abelian case. Here we focus on independence of the chosen comonad: conditions for homology to depend on the induced class of projectives only.


Limit theorems for conditioned multitype Dawson-Watanabe processes and Feller diffusions

Abstract: A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every finite time interval, its distribution is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. Several results on the long time behavior of the conditioned mass process|the conditioned multitype Feller branching diffusion are then proved. The general case is first considered, where the mutation matrix which models the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are analyzed too.


SUSY structures on deformed supermanifolds

Abstract: We construct a geometric structure on deformed supermanifolds as a certain subalgebra of the vector fields. In the classical limit we obtain a decoupling of the infinitesimal odd and even transformations, whereas in the semiclassical limit the result is a representation of the supersymmetry algebra. In the case of mass preserving structure we describe all high energy corrections to this algebra.


Proper holomorphic disks in the complement of varieties in \C^2

Abstract: For any closed analytic set X in C^2 there exists a proper holomorphic embedding of the unit disk into C^2 such that the image avoids X.


Cohomology of Split Group Extensions and Characteristic Classes

Abstract: There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochischild-Serre spectral sequence of an extension of an integral lattice L by a group G. These characteristic classes exist in a given page of the spectral sequence provided the differentials in the previous pages are all zero. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.


Singular measures of circle homeomorphisms with two break points

Abstract: Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.


On the Hausdorff dimension of invariant measures of weakly contracting on average measurable IFS

Abstract: We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is ergodic. We also prove that it is ergodic iff the related skew product is.


Degenerating families of dendrograms

Abstract: Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist $p$-adic representation of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the $p$-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are $p$-adic parameter spaces of (families of) dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.


Exchangeable Random Networks

Abstract: We introduce and study a class of exchangeable random graph ensembles. They can be used as statistical null models for empirical networks, and as a tool for theoretical investigations. We provide general theorems that carachterize the degree distribution of the ensemble graphs, together with some features that are important for applications, such as subgraph distributions and kernel of the adjacency matrix. These results are used to compare to other models of simple and complex networks. A particular case of directed networks with power-law out--degree is studied in more detail, as an example of the flexibility of the model in applications.


Parallel Tiled QR Factorization for Multicore Architectures

Abstract: As multicore systems continue to gain ground in the High Performance Computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents an algorithm for the QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in an out of order execution of the tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with the LAPACK algorithm for QR factorization where parallelism can only be exploited at the level of the BLAS operations.


Hodge metrics and the curvature of higher direct images

Abstract: Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.


Renault's Equivalence Theorem for Groupoid Crossed Products

Abstract: We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a detailed treatment of the Brauer semigroup for a locally Hausdorff, locally compact groupoid.


Kuranishi homology and Kuranishi cohomology

Abstract: A Kuranishi space is a topological space with a Kuranishi structure, defined by Fukaya and Ono. Kuranishi structures occur naturally on moduli spaces of J-holomorphic curves in symplectic geometry. Let Y be an orbifold and R a commutative ring or Q-algebra. We define two kinds of Kuranishi homology KH_*(Y;R). The chain complex KC_*(Y;R) defining KH_*(Y;R) is spanned over R by [X,f,G], for X a compact oriented Kuranishi space with corners, f : X --> Y smooth, and G "gauge-fixing data" which makes Aut(X,f,G) finite. Our main result is that these are isomorphic to singular homology. We define Poincare dual Kuranishi cohomology, isomorphic to compactly-supported cohomology. We define five kinds of Kuranishi (co)bordism spanned by isomorphism classes[X,f] for X a compact oriented Kuranishi space without boundary and f : X --> Y smooth. They are new topological invariants, and we show they are very large. These theories are powerful new tools in symplectic geometry. Defining virtual cycles and chains for moduli spaces of J-holomorphic curves is trivial in Kuranishi (co)homology. There is no need to perturb moduli spaces, and no problems with transversality. This gives major simplifications in Lagrangian Floer cohomology. We define new Gromov-Witten type invariants in Kuranishi bordism, over Z not Q. We sketch how these may be used to prove the integrality conjecture for Gopakumar-Vafa invariants. This paper is surveyed in arXiv:0710.5634.


Origin of the numerals, Zero concept

Abstract: The partisans of the hypothesis of the Indian origin of the numerals create confusion between the history of the Indian mathematics and the history of our modern numerals. To argue the thesis of the Indian origin of the numbers they confound between: the "intuitive zero " of Brahmagupta, that means ''nothing'' and which is the difference of two equal numbers, the "numeral zero" used in the representation of the numbers and the "mathematical zero" defined by the modern mathematicians. "Sifr" designate the "numeral zero" and "Shunya" designate the "intuitive zero". The word "Sifr" is not a traduction of the word "Shunya" and does not derive from the Indian word "Shunya", since the word "Sifr" and its derivatives existed in Arabic long before the appearance of zero itself The facts that the "intuitive zero" and the "mathematical zero" are represented currently by the "numerals zero" symbol "0" are only consequences of the representation of the numbers by the "Ghubari" numerals


Trigonometric Series via Laplace Transforms

Abstract: The author's method (math-ph/9804010) that uses the Laplace transform to find exact values for a large class of convergent series is extended to trigonometric series.


Exceptional representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

Abstract: In this paper, we study the set of $\Delta$-filtered modules of quasi-hereditary algebras arising from quotients of the double of quivers of type $A$. Our main result is that for any fixed $\Delta$-dimension vector, there is a unique (up to isomorphism) exceptional $\Delta$-filtered module. We then apply this result to show that there is always an open adjoint orbit in the nilpotent radical of a seaweed Lie algebra in $\mathrm{gl}_{n}(\field)$, thus answering positively in this $\mathrm{gl}_{n}(\field)$ case to a question raised independently by Michel Duflo and Dmitri Panyushev. An example of a seaweed Lie algebra in a simple Lie algebra of type $E_{8}$ not admitting an open orbit in its nilpotent radical is given.


On the dihedral n-body problem

Abstract: Consider n=2l>=4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle pi around two secant lines in space meeting at an angle of pi/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special $n$-body problem with three degrees of freedom, which is a kind of generalisation of Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds.


Sur la cat\'egorie des bimodules de Soergel

Abstract: The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La cat\'egorie B de Soergel d'un syst\`eme de Coxeter (W,S) est une cat\'egorie de bimodules sur une alg\`ebre de polyn\^omes sur laquelle W agit. C'est une cat\'egorification de l'alg\`ebre de Hecke de (W,S). Dans cet article nous donnons une description combinatoire des espaces de morphismes dans B. En corollaire, on obtient une description analogue des morphismes dans O_0-proj, o\`u O_0 est le bloc principal de la cat\'egorie O de BGG.


Algebras associated to acyclic directed graphs

Abstract: We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.


An equivalent condition for a uniform space to be coverable

Abstract: We prove that an equivalent condition for a uniform space to be coverable is that the images of the natural projections in the fundamental inverse system are uniformly open in a certain sense. As corollaries we (1) obtain a concrete way to find covering entourage, (2) correct an error in [3] and (3) show that coverable is equivalent to chain connected and uniformly joinable in the sense of arXiv:0706.3937.


Covering R-trees

Abstract: We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of the fundamental group of X. In particular, the Sierpin'ski gasket and carpet, and the Menger sponge all have the same covering R-tree, which is complete and has at each point valency equal to the continuum. This latter R-tree is of particular interest because it is "universal" in at least two senses: First, every R-tree of valency at most the continuum can be isometrically embedded in it. Second, every Peano continuum is the image of it via an open light mapping. We provide a sketch of our previous construction of the uniform universal cover in the special case of inner metric spaces, the properties of which are used in the proof.


Two Analogs of Intrinsically Linked Graphs

Abstract: A graph G is intrinsically S^1-linked if for every embedding of the vertices of G into S^1, vertices that form the endpoints of two disjoint edges in G form a non-split link in the embedding. We show that a graph is intrinsically S^1-linked if and only if it is not outer-planar. A graph is outer-flat if it can be embedded in the 3-ball such that all of its vertices map to the boundary of the 3-ball, all edges to the interior, and every cycle bounds a disk in the 3-ball that meets the graph only along its boundary. We show that a graph is outer-flat if and only if it is planar.


Prime Ideals of q-Commutative Power Series Rings

Abstract: We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters).


On the type of triangle groups

Abstract: We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.


Isoperimetric inequalities for eigenvalues of triangles

Abstract: Lower bounds estimates are proved for the first eigenvalue for the Dirichlet Laplacian on arbitrary triangles using various symmetrization techniques. These results can viewed as a generalization of P\'olya's isoperimetric bounds. It is also shown that amongst triangles, the equilateral triangle minimizes the spectral gap and (under additional assumption) the ratio of the first two eigenvalues. This last result resembles the Payne-P\'olya-Weinberger conjecture proved by Ashbaugh and Benguria.


Deformation bicomplex of module-algebras

Abstract: The deformation bicomplex of a module-algebra over a bialgebra is constructed. It is then applied to study algebraic deformations in which both the module structure and the algebra structure are deformed. The cases of module-coalgebras, comodule-(co)algebras, and (co)module-bialgebras are also considered.


GK-dimension of birationally commutative surfaces

Abstract: Let k be an algebraically closed field, let K/k be a finitely generated field extension of transcendence degree 2 with automorphism sigma, and let A be an N-graded subalgebra of Q = K[t; sigma] with A_n finite dimensional over k for all n. Then if A is big enough in Q in an appropriate sense, we prove that GKdim A = 3,4,5 or is infinite, with the exact value depending only on the geometric properties of sigma. The proof uses techniques in the birational geometry of surfaces which are of independent interest.


The Baum-Connes assembly map and the generalized Bass conjecture

Abstract: We show that the image of Connes-Karoubi-Chern character, restricted to the image of the Baum-Connes assembly map in the Bott-periodized topological K-theory of the complex group algebra, lies in the elliptic summand of the (periodic) cyclic homology of the group algebra. This implies that for any (weighted) ell-1 completion of the group algebra, rational surjectivity of the Baum-Connes assembly map implies the generalized Bass conjecture for that algebra.


Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture

Abstract: By deploying dense subalgebras of $\ell^1(G)$ we generalize the Bass conjecture in terms of Connes' cyclic homology theory. In particular, we propose a stronger version of the $\ell^1$-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy-bound property and nilpotent periodicity property, satisfy the $\ell^1$-Stronger-Bass Conjecture. Moreover, we determine the conjugacy-bound for relatively hyperbolic groups and compute the cyclic cohomology of the $\ell^1$-algebra of any discrete group.


The Laplace transform of the digamma function: an integral due to Glasser, Manna and Oloa

Abstract: We provide an analytic expression for the Laplace transform of the digamma fuction. This complements work of L. Glasser, D. Manna and O. Oloa on this question. The Laplace transform is continuous in the Laplace-variable a. The derivative admits a jump at a= ln 2.


High frequency dispersive estimates in dimension two

Abstract: We prove dispersive estimates at high frequency in dimension two for both the wave and the Schrodinger groupes for a very large class of real-valued potentials.


p-adic Dedekind and Hardy-Berndt type sum related to Volkenborn Integral on Z_p

Abstract: The purpose of this paper is to construct p-adic Dedekind sums and Hardy-Berndt type sums. We also construct generating function of the twisted Bernoulli polynomials and functions. Furthermore, we give some discussions on elliptic analogue of the Apostol-Dedekind sums.


On the p-adic Beilinson conjecture for number fields

Abstract: We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.


$S_t^1\times S_s^1$-valued lightcone Gauss map of a Lorentzian surface in semi-Euclidean 4-space

Abstract: We define the notions of $S_t^1\times S_s^1$-valued lightcone Gauss maps, lightcone pedal surface and Lorentzian lightcone height function of Lorentzian surface in semi-Euclidean 4-space and established the relationships between singularities of these objects and geometric invariants of the surface as applications of standard techniques of singularity theory for the Lorentzian lightcone height function.


On singular Calogero-Moser spaces

Abstract: Using combinatorial properties of complex reflection groups, we show that the generalised Calogero-Moser space associated to the centre of the corresponding rational Cherednik algebra is singular for all values of its deformation parameter c if and only if the group is different from the wreath product $S_n\wr C_m$ and the binary tetrahedral group. This result and a theorem of Ginzburg and Kaledin imply that there does not exist a symplectic resolution of the singular symplectic variety h+h*/W outside of these cases; conversely we show that there exists a symplectic resolution for the binary tetrahedral group (Hilbert schemes provide resolutions for the wreath product case).


A Cartan-Eilenberg approach to Homotopical Algebra

Abstract: In this paper we propose an approach to homotopical algebra where the basic ingredient is a category with two classes of distinguished morphisms: strong and weak equivalences. These data determine the cofibrant objects by an extension property analogous to the classical lifting property of projective modules. We define a Cartan-Eilenberg category as a category with strong and weak equivalences such that there is an equivalence between its localization with respect to weak equivalences and the localised category of cofibrant objets with respect to strong equivalences. This equivalence allows us to extend the classical theory of derived additive functors to this non additive setting. The main examples include Quillen model categories and functor categories with a triple, in the last case we find examples in which the class of strong equivalences is not determined by a homotopy relation. Among other applications, we prove the existence of filtered minimal models for \emph{cdg} algebras over a zero-characteristic field and we formulate an acyclic models theorem for non additive functors.


Elliptic Gauss Sums and Hecke L-values at s=1

Abstract: The rationality of the elliptic Gauss sum coefficient is shown. The following is a specific case of our argument. Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and \varpi=2.62205... is the real period of the elliptic curve y^2=x^3-x, so that f(u) is an elliptic function relative to the period lattice Z[i]. Let \pi be a primary prime of Z[i] such that norm(\pi)\equiv 13\mod 16. Let S be the quarter set mod \pi consisting of quartic residues. Let us define G(\pi):=\sum_{\nu\in S} f(\nu/\pi) and \tilde{\pi}:=\prod_{\nu\in S} f(\nu/\pi). The former G(\pi) is a typical example of elliptic Gauss sum; the latter is regarded as a canonical 4-th root of -\pi: (\tilde{\pi})^4=-\pi. Then we have Theorem: G(\pi)/(\tilde{\pi})^3 is a rational odd integer. G(\pi) appears naturally in the central value of Hecke L associated to the quartic residue character mod \pi, and our proof is based on the functional equation of L and an explicit formula of the root number. In fact, the latter is nothing but the Cassels-Matthews formula on the quartic Gauss sum.


Exponential inequalities for self-normalized martingales with applications

Abstract: We propose several exponential inequalities for self-normalized martingales similar to those established by De la Pe\~{n}a. The keystone is the introduction of a new notion of random variable heavy on left or right. Applications associated with linear regressions, autoregressive and branching processes are also provided.


Relating two Hopf algebras built from an operad

Abstract: Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the $\NAP$ operad, where the incidence Hopf algebra is identified with the Connes-Kreimer Hopf algebra of rooted trees.


Scaling limits for random fields with long-range dependence

Abstract: This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density $\lambda$ of the sets grows to infinity and the mean volume $\rho$ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which $\lambda$ and $\rho$ are scaled. If $\lambda$ grows much faster than $\rho$ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.


Malliavin calculus and Clark-Ocone formula for functionals of a square-integrable L\'evy process

Abstract: In this paper, we construct a Malliavin derivative for functionals of square-integrable L\'evy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a L\'evy process.


On Categorical Theory-Building: Beyond the Formal

Abstract: I propose a notion of theory motivated by Category theory.


Ergodic properties of Poissonian ID processes

Abstract: We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its L\'{e}vy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects.


p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

Abstract: The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.


Parshin Residues via Coboundary Operators

Abstract: The article consist of two main parts: an analog of the Leray Theory for Singular Varieties and its application to the Theory of Parshin's Residues. The first part is independent from the second. It uses the theory of Whitney stratifications. The second part is an application of the first. In particular, a geometric and very transparent proof of the Parshin's Reciprocity Law for residues is given.


A hermitian analogue of the Broecker-Prestel theorem

Abstract: The Broecker-Prestel local-global principle characterizes weak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the henselizations and isotropy over the real closures of that field. A hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of pythagorean index at most two, and algebras over SAP and ED fields.


Finite depth and Jacobson-Bourbaki correspondence

Abstract: We introduce a notion of depth three tower of three rings C < B < A with depth two ring extension A | B recovered when B = C. If A = \End B_C and B | C is a Frobenius extension, this captures the notion of depth three for a Frobenius extension in arXiv:math/0107064 and arXiv:math/0108067, such that if B | C is depth three, then A | C is depth two (a phenomenon of finite depth subfactors, see arXiv:math/0006057). We provide a similar definition of finite depth Frobenius extension with embedding theorem utilizing a depth three subtower of the Jones tower. If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring \End {}_BA_C and coring (A \o_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.


On Landau--Ginzburg models for Fano varieties

Abstract: We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.


A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics

Abstract: We discuss how matrix-free/timestepper algorithms can efficiently be used with dynamic non-Newtonian fluid mechanics simulators in performing systematic stability/bifurcation analysis. The timestepper approach to bifurcation analysis of large scale systems is applied to the plane Poiseuille flow of an Oldroyd-B fluid with non-monotonic slip at the wall, in order to further investigate a mechanism of extrusion instability based on the combination of viscoelasticity and nonmonotonic slip. Due to the nonmonotonicity of the slip equation the resulting steady-state flow curve is nonmonotonic and unstable steady-states appear in the negative-slope regime. It has been known that self-sustained oscillations of the pressure gradient are obtained when an unstable steady-state is perturbed [Fyrillas et al., Polymer Eng. Sci. 39 (1999) 2498-2504]. Treating the simulator of a distributed parameter model describing the dynamics of the above flow as an input-output black-box timestepper of the state variables, stable and unstable branches of both equilibrium and periodic oscillating solutions are computed and their stability is examined. It is shown for the first time how equilibrium solutions lose stability to oscillating ones through a subcritical Hopf bifurcation point which generates a branch of unstable limit cycles and how the stable periodic solutions lose their stability through a critical point which marks the onset of the unstable limit cycles. This implicates the coexistence of stable equilibria with stable and unstable periodic solutions in a narrow range of volumetric flow rates.


Conformal paracontact curvature and the local flatness theorem

Abstract: A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature if and only if the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan-Chern-Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in the complex vector space is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.


Nonfibered knots and representation shifts

Abstract: The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from symbolic dynamics.


Binary Models for Marginal Independence

Abstract: Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach taken is graphical and draws on analogies to multivariate Gaussian models for marginal independence. For the graphical model representation we use bi-directed graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. Finally we consider combining these models with symmetry restrictions.


Equivariant Satake category and Kostant-Whittaker reduction

Abstract: We explain (following V. Drinfeld) how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra equivariant with respect to the adjoint action, due to some old results of V. Ginzburg. The global cohomology functor corresponds under this identification to restricti on to the Kostant slice. We extend this description to loop rotation equivariant derived category, linking it to Harish-Chandra bimodules for the Langlands dual Lie algebra, so that the global cohomology functor corresponds to the quantum Kostant-Whittaker reduction of a Harish-Chandra bimodule. We derive a conjecture from math.AG/0306413 which identifies the loop-rotation equivariant homology of the affine Grassmannian with quantized completed Toda lattice.


$N_{\p}$-type quotient modules on the torus

Abstract: Structure of the quotient modules in $\hh$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{\p}$-type quotient modules, namely, quotient modules of the form $\hh\ominus [z-\p]$, where $\p (w)$ is a function in the classical Hardy space $H^2(\G)$ and $[z-\p]$ is the submodule generated by $z-\p (w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{\p}$-type quotient module is its close connections with some classical single variable operator theories.


Approximate reduction of dynamical systems

Abstract: The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with mechanical systems with symmetry--which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.


Foliations on quaternion CR-submanifolds

Abstract: The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of a quaternion K\"{a}hler manifold.


Occupation Statistics of Critical Branching Random Walks in Two or Higher Dimensions

Abstract: Consider a critical nearest neighbor branching random walk on the $d$-dimensional integer lattice initiated by a single particle at the origin. Let $G_{n}$ be the event that the branching random walk survives to generation $n$. We obtain limit theorems conditional on the event $G_{n}$ for a variety of occupation statistics: (1) Let $V_{n}$ be the maximal number of particles at a single site at time $n$. If the offspring distribution has finite $\alpha$th moment for some integer $\alpha\geq 2$, then in dimensions 3 and higher, $V_n=O_p(n^{1/\alpha})$; and if the offspring distribution has an exponentially decaying tail, then $V_n=O_p(\log n)$ in dimensions 3 and higher, and $V_n=O_p((\log n)^2)$ in dimension 2. Furthermore, if the offspring distribution is non-degenerate then $P(V_n\geq \delta \log n | G_{n})\to 1$ for some $\delta >0$. (2) Let $M_{n} (j)$ be the number of multiplicity-$j$ sites in the $n$th generation, that is, sites occupied by exactly $j$ particles. In dimensions 3 and higher, the random variables $M_{n} (j)/n$ converge jointly to multiples of an exponential random variable. (3) In dimension 2, the number of particles at a "typical" site (that is, at the location of a randomly chosen particle of the $n$th generation) is of order $O_p(\log n)$, and the number of occupied sites is $O_p(n/\log n)$.


Symmetry groups of non-simply-connected four-manifolds

Abstract: Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $\chi(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically trivial, locally linear, effective action on $M$, then $G$ must be cyclic. With additional assumptions to ensure orientability of some components of the singular set (e.g. if $G$ acts by symplectic symmetries, or preserving a spin structure), we also rule out $C_2 \times C_2$ actions. The proofs use equivariant cohomology, localization, and a careful study of the first cohomology groups of the (potential) singular set.


Modules of constant Jordan type

Abstract: We introduce the class of modules of constant Jordan type for a finite group scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.


The complex Busemann-Petty problem on sections of convex bodies

Abstract: The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$


Performance Evaluation of a Multi-Agent Risk-Sensitive Tracking System

Abstract: In this paper, we consider a simple linear exponential quadratic Gaussian (LEQG) tracking problem for a multi-agent system. We study the dynamical behaviors of the group as we vary the risk-sensitivity parameter, comparing in particular the risk averse case to the LQG case. Then we consider the evolution of the performance per agent as the number of agents in the system increases. We provide some analytical as well as simulation results. In general, more agents are beneficial only if noisy agent dynamics and/or imperfect measurements are considered. The critical value of the risk sensitivity parameter above which the cost becomes infinite increases with the number of agents. In other words, for a fixed positive value of this parameter, there is a minimum number of agents above which the cost remains finite.


Semifinite spectral triples associated with graph C*-algebras

Abstract: We review the recent construction of semifinite spectral triples for graph C^*-algebras. These examples have inspired many other developments and we review some of these such as the relation between the semifinite index and the Kasparov product, examples of noncommutative manifolds, and an index theorem in twisted cyclic theory using a KMS state.


Stochastic evolution equations for nonlinear filtering of random fields in the presence of fractional Brownian sheet observation noise

Abstract: The problem of nonlinear filtering of a random field observed in the presence of a noise, modeled by a persistent fractional Brownian sheet of Hurst index $(H_1,H_2)$ with $0.5<H_1,H_2<1$, is studied and a suitable version of the Bayes' formula for the optimal filter is obtained. Two types of spatial "fractional" analogues of the Duncan-Mortensen-Zakai equation are also derived: one tracks evolution of the unnormalized optimal filter along an arbitrary "monotone increasing" (in the sense of partial ordering in $\mathbb{R}^2$) one-dimensional curve in the plane, while the other describes dynamics of the filter along the paths that are truly two-dimensional. Although the paper deals with the two-dimensional parameter space, the presented approach and results extend to $d$-parameter random fields with arbitrary $d\geq 3$.


Cha\^{i}nes de Markov Constructives Index\'{e}es par Z

Abstract: Nous \'{e}tudions les cha\^{{\i}}nes de Markov $(X_n)_{n\in\mathbf{Z}}$ gouvern\'{e}es par une relation de r\'{e}currence de la forme $X_{n+1}=f(X_n,V_{n+1})$, o\`{u} $(V_n)_{n\in\mathbf{Z}}$ est une suite de variables al\'{e}atoires ind\'{e}pendantes et de m\^{e}me loi telle pour tout $n\in \mathbf{Z}$, $V_{n+1}$ est ind\'{e}pendante de la suite $((X_k,V_k))_{k\le n}$. L'objet de l'article est de donner une condition n\'{e}cessaire et suffisante pour que les innovations $(V_n)_{n\in\mathbf{Z}}$ d\'{e}terminent compl\`{e}tement la suite $(X_n)_{n\in \mathbf{Z}}$ et de d\'{e}crire l'information manquante dans le cas contraire.


The C*-algebras qA\otimes K and S^2A\otimes K are asymptotically equivalent

Abstract: Let $A$ be a separable $C^*$-algebra. We prove that its stabilized second suspension $S^2A\otimes \mathcal K$ and the $C^*$-algebra $qA\otimes \mathcal K$ constructed by Cuntz in the framework of his picture of KK-theory are asymptotically equivalent. This means that there exist asymptotic morphisms from each to the other whose compositions are homotopic to the identity maps. This result yields an easy description of the natural transformation from KK-theory to E-theory. One more corollary is the following. T. Loring ([3]) proved that any asymptotic morphism from $\qC$ to any $C^*$-algebra $B$ is homotopic to a $\ast$-homomorphism. We prove that the same is true when $\C$ is replaced by any nuclear $C^*$-algebra $A$ and when $B$ is stable.


Rational Landen transformations on the real line

Abstract: The rational Landen transformation is a map on the space of coefficients of a rational integrand that preserves the value of the integral. We provide a family of these transformations that apply to rational integrands on the whole line. Given an integer m, these transformations produce a numerical scheme to evaluate the integral that is of order m.


Filtration shrinkage by level-crossings of a diffusion

Abstract: We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points $x_1<...<x_N$ in $\mathbb{R}$, the region indicator function $R(x)$ assumes the value $i$ if $x\in(x_{i-1},x_i]$. We take $\mathbb{F}$ to be the filtration generated by $(R(X_t))_{t\geq0}$, where $X$ is a diffusion with infinitesimal generator $\mathcal{A}$. We prove a martingale representation theorem for $\mathbb{F}$ in terms of stochastic integrals with respect to $N$ random measures whose compensators have a simple form given in terms of certain L\'{e}vy measures $F^{j\pm}_i$, which are related to the differential equation $\mathcal{A}u=\lambda u$.


Alternative parametrizations and reference priors for decomposable discrete graphical models

Abstract: For a given discrete decomposable graphical model, we identify several alternative parametrizations, and construct the corresponding reference priors for suitable groupings of the parameters. Specifically, assuming that the cliques of the graph are arranged in a perfect order, the parameters we consider are conditional probabilities of clique-residuals given separators, as well as generalized log-odds-ratios. We also consider a parametrization associated to a collection of variables representing a cut for the statistical model. The reference priors we obtain do not depend on the order of the groupings, belong to a conjugate family, and are proper.


Intrinsic tests for the equality of two correlated proportions

Abstract: Correlated proportions arise in longitudinal (panel) studies. A typical example is the ``opinion swing'' problem: ``Has the proportion of people favoring a politician changed after his recent speech to the nation on TV?''. Since the same group of individuals is interviewed before and after the speech, the two proportions are correlated. A natural null hypothesis to be tested is whether the corresponding population proportions are equal. A standard Bayesian approach to this problem has already been considered in the literature, based on a Dirichlet prior for the cell-probabilities of the underlying two-by-two table under the alternative hypothesis, together with an induced prior under the null. In lack of specific prior information, a diffuse (e.g. uniform) distribution may be used. We claim that this approach is not satisfactory, since in a testing problem one should make sure that the prior under the alternative be adequately centered around the region specified by the null, in order to obtain a fair comparison between the two hypotheses. Following an intrinsic prior methodology, we develop two strategies for the construction of a collection of objective priors increasingly peaked around the null. We provide a simple interpretation of their structure in terms of weighted imaginary sample scenarios. We illustrate our method by means of three examples, carrying out sensitivity analysis and providing comparison with existing results.


Some tensor products

Abstract: We define the tensor product of 1-motives with motives of weight 0 and we construct explicitely the 1-motive underlying the quotient M_1 \otimes M_2 / W_{-3}(M_1 \otimes M_2).


The growth of additive processes

Abstract: Let $X_t$ be any additive process in $\mathbb{R}^d.$ There are finite indices $\delta_i, \beta_i, i=1,2$ and a function $u$, all of which are defined in terms of the characteristics of $X_t$, such that \liminf_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>\delta_1$, \cr\infty, \quad if $\eta<\delta_2$,} \limsup_{t\to0}u(t)^{-1/\eta}X_t^*= \cases{0, \quad if $\eta>\beta_2$, \cr\infty, \quad if $\eta<\beta_1$,}\qquad {a.s.}, where $X_t^*=\sup_{0\le s\le t}|X_s|.$ When $X_t$ is a L\'{e}vy process with $X_0=0$, $\delta_1=\delta_2$, $\beta_1=\beta_2$ and $u(t)=t.$ This is a special case obtained by Pruitt. When $X_t$ is not a L\'{e}vy process, its characteristics are complicated functions of $t$. However, there are interesting conditions under which $u$ becomes sharp to achieve $\delta_1=\delta_2$, $\beta_1=\beta_2.$


Maximal Arithmetic Progressions in Random Subsets

Abstract: Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2)).


On elliptic differential operators with shifts: II. The cohomological index formula

Abstract: This paper is a continuation of arXiv:0706.3511, where we obtained a local index formula for matrix elliptic operators with shifts. Here we establish a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators.


Manifolds with nonnegative isotropic curvature

Abstract: We prove that if $(M^n,g)$, $n \ge 4$, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature, then one of the following possibilities hold: (i) $M$ admits a metric with positive isotropic curvature (ii) $(M,g)$ is isometric to a locally symmetric space (iii) $(M,g)$ is K\"ahler and biholomorphic to $\C P^\frac {n}{2}$. (iv) $(M,g)$ is quaternionic-K\"ahler. This is implied by the following result: Let $(M^{2n},g)$ be a compact, locally irreducible K\"ahler manifold with nonnegative isotropic curvature. Then either $M$ is biholomorphic to $\C P^n$ or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of S. Brendle and R. Schoen on the Ricci flow.


Knot colouring polynomials

Abstract: This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \pi_K. Given a finite group G and an element x in G, we consider the set of representations \rho from \pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses. We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.


A Topological Characterization Of Knots and Links Arising From Site-Specific Recombination

Abstract: We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2,m)-torus knot or link substrate. We show that all knotted or linked products fall into a single family, and prove that the size of this family grows linearly with the cube of the minimum number of crossings. Additionally, we prove that the only possible products of an unknot substrate are either clasp knots and links or (2,m)-torus knots and links. Finally, in the (common) case of (2,m)-torus knot or link substrates whose products have minimal crossing number m+1, we prove that the types of products are tightly prescribed, and use this to examine previously uncharacterized experimental data.


Multivariate normal approximation in geometric probability

Abstract: Consider a measure $\mu_\lambda = \sum_x \xi_x \delta_x$ where the sum is over points $x$ of a Poisson point process of intensity $\lambda$ on a bounded region in $d$-space, and $\xi_x$ is a functional determined by the Poisson points near to $x$, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the $\mu_\lambda$-measures (suitably scaled and centred) of disjoint sets in $R^d$ are asymptotically independent normals as $\lambda \to \infty$; here we give an $O(\lambda^{-1/(2d + \epsilon)})$ bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.