The t-singular homology of orbifolds

For an orbifold M we define a new homology group, called t-singular homology group t-Hq(M) by using singular simplicies intersecting ‘transversely’ with ΣM. The rightness of this homology group is ensured by the facts that the 1-dimensional homology group t-H₁(M) is isomorphic to the abelianization of the orbifold fundamental group π₁(M,x₀). If M is a manifold, t-Hq(M) coincides with the usual singular homology group. We prove that it is a ‘b-homotopy’ invariant of orbifolds and develop many algebraic tools for the calculations. Consequently we calculate the t-singular homology groups of several orbifolds.     

The ws-singular cohomology of orbifolds

We define a cohomology with integral coefficients of an orbifold M, which we call the ws-singular cohomology ws-Hq(M) of M.     

The duality theorem between the t-singular homology and the ws-singular cohomology of orbifolds

In the present paper, we prove that for an n-dimensional compact orbifold with an s-homological orientation, the duality of the ws-singular cohomology group and the t-singular homology group holds. The key tools are “the t-modification of the cap product” for giving the duality homomorphism and “the Convex Suborbifold Theorem” for extending the local duality isomorphism to the global one. The duality theorem proved in the present paper is a naturally required consequence of the preceding works of the authors.     

An Upper Bound for the Cardinality of an <Emphasis Type="Italic">s</Emphasis>-Distance Set in Euclidean Space

In this paper we show that if X is an s-distance set in ^m and X is on p concentric spheres then Moreover if X is antipodal, then .     

GENERALISED IDEALS OF RINGS

Abstract

Using a general definition of a regularity for rings, F- and F- qausi-ideals of a ring are defined. These concepts are shown to be generalizations of ideals or one-sided ideals of a ring. An F-semi prime F—(F-quasi-) ideal of a ring R is also defined. F-regular rings are characterized in terms of F-semi prime F- (F-quasi-) ideals for a large class of polynomial regularities including some well known regularities. A more general characterization of the prime radical β(R) of a ring are given in terms of F—(F-quasi-) ideals.     

Homoclinic and heteroclinic orbits for a class of Hamiltonian systems

The existence of a rich structure of homoclinic and heteroclinic solutions is established for a family of Hamiltonian systems that serve as a simpler model for the multiple pendulum system. The proof is based on recently developed arguments from the calculus of variations that have proved useful in finding actual solutions of an equation near approximate solution.This research was sponsorted in part by the National Science Foundation under grant #MCS-8110556 and the U. S. Army Research Office under Contract #DAAL03-87-K-0043. Any reproduction for the purpose of the United States Government is permitted.     

STRONGLY SEQUENTIALLY CONTINUOUS FUNCTIONS

Abstract

Given a topological abelian group G, we study the class of strongly sequentially continuous functions on G. Strong sequential continuity is a property intermediate between sequential continuity and uniform sequential continuity, which appeared naturally in the study of smooth functions on Banach spaces. In this paper, we shall mainly concentrate on the gap between strong sequential continuity and uniform sequential continuity. It turns out that if G has some completeness property—for example, if it is completely metrizable—then all strongly sequentially continuous functions on G are uniformly sequentially continuous. On the other hand, we exhibit a large and natural class of groups for which the two notions differ. This class is defined by a property reminiscent of the classical Dirichlet theorem; it includes all dense sugroups of R generated by an increasing sequence of Dirichlet sets, and groups of the form (X, w), where X is a separable Banach space failing the Schur property. Finally, we show that the family of bounded, real-valued strongly sequentially continuous functions on G is a closed subalgebra of l∞(G).     

On topological spaces of orders

We study the space of linear orders on a given set X, denoted by Op(X), endowed with the topology of pointwise convergence. We show, in particular, that if |X|=ω₁ or |X|=ω₀ then Op(X) is homeomorphic to ω₁² and ω₀², respectively.     

Hochschild cohomology and Atiyah classes

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich.Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.     

Mathematical statistics and problems of ancient chronology. A new approach

New methods of processing statistically quantitative textual information of a narrative character are introduced. The methods can be used to recognize dependent and independent texts among large collections of texts. The methods are applied to the problem of correct dating of the events in ancient chronology. These results induce conjecture on the redating of some historical events.     

Rao distances

We determine Riemannian distances between a large class of multivariate probability densities with the same mean, where the Riemannian metric is induced by a weighted Fisher information matrix. We reduce the evaluation of distances to quadrature and in some cases give closed form expressions.     

Structured total least norm and approximate GCDs of inexact polynomials

The determination of an approximate greatest common divisor (GCD) of two inexact polynomials f=f(y)$f=f(y)$ and g=g(y)$g=g(y)$ arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation S^*(f,g)$S^{*}(f,g)$ of the Sylvester resultant matrix S(f,g)$S(f,g)$. In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of S(f,g)$S(f,g)$, and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix S^*(f,g)$S^{*}(f,g)$, and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented.     

Robustness of nonuniform behavior for discrete dynamics

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we establish the robustness of a nonuniform dichotomy, in the sense that the existence of such a dichotomy for a given linear cocycle persists under sufficiently small linear perturbations. The robustness result for the nonuniform contractions is also established.     

Ramification of rough paths

The stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended Chen's multiplicative property involves the Dürr-Connes-Kreimer coproduct on rooted trees. This turns out to be the natural setting for a non-geometric theory of rough paths.     

Nonuniform exponential contractions and Lyapunov sequences

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict Lyapunov sequences. We note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. We also obtain “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. We also show how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. Moreover, we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.     

Lyapunov functions for trichotomies with growth rates

We consider linear equations x^′=A(t)x that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates ecρ(t) determined by a function ρ(t). For example, the usual exponential behavior with ρ(t)=t is included as a very special case, and when ρ(t)=logt we obtain a polynomial behavior. We emphasize that we also consider the general case of nonuniform exponential behavior, which corresponds to the existence of what we call a ρ-nonuniform exponential trichotomy. This is known to occur in a large class of nonautonomous linear equations. Our main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a ρ-nonuniform exponential trichotomy. This includes criteria for the existence of a ρ-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions we show that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomies in terms of strict Lyapunov functions, we establish the robustness of ρ-nonuniform exponential trichotomies under sufficiently small linear perturbations. We emphasize that in comparison with former works, our proof of the robustness is much simpler even when ρ(t)=t.     

On a differential equation characterizing Euclidean spheres

A characterization of Euclidean spheres out of complete Riemannian manifolds is made by certain vector fields on complete Riemannian manifolds satisfying a partial differential equation on vector fields.     

On dualizing a multivariable Poisson summation formula

It is known [7] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function ø of one variable into a function and its cosine transform in a generalized sense. The present work presents conditions on ø for which the transform relation holds in the classical sense, and extends this result to a class of generalizations of the Poisson formula in any number of dimensions.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.