Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.     

Discrete Differential Calculus on Simplicial Complexes and Constrained Homology*

Let V be a finite set. Let K be a simplicial complex with its vertices in V .In this paper, the author discusses some differential calculus on V . He constructs some constrained homology groups of K by using the differential calculus on V . Moreover, he defines an independence hypergraph to be the complement of a simplicial complex in the complete hypergraph on V . Let L be an independence hypergraph with its vertices in V .He constructs some constrained cohomology groups of L by using the differential calculus on V .     

On Quillen?s Theorem A for posets

A theorem of McCord of 1966 and Quillen?s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads then to stronger formulations of the simplicial version of Quillen?s Theorem A, the Nerve Lemma and other known results. In particular we establish a conjecture of Kozlov on the simple homotopy type of the crosscut complex and we improve a well-known result of Cohen on contractible mappings.     

The topological structure of maximal lattice free convex bodies: The general case

Given a genericm × n matrixA, the simplicial complexK(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax b}. The main result of this paper is that the topological space associated withK(A) is homeomorphic withR ^m–1 . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by NSF grant SES-9121936 and the program in Discrete Mathematics at Yale University.Partially supported by the Hungarian NSF grant 1909 and the program in Discrete Mathematics at Yale University.     

Regenerative cascade homotopies for solving polynomial systems

A key step in the numerical computation of the irreducible decomposition of a polynomial system is the computation of a witness superset of the solution set. In many problems involving a solution set of a polynomial system, the witness superset contains all the needed information. Sommese and Wampler gave the first numerical method to compute witness supersets, based on dimension-by-dimension slicing of the solution set by generic linear spaces, followed later by the cascade homotopy of Sommese and Verschelde. Recently, the authors of this article introduced a new method, regeneration, to compute solution sets of polynomial systems. Tests showed that combining regeneration with the dimension-by-dimension algorithm was significantly faster than naively combining it with the cascade homotopy. However, in this article, we combine an appropriate randomization of the polynomial system with the regeneration technique to construct a new cascade of homotopies for computing witness supersets. Computational tests give strong evidence that regenerative cascade is superior in practice to previous methods.     

An Alexandrov topology for maximal Cohen-Macaulay modules

Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.     

Extending Persistence Using Poincaré and Lefschetz Duality

Persistent homology has proven to be a useful tool in a variety of contexts, including the recognition and measurement of shape characteristics of surfaces in ℝ³. Persistence pairs homology classes that are born and die in a filtration of a topological space, but does not pair its actual homology classes. For the sublevelset filtration of a surface in ℝ³, persistence has been extended to a pairing of essential classes using Reeb graphs. In this paper, we give an algebraic formulation that extends persistence to essential homology for any filtered space, present an algorithm to calculate it, and describe how it aids our ability to recognize shape features for codimension 1 submanifolds of Euclidean space. The extension derives from Poincaré duality but generalizes to nonmanifold spaces. We prove stability for general triangulated spaces and duality as well as symmetry for triangulated manifolds. An erratum to this article can be found at     

Weak Dimension of FP-injective Modules Over Chain Rings

It is proven that the weak dimension of each FP-injective module over a chain ring which is either Archimedean or not semicoherent is less or equal to 2. This implies that the projective dimension of any countably generated FP-injective module over an Archimedean chain ring is less or equal to 3.     

Test sets for integer programs

In this paper I discuss various properties of the simplicial complex of maximal lattice free bodies associated with a matrixA. If the matrix satisfies some mild conditions, and isgeneric, the edges of the complex form the minimal test set for the family of integer programs obtained by selecting a particular row ofA as the objective function, and using the remaining rows to impose constraints on the integer variables.     

Fixed point properties for nonexpansive representations of topological semigroups

In this paper, we study a fixed point and a nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings on a nonempty subset of a Hilbert space.     

同伦分析方法：一种新的求解非线性问题的近似解析方法

本文描述了一种称为“同伦分析方法”（ＨＡＭ）的新的求解非线性问题的近似解析方法之基本思想·不同于摄动展开方法，“同伦分析方法”的有效性不依赖于所研究的非线性方程中是否含有小参数·因此，该方法提供了一个强有力的分析非线性问题的新工具·作为示例，我们应用一个典型的非线性问题来说明该方法的有效性及其巨大潜力·     

The Orlicz-Pettis theorem for topological Riesz spaces

A finitely additive vector measure from a -ring to a Riesz space is countably additive (exhaustive) for all Hausdorff Lebesgue topologies on the range space, or for none of them. In particular, subseries convergent series are the same for all Hausdorff Lebesgue topologies on a Riesz space.

     

A Chain Rule for Upper Semicontinuous (CF)-Mappings

A chain rule for calculating convexificators of composite functions of the type f = h∘g, with the inner factor g being a transformation of , is proposed. The proof is based on a double application of a mean value theorem for (CF)-mappings due to V.F. Demyanov and V. Jeyakumar (see [4]), along with a stability property for the support of a certain ɛ-perturbation of (CF)-mappings.     

上证指数、深圳指数预测的马尔柯夫链预测模糊模型

介绍马尔柯夫链预测法的模糊模型 ,应用模糊模型对证券指数进行预测的实证研究     

Digital topological method for computing genus and the Betti numbers

This paper concerns with computation of topological invariants such as genus and the Betti numbers. We design a linear time algorithm that determines such invariants for digital spaces in 3D. These computations could have applications in medical imaging as they can be used to identify patterns in 3D image.Our method is based on cubical images with direct adjacency, also called (6,26)-connectivity images in discrete geometry. There are only six types of local surface points in such a digital surface. Two mathematical ingredients are used. First, we use the Gauss-Bonnett Theorem in differential geometry to determine the genus of 2-dimensional digital surfaces. This is done by counting the contribution for each of the six types of local surface points. The new formula derived in this paper that calculates genus is g=1+(|M₅|+2⋅|M₆|−|M₃|)/8 where Mi indicates the set of surface-points each of which has i adjacent points on the surface. Second, we apply the Alexander duality to express the homology groups of a 3D manifold in the usual 3D space in terms of the homology groups of its boundary surface.While our result is stated for digital spaces, the same idea can be applied to simplicial complexes in 3D or more general cell complexes.     

Unbalanced digit sets and the closest choice strategy for minimal weight integer representations

An algorithm is presented that produces an optimal radix-2 representation of an input integer n using digits from the set , where ℓ ≤ 0 and u ≥ 1. The algorithm works by scanning the digits of the binary representation of n from left-to-right (i.e., from most-significant to least-significant); further, the algorithm is of the online variety in that it needs to scan only a bounded number of input digits before giving an output digit (i.e., the algorithm produces output before scanning the entire input). The output representation is optimal in the sense that, of all radix-2 representations of n with digits from D _ℓ,u , it has as few nonzero digits as possible (i.e., it has minimal weight). Such representations are useful in the efficient implementation of elliptic curve cryptography. The strategy the algorithm utilizes is to choose an integer of the form d 2 ⁱ , where , that is closest to n with respect to a particular distance function. It is possible to choose values of ℓ and u so that the set D _ℓ,u is unbalanced in the sense that it contains more negative digits than positive digits, or more positive digits than negative digits. Our distance function takes the possible unbalanced nature of D _ℓ,u into account.