On estimates for integral solutions of linear inequalities

Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem.     

Alternative positional FEM applied to thermomechanical impact of truss structures

This paper presents a formulation to deal with dynamic thermomechanical problems by the finite element method. The proposed methodology is based on the minimum potential energy theorem written regarding nodal positions, not displacements, to solve the mechanical problem. The thermal problem is solved by a regular finite element method. Such formulation has the advantage of being simple and accurate. As a solution strategy, it has been used as a natural split of the thermomechanical problem, usually called isothermal split or isothermal staggered algorithm. Usual internal variables and the additive decomposition of the strain tensor have been adopted to model the plastic behavior. Four examples are presented to show the applicability of the technique. The results are compared with other authors’ numerical solutions and experimental results.     

Persistency of linear programming relaxations for the stable set problem

The Nemhauser–Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.

     

Evaluations of Parameters for the Optimization of S.S.O.R. and A.D.I. Preconditioning

The main goal of this paper is to give two ways to estimate the needed parameters in order to obtain the condition number of S.S.O.R. preconditioned matrices, namely, the algebraic matricial formulation of convexity Riesz theorem and the tridiagonal Fourier analysis. The improvement with respect to Axelsson's approach is explicitly given. Estimations of the condition number in the case of A.D.I. preconditioning is also considered.     

一类初等微分几何定理机器证明的算法与实现

空间曲面上的曲线论是初等微分几何的重要部分．作者提出了一种以外微分运算和向量计算为主要工具,可以进行有关曲面上曲线局部性质的定理机器证明的算法．该算法结合了曲面上的活动标架,曲面上曲线的测地标架和曲线自身的Frenet标架,在Maple 9下得到实现．对20个例子进行的测试表明,由该算法生成的自动证明简短可读．     

A modified Chan-Vese model and its theoretical proof

We present a modified Chan-Vese functional and give its theoretical proof. By using the geometric heat flow method to all the Euler-Lagrange equations, a system of evolution equations in level set formulation is derived. We study the existence of solution to this system by Schauder fixed point theorem and the implicit function theorem in Banach space. This variational formulation can detect interior and exterior boundaries of desired object(s) in color images.     

Fractional extensions of some boundary value problems in oil strata

In the present paper, we solve three boundary value problems related to the temperature field in oil strata — the fractional extensions of the incomplete lumped formulation and lumped formulation in the linear case and the fractional generalization of the incomplete lumped formulation in the radial case. By using the Caputo differintegral operator and the Laplace transform, the solutions are obtained in integral forms where the integrand is expressed in terms of the convolution of some auxiliary functions of Wright function type. A generalization of the Laplace transform convolution theorem, known as Efros’ theorem is widely used.     

The invariant subspace problem

A fundamental problem is to determine whether every bounded linear transformation in Hilbert space has a nontrivial invariant subspace. A formal proof [1] of the existence of invariant subspaces is given by the theory of square summable power series [2] in its vector formulation [3]. A determination of extreme points of a convex set remains for the justification of the formal argument. A characterization of extreme points which implies the existence of invariant subspaces has been conjectured [4]. New information is obtained from a localization of the theory of square summable power series [5] which allows the formulation of extreme point problems which are closely related because of the Carathéodory-Fejér extension theorem [6]. The conjectured characterization of extreme points is shown to be false. Extreme points need not have the properties required for the construction of invariant subspaces.     

General approach to the development of mathematical models of nondestructive stress testing. II. Physical model and equations of local relation between stresses and their initial distribution

Within the framework of a physical model of free strain, we propose a method for reducing nonlinear equations of state of a deformable body to equations that characterize the local influence of initial stresses on the current state of the body. The method is based on the constructive mathematical formulation of the principle of initial independence of the strained state and the general theorem on the elasticity potential.     

A kinetic approach to comparison properties for degenerate parabolic-hyperbolic equations with boundary conditions

We study the comparison principle for degenerate parabolic-hyperbolic equations with initial and nonhomogeneous boundary conditions. We prove a comparison theorem for any entropy sub- and supersolution. The L¹ contractivity and, therefore, uniqueness of entropy solutions has been obtained so far by some authors, but it seems that any comparison theorem is not proven. The method used there is the doubling variable technique due to Kru?kov. Our method is based upon the kinetic formulation and the kinetic techniques. By developing the kinetic techniques for degenerate parabolic-hyperbolic equations with boundary conditions, we can obtain a comparison property which obviously extends the L¹ contractive property.     

The theorem of the alternative,the key-theorem,and the vector-maximum problem

Consequences of a general formulation of the theorem of the alternative are exploited.     

Interpolation and Sampling: E.T. Whittaker, K. Ogura and Their Followers

The classical sampling theorem has often been attributed to E.T.?Whittaker, but this attribution is not strictly valid. One must carefully distinguish, for example, between the concepts of sampling and of interpolation, and we find that Whittaker worked in interpolation theory, not sampling theory. Again, it has been said that K.?Ogura was the first to give a properly rigorous proof of the sampling theorem. We find that he only indicated where the method of proof could be found; we identify what is, in all probability, the proof he had in mind. Ogura states his sampling theorem as a ??converse of Whittaker??s theorem??, but identifies an error in Whittaker??s work. In order to study these matters in detail we find it necessary to make a complete review of the famous 1915 paper of E.T. Whittaker, and two not so well known papers of Ogura dating from 1920. Since the life and work of Ogura is practically unknown outside Japan, and there he is usually regarded only as an educationalist, we present a detailed overview together with a list of some 70 papers of his which we had to compile. K.?Ogura is presented in the setting of mathematics in Japan of the early 20th century. Finally, because many engineering textbooks refer to Whittaker as a source for the sampling theorem, we make a very brief review of some early introductions of sampling methods in the engineering context, mentioning H.?Nyquist, K.?Küpfmüller, V.?Kotel??nikov, H.?Raabe, C.E. Shannon and I.?Someya.     

字定理在Heegaard分解方面的一个应用

The word theorem states that x can be denoted as a rotation inserting word of A if x is in the normal closure of A in F(X). As an application of the theorem, in this note a condition that guarantees reducing the genus of Heegaard splitting of 3-manifolds is given. This leads Poincare conjecture to a new formulation.     

关于C.Fefferman定理与全纯映射的边界扩充

C．Fefferman定理证明了光滑有界强拟凸域之间的双全纯映射可以光滑延拓到边界，这个结果已经被推广到各种情形．其中Bell和Catlin以及Diederich和Fornaess独立地将其推广到拟凸域的逆紧全纯映射．本文较全面地综述了C．Fefferman定理的推广情况以及Bergman投射的边界正则性问题，同时对如何去掉Bell和Catlin以及Diederich和Fornaess定理条件中的拟凸性给出一个新观察，提出一个解决方向并且说明在具体情况下这个新观察确实是可以提供答案的．     

A Jordan surface theorem for three-dimensional digital spaces

Many applications of digital image processing now deal with three-dimensional images (the third dimension can be time or a spatial dimension). In this paper we develop a topological model for digital three space which can be useful in this context. In particular, we prove a digital, three-dimensional, analogue of the Jordan curve theorem. (The Jordan curve theorem states that a simple closed curve separates the real plane into two connected components.) Our theorem here is a digital topological formulation of the Jordan-Brouwer theorem about surfaces that separate three-dimensional space into two connected components.     

A theorem of Latyshev

A new more visible formulation of a well-known theorem of Latyshev on the simplest identity in a nonmatrix manifold of associative algebras is presented, and a short proof of this theorem is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 12–16, 1983.     

A finitization of Littlewood's Tauberian theorem and an application in Tauberian remainder theory

In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former.     

Extended formulations in combinatorial optimization

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By “perfect formulation”, we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called “compact extended formulations”. We survey various tools for deriving and studying extended formulations, such as Fourier’s procedure for projection, Minkowski-Weyl’s theorem, Balas’ theorem for the union of polyhedra, Yannakakis’ theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set, and we present the proof of Fiorini, Massar, Pokutta, Tiwary and de Wolf of an exponential lower bound for the cut polytope. We also present Bienstock’s approximate compact extended formulation for the knapsack problem, Goemans’ result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.     

A relative interpolation theorem for infinitary universal Horn logic and its applications

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic.