Parity in knot theory and graph-links

The present monograph is devoted to low-dimensional topology in the context of two thriving theories: parity theory and theory of graph-links, the latter being an important generalization of virtual knot theory constructed by means of intersection graphs. Parity theory discovered by the second-named author leads to a new perspective in virtual knot theory, the theory of cobordisms in two-dimensional surfaces, and other new domains of topology. Theory of graph-links highlights a new combinatorial approach to knot theory.     

Beyond elementary catastrophe theory

We give a brief discussion of the relations between elementary catastrophe theory, general catastrophe theory, singularity theory, bifurcation theory, and topological dynamics. This is intended to clarify the status, and potential applicability, of “catastrophe theory,” a phrase used by different authors and at different times with different meanings. Catastrophe theory has often been criticized for (supposed) applicability only to gradient systems of differential equations; but properly speaking this criticism can apply only to the elementary version of the theory (where it is in any case wrong). Roughly speaking, elementary catastrophe theory deals with the singularities of real-valued functions, general catastrophe theory with singularities of flows. Between these lies singularity theory, which deals with vector-valued functions. All relate strongly to bifurcation theory and topological dynamics. The issue is more subtle than it appears to be, and we describe an example where elementary catastrophe theory has been used to solve a long-standing problem about nongradient flows: degenerate Hopf bifurcation.     

On the decision problem for theories of finite models

An infinite extension of the elementary theory of Abelian groups is constructed, which is proved to be decidable, while the elementary theory of its finite models is shown to be undecidable. Tarski’s proof of undecidability for the elementary theory of Abelian cancellation semigroups is presented in detail. Szmielew’s proof of the decidability of the elementary theory of Abelian groups is used to prove the decidability of the elementary theory of finite Abelian groups, and an axiom system for this theory is exhibited. It follows that the elementary theory of Abelian cancellation semigroups, while undecidable, has a decidable theory of finite models.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

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Bitopological spaces

The paper is, in essence, a monograph devoted to the theory of bitopological spaces and its applications. Not exhausting the entire subject, it reflects basic ideas and methods of the theory. The Introduction gives an idea of the origins of the basic notions, contents, methods, and problems both of the classical (in the spirit of Kelly) and of the general theory of bitopological spaces. The classical theory is described rather schematically in Chapter I, only the theory of extensions of topological and bitopological spaces and the theory of completion of uniform spaces are presented in more detail. The main focus is on the general theory of bitopological spaces (Chapter II). Notions, methods, and results presented here have no analogues in the classical theory. As applications, foundations of the theory of bitopological manifolds, in particular, bitopologically represented piecewise linear manifolds (Chapter III), and the foundations of the theory of bitopological groups are presented (Chapter IV). Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 242, 1997, pp. 7–216. Translated by A. A. Ivanov.     

Strong shape theory and resolutions

It is shown that the strong shape theory of compact metrizable spaces extends to a theory for all topological spaces. The extension resembles the inverse systems approach to shape theory of Marde?i? and Segal. Fundamental roles are played by the Steenrod homotopy theory of Edwards and Hastings and the theory of ANR-resolutions due to Marde?i?.     

横观各向同性板的弹性理论和弹性改进理论及一种新的厚板理论

本文从横观各向同性体弹性力学位移形式的基本方程出发,考虑板面承受横向荷载,建立了横观各向同性板弯曲的弹性理论.并由此建立了一个在板的每边能满足三个边界条件的弹性改进理论和一种新的厚板理论.文中求得了周边简支多边形板的弹性改进理论解,数值结果与三维弹性理论精确解的结果非常接近.新的厚板理论和以往的中厚板理论的系统比较表明,我们提出的厚板理论最靠近弹性理论的结果.     

应用图论分析与最优化理论来数据挖掘大规模水牛普里昂蛋白结构数据

图论、最优化理论显然在蛋白质结构的研究中大有用场. 首先, 调查/回顾了研究蛋白质结构的所有图论模型. 其后, 建立了一个图论模型: 让蛋白质的侧链来作为图的顶点, 应用图论的诸如团、 $k$-团、 社群、 枢纽、聚类等概念来建立图的边. 然后, 应用数学最优化的现代摩登数据挖掘算法/方法来分析水牛普里昂蛋白结构的大数据. 成功与令人耳目一新的数值结果将展示给朋友们.     

Lie algebras and recurrence relations I

We study representations of the Heisenberg-Weyl algebra and a variety of Lie algebras, e.g., su(2), related through various aspects of the spectral theory of self-adjoint operators, the theory of orthogonal polynomials, and basic quantum theory. The approach taken here enables extensions from the one-variable case to be made in a natural manner. Extensions to certain infinite-dimensional Lie algebras (continuous tensor products, q-analogs) can be found as well. Particularly, we discuss the relationship between generating functions and representations of Lie algebras, spectral theory for operators that lead to systems of orthogonal polynomials and, importantly, the precise connection between the representation theory of Lie algebras and classical probability distributions is presented via the notions of quantum probability theory. Coincidentally, our theory is closed connected to the study of exponential families with quadratic variance in statistical theory.     

Finite element analysis of arches undergoing large rotations - I: Theoretical comparison

In this two part paper, the first part deals with five different nonlinear theories applicable to the analysis of arches in the context of solving the large displacement and the large rotation problem. These theories include, classical theory, first-order shear deformation theory, third-order shear deformation theory, modified classical theory and the Donnell-type theory. All the theories are developed using the Total Lagrangian approach. Simplifications and assumptions used in each of the theory are discussed. Explicit strain displacement gradient relations and element independent equilibrium equations in terms of displacement gradients are given for all the theories. Limitations of each of theory are discussed. In the second part of this paper, application of these theories for the classification of arch geometries is considered.     

An Iterative Analytical Theory in the Mechanics of Layered Composite Systems

An iterative analytical theory in the mechanics of layered composite systems is developed. The prehistory of the nonclassical theory of layered systems is presented. The division of this theory into two principal directions - discrete-structural and continuous-structural - is mentioned. The basic iterative Ambartsumyan theory, which belongs to the second direction, is described. The formation of the generalized iteration theory of first approximation is shown. In this theory, the disagreement between the kinematic and static models is removed, i.e., a generalization of these models is realized. The theory of second approximation is described. An iterative principle is presented for the formation of a higher-approximation nonclassical theory. Based on this principle, theories of anisotropic composite shallow shells, plates, and beams are formulated. Comparative calculation results for different layered composite systems are presented.     

A consistent second-order plate theory for monotropic material

Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropic homogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accurate mechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elastic deformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff's plate theory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler's theory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. Details of the derivation of the theory will be published in a forthcoming paper [3]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relativistically Covariant Quantum Field Theory of the Maslov Complex Germ

We consider an explicitly covariant formulation of the quantum field theory of the Maslov complex germ (semiclassical field theory) in the example of a scalar field. The main object in the theory is the “semiclassical bundle” whose base is the set of classical states and whose fibers are the spaces of states of the quantum theory in an external field. The respective semiclassical states occurring in the Maslov complex germ theory at a point and in the theory of Lagrangian manifolds with a complex germ are represented by points and surfaces in the semiclassical bundle space. We formulate semiclassical analogues of quantum field theory axioms and establish a relation between the covariant semiclassical theory and both the Hamiltonian formulation previously constructed and the axiomatic field theory constructions Schwinger sources, the Bogoliubov S-matrix, and the Lehmann-Symanzik-Zimmermann R-functions. We propose a new covariant formulation of classical field theory and a scheme of semiclassical quantization of fields that does not involve a postulated replacement of Poisson brackets with commutators.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 492–512, September, 2005.     

Gauge theory of the liquid-glass transition in static and dynamical approaches

We propose static and dynamical formulations of the liquid-glass transition theory based on the glass gauge theory and the fluctuation theory of phase transitions. In accordance with the proposed theory, the liquid-glass transition is an unattainable second-order phase transition blocked by a premature critical slowing of the gauge field relaxation caused by the system frustration. We show that the proposed theory qualitatively agrees well with experimental data.     

狄氏型及其在数学物理中的应用

简介狄氏型理论。该理论已成为紧密联系解析位势论与马氏过程理论的强有力数学工具,并因此在数学与物理中有许多应用。     

Local set theory

In 1945, Eilenberg and MacLane introduced the new mathematical notion of category. Unfortunately, from the very beginning, category theory did not fit into the framework of either Zermelo—Fraenkel set theory or even von Neumann—Bernays—Gödel set-class theory. For this reason, in 1959, MacLane posed the general problem of constructing a new, more flexible, axiomatic set theory which would be an adequate logical basis for the whole of naïve category theory. In this paper, we give axiomatic foundations for local set theory. This theory might be one of the possible solutions of the MacLane problem.     

Frame quasi-uniformities

This paper presents an entourage-like theory of quasi-uniformities for frames. The theory comprises the theory of uniformities for frames as well as the classical theory of quasi-uniformities for spaces.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples.     

Random graphs in the monadic theory of order

We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line). Received: 18 July 1996