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.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

ＡＮＤＲＥＩＫＨＲＥＮＮＩＫＯＶ（ＤｅｐａｒｔｍｅｎｔｏｆＨｉｇｈＭａｔｈｅｍａｔｉｃｓ，ＭｏｓｃｏｗＩｎｓｔｉｔｕｔｅｏｆＥｌｅｃｔｒｏｎｉｃＥｎｇｉｎｅｅｒｉｎｇ，１０３４９８，Ｍｏｓｃｏｗ，Ｋ－４９８，Ｒｕｓｓｉａｎ）（Ｔｈｉｓｗｏｒｋｉｓｓｕ...     

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.     

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?.     

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.     

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.     

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)     

Halfway to Infinity: Systems Theorizing for the Practitioners?

Systems theory is seen as stressing the rational mind (what we think about) to the detriment of the transactional mind (which we use day-to-day). This stress can be harmful to practitioners who must work in practical, transactional worlds. Rather than attempting to be infinitely rational, we propose a position halfway to infinity-rational and transactional combined.Much of systems theory is pleasing to hear and difficult to contradict. Much of the theory is about design of systems, but it is in the often non-rational process of implementation that the real system emerges. Also the theory tells us about systems but not how to identify the `correct' system that our O.R. efforts are impacting. The theory also tells about complexity, but the tools presented for handling it do not seem related to the theory.The halfway position is to build a body of contingent theory from reflection on experience, using systems theory as a framework for thinking rather than a guide for action.     

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.     

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。     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Interactions between party members and local party activists: A formal model and its empirical application*

Data taken from two surveys among party members and local party activists are used to test a formal theory of intra‐party interaction at a low regional level. For this purpose a miniature version of legitimation theory is axiomatized according to Joseph D. Sneed's proposals. A possible partial model for this miniature theory is defined, terms theoretical in this theory are introduced to make up a possible model of the theory, and several variants of the model of this theory are discussed together with their empirical claims. The axiomatized theory does not only allow the interpretation of empirical correlations among attitudinal variables as effects of interactions between persons bearing those attitudes, but also compels to consider nonlinear interactions as well.     

Geometry as seen by string theory

This is an introductory review of the topological string theory from physicist’s perspective. I start with the definition of the theory and describe its relation to the Gromov–Witten invariants. The BCOV holomorphic anomaly equations, which generalize the Quillen anomaly formula, can be used to compute higher genus partition functions of the theory. The open/closed string duality relates the closed topological string theory to the Chern–Simons gauge theory and the random matrix model. As an application of the topological string theory, I discuss the counting of bound states of D-branes.     

New Formulas Related to Analytic Number Theory and Their Applications in Statistical Physics

Since the deep paper by Bohr and Kalckar in 1938, it has been known that the Ramanujan formula in number theory is related to statistical physics and nuclear theory. From the early 1970s, there have been attempts to generalize number theory from the space of integers to the space of rational numbers, i.e., to construct a so-called analytic number theory. In statistical physics, we consider parameters such as the volume V, temperature T, and chemical potential μ, which are not integers and are consequently related to analytic number theory. This relation to physical concepts leads us to seek new relations in analytic number theory, and these relations turn out to be useful in statistical physics.     

关于反拟阵之间的映射

In this paper, the definitions of the most common and elementary mappings between matroids are extended to antimatroids first. Then the poset theory is used to find out the flats of an antimatroid and obtain all of strong maps for a given antimatroid. Besides, the poset theory is also used to deal with the relationships among the mappings between antimatroids. All the discussion is connected with poset theory. This claims that poset theory is an important tool for the study of antimatroid theory.     

代数表示论简介与综述

代数表示论是本世纪七十年代初兴起的代数学的一个新的分支。它的基本内容是研究一个Ａｒｔｉｎ代数上的模范畴。由于各国代数学家的共同努力，这一理论于最近二十年代有了异常迅猛的发展并逐步趋于完善。本文介绍了代数表示论的理论基础；几乎可裂序列；箭图和赋值箭图的表示；Ｃｏｘｅｔｅｒ函子；ＡＲ－箭图的覆盖及代数的Ｇａｌｏｉｓ覆盖。并简单介绍了在代数表示论中普遍应用的工具：Ｔｉｌｔ理论，以及著名的Ｄｐｏｚｇ定理证     

Categorical set theory: A characterization of the category of sets

We consider Zermelo-Fraenkel set theory ZF and the theory ETS(ZF) of the elementary topos of ZF sets, which is an extension of Lawvere-Tierney's theory of elementary topoi, and prove that the theory ETS(ZF) characterizes the category (topos) of ZF-sets in the following sense. The category (topos) of ZF sets satisfies the axioms of ETS(ZF), and conversely we can define within topos theory ETS(ZF) the model of set-objects in which the ZF axiloms hold, and, furthermore, the model of set-objects in the topos of ZF sets is “equivalent” to set theory ZF and the topos of set-objects in ETS(ZF) is “logical equivalent” to topos theory ETS(ZF). Actually, the corresponding result for a weak set theory Z and the theory ETS(Z) of the elementary topos of Z sets is proved. Adding further axioms (axiom of choice, continuum hypothesis, etc.), various results of the same character are obtained.The construction of the model of set-objects uses a universal mapping property of transitive sets, which enables us to introduce transitive-set-objects in any elementary topos and to prove their basic properties.     

Normal subsystems of fusion systems

The notion of a fusion system was first defined and exploredby Puig in the context of modular representation theory. Later,Broto, Levi, and Oliver significantly extended the theory offusion systems as a tool in homotopy theory. In this paper webegin a program to establish a local theory of fusion systemssimilar to the local theory of finite groups. In particular,we define the notion of a normal subsystem of a saturated fusionsystem, and prove some basic results about normal subsystemsand factor systems.     

Toposes for semigroups: an invitation

This is an expository paper in which we explain the basic ideas of topos theory in connection with semigroup theory. We focus mainly on the classifying topos of an inverse semigroup or pseudogroup, and to some extent on creating a dictionary between the language of semigroups and topos theory. We begin with the algebraic theory having to do with inverse semigroups, and then turn to an analysis of pseudogroups using sheaf theory. Our work includes some new material on wide semigroup homomorphisms and their geometric morphisms.