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 theory for nonlinear soft webs

In this paper, a fundamental theory for deformable webs not resisting any compressive membrane forces is developed through a direct derivation on the deformed configuration. In order to better describe the deformable webs, the classification of deformable webs is presented. A theory for the non-continuum elastic network webs consisting of many deformable cables is presented from the deformable cable theory, and then a theory for fabric deformable webs without relative sliding is developed as well. Finally, a nonlinear theory for continuous deformable webs is presented on the deformed configuration. The local criteria for the existence of such deformable webs are presented through the definitions, and such criteria are very significant for the wrinkling stability of the deformable webs. A deformable web possessing the local wrinkling is an unsolved problem in numerical computations. The theory for fabric webs with relative motions needs to be further developed. Herein the fundamental theory for deformable webs is presented only, and numerical examples will be presented in sequel. Such a theory of deformable webs can be applied to textile or other soft materials and bio-membranes.     

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.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

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关于mr(2,q)的一个新值

编码理论中关于寻找某一线性码的最大长度涉及到有限射影空间中关于t-blockingsets,(k,r-ares和caps集所含元素的个数的问题,本文研究了(k,r)-arc集的元素的个数,找到了使得(k,r)-arc集存在的最大k值,即mr(2,q)的一个新值,丰富了编码理论的相关内容。     

Similarity of operator blocks and canonical forms. I. General results,feedback equivalence and kronecker indices

In the present paper the problem of classifying blocks of matrices up to similarity is considered. The notion of block similarity used here is a natural generalization of similarity for matrices. The invariants are described and canonical forms are given. This theory of block-similarity provides a general framework, which includes the state feedback theory for systems, the theory of Kronecker equivalence and a similarity theory for non-everywhere defined operators. New applications, in particular to factorization problems, are also obtained.     

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.     

On unbounded probability theory

The value of the empirical expectation coincides with that of the mean energy of an ideal Bose gas for one particle. The exact mathematical identity for these quantities makes it possible to carry over the concept of temperature corresponding to the mean energy to an unboundedly increasing sequence of random values for a new unbounded probability theory and for a generalization of Kolmogorov complexity theory. The notion of spectral gap, which was introduced in superconductivity theory, is carried over to unbounded probability theory.     

An extended beam theory for smart materials applications part I: Extended beam models,duality theory,and finite element simulations

An extended theory for elastic and plastic beam problems is studied. By introducing new dependent and independent variables, the standard Timoshenko beam model is extended to take account of shear variation in the lateral direction. The dynamic governing equations are established via Hamilton's principle, and existence and uniqueness results for the solution of the static problem are proved. Using the theory of convex analysis, the duality theory for the extended beam model is developed. Moreover, the extended theory for rigid-perfectly plastic beams is also established. Based on the extended model, a finite-element method is proposed and numerical results are obtained indicating the usefulness of the extended theory in applications.The work of the first author was supported in part by National Science Foundation under Grant DMS9400565.     

Non-Deterministic Closure Theory and Universal Arrows

Traditional closure theory discusses the closure operations on orders with graph-theoretic methods, or the reflectors on skeletal categories with category-theoretic methods. Both approaches are confined, like most of classical mathematics, to total and deterministic operations. So traditional closure theory makes it possible to define the semantics of the while-do commands only for terminating and deterministic programming. This paper outlines a closure theory for relations which transcend totality and determinism. For the sake of conciseness, the language used is that of graph theory but the methods are category-theoretic and some hints are offered for a possible translation into the language of category theory. Our basic idea is that closure relations consist of universal arrows in the sense of category theory. The new closure theory is appropriate for defining a semantics of the while-do commands both for terminating, deterministic programming and for non-terminating, non-deterministic programming.     

Normal forms of linear second order partial differential equations on the plane

The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recent results in the theory, is included in the paper in the parametric form together with proof. There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.     

One-dimensional nil-DAHA and Whittaker functions II

This work is devoted to the theory of nil-DAHA for the root system A₁ and its applications to symmetric and nonsymmetric (spinor) global q-Whittaker functions, integrating the q-Toda eigenvalue problem and its Dunkl-type nonsymmetric version. The spinor global functions extend the symmetric ones to the case of all Demazure characters (not only those for dominant weights); the corresponding Gromov–Witten theory is not known. The main result of the paper is a complete algebraic theory of these functions in terms of induced modules of the core subalgebra of nil-DAHA. It is the first instance of the DAHA theory of canonical-crystal bases, quite non-trivial even for A₁. As the second part of this work, this paper is mainly devoted to the theory of the core subalgebra of nil-DAHA, its induced modules and their applications to the nonsymmetric global Whittaker functions. The first part was about the analytic aspects of our construction and a general algebraic theory of nil-DAHA for A₁.     

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.     

Morse-type index theory for flows and periodic solutions for Hamiltonian Equations

An index theory for flows is presented which extends the classical Morse theory for gradient flows on compact manifolds. The theory is used to prove a Morse-type existence statement for periodic solutions of a time-dependent (periodic in time) and asymptotically linear Hamiltonian equation.     

Linear inequalities,mathematical programming and matrix theory

A survey is made of solvability theory for systems of complex linear inequalities.This theory is applied to complex mathematical programming and stability and inertia theorems in matrix theory.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.     

A minimalist two-level foundation for constructive mathematics

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin.One level is given by an intensional type theory, called Minimal type theory. This theory extends a previous version with collections.The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model.This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofs-as-programs” paradigm and acts as a programming language.     

Strongly nonlinear internal soliton load on a small vertical circular cylinder in two-layer fluids

The solutions of MCC theory are used to investigate larger-amplitude strongly nonlinear internal soliton load on a small surface-piercing circular cylinder in two-layer fluids. By comparing the wave profiles and instantaneous horizontal velocities calculated by MCC theory with those of KdV theory and experimental data, we verify the validity of MCC theory for larger-amplitude strongly nonlinear internal soliton. The accelerations are computed, and then force and torque on a small cylinder are estimated based on Morison’s formula for both MCC and KdV theories. Computed results show that the internal soliton force and torque become more and more large and wide with the increase of amplitude for MCC theory. The location of torque crest calculated by MCC theory departs from origin (moving to the right) as the amplitude grows and whenever the inertial term is included or not, the wave forces computed based on the two theories both have small discrepancies for the same amplitude, but when the inertial term is included, the torque obtained by MCC theory will be much larger and the torque obtained by KdV still have a small discrepancy. The reasons are presented in detail. The internal wave force will be underestimated if the traditional KdV theory is used. Therefore, ocean engineers should consider the large-amplitude strongly nonlinear internal soliton load on marine construct carefully.     

The theory of micropolar thin elastic shells

A boundary-value problem of the three-dimensional micropolar, asymmetric, moment theory of elasticity with free rotation is investigated in the case of a thin shell. It is assumed that the general stress-strain state (SSS) is comprised of an internal SSS and boundary layers. An asymptotic method of integrating a three-dimensional boundary-value problem of the micropolar theory of elasticity with free rotation is used for their approximate determination. Three different asymptotics are constructed for this problem, depending on the values of the dimensionless physical parameters. The initial approximation for the first asymptotics leads to the theory of micropolar shells with free rotation, the approximation for the second leads to the theory of micropolar shells with constrained rotation and the approximation for the third asymptotics leads to the so-called theory of micropolar shells “with a small shear stiffness”. Micropolar boundary layers are constructed. The problem of the matching of the internal problem and the boundary-layer solutions is investigated. The two-dimensional boundary conditions for the above-mentioned theories of micropolar shells are determined.     

Static and Dynamic Theories of Liquid Crystals

The study of liquid crystals gives rise to many fascinating but difficult mathematical problems. The purpose of this paper is to briefly summarize some recent advances, as well as to describe the present state of art of the theory of liquid crystals. For the static theory, we emphasis on the theory of defects and the theory of Smectic A materials. We will also study the Ericksen-Leslie theory for the liquid crystal flow. The well-posedness as well as the motion of the defects will be discussed.