加强数学建模课程建设 推进优质课程资源共享

本文介绍了高教社在支持全国大学生数学建模竞赛的过程中,致力于数学建模教材及相关资料的出版;搭建各级各类交流研讨平台,扩大数学建模的影响力;与组委会共同组织数学建模类课程开设情况的调查;共同建设中国高校数学建模课程中心,创新出版模式,积极推进数学建模优质课程资源共享,为促进数学建模课程走进更多的高校所做的探索与实践.     

Krasnosel'skii-type characterizations for a cone

Three combinatorial criteria for cones involving the concepts of R-visibility and clear R-visibility introduced for sets in Euclidean space are established. They are direct analogues of Krasnosel'skii-type theorems existing for starshaped sets.The author is with the Department of Mathematics, University of Notre Dame, Indiana, on leave from the Mathematical Institute of the Polish Academy of Sciences.     

The duality of convex functions and Cesari's property (Q)

We examine the role of property (Q) in optimal control and the calculus of variations and, using lattice theoretical considerations, obtain a characterization of property (Q) in terms of Hamiltonian and conjugate functions. Connections with previous work of Rockafellar, Moreau, Brøndsted, De Giorgi, and others are established.This paper was presented at the Conference on Mathematical Questions of Control, sponsored by the International Banach Centre of the Polish Academy of Sciences, Zakopane, Poland, 1974.     

A Semilattice of Degrees of Computable Metrics

Siberian Mathematical Journal - Under study is the ordering $ {\mathcal{CM}}_{c}({\mathbf{X}}) $ of $ c $ -degrees of computable metrics on a Polish space  $ {\mathbf{X}} $ with...     

On the theory of motion of nonholonomic systems. The reducing-multiplier theorem

This classical paper by S.A. Chaplygin presents a part of his research in non-holonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the “reducing-multiplier method”. The method is illustrated on two concrete problems from non-holonomic mechanics. This paper produced a considerable effect on the further development of the Russian non-holonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world. This publication contributes to the series of RCD translations of Chaplygin’s scientific heritage. In 2002 we published two of his works (both cited in this one) in the special issue dedicated to non-holonomic mechanics (RCD, Vol. 7, no. 2). These translations along with translations of his other two papers on hydrodynamics (RCD, Vol. 12, nos. 1,2) aroused considerable interest and are broadly cited by modern researches. Originally published in: Matematicheskiĭ sbornik (Mathematical Collection), 1911, vol. 28, issue 1. The content of §§ 2 and 3 of this study was presented at the session of the Moscow Mathematical Society on December 11, 1906.     

Determining dimension of the kernel of a cone

Combinatorial conditions of Krasnosel'skii-type involving concepts ofR-visibility and clearR-visibility are given to ensure that the dimension of theR-kernel of a proper subset ofR ^d is greater than or equal tok, 0<-k<d.The author is with the Department of Mathematics of the University of Notre Dame Indiana, on leave from the Mathematical Institute of the Polish Academy of Sciences.     

On a conjecture of Furstenberg

The Hausdorff dimension of the set of numbers which can be written using digits 0, 1,t in base 3 is estimated. For everyt irrational a lower bound 0.767 … is found. Support from NSF grant DMS-0072312 and Penn State in the form of a sabbatical leave is acknowledged. This work was partly done when the author was visiting the Mathematical Institute of the Polish Academy of Sciences in Warsaw.     

Pairwise-balanced,variance-balanced and resistant incomplete block designs revisited

A general solution is derived to the problem of characterizing block designs that are simultaneously pairwise-balanced and variance-balanced. Applications of the characterizations obtained to some problems concerned with the local resistance of BIB designs are presented.This research was begun while the second author was visiting the Department of Mathematical and Statistical Methods, Academy of Agriculture in Pozna, Poland.The work of this author was partially supported by the Polish Academy of Sciences Grant No. CPBP 01.01.2/2.     

On Projective Mappings

Mathematical Notes - Let X,Y be Polish spaces, and let $\mathcal{B}_k $ be the σ-algebra generated by the projective class $L_{2k + 1} $ . A mapping $f:X \mapsto Y$ is called $k$...     

Intersection formulae for the kernel of a cone

A subset S of Euclidean space is called a cone if it is the union of a set of halflines having the same endpoint called the apex of the cone, and the set of all such apices is denoted by ker _R S and called the R-kernel or, when it does not lead to any confusion with the kernel of a starshaped set, simply the kernel of S. ker _R S is shown to be the intersection of a family of flats passing through some selected boundary points of S. Three independent formulae of this type are established, respectively: for an arbitrary proper subset S, for S closed, and for S closed connected and nonconvex.The author is with the Department of Mathematics, University of Notre Dame, Indiana, on leave from the Mathematical Institute of the Polish Academy of Sciences.     

Investigation of the mobile platform motion optimal control at energy performance index

The paper presents a method of optimal control of the mobile robot on example of the 2-wheel mobile platform. The robot investigated has non-holonomic constraints. Mathematical model of 2-wheel mobile platform is characterized by finite number of nonlinear differential equations. The presented method of control is minimizing velocity and position errors. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness of Polish group topology

We show the limits of Mackey's theorem applied to identity sets to prove that a given group has a unique Polish group topology.Verbal sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general.A Polish group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.     

Objective function approximations in mathematical programming

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the best approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained.This paper was partially supported by the National Science Foundation and by the Office of Naval Research, and was the basis for a plenary lecture delivered at the IX International Symposium on Mathematical Programming in Budapest, Hungary, August 1976.     

The strength of Mac Lane set theory

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks.     

Comparing constraint programming and mathematical programming approaches to discrete optimisation—the change problem

This paper is aimed at researchers and practitioners in Operational Research who are interested in the new field of Constraint Programming/Constraint Logic Programming. Due to differing terminology and problem representation they might have found it difficult to access the field. The paper focuses on discrete optimisation problems. The first part lists frequently used terms in Constraint Programming (CP), contrasting them with their counterparts in Mathematical Programming (MP). The second part explains some of the most important concepts and techniques in more detail by comparing the CP and MP implementations of a small example problem, the ‘Change Problem’. It includes an overview of the respective results. In conclusion a more generalised comparison of CP and MP techniques is given.     

A framework for statistical software development, maintenance, and publishing within an open-access business model

There are several fundamental problems with statistical software development in the academic community. In addition, the development and dissemination of academic software will become increasingly difficult due to a variety of reasons. To solve these problems, a new framework for statistical software development, maintenance, and publishing is proposed: it is based on the paradigm that academic and commercial software should be both cost-effectively created, maintained and published with Marketing Principles in mind. The framework has been seamlessly integrated into a highly successful website () that operates as a provider of free web-based statistical software. Finally it is explained how the R framework provides a platform for the development of a true compendium publishing system.     

Wide area telecommunication network design: application to the Alberta SuperNet

This article proposes a solution methodology for the design of a wide area telecommunication network. This study is motivated by the Alberta SuperNet project, which provides broadband Internet access to 422 communities across Alberta. There are two components to this problem: the network design itself, consisting of selecting which links will be part of the solution and which nodes should house shelters; and the loading problem which consists of determining which signal transport technology should be installed on the selected edges of the network. Mathematical models are described for these two subproblems. A tabu search algorithm heuristic is developed and tested on randomly generated instances and on Alberta SuperNet data.     

On the theory of linear systems of ordinary differential equations

Author's abstract of dissertation for the attainment of the scientific degree of Doctor of the Physical and Mathematical Sciences. Dissertation defended May 24, 1968 at the sitting of the Council of the Mechanical and Mathematical Faculty of the M. V. Lomonosov State University.Official opponents: Doctor of the Physical and Mathematical Sciences D. V. Anosov, Doctor of the Physical and Mathematical Sciences Professor R. é Vinograd, and Doctor of the Physical and Mathematical Sciences Professor Ya. G. Sinai.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 483–490, October, 1968.     

On a Roelcke-precompact Polish group that cannot act transitively on a complete metric space

We study when a continuous isometric action of a Polish group on a complete metric space is, or can be, transitive. Our main results consist of showing that for certain Polish groups, namely Aut* (μ) and Homeo+ [0, 1], such an action can never be transitive (unless the space acted upon is a singleton).We also point out that in all known examples, this pathology coincides with the pathology of Polish groups that are not closed permutation groups and yet have discrete uniform distance, asking whether there is a relation. We conclude with a general characterisation/classification of transitive continuous isometric actions of a Roelcke-precompact Polish group on a complete metric space. In particular, the morphism from a Roelcke-precompact Polish group to its Bohr compactification is surjective.     

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

Tomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.