Harmonic Maps in Connection of Phase Transitions with Higher Dimensional Potential Wells

This is in the sequel of authors' paper [Lin, F. H., Pan, X. B. and Wang, C. Y.,Phase transition for potentials of high dimensional wells, Comm. Pure Appl. Math., 65(6),2012, 833–888] in which the authors had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition.The authors also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg's work(in 1989) on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.     

Non-abelian Hopf cohomology II—The general case

We introduce and study non-abelian cohomology sets of Hopf algebras with coefficients in Hopf comodule algebras. We prove that these sets generalize as well Serre's non-abelian group cohomology theory as the cohomological theory constructed by the authors in a previous article. We establish their functoriality and compute explicit examples. Further we classify Hopf torsors.     

Integral equations and potential-theoretic type integrals of orthogonal polynomials

Abel's theorem is used to solve the Fredholm integral equations of the first kind with Gauss's hypergeometric kernel. The case that the kernel takes general form is discussed in detail and the solution is given. Here, the works of some authors are considered as special cases. The discussion focuses on the three dimensional contact problems in the theory of elasticity with general kernel.     

Further bounds for waiting time distributions and their means in the case ρ<1

In the paper exponential bounds for the instationary and stationary waiting time d.f.'s in GI/GI/1/∞, derived by several authors partly using martingale theory arguments, are improved with the help of monotonicity properties. Further it is shown that Ne?eptjrenko's bound for the rate of convergence of the mean waiting time in single server queues Is also valid for many server queues with ρ < 1.     

On Measuring the Contribution of OR

Cost-benefit analysis offers a common-sense approach to measuring the contribution of OR but encounters difficulties which have been recognized by Ackoff. Ackoff's concept of adaptivity appears to offer a neat solution to those difficulties but can be seen to provide only a methodology and to lack a background theory. The authors argue that: in general the development of background theory should precede the development of a methodology; for OR the background theory required is one of decision-making; and that the social sciences provide the resource from which an adequate theory of decision-making can be developed.     

具有两个p'维非线性不可约特征标的非可解群

McKay猜想是有限群表示理论中的一个重要问题.本文考虑了具有两个p'维不可约特征标的非可解群群G,并证明了McKay猜想对此类群成立.更进一步,当p为奇素数时,下面结果成立:p=3,G≌PSL_2(7)或存在一个正规2-群N,满足G/N≌PSL_2(5).     

Properties of measures of information in evidence and possibility theories

An overview of information measures recently introduced by several authors in the setting of Shafer's theory of evidence is proposed. New results pertaining to additivity and monotonicity properties of these measures of information are presented. The interpretation of each measure of information as opposed to others is discussed. The potential usefulness of measures of specificity or imprecision is suggested, and a ‘principle of minimal specificity' is stated for the purpose of reconstructing a body of evidence from incomplete knowledge.     

On the Dirichlet's box principle

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong.     

Direct and Inverse Scattering Problem for Canonical Systems with a Strictly Pseudo – Exponential Potential

Explicit formulas are given for the solutions of the direct and inverse scattering problems for a canonical differential system with a strictly pseudo–exponential potential. The proofs are self–contained and employ state space techniques from mathematical system theory. The paper supplements an earlier paper of the first two authors where explicit formulas were given using Marchenko's approach, and an earlier paper of the last three authors where self–contained proofs were given for the corresponding direct and inverse spectral problems. Two types of factorizations of the scattering matrix function appear and connections between them are considered.     

Inverse problems for Sturm–Liouville operators on a star-shaped graph with mixed spectral data

The partial inverse spectral problem for Sturm–Liouville operators on a star-shaped graph was studied. The authors showed that if the potentials but one were known a priori, then the unknown potential on the whole interval can be uniquely determined by part of information of the potential and part of eigenvalues. The methods employed rest on the Weyl's m-function and theory concerning densities of zeros of entire functions.     

Rosenthal’s Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations

In this paper, the authors generalize the concept of asymptotically almost \linebreak negatively associated random variables from the classic probability space to the upper expectation space. Within the framework, the authors prove some different types of Rosenthal''s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal''s inequalities.     

Social percolation in relations between activists and supporters

This article is a study of the relations between activists with a strong belief and their passive supporters. In order to achieve a fuller understanding of these relations the authors make use of bond percolation theory. The theories hold that when the size of a group's communication in a given society exceeds a critical threshold; that is, when the size of the cluster of supporters' bonds becomes practically infinite within the society by virtue of the work of activists, subsequently supporters and activists come to hold key positions in society. A graph is generated indicating the probability of activists' bonds with respect to the probability of supporters' bonds within a lattice, when the biggest cluster of supporters' bonds becomes infinite. Simulations have been undertaken, and when a case is found in which infinite clusters emerge, this value of the probability of both supporters' and activists' bonds is isolated. A graph can then be generated a graph on which to plot these values as points. After that, these results can be discussed at length and some conclusions are derived from them. © 2006 Wiley Periodicals, Inc. Complexity 11: 51–56, 2006     

Properties of measures of information in evidence and possibility theories

An overview of information measures recently introduced by several authors in the settting of Shafer's theory of evidence is proposed. New results pertaining to additivity and monotonicity properties of these measures of information are presented. The interpretation of each measure of information as opposed to others is discussed. The potential usefulness of measures of specificity or imprecision is suggested, and a ‘principle of minimal specificity’ is stated for the purpose of reconstructing a body of evidence from incomplete knowledge.     

On the importance of mathematical play

A number of existing theories and proposals for the meaning and characteristics of ‘play’ are considered before the authors suggest six characteristics of mathematical play, including the idea that it is not confined to childhood. Previous studies provide evidence for relating play to cognitive gain while the place of mathematical play in research activities is illustrated by describing a mathematician's approach to a number investigation from the classroom-The Six Circles. The problem-solving process for the Six Circles and observations of students solving calculator and integration problems are analysed in relation to theories of play and cognitive gain and also considered from the perspective of the students' experience. Piaget's theory for the assimilation and accommodation of new information and Davis's view of play as ‘space to support learning’ are reflected in the authors' rationale for suggesting that open questions and mathematical play provide opportunities for students to develop their own conjectures, with no threat of failure, and provide a foundation for mathematical learning. Some difficulties of implementing a ‘play’ approach in the classroom are discussed and further research questions proposed.     

Numerical analysis of laminated shells under in-plane axial compression

The authors of the paper present numerical stability analysis of axially compressed laminated shells. The analysis is carried out by using the commercial code NX-Nastran and authors' own program. In both cases the Equivalent Single Layer approach is adopted, in which the laminated shell is treated as an orthotropic single layer panel with the resultant stiffness calculated for the multilayered cross-section. Since the distribution of displacement in the thickness direction is assumed to be linear (First Order Shear Deformation), the shell kinematics is consistent with the Reissner-Mindlin type theory. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithmic hypergraph regularity lemma

Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.     

A characterization of an element of best simultaneous approximation

Deutsch [4] has suggested that some problems of best simultaneous approximation might profitably be viewed as problems of best approximation in an appropriate product space. A few authors have touched upon this approach; none, however, have pursued it consistently or developed a complete problem along such a line, even in the simplest of cases. In this paper, we show that Deutsch's suggestion can easily be carried out using known results from approximation theory to establish existence, uniqueness, and characterization results. An algorithm guaranteed to converge strongly to the element of best simultaneous approximation under certain circumstances is also proposed.     

Applications of location analysis in agriculture: a survey

This article presents a survey of applications of operations research-based techniques to location problems in agriculture. From the early work in the field to more recent applications, researchers have successfully applied, adapted, and improved theoretical location models to fulfil the field's specific needs. This article discusses earlier work in agricultural location theory starting with the seminal work by von Thünen. It then introduces diverse and innovative applications of location models from the 1950s and 1960s. The authors then identify distinguishing features of location problems in agriculture, and discuss their impact on the analysis. In order to illustrate these features and the variety of operations research-based applications to agricultural location, the article turns to six real-world examples. It then considers, in Conclusion, some of the directions that agricultural location theory may take in the future.