Updating the hamiltonian problem—A survey

This is intended as a survey article covering recent developments in the area of hamiltonian graphs, that is, graphs containing a spanning cycle. This article also contains some material on related topics such as traceable, hamiltonian-connected and pancyclic graphs and digraphs, as well as an extensive bibliography of papers in the area.     

Developing reflectiveness in mathematics classrooms— An aim to be reached in several ways

The importance of reflection has been emphasized in many conceptions of mathematics education and mathematical literacy. Beyond that, this article emphasizes the importance of reflectiveness, not as an activity or an ability, but as an attitude to be aimed at. The article shows examples and ways to approach this big aim of reflectiveness. It is not only a matter of initiating reflection on different levels, but also a matter of developing an individual disposition for questioning. For this, sense reflections and self reflections play a key role.     

Quantaloids,enriched categories and automata theory

This article is intended to be an survey article outlining how the theory of quantaloids and categories enriched in them provides an effective means of analyzing both automata and tree automata. The emphasis is on the unification of concepts and how categorical methods provide insight into various calculations and theorems, both illuminating the original presentation as well as yielding conceptually simpler proofs. Proofs will be omitted and the emphasis is on providing the reader (even a relatively inexperienced one) with an understanding of the basic constructions and results.     

The sound of music and its link with mathematics

This article discusses a relatively unnoticed application ofmathematics by describing its connection with an aspect of music,in particular, the musical scales. Stemming from a problem foundin a Year 9 mathematics textbook commonly used in Singapore,the article illustrates the role of mathematics in musical scalesby first considering the frequency ratios of consecutive musicalnotes in the ‘just scale’ and secondly explaininghow an anomaly in the ‘just scale’ caused by theuneven frequency ratios is resolved with the help of mathematics,thereby leading to the development of the well-tempered scale.The article ends with an exploration of the frequency ratiosof consecutive musical notes in the well-tempered scale. Notonly does the article aim to broaden the teachers’ horizonswith such an introduction to the mathematical aspect of music,it also hopes to enrich their mathematical experiences as well.     

Ways to square the parabola – a commented picture gallery

The mathematical literature knows several quite different ways to determine the area of a parabolic segment. In this article they are brought together, compared and varied. Finally two new proofs are added to the collection. Each proof is displayed by an expressive figure. A colorful compilation of those figures accompanies the article as an appendix.     

Embodied Cognition and the Mathematical Emotional Orientation

The common image of mathematics is one of a solitary and unemotional activity. In contrast, this article views both emotions and social discourse as central to mathematical activity. What defines mathematics is a set of shared preferences that has been referred to as an emotional orientation. This article describes the key features of emotional orientations, their bodily bases, and their relation to language and social contexts. The concept of a mathematical emotional orientation provides an interpretive frame that is informative and useful for analyzing mathematical activity in educational research and practice.     

Embodied Cognition and the Mathematical Emotional Orientation

The common image of mathematics is one of a solitary and unemotional activity. In contrast, this article views both emotions and social discourse as central to mathematical activity. What defines mathematics is a set of shared preferences that has been referred to as an emotional orientation. This article describes the key features of emotional orientations, their bodily bases, and their relation to language and social contexts. The concept of a mathematical emotional orientation provides an interpretive frame that is informative and useful for analyzing mathematical activity in educational research and practice.     

Infinitesimal group schemes as iterative differential Galois groups

This article is concerned with Galois theory for iterative differential fields (ID-fields) in positive characteristic. More precisely, we consider purely inseparable Picard-Vessiot extensions, because these are the ones having an infinitesimal group scheme as iterative differential Galois group. In this article we prove a necessary and sufficient condition to decide whether an infinitesimal group scheme occurs as Galois group scheme of a Picard-Vessiot extension over a given ID-field or not. In particular, this solves the inverse ID-Galois problem for infinitesimal group schemes. Furthermore, this gives a tool to tell whether all purely inseparable ID-extensions are in fact Picard-Vessiot extensions.     

Numerical computations of conductivity over agglomerated continuum percolation models

In order to clarify how the percolation theory governs the conductivities in real materials which consist of small conductive particles, e.g., nanoparticles, with random configurations in an insulator, we numerically investigate the conductivities of continuum percolation models consisting of overlapped particles using the finite difference method as a sequel of our previous article [S. Matsutani, Y. Shimosako, Y. Wang, Int. J. Mod. Phys. C 21 (2010) 709–729]. As the previous article showed the shape effect of each particle by handling different aspect ratios of spheroids, in this article we numerically show influences of the agglomeration of the particles on conductivities after we model the agglomerated configuration by employing a simple numerical algorithm which simulate an agglomerated configuration of particles by a natural parameter. We conclude that the dominant agglomeration effect on the conductivities can be interpreted as the size effect of an analyzed region. We also discuss an effect of shape of the agglomerated clusters on its universal property.     

Strategies and flexibility in mental calculation

This article focuses on issues around flexibility in mental calculation. When calculating a subtraction or multiplication mentally, it is desirable for students to use a method that suits the problem—to be flexible—but is it fair to describe this as “strategic flexibility”? What exactly is it that happens when someone ‘decides’ to tackle a mental calculation in a particular way? In the article, I make some distinctions among strategies and strategic flexibility in mental calculation to explore and try to clarify the terms and associated ideas. At the heart of the article is an argument that the model of strategic choice is not able to give an adequate account of strategic flexibility in all aspects of mental calculation, and an alternative is proposed.     

An elementary proof of the Karush–Kuhn–Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints

We present an elementary proof of the Karush–Kuhn–Tucker theorem for the problem with a finite number of nonlinear inequality constraints in normed linear spaces under the linear independence constraint qualification. Most proofs in the literature rely on advanced concepts and results such as the convex separation theorem and Farkas, lemma. By contrast, the proofs given in this article, including a proof of the lemma, employ only basic results from linear algebra. The lemma derived in this article represents an independent theoretical result.     

Global optimization in metric spaces with partial orders

The main objective of this article is to resolve an optimization problem in the setting of a metric space that is endowed with a partial order. In fact, given non-empty subsets A and B of a metric space that is equipped with a partial order, and a non-self mapping S: A?→?B, this article explores the existence of an optimal approximate solution, known as a best proximity point of the mapping S, to the equation Sx?=?x, where S is a proximally increasing, ordered proximal contraction. This article exhibits an algorithm for determining such an optimal approximate solution. Moreover, the result elicited in this article subsumes a fixed point theorem, due to Nieto and Rodriguez-Lopez, in the setting of a metric space with a partial order.     

The Mathematics of Internet Search Engines

This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.     

在金融危机中企业受波击的数学模型

建立了在金融危机(或经济危机)中企业受波击的微分方程模型,并对模型进行了求解和经济解释.     

Supporting and separating subsets for invex bodies

Invex bodies represent an important class of bodies which are considered as a generalization of convex bodies.In this article,the author studies the supporting for this class of bodies as well as the separating subsets of two bodies.     

Valuations,intersections et fonctions de Belyi-quelques remarques sur la construction des hauteurs

We present in this article several possibilities to approach the height of an algebraic curve defined over a number field: as an intersection number via the Arakelov theory, as a limit point of the heights of its algebraic points and, finally, using the minimal degree of Belyi functions.     

On some infinite sums of iterated radicals

The general purpose of this article is to shed some light on the understanding of real numbers, particularly with regard to two issues: the real number as the limit of a sequence of rational numbers and the development of irrational numbers as a continued fraction. By generalizing the expression of the golden ratio in the form of the limit of two particular sequences, a new characterization of this number will appear. In that process, an infinite sum of iterated radicals is obtained. Based on that result, this article will then proceed to analyse that sum. The conditions under which the infinite sum yields an integer will be inspected, thereby calculating the value of the sum. After that, a method is established to develop some algebraic irrationals as a continued fraction. Finally, the results will be applied to an infinite difference of iterated radicals.     

Information Security, Mathematics, and Public-Key Cryptography

Public-key cryptography is today recognized as an important tool in the provision of information security. This article gives an overview of the field on the occasion of its 22nd birthday.