Bit by bit, the discoverers decided what it was they had discovered

Conclusion  Is it possible to answer the questionWhat is the “birthday” of elliptic functions? Yes, but far from uniquely. But does the overabundance of possible answers occasioned by the inherent n?ivety of the question mean that such lines of inquiry are pointless for the historian? Can questions regarding the temporal origins of mathematical areas and the research to which they lead ever be useful or instructive?     

Dialogue in Mathematical Correspondence by School Students

Can school students use correspondence about solutions of mathematics problems for educational purposes? The answer is yes. In this article we shall present a correspondence approach used in work with mathematically gifted students in Bulgaria, an approach in which the dialogue and the discussion plan are carefully organized.     

Acute triangulation of a triangle in a general setting revisited

Axiom A16 from Pambuccian (Can. Math. Bull. 53, 534?C541, 2010) is shown to be superfluous as it depends on axioms A1?CA15. This provides a surprisingly simple axiom system in which the acute triangulation with seven triangles can be proved for any triangle, consisting only of A1?CA15 in Pambuccian (Can. Math. Bull. 53, 534?C541, 2010).     

Do manifolds have little symmetry?

This note surveys certain aspects (including recent results) of the following problem stated by F. Raymond and R. Schultz: “It is generally felt that a manifold ‘chosen at random’ will have little symmetry. Can this intuitive notion be made more precise? Does there exist a closed simply connected manifold on which no finite group acts effectively?” Dedicated to Albrecht Dold and Edward Fadell     

The contribution of critical realism as an underpinning philosophy for OR/MS and systems

Many issues are under debate as to the philosophical nature of OR/MS: is it science or technology? Is it natural or social science? Can it be realist as well as being interpretivist? There are also many debates within the philosophy of science itself. This paper proposes that a particular account of the philosophy of science, known as ‘critical realism’, is especially suitable as an underpinning of OR/MS. The structure of the argument of this paper is to outline the main positions within the philosophy of science and highlight their problems, especially from the point of view of OR/MS; then to introduce critical realism and to show how it addresses these problems and how it is particularly appropriate for OR/MS; and finally to illustrate this by considering examples of various practical OR methods.     

Generalization of an Existence Theorem for Variational Inequalities

By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.     

Optimizing over the first Chvátal closure

How difficult is, in practice, to optimize exactly over the first Chvátal closure of a generic ILP? Which fraction of the integrality gap can be closed this way, e.g., for some hard problems in the MIPLIB library? Can the first-closure optimization be useful as a research (off-line) tool to guess the structure of some relevant classes of inequalities, when a specific combinatorial problem is addressed? In this paper we give answers to the above questions, based on an extensive computational analysis. Our approach is to model the rank-1 Chvátal-Gomory separation problem, which is known to be NP-hard, through a MIP model, which is then solved through a general-purpose MIP solver. As far as we know, this approach was never implemented and evaluated computationally by previous authors, though it gives a very useful separation tool for general ILP problems. We report the optimal value over the first Chvátal closure for a set of ILP problems from MIPLIB 3.0 and 2003. We also report, for the first time, the optimal solution of a very hard instance from MIPLIB 2003, namely nsrand-ipx, obtained by using our cut separation procedure to preprocess the original ILP model. Finally, we describe a new class of ATSP facets found with the help of our separation procedure.     

The Role of Emotions in Multi-Organizational Decision-Making

This paper addresses two questions. (i) What is the role of emotions in decision-making? (ii) Can emotions be attributed to organizations, and is their role in decisions made between organizations the same as in decisions made between individual people? The answers given to question (ii) are yes and yes. The proposed answer to question (i) is based upon a game-theoretic approach, which considers how emotions may be of strategic benefit to game players.     

Construction of covering maps between orientable surfaces

Consider the following problem. Can a set of numbers be realized as boundary covering indices of a covering map between surfaces? How to realize them? A set of equivalent criteria for this problem are found, which can be checked by a finite algorithm. Furthermore, the algorithm will also construct a solution if such one exists. As an application, a well_known group of necessary conditions are shown to be not sufficient in infinitely many cases, while in most cases, numbers satisfying them are realizable.     

A two-coloring of Cartesian products

We shall color the Cartesian product ω × ω₁with two colors. Can an infinite subset A ?ω and an uncountable subset B ?ω₁ be found such that the product A × B can be one-colored? This problem proves to be unsolvable in ZFC.     

How Boots Befooled the King: Wisdom, Truth, and the Stoics

Can the wise person be fooled? The Stoics take a very strong view on this question, holding that the wise person (or sage) is never deceived and never believes anything that is false. This seems to be an implausibly strong claim, but it follows directly from some basic tenets of the Stoic cognitive and psychological world-view. In developing an account of what wisdom really requires, I will explore the tenets of the Stoic view that lead to this infallibilism about wisdom, and show that many of the elements of the Stoic picture can be preserved in a more plausible fallibilist approach. Specifically, I propose to develop a Stoic fallibilist virtue epistemology that is based on the Stoic model of the moral virtues. This model of the intellectual virtues will show that (in keeping with a folk distinction) the wise person is never befooled, though that person might be fooled.     

Local rules and global order, or aperiodic tilings

Can local rules impose a global order? If yes, when and how? This is a philosophical question that could be asked in many cases. How does local interaction of atoms create crystals (or quasicrystals)? How does one living cell manage to develop into a pine cone whose seeds form spirals (and the number of spirals     

Formal Integrability Criteria for Nonlinear Partial Difference Equations

In recent years, there has been a substantial growth of interest in discrete differential geometry. Can results in formal differential geometry also be discretized for nonlinear partial difference equations? Using some techniques in exterior difference calculus, we investigate the geometric property of discrete Jet bundles. Further, we define the formal integrability and discrete Spencer cohomology for nonlinear partial difference equations, and give the formal integrability criteria.     

Exploring the dynamics of journal citations: Modelling with s-curves

This paper reports on an exploratory analysis of the behaviour of citations for management science papers over a 14-year period. Citations often display s-curve type behaviour: beginning slowly, rising in response to previous citations, and then declining as the material becomes obsolete. Within the context of citation research such functions are known as obsolescence functions. The paper addresses three specific questions: (i) can collections of papers from the same journal all be modelled using the same obsolescence function? (ii) Can we identify specific patterns of behaviour such as ‘sleeping beauties’ or ‘shooting stars’? (iii) Can we predict the number of future citations from the pattern of behaviour in the first few years? Over 600 papers published in six leading management science journals are analysed using a variety of s-curves.     

Minimal Simplicial Dissections and Triangulations of Convex 3-Polytopes

This paper addresses three questions related to minimal triangulations of a three-dimensional convex polytope P . • Can the minimal number of tetrahedra in a triangulation be decreased if one allows the use of interior points of P as vertices? • Can a dissection of P use fewer tetrahedra than a triangulation? • Does the size of a minimal triangulation depend on the geometric realization of P ? The main result of this paper is that all these questions have an affirmative answer. Even stronger, the gaps of size produced by allowing interior vertices or by using dissections may be linear in the number of points. Received August 16, 1999, and in revised form February 29, 2000. Online publication May 19, 2000.     

Spaces from pieces II

In Parts II to IV, we are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T₀, T₁, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even when they are not really needed).     

Spaces from pieces IV

We are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T₀, T₁, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even they are not really needed).     

Envelopes of holomorphy of Jordan curves in ℂn

A comprehensive answer is given to two related questions: (i) Can continuous functions be approximated by holomorphic ones on Jordan curves in ?ⁿ (A.G. Vitushkin, 1964)? (ii) Is any Jordan curve in ?ⁿ holomorphically convex (E.L. Stout, 1970)?     

Solutions to two problems of Dénes and Keedwell on row-complete latin squares

In a recent book, Dénes and Keedwell pose several questions concerning row-complete latin squares. Two of these questions are: (1) Is every row-complete latin square the Cayley table of a group? (2) Can every row-complete latin square be made column-complete by a suitable reordering of its rows? In the present paper, both questions are shown to have negative answers.     

Mathematical notation and the use of functions (Poster Presentation)

In contrast to natural languages, mathematical notation is accepted as being exceptionally precise. It shall make mathematical statements unambiguous, it shall allow formal manipulation, it is model for programming languages, computer algebra systems and machine provers. However, what is traditional notation and is it indeed as precise as expected? We discuss some examples of notation which require caution. How are they adapted in computer algebra systems? Can we improve them somehow? What can we learn from functional programming? (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)