On the core: complement-reduced game and max-reduced game

This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to “complement-reduced game” and converse consistency with respect to “max-reduced game”. The other is based on consistency with respect to “max-reduced game” and weak converse consistency with respect to “complement-reduced game”. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non-emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.     

Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete

The stability ofL ²-eigenvalues and associated eigenspaces of singular second order differential operators of Schrödinger-type is shown for asymptotic perturbations of the coefficients and the domain of definition. The perturbations involved are more general than those studied in [3] and [5], because we do not postulate the convergence of the coefficients “from above” or of the domains “from inside” or “from outside”. Moreover, the domain of definition is allowed to be perturbed in its interior. The underlying abstract perturbation theory was established in a previous paper [9].     

Hypernumbers and quantum field theory with a summary of physically applicable hypernumber arithmetics and their geometries

Computing in hypernumber arithmetics is discussed, and specifically that of M-algebra, which includes the operations of complex, quaternion, and Cayley numbers (octaves or octonions) as subsets of itself. It is shown that modern quantum gravitation theory requires minimally the 16-dimensional space of the author's M-arithmetic (announced in Appl. Math. and Comput., 1976, p. 211 f. and 1978, p. 45 f.), which has 4320 units (including positive and negative) rather than the mere 2 units of ordinary or “real” arithmetic, the 4 units of complex arithmetic, the 24 units of quaternion arithmetic, or the 240 units of octonion or octave arithmetic. Thus computer programming is the natural tool for computations in advanced quantum physics. It turns out that more than three kinds of i-type hypernumbers and more than three kinds of the \Ge-type are needed to ensure the necessary nilpotent and noncommutative algebra required in unified field theory. It is also shown that “more than three” here means “at least seven”; and it turns out that a 16-dimensional arithmetic is needed for such computations. The following paper contextualizes, characterizes, and specifies that arithmetic as the apex of a hierarchy susceptible of clear geometric definition. And the hypernumbers needed in quantized unified field theory are specified.     

Computer Algebra and Bifurcations

In this paper we will present the family of Newton algorithms. From the computer algebra point of view, the most basic of them is well known for the local analysis of plane algebraic curves f(x,y)=0 and consists in expanding y as Puiseux series in the variable x. A similar algorithm has been developped for multi-variate algebraic equations and for linear differential equations, using the same basic tools: a “regular” case, associated with a “simple” class of solutions, and a “simple” method of calculus of these solutions; a Newton polygon; changes of variable of type ramification; changes of unknown function of two types y=ct ^μ+? or y=exp?(c/t ^μ)?. Our purpose is first to define a “regular” case for nonlinear implicit differential equations f(t,y,y′)=0. We will then apply the result to an explicit differential equation with a parameter y′=f(y,α) in order to make a link between the expansions of the solutions obtained by our local analysis and the classical theory of bifurcations.     

A “strange” vector-valued quantum modular form

Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of ${\mathbb{Q}}$ Q , and whose obstruction to modularity is some analytically “nice” function. Motivated by Zagier’s example of the quantum modularity of Kontsevich’s “strange” function F(q), we revisit work of Andrews, Jiménez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly “strange”.     

Functions and Sets of Smooth Substructure: Relationships and Examples

The past decade has seen the introduction of a number of classes of nonsmooth functions possessing smooth substructure, e.g., “amenable functions”, “partly smooth functions”, and “g ° F decomposable functions”. Along with these classes a number of structural properties have been proposed, e.g., “identifiable surfaces”, “fast tracks”, and “primal-dual gradient structures”. In this paper we examine the relationships between these various classes of functions and their smooth substructures. In the convex case we show that the definitions of identifiable surfaces, fast tracks, and partly smooth functions are equivalent. In the nonconvex case we discuss when a primal-dual gradient structure or g ° F decomposition implies the function is partly smooth, and vice versa. We further provide examples to show these classes are not equal.     

Syzygy and Torsionless Modules with Respect to a Semidualizing Module

This paper aims at generalizing the notions of “Auslander dual”, “k-torsionless” and “k-th syzygy” modules to the relative setting with respect to a semidualizing module. It is shown that the “relative” Auslander dual shares many nice properties with the Auslander dual originally introduced by Auslander and Bridger. It is also shown the interplay between the “relative” version of “k-torsionless” and “k-th syzygy” modules parallels that of k-torsionless and k-th syzygy modules.     

The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations

In this paper, we address some fundamental issues concerning “time marching” numerical schemes for computing steady state solutions of boundary value problems for nonlinear partial differential equations. Simple examples are used to illustrate that even theoretically convergent schemes can produce numerical steady state solutions that do not correspond to steady state solutions of the boundary value problem. This phenomenon must be considered in any computational study of nonunique solutions to partial differential equations that govern physical systems such as fluid flows. In particular, numerical calculations have been used to “suggest” that certain Euler equations do not have a unique solution. For Burgers' equation on a finite spatial interval with Neumann boundary conditions the only steady state solutions are constant (in space) functions. Moreover, according to recent theoretical results, for any initial condition the corresponding solution to Burgers' equation must converge to a constant as t → ∞. However, we present a convergent finite difference scheme that produces false nonconstant numerical steady state “solutions.” These erroneous solutions arise out of the necessary finite floating point arithmetic inherent in every digital computer. We suggest the resulting numerical steady state solution may be viewed as a solution to a “nearby” boundary value problem with high sensitivity to changes in the boundary conditions. Finally, we close with some comments on the relevance of this paper to some recent “numerical based proofs” of the existence of nonunique solutions to Euler equations and to aerodynamic design.     

System dynamics and microworlds for policymakers

In the past ten years, system dynamics has become more accessible to policymakers and to the academic community. The paper reviews four major developments in the subject that have brought about this change. There have been improvements in the symbols and software used to map and model system structure. New ideas have been adopted from behavioural decision theory which help to transfer policymakers' knowledge into computer models. There have been improvements in methods of simulation analysis that enable modelers and model users to gain better insight into dynamic behaviour. Greater emphasis has been placed on small transparent models, on games and on dialogue between ‘mental models’ and computer simulations. Together these developments allow modelers to create computer-based learning environments (or microworlds) for policymakers to ‘play-with’ their knowledge of business and social systems and to debate policy and strategy change. The paper concludes with some thoughts on future research.     

Stochastic viscosity solutions for nonlinear stochastic partial differential equations. Part I

This paper, together with the accompanying work (Part II, Stochastic Process. Appl. 93 (2001) 205–228) is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential equations. We introduce a definition of stochastic viscosity solution in the spirit of its deterministic counterpart, with special consideration given to the stochastic integrals. We show that a stochastic PDE can be converted to a PDE with random coefficients via a Doss–Sussmann-type transformation, so that a stochastic viscosity solution can be defined in a “point-wise” manner. Using the recently developed theory on backward/backward doubly stochastic differential equations, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman–Kac formula. Some properties of the stochastic viscosity solution will also be studied in this paper. The uniqueness of the stochastic viscosity solution will be addressed separately in Part II where the relation between the stochastic viscosity solution and the ω-wise, “deterministic” viscosity solution to the PDE with random coefficients will be established.     

Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

The Fundamental Pro-groupoid of an Affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don’t remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π ₁(spec(R)), i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π ₁(spec(R)) in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of “commutative 2-ring” that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π ₁ for the corresponding “affine 2-schemes.” These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not “true” groups but only profinite groups, and one cannot hope to recover more: the “Tannakian” functor represented by the étale fundamental group of a scheme preserves finite products but not all products.     

Artificial intelligence: Debates about its use and abuse

This paper is concerned with the question, “Is what a stored-program digital computer does thinking-in the full human sense of the term?” Several current controversies are examined, including the meaning and usefulness of the Turing test to determine “intelligence.” The Lucas controversy of the early 1960s is taken up, dealing with the philosophical issues related to the man-versus-machine debate, and Dreyfus' ideas against Machine Intelligence are explored. Searle's ideas in opposition to the validity of the Turing test are described, as are various interpretations of the Chinese room thought-experiment and its relation to real “thought”. Weizenbaum's opposition to the “information-processing model of man” is also developed. The paper concludes with a comparison of the 19th-century debates over Darwinian Evolution and those in this century over Artificial Intelligence.     

Problems and methods with multiple objective functions

LetA be a set of feasible alternatives or decisions, and supposen different indices, measures, or objectives are associated with each possible decision ofA. How can a “best” feasible decision be made? What methods can be used or experimented with to reach some decision? The purpose of this paper is to attempt a synthesis of the main approaches to this problem which have been studied to date. Four different classes of approaches are distinguished: (1) aggregation of multiple objective functions into a unique function defining a complete preference order; (2) progressive definition of preference together with exploration of the feasible set; (3) definition of a partial order stronger than the product of then complete orders associated with then objective functions; and (4) maximum reduction of uncertainty and incomparability.     

A reconsideration of Legendre-Jacobi symbols

The classical definition of the Jacobi symbol (a:b) was badly conceived for negative values of b. Alternative useful definitions of (a:?1) are proposed here. This is an elaboration of a point in the article “Spinor genera of binary quadratic forms” in this issue.     

Defining a mathematical research school: the case of algebra at the University of Chicago, 1892–1945

Historians of science have long considered the concept of the “research school” as a potent analytical construct for understanding the development of the laboratory sciences. Unfortunately, their definitions fall short in the case of mathematics. Here, a definition of “mathematical research school” is proposed in the context of a case study of algebraic work associated with the University of Chicago's Department of Mathematics from the University's founding in 1892 through 1945.