Media theory: Representations and examples

In this paper we develop a representational approach to media theory. We construct representations of media by well-graded families of sets and partial cubes and establish the uniqueness of these representations. Two particular examples of media are also described in detail.     

Some examples in PI ring theory

Several examples are constructed, including a finite ring which cannot be embedded in matrices over any commutative ring, a semiprime PI ring with no classical ring of quotients, an example showing that the property of having all regular elements invertible is not inherited by matrix ringsM _n(R), and a prime PI ringR with an idempotente such thatR/ReR has finitely generated projective modules not induced by any finitely-generated projective R-module. Most of this work was done while the author was a guest of the University of Leeds’ Ring Theory Year (1972–1973), with the support of an Alfred P. Sloan Fellowship, and under the stimulating influence of Lance W. Small.     

Multiplier Hopf algebroids: Basic theory and examples

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discusses invertibility of the antipode, and presents some examples and special cases.     

Combinatorial examples in the theory of AF-algebras

We give examples of ramification graphs having combinatorial origin. These graphs determine the group of dimensions and traces of the corresponding AF-algebras.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akad. Nauk SSSR, Vol. 172, pp. 55–67, 1989.     

Some new examples in the theory of quadratic forms

We construct a 6-dimensional anisotropic quadratic form and a 4-dimensional quadratic form over some fieldF such that becomes isotropic over the function field but every proper subform of is still anisotropic over . It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form with trivial determinant such that the index of the Clifford invariant of is 4 but can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest. Received November 11, 1997; in final form June 24, 1999 / Published online May 8, 2000     

Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

Conservative parabolic problems: Nondegenerated theory and degenerated examples from population dynamics

We consider partial differential equations of drift‐diffusion type in the unit interval, supplemented by either 2 conservation laws or by a conservation law and a further boundary condition. We treat 2 different cases: (1) uniform parabolic problems and (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, brings new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either 1 or 2 absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit. Two prototypical problems from population dynamics are treated in detail. For these problems, we show that the structure of measure‐valued solutions is such that they are absolutely continuous in the interior. However, they will also include Dirac masses at the degenerated boundaries, which appear, irrespective of the regularity of the initial data, at time t=0⁺. The time evolution of these singular masses is also explicitly described and, as a by‐product, uniqueness of this measure solution is obtained.     

Definitions and examples in elementary calculus: the case of monotonicity of functions

This paper is devoted to the investigation of students’ understanding and handling of examples in the framework of an example-based introductory mathematics undergraduate course. The plan of the course included a wide use of graphs in standard lectures, tutoring sessions as well as in examinations. This study deals with the notion of increasing function, which has been introduced by means of both the standard definition and a range of examples and non-examples, most often conveyed through graphs. We have analysed students’ interpretations of the notion of increasing function as they applied them in a set of written examination tests. The data gathered have been completed by a number of interviews of students whose answers were difficult to interpret. The outcomes underline the importance of linguistic and semiotic competence and suggest that the design of innovative teaching paths should take care of the linguistic and semiotic skills needed to handle the representations involved.     

Construction of polynomials of a special kind and examples of their application in the theory of power series and in the wavelet theory

In this paper we construct polynomials of a special type. We consider examples of their application for studying the convergence of special power series, for determining the upper and lower Riesz bounds for a basis consisting of B-splines, and for studying the convergence of a sequence of Battle-Lemarié scaling functions.     

Competition and contracting in service industries

In service industries with congestion effects, two very different contractual structures are commonly observed, depending on whether or not firms choose to offer a guaranteed service level. We analyze the impact of these choices on market outcomes in oligopolistic industries. Our results highlight how different contractual agreements change the intensity of price competition in service industries. Broadly speaking, we show that competition is intensified when firms choose to offer service level guarantees.