Reducing belief simpliciter to degrees of belief

Is it possible to give an explicit definition of belief (simpliciter) in terms of subjective probability, such that believed propositions are guaranteed to have a sufficiently high probability, and yet it is neither the case that belief is stripped of any of its usual logical properties, nor is it the case that believed propositions are bound to have probability 1? We prove the answer is ‘yes’, and that given some plausible logical postulates on belief that involve a contextual “cautiousness” threshold, there is but one way of determining the extension of the concept of belief that does the job. The qualitative concept of belief is not to be eliminated from scientific or philosophical discourse, rather, by reducing qualitative belief to assignments of resiliently high degrees of belief and a “cautiousness” threshold, qualitative and quantitative belief turn out to be governed by one unified theory that offers the prospects of a huge range of applications. Within that theory, logic and probability theory are not opposed to each other but go hand in hand.     

ESSENTIALLY ALGEBRAIC CATEGORIES

Abstract

Various familiar concepts of algebraic categories (such as primitive and quasiprimitive categories of algebras as well as varietal, monadic, regular monadic, and regular categories) are contrasted with each other and with the new notion of essentially algebraic categories.

The new notion has pleasant features and covers several cases of an apparently algebraic nature, which are not covered by any of the older concepts.     

On two epireflective subcategories of the category of 2T 0-fuzzy bitopological spaces

In the space of absolutely summable sequences the total variation of a sequence becomes a natural norm, and is a measure of smoothness. This norm is preserved by the so-called LU LU-smoothers, in that the variation of the image plus the variation of the difference is equal to the variation of the original sequence. This surprising property is useful for strategies for the removal of impulsive and other random noise. Strong shape preserving properties of these smoothers are derived in the process of proving this. The results shed light on similar properties of the popular median smoothers, as well as their sometimes enigmatic behaviour.     

CHARACTERISATIONS OF OPERATORS OF LOWER SEMI-FREDHOLM TYPE IN NORMED LINEAR SPACES

Abstract

Let X and Y be normed linear spaces. A linear operator T: D(T) ? X → Y is called an F-operator if its adjoint T′: D(T) ? Y′ → D(T)' is a φ+ -operator, i.e. has closed range and finite dimensional-kernel. Characterisations of an F_-operator T are obtained in the general case and in the case when T is closable. Unbounded strictly cosingular operators are defined and shown to belong to the class of F_ -admissible pertubations whenever Y is complete.     

Graph Pairs and their Entropies: Modularity Problems

Received June 24, 1997     

Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations

Summary. The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this. Received May 25, 1998 / Revised version received August 31, 1999 / Published online July 12, 2000