A general theory of self-similarity

A little-known and highly economical characterization of the real interval [0,1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Δn may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations.This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.     

A general theory of convexity

A generalization of the usual notion of convexity is developed. It is shown that for certain applications some of the postulates of the generalized theory must be relaxed. The independence of the postulates is discussed and representation theorems are given. A topology which is compatible with the convexity structure is constructed.     

Formulation of hybrid Trefftz finite element method for elastoplasticity

The present investigation provides a hybrid Trefftz finite element approach for analysing elastoplastic problems. A dual variational functional is constructed and used to derive hybrid Trefftz finite element formulation for elastoplasticity of bulky solids. The formulation is applicable to either strain hardening or elastic-perfectly plastic materials. A solution algorithm based on initial stress formulation is introduced into the new element model. The performance of the proposed element model is assessed by three examples and comparison is made with results obtained by other approaches. The hybrid Trefftz finite element approach is demonstrated to be particularly suited for nonlinear analysis of two-dimensional elastoplastic problems.     

The Trefftz method using fundamental solutions for biharmonic equations

In this paper, the Trefftz method of fundamental solution (FS), called the method of fundamental solution (MFS), is used for biharmonic equations. The bounds of errors are derived for the MFS with Almansi’s fundamental solutions (denoted as the MAFS) in bounded simply connected domains. The exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions, respectively. The stability analysis of the MAFS is also made for circular domains. Numerical experiments are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions (MPS) is always superior to the MFS and the MAFS, based on numerical examples. However, if such singular particular solutions near the singular points do not exist, the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points. Based on the computed results, the MFS using the greedy adaptive techniques may provide more accurate solutions for singularity problems. Moreover, the numerical solutions by the MAFS with Almansi’s FS are slightly better in accuracy and stability than those by the traditional MFS. Hence, the MAFS with the AFS is recommended for biharmonic equations due to its simplicity.     

A general theory of almost factoriality

Let R be a commutative integral domain with 1. The non-zero elements a,b of R may be calledv-coprime if aRbR=abR. A Krull domain is calledalmost factorial if for all f,g in R there is nN such that fⁿRgⁿR is principal. From this it is easy to establish that if R is almost factorial then for all x in R there is nN such that xⁿ=p₁p₂...p_r where p_i are mutually v-coprime primary elements and that this expression is unique. In this article we drop the requirement that R be Krull and replace the primary elements by elements calledprime blocks and develope a theory of almost factoriality, a special case of which is the theory of almost factorial Krull domains.     

A general theory of hypersurface potentials

Summary A general theory of hypersurface potentials in n-dimensional space is proposed. Not only smooth densities but also potentials generated either by L ¹ functions or by measures are considered.     

Belief and probability: A general theory of probability cores

This paper considers varieties of probabilism capable of distilling paradox-free qualitative doxastic notions (e.g., full belief, expectation, and plain belief) from a notion of probability taken as a primitive. We show that core systems, collections of nested propositions expressible in the underlying algebra, can play a crucial role in these derivations. We demonstrate how the notion of a probability core can be naturally generalized to high probability, giving rise to what we call a high probability core, a notion that when formulated in terms of classical monadic probability coincides with the notion of stability proposed by Hannes Leitgeb [32]. Our work continues by one of us in collaboration with Rohit Parikh [7]. In turn, the latter work was inspired by the seminal work of Bas van Fraassen [46]. We argue that the adoption of dyadic probability as a primitive (as articulated by van Fraassen [46]) admits a smoother connection with the standard theory of probability cores as well as a better model in which to situate doxastic notions like full belief. We also illustrate how the basic structure underlying a system of cores naturally leads to alternative probabilistic acceptance rules, like the so-called ratio rule initially proposed by Isaac Levi [34].Core systems in their various guises are ubiquitous in many areas of formal epistemology (e.g., belief revision, the semantics of conditionals, modal logic, etc.). We argue that core systems can also play a natural and important role in Bayesian epistemology and decision theory. In fact, the final part of the article shows that probabilistic core systems are naturally derivable from basic decision-theoretic axioms which incorporate only qualitative aspects of core systems; that the qualitative aspects of core systems alone can be naturally integrated in the articulation of coherence of primitive conditional probability; and that the guiding idea behind the primary qualitative features of a core system gives rise to the formulation of lexicographic decision rules.     

A Trefftz/MFS mixed-type method to solve the Cauchy problem of the Laplace equation

By using the multiple-scale Trefftz method (MSTM) to solve the Cauchy problem of the Laplace equation in an arbitrary bounded domain, we may lose the accuracy several orders when the noise being imposed on the specified Cauchy data is quite large. In addition to the linear equations obtained from the MSTM, the fundamental solutions play as the test functions being inserted into a derived boundary integral equation. Therefore, after merely supplementing a few linear equations in the mixed-type method (MTM), which is a well organized combination of the Trefftz method and the method of fundamental solutions (MFS), we can improve the ill-conditioned behavior of the linear equations system and hence increase the accuracy of the solution for the Cauchy problem significantly, as explored by two numerical examples.     

A general theory of fuzzy plausibility measures

The purpose of this paper is to present a fuzzification of probability theory, or more precisely to give a fuzzification of plausibility measures first introduced by Shafer in 1976. Although plausibility measures include probability measures as well as possibility measures, it is a typical result of this theory that only a fuzzification of possibility measures is attainable, while a fuzzification of probability measures seems to be impossible. Moreover with regard to fuzzy plausibility measures we specify a concept of mean values and entropies, which can be considered as a direct generalization of the classical notions of mean value and entropy based upon probability measures.     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains

A highly accurate new solver is developed to deal with the Dirichlet problems for the 2D Laplace equation in the doubly connected domains. We introduce two circular artificial boundaries determined uniquely by the physical problem domain, and derive a Dirichlet to Dirichlet mapping on these two circles, which are exact boundary conditions described by the first kind Fredholm integral equations. As a direct result, we obtain a modified Trefftz method equipped with two characteristic length factors, ensuring that the new solver is stable because the condition number can be greatly reduced. Then, the collocation method is used to derive a linear equations system to determine the unknown coefficients. The new method possesses several advantages: mesh‐free, singularity‐free, non‐illposedness, semi‐analyticity of solution, efficiency, accuracy, and stability. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A general theory of measures for nonnegative matrices

By a measure μ on the set N of m × n nonnegative matrices we mean that μ is a function from N to the nonnegative reals such that (i) μ(λA)=λμ(A) for all nonnegative λ and all A ∈ N, and (ii) μ(A + B) ? μ(A) for all A,B ? N. This paper develops a theory of such measures and shows how this theory can be applied to particular problems.     

A general theory of some positive dependence notions

A general theory of concepts of positive dependence, which are weaker than association but stronger than orthant dependence, is developed. A random vector X is associated if and only if $\text{P(}\text{X}\text{∈ A ? B) ≥ P(}\text{X}\text{∈ A) P(}\text{X}\text{∈ B)}$ for all open upper sets A and B. By requiring the above inequality to hold only for some open upper sets A and B various notions of positive dependence which are weaker than association are obtained. First a general theory is given and then the results are specialized to some concepts of a particular interest. Various properties and interrelationships are derived and some applications are discussed.     

A general cone decomposition theory based on efficiency

LetK ₁ andK ₂ be two convex cones in some common vector space. This paper is concerned with the question of finding a good decomposition, with respect toK ₁ andK ₂, of a given element of the Minkowski sumK ₁ +K ₂. We propose the criterion of efficiency as a measure for the quality of a decomposition. This criterion allows us to set up a framework from which a general cone decomposition theory is then derived.Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), under project PS89-0058.     

A general Galois theory for cofunctions and corelations

Cofunctions and corelations are introduced and a corresponding Galois connection is studied in analogy to operations and relations in universal algebra. The Galois closed sets are characterized by local closures of clones of cofunctions and corelations, respectively. As an application concrete characterization problems are considered e.g. for strong bisimulations.     

Trefftz energy method for solving the Cauchy problem of the Laplace equation

The inverse Cauchy problem of Laplace equation is hard to solve numerically, since it is highly ill-posed in the Hadamard sense. With this in mind, we propose a natural regularization technique to overcome the difficulty. In the linear space of the Trefftz bases for solving the Laplace equation, we introduce a novel concept to construct the Trefftz energy bases used in the numerical solution for the inverse Cauchy problem of the Laplace equation in arbitrary star plane domain. The Trefftz energy bases not only satisfy the Laplace equation but also preserve the energy, whose performance is better than the original Trefftz bases. We test the new method by two numerical examples.