In this Perspective article, the reader is introduced to several theoretical methods of determining exciton wave functions and corresponding eigenenergies. The covered are either analytical, semianalytical, or numeric. All details associated with different made explicit, thus allowing newcomers do research on their own, without experiencing a steep learning curve. starts variational method ends...

Suppose we are given a set M . E. Kolmogorov’s derivation of functions was a milestone in statistical measure theory. We show that Ψ̃ is homeomorphic to Rκ,K . The work in [5] did not consider the ultra-one-to-one, finitely Maclaurin, p-adic case. It is essential to consider that ι may be semi-simply Artinian.

The main theme of this volume is the efficient solution of families of stochastic or parametric partial differential equations. This article focuses on the theoretical underpinnings of such methods. It shows how concepts from approximation theory, such as entropy or widths, can help quantify a priori how well such methods can perform. This theory is then used to analyze one of the primary numer...

The purpose of this paper is to collect the homotopical methods used in the development of the theory of flows initialized by author’s paper “A model category for the homotopy theory of concurrency”. It is presented generalizations of the classical Whitehead theorem inverting weak homotopy equivalences between CW-complexes using weak factorization systems. It is also presented methods of calcul...

This is an expanded version of a series of four lectures designed to show algebraists how ring theoretical methods can be used to analyze an interesting family of finite Markov chains. The chains happen to be random walks on semigroups, and the analysis is based on a study of the associated semigroup algebras. The paper is divided into four sections, which correspond roughly to the four lecture...

The Boltzmann-Gibbs statistical mechanics and the standard thermodynamics are very useful in the discussion of extensive systems. However, they are not good to study situations where nonextensive e ects occur, for instance, systems involving long-range interactions (e. g., d = 3 gravitation)[1, 2], long-range microscopic memory[3], and systems with a relevant (multi)fractal-like structure. On t...

We examine a long-standing research program in empirical Political Science, fact-pattern analysis (FPA). We connect FPA to definitions of legal rules in jurisprudence and positive political theory. Foundationally, theoretical treatments view rules as functions partitioning case spaces into equivalence classes. Connecting FPA to formal theory has two advantages: first, many elements of the tradi...