Prime Ideal Theorems and systems of finite character

summary:\font\jeden=rsfs7 \font\dva=rsfs10 We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if \text{\jeden S} is a system of finite character then so is the system of all collections of finite subsets of \bigcup \text{\jeden S} meeting a common member of \text{\jeden S}), the Finite Cutset Lemma (a finitary version of the Teichm"uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma

Quark confinement: dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang-Mills theory

The purpose of this paper is to review the recent progress in understanding quark confinement. The emphasis of this review is placed on how to obtain a manifestly gauge-independent picture for quark confinement supporting the dual superconductivity in the Yang-Mills theory, which should be compared with the Abelian projection proposed by 't Hooft. The basic tools are novel reformulations of the Yang-Mills theory based on change of variables extending the decomposition of the $SU(N)$ Yang-Mills field due to Cho, Duan-Ge and Faddeev-Niemi, together with the combined use of extended versions of the Diakonov-Petrov version of the non-Abelian Stokes theorem for the $SU(N)$ Wilson loop operator. Moreover, we give the lattice gauge theoretical versions of the reformulation of the Yang-Mills theory which enables us to perform the numerical simulations on the lattice. In fact, we present some numerical evidences for supporting the dual superconductivity for quark confinement. The numerical simulations include the derivation of the linear potential for static interquark potential, i.e., non-vanishing string tension, in which the "Abelian" dominance and magnetic monopole dominance are established, confirmation of the dual Meissner effect by measuring the chromoelectric flux tube between quark-antiquark pair, the induced magnetic-monopole current, and the type of dual superconductivity, etc. In addition, we give a direct connection between the topological configuration of the Yang-Mills field such as instantons/merons and the magnetic monopole.Comment: 304 pages; 62 figures and 13 tables; a version published in Physics Reports, including corrections of errors in v

Some topics in set theory

This thesis is divided into two parts. In the first of these we consider Ackermann-type set theories and many of our results concern natural models. We prove a number of results about the existence of natural models of Ackermann's set theory, A, and applications of this work are shown to answer several questions raised by Reinhardt in [56]. A+ (introduced in [56]) is another Ackermann-type set theory and we show that its set theoretic part is precisely ZF. Then we introduce the notion of natural models of A + and show how our results on natural models of A extend to these models. There are a number of results about other Ackermann-type set theories and some of the work which was already known for ZF is extended to A. This includes permutation models, which are shown to answer another of Reinhardt's questions. In the second part we consider the different approaches to set theory; dealing mainly with the more philosophical aspects. We reconsider Cantor's work, suggest that it has frequently been misunderstood and indicate how quasi-constructive set theories seem to use a definite part of Cantor's earlier ideas. Other approaches to set theory are also considered and criticised. The section on NF includes some more technical observations on ordered pairs. There is also an appendix, in which we outline some results on extended ordinal arithmetic.<p

Issues in commonsense set theory

The success of set theory as a foundation for mathematics inspires its use in artificial intelligence, particularly in commonsense reasoning. In this survey, we briefly review classical set theory from an AI perspective, and then consider alternative set theories. Desirable properties of a possible commonsense set theory are investigated, treating different aspects like cumulative hierarchy, self-reference, cardinality, etc. Assorted examples from the ground-breaking research on the subject are also given. © 1995 Kluwer Academic Publishers

Carrizozo Outlook, 12-09-1921

https://digitalrepository.unm.edu/c_outlook_news/1302/thumbnail.jp

Influence of non-local diffusion in avascular tumour growth

The availability and evolution of chemical agents play an important role in the growth of a tumour and, therefore, the mathematical description of their consumption is of special interest. Usually, Fick’s law of diffusion is adopted for describing the local character of the evolution of chemicals. However, in a highly complex, heterogeneous medium, as is a tumour, the progression of chemical species could be influenced by non-local interactions. In this respect, our goal is to investigate the influence of such types of diffusion on the growth of a tumour in the avascular stage. For our purposes, we consider a diffusion equation for the evolution of the chemical agents that accounts for the existence of non-local interactions in a non-Fickean manner, and that involves notions of fractional calculus. In particular, the introduction of derivatives or integrals of fractional type of order α ∈ ℝ has proven to be an effective mathematical tool in the description of various non-local phenomena. To achieve our goals, we adopt part of the modelling assumptions outlined in previous works, in which the growth of a tumour is described in terms of mass transfer among the tumour’s constituents and structural changes that occur in the tumour itself in response to growth. The latter ones are characterised by means of the Bilby–Kröner–Lee decomposition of the deformation gradient tensor. We perform numerical simulations, whose results indicate the relevance of embracing a fractional framework in modelling tumour growth. Specifically, the real parameter α ‘dominates’ the way in which the tumour grows, since it permits the modelling of a variety of growth patterns ranging from the standard growth to no growth at all