“Further on in this paper we investigate the significance of the concept of continuity, concluding that it should not be retained as a fundamental characteristic of space in general and of spacetime(s) in particular.” (p. 1428) #Auffray #continuity #spacetime
“Paradoxien rufen Staunen, Verwirrung und die Lust am Außergewöhnlichen hervor. Aber nicht nur das: Es sind Paradoxien, die bis heute auf Grundprobleme der Philosophie, der Mathematik sowie der Naturwissenschaften hinweisen und uns zu revolutionären Lösungsvorschlägen herausfordern.
Einige Paradoxien markieren dabei vielleicht sogar unüberwindbare Grenzen unseres Wissens. Dieser Band stellt eine Reihe der wichtigsten Paradoxien – Paradoxien der Wahrheit, des Infiniten, der Bestätigung, der Vagheit, der Quantenmechanik, der Zeit, des Visuellen und des Auditiven – sowie Überlegungen zu allgemeinen Lösungswegen aus einer analytisch-philosophischen Perspektive vor. Dabei richtet er sich an interessierte Einsteiger in die Thematik, ohne den Gegenstand dabei zu sehr zu verkürzen.” #Bauer #Damschen #Siebel #Paradoxien
“We deal with atoms and bonds: a bond is a (perhaps directed) path between two atoms. An n-stepped walk is an ordered connected path along n consecutive bonds, each step being in the permitted direction of a bond (if the bond in question is directed) and starting from the atom reached by the previous step (if any). Wn(A) denotes a typical n-stepped walk starting from an atom A. A walk is self-avoiding if it visits no atom more than once. Sn(A) denotes a self-avoiding Wn(A). Two walks are distinct if they do not utilize the same set of bonds, with due regard to order. fA(n, r) denotes the number of distinct Wn(A), each of which can be broken into r or fewer self-avoiding subwalks. In particular we write fA(n) =fA(n, 1) for the number of distinct Sn(A). Two atoms A and B are outlike if fA(n) = fB{n) for all n. An outlike class is a class of pairwise outlike atoms. A crystal is an infinite set of atoms and bonds satisfying the three postulates” (p. 642) #Hammersley #atom #bond #walk #crystal
“The study of networks, in the form of mathematical graph theory, is one of the fundamental pillars of discrete mathematics.” (p. 2) #Newman #networks #GraphTheory #mathematics
“Suppose we immerse a large porous stone in a bucket of water. What is the probability that the centre of the stone is wetted? In formulating a simple stochastic model for such a situation, Broadbent and Hammersley (1957) gave birth to the ‚percolation model‘.” (p. 1) #Grimmett #Broadbent #Hammersley #percolation