A note on alternative digital topologies

AbstractThis paper contains a brief outline—in a form intended to clarify their interrelation—of four theories devised for the analysis of the topological attributes of a digital image. They are (i) Rosenfeld's combinatorial theory (to which he gave the name “digital topology”), (ii) the generalization of this due to Kong and Roscoe, (iii) Khalimsky, Kopperman and Meyer's theory of sets embedded in a locally finite space and (iv) the theory which results from choosing a specific model for the process by which the digital image is produced

Causal categories: relativistically interacting processes

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a `causal category'. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure

Metrization Theorem for Space-Times: From Urysohn’s Problem Towards Physically Useful Constructive Mathematics

To Yuri Gurevich, in honor of his enthusiastic longtime quest for efficiency and constructivity. Abstract. In the early 1920s, Pavel Urysohn proved his famous lemma (sometimes referred to as “first non-trivial result of point set topology”). Among other applications, this lemma was instrumental in proving that under reasonable conditions, every topological space can be metrized. A few years before that, in 1919, a complex mathematical theory was experimentally proven to be extremely useful in the description of real world phenomena: namely, during a solar eclipse, General Relativity theory – that uses pseudo-Riemann spaces to describe space-time – was (spectacularly) experimentally confirmed. Motivated by this success, Urysohn started working on an extension of his lemma and of the metrization theorem to (causality-)ordered topological spaces and corresponding pseudo-metrics. After Urysohn’s early death in 1924, this activity was continued in Russia by his student Vadim Efremovich, Efremovich’