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A model of computation is abstract if, when applied to any algebra, the resulting programs for computable functions and sets on that algebra are invariant under isomorphisms, and hence do not depend on a representation for the algebra.…
Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such…
We consider fixed-point models for topological phases of matter formulated as discrete path integrals in the language of tensor networks. Such zero-correlation length models with an exact notion of topological invariance are known in the…
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (\`a la Gol'dshtein-Troyanov) induces - under suitable locality assumptions - a first-order differential structure.
This chapter aims to provide a clear and understandable picture of constructive semigroups with apartness in Bishop's style of constructive mathematics, BISH. Our theory is partly inspired by the classical case, but it is distinguished from…
We formulate a definition of the existence property that works with "structural" set theories, in the mode of ETCS (the elementary theory of the category of sets). We show that a range of structural set theories, when formulated using…
In this paper, first we obtain some new and interesting results on projective modules and on the upper topology of an ordinal number. Then it is shown that the rank map of a locally of finite type projective module is continuous with…
Recent evidence indicates that the abundance of recurring elementary interaction patterns in complex networks, often called subgraphs or motifs, carry significant information about their function and overall organization. Yet, the…
The processes of constructing some graphs from others using binary operations of union with intersection (gluing) are studied. For graph classes closed with respect to gluing operations the elemental and operational bases are introduced.…
A family of locally equivalent models is considered. They can be taken as a generalization to $d+1$ dimensions of the Topological Massive and ``Self-dual'' models in 2+1 dimensions. The corresponding 3+1 models are analized in detail. It is…
We describe the local and global structure of the fixed locus for the action of a rational function on the Berkovich projective line over a complete nontrivially-valued algebraically closed nonarchimedean field. This includes a bound for…
Here we look at some geometric properties related to connectedness and topological dimension 0, especially in connection with norms on vector spaces over fields with absolute value functions, which may be non-archimedian.
A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies…
Hyperspaces form a powerful tool in some branches of mathematics: lots of fractal and other geometric objects can be viewed as fixed points of some functions in suitable hyperspaces - as well as interesting classes of formal languages in…
The topic of fixed points in digital metric spaces has drawn yet more publications with assertions that are incorrect, incorrectly proven, trivial, or incoherently stated. We discuss publications with bad assertions concerning fixed points…
Cantor's ordinal numbers, a powerful extension of the natural numbers, are a cornerstone of set theory. They can be used to reason about the termination of processes, prove the consistency of logical systems, and justify some of the core…
An example of constructive (in A.A.Markov's sense) real-valued function, which is integrable by Riemann, but is not integrable by Darboux, is constructed.
This paper explores the hard problem of consciousness from a different perspective. Instead of drawing distinctions between the physical and the mental, an exploration of a more foundational relationship is examined: the relationship…
We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in…
This paper is one in a series that investigates topological measures on locally compact spaces. A topological measure is a set function which is finitely additive on the collection of open and compact sets, inner regular on open sets, and…