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Many physical systems share the property of scale invariance. Most of them show ordinary power-law scaling, where quantities can be expressed as a leading power law times a scaling function which depends on scaling-invariant ratios of the…
Scaling describes how a given quantity $Y$ that characterizes a system varies with its size $P$. For most complex systems it is of the form $Y\sim P^\beta$ with a nontrivial value of the exponent $\beta$, usually determined by regression…
We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and…
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and…
Cities are often compared through scaling laws, usually expressed as power-law relations between population size and aggregate urban quantities related to infrastructure, socioeconomic activity, or environmental impacts. These laws are…
Different types of reasoning impose different structural demands on representational systems, yet no systematic account of these demands exists across psychology, AI, and philosophy of mind. I propose a framework identifying four structural…
We give a thoroughful explanation of the general properties of different, general scales, corresponding to different (all possible) mathematical functions f(x), we mention and analyse many examples. These observations and statements might…
The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value…
Cities are some of the most intricate and advanced creations of humanity. Most objects in cities are perfectly synchronised to coordinate activities such as jobs, education, transportation, entertainment, and waste management. Although each…
We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an…
We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…
The prevalence of many urban phenomena changes systematically with population size. We propose a theory that unifies models of economic complexity and cultural evolution to derive urban scaling. The theory accounts for the difference in…
Scaling laws are powerful summaries of the variations of urban attributes with city size. However, the validity of their universal meaning for cities is hampered by the observation that different scaling regimes can be encountered for the…
Classification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic,…
Consider complex semisimple Lie algebras of a given dimension specified by their structure constants. We describe a finite collection of rational functions in the structure constants that form a complete set of invariants: two sets of…
We provide a treatment of isomorphism within a set-theoretic formulation of dependent type theory. Type expressions are assigned their natural set-theoretic compositional meaning. Types are divided into small and large types --- sets and…
We introduce the notion of quantum duplicates of an (associative, unital) algebra, motivated by the problem of constructing toy-models for quantizations of certain configuration spaces in quantum mechanics. The proposed (algebraic) model…
We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and…
We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being…
Individual scores on common factors are required in some applied settings (e.g., business and marketing settings). Common factors are based on reflective indicators, but their scores cannot unambiguously be determined. Therefore, factor…