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We study Pythagorean hyperplane arrangements, originally defined by Zaslavsky. In this first part of a series on such arrangements, we introduce a new notion of genericity for such arrangements. Using this notion we construct an auxiliary…
Similarities between entities occur frequently in many real-world scenarios. For over a century, researchers in different fields have proposed a range of approaches to measure the similarity between entities. More recently, inspired by…
We investigate various topological spaces and varieties which can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the…
General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not…
We characterize Morita equivalence of theories in the sense of Johnstone in terms of a new syntactic notion of a common definitional extension developed by Barrett and Halvorson for cartesian, regular, coherent, geometric and first-order…
Mixture models have been around for over 150 years, as an intuitively simple and practical tool for enriching the collection of probability distributions available for modelling data. In this chapter we describe the basic ideas of the…
A fundamental theme in automata theory is regular languages of words and trees, and their many equivalent definitions. Salvati has proposed a generalization to regular languages of simply typed $\lambda$-terms, defined using denotational…
Recently, J.~T.~Denniston, A.~Melton, and S.~E.~Rodabaugh introduced a lattice-valued analogue of the concept of institution of J.~A.~Goguen and R.~M.~Burstall, comparing it, moreover, with the (lattice-valued version of the) notion of…
We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $\mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show…
Tube formulas refer to the study of volumes of $r$ neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a…
Multisorted modules, equivalently representations of quivers, equivalently additive functors on preadditive categories, encompass a wide variety of additive structures. In addition, every module has a natural and useful multisorted…
This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…
Rational inference relations were introduced by Lehmann and Magidor as the ideal systems for drawing conclusions from a conditional base. However, there has been no simple characterization of these relations, other than its original…
This article presents a theory of modules with iterative connection. This theory is a generalisation of the theory of modules with connection in characteristic zero to modules over rings of arbitrary characteristic. We show that these…
Valuation algebras abstract a large number of formalisms for automated reasoning and enable the definition of generic inference procedures. Many of these formalisms provide some notions of solutions. Typical examples are satisfying…
It has been reliably shown that the similarity of word embeddings obtained from popular neural models such as BERT approximates effectively a form of semantic similarity of the meaning of those words. It is therefore natural to wonder if…
Le but cette note est de tenter d'expliquer les liens \'etroits qui unissent la th\'eorie des empilements de cercles et des modules combinatoires, et de comparer les approches \`a la conjecture de J.W. Cannon qui en d\'ecoulent. ???? The…
When we represent real-world systems as networks, the directions of links often convey valuable information. Finding module structures that respect link directions is one of the most important tasks for analyzing directed networks. Although…
Despite the success of language models using neural networks, it remains unclear to what extent neural models have the generalization ability to perform inferences. In this paper, we introduce a method for evaluating whether neural models…
A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of…