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There exist several simple representations of uncertainty that are easier to handle than more general ones. Among them are random sets, possibility distributions, probability intervals, and more recently Ferson's p-boxes and Neumaier's…
Crisp and $L$-fuzzy ambiguous representations of closed subsets of one space by closed subsets of another space are introduced. It is shown that, for each pair of compact Hausdorff spaces, the set of (crisp or $L$-fuzzy) ambiguous…
In this paper we address the issue of existence of cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for $p$-adic groups known by the first…
Quasiprobability representations are well-established tools in quantum information science, with applications ranging from the classical simulability of quantum computation to quantum process tomography, quantum error correction, and…
We claim that human mathematics is only a limited part of the consequences of the chosen basic axioms. Properly human mathematics varies with time but appears to have universal features which we try to analyze. In particular the functioning…
This paper surveys basic properties of finite presentation in groups, Lie algebras and rings. It includes some new results and also new, more elementary proofs, of some results that are already in the literature. In particular, we discuss…
A model of knowledge representation is described in which propositional facts and the relationships among them can be supported by other facts. The set of knowledge which can be supported is called the set of cognitive units, each having…
We show that the spherical subalgebra of the rational Cherednik algebra associated to the wreath product of a symmetric group and a cyclic group is isomorphic to a quotient of the ring of invariant differential operators on a space of…
These are lecture notes of a minicourse given by the first author at the Summer School on Quantization at the University of Notre Dame in June 2011. The notes were written up and expanded by the second author who took the liberty of adding…
This note imparts heuristic arguments and theorectical evidences that contradict the abc conjecture over the rational numbers. In addition, the rudimentary datails for transforming this problem into the doimain of equidistribution theory…
Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We…
This article is intended as a compendium and guide to the variety of Bell Inequality derivations that have appeared in the literature in recent years, classifying them into six broad categories, revealing the underlying, often hidden,…
The theory of modular forms and spherical harmonic analysis are applied to establish new best bounds towards the counting and equidistribution of rational points on spheres and other higher dimensional ellipsoids, in what may be viewed as a…
We find the $\ell$-weights and the $\ell$-weight vectors for the highest $\ell$-weight $q$-oscillator representations of the positive Borel subalgebra of the quantum loop algebra $U_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values…
A brief review of a superanalysis over real and $p$-adic superspaces is presented. Adelic superspace is introduced and an adelic superanalysis, which contains real and $p$-adic superanalysis, is initiated.
A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is…
We introduce and study semi-equational and weakly semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong…
Non notherian Formal schemes of perfectoid type (for example $\mathbb{Z}_p[p^{1/p^\infty}]\langle X^{1/p^\infty} \rangle$ along with its multivariate version) with rational degree are constructed and are shown to be admissible. These formal…
Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted…
There has been considerable recent interest in explainability in AI, especially with black-box machine learning models. As correctly observed by the planning community, when the application at hand is not a single-shot decision or…