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In this article we introduce Variable exponent Fock spaces and study some of their basic properties such as the boundedness of evaluation functionals, density of polynomials, boundedness of a Bergman-type projection and duality.
We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure,…
In this paper we present the following two results: we give an explicit description of the space of orderings of the field Q(x) as an inverse limit of finite spaces of orderings and we provide a new, simple proof of the fact that the class…
Figuring out the physical rationale behind natural selection of quantum theory is one of the most acclaimed quests in quantum foundational research. This pursuit has inspired several axiomatic initiatives to derive mathematical formulation…
We provide, for any regular uncountable cardinal $\kappa$, a new argument for Pincus' result on the consistency of $\mathrm{ZF}$ with the higher dependent choice principle $\mathrm{DC}_{<\kappa}$ and the ordering principle in the presence…
Based on set theoretic ordering properties, a general formulation for constructing a pair of convertibility monotones, which are generalizations of distillable entanglement and entanglement cost, is presented. The new pair of monotones do…
We address some fundamental problems concerning the structure of idealistic equivalence relations. In particular, we show that, under analytic determinacy, there are continuum many idealistic analytic equivalence relations that are not…
Within the framework of Zermelo-Fraenkel set theory without the Axiom of Choice, we establish equivalents to the assertion "the union of a countable collection of finite sets is countable" in the context of metric spaces, probability…
We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent…
We consider expressions built up from binary relation names using the operators union, composition, and set difference. We show that it is undecidable to test whether a given such expression $e$ is finitely satisfiable, i.e., whether there…
We study selective and game-theoretic versions of properties like the ccc, weak Lindel\"ofness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal…
We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal…
We study partitions of Fra\"{\i}ss\'{e} limits of classes of finite relational structures where the partitions are encoded by infinite binary sequences which are random in the sense of Kolmogorov, Chaitin and Solomonoff. It is shown that…
We first introduce the notion of Hamiltonian structure for a partial difference equation. Then we construct some infinite quivers, and realize the discrete KdV equation, the Hirota-Miwa equation and its various reductions as the mutation…
We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain which can be compared wrt. equality. As the satisfiability problem for this logic is undecidable in general, in…
We obtain multiplicity results for a class of first-order superquadratic Hamiltonian systems and a class of indefinite superquadratic elliptic systems which lead to the study of strongly indefinite functionals. There is no assumption to the…
In this paper, we study weakly nonlinear boundary value problems on infinite intervals. For such problems, we provide criteria for the existence of solutions as well as a qualitative description of the behavior of solutions depending on a…
A preferential arrangement of a finite set is an ordered partition. Associated with each such ordered partition is a chain of subsets or blocks endowed with a linear order. The chain may be split into sections by the introduction of a…
In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the…
It is often claimed that analysis with infinitesimals requires more substantial use of the Axiom of Choice than traditional elementary analysis. The claim is based on the observation that the hyperreals entail the existence of nonprincipal…