Related papers: First-order definability of Darmon points in numbe…

In this paper we study the existence and uniqueness of fixed points of a class of mappings defined on complete, (sequentially compact) cone metric spaces, without continuity conditions and depending on another function.

The integral of a function $f$ defined on a symmetric space $M \simeq G/K$ may be expressed in the form of a determinant (or Pfaffian), when $f$ is $K$-invariant and, in a certain sense, a tensor power of a positive function of a single…

We study first-order logic (FO) over the structure consisting of finite words over some alphabet $A$, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is…

Definite descriptions are first-order expressions that denote unique objects. In this paper, we propose a second-order counterpart, designed to refer to unique relations between objects. We investigate this notion within the framework of…

In 2016 Ananyan and Hochster proved Stillman's conjecture by showing the existence of a uniform upper bound on the length of an $R_\eta$-sequence containing fixed $n$ forms of degree at most $d$ in polynomial rings over a field. This result…

Let $K\subseteq \mathbb{R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\mathbb{R}_{\exp}$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is…

Over a number field $K$, a celebrated result of Silverman states that if $\varphi(z)\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit…

The decomposition of arbitrary unitary transformations into sequences of simpler, physically realizable operations is a foundational problem in quantum information science, quantum control, and linear optics. We establish a 1D Quantum Field…

First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…

The general theme of this note is illustrated by the following theorem: Theorem 1. Suppose $K$ is a compact set in the complex plane and 0 belongs to the boundary $\partial K$. Let ${\cal A}(K)$ denote the space of all functions $f$ on $K$…

A new density matrix and corresponding quantum kinetic equations are introduced for fermions undergoing coherent evolution either in time (coherent particle production) or in space (quantum reflection). A central element in our derivation…

Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a general framework which allows for partial generalizations of…

We rewrite simplicially the standard definitions of a complete first order theory, a model of it, and various characterisations of stability of a complete first order theory. In our reformulations the simplicial language replaces the…

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent…

Quantifier-elimination or model-completeness of the affine part of some classical first order theories are proved.

Turing machines and spin models share a notion of universality according to which some simulate all others. Is there a theory of universality that captures this notion? We set up a categorical framework for universality which includes as…

For a prime ideal $\mathfrak{P}$ of the ring of integers of a number field $K$, we give a general definition of $\mathfrak{P}$-adic continued fraction, which also includes classical definitions of continued fractions in the field of…

Let $\psi$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $\psi$ by $F$, provided $\psi$ is…

Let S be a subset of the unit disk, and let F(s) denote the class of completely multiplicative functions f such that f(p) is in S for all primes p. The authors' main concern is which numbers arise as mean-values of functions in F(s). More…

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection…