Related papers: A construction of a $\lambda$- Poisson generic seq…
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski…
A strongly polynomial sequence of graphs $(G_n)$ is a sequence $(G_n)_{n\in\mathbb{N}}$ of finite graphs such that, for every graph $F$, the number of homomorphisms from $F$ to $G_n$ is a fixed polynomial function of $n$ (depending on $F$).…
We define a simple rule that allows to describe sequences of projective measurements for a broad class of generalized probabilistic models. This class embraces quantum mechanics and classical probability theory, but, for example, also the…
In this note the notion of Poisson brackets in Kontsevich's "Lie World" is developed. These brackets can be thought of as "universally" defined classical Poisson structures, namely formal expressions only involving the structure maps of a…
We consider sequences of generalized Bell numbers B(n), n=0,1,... for which there exist Dobinski-type summation formulas; that is, where B(n) is represented as an infinite sum over k of terms P(k)^n/D(k). These include the standard Bell…
We study growth rates of generalised Fibonacci sequences of a particular structure. These sequences are constructed from choosing two real numbers for the first two terms and always having the next term be either the sum or the difference…
We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is…
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order…
Out-of-distribution generalization capabilities of sequence-to-sequence models can be studied from the lens of two crucial forms of generalization: length generalization -- the ability to generalize to longer sequences than ones seen during…
We introduce the \emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={\mathcal P} (P, \, Q )$ satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional…
If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the…
Ordering identities in the Weyl-Heisenberg algebra generated by single-mode boson operators are investigated. A boson string composed of creation and annihilation operators can be expanded as a linear combination of other such strings, the…
A generalization of numeration system in which the set N of the natural numbers is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. Here we show that if P belonging to Q[x]…
The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches…
Given a finite group with a generating subset there is a well-established notion of length for a group element given in terms of its minimal length expression as a product of elements from the generating set. Recently, certain quantities…
We show the existence of and explicitly construct generic polynomials for various groups, over fields of positive characteristic. The methods we develop apply to a broad class of connected linear algebraic groups defined over finite fields…
Let $T_\lambda$ be a Galton--Watson tree with Poisson($\lambda$) offspring, and let $A$ be a tree property. In this paper, are concerned with the regularity of the function $\mathbb{P}_\lambda(A):= \mathbb{P}(T_\lambda \vdash A)$. We show…
Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of…
In the present paper we generate binary pseudorandom sequences using generalized polynomials. A generalized polynomial is a function in whose description we not only allow addition and product (as it is the case in usual polynomials) but…
For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…