Related papers: On two conjectures regarding generalized sequence …
We examine a generalization of the binomial distribution associated with a strictly increasing sequence of numbers and we prove its Poisson-like limit. Such generalizations might be found in quantum optics with imperfect detection. We…
In this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of…
We consider first-order logics of sequences ordered by the subsequence ordering, aka sequence embedding. We show that the \Sigma_2 theory is undecidable, answering a question left open by Kuske. Regarding fragments with a bounded number of…
We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of…
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…
A notion of rank developed previously by the author is used to describe two correspondences which classify small unitary representations of split real forms of $E_6$ and $E_7$. The case of small principal series is studied in detail.
In this note we present a method to bound gaps between primes via the divergence of the series of reciprocals of the prime numbers, a consequence of a version of the Bertrand's test for convergence of series of positive numbers and a…
This is an exposition of recent developments in the theory of bounded differences between primes. Readers are expected to be beginners of analytic number theory. The present text is a substantially improved and augmented version of the one…
In this article we study in depth the Dirichlet theorem, which states that if a, b are relative prime integers, the sequence p = an + b contains infinite prime numbers, we simplify and generalize this theorem, we enunciate some special…
We introduce the notion of composition series of triangulated categories, which generalizes full exceptional sequences. The lengths of composition series yield invariants for triangulated categories. We study composition series of derived…
Six conjectures on pairs of consecutive primes are listed in this paper, together with examples for each case.
We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.
In paper on a classification of Lehmer triples, Juricevic conjectured that there are infinitely many primes of special form. We disprove one of his conjectures and consider the other one.
In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $\delta$-discretized…
We investigate the decidability of the monadic second-order (MSO) theory of the structure $\langle \mathbb{N};<,P_1, \ldots,P_d \rangle$, for various unary predicates $P_1,\ldots,P_d \subseteq \mathbb{N}$. We focus in particular on…
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the…