Related papers: On $k$-free numbers over Beatty sequences
Let $\Delta$ be simplicial complex and let $k[\Delta]$ denote the Stanley--Reisner ring corresponding to $\Delta$. Suppose that $k[\Delta]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity…
In this paper we express the difference of two complementary Beatty sequences, as the sum of two Beatty sequences closely related to them. In the process we introduce a new Algorithm that generalizes the well known Minimum Excluded…
We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or…
In this paper we prove parts of a conjecture of Herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded $k$-th syzygy module over the polynomial ring. If in addition the module is…
Let $\Delta$ be simplicial complex and let $k[\Delta]$ denote the Stanley--Reisner ring corresponding to $\Delta$. Suppose that $k[\Delta]$ has a pure free resolution. Then we describe the Betti numbers and the Hilbert--Samuel multiplicity…
This paper is devoted to establishing several new formulas relating Bernoulli and Stirling numbers of both kinds.
The paper derives an equation for the Cauchy transform of the solution of a free stochastic differential equation (SDE). This new equation is used to solve several particular examples of free SDEs.
In this paper we present many congruences for several Ap\'ery-like sequences.
In this note, we study the problem on the existence of non-cyclic free subgroups of the skew group algebra of a locally finite group over a field.
We introduce a model-complete theory which completely axiomatizes the structure $Z_{\alpha}=(Z, +, 0, 1, f)$ where $f : x \to \lfloor{\alpha} x \rfloor $ is a unary function with $\alpha$ a fixed transcendental number. When $\alpha$ is…
We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.
The object of observation in present paper is statistical independence of real sequences and its description as independence with re spect to certain class of densities.
In a recent paper, one of us posed three open problems concerning squarefree arithmetic progressions in infinite words. In this note we solve these problems and prove some additional results.
Let $k \ge 2$ and $\alpha_1, \beta_1, ..., \alpha_k, \beta_k$ be reals such that the $\alpha_i$'s are irrational and greater than 1. Suppose further that some ratio $\alpha_i/\alpha_j$ is irrational. We study the representations of an…
We provide an alternative proof that the finite rational linear combination of radicals, under certain constraint, are linearly independent over $\mathbb{Q}$.
In this paper we study ascending chain conditions in a free Baxter algebra by making use of explicit constructions of free Baxter algebras that were obtained recently. We investigate ascending chain conditions both for ideals and for Baxter…
Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that…
We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.
This paper describes a simple method for estimating lower bounds on the number of classes of equivalence for a special kind of integer sequences, called division sequences. The method is based on adding group structure to classes of…
The paper containes a classification of consistent free string models in physical dimensions and a brief discussion of recent results concerning relations between various models.