相关论文: On rational definite summation
In this manuscript, we investigate some properties of certain counting functions, associated to the ergodic sums computed along the periodic orbits of the skew-product map, related to a finitely generated rational semigroup. To be precise,…
Given $r \in \mathbb{N}$, define the function $S_{r}: \mathbb{N} \rightarrow \mathbb{Q}$ by $S_{r}(n)=\displaystyle \sum_{k=0}^{n} \frac{k}{k+r} \binom{n}{k}$. In $2015$, the second author conjectured that there are infinitely many $r \in…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of $S$-unit groups of orders. The algorithm is then used for some…
The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to C_p be the p-adic polylogarithms defined by Coleman. Then a certain linear…
We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.
Zaremba's Conjecture concerns the formation of continued fractions with partial quotients restricted to a given alphabet. In order to answer the numerous questions that arrive from this conjecture, it is best to consider a semi-group, often…
We solve multiple conjectures by Byszewski and Ulas about the sum of base $b$ digits function. In order to do this, we develop general results about summations over the sum of digits function. As a corollary, we describe an unexpected new…
We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
The characteristic properties of Artin's denominators in Hilbert's 17th problem are obtained. It is proved that numerators of partial derivative of rational real function from the Nevanlinna class are SOS polynomials.
We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two row arrays of multisets. This generalization leads to new enumerative results that have representation theoretic interpretations as decompositions of centralizer…
In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we…
The aim of this paper is to prove wordlessly the sum formula of $1^{k}+2^{k}+\ldots +n^{k}$, $k\in\{1,2,3\}$.
We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable…
We show that each rational number $r$, $0\le r<1$, occurs as the fractional part of a Dedekind sum $S(m,n)$.
We consider Birkhoff sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let $S_N= S_N(\alpha,x)$ be the N^th non-renormalized Birkhoff sum, where $x in [0,1)$ is the initial point,…
Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…
In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an…
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…