Related papers: The period length of Euler's number e

A theory of spline quadrature rules for arbitrary continuity class in a closed interval $[a, b]$ with arbitrary nonuniform subintervals based on semi-classical orthogonal Jacobi polynomials is proposed. For continuity class $c \ge 2$ this…

Let $E_k$ be the set of positive integers having exactly $k$ prime factors. We show that almost all intervals $[x,x+\log^{1+\varepsilon} x]$ contain $E_3$ numbers, and almost all intervals $[x,x+\log^{3.51} x]$ contain $E_2$ numbers. By…

A simple method called symbolic representation for piecewise linear functions on the real line is introduced and used to compute the numbers of periodic points of all periods for some such functions. Since, for every positive integer m, the…

We determine the maximal length of the period of a periodic word defined by $n$ restrictions. It happens to be the corresponding Fibonacci number.

Kosaraju in ``Computation of squares in a string'' briefly described a linear-time algorithm for computing the minimal squares starting at each position in a word. Using the same construction of suffix trees, we generalize his result and…

Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant \theta_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients…

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher…

We develop a simple method to obtain approximate analytical expressions for the period of a particle moving in a given potential. The method is inspired to the Linear Delta Expansion (LDE) and it is applied to a large class of potentials.…

Let $k\ge 0,a\ge 1$ and $b\ge 0$ be integers. We define the arithmetic function $g_{k,a,b}$ for any positive integer $n$ by $g_{k,a,b}(n):=\frac{(b+na)(b+(n+1)a)...(b+(n+k)a)} {{\rm lcm}(b+na,b+(n+1)a,...,b+(n+k)a)}.$ Letting $a=1$ and…

In this paper, we obtain analogues of Jacobi's derivative formula in terms of the theta constants with rational characteristics. For this purpose, we use the arithmetic formulas of the number of representations of a natural number…

We begin the systematic study of the spectral theory of periodic Jacobi matrices on trees including a formal definition. The most significant result that appears here for the first time is that these operators have no singular continuous…

The Fibonacci sequence defined by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ has a shortest period length of $4\cdot 3^{k-1}$ modulo $3^k$ for every $k\in\mathbb{N}$. In 2011, Bundschuh and Bundschuh \cite{bundschuh3} gave the frequencies…

The goal of this paper is to give a simple proof of the convergence to time-periodic states of the solutions of time-periodic Hamilton-Jacobi equations on the circle with convex Hamiltonian. Note that the period of the limiting solutions…

For any positive integer $r$, the $r$-Fubini number with parameter $n$, denoted by $F_{n,r}$, is equal to the number of ways that the elements of a set with $n+r$ elements can be weak ordered such that the $r$ least elements are in distinct…

We characterize the structure of the periods of a neuronal recurrence equation. Firstly, we give a characterization of k-chains in 0-1 periodic sequences. Secondly, we characterize the periods of all cycles of some neuronal recurrence…

We explore some connections between moments of rescaled little q-Jacobi polynomials, q-analogues of values at negative integers for some Dirichlet series, and the q-Eulerian polynomials of wreath products of symmetric groups.

For a periodic function $f$ with bounded variation and integral zero on its period interval, we show that $\sum_{k=1}^\infty c_k^2 (\log\log k)^\gamma <\infty$, $\gamma>4$ implies the almost everywhere convergence of $\sum_{k=1}^\infty c_k…

We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 \le j \le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard "sprinkling"…

Aim of this paper is to investigate the existence of periodic solutions of a nonlinear planar autonomous system having a limit cycle x_0 of least period T_0>0 when it is perturbed by a small parameter, T_1-periodic, perturbation. In the…

Given a polynomial $f\in\mathbb{C}[x]$, we consider the family of superelliptic curves $y^d=f(x)$ and their Jacobians $J_d$ for varying integers $d$. We show that for any integer $g$ the number of abelian varieties up to isogeny of…