相关论文: A remark on perturbations of sine and cosine sums

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this…

Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \geq 1$, a symmetrically constrained composition $\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

Given lacunary sequence of integers, $n_k$, $n_{k+1}/n_k>\lambda>1$, we define a new sequence $\{m_k\}$ formed by all possible $l$-wise sums $\pm n_{k_1}\pm n_{k_2}\pm \ldots\pm n_{k_l}$. We prove if $\lambda>\lambda_l$, then any series…

We study the instabilities of a harmonic oscillator subject to additive and dichotomous multiplicative noise, focussing on the dependance of the instability threshold on the mass. For multiplicative noise in the damping, the instability…

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such…

It is shown that different ways of interacting strings formed in high energy nucleus-nucleus collisions cause a different strength of the chaoticity parameter lambda of Bose-Einstein correlations. In particular, in the case of percolation…

Let $\left\{a_1, \dots, a_n\right\} \subset \mathbb{N}$ be a set of positive integers, $a_n$ denoting the largest element, so that for any two of the $2^n$ subsets the sum of all elements is distinct. Erd\H{o}s asked whether this implies…

We consider Cantor measures on the line, with contraction factor $N^{-1}=p^{-\alpha}$ (where $p$ a positive prime, $\alpha$ a positive integer) and $m$ positive integer digits lying in distinct residue classes modulo $N$. We obtain a…

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j]…

The discrete multiscale analysis for boundary value problems in nonlinear discrete systems leads to a first order discrete modulational instability above a threshold amplitude for wave numbers beyond the zero of group velocity dispersion.…

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct…

A length $n$ cosine sum is an expression of the form $\cos a_1\theta + \cdots + \cos a_n\theta$ where $a_1 < \cdots < a_n$ are positive integers, and a length $n$ Newman polynomial is an expression of the form $z^{a_1} + \cdots + z^{a_n}$…

Let $F_n$ be an $n$ by $n$ symmetric matrix whose entries are bounded by $n^{\gamma}$ for some $\gamma>0$. Consider a randomly perturbed matrix $M_n=F_n+X_n$, where $X_n$ is a random symmetric matrix whose upper diagonal entries $x_{ij}$…

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations.…

We present results on the stability of quantum systems consisting of a negative charge $-q_1$ with mass $m_{1}$ and two positive charges $q_2$ and $q_3$, with masses $m_{2}$ and $m_{3}$, respectively. We show that, for given masses $m_{i}$,…

Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are…

Let $\Lambda$ be the von Mangoldt function and $r_{Q}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}\Lambda\left(m_{1}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares (that we will…

We derive the ultimate bounds on the performance of nonlinear measurement schemes in the presence of noise. In particular, we investigate the precision of the second-order estimation scheme in the presence of the two most detrimental types…

We consider the asymptotics of sums of the form $$ \frac1{F_n^\sigma} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(\pi m/F_n)}\right|^\sigma} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(\pi F_{n-1}m/F_n)}\right|^\sigma} $$ where $(F_n)_{n \in…