Related papers: Small values of signed harmonic sums

Given $n$ length-$\ell$ strings $S =\{s_1, ..., s_n\}$ over a constant size alphabet $\Sigma$ together with parameters $d$ and $k$, the objective in the {\em Consensus String with Outliers} problem is to find a subset $S^*$ of $S$ of size…

This paper provides bounds for the number of terms, denoted by $f$, of a harmonic sum with the condition that it starts from any arbitrary unit fraction $\frac{1}{m}$, $m > 1$, until another unit fraction $\frac{1}{m+f-1}$ such that the sum…

Let $X$ be the constrained random walk on $\mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $\lambda$, $\mu_1$, $\mu_2$,...,$\mu_d$, where $\{e_1,e_2,..,e_d\}$ are the standard…

A closed string $u$ is either of length one or contains a border that occurs only as a prefix and as a suffix in $u$ and nowhere else within $u$. In this paper, we present fast $\mathcal{O}(n\log n)$ time algorithms to compute all…

Let $n$ and $t$ be positive integers with $t\geq 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted $\tau_k(n)$, is defined as follows:…

For any two partitions $\lambda$ and $\mu$ of a positive integer $N$, let $\chi_{\lambda}(\mu)$ be the value of the irreducible character of the symmetric group $S_{N}$ associated with $\lambda$, evaluated at the conjugacy class of elements…

We show that under the assumption of a 24-term version of Fermat's Last Theorem, there exists an absolute constant c > 0 such that if S is a set of n > n_0 positive integers satisfying |S.S| < n^(1+c), then the sumset S.S satisfies |S+S| >>…

For $A\subseteq\mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$ whose $A$-weighted sum is zero.…

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…

For any norms $N_1,\ldots,N_m$ on $\mathbb{R}^n$ and $N(x) := N_1(x)+\cdots+N_m(x)$, we show there is a sparsified norm $\tilde{N}(x) = w_1 N_1(x) + \cdots + w_m N_m(x)$ such that $|N(x) - \tilde{N}(x)| \leq \epsilon N(x)$ for all $x \in…

In 1948, Erd\"{o}s and Straus formulated a conjecture : for any positive integer $n>2$, there exist positive integers $n_1,n_2$ and $n_3$ such that…

We show that the distance in total variation between $(\mathrm{Tr}\ U, \frac{1}{\sqrt{2}}\mathrm{Tr}\ U^2, \cdots, \frac{1}{\sqrt{m}}\mathrm{Tr}\ U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic…

In this paper we consider a variation of the classical Tur\'{a}n-type extremal problems. Let $S$ be an $n$-term graphical sequence, and $\sigma(S)$ be the sum of the terms in $S$. Let $H$ be a graph. The problem is to determine the smallest…

The Ulam sequence is defined as $a_1 =1, a_2 = 2$ and $a_n$ being the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives $$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47,…

A natural variant of Lehmer's conjecture that the Ramanujan $\tau$-function never vanishes asks whether, for any given integer $\alpha$, there exist any $n \in \mathbb{Z}^+$ such that $\tau(n) = \alpha$. A series of recent papers excludes…

Let $j(z)$ be the modular $j$-invariant function. Let $\tau$ be an algebraic number in the complex upper half plane $\mathbb{H}$. It was proved by Schneider and Siegel that if $\tau$ is not a CM point, i.e.,…

Let $n > 2$, $\gamma > \frac{n-1}{n-2}$, and $\lambda \in \mathbb{R}$. We prove that if $M$ and $N$ are two smooth $n$-manifolds that admit a complete Riemannian metric satisfying \[ -\gamma\Delta + \mathrm{Ric} > \lambda, \] then the…

Koiran's real $\tau$-conjecture claims that the number of real zeros of a structured polynomial given as a sum of $m$ products of $k$ real sparse polynomials, each with at most $t$ monomials, is bounded by a polynomial in $m,k,t$. This…

We present an extension of the Prouhet-Tarry-Escott problem by demonstrating that signed sums of noninteger powers of consecutive integers can be made arbitrarily close to zero.

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|,…