Related papers: On a conjecture by Boyd
This is the translation of Leonhard Euler's paper "De Seriebus divergentibus" written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-\text{etc.}$ and uses…
An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the…
For any $m,n\in\mathbb{N}=\{0,1,2\ldots\}$, the truncated hypergeometric series ${}_{m+1}F_m$ is defined by $$ {}_{m+1}F_m\bigg[\begin{matrix}x_0&x_1&\ldots&x_m\\…
Recently, the higher order Tur\'{a}n inequalities for the Boros-Moll sequences $\{d_\ell(m)\}_{\ell=0}^m$ were obtained by Guo. In this paper, we show a different approach to this result. Our proof is based on a criterion derived by Hou and…
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed.…
Baader, J\"org, and Parlier recently established an upper bound for the crossing number of curve systems of size $m\asymp g^{1+\alpha}$ on a genus $g$ surface, obtaining a leading coefficient of $9/4=2.25$. Their construction relies on…
For a partition $\nu$, let $\lambda,\mu\subseteq \nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. Butler conjectured that the divided difference…
We prove some results conjectured by Zhi-Hong Sun regarding the value $\mod p$ of $\varepsilon_d^{\frac{p-1}4}$, where $\varepsilon_d$ is a unit of norm $-1$ in some fields $\mathbb Q(\sqrt d)$, with $\left(\frac{-1}p\right) =\left(\frac…
We prove that at least $\left( \dfrac{(1+\epsilon)2m}{N-1}+1+\epsilon \right)^N$, where $0\leqslant \epsilon <1$, many general points, satisfy Demailly's conjecture. Previously, it was known to be true for at least $(2m+2)^N$ many general…
We prove log-concavity of the lengths of the top rows of Young diagrams under Poissonized Plancherel measure. This is the first known positive result towards a 2008 conjecture of Chen that the length of the top row of a Young diagram under…
The Mahler measure of a function on the real d-torus is its geometric mean over the torus. It appears in number theory, ergodic theory and other fields. The Fuglede-Kadison determinant is defined in the context of von Neumann algebra theory…
A structured approach for the Collatz conjecture is presented using just the odd integers that are, in turn, divided into categories based on the roles they play such as Starter, Intermediary and Terminal. The expression 4x+1 is used as a…
Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old…
Pippenger ([Pippenger, 1977]) showed the existence of $(6m,4m,3m,6)$-concentrator for each positive integer $m$ using a probabilistic method. We generalize his approach and prove existence of $(6m,4m,3m,5.05)$-concentrator (which is no…
It has been long congectured that the crossing number of $C_m\times C_n$ is $(m-2)n$ for $2<m<=n$. In this paper we proved that conjecture is true for all but finitely many $n$ for each $m$. More specifically we proved conjecture for…
In this paper, the recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original.…
The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler…
In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication…
An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain "strange" symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and…
We give a new and simple proof of a theorem of Garza estimating the height (or Mahler measure) of an algebraic number with real conjugates.