Related papers: Sharp double inequality for complete elliptic inte…

This note is concerned with some essential properties (optimal isoperimetry, first variation, and monotonicity formula) of the so-called $[0,1)\ni\gamma$-torsional rigidity $\mathcal{T}_{\gamma,\mathsf{g}}$ on a complete Riemannian…

We use complex contour integral techniques to study the entropy H and subentropy Q as functions of the elementary symmetric polynomials, revealing a series of striking properties. In particular for these variables, derivatives of -Q are…

In this note I prove the following property of Herglotz functions, which to my knowledge is new: For a Herglotz function $h(z)$ and a real number $r \in \mathbb R$ define a Herglotz function $g_r(z) = (r - h(z))^{-1}.$ Let $\mu_r^{(s)}$ be…

We describe algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for…

Let $I_{\nu}(x)$ and $K_{\nu}(x)$ be the first and second kind modified Bessel functions. It is shown that the nullclines of the Riccati equation satisfied by $x^{\alpha} \Phi_{i,\nu}(x)$, $i=1,2$, with…

We produce an estimate for the $K$-Bessel function $K_{r + i t}(y)$ with positive, real argument $y$ and of large complex order $r+it$ where $r$ is bounded and $t = y \sin \theta$ for a fixed parameter $0\leq \theta\leq \pi/2$ or $t= y…

The Kerr-Schild form provides a natural way of realizing the classical double copy that relates exact solutions in general relativity to exact solutions in gauge theory. In this paper, we examine the asymptotic structure of Kerr-Schild…

We compute the $2$-completed integral motivic homology, effective algebraic K-theory, and very effective hermitian K-theory of the geometric classifying space of the cyclic group of order two over algebraically closed fields, the real…

Gabcke proved a new integral expression for the auxiliary Riemann function \[\mathop{\mathcal R}(s)=2^{s/2}\pi^{s/2}e^{\pi i(s-1)/4}\int_{-\frac12\searrow\frac12} \frac{e^{-\pi i u^2/2+\pi i u}}{2i\cos\pi u}U(s-\tfrac12,\sqrt{2\pi}e^{\pi…

We investigate the elliptic integrable model introduced by Deguchi and Martin, which is an elliptic extension of the Perk-Schultz model. We introduce and study a class of partition functions of the elliptic model by using the…

We propose an algorithm that calculates isogenies between elliptic curves defined over an extension $K$ of $\mathbb{Q}_2$. It consists in efficiently solving with a logarithmic loss of $2$-adic precision the first order differential…

We consider the sums $S(k)=\sum_{n=0}^{\infty}\frac{(-1)^{nk}}{(2n+1)^k}$ and $\zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we…

This paper provides some explicit formulas related to addition theorems for elliptic integrals $\int_0^x dt/R(t)$, where $R(t)$ is the square root from a polynomial of degree 4. These integrals are related to complex elliptic genera and are…

We show that a master integrand basis exists for all planar, two-loop amplitudes in massless four-dimensional theories which is fully stratified by rigidity -- with each integrand being either pure and strictly polylogarithmic or (pure and)…

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where…

In 2021, J.~Agler and J.~E. McCarthy proposed a two-step programme toward the celebrated Krzy\.z conjecture. The first step is to prove an entropy conjecture for polynomials whose zeros all lie on the unit circle; the second is to establish…

The function $Q(x):=\sum_{n\ge 1} (1/n) \sin(x/n)$ was introduced by Hardy and Littlewood [10] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a…

Let $k\ge 2$ be an integer and let $A$ be a set of nonnegative integers. The representation function $R_{A,k}(n)$ for the set $A$ is the number of representations of a nonnegative integer $n$ as the sum of $k$ terms from $A$. Let $A(n)$…

We revisit the classical integrals introduced by Coxeter, not to recalculate their well-known exact values, but to use them as a tool to derive elliptic integral identities. By embedding Coxeter's first integral into a one-parameter family…

For all $n > k \ge 1$, we give formulas for the nullity $N(n,k)$ of the $n \times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals have all entries $1$ and whose remaining superdiagonals have all entries $0$. This is…