Related papers: On the Counting Function of Elliptic Carmichael Nu…
In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of…
In this note, we connect the $n$-torsions of the Picard group of an elliptic surface to the $n$-divisibility of the class group of torsion fields for a given integer $n>1$. We also connect the $n$-divisibility of the Selmer group to that of…
We review certain classes of iterated integrals that appear in the computation of Feynman integrals that involve elliptic functions. These functions generalise the well-known class of multiple polylogarithms to elliptic curves and are…
We give some bounds on edge Folkman numbers.
Entropy numbers are an important tool for quantifying the compactness of operators. Besides establishing new upper bounds on the entropy numbers of diagonal operators $D_\sigma$ from $\ell_p$ to $\ell_q$, where $p\not=q$, we investigate the…
We develop new tools leading, for each integer $n\ge 4$, to a significantly improved upper bound for the uniform exponent of rational approximation $\widehat{\lambda}_n(\xi)$ to successive powers $1,\xi,\dots,\xi^n$ of a given real…
For a polarized complex Abelian variety A, of dimension g>1, we study the function N_A(t) counting the number of elliptic curves in A with degree bounded by t. We describe elliptic curves as solutions of Diophantine equations which, at…
Upper bounds on the maximum number of codewords in a binary code of a given length and minimum Hamming distance are considered. New bounds are derived by a combination of linear programming and counting arguments. Some of these bounds…
Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an…
Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_\Lambda f(n) := \sup_{\lambda \in \Lambda} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2\pi i \lambda p}}{p} |,\] where…
A numerical procedure providing guaranteed two-sided bounds on the effective coefficients of elliptic partial differential operators is presented. The upper bounds are obtained in a standard manner through the variational formulation of the…
Deng (arXiv:math/9812082) gave an asymptotic formula for the number of rational points on a weighted projective space over a number field with respect to a certain height function. We prove a generalization of Deng's result involving a…
The Jacobsthal function has aroused interest in various contexts in the past decades. We review several algorithmic ideas for the computation of Jacobsthal's function for primorial numbers and discuss their practicability regarding…
We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and…
We introduce an algorithm that can be used to compute the canonical height of a point on an elliptic curve over the rationals in quasi-linear time. As in most previous algorithms, we decompose the difference between the canonical and the…
We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term.…
In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x…
In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing the result in [BL] and the arguments in [DcFLVW], we present an elementary…
An upper estimate for the Lempert function of any $C^{1+\epsilon}$-smooth bounded domain in $\Bbb C^n$ is found in terms of the boundary distance.
The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $\pi(x)$ function and Riemann's hypothesis.