scholarly journals New formulations for the Kissing Number Problem

2007 ◽  
Vol 155 (14) ◽  
pp. 1837-1841 ◽  
Author(s):  
Sergei Kucherenko ◽  
Pietro Belotti ◽  
Leo Liberti ◽  
Nelson Maculan
Keyword(s):  
Kissing Number ◽  
Number Problem

scholarly journals A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

2018 ◽  
Vol 28 (1) ◽  
pp. 100-129 ◽  
Author(s):  
JIANGE LI ◽  
MOKSHAY MADIMAN
Keyword(s):  
Hilbert Spaces ◽  
Vector Spaces ◽  
Combinatorial Approach ◽  
Kissing Number ◽  
Small Ball ◽  
Small Ball Probabilities ◽  
Packing Problems ◽  
Number Problem ◽  
Probabilistic Inequalities ◽  
Independent Identically Distributed

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

The Kissing Number Problem: A New Result from Global Optimization

2004 ◽  
Vol 17 ◽  
pp. 203-207 ◽  
Author(s):  
Leo Liberti ◽  
Nelson Maculan ◽  
Sergei Kucherenko
Keyword(s):  
Global Optimization ◽  
Kissing Number ◽  
Number Problem

Cryptanalysis of elliptic curve hidden number problem from PKC 2017

2019 ◽  
Vol 88 (2) ◽  
pp. 341-361
Author(s):  
Jun Xu ◽  
Lei Hu ◽  
Santanu Sarkar
Keyword(s):  
Elliptic Curve ◽  
Number Problem ◽  
Hidden Number Problem

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Target Space ◽  
Large Set ◽  
Hecke Eigenvalues ◽  
Artin's Conjecture ◽  
Number Problem ◽  
Novel Method ◽  
Hecke Modules ◽  
Primitive Roots

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

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