Partition d'un entier

Résumé

En mathématiques, une partition d'un entier (parfois aussi appelée partage d'un entier) est une décomposition de cet entier en une somme d'entiers strictement positifs (appelés parties ou sommants), à l'ordre près des termes (à la différence du problème de composition tenant compte de l'ordre des termes). Une telle partition est en général représentée par la suite des termes de la somme, rangés par ordre décroissant. Elle est visualisée à l'aide de son diagramme de Ferrers, qui met en évidence la notion de partition duale ou conjuguée. Pour un entier naturel fixé, l'ensemble de ses partitions est fini et muni d'un ordre lexicographique. La suite du nombre de partitions des entiers naturels successifs est déterminée par un algorithme récursif. Hardy et Ramanujan en ont donné un développement asymptotique en 1918, puis Hans Rademacher en a donné une formule exacte en 1937. Elle est répertoriée comme . vignette|droite|Diagrammes de Ferrers des partitions des entiers jusqu'à 8. Pré

Source officielle

https://fr.wikipedia.org/wiki/Partition_d'un_entier

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Fluctuation estimates for the multi-cell formula in stochastic homogenization of partitions

Matthias Ruf

In this paper we derive quantitative estimates in the context of stochastic homogenization for integral functionals defined on finite partitions, where the random surface integrand is assumed to be stationary. Requiring the integrand to satisfy in addition a multiscale functional inequality, we control quantitatively the fluctuations of the asymptotic cell formulas defining the homogenized surface integrand. As a byproduct we obtain a simplified cell formula where we replace cubes by almost flat hyperrectangles.

2022

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Forbidden induced subposets of given height

Istvan Tomon

Let P be a partially ordered set. The function La* (n, P) denotes the size of the largest family F subset of 2([n]) that does not contain an induced copy of P. It was proved by Methuku and Palvolgyi that there exists a constant C-P (depending only on P) such that La*(n,P) < C-P(left perpendicular n/2 right perpendicular n). However, the order of the constant C-P following from their proof is typically exponential in vertical bar P vertical bar. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer h there exists a constant c(h) such that if P has height at most h, then La* (n, P) = 2 vertical bar P vertical bar, then vertical bar F vertical bar

ACADEMIC PRESS INC ELSEVIER SCIENCE2019

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In this thesis we study a number of problems in Discrete Combinatorial Geometry in finite spaces. The contents in this thesis are structured as follows: In Chapter 1 we will state the main results and the notations which will be used throughout the thesis. Chapter 2 is a version of the paper entitled "Sumsets of the distance sets in finite spaces", which has been submitted for publication, (2017). Chapter 3 is a version of the paper entitled "Three-variable expanding polynomials and higher-dimensional distinct distances", which has been submitted for publication, co-authored with L. A. Vinh and de Zeeuw. The author was one of the main investigators of this chapter. Chapter 4 is a postprint version of the paper entitled "Distinct distances on regular varieties over finite fields", Journal of Number Theory, 173(2017), 602-613, co-authored with D. D. Hieu. The author was one of the main investigators of this chapter. Chapter 5 is a postprint version of the paper entitled " Incidences between points and generalized spheres over finite fields and related problems", Forum Mathematicum, Volume 29, Issue 2 (Mar 2017), co-authored with N. D. Phuong and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 6 is a version of the paper entitled "Distinct spreads in finite spaces", which has been submitted for publication, co-authored with B. Lund and L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 7 is a version of the paper entitled "Paths in pseudo-random graphs", which has been submitted for publication, co-authored with L. A. Vinh. The author was one of the main investigators of this chapter. Chapter 8 is a version of the paper entitled "Conditional expanding bounds for two-variable functions over arbitrary fields", which has been submitted for publication, co-authored with Hossein Nassajian Mojarrad. The author was one of the main investigators of this chapter. Chapter 9 is a postprint version of the paper entitled "A Szemeredi-Trotter type theorem, sum-product estimates in finite quasifields, and related results", Journal of Combinatorial Theory Series A, 147(2017), 55-74, co-authored with Michael Tait, Craig Timmons, Le Anh Vinh. The author was one of the main investigators of this chapter. The content of this chapter also appears in Michael Tait's Phd thesis. In Chapter 10, we will mention some open problems on Erd\H{o}s distinct distances problem and generalizations.

EPFL2017

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EPFL2017