Search results for "combinatoric"
showing 10 items of 1776 documents
A note on Kakeya sets of horizontal and SL(2) lines
2022
We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal and conical lines in $\mathbb{R}^{3}$, combined with recent work on restricted families of projecti…
Le cône diamant symplectique
2009
Resume Si n + est le facteur nilpotent d'une algebre semi-simple g , le cone diamant de g est la description combinatoire d'une base d'un n + module indecomposable naturel. Cette notion a ete introduite par N.J. Wildberger pour sl ( 3 ) , le cone diamant de sl ( n ) est decrit dans Arnal (2006) [2] , celui des algebres semi-simples de rang 2 dans Agrebaoui (2008) [1] . Dans cet article, nous generalisons ces constructions au cas des algebres de Lie sp ( 2 n ) . Les tableaux de Young semi-standards symplectiques ont ete definis par C. De Concini (1979) [4] , ils forment une base de l'algebre de forme de sp ( 2 n ) . Nous introduisons ici la notion de tableaux de Young quasi standards symplec…
Complex group algebras of finite groups: Brauer's Problem 1
2007
Abstract Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m . We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.
Codimensions of algebras and growth functions
2008
Abstract Let A be an algebra over a field F of characteristic zero and let c n ( A ) , n = 1 , 2 , … , be its sequence of codimensions. We prove that if c n ( A ) is exponentially bounded, its exponential growth can be any real number >1. This is achieved by constructing, for any real number α > 1 , an F-algebra A α such that lim n → ∞ c n ( A α ) n exists and equals α. The methods are based on the representation theory of the symmetric group and on properties of infinite Sturmian and periodic words.
An application of neural networks to natural scene segmentation
2006
This paper introduces a method for low level image segmentation. Pixels of the image are classified corresponding to their chromatic features.
Coprime actions and correspondences of Brauer characters
2017
We prove several results giving substantial evidence in support of the conjectural existence of a Glauberman–Isaacs bijection for Brauer characters under a coprime action. We also discuss related bijections for the McKay conjecture.
Exhaustive generation for permutations avoiding (colored) regular sets of patterns
2019
Abstract Despite the fact that the field of pattern avoiding permutations has been skyrocketing over the last two decades, there are very few exhaustive generating algorithms for such classes of permutations. In this paper we introduce the notions of regular and colored regular set of forbidden patterns, which are particular cases of right-justified sets of forbidden patterns. We show the (colored) regularity of several sets of forbidden patterns (some of them involving variable length patterns) and we derive a general framework for the efficient generation of permutations avoiding them. The obtained generating algorithms are based on succession functions, a notion which is a byproduct of t…
Combinatorial Gray codes for classes of pattern avoiding permutations
2007
The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, Schr\"oder, Pell, even index Fibonacci numbers and the central binomial coefficients. Consequently, this provides Gray codes for $\s_n(\tau)$ for all $\tau\in \s_3$ and the obtained Gray codes have distances 4 and 5.
Combinatorics of generalized Bethe equations
2012
A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over \({\mathbb{Z}^M}\), and on the other hand, they count integer points in certain M-dimensional polytopes.
The Reconstruction of Polyominoes from Approximately Orthogonal Projections
2001
The reconstruction of discrete two-dimensional pictures from their projection is one of the central problems in the areas of medical diagnostics, computer-aided tomography, pattern recognition, image processing, and data compression. In this note, we determine the computational complexity of the problem of reconstruction of polyominoes from their approximately orthogonal projections. We will prove that it is NP-complete if we reconstruct polyominoes, horizontal convex polyominoes and vertical convex polyominoes. Moreover we will give the polynomial algorithm for the reconstruction of hv-convex polyominoes that has time complexity O(m3n3).