Search results for "Combinatorics"

Some characterizations of algebras with involution with polynomial growth of their codimensions

2018

Let A be an associative algebra endowed with an involution ∗ of the first kind and let c ∗n (A) denote the sequence of ∗-codimensions of A. In this paper, we are interested in algebras with involution such that the ∗-codimension sequence is polynomially bounded. We shall prove that A is of this kind if and only if it satisfies the same identities of a finite direct sum of finite dimensional algebras with involution A i , each of which with Jacobson radical of codimension less than or equal to one in A i . We shall also relate the condition of having polynomial codimension growth with the sequence of cocharacters and with the sequence of colengths. Along the way, we shall show that the multi…

Normalizing biproportional methods

2002

International audience; Biproportional methods are used to update matrices: the projection of a matrix Z to give it the column and row sums of another matrix is R Z S, where R and S are diagonal and secure the constraints of the problem (R and S have no signification at all because they are not identified). However, normalizing R or S generates important mathematical difficulties: it amounts to put constraints on Lagrange multipliers, non negativity (and so the existence of the solution) is not guaranteed at equilibrium or along the path to equilibrium.

Khovanov–Rozansky homology for embedded graphs

2011

Vertical Representation of C∞-words

2015

International audience; We present a new framework for dealing with C∞-words, based on their left and right frontiers. Thisallows us to give a compact representation of them, and to describe the set of C∞-words throughan infinite directed acyclic graph G. This graph is defined by a map acting on the frontiers ofC∞-words. We show that this map can be defined recursively and with no explicit reference toC∞-words. We then show that some important conjectures on C∞-words follow from analogousstatements on the structure of the graph G.

Some Remarks on Differentiable Sequences and Recursivity

2010

International audience; We investigate the recursive structure of differentiable sequences over the alphabet {1, 2}. We derive a recursive formula for the (n + 1)-th symbol of a differentiable sequence, which yields to a new recursive formula for the Kolakoski sequence. Finally, we show that the sequence of absolute differences of consecutive symbols of a differentiable sequence u is a morphic image of the run-length encoding of u.

Trace cocharacters and the Kronecker products of Schur functions

2003

Abstract It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra Mr(F) of r×r matrices over a field F of characteristic zero are given by TCr,n=∑λ∈Λr(n)χλ⊗χλ where χλ⊗χλ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λr(n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ=(λ1⩽⋯⩽λk) with k⩽r. We study the behavior of the sequence of trace cocharacters TCr,n. In particular, we study the behavior of the coefficient of χ(ν,n−m) in TCr,n as a function of n where ν=(ν1⩽⋯⩽νk) i…

Lattices of Jordan algebras

2010

AbstractCommutative Jordan algebras play a central part in orthogonal models. The generations of these algebras is studied and applied in deriving lattices of such algebras. These lattices constitute the natural framework for deriving new orthogonal models through factor aggregation and disaggregation.

Quantitative uniqueness estimates for pp-Laplace type equations in the plane

2016

Abstract In this article our main concern is to prove the quantitative unique estimates for the p -Laplace equation, 1 p ∞ , with a locally Lipschitz drift in the plane. To be more precise, let u ∈ W l o c 1 , p ( R 2 ) be a nontrivial weak solution to div ( | ∇ u | p − 2 ∇ u ) + W ⋅ ( | ∇ u | p − 2 ∇ u ) = 0  in  R 2 , where W is a locally Lipschitz real vector satisfying ‖ W ‖ L q ( R 2 ) ≤ M for q ≥ max { p , 2 } . Assume that u satisfies certain a priori assumption at 0. For q > max { p , 2 } or q = p > 2 , if ‖ u ‖ L ∞ ( R 2 ) ≤ C 0 , then u satisfies the following asymptotic estimates at R ≫ 1 inf | z 0 | = R sup | z − z 0 | 1 | u ( z ) | ≥ e − C R 1 − 2 q log R , where C > 0 depends …

The doodle of a finitely determined map germ from R2 to R3

2009

Let f:U⊂R2→R3 be a representative of a finitely determined map germ f:(R2,0)→(R3,0). Consider the curve obtained as the intersection of the image of the mapping f with a sufficiently small sphere Sϵ2 centered at the origin in R3, call this curve the associated doodle of the map germ f. For a large class of map germs the associated doodle has many transversal self-intersections. The topological classification of such map germs is considered from the point of view of the associated doodles.

Problemas históricos y dificultades de los estudiantes en la conceptualización de sustancia compuesto químico

2008

Este trabajo ofrece un análisis histórico sobre los problemas que tuvo que resolver la ciencia hasta llegar a la construcción de los conceptos macroscópicos de sustancia y compuesto químico en el contexto de la teoría daltoniana. Por otra parte, en él se muestran algunas de las dificultades de comprensión que estos conceptos ofrecen a los estudiantes. Para determinarlas, se realiza un estudio transversal con alumnos de 15 a 18 años, lo que permite evaluar el significado que otorgan a la idea de sustancia, al tiempo que se constata la necesidad de su comprensión para poder entender los cambios químicos. Por último, se plantea la existencia de ciertas semejanzas entre las ideas sobre la compo…