Search results for "Combinatorics"
showing 10 items of 1770 documents
On the ∗-cocharacter sequence of 3×3 matrices
2000
Abstract Let M 3 (F) be the algebra of 3×3 matrices with involution * over a field F of characteristic zero. We study the ∗ -polynomial identities of M 3 (F) , where ∗=t is the transpose involution, through the representation theory of the hyperoctahedral group B n . After decomposing the space of multilinear ∗ -polynomial identities of degree n under the B n -action, we determine which irreducible B n -modules appear with non-zero multiplicity. In symbols, we write the nth ∗ -cocharacter χ n (M 3 (F),*)=∑ r=0 n ∑ λ⊢r,h(λ)⩽6 μ⊢n−r,h(μ)⩽3 m λ,μ χ λ,μ , where λ and μ are partitions of r and n−r , respectively, χ λ,μ is the irreducible B n -character associated to the pair (λ,μ) and m λ,μ ⩾0 i…
Fixed points and completeness on partial metric spaces
2015
Recently, Suzuki [T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136 (2008), 1861-1869] proved a fixed point theorem that is a generalization of the Banach contraction principle and characterizes the metric completeness. Paesano and Vetro [D. Paesano and P. Vetro, Suzuki's type characterizations of completeness for partial metric spaces and fixed points for partially ordered metric spaces, Topology Appl., 159 (2012), 911-920] proved an analogous fixed point result for a selfmapping on a partial metric space that characterizes the partial metric 0-completeness. In this paper we prove a fixed point result for a new class of…
Ordinary and graded cocharacter of the Jordan algebra of 2x2 upper triangular matrices
2014
Abstract Let F be a field of characteristic zero and U J 2 ( F ) be the Jordan algebra of 2 × 2 upper triangular matrices over F . In this paper we give a complete description of the space of multilinear graded and ordinary identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . For every Z 2 -grading of U J 2 ( F ) we compute the multiplicities in the graded cocharacter sequence and furthermore we compute the ordinary cocharacter.
Fine and Wilf's Theorem for Three periods and a Generalization of Sturmian Words
1999
AbstractWe extend the theorem of Fine and Wilf to words having three periods. We then define the set 3-PER of words of maximal length for which such result does not apply. We prove that the set 3-PER and the sequences of complexity 2n + 1, introduced by Arnoux and Rauzy to generalize Sturmian words, have the same set of factors.
A note on the packing of two copies of some trees into their third power
2003
Abstract It is proved in [1] that if a tree T of order n is not a star, then there exists an edge-disjoint placement of two copies of this tree into its fourth power. In this paper, we prove the packing of some trees into their third power.
Graphs of stable maps from closed surfaces to the projective plane
2018
Abstract We describe how to attach a weighted graph to each stable map from closed surfaces to projective plane and prove that any weighted graph with non negatively weighted vertices is the graph of some stable map from a closed surface to the projective plane.
A Polynomial Quantum Query Lower Bound for the Set Equality Problem
2004
The set equality problem is to tell whether two sets A and B are equal or disjoint under the promise that one of these is the case. This problem is related to the Graph Isomorphism problem. It was an open problem to find any ω(1) query lower bound when sets A and B are given by quantum oracles. We will show that any error-bounded quantum query algorithm that solves the set equality problem must evaluate oracles \(\Omega(\sqrt[5]{\frac{n}{\ln n}})\) times, where n=|A|=|B|.
The Monadic Quantifier Alternation Hierarchy over Grids and Graphs
2002
AbstractThe monadic second-order quantifier alternation hierarchy over the class of finite graphs is shown to be strict. The proof is based on automata theoretic ideas and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. Considering grids where the width is a function of the height, we prove that the difference between the levels k+1 and k of the monadic hierarchy is witnessed by a set of grids where this function is (k+1)-fold exponential. We then transfer the hierarchy result to the class of directed (or undirected) graphs, using an encoding technique called strong reduction. It is notable that one can obtain sets of graphs which occur arbitrar…
Standard polynomials are characterized by their degree and exponent
2011
Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .
The surjective hull of a polynomial ideal
2016
The aim of this paper is the study of surjective ideals of homogeneous polynomials between Banach spaces. To do so we define the surjective hull of a polynomial ideal and prove the main properties of this hull procedure. For a more comprehensive theory, new lifting properties of homogeneous polynomials are proved and applied to the description of the surjective hulls of the ideals of I-bounded polynomials and of composition polynomials ideals. Several applications are provided.