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combinatorial structures and Lie algebras of upper triangular matrices

APPLIED MATHEMATICS LETTERS 2012年第3期25卷 514-519页

作者： Ceballos, M. Nunez, J. Tenorio, A. F. Univ Pablo Olavide Escuela Politecn Super Dpto Econ Metodos Cuantitativos & Ha Econ Seville 41013 Spain Univ Seville Fac Matemat Dept Geometria & Topol E-41080 Seville Spain

This work shows how to associate the Lie algebra h(n), of upper triangular matrices, with a specific combinatorial structure of dimension 2, for n is an element of N. The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras. (C) 2011 Elsevier Ltd. All rights reserved.

关键词： combinatorial structures Maximal abelian dimension Solvable Lie algebras Abelian subalgebras Faithful matrix representation

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Random combinatorial structures: the convergent case

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JOURNAL OF combinatorial THEORY SERIES A 2005年第2期109卷 203-220页

作者： Barbour, AD Granovsky, BL Angew Math CH-8057 Zurich Switzerland Technion Israel Inst Technol Dept Math IL-32000 Haifa Israel

This paper studies the distribution of the component spectrum of combinatorial structures such as uniform random forests, in which the classical generating function for the numbers of (irreducible) elements of the different sizes converges at the radius of convergence;here, this property is expressed in terms of the expectations of independent random variables Z(j), j >= 1, whose joint distribution, conditional on the event that Sigma(n)(j)(=1) jZ(j) = n, gives the distribution of the component spectrum for a random structure of size n. For a large class of such structures, we show that the component spectrum is asymptotically composed of Z(j) components of small sizes j, j >=, 1, with the remaining part, of size close to n, being made up of a single, giant component. (c) 2004 Elsevier Inc. All rights reserved.

关键词： combinatorial structures giant component conditioning relation coagulation-fragmentation

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Study of Lie algebras by using combinatorial structures

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LINEAR ALGEBRA AND ITS APPLICATIONS 2012年第2期436卷 349-363页

作者： Ceballos, Manuel Nunez, Juan Tenorio, Angel F. Univ Seville Fac Matemat Dept Geometria & Topol E-41080 Seville Spain Univ Pablo de Olavide Dpto Econ Metodos Cuantitat & Ha Econ Escuela Politecn Super Seville 41013 Spain

In this paper, we study the structure and properties of those n-dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of n vertices. Our main goal is to obtain criteria determining when a Lie algebra is associated with some of combinatorial structures considered in this paper, as well as to study the properties of those structures in order to use them as a tool for classifying the types of Lie algebras associated with them. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Lie algebras combinatorial structures Triangular configurations Complete graphs Digraphs

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Study of Lie algebras by using combinatorial structures

Study of Lie algebras by using combinatorial structures

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Conference on Applied Linear Algebra in Honor of Hans Schneider (ALA)

关键词： Lie algebras combinatorial structures Triangular configurations Complete graphs Digraphs

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New results and applications of superimposed codes (and related combinatorial structures) to the design of efficient group testing procedures

New results and applications of superimposed codes (and rela...

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IEEE International Symposium on Information Theory

作者： De Bonis, A Vaccaro, U Univ Salerno Dipartimento Informat & Applicaz I-84081 Baronissi SA Italy

ISBN: (纸本)0780382803

Let s be the number of unknown positive elements in a population of n members, 2 less than or equal to s < n. We aim at finding all the s positive elements by testing group of members of the population, under the constraint that a group tests positive if and only if it contains exactly one positive element. This model was considered in [1, 2]. We provide tight upper and lower bounds on the optimal number of tests needed to solve above group testing problem, improving on [6]. Instrumental to our results are new and improved bounds for generalized superimposed codes in the sense of [3, 5].

关键词： encoding group theory superimposed codes combinatorial structures group testing procedures upper-lower bounds

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Construction of some hypergroups from combinatorial structures

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浙江大学学报(英文版) 2003年第1期4卷 76-79页

作者： Ali Reza Ashrafi Ahmad Reza Eslami-Harandi Department of Mathematics Faculty of Science University of Kashan Kashan Iran

Jajcay's studies(1993;1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme(G), the stabilizer of the identity e∈G in the group Sym(G). We prove that (Syme(G) ,⊙) is a hypergroup and characterize the subhypergroups of this hypergroup. Finally, we show that the set of all subhypergroups of Syme(G) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut(G).

关键词： Finite group Rotary closed subgroup Hypergroup Sub-hypergroup combinatorial structures

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Construction of some hypergroups from combinatorial structures

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2003年第1期 77-80页

作者： Ali Reza Ashrafi,Ahmad RezaEslami-Harandi Department of Mathematics Faculty of Science University of Kashan Department of Mathematics Faculty of Science University of Kashan Kashan Iran Kashan Iran

Jajcay's studies(1993;1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Sym e(G) , the stabilizer of the identity e∈G in the group Sym(G) . We prove that (Sym e(G) ,⊙) is a hypergroup and characterize the subhypergroups of this hypergroup. Finally, we show that the set of all subhypergroups of Sym e(G) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut(G) .

关键词： Finite group Rotary closed subgroup Hypergroup Sub hypergroup combinatorial structures

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Construction of some hypergroups from combinatorial structures

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Journal of Zhejiang University Science 2003年第1期4卷 76-79页

作者： AliRezaAshrafi AhmadRezaEslami-Harandi DepartmentofMathematics FacultyofScienceUniversityofKashanKashanlran

Jajcay's studies( 1993 ; 1994) on the automorphism groups of Cayley maps yielded a new product of groups, which he called, rotary product. Using this product, we define a hyperoperation ⊙ on the group Syme (G) , the stabilizer of the identity e ∈ G in the group Sym (G) . We prove that ( Syme (G) , ⊙) is a hypergroup and characterize the subhypergroups of this ***, we show that the set of all subhypergroups of Syme ( G ) constitute a lattice under ordinary join and meet and that the minimal elements of order two of this lattice is a subgroup of Aut (G) .

关键词： Finite group Rotary closed subgroup Hypergroup Sub hypergroup combinatorial structures

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A Parametric Class of Mutually Unbiased Bases Using Resolvable Block Designs 25th

A Parametric Class of Mutually Unbiased Bases Using Resolvab...

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25th International Conference on Cryptology in India

作者： Kumar, Ajeet Kumar, Rakesh Maitra, Subhamoy Indian Stat Inst Appl Stat Unit 203 BT Rd Kolkata 700108 India

ISBN: (纸本)9783031803079;9783031803086

Mutually Unbiased Bases (MUBs) have important applications in several domains, particularly in Quantum Cryptography where the bases are used in designing the protocols related to Quantum Key Distribution (QKD). In this paper we consider parameterization of MUBs so that one can explore several classes of them for various applications. For dimension d = s(2), we present the construction of affine-parametric classes with MOLS(s) + 2 many MUBs, where MOLS(s) is the number of Mutually Orthogonal Latin Squares of dimension s. If s is a power of prime, then MOLS(s) = s - 1, and the number of MUBs will be s + 1. Considering the first one to be the identity matrix, in our construction, each of the rest MOLS(s) + 1 MUBs will have at least s(s - 1) free parameters, that cannot be absorbed by a global unitary operation. In comparison to Goyeneche et al.'s paper (2015), our result produces larger number of MUBs as well as free parameters in most of the cases. This can help in exploring various choices of MUBs in the protocols for higher dimensional QKDs and other applications of MUBs related to quantum information.

关键词： Quantum Cryptography Quantum Key Distribution combinatorial structures Mutually Unbiased Bases Parallel class Resolvable block design

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Communication and location discovery in geometric ring networks

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INFORMATION AND COMPUTATION 2019年 266卷 19-48页

作者： Gasieniec, Leszek Jurdzinski, Tomasz Martin, Russell Stachowiak, Grzegorz Univ Liverpool Dept Comp Sci Ashton BldgAshton St Liverpool L69 3BX Merseyside England Univ Wroclaw Inst Comp Sci Przesmyckiego 20 PL-50383 Wroclaw Poland

We study a distributed coordination mechanism for uniform agents located on a circle. The agents perform their actions in synchronized rounds. At the beginning of each round an agent chooses the direction of its movement from clockwise, anticlockwise, or idle, and moves at unit speed during this round. Agents are not allowed to overpass, i.e., when an agent collides with another it instantly starts moving with the same speed in the opposite direction (without exchanging any information with the other agent). However, at the end of each round each agent has access to limited information regarding its trajectory of movement during this round. We assume that n mobile agents are initially located on a circle unit circumference at arbitrary but distinct positions unknown to other agents. The agents are equipped with unique identifiers from a fixed range. The location discovery task to be performed by each agent is to determine the initial position of every other agent. Our main result states that, if the only available information about movement in a round is limited to distance between the initial and the final position, then there is a superlinear lower bound on time needed to solve the location discovery problem. Interestingly, this result corresponds to a combinatorial symmetry breaking problem, which might be of independent interest. If, on the other hand, an agent has access to the distance to its first collision with another agent in a round, we design an asymptotically efficient and close to optimal solution for the location discovery problem. Assuming that agents are anonymous (there are no IDs distinguishing them), our solution applied to randomly chosen IDs from appropriately chosen range gives an (almost) optimal algorithm, improving upon the complexity of previous randomized results. (C) 2018 Published by Elsevier Inc.

关键词： Mobile robots Distributed algorithms Location discovery Boundary patrolling combinatorial structures

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