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## Determination of (0,2)-regular sets in graphs

Pacheco, Maria F.; Cardoso, Domingos Moreira; Luz, Carlos J.
Fonte: Universidade de Aveiro Publicador: Universidade de Aveiro
Tipo: Conferência ou Objeto de Conferência
ENG
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An eigenvalue of a graph is main iff its associated eigenspace is not orthogonal to the all-one vector j. The main characteristic polynomial of a graph G with p main distinct eigenvalues is

## Validação cruzada com correção de autovalores e regressão isotônica nos modelos de efeitos principais aditivos e interação multiplicativa; Cross-validation with eigenvalue correction and isotonic regression in the additive main effect and multiplicative interaction model

PIOVESAN, Pamela; ARAÚJO, Lucio Borges de; DIAS, Carlos Tadeu dos Santos
Fonte: Universidade Federal de Santa Maria Publicador: Universidade Federal de Santa Maria
Tipo: Artigo de Revista Científica
POR
Relevância na Pesquisa
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Neste trabalho é apresentada a aplicação dos modelos AMMI com o propósito de analisar a interação entre genótipo e ambiente em experimentos agronômicos multiambientais com dados balanceados. São apresentados dois métodos de validação cruzada e o aperfeiçoamento desses métodos por meio da correção de autovalores, sendo estes ordenados por meio da regressão isotônica. É realizado um estudo comparativo entre esses métodos por meio de dados reais. Os resultados mostram para esse conjunto de dados que o método de EASTMENT & KRZANOWSKI (1982) seleciona um modelo mais parcimonioso. Além disso, quando esse método é aperfeiçoado com a correção dos autovalores, o número de componentes não se altera. O método de GABRIEL (2002) seleciona um maior número de termos no modelo, e, quando se aplica a correção de autovalores, o número de termos diminui. O aperfeiçoamento desses métodos por meio da correção de autovalores traz um grande benefício para o pesquisador do ponto de vista prático, uma vez que a seleção do número de termos multiplicativos representa um ganho do número de blocos (ou repetições), quando se utiliza o modelo AMMI em vez do modelo completo, sendo, portanto, melhor utilizar o modelo AMMI com correção dos autovalores e selecionar o número de componentes por meio do método de Eastment e Krzanowski.; This paper presents an application of AMMI models - Additive Main effects and Multiplicative Interaction model - for a thorough study about the effect of the interaction between genotype and environment in multi-environments experiments with balanced data. Two methods of crossed validation are presented and the improvement of these methods through the correction of eigenvalues...

## Distribuição empírica dos autovalores associados à matriz de interação dos modelos AMMI pelo método bootstrap não-paramétrico; Empirical distribution of eigenvalues associated with the interaction matrix of the AMMI models for non-parametric bootstrap method

Hongyu, Kuang
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 25/01/2012 PT
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A interação genótipos ambientes (G E) foi definido por Shelbourne (1972) como sendo a variação entre genótipos em resposta a diferentes condições ambientais. Sua magnitude na expressão fenotípica do caráter pode reduzir a correlação entre fenótipo e genótipo, in acionando a variância genética e, por sua vez, parâmetros dependentes desta, como herdabilidade e ganho genético com a seleção. Estudos sobre a adaptabilidade e a estabilidade fenotípica permitem particularizar os efeitos da interação GE ao nível de genótipo e ambiente, identificando a contribuição relativa de cada um para a interação total. Varias metodologias estatísticas têm sido propostas para a interpretação da interação G E proveniente de um grupo de cultivares testados em vários ambientes. Entre essas metodologias destaca-se os modelos AMMI (Additive Main Eects and Multiplicative Interaction Model), que vem ganhando grande aplicabilidade nos últimos anos. O modelo AMMI e um método uni-multivariado, que engloba uma analise de variância para os efeitos principais, que são os efeitos dos genótipos (G) e os ambientes (E) e para os efeitos multiplicativos (interação genótipo ambiente), para a qual utiliza-se a decomposição em valor singular (DVS). Essa técnica multivariada baseia-se no uso dos autovalores e autovetores provenientes da matriz de interação G E. Araujo e Dias (2005) verificaram o problema de superestimação e subestimação de autovalores estimados da maneira convencional. Efron(1979) propôs uma técnica de simulação numérica chamada Bootstrap para avaliar tais incertezas. O método Bootstrap consiste em uma técnica de reamostragem que permite aproximar a distribuição de uma função das observações a partir da distribuição empírica dos dados. Por meio desse método...

## O espectro de grafos threshold e aplicações

Tura, Fernando Colman
Fonte: Universidade Federal do Rio Grande do Sul Publicador: Universidade Federal do Rio Grande do Sul
Tipo: Tese de Doutorado Formato: application/pdf
POR
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Nesta tese de doutorado estudamos uma classe de grafos denominada threshold. Iniciamos apresentando algumas caracterizações dos grafos threshold e definindo-os de uma forma apropriada para o nosso propósito. Mais especificamente, estudamos o espectro dos grafos threshold. Para isso apresentamos alguns resultados previamente conhecidos, como por exemplo, em relação à matriz de adjacência, uma redução para o cálculo do polinômio característico e a multiplicidade dos autovalores não principais. Desenvolvemos um algoritmo que constrói uma matriz diagonal D congruente a A + xI , onde A é a matriz de adjacência de um grafo threshold, x é um número real e I é a matriz identidade. Como aplicação, determinamos quantos autovalores de um grafo threshold G pertencem a um intervalo real (a, b]. A implementação do nosso algoritmo depende apenas da sequência binária de um grafo threshold e sua complexidade é de ordem O(n). Tal algoritmo é facilmente adaptado para a matriz distância Θ de um grafo threshold G. Nesta tese apresentamos aplicações desse algoritmo, como a simplicidade dos autovalores principais, a minimalidade do menor autovalor de um threshold, exibindo uma fórmula para esse menor autovalor, uma ordenação dos grafos threshold via índice...

## Main eigenvalues and (κ, τ)-regular sets

Cardoso, D.M.; Sciriha, I.; Zerafa, C.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
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A (κ, τ)-regular set is a subset of the vertices of a graph G, inducing a κ-regular subgraph such that every vertex not in the subset has τ neighbors in it. A main eigenvalue of the adjacency matrix A of a graph G has an eigenvector not orthogonal to the all-one vector j. For graphs with a (κ, τ)-regular set a necessary and sufficient condition for an eigenvalue be non-main is deduced and the main eigenvalues are characterized. These results are applied to the construction of infinite families of bidegreed graphs with two main eigenvalues and the same spectral radius (index) and some relations with strongly regular graphs are obtained. Finally, the determination of (κ, τ)-regular sets is analyzed. © 2009 Elsevier Inc. All rights reserved.; CEOC; FCT; FEDER/POCI 2010; University of Malta

## Eigenvalues of a H-generalized join graph operation constrained by vertex subsets

Cardoso, Domingos M.; Martins, Enide A.; Robbiano, Maria; Rojo, Oscar
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
36.05%
A generalized H-join operation of a family of graphs G1, . . . , Gp, where H has order p, constrained by a family of vertex subsets Si ⊆V(Gi), i = 1, . . . , p, is introduced. When each vertex subset Si is (ki, τi)-regular, it is deduced that all non-main adjacency eigenvalues of Gi , different from ki−τi , remain as eigenvalues of the graph G obtained by this operation. If each Gi is ki-regular and all the vertex subsets are such that Si = V(Gi), the H-generalized join constrained by these vertex subsets coincides with the H-join operation. Furthermore, some applications on the spread of graphs are presented.

## A restarted Lanczos process for computing a large percentage of the eigenvalues of a symmetric matrix

Zuo, Wei
Fonte: Universidade Rice Publicador: Universidade Rice
ENG
Relevância na Pesquisa
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The Lanczos algorithm is a well known technique for approximating a few eigenvalues and corresponding eigenvectors of a large-scale real symmetric matric arising in many scientific and engineering applications. In this thesis a modified restarted k-step Lanczos algorithm is presented. The basic Lanczos process suffers from numerical difficulties such as large storage requirements and loss of orthogonality among the basis vectors. The current restarted Lanczos method is designed to overcome these difficulties by fixing the number of steps in the Lanczos process at a prespecified value k, where k is modest and proportional to the total number of eigenvalues of interest. However, it is possible that the total number of desired eigenvalues may not be moderate. The main difference between this restart scheme and other existing schemes is that the prescribed value k in our algorithm is only a reasonable number. It is independent of the total number of desired eigenvalues. In other words, this algorithm may compute a large percentage of eigenvalues and associated eigenvectors of a large symmetric matrix with significantly reduced storage requirement. Strategies for implementing the algorithm on parallel distributed-memory machines are presented. The efficiency of this algorithm is justified by computational results for the electronic structure problem in quantum chemistry.

## Behavior Modes, Pathways and Overall Trajectories: Eigenvector and Eigenvalue Analysis of Dynamic Systems

Goncalves, Paulo
Fonte: Camrbidge, MA; Alfred P. Sloan School of Management, Massachusetts Institute of Technology Publicador: Camrbidge, MA; Alfred P. Sloan School of Management, Massachusetts Institute of Technology
Tipo: Trabalho em Andamento
EN_US
Relevância na Pesquisa
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One of the most fundamental principles in system dynamics is the premise that the structure of the system will generate its behavior. Such philosophical position has fostered the development of a number of formal methods aimed at understanding the causes of model behavior. To most in the field of system dynamics, behavior is commonly understood as modes of behavior (e.g., exponential growth, exponential decay, and oscillation) because of their direct association with the feedback loops (e.g., reinforcing, balancing, and balancing with delays, respectively) that generate them. Hence, traditional research on formal model analysis has emphasized which loops cause a particular “mode” of behavior, with eigenvalues representing the most important link between structure and behavior. The main contribution of this work arises from a choice to focus our analysis in the overall trajectory of a state variable – a broader definition of behavior than that of a specific behavior mode. When we consider overall behavior trajectories, contributions from eigenvectors are just as central as those from eigenvalues. Our approach to understanding model behavior derives an equation describing overall behavior trajectories in terms of both eigenvalues and eigenvectors. We then use the derivatives of both eigenvalues and eigenvectors with respect to link (or loop) gains to measure how they affect overall behavior trajectories over time. The direct consequence of focusing on behavior trajectories is that system dynamics researchers' reliance on eigenvalue elasticities can be seen as too-narrow a focus on model behavior – a focus that has excluded the short term impact of a change in loop (or link) gain in its analysis.

## Tricyclic graphs with exactly two main eigenvalues

Huang, He; Deng, Hanyuan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.

## Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods

Hu, Jun; Yang, Xueqin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/12/2014
Relevância na Pesquisa
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In this paper, we analyze the lower bound property of the discrete eigenvalues by the rectangular Morley elements of the biharmonic operators in both two and three dimensions. The analysis relies on an identity for the errors of eigenvalues. We explore a refined property of the canonical interpolation operators and use it to analyze the key term in this identity. In particular, we show that such a term is of higher order for two dimensions, and is negative and of second order for three dimensions, which causes a main difficulty. To overcome it, we propose a novel decomposition of the first term in the aforementioned identity. Finally, we establish a saturation condition to show that the discrete eigenvalues are smaller than the exact ones. We present some numerical results to demonstrate the theoretical results.

## Tricyclic graphs with exactly two main eigenvalues

Fan, Xiaoxia; Luo, Yanfeng
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/12/2010
Relevância na Pesquisa
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An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected tricyclic graphs with exactly two main eigenvalues are determined.; Comment: 16 pages, 4 figures

## Eigenvalues of collapsing domains and drift Laplacians

Lu, Zhiqin; Rowlett, Julie
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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By introducing a weight function to the Laplace operator, Bakry and \'Emery defined the "drift Laplacian" to study diffusion processes. Our first main result is that, given a Bakry-\'Emery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new relationship between Dirichlet eigenvalues of domains in $\R^n$ and Neumann eigenvalues of domains in $\R^{n+1}$ and a new maximum principle. Using our main result and maximum principle, we are able to generalize \emph{all the results in Riemannian geometry based on gradient estimates to Bakry-\'Emery manifolds}.

## Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)

Burda, Z.; Jarosz, A.; Livan, G.; Nowak, M. A.; Swiech, A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/03/2011
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We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These densities are encoded in the form of the so called M-transforms, for which polynomial equations are found. We exploit the methods of planar diagrammatics, enhanced to the non-Hermitian case, and free random variables, respectively; both are described in the appendices. As particular results of these two main equations, we find the singular behavior of the spectral densities near zero. Moreover, we propose a finite-size form of the spectral density of the product close to the border of its eigenvalues' domain. Also, led by the striking similarity between the two main equations, we put forward a conjecture about a simple relationship between the eigenvalues and singular values of any non-Hermitian random matrix whose spectrum exhibits rotational symmetry around zero.; Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical Physics and Information Theory," September 26-30, 2010, Krakow, Poland

## Relation between the eigenfrequencies of Bogoliubov excitations of Bose-Einstein condensates and the eigenvalues of the Jacobian in a time-dependent variational approach

Kreibich, Manuel; Main, Jörg; Wunner, Günter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We study the relation between the eigenfrequencies of the Bogoliubov excitations of Bose-Einstein condensates, and the eigenvalues of the Jacobian stability matrix in a variational approach which maps the Gross-Pitaevskii equation to a system of equations of motion for the variational parameters. We do this for Bose-Einstein condensates with attractive contact interaction in an external trap, and for a simple model of a self-trapped Bose-Einstein condensate with attractive 1/r interaction. The stationary solutions of the Gross-Pitaevskii equation and Bogoliubov excitations are calculated using a finite-difference scheme. The Bogoliubov spectra of the ground and excited state of the self-trapped monopolar condensate exhibits a Rydberg-like structure, which can be explained by means of a quantum defect theory. On the variational side, we treat the problem using an ansatz of time-dependent coupled Gaussians combined with spherical harmonics. We first apply this ansatz to a condensate in an external trap without long-range interaction, and calculate the excitation spectrum with the help of the time-dependent variational principle. Comparing with the full-numerical results, we find a good agreement for the eigenfrequencies of the lowest excitation modes with arbitrary angular momenta. The variational method is then applied to calculate the excitations of the self-trapped monopolar condensates...

## Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues

Li, Shuchao; Yang, Xue
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/04/2013
Relevância na Pesquisa
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The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Chen and Huang [L. Chen, Q.X. Huang, Trees, unicyclic graphs and bicyclic graphs with exactly two $Q$-main eigenvalues, submitted for publication] characterized all trees, unicylic graphs and bicyclic graphs with exactly two main $Q$-eigenvalues, respectively. As a continuance of it, in this paper, all tricyclic graphs with exactly two $Q$-main eigenvalues are characterized.; Comment: 25 pages;6 figures

## On the Main Signless Laplacian Eigenvalues of a Graph

Deng, Hanyuan; Huang, He
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 29/08/2012
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A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, we first give the necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues, and then characterize the trees and unicyclic graphs with exactly two main signless Laplacian eigenvalues, respectively.

## Bicyclic graphs with exactly two main signless Laplacian eigenvalues

Huang, He; Deng, Hanyuan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.04%
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.

## The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods

Hu, Jun; Huang, Yunqing; Lin, Qun
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we first show abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. As one application, we show that this condition hold for most nonconforming elements in literature. As another important application, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally...

## On Eigenvalues of Random Complexes

Gundert, Anna; Wagner, Uli
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.12%
We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails...

## Count of eigenvalues in the generalized eigenvalue problem

Chugunova, M.; Pelinovsky, D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/02/2006
Relevância na Pesquisa
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We address the count of isolated and embedded eigenvalues in a generalized eigenvalue problem defined by two self-adjoint operators with a positive essential spectrum and a finite number of isolated eigenvalues. The generalized eigenvalue problem determines spectral stability of nonlinear waves in a Hamiltonian dynamical system. The theory is based on the Pontryagin's Invariant Subspace theorem in an indefinite inner product space but it extends beyond the scope of earlier papers of Pontryagin, Krein, Grillakis, and others. Our main results are (i) the number of unstable and potentially unstable eigenvalues {\em equals} the number of negative eigenvalues of the self-adjoint operators, (ii) the total number of isolated eigenvalues of the generalized eigenvalue problem is {\em bounded from above} by the total number of isolated eigenvalues of the self-adjoint operators, and (iii) the quadratic form defined by the indefinite inner product is strictly positive on the subspace related to the absolutely continuous part of the spectrum of the generalized eigenvalue problem. Applications to solitons and vortices of the nonlinear Schr\"{o}dinger equations and solitons of the Korteweg--De Vries equations are developed from the general theory.; Comment: 30 pages...