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The graph p-Laplacian eigenvalue problem
Full text linkIn this thesis we discuss the graph p-Laplacian eigenvalue problem. In particular, after reviewing the state of the art, we present new results on the nodal domain count of the p-Laplacian eigenpairs, on the graph infinity-Laplacian eigenproblem, and on the computation of the p-Laplacian eigenpairs. Concerning the nodal domain count, we prove that the number of nodal domains induced by a p-Laplacian eigenfunction can be bounded, both from above and below, in terms of the position of the corresponding eigenvalue in the variational spectrum. Moreover, we prove that on trees the variational spectrum exhausts the entire spectrum, and the number of nodal domains induced by an everywhere nonzero eigenfunction is equal to the variational index of the corresponding eigenvalue. These results allow us to derive, from the p-Laplacian spectrum, topological information about the graph. Indeed, when p is equal to 1 and infinity, the p-Laplacian eigenvalues approximate the Cheeger constants and the packing radii of the graph, respectively. The study of the infinity-Laplacian eigenproblem is another major contribution of this thesis. In particular, we compare different formulations of this degenerate eigenproblem. In the first case we study the infinity-eigenpairs as solutions of the limiting eigenvalue equation, in the second case we define the infinity-eigenpairs as generalized critical points of the infinity-Rayleigh quotient. Then, we relate the infinity variational eigenvalues to the packing radii of the graph. Here, among other things, we prove that the first and the second infinity variational eigenvalues are exactly equal to the first and the second packing radii of the graph. Finally, we present a novel approach to compute the p-Laplacian eigenpairs both in the smooth case 2<p<infinity and in the degenerate case p=infinity. To this end, we observe that the p-Laplacian eigenvalue problem, both for 2<p<infinity and p=infinity, can be reformulated as a constrained linear weighted-Laplacian eigenvalue problem. Based on this remark, we introduce a family of energy functions whose domain is the space of positive measures on the edges and on the nodes of the graph. Then, we first prove that the unique saddle point of the first energy function corresponds to the unique first eigenpair of the p-Laplacian. Second, we prove that smooth saddle points of the k-th energy function correpond to p-Laplacian eigenpairs (f,lambda), such that the Morse index of the p-Rayleigh quotient in f is equal to k. Based on such results, we introduce gradient flows suited to compute saddle points of the proposed energy functions and we discuss the results of their numerical integration. Practically, the integration of the gradient flows, at each step, requires only the computation of a linear eigenpair. Hence we are able to use all the theoretical and numerical advantages of the linear setting to overcome the difficulties of solving a nonlinear equation. However, the theoretical study of the gradient flows remains an open problem, which deserves a future in-depth study.In this thesis we discuss the graph p-Laplacian eigenvalue problem. In particular, after reviewing the state of the art, we present new results on the nodal domain count of the p-Laplacian eigenpairs, on the graph infinity-Laplacian eigenproblem, and on the computation of the p-Laplacian eigenpairs. Concerning the nodal domain count, we prove that the number of nodal domains induced by a p-Laplacian eigenfunction can be bounded, both from above and below, in terms of the position of the corresponding eigenvalue in the variational spectrum. Moreover, we prove that on trees the variational spectrum exhausts the entire spectrum, and the number of nodal domains induced by an everywhere nonzero eigenfunction is equal to the variational index of the corresponding eigenvalue. These results allow us to derive, from the p-Laplacian spectrum, topological information about the graph. Indeed, when p is equal to 1 and infinity, the p-Laplacian eigenvalues approximate the Cheeger constants and the packing radii of the graph, respectively. The study of the infinity-Laplacian eigenproblem is another major contribution of this thesis. In particular, we compare different formulations of this degenerate eigenproblem. In the first case we study the infinity-eigenpairs as solutions of the limiting eigenvalue equation, in the second case we define the infinity-eigenpairs as generalized critical points of the infinity-Rayleigh quotient. Then, we relate the infinity variational eigenvalues to the packing radii of the graph. Here, among other things, we prove that the first and the second infinity variational eigenvalues are exactly equal to the first and the second packing radii of the graph. Finally, we present a novel approach to compute the p-Laplacian eigenpairs both in the smooth case 2<p<infinity and in the degenerate case p=infinity. To this end, we observe that the p-Laplacian eigenvalue problem, both for 2<p<infinity and p=infinity, can be reformulated as a constrained linear weighted-Laplacian eigenvalue problem. Based on this remark, we introduce a family of energy functions whose domain is the space of positive measures on the edges and on the nodes of the graph. Then, we first prove that the unique saddle point of the first energy function corresponds to the unique first eigenpair of the p-Laplacian. Second, we prove that smooth saddle points of the k-th energy function correpond to p-Laplacian eigenpairs (f,lambda), such that the Morse index of the p-Rayleigh quotient in f is equal to k. Based on such results, we introduce gradient flows suited to compute saddle points of the proposed energy functions and we discuss the results of their numerical integration. Practically, the integration of the gradient flows, at each step, requires only the computation of a linear eigenpair. Hence we are able to use all the theoretical and numerical advantages of the linear setting to overcome the difficulties of solving a nonlinear equation. However, the theoretical study of the gradient flows remains an open problem, which deserves a future in-depth study
Graph -Laplacian Eigenpairs as Saddle Points of a Family of Spectral Energy Functions
No full textWe address the problem of computing the graph p-Laplacian eigenpairs for p ξ (2, ∞). We propose a reformulation of the graph p-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and discuss theoretical and computational advantages. We provide a correspondence between p-Laplacian eigenpairs and linear eigenpairs of a constrained generalized weighted Laplacian eigenvalue problem. As a result, we can assign an index to any p-Laplacian eigenpair that matches the Morse index of the p-Rayleigh quotient evaluated at the eigenfunction. In the second part of the paper, we introduce a class of spectral energy functions that depend on edge and node weights. We prove that differentiable saddle points of the kth energy function correspond to p-Laplacian eigenpairs having index equal to k. Moreover, the first energy function is proved to possess a unique saddle point which corresponds to the unique first p-Laplacian eigenpair. Finally, we develop novel gradient-based numerical methods suited to compute p-Laplacian eigenpairs for any p ξ (2, ∞) and present some experiments
The Jordan and Frobenius pairs of the inverse
No full textGiven a matrix there exists a nonsingular matrix such that , where is a very sparse matrix with a diagonal block structure,
known as Jordan canonical form (JCF) of . Assume that is nonsingular and that and are given. How to obtain and
such that and is the JCF of ? Curiously, the answer involves the Pascal matrix. For the Frobenius canonical form (FCF), where blocks are companion matrices, the analogous question has a very simple answer. Jordan blocks and companion are non-derogatory lower Hessenberg matrices. The answers to the two questions will be obtained by solving two linear matrix equations involving these matrices
THE GRAPH ∞-LAPLACIAN EIGENVALUE PROBLEM
No full textAbstract. We analyze various formulations of the ∞-Laplacian eigenvalue problem on graphs, comparing their properties and highlighting their respective advantages and limitations. First, we investigate the graph ∞-eigenpairs arising as limits of p-Laplacian eigenpairs, extending key results from the continuous setting to the discrete domain. We prove that every limit of p-Laplacian eigen-pair, for p going to ∞, satisfies a limit eigenvalue equation and establish that the corresponding eigenvalue can be bounded from below by the packing radius of the graph, indexed by the number of nodal domains induced by the eigenfunction. Additionally, we show that the limits, for p going to ∞, of the variational p-Laplacian eigenvalues are bounded both from above and from below by the packing radii, achieving equality for the smallest two variational eigenvalues and corresponding packing radii of the graph. In the second part of the paper, we introduce generalized ∞-Laplacian eigenpairs as generalized critical points and values of the ∞-Rayleigh quotient. We prove that the generalized variational ∞-eigenvalues satisfy the same upper bounds in terms of packing radii as the limit of the variational eigenvalues, again with equality holding between the smallest two ∞-variational eigenvalues and the first and second packing radii of the graph. Moreover, we establish that any solution to the limit eigenvalue equation is also a generalized eigenpair, while any generalized eigenpair satisfies the limit eigenvalue equation on a suitable subgraph
Algebras closed by J-Hermitianity in displacement formulas
No full textWe introduce the notion of -Hermitianity of a matrix, as a generalization of Hermitianity, and, more generally, of closure by -Hermitianity of a set of matrices. Many well known algebras, like upper and lower triangular Toeplitz, Circulants and matrices, as well as certain algebras that have dimension higher than the matrix order, turn out to be closed by -Hermitianity. As an application, we generalize some theorems about displacement decompositions presented in [1, 2], by assuming the matrix algebras involved closed by -Hermitianity. Even if such hypothesis on the structure is not necessary in the case of algebras generated by one matrix, as it has been proved in [3], our result is relevant because it could yield new low complexity displacement formulas involving not one-matrix-generated commutative algebras
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
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