Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems

[EN] In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos-based method for computing all real eigenvalues contained in a given interval of large-scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory-efficient by representing the computed pseudo-Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.Agencia Estatal de Investigacion, Grant/Award Number: TIN2016-75985-PCampos, C.; Román Moltó, JE. (2020). Inertia-based spectrum slicing for symmetric quadratic eigenvalue problems. 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A new framework for consensus for discrete-time directed networks of multi-agents with distributed delays

Copyright @ 2012 Taylor & FrancisIn this article, the distributed consensus problem is considered for discrete-time delayed networks of dynamic agents with fixed topologies, where the networks under investigation are directed and the time-delays involved are distributed time delays including a single or multiple time delay(s) as special cases. By using the invariance principle of delay difference systems, a new unified framework is established to deal with the consensus for the discrete-time delayed multi-agent system. It is shown that the addressed discrete-time network with arbitrary distributed time delays reaches consensus provided that it is strongly connected. A numerical example is presented to illustrate the proposed methods.This work was supported in part by City University of Hong Kong under Grant 7008114, the Royal Society of the UK, the National Natural Science Foundation of China under Grants 60774073 and 61074129, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2010313

A differentiable characterization of local contractions on Banach spaces

This note provides a differentiable characterization of local contractions on an arbitrary Banach space. As a corollary, a refinement to Ostrowski’s sufficient condition for local convergence in finite spaces is obtained, which applies to many models, e.g. in economics, ecology or game theory, where one has an interest in fixed point iterations and local stability of discrete dynamic processes. We show that for the local contraction property to hold, continuity of the derivative at the fixed point is indispensable

When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices?

The mapping (Formula presented.) , where the matrices (Formula presented.) are skew-Hamiltonian with respect to transposition, is studied. Let (Formula presented.) be the range of (Formula presented.) : we give an implicit characterization of (Formula presented.) , obtaining results that find an application in algebraic geometry. Namely, they are used in [R. Abuaf and A. Boralevi, Orthogonal bundles and skew-Hamiltonian matrices, Submitted] to study orthogonal vector bundles. We also give alternative and more explicit characterizations of (Formula presented.) for (Formula presented.). Moreover, we prove that for (Formula presented.) , the complement of (Formula presented.) is nowhere dense in the set of (Formula presented.) -dimensional Hamiltonian matrices, denoted by (Formula presented.) , implying that almost all matrices in (Formula presented.) are in (Formula presented.) for (Formula presented.). Finally, we show that (Formula presented.) is never surjective as a mapping from (Formula presented.) to (Formula presented.) , where (Formula presented.) is the set of (Formula presented.) -dimensional skew-Hamiltonian matrices. Along the way, we discuss the connections of this problem with several existing results in matrix theory

Random Surfing Without Teleportation

In the standard Random Surfer Model, the teleportation matrix is necessary to ensure that the final PageRank vector is well-defined. The introduction of this matrix, however, results in serious problems and imposes fundamental limitations to the quality of the ranking vectors. In this work, building on the recently proposed NCDawareRank framework, we exploit the decomposition of the underlying space into blocks, and we derive easy to check necessary and sufficient conditions for random surfing without teleportation.Comment: 13 pages. Published in the Volume: "Algorithms, Probability, Networks and Games, Springer-Verlag, 2015". (The updated version corrects small typos/errors

Privacy-Preserving Patient Similarity Learning in a Federated Environment: Development and Analysis

Background: There is an urgent need for the development of global analytic frameworks that can perform analyses in a privacy-preserving federated environment across multiple institutions without privacy leakage. A few studies on the topic of federated medical analysis have been conducted recently with the focus on several algorithms. However, none of them have solved similar patient matching, which is useful for applications such as cohort construction for cross-institution observational studies, disease surveillance, and clinical trials recruitment. Objective: The aim of this study was to present a privacy-preserving platform in a federated setting for patient similarity learning across institutions. Without sharing patient-level information, our model can find similar patients from one hospital to another. Methods: We proposed a federated patient hashing framework and developed a novel algorithm to learn context-specific hash codes to represent patients across institutions. The similarities between patients can be efficiently computed using the resulting hash codes of corresponding patients. To avoid security attack from reverse engineering on the model, we applied homomorphic encryption to patient similarity search in a federated setting. Results: We used sequential medical events extracted from the Multiparameter Intelligent Monitoring in Intensive Care-III database to evaluate the proposed algorithm in predicting the incidence of five diseases independently. Our algorithm achieved averaged area under the curves of 0.9154 and 0.8012 with balanced and imbalanced data, respectively, in ??-nearest neighbor with ??=3. We also confirmed privacy preservation in similarity search by using homomorphic encryption. Conclusions: The proposed algorithm can help search similar patients across institutions effectively to support federated data analysis in a privacy-preserving manner

On the Maxwell-Stefan approach to multicomponent diffusion

We consider the system of Maxwell-Stefan equations which describe multicomponent diffusive fluxes in non-dilute solutions or gas mixtures. We apply the Perron-Frobenius theorem to the irreducible and quasi-positive matrix which governs the flux-force relations and are able to show normal ellipticity of the associated multicomponent diffusion operator. This provides local-in-time wellposedness of the Maxwell-Stefan multicomponent diffusion system in the isobaric, isothermal case.Comment: Based on a talk given at the Conference on Nonlinear Parabolic Problems in Bedlewo, Mai 200

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T=c1T1+c2T2+c3T3- c4(T1T2+T3T1+T2T3), where T1, T2, T3 are three n x n tripotent matrices and c1, c2, c3, c4 are complex numbers with c1, c2, c3 nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.We would like to thank the referee for his/her careful reading. The first author was supported by the Vicerrectorado de Investigacion U.P.V. PAID 06-2010-2285.Benítez López, J.; Sarduvan, M.; Ülker, S.; Özdemir, H. (2013). On nonsingularity of combinations of three group invertible matrices and three tripotent matrices. Linear and Multilinear Algebra. 61(4):463-481. https://doi.org/10.1080/03081087.2012.689986S463481614Baksalary, J. K., & Baksalary, O. M. (2004). Nonsingularity of linear combinationsof idempotent matrices. Linear Algebra and its Applications, 388, 25-29. doi:10.1016/j.laa.2004.02.025Baksalary, J. K., Baksalary, O. M., & Özdemir, H. (2004). A note on linear combinations of commuting tripotent matrices. Linear Algebra and its Applications, 388, 45-51. doi:10.1016/j.laa.2004.01.011Benítez, J., Liu, X., & Zhu, T. (2010). Nonsingularity and group invertibility of linear combinations of twok-potent matrices. Linear and Multilinear Algebra, 58(8), 1023-1035. doi:10.1080/03081080903207932Benítez, J., & Thome, N. (2006). {k}-Group Periodic Matrices. SIAM Journal on Matrix Analysis and Applications, 28(1), 9-25. doi:10.1137/s0895479803437384Gross, J., & Trenkler, G. (2000). Nonsingularity of the Difference of Two Oblique Projectors. SIAM Journal on Matrix Analysis and Applications, 21(2), 390-395. doi:10.1137/s0895479897320277Horn, R. A., & Johnson, C. R. (1985). Matrix Analysis. doi:10.1017/cbo9780511810817Koliha, J. J., & Rakočević, V. (2006). The nullity and rank of linear combinations of idempotent matrices. Linear Algebra and its Applications, 418(1), 11-14. doi:10.1016/j.laa.2006.01.011Koliha, J. ., Rakočević, V., & Straškraba, I. (2004). The difference and sum of projectors. Linear Algebra and its Applications, 388, 279-288. doi:10.1016/j.laa.2004.03.008Liu, X., Wu, S., & Benítez, J. (2011). On nonsingularity of combinations of two group invertible matrices and two tripotent matrices. Linear and Multilinear Algebra, 59(12), 1409-1417. doi:10.1080/03081087.2011.558843Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.9780898719512Mitra, S. K. (1987). On group inverses and the sharp order. Linear Algebra and its Applications, 92, 17-37. doi:10.1016/0024-3795(87)90248-5Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Zhang, F. (1999). Matrix Theory. Universitext. doi:10.1007/978-1-4757-5797-2Zuo, K. (2010). Nonsingularity of the difference and the sum of two idempotent matrices. Linear Algebra and its Applications, 433(2), 476-482. doi:10.1016/j.laa.2010.03.01