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Localization and delocalization for heavy tailed band matrices

Florent Benaych-Georges, Sandrine Péché (2014)

Annales de l'I.H.P. Probabilités et statistiques

We consider some random band matrices with band-width N μ whose entries are independent random variables with distribution tail in x - α . We consider the largest eigenvalues and the associated eigenvectors and prove the following phase transition. On the one hand, when α l t ; 2 ( 1 + μ - 1 ) , the largest eigenvalues have order N ( 1 + μ ) / α , are asymptotically distributed as a Poisson process and their associated eigenvectors are essentially carried by two coordinates (this phenomenon has already been remarked for full matrices by Soshnikov...

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A . The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X . Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A , generalizing previous results on Perron-Frobenius properties of matrices with...

L'octogone régulier et la signature des formes quadratiques entières non singulières

Catherine Bailly, Maria de Jesus Cabral (2003)

Annales de l’institut Fourier

La formule généralisant la loi de réciprocité quadratique de Legendre et exprimant le reste par huit de la signature d'une forme quadratique entière non dégénérée à l'aide d'une somme de Gauss est attribuée par Milnor à Milgram, la faisant remonter à Braun. Le formalisme de Witt la réduit au cas de dimension 1 que Chandrasekharan attribue à Cauchy et Kronecker. Braun soulignait que les preuves de ces formules nécessitent des moyens d'analyse. Une propriété métrique de l'octogone...

Log-majorizations and norm inequalities for exponential operators

Fumio Hiai (1997)

Banach Center Publications

Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii-Wielandt and the Gelfand-Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden-Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.

Low rank Tucker-type tensor approximation to classical potentials

B. Khoromskij, V. Khoromskaia (2007)

Open Mathematics

This paper investigates best rank-(r 1,..., r d) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝd. Super-convergence of the best rank-(r 1,..., r d) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer...

Lower bounds for the largest eigenvalue of the gcd matrix on { 1 , 2 , ⋯ , n }

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

Consider the n × n matrix with ( i , j ) ’th entry gcd ( i , j ) . Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n / n . Because s n cannot be expressed algebraically as a function of n , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that λ n > 6 π - 2 n log n for all n . If n is large enough, this follows from F. Balatoni (1969).

Low-rank tensor representation of Slater-type and Hydrogen-like orbitals

Martin Mrovec (2017)

Applications of Mathematics

The paper focuses on a low-rank tensor structured representation of Slater-type and Hydrogen-like orbital basis functions that can be used in electronic structure calculations. Standard packages use the Gaussian-type basis functions which allow us to analytically evaluate the necessary integrals. Slater-type and Hydrogen-like orbital functions are physically more appropriate, but they are not analytically integrable. A numerical integration is too expensive when using the standard discretization...

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