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Convex Analysisの二,三の進展について

The fundamental theorems on the asymptotic behavior of eigenvalues, inverses, and products of banded Toeplitz matrices and Toeplitz matrices with absolutely summable elements are derived in a tutorial manner. Mathematical elegance and generality are sacrificed for conceptual simplicity and insight in the hope of making these results available to engineers lacking either the background or endurance to attack the mathematical literature on the subject. By limiting the generality of the matrices considered, the essential ideas and results can be conveyed in a more intuitive manner without the mathematical machinery required for the most general cases. As an application the results are applied to the study of the covariance matrices and their factors of linear models of discrete time random processes.

/pdf/toeplitz-and-circulant-matrices-3e6lj1a9jc.pdf

Toeplitz and circulant matrices

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

: Real Hardy spaces and BMO are well-known function spaces used in harmonic analysis and PDE, and have given rise to many related function spaces. This talk will focus on a few of these variants and review results on duality, boundedness of operators, and extension domains.

Southeastern Analysis Meeting 39: Titles and Abstracts Plenary and Semi-Plenary Talks Hardy spaces, BMO and related function spaces

Abstract This is an expository paper embarking on the asymptotic behavior of the entries of the inverses of positive definite symmetric Toeplitz matrices as the matrix dimension goes to infinity. We consider the behavior of the entries in neighborhoods of the four corners as well as the density of the distribution of the entries over all of the inverse matrix. 

https://link.springer.com/content/pdf/10.1007/s44146-022-00007-0.pdf

Entries of the inverses of large positive definite Toeplitz matrices

We present a novel Newton method for canonical Wiener–Hopf and spectral factorization of matrix polynomials. The initial vector results from solving a block Toeplitz-like system, and the Jacobi matrix governing the Newton iteration has nice structural and numerical properties. The local quadratic convergence of the method is proved and was tested numerically. For scalar polynomials of degree 20000, a superfast version of the method implemented on a laptop typically reqired about half a minute to produce an initial vector and then performed the Newton iteration within one second. In the matrix case, the method worked reproachless on a laptop with 8 Gigabyte RAM if the degree of the polynomial times the squared matrix dimension did not exceed 20000. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

Canonical Wiener–Hopf and spectral factorization of large‐degree matrix polynomials

Given a compact Hausdorff spaceX, we may associate with every continuous mapa: X → X a composition operatorC

 a
 onC(X) by the rule(C

 a
 
f)(x) = f(a(x)). We describe all self-mapsa for whichC

 a
 is an algebraic operator or an essentially algebraic operator (i.e. an operator algebraic modulo compact operators), determine the characteristic polynomialp

 a
 
(z) and the essentially characteristic polynomialq

 a
 
(z) in these cases and show how the connectivity ofX may be characterized in terms of the quotientsp

 a
 
(z)/q

 a
 
(z).

Algebraic and essentially algebraic composition operators on C(X)

There are infinitely many finite groups of orthogonal matrices all orbits of which, including those of irrational vectors, span lattices, that is, discrete additive subgroups of the underlying Euclidean space. We show that, both up to isomorphism and up to orthogonal similarity, exactly eight of these groups are irreducible: the two trivial groups in one dimension, the cyclic groups of orders 3, 4, 6 in two dimensions, and the quaternion, binary dihedral, binary tetrahedral groups in four dimensions.

Albrecht Böttcher

Papers

Southeastern Analysis Meeting 39: Titles and Abstracts Plenary and Semi-Plenary Talks Hardy spaces, BMO and related function spaces

Entries of the inverses of large positive definite Toeplitz matrices

Canonical Wiener–Hopf and spectral factorization of large‐degree matrix polynomials

Algebraic and essentially algebraic composition operators on C(X)

Groups of Orthogonal Matrices All Orbits of Which Generate Lattices