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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.

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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)

On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line

We present conditions that allow us to pass from the convergence of probability measures in distribution to the uniform convergence of the associated quantile functions. Under these conditions, one can in particular pass from the asymptotic distribution of collections of real numbers, such as the eigenvalues of a family of n-by-n matrices as n goes to infinity, to their uniform approximation by the values of the quantile function at equidistant points. For Hermitian Toeplitz-like matrices, convergence in distribution is ensured by theorems of the Szegő type. Our results transfer these convergence theorems into uniform convergence statements.

From convergence in distribution to uniform convergence

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

Wiener–Hopf and spectral factorization of real polynomials by Newton’s method

This paper is concerned with the spectra of Toeplitz operators with piecewise continuous symbols and with the symbol calculus for singular integral operators with piecewise continuous coefficients on L P (Γ) where 1 < p < ∞ and Γ is a Carleson Jordan curve. It is well known that piecewise smooth curves lead to the appearance of circular arcs in the essential spectra of Toeplitz operators, and only recently the authors discovered that certain Carleson curves metamorphose these circular arcs into logarithmic double-spirals. In the present paper we dispose of the matter by determining the local spectra produced by a general Carleson curve. These spectra are of a qualitatively new type and may, in particular, be heavy sets — until now such a phenomenon has only be observed for spaces with general Muckenhoupt weights.

Albrecht Böttcher

Papers

On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line

From convergence in distribution to uniform convergence

Exploration of Toeplitz-like matrices with unbounded symbols is not a purely academic journey

Wiener–Hopf and spectral factorization of real polynomials by Newton’s method

Toeplitz and Singular Integral Operators on General Carleson Jordan Curves