Albrecht Böttcher

Technische Universität Chemnitz, Karl-Marx-Stadt, Saxony, Germany

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Publications (219)93.31 Total impact

  • Albrecht Böttcher · Fuad Kittaneh
    [Show abstract] [Hide abstract] ABSTRACT: The paper concerns the limit of the zero set Sn, of the Fibonacci-type polynomials G(n)(x) = G(n)(a,b,x) given by G(0)(x) = a, G(1)(x) = x+b, and G(n+2) (x) = XG(n+1) (x) G(n)(x). It presents an explicit set L consisting of at most two points such that S-n converges to the union of [-2i, 2i] and L in the Hausdorff metric. The proof is based on the asymptotic analysis of the eigenvalues of special non Hermitian perturbations of Hermitian Toeplitz matrices. (c) 2016 Elsevier Inc. All rights reserved.
    No preview · Article · Jun 2016 · Journal of Number Theory
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    [Show abstract] [Hide abstract] ABSTRACT: The paper is devoted to the structure and the asymptotics of the eigenvector matrix of Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The results extend existing results on banded Toeplitz matrices to full Toeplitz matrices with temperate decay of the entries in the first row and column. We establish formulas for both the exact and the asymptotic computation of arbitrarily prescribed individual components of arbitrarily prescribed individual eigenvectors. These formulas are in terms of the Wiener–Hopf factorization of a function which depends solely on the symbol and the corresponding eigenvalue or even only on an approximation to this eigenvalue. The paper does not aim at providing numerical algorithms. Its main purpose is rather to reveal the structure underneath the eigenvectors, which are shown to be the sum of two harmonics and certain edge corrections.
    Full-text · Article · Mar 2016 · Linear Algebra and its Applications
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    [Show abstract] [Hide abstract] ABSTRACT: We consider the well-known Rosenbloom-Tsfasman function field lattices in the special case of Hermitian function fields. We show that in this case the resulting lattices are generated by their minimal vectors, provide an estimate on the total number of minimal vectors, and derive properties of the automorphism groups of these lattices. Our study continues previous investigations of lattices coming from elliptic curves and finite Abelian groups. The lattices we are faced with here are more subtle than those considered previously, and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of Gerhard Hiss on the factorization of functions with particular divisor support into lines and their inverses.
    Full-text · Article · Feb 2016 · Journal of Algebra
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    [Show abstract] [Hide abstract] ABSTRACT: We consider lattices generated by finite Abelian groups. The main result says that such a lattice is strongly eutactic, which means the normalized minimal vectors of the lattice form a spherical 2-design, if and only if the group is of odd order or if it is a power of the group of order 2. This result also yields a criterion for the appropriately normalized minimal vectors to constitute a uniform normalized tight frame.
    Full-text · Article · Oct 2015
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    Johan Manuel Bogoya · Albrecht Boettcher · Egor A. Maximenko
    [Show abstract] [Hide abstract] ABSTRACT: 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 Szeg\H{o} type. Our results transfer these convergence theorems into uniform convergence statements.
    Full-text · Article · Sep 2015 · Boletín de la Sociedad Matemática Mexicana
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    Albrecht Böttcher · Danny Kowerko · Roland K.O. Sigel
    Full-text · Dataset · Apr 2015
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    Albrecht Böttcher · Danny Kowerko · Roland K.O. Sigel
    Full-text · Dataset · Apr 2015
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    Albrecht Böttcher · Danny Kowerko · Roland K O Sigel
    [Show abstract] [Hide abstract] ABSTRACT: The analysis of thermal melting curves requires the knowledge of equations for the temperature dependence of the relative fraction of folded and unfolded components. To implement these equations as standard tools for curve fitting, they should be as explicit as possible. From the van't Hoff formalism it is known that the equilibrium constant and hence the folded fraction is a function of the absolute temperature, the van't Hoff transition enthalpy, and the melting temperature. The work presented here is devoted to the mathematically self-contained derivation and the listing of explicit equations for the folded fraction as a function of the thermodynamic parameters in the case of arbitrary molecularities. Part of the results are known, others are new. It is in particular shown for the first time that the folded fraction is the composition of a universal function which depends solely on the molecularity and a dimensionless function which is governed by the concrete thermodynamic regime but is independent of the molecularity. The results will prove useful for extracting the thermodynamic parameters from experimental data on the basis of regression analysis. As supporting information, open-source Matlab scripts for the computer implementation of the equations are provided. Copyright © 2015 Elsevier B.V. All rights reserved.
    Full-text · Article · Apr 2015 · Biophysical Chemistry
  • J.M. Bogoya · A. Böttcher · S.M. Grudsky · E.A. Maximenko
    [Show abstract] [Hide abstract] ABSTRACT: The collective behavior of the singular values of large Toeplitz matrices is described by the Avram–Parter theorem. In the case of Hermitian matrices, the Avram–Parter theorem is equivalent to Szegő’s theorem on the eigenvalues. The Avram–Parter theorem in conjunction with an improvement made by Trench implies estimates in the mean between the singular values and the appropriately ordered absolute values of the symbol. The purpose of this paper is twofold. Under natural hypotheses, we first strengthen the known estimates in the mean to estimates in the maximum norm, thus turning from collective results on the singular values to results on individual singular values. Secondly, we want to emphasize that the use of the quantile function eases the proofs and statements of results significantly and provides a promising language for forthcoming research into higher order asymptotics for individual singular values.
    No preview · Article · Mar 2015 · Journal of Approximation Theory
  • J.M. Bogoya · A. Böttcher · S.M. Grudsky · E.A. Maximenko
    [Show abstract] [Hide abstract] ABSTRACT: The paper presents higher-order asymptotic formulas for the eigenvalues of large Hermitian Toeplitz matrices with moderately smooth symbols which trace out a simple loop on the real line. The formulas are established not only for the extreme eigenvalues, but also for the inner eigenvalues. The results extend and make more precise existing results, which so far pertain to banded matrices or to matrices with infinitely differentiable symbols. Also given is a fixed-point equation for the eigenvalues which may be solved numerically by an iteration method.
    No preview · Article · Feb 2015 · Journal of Mathematical Analysis and Applications
  • Albrecht Böttcher · Martin Halwass
    [Show abstract] [Hide abstract] ABSTRACT: 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)
    No preview · Article · Dec 2014 · PAMM
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    [Show abstract] [Hide abstract] ABSTRACT: The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher–Hartwig symbols. In that case we establish formulas for pure single Fisher–Hartwig singularities and for Hermitian matrices induced by general Fisher–Hartwig symbols.
    Full-text · Article · Nov 2014 · Journal of Functional Analysis
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    [Show abstract] [Hide abstract] ABSTRACT: The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig symbols.
    Full-text · Article · Jun 2014
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    [Show abstract] [Hide abstract] ABSTRACT: This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In case the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by Min Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.
    Full-text · Article · Jun 2014 · SIAM Journal on Discrete Mathematics
  • Albrecht Böttcher
    [Show abstract] [Hide abstract] ABSTRACT: We consider the determinants of compressions of Toeplitz operators to finite-dimensional model spaces and establish analogues of the Borodin-Okounkov formula and the strong Szegő limit theorem in this setting.
    No preview · Article · Mar 2014 · Integral Equations and Operator Theory
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    Albrecht Böttcher · Martin Halwass
    [Show abstract] [Hide abstract] ABSTRACT: The paper presents a novel Newton method for constructing canonical Wiener-Hopf factorizations of complex matrix polynomials and spectral factorizations of positive definite matrix polynomials. The factorizations are the ones needed for discrete-time linear systems and hence with respect to the unit circle. The Jacobi matrix is analyzed, and the convergence of the method is proved and tested numerically. A new class of highly ill-conditioned test polynomials is introduced, and the method is shown to manifest its very good performance also in this critical setting.
    Preview · Article · Dec 2013 · The electronic journal of linear algebra ELA
  • Albrecht Böttcher · Ilya Spitkovsky
    No preview · Chapter · Dec 2013
  • A. Böttcher · I.M. Spitkovsky
    [Show abstract] [Hide abstract] ABSTRACT: Let PP and QQ be idempotents in a real or complex associative algebra and consider the list of products P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,…P,Q,PQ,QP,PQP,QPQ,PQPQ,QPQP,…. The number of factors is called the order of the product. We say that PP and QQ are tightly coupled if the list contains two products which take the same value and whose orders differ by at most 11. The main result of the paper is the classification of all algebras which are generated by two tightly coupled idempotents. In other words, we provide a list of algebras such that every algebra generated by two tightly coupled idempotents is isomorphic to exactly one algebra of the list. For example, it follows that up to isomorphisms there are exactly 16 copies of such algebras in which the equality PQP=PQPQPQP=PQPQ holds.
    No preview · Article · Aug 2013 · Linear Algebra and its Applications
  • Albrecht Böttcher
    [Show abstract] [Hide abstract] ABSTRACT: Using results on block Toeplitz matrices, in particular the Szegő–Widom limit theorem and Gohberg and Feldman’s convergence theorem for the finite section method, we compute the spontaneous magnetization of an Ising model. In this way, the transition from the ferromagnetic phase to the paramagnetic phase may be understood as the transition from invertibility to non-invertibility of a certain block Toeplitz operator associated with the model.
    No preview · Article · Aug 2013 · Linear Algebra and its Applications
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    Albrecht Boettcher
    [Show abstract] [Hide abstract] ABSTRACT: Let H(N) denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in H(N) whose largest coefficients are as small as possible and also for polynomials in H(N) with minimal sum of the coefficients. Let h(N) and s(N) denote these minimal values. Using Fekete's subadditive lemma we show that the N-th square roots of h(N) and s(N) have a limit as N goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.
    Preview · Article · Aug 2013 · Computational Methods and Function Theory