Albrecht Böttcher

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

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Publications (182)84.14 Total impact

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    Albrecht Böttcher · Danny Kowerko · Roland K.O. Sigel
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    Albrecht Böttcher · Danny Kowerko · Roland K.O. Sigel
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    Albrecht Böttcher · Danny Kowerko · Roland K O Sigel
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    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.
    Biophysical Chemistry 04/2015; 202. DOI:10.1016/j.bpc.2015.04.001 · 2.32 Impact Factor
  • J.M. Bogoya · A. Böttcher · S.M. Grudsky · E.A. Maximenko
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    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.
    Journal of Approximation Theory 03/2015; 196:75--100. DOI:10.1016/j.jat.2015.03.003 · 0.90 Impact Factor
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    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.
  • J.M. Bogoya · A. Böttcher · S.M. Grudsky · E.A. Maximenko
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    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.
    Journal of Mathematical Analysis and Applications 02/2015; 422(2). DOI:10.1016/j.jmaa.2014.09.057 · 1.12 Impact Factor
  • Albrecht Böttcher · Martin Halwass
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    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)
    PAMM 12/2014; 14(1). DOI:10.1002/pamm.201410389
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    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.
    Journal of Functional Analysis 11/2014; 268(1). DOI:10.1016/j.jfa.2014.10.023 · 1.15 Impact Factor
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    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.
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    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.
    SIAM Journal on Discrete Mathematics 06/2014; 29(1). DOI:10.1137/140982520 · 0.58 Impact Factor
  • Albrecht Böttcher
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    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.
    Integral Equations and Operator Theory 03/2014; 3(3). DOI:10.1007/s00020-013-2118-5 · 0.58 Impact Factor
  • A. Böttcher · I. M. Spitkovsky
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    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.
    Linear Algebra and its Applications 08/2013; 439(3):538–551. DOI:10.1016/j.laa.2012.07.020 · 0.98 Impact Factor
  • Albrecht Böttcher
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    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.
    Linear Algebra and its Applications 08/2013; 439(3):675–685. DOI:10.1016/j.laa.2012.09.025 · 0.98 Impact Factor
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    Albrecht Boettcher
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    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.
    08/2013; 14(1). DOI:10.1007/s40315-014-0061-3
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    Albrecht Böttcher
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    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 Szeg\"o limit theorem in this setting.
  • Albrecht Böttcher · Martin Halwass
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    ABSTRACT: We present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle, respectively. The approach is based on solving a nonlinear system by Newton’s method. The Jacobi matrix is a polynomial of a companion matrix and thus a Bezoutian times the inverse of a triangular Toeplitz matrix. After getting deeper understanding of the displacement structure of the Bezoutian, each Newton step can be reduced to a few applications of discrete Fourier or cosine transforms and the LU-decomposition of a conceived linear system with a Cauchy-like matrix. These structural features are employed to design a fast algorithm, which shows excellent numerical behavior. For example, in the case of the extremely ill-conditioned test polynomial (2zn+∑k=0n-1zk)(2+∑k=1nzk) of the degree 2n=200002n=20000, we obtain the factorization after 22 Newton steps with an error of 3.4·10-83.4·10-8 in the Euclidean norm, the execution time on a laptop being a couple of minutes. The local quadratic convergence of our Newton method is proved and explicit convergence balls are given on the basis of estimates for the Lipschitz constant which occurs in Kantorovich’s theorem. We also design and test two superfast versions of our method, which, for polynomials of degree 20000, typically need about half a minute to produce an initial coefficient vector and then perform the Newton iteration within 1 s.
    Linear Algebra and its Applications 06/2013; 438(12):4760–4805. DOI:10.1016/j.laa.2013.02.020 · 0.98 Impact Factor
  • Albrecht Böttcher · Ilya M. Spitkovsky
  • Albrecht Böttcher · Martin Halwass
    The electronic journal of linear algebra ELA 01/2013; · 0.89 Impact Factor
  • Albrecht Böttcher · Albrecht Pietsch
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    ABSTRACT: In the theory of traces on operator ideals, it is desirable to treat not only the complex case. Several proofs become much easier when the underlying operators are represented by real matrices. Motivated by this observation, we prove two theorems which, to the best of our knowledge, are not available in the real setting: (1) every operator is a finite linear combination of orthogonal operators, and (2) every skew-symmetric compact operator S is a commutator [A, T], where certain properties of S are inherited to T. In our opinion, theses results are interesting for their own sake. They will also be used in future studies of trace theory by the second-named author.
    Integral Equations and Operator Theory 12/2012; 74(4). DOI:10.1007/s00020-012-1999-z · 0.58 Impact Factor
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    ABSTRACT: The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper confirms and makes more precise a conjecture by Dai, Geary, and Kadanoff of 2009 in a particular case.
    Linear Algebra and its Applications 05/2012; 436(9):3480–3492. DOI:10.1016/j.laa.2011.12.012 · 0.98 Impact Factor