A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

Variance bounds, with an application to norm bounds for commutators

Murthy and Sethi [M.N. Murthy, V.K. Sethi, Sankhya Ser. B 27 (1965) 201-210] gave a sharp upper bound on the variance of a real random variable in terms of the range of values of that variable. We generalise this bound to the complex case and, more importantly, to the matrix case. In doing so, we make contact with several geometrical and matrix analytical concepts, such as the numerical range, and introduce the new concept of radius of a matrix.We also give a new and simplified proof for a sharp upper bound on the Frobenius norm of commutators recently proven by Böttcher and Wenzel [A. Böttcher, D. Wenzel, The Frobenius norm and the commutator, Linear Algebra Appl. 429 (2008) 1864-1885] and point out that at the heart of this proof lies exactly the matrix version of the variance we have introduced. As an immediate application of our variance bounds we obtain stronger versions of Böttcher and Wenzel’s upper bound.     

The propensity to disrupt and the disruption nucleolus of a characteristic function game

Gately [1974] recently introduced the concept of an individual player's “propensity to disrupt” a payoff vector in a three-person characteristic function game. As a generalisation of this concept we propose the “disruption nucleolus” of ann-person game. The properties and computational possibilities of this concept are analogous to those of the nucleolus itself. Two numerical examples are given.     

On Hill's Bipotential

A new formulation of the constitutive law of materials based on the concept of bipotential was introduced by de Saxcé. This concept is able to describe non-associated laws and to suggest robust numerical algorithms. Materials that admit a bipotential to model their behaviour constitute the class of Implicit Standard Materials (ISM). This class contains, as a sub-class, the Generalized Standard Materials (GSM). In this paper, we present a bipotential expressing the coaxiality of two tensors with preservation of the order of the eigenvalues. We call it Hill's bipotential. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept. The authors acknowledge support through DFG Research Center “Mathematics for Key Technologies” (FZT 86) in Berlin.     

Spectral approximation in L(H)

We introduce the concept of numerical range approximant which is related to that of a spectral approximant. We obtain a variant of Bouldin result on spectral approximation of normal operators [3] and an analogous result for numerical range approximation . We also extend Maher's result on partially isometric approximation [9] and Halmos' on positive approximation [6].     

On Positivity and Roots in Operator Algebras

In earlier papers the second author and Charles Read have introduced and studied a new notion of positivity for operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. The present paper consists of complements to some facts in the just mentioned papers, concerning this notion of positivity. For example we prove a result on the numerical range of products of the roots of commuting operators with numerical range in a sector.     

An extension of the Krylov method for calculating the coefficients of the minimal polynomial

The concept of a k-minimal polynomial of an operator is introduced, and a method for approximate calculation of the coefficients of this polynomial is proposed. The method uses the calculated values of certain functionals on iterations of the operator. Special features emerging when the algorithm is used in combination with the Monte Carlo method are discussed, and numerical results are given.     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

The calculation of the distance to a nearby defective matrix

The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill‐conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter‐dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part II: comparison with experimental evidence

In dell’Isola et al. (Zeitschrift für Angewandte Math und Physik 66(6):3473–3498, 2015, Proc R Soc Lond A Math Phys Eng Sci 472(2185):1–23, 2016) pantographic sheets are proposed as a basic constituent for a novel metamaterial. In Part I, see Turco et al. (Zeitschrift für Angewandte Math und Physik, doi:10.1007/s00033-016-0713-4, 2016), two different numerical models are applied in order to design an experimental setup aimed to prove the effectiveness of introduced concept. The aim of this paper is to prove that the Hencky-type model introduced for planar pantographic sheets allows for the correct prediction, in a large range of imposed displacements, of the experimental measurements concerning specimens undergoing coupled bending and extensional deformations. The four-parameter numerical model introduced is shown to have a large range of applicability: Indeed without changing the values of the material parameters previously attributed in simple extensional tests to a specific specimen by a best-fit procedure, it is possible to forecast its behavior in all the considered type of imposed deformations. The measurements performed include the determination of reactive forces exerted by used hard devices, and the numerical modeling is able to predict very carefully quantitatively and qualitatively also this complex aspect of phenomenology, where previously attempted models seem to have failed.     

A one-dimensional model for dispersive wave turbulence

Summary A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|^−1/3) spectrum compared with the steeper (|k|^−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.     

Simultaneous partial Padé approximants

In this paper the concept of partial Padé approximation, introduced by Claude Brezinski, is generalised to the case of simultaneous rational approximation with common denominator. The use of information about known poles and/or zeros, can lead to approximants with a better numerical behaviour than in the case of ordinary simultaneous Padé approximation.     

线性约束下的共轭投影变尺度法及其超线性收敛性

§1.引言 对于线性约束非线性规划问题,自从Zoutendijk于1960年提出容许方向法以来,相继出现了很多可行方向法,特别是Rosen和Goldfarb的梯度投影法引人注目。很多作者对他们的方法进行了各种形式的改进,把线性约束的情形推广到非线性约束的情形,从凸规划的可行方向法发展到非凸规划的可行方向法,通过引进ε-有效约束集的概念,从     

STUDENT TEACHERS’ UNDERSTANDING OF RATIONAL NUMBER: PART-WHOLE AND NUMERICAL CONSTRUCTS

Previous research has shown that secondary school students’ understanding of fractions is dominated by the part-whole concept to the possible detriment of their understanding of a fraction as a number in its own right. The present paper reports on an investigation into the understanding of intending primary teachers in this area. Four representatives of a cohort of sixty students on a PGCE course specialising in the lower primary age range were asked detailed questions probing their knowledge of fractions. The conclusion was that the part-whole concept dominates. All of the students were familiar with the numerical concept from their work on the PGCE course, but they reverted to the more familiar part-whole ideas in attempting to solve problems.     

Birkhoff-James approximate orthogonality sets and numerical ranges

In this paper, the notion of Birkhoff-James approximate orthogonality sets is introduced for rectangular matrices and matrix polynomials. The proposed definition yields a natural generalization of standard numerical range and q-numerical range (and also of recent extensions), sharing with them several geometric properties.     

Graph Collapsing in Shortest Path Auction Algorithms

In this paper we consider the problem of finding a shortest path from a source node to a fixed target node (SSP) or to all the nodes (SPT) on a directed graph. A family of algorithms which derives from the known auction algorithm is introduced. The key feature of these algorithms is based on topological transformations operated on the graphs that replace an optimal sub-path with a single arc of the same length (graph collapsing concept). The same idea is applied both to the standard auction algorithm and to a modified version of the algorithm. In the last mentioned case a good saving in computation cost is obtained as shown by the reported numerical examples.     

一个改进的解线性规划问题的熵函数法

本文将有效因子的概念引入到Shannon熵的信息结构中,提出了一个改进的解线规划问题的熵函数法,随后的理论结果和数值例子表明了本文提出了的方法是有效的。     

Square-root Householder subspace tracking

A class of singular value decomposition (SVD)-type subspace trackers based on the overdetermined row-Householder principle is introduced. These algorithms are maximally fast with a dominant operations count of 3Nr multiplications per time update. They can be regarded as square-root forms of previously introduced conventional fast subspace trackers and offer interesting features such as perfectly orthonormal basis estimates, lowest dynamic range requirements, and highest numerical robustness and stability. Several variants of the method are proposed and studied experimentally.     

On the numerical range in tropical algebra

ABSTRACT

We introduce the concept of numerical range in tropical linear algebra, explore its properties and show in particular that it contains all the tropical eigenvalues. An explicit formula is obtained for the tropical numerical range and an application in the asymptotic analysis of condition number of symmetric matrices is proposed.