The Algebraic Theory of Matrix Polynomials

A matrix S is a solvent of the matrix polynomial M(X)=A₀Xᵐ +...+ Am if M(S)=O where A, X, and S are square matrices. In this paper we develop the algebraic theory of matrix polynomials and solvents. We define division and interpolation, investigate the properties of block Vandermonde matrices, and define and study the existence of a complete set of solvents. We study the relation between the matrix polynomial problem and the lambda-matrix problem, which is to find a scalar A₀λᵐ + A₁λᵐ⁻¹ +...+ Am is singular. In a future paper we extend Traub’s algorithm for calculating zeros of scalar polynomials to matrix polynomials and establish global convergence properties of this algorithm for a class of matrix polynomials

Algorithms for Solvents of Matrix Polynomials

In an earlier paper we developed the algebraic theory of matrix polynomials. Here we introduce two algorithms for computing "dominant" solvents. Global convergence of the algorithms under certain conditions is established

A Variable Metric Variant of the Karmarkar Algorithm for Linear Programming

The most time-consuming part of the Karmarkar algorithm for linear programming is the projection of a vector onto the nullspace of a matrix that changes at each iteration. We present a variant of the Karmarkar algorithm that uses standard variable-metric techniques in an innovative way to approximate this projection. In limited tests, this modification greatly reduces the number of matrix factorizations needed for the solution of linear programming problems

A curvilinear search using tridiagonal secant updates for unconstrained optimization

The idea of doing a curvilinear search along the Levenberg- Marquardt path s(μ) = - (H + μI)⁻¹g always has been appealing, but the cost of solving a linear system for each trial value of the parameter y has discouraged its implementation. In this paper, an algorithm for searching along a path which includes s(μ) is studied. The algorithm uses a special inexpensive QTcQT to QT₊QT Hessian update which trivializes the linear algebra required to compute s(μ). This update is based on earlier work of Dennis-Marwil and Martinez on least-change secant updates of matrix factors. The new algorithm is shown to be local and q-superlinearily convergent to stationary points, and to be globally q-superlinearily convergent for quasi-convex functions. Computational tests are given that show the new algorithm to be robust and efficient.Facultad de Ciencias Exacta

Spinodal Decomposition in a Binary Polymer Mixture: Dynamic Self Consistent Field Theory and Monte Carlo Simulations

We investigate how the dynamics of a single chain influences the kinetics of early stage phase separation in a symmetric binary polymer mixture. We consider quenches from the disordered phase into the region of spinodal instability. On a mean field level we approach this problem with two methods: a dynamical extension of the self consistent field theory for Gaussian chains, with the density variables evolving in time, and the method of the external potential dynamics where the effective external fields are propagated in time. Different wave vector dependencies of the kinetic coefficient are taken into account. These early stages of spinodal decomposition are also studied through Monte Carlo simulations employing the bond fluctuation model that maps the chains -- in our case with 64 effective segments -- on a coarse grained lattice. The results obtained through self consistent field calculations and Monte Carlo simulations can be compared because the time, length, and temperature scales are mapped onto each other through the diffusion constant, the chain extension, and the energy of mixing. The quantitative comparison of the relaxation rate of the global structure factor shows that a kinetic coefficient according to the Rouse model gives a much better agreement than a local, i.e. wave vector independent, kinetic factor. Including fluctuations in the self consistent field calculations leads to a shorter time span of spinodal behaviour and a reduction of the relaxation rate for smaller wave vectors and prevents the relaxation rate from becoming negative for larger values of the wave vector. This is also in agreement with the simulation results.Comment: Phys.Rev.E in prin

Novel genetic loci associated with hippocampal volume

The hippocampal formation is a brain structure integrally involved in episodic memory, spatial navigation, cognition and stress responsiveness. Structural abnormalities in hippocampal volume and shape are found in several common neuropsychiatric disorders. To identify the genetic underpinnings of hippocampal structure here we perform a genome-wide association study (GWAS) of 33,536 individuals and discover six independent loci significantly associated with hippocampal volume, four of them novel. Of the novel loci, three lie within genes (ASTN2, DPP4 and MAST4) and one is found 200 kb upstream of SHH. A hippocampal subfield analysis shows that a locus within the MSRB3 gene shows evidence of a localized effect along the dentate gyrus, subiculum, CA1 and fissure. Further, we show that genetic variants associated with decreased hippocampal volume are also associated with increased risk for Alzheimer's disease (rg =-0.155). Our findings suggest novel biological pathways through which human genetic variation influences hippocampal volume and risk for neuropsychiatric illness

An Observational Overview of Solar Flares

We present an overview of solar flares and associated phenomena, drawing upon a wide range of observational data primarily from the RHESSI era. Following an introductory discussion and overview of the status of observational capabilities, the article is split into topical sections which deal with different areas of flare phenomena (footpoints and ribbons, coronal sources, relationship to coronal mass ejections) and their interconnections. We also discuss flare soft X-ray spectroscopy and the energetics of the process. The emphasis is to describe the observations from multiple points of view, while bearing in mind the models that link them to each other and to theory. The present theoretical and observational understanding of solar flares is far from complete, so we conclude with a brief discussion of models, and a list of missing but important observations.Comment: This is an article for a monograph on the physics of solar flares, inspired by RHESSI observations. The individual articles are to appear in Space Science Reviews (2011

Surrogate Modelling and Space Mapping for Engineering Optimization: A Summary of the Danish Technical University November 2000 Workshop

This is intended to be an outline of the topics presented in the November 16-18, 2000 workshop held at the Danish Technical University in Lyngby outside Copenhagen. The focus of this workshop was on the construction of inexpensive models and their use with optimization to support engineering decision making. The purpose of this summary is to organize the various topics presented there into a unified whole. An additional purpose is to give an intuitive introduction to the promising concept of space mapping as a way to leverage a cheap coarse model into an effective optimization surrogate for a more detailed model