Showing papers on "Matrix (mathematics) published in 2002"

The Bethe-Ansatz for N=4 Super Yang-Mills

Abstract: We derive the one loop mixing matrix for anomalous dimensions in N=4 Super Yang-Mills. We show that this matrix can be identified with the Hamiltonian of an integrable SO(6) spin chain with vector sites. We then use the Bethe ansatz to find a recipe for computing anomalous dimensions for a wide range of operators. We give exact results for BMN operators with two impurities and results up to and including first order 1/J corrections for BMN operators with many impurities. We then use a result of Reshetikhin's to find the exact one-loop anomalous dimension for an SO(6) singlet in the limit of large bare dimension. We also show that this last anomalous dimension is proportional to the square root of the string level in the weak coupling limit.

Solving Linear Rational Expectations Models

Abstract: We describe methods for solving general linear rational expectations models in continuous or discrete timing with or without exogenous variables. The methods are based on matrix eigenvalue decompositions.

Iterative oblique projection onto convex sets and the split feasibility problem

Abstract: Let C and Q be nonempty closed convex sets in R N and R M , respectively, and A an M by N real matrix. The split feasibility problem (SFP) is to find x ∈ C with Ax ∈ Q ,i f suchx exist. An iterative method for solving the SFP, called the CQ algorithm, has the following iterative step: x k+1 = PC (x k + γ A T (PQ − I )Ax k ), where γ ∈ (0, 2/L) with L the largest eigenvalue of the matrix A T A and PC and PQ denote the orthogonal projections onto C and Q, respectively; that is, PC x minimizesc − x� ,o ver allc ∈ C.T heCQ algorithm converges to a solution of the SFP, or, more generally, to a minimizer ofPQ Ac − Acover c in C, whenever such exist. The CQ algorithm involves only the orthogonal projections onto C and Q, which we shall assume are easily calculated, and involves no matrix inverses. If A is normalized so that each row has length one, then L does not exceed the maximum number of nonzero entries in any column of A, which provides a helpful estimate of L for sparse matrices. Particular cases of the CQ algorithm are the Landweber and projected Landweber methods for obtaining exact or approximate solutions of the linear equations Ax = b ;t healgebraic reconstruction technique of Gordon, Bender and Herman is a particular case of a block-iterative version of the CQ algorithm. One application of the CQ algorithm that is the subject of ongoing work is dynamic emission tomographic image reconstruction, in which the vector x is the concatenation of several images corresponding to successive discrete times. The matrix A and the set Q can then be selected to impose constraints on the behaviour over time of the intensities at fixed voxels, as well as to require consistency (or near consistency) with measured data.

Computing the nearest correlation matrix—a problem from finance

Abstract: Given a symmetric matrix what is the nearest correlation matrix, that is, the nearest symmetric positive semidefinite matrix with unit diagonal? This problem arises in the finance industry, where the correlations are between stocks. For distance measured in two weighted Frobenius norms we characterize the solution using convex analysis. We show how the modified alternating projections method can be used to compute the solution for the more commonly used of the weighted Frobenius norms. In the finance application the original matrix has many zero or negative eigenvalues; we show that for a certain class of weights the nearest correlation matrix has correspondingly many zero eigenvalues and that this fact can be exploited in the computation.

General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference

Abstract: The optical response of coherent thin-film multilayers is often represented with Fresnel coefficients in a 2 x 2 matrix configuration. Here the usual transfer matrix was modified to a generic form, with the ability to use the absolute squares of the Fresnel coefficients, so as to include incoherent (thick layers) and partially coherent (rough surface or interfaces) reflection and transmission. The method is integrated by use of models for refractive-index depth profiling. The utility of the method is illustrated with various multilayer structures formed by ion implantation into Si, including buried insulating and conducting layers, and multilayers with a thick incoherent layer in an arbitrary position.

Plasma protein matrices and methods for their preparation

Abstract: A freeze dried biocompatible matrix comprising plasma proteins, useful as implants for tissue engineering as well as in biotechnology, and methods of producing said matrix are provided. Mechanical and physical parameters can be controlled by use of auxiliary components or additives which may be removed after the matrix is formed in order to improve the biological properties of the matrix. The matrices according to the present invention may be used clinically, per se or as a cell-bearing implant.

A subspace approach to balanced truncation for model reduction of nonlinear control systems

Abstract: In this paper, we introduce a new method of model reduction for nonlinear control systems. Our approach is to construct an approximately balanced realization. The method requires only standard matrix computations, and we show that when it is applied to linear systems it results in the usual balanced truncation. For nonlinear systems, the method makes use of data from either simulation or experiment to identify the dynamics relevant to the input}output map of the system. An important feature of this approach is that the resulting reduced-order model is nonlinear, and has inputs and outputs suitable for control. We perform an example reduction for a nonlinear mechanical system.

Comparative quantitative genetics : evolution of the G matrix

Abstract: Quantitative genetics provides one of the most promising frameworks with which to unify the fields of macroevolution and microevolution. The genetic variance–covariance matrix ( G ) is crucial to quantitative genetic predictions about macroevolution. In spite of years of study, we still know little about how G evolves. Recent studies have been applying an increasingly phylogenetic perspective and more sophisticated statistical techniques to address G matrix evolution. We propose that a new field, comparative quantitative genetics, has emerged. Here we summarize what is known about several key questions in the field and compare the strengths and weaknesses of the many statistical and conceptual approaches now being employed. Past studies have made it clear that the key question is no longer whether G evolves but rather how fast and in what manner. We highlight the most promising future directions for this emerging field.

A recurrent neural network for solving Sylvester equation with time-varying coefficients

Abstract: Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network. Simulation results of time-varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network.

Non-orthogonal joint diagonalization in the least-squares sense with application in blind source separation

Abstract: Approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation (BSS) algorithms. A common measure of the attained diagonalization of the set is the weighted least-squares (WLS) criterion. However, most well-known algorithms are restricted to finding an orthogonal diagonalizing matrix, relying on a whitening phase for the nonorthogonal factor. Often, such an approach implies unbalanced weighting, which can result in degraded performance. We propose an iterative alternating-directions algorithm for minimizing the WLS criterion with respect to a general (not necessarily orthogonal) diagonalizing matrix. Under some mild assumptions, we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations. We distinguish between Hermitian and symmetrical problems. Using BSS simulations results, we demonstrate the improvement in estimating the mixing matrix, resulting from the relaxation of the orthogonality restriction.

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

Abstract: This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite nonzero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.

Schwinger-Keldysh Propagators from AdS/CFT Correspondence

Abstract: We demonstrate how to compute real-time Green's functions for a class of finite temperature field theories from their AdS gravity duals. In particular, we reproduce the two-by-two Schwinger-Keldysh matrix propagator from a gravity calculation. Our methods should work also for computing higher point Lorentzian signature correlators. We elucidate the boundary condition subtleties which hampered previous efforts to build a Lorentzian-signature AdS/CFT correspondence. For two-point correlators, our construction is automatically equivalent to the previously formulated prescription for the retarded propagator.

Fast curvature matrix-vector products for second-order gradient descent

Abstract: We propose a generic method for iteratively approximating various second-order gradient steps--Newton, Gauss-Newton, Levenberg-Marquardt, and natural gradient--in linear time per iteration, using special curvature matrix-vector products that can be computed in O(n). Two recent acceleration techniques for on-line learning, matrix momentum and stochastic meta-descent (SMD), implement this approach. Since both were originally derived by very different routes, this offers fresh insight into their operation, resulting in further improvements to SMD.

Correcting the Colour-Dipole Cascade Model with Fixed Order Matrix Elements

Abstract: An algorithm is presented in which the Colour-Dipole Cascade Model as implemented in the Ariadne program is corrected to match the fixed order tree-level matrix elements for e e(+) --> n jets. The result is a full parton level generator for e e(+) annihilation where the generated states are correct on tree-level to fixed order in alpha(s) and to all orders with modified leading logarithmic (MLLA) accuracy. In this paper, matrix elements are used up to second order in alpha(s), but the scheme is applicable also for higher orders. A strategy for also including exact virtual corrections to fixed order is suggested and the possibility to extend the scheme to hadronic collisions is discussed.

Regularization of geopotential determination from satellite data by variance components

Abstract: Different types of present or future satellite data have to be combined by applying appropriate weighting for the determination of the gravity field of the Earth, for instance GPS observations for CHAMP with satellite to satellite tracking for the coming mission GRACE as well as gradiometer measurements for GOCE In addition, the estimate of the geopotential has to be smoothed or regularized because of the inversion problem It is proposed to solve these two tasks by Bayesian inference on variance components The estimates of the variance components are computed by a stochastic estimator of the traces of matrices connected with the inverse of the matrix of normal equations, thus leading to a new method for determining variance components for large linear systems The posterior density function for the variance components, weighting factors and regularization parameters are given in order to compute the confidence intervals for these quantities Test computations with simulated gradiometer observations for GOCE and satellite to satellite tracking for GRACE show the validity of the approach

Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

Abstract: Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left-right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.

Large N and double scaling limits in two dimensions

Abstract: Recently, the author has constructed a series of four dimensional non-critical string theories with eight supercharges, dual to theories of light electric and magnetic charges, for which exact formulas for the central charge of the space-time supersymmetry algebra as a function of the world-sheet couplings were obtained. The basic idea was to generalize the old matrix model approach, replacing the simple matrix integrals by the four dimensional matrix path integrals of N=2 supersymmetric Yang-Mills theory, and the Kazakov critical points by the Argyres-Douglas critical points. In the present paper, we study qualitatively similar toy path integrals corresponding to the two dimensional N=2 supersymmetric non-linear sigma model with target space CP^n and twisted mass terms. This theory has some very strong similarities with N=2 super Yang-Mills, including the presence of critical points in the vicinity of which the large n expansion is IR divergent. The model being exactly solvable at large n, we can study non-BPS observables and give full proofs that double scaling limits exist and correspond to universal continuum limits. A complete characterization of the double scaled theories is given. We find evidence for dimensional transmutation of the string coupling in some non-critical string theories. We also identify en passant some non-BPS particles that become massless at the singularities in addition to the usual BPS states.

Matrix elements and parton showers in hadronic interactions.

Abstract: A method is suggested to combine tree level QCD matrix elements for the production of multi jet final states and the parton shower in hadronic interactions. The method follows closely an algorithm developed recently for the case of e+e− annihilations \cite{Catani:2001cc}.

Ab initio molecular dynamics in a finite homogeneous electric field

Abstract: We treat homogeneous electric fields within density functional calculations with periodic boundary conditions. A nonlocal energy functional depending on the applied field is used within an ab initio molecular dynamics scheme. The reliability of the method is demonstrated in the case of bulk MgO for the Born effective charges, and the high- and low-frequency dielectric constants. We evaluate the static dielectric constant by performing a damped molecular dynamics in an electric field and avoiding the calculation of the dynamical matrix. Application of this method to vitreous silica shows good agreement with experiment and illustrates its potential for systems of large size.

Adaptive synthesis and design of resonator filters with source/load-multiresonator coupling

Abstract: The paper presents a universal and comprehensive synthesis technique of coupled resonator filters with source/load-multiresonator coupling. The approach is based on repeated analyses of a circuit with the desired topology; no similarity transformation is needed. Restrictions imposed by the implementation on the coupling coefficients such as signs and orders of magnitudes are straightforwardly handled within this technique. The technique is then used to synthesize and design filters with full or almost full coupling matrices by selecting, among the infinite number of solutions, the matrix that corresponds to the actual implementation. In such cases, analytical techniques and those based on similarity transformations cannot be used since they provide no mechanism to constrain individual coupling coefficients in order to discriminate between two full coupling matrices, which are both solutions to the synthesis problem. Using the technique described in this paper, a filter designer can extract the coupling matrix of a filter of arbitrary order and topology while enforcing relevant constraints. There is no need to master all the different existing similarity-transformation-based techniques and the topologies to which they are applicable. For the first time, detailed investigations of parasitic coupling effects, for either compensation or utilization, are made possible. The method is applied to the synthesis of a variety of filters, some of which are then designed and built and their response measured.

Products of random matrices.

Abstract: We derive analytic expressions for infinite products of random 2 x 2 matrices. The determinant of the target matrix is log-normally distributed, whereas the remainder is a surprisingly complicated function of a parameter characterizing the norm of the matrix and a parameter characterizing its skewness. The distribution may have importance as an uncommitted prior in statistical image analysis.

The MUSIC-algorithm and the factorization method in inverse scattering theory for inhomogeneous media

Abstract: We consider the scattering of time-harmonic plane waves by an inhomogeneous medium. The far field patterns u? of the scattered waves depend on the index of refraction 1 + q, the frequency, and directions and of observation and incidence, respectively. The inverse problem which is studied in this paper is to determine the support ? of q from the knowledge of u? (, ) for all , where the frequency is fixed (and known). Our new approach is based on the far field operator F which is the integral operator with kernel u? (, ). It depends on the data only and is therefore known (at least approximately). The MUSIC algorithm in signal processing uses the discrete version of F, i.e. the matrix F = (u? ( i, j)) N?N, and determines the locations of the point scatterers. The key idea in both cases is to factorize F and F in the forms where the operator S and the matrix S are 'more explicit' than F and F, respectively, and T, T are suitable isomorphisms. In a first theoretical result we show that the ranges of S and F# coincide, where F# is some suitable combination of the real and imaginary parts of F. In the finite dimensional case a simple argument from matrix theory yields that the ranges of S and F coincide. Since F# is known from the data we can decide for every function on the unit sphere whether it belongs to the range of S or not. We apply this test to the far field patterns of point sources and arrive at an explicit test whether a point z belongs to ? or not. We will demonstrate that this method also leads to a fast visualization of the obstacle.

Computer code for rational approximation of frequency dependent admittance matrices

Abstract: This paper deals with the problem of approximating with rational functions a matrix whose frequency-dependent elements have been obtained from calculations or from measurements. Based on a previously developed technique (vector fitting), a set of callable routines have been written in the MATLAB language. These routines allow for rational approximation with a common set of stable poles, automatic selection of initial poles, passivity enforcement, and creation of an equivalent electrical network which can be imported into ATP-EMTP. Usage of sparse arithmetic permits the computer code to handle large systems. The methodology is demonstrated by application to a frequency-dependent network equivalent of a radial distribution network (phase domain), for which the accuracy is validated in both the frequency domain and the time domain. The computer code is in the public domain and is available from the author.

Nanocomposite materials with engineered properties

Abstract: The invention relates to a nanocomposite material. The nanocomposite material comprises a matrix material and a plurality of quantum dots dispersed in the matrix material. The nanocomposite material has a nonlinear index of refraction γ that is at least 10 −9 cm 2 /W when irradiated with light having a wavelength λ between approximately 3×10 −5 cm and 2×10 −4 cm.