Calculation of transmission probability by solving an eigenvalue problem

NASA Astrophysics Data System (ADS)

Bubin, Sergiy; Varga, Kálmán

2010-11-01

The electron transmission probability in nanodevices is calculated by solving an eigenvalue problem. The eigenvalues are the transmission probabilities and the number of nonzero eigenvalues is equal to the number of open quantum transmission eigenchannels. The number of open eigenchannels is typically a few dozen at most, thus the computational cost amounts to the calculation of a few outer eigenvalues of a complex Hermitian matrix (the transmission matrix). The method is implemented on a real space grid basis providing an alternative to localized atomic orbital based quantum transport calculations. Numerical examples are presented to illustrate the efficiency of the method.

Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar

2016-06-15

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. -- Highlights: •Symmetric quadratic operators aremore » useful models for many physical applications. •Any such operator exhibits a pseudo-Hermitian matrix representation. •Its eigenvalues are the natural frequencies of the Hamiltonian operator. •The eigenvalues may be real or complex and describe a phase transition.« less

Rich structure in the correlation matrix spectra in non-equilibrium steady states

NASA Astrophysics Data System (ADS)

Biswas, Soham; Leyvraz, Francois; Monroy Castillero, Paulino; Seligman, Thomas H.

2017-01-01

It has been shown that, if a model displays long-range (power-law) spatial correlations, its equal-time correlation matrix will also have a power law tail in the distribution of its high-lying eigenvalues. The purpose of this paper is to show that the converse is generally incorrect: a power-law tail in the high-lying eigenvalues of the correlation matrix may exist even in the absence of equal-time power law correlations in the initial model. We may therefore view the study of the eigenvalue distribution of the correlation matrix as a more powerful tool than the study of spatial Correlations, one which may in fact uncover structure, that would otherwise not be apparent. Specifically, we show that in the Totally Asymmetric Simple Exclusion Process, whereas there are no clearly visible correlations in the steady state, the eigenvalues of its correlation matrix exhibit a rich structure which we describe in detail.

Rich structure in the correlation matrix spectra in non-equilibrium steady states.

PubMed

Biswas, Soham; Leyvraz, Francois; Monroy Castillero, Paulino; Seligman, Thomas H

2017-01-17

It has been shown that, if a model displays long-range (power-law) spatial correlations, its equal-time correlation matrix will also have a power law tail in the distribution of its high-lying eigenvalues. The purpose of this paper is to show that the converse is generally incorrect: a power-law tail in the high-lying eigenvalues of the correlation matrix may exist even in the absence of equal-time power law correlations in the initial model. We may therefore view the study of the eigenvalue distribution of the correlation matrix as a more powerful tool than the study of spatial Correlations, one which may in fact uncover structure, that would otherwise not be apparent. Specifically, we show that in the Totally Asymmetric Simple Exclusion Process, whereas there are no clearly visible correlations in the steady state, the eigenvalues of its correlation matrix exhibit a rich structure which we describe in detail.

Spectrum of walk matrix for Koch network and its application

NASA Astrophysics Data System (ADS)

Xie, Pinchen; Lin, Yuan; Zhang, Zhongzhi

2015-06-01

Various structural and dynamical properties of a network are encoded in the eigenvalues of walk matrix describing random walks on the network. In this paper, we study the spectra of walk matrix of the Koch network, which displays the prominent scale-free and small-world features. Utilizing the particular architecture of the network, we obtain all the eigenvalues and their corresponding multiplicities. Based on the link between the eigenvalues of walk matrix and random target access time defined as the expected time for a walker going from an arbitrary node to another one selected randomly according to the steady-state distribution, we then derive an explicit solution to the random target access time for random walks on the Koch network. Finally, we corroborate our computation for the eigenvalues by enumerating spanning trees in the Koch network, using the connection governing eigenvalues and spanning trees, where a spanning tree of a network is a subgraph of the network, that is, a tree containing all the nodes.

Multicomponent diffusion in basaltic melts at 1350 °C

NASA Astrophysics Data System (ADS)

Guo, Chenghuan; Zhang, Youxue

2018-05-01

Nine successful diffusion couple experiments were conducted in an 8-component SiO2-TiO2-Al2O3-FeO-MgO-CaO-Na2O-K2O system at ∼1350 °C and at 1 GPa, to study multicomponent diffusion in basaltic melts. At least 3 traverses were measured to obtain diffusion profiles for each experiment. Multicomponent diffusion matrix at 1350 °C was obtained by simultaneously fitting diffusion profiles of diffusion couple experiments. Furthermore, in order to better constrain the diffusion matrix and reconcile mineral dissolution data, mineral dissolution experiments in the literature and diffusion couple experiments from this study, were fit together. All features of diffusion profiles in both diffusion couple and mineral dissolution experiments were well reproduced by the diffusion matrix. Diffusion mechanism is inferred from eigenvectors of the diffusion matrix, and it shows that the diffusive exchange between network-formers SiO2 and Al2O3 is the slowest, the exchange of SiO2 with other oxide components is the second slowest with an eigenvalue that is only ∼10% larger, then the exchange between divalent oxide components and all the other oxide components is the third slowest with an eigenvalue that is twice the smallest eigenvalue, then the exchange of FeO + K2O with all the other oxide components is the fourth slowest with an eigenvalue that is 5 times the smallest eigenvalue, then the exchange of MgO with FeO + CaO is the third fastest with an eigenvalue that is 6.3 times the smallest eigenvalue, then the exchange of CaO + K2O with all the other oxide components is the second fastest with an eigenvalue that is 7.5 times the smallest eigenvalue, and the exchange of Na2O with all other oxide components is the fastest with an eigenvalue that is 31 times the smallest eigenvalue. The slowest and fastest eigenvectors are consistent with those for simpler systems in most literature. The obtained diffusion matrix was successfully applied to predict diffusion profiles during mineral dissolution in basaltic melts.

Two-faced property of a market factor in asset pricing and diversification effect

NASA Astrophysics Data System (ADS)

Eom, Cheoljun

2017-04-01

This study empirically investigates the test hypothesis that a market factor acting as a representative common factor in the pricing models has a negative influence on constructing a well-diversified portfolio from the Markowitz mean-variance optimization function (MVOF). We use the comparative correlation matrix (C-CM) method to control a single eigenvalue among all eigenvalues included in the sample correlation matrix (S-CM), through the random matrix theory (RMT). In particular, this study observes the effect of the largest eigenvalue that has the property of the market factor. According to the results, the largest eigenvalue has the highest explanatory power on the stock return changes. The C-CM without the largest eigenvalue in the S-CM constructs a more diversified portfolio capable of improving the practical applicability of the MVOF. Moreover, the more diversified portfolio constructed from this C-CM has better out-of-sample performance in the future period. These results support the test hypothesis for the two-faced property of the market factor, defined by the largest eigenvalue.

A New Measure of Wireless Network Connectivity

DTIC Science & Technology

2014-10-31

matrix QG. From Lemma 1, QG is a non-zero nonnegative matrix. Thus from the Perron - Frobenius Theorem, [24], its largest magni- tude eigenvalue, known as...the Perron - Frobenius eigenvalue is real and positive. Further as QG is symmetric, all its eigenval- ues are real, and its largest magnitude...eigenvalue λmax(QG) is also its largest singular value. Also from the Perron - Frobenius Theorem, should the network be connected, i.e. QG is positive as opposed

A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems

NASA Astrophysics Data System (ADS)

Babaee, Hessam; Choi, Minseok; Sapsis, Themistoklis P.; Karniadakis, George Em

2017-09-01

We develop a new robust methodology for the stochastic Navier-Stokes equations based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) methods [1-3]. Both approaches are variants of a generalized Karhunen-Loève (KL) expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics, hence, capturing the low-dimensional structure of the solution. The DO and BO formulations are mathematically equivalent [3], but they exhibit computationally complimentary properties. Specifically, the BO formulation may fail due to crossing of the eigenvalues of the covariance matrix, while both BO and DO become unstable when there is a high condition number of the covariance matrix or zero eigenvalues. To this end, we combine the two methods into a robust hybrid framework and in addition we employ a pseudo-inverse technique to invert the covariance matrix. The robustness of the proposed method stems from addressing the following issues in the DO/BO formulation: (i) eigenvalue crossing: we resolve the issue of eigenvalue crossing in the BO formulation by switching to the DO near eigenvalue crossing using the equivalence theorem and switching back to BO when the distance between eigenvalues is larger than a threshold value; (ii) ill-conditioned covariance matrix: we utilize a pseudo-inverse strategy to invert the covariance matrix; (iii) adaptivity: we utilize an adaptive strategy to add/remove modes to resolve the covariance matrix up to a threshold value. In particular, we introduce a soft-threshold criterion to allow the system to adapt to the newly added/removed mode and therefore avoid repetitive and unnecessary mode addition/removal. When the total variance approaches zero, we show that the DO/BO formulation becomes equivalent to the evolution equation of the Optimally Time-Dependent modes [4]. We demonstrate the capability of the proposed methodology with several numerical examples, namely (i) stochastic Burgers equation: we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues; (ii) stochastic Kovasznay flow: we examine the method in the presence of a singular covariance matrix; and (iii) we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen DO/BO modes are active.

Integrability and conformal data of the dimer model

NASA Astrophysics Data System (ADS)

Morin-Duchesne, Alexi; Rasmussen, Jørgen; Ruelle, Philippe

2016-04-01

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a c=-2 description. Using Lieb’s transfer matrix and its description in terms of the Temperley-Lieb algebra {{TL}}n at β =0, we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analyzed in the scaling limit, and the result for {L}0-\\frac{c}{24} is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of {{TL}}n and are found to yield a c=-2 realization of the Virasoro algebra, familiar from fermionic bc ghost systems. In this realization, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the c=-2 conformal integrals of motion. Consistent with the expression for {L}0-\\frac{c}{24} obtained from the transfer matrix, we also construct higher Virasoro modes with c = 1 and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the c = 1 integrals of motion. Although this indicates that Lieb’s transfer matrix description is incompatible with the c = 1 interpretation, it does not rule out the existence of an alternative, c = 1 compatible, transfer matrix description of the dimer model.

Gravitational lensing by eigenvalue distributions of random matrix models

NASA Astrophysics Data System (ADS)

Martínez Alonso, Luis; Medina, Elena

2018-05-01

We propose to use eigenvalue densities of unitary random matrix ensembles as mass distributions in gravitational lensing. The corresponding lens equations reduce to algebraic equations in the complex plane which can be treated analytically. We prove that these models can be applied to describe lensing by systems of edge-on galaxies. We illustrate our analysis with the Gaussian and the quartic unitary matrix ensembles.

Spectral properties of Google matrix of Wikipedia and other networks

NASA Astrophysics Data System (ADS)

Ermann, Leonardo; Frahm, Klaus M.; Shepelyansky, Dima L.

2013-05-01

We study the properties of eigenvalues and eigenvectors of the Google matrix of the Wikipedia articles hyperlink network and other real networks. With the help of the Arnoldi method, we analyze the distribution of eigenvalues in the complex plane and show that eigenstates with significant eigenvalue modulus are located on well defined network communities. We also show that the correlator between PageRank and CheiRank vectors distinguishes different organizations of information flow on BBC and Le Monde web sites.

Numerical solution of quadratic matrix equations for free vibration analysis of structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.

1975-01-01

This paper is concerned with the efficient and accurate solution of the eigenvalue problem represented by quadratic matrix equations. Such matrix forms are obtained in connection with the free vibration analysis of structures, discretized by finite 'dynamic' elements, resulting in frequency-dependent stiffness and inertia matrices. The paper presents a new numerical solution procedure of the quadratic matrix equations, based on a combined Sturm sequence and inverse iteration technique enabling economical and accurate determination of a few required eigenvalues and associated vectors. An alternative procedure based on a simultaneous iteration procedure is also described when only the first few modes are the usual requirement. The employment of finite dynamic elements in conjunction with the presently developed eigenvalue routines results in a most significant economy in the dynamic analysis of structures.

The wasteland of random supergravities

NASA Astrophysics Data System (ADS)

Marsh, David; McAllister, Liam; Wrase, Timm

2012-03-01

We show that in a general {N} = {1} supergravity with N ≫ 1 scalar fields, an exponentially small fraction of the de Sitter critical points are metastable vacua. Taking the superpotential and Kähler potential to be random functions, we construct a random matrix model for the Hessian matrix, which is well-approximated by the sum of a Wigner matrix and two Wishart matrices. We compute the eigenvalue spectrum analytically from the free convolution of the constituent spectra and find that in typical configurations, a significant fraction of the eigenvalues are negative. Building on the Tracy-Widom law governing fluctuations of extreme eigenvalues, we determine the probability P of a large fluctuation in which all the eigenvalues become positive. Strong eigenvalue repulsion makes this extremely unlikely: we find P ∝ exp(- c N p ), with c, p being constants. For generic critical points we find p ≈ 1 .5, while for approximately-supersymmetric critical points, p ≈ 1 .3. Our results have significant implications for the counting of de Sitter vacua in string theory, but the number of vacua remains vast.

Recurrence quantity analysis based on matrix eigenvalues

NASA Astrophysics Data System (ADS)

Yang, Pengbo; Shang, Pengjian

2018-06-01

Recurrence plots is a powerful tool for visualization and analysis of dynamical systems. Recurrence quantification analysis (RQA), based on point density and diagonal and vertical line structures in the recurrence plots, is considered to be alternative measures to quantify the complexity of dynamical systems. In this paper, we present a new measure based on recurrence matrix to quantify the dynamical properties of a given system. Matrix eigenvalues can reflect the basic characteristics of the complex systems, so we show the properties of the system by exploring the eigenvalues of the recurrence matrix. Considering that Shannon entropy has been defined as a complexity measure, we propose the definition of entropy of matrix eigenvalues (EOME) as a new RQA measure. We confirm that EOME can be used as a metric to quantify the behavior changes of the system. As a given dynamical system changes from a non-chaotic to a chaotic regime, the EOME will increase as well. The bigger EOME values imply higher complexity and lower predictability. We also study the effect of some factors on EOME,including data length, recurrence threshold, the embedding dimension, and additional noise. Finally, we demonstrate an application in physiology. The advantage of this measure lies in a high sensitivity and simple computation.

On the cross-stream spectral method for the Orr-Sommerfeld equation

NASA Technical Reports Server (NTRS)

Zorumski, William E.; Hodge, Steven L.

1993-01-01

Cross-stream models are defined as solutions to the Orr-Sommerfeld equation which are propagating normal to the flow direction. These models are utilized as a basis for a Hilbert space to approximate the spectrum of the Orr-Sommerfeld equation with plane Poiseuille flow. The cross-stream basis leads to a standard eigenvalue problem for the frequencies of Poiseuille flow instability waves. The coefficient matrix in the eigenvalue problem is shown to be the sum of a real matrix and a negative-imaginary diagonal matrix which represents the frequencies of the cross-stream modes. The real coefficient matrix is shown to approach a Toeplitz matrix when the row and column indices are large. The Toeplitz matrix is diagonally dominant, and the diagonal elements vary inversely in magnitude with diagonal position. The Poiseuille flow eigenvalues are shown to lie within Gersgorin disks with radii bounded by the product of the average flow speed and the axial wavenumber. It is shown that the eigenvalues approach the Gersgorin disk centers when the mode index is large, so that the method may be used to compute spectra with an essentially unlimited number of elements. When the mode index is large, the real part of the eigenvalue is the product of the axial wavenumber and the average flow speed, and the imaginary part of the eigen value is identical to the corresponding cross-stream mode frequency. The cross-stream method is numerically well-conditioned in comparison to Chebyshev based methods, providing equivalent accuracy for small mode indices and superior accuracy for large indices.

EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.

PubMed

Tal-Ezer, Hillel

2016-05-19

Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler.

The critical boundary RSOS M(3,5) model

NASA Astrophysics Data System (ADS)

El Deeb, O.

2017-12-01

We consider the critical nonunitary minimal model M(3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz ( TBA) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A 4 restricted solid-on-solid Forrester-Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the ( r, s) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of ( m, n) systems.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bai, Zhaojun; Yang, Chao

What is common among electronic structure calculation, design of MEMS devices, vibrational analysis of high speed railways, and simulation of the electromagnetic field of a particle accelerator? The answer: they all require solving large scale nonlinear eigenvalue problems. In fact, these are just a handful of examples in which solving nonlinear eigenvalue problems accurately and efficiently is becoming increasingly important. Recognizing the importance of this class of problems, an invited minisymposium dedicated to nonlinear eigenvalue problems was held at the 2005 SIAM Annual Meeting. The purpose of the minisymposium was to bring together numerical analysts and application scientists to showcasemore » some of the cutting edge results from both communities and to discuss the challenges they are still facing. The minisymposium consisted of eight talks divided into two sessions. The first three talks focused on a type of nonlinear eigenvalue problem arising from electronic structure calculations. In this type of problem, the matrix Hamiltonian H depends, in a non-trivial way, on the set of eigenvectors X to be computed. The invariant subspace spanned by these eigenvectors also minimizes a total energy function that is highly nonlinear with respect to X on a manifold defined by a set of orthonormality constraints. In other applications, the nonlinearity of the matrix eigenvalue problem is restricted to the dependency of the matrix on the eigenvalues to be computed. These problems are often called polynomial or rational eigenvalue problems In the second session, Christian Mehl from Technical University of Berlin described numerical techniques for solving a special type of polynomial eigenvalue problem arising from vibration analysis of rail tracks excited by high-speed trains.« less

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

NASA Technical Reports Server (NTRS)

Beam, Richard M.; Warming, Robert F.

1991-01-01

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Bethe states of the trigonometric SU(3) spin chain with generic open boundaries

NASA Astrophysics Data System (ADS)

Sun, Pei; Xin, Zhirong; Qiao, Yi; Wen, Fakai; Hao, Kun; Cao, Junpeng; Li, Guang-Liang; Yang, Tao; Yang, Wen-Li; Shi, Kangjie

2018-06-01

By combining the algebraic Bethe ansatz and the off-diagonal Bethe ansatz, we investigate the trigonometric SU (3) model with generic open boundaries. The eigenvalues of the transfer matrix are given in terms of an inhomogeneous T - Q relation, and the corresponding eigenstates are expressed in terms of nested Bethe-type eigenstates which have well-defined homogeneous limit. This exact solution provides a basis for further analyzing the thermodynamic properties and correlation functions of the anisotropic models associated with higher rank algebras.

The ELPA library: scalable parallel eigenvalue solutions for electronic structure theory and computational science.

PubMed

Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H

2014-05-28

Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem sizes arising in the field of electronic structure theory is demonstrated for current high-performance computer architectures such as Cray or Intel/Infiniband. For a matrix of dimension 260,000, scalability up to 295,000 CPU cores has been shown on BlueGene/P.

Field patterns without blow up

NASA Astrophysics Data System (ADS)

Mattei, Ornella; Milton, Graeme W.

2017-09-01

Field patterns, first proposed by the authors in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819), are a new type of wave propagating along orderly patterns of characteristic lines which arise in specific space-time microstructures whose geometry in one spatial dimension plus time is somehow commensurate with the slope of the characteristic lines. In particular, in Milton and Mattei (2017 Proc. R. Soc. A 473 20160819) the authors propose two examples of space-time geometries in which field patterns occur: they are two-phase microstructures in which rectangular space-time inclusions of one material are embedded in another material. After a sufficiently long interval of time, field patterns have local periodicity both in time and space. This allows one to focus only on solving the problem on the discrete network on which a field pattern lives and to define a suitable transfer matrix that, given the solution at a certain time, provides the solution after one time period. For the aforementioned microstructures, many of the eigenvalues of this { P }{ T }-symmetric transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. However, there are also modes that blow up exponentially with time coupled with modes that decrease exponentially with time. The question arises as to whether there are space-time microstructures such that the transfer matrix only has eigenvalues on the unit circle, so that there are no growing modes (modes that blow-up)? The answer is found here, where we see that certain space-time checkerboards have the property that all the modes are propagating modes, within a certain range of the material parameters. Interestingly, when there is no blow-up, the waves generated by an instantaneous disturbance at a point look like shocks with a wake of oscillatory waves, whose amplitude, very remarkably, does not tend to zero away from the wave front.

Fast Eigensolver for Computing 3D Earth's Normal Modes

NASA Astrophysics Data System (ADS)

Shi, J.; De Hoop, M. V.; Li, R.; Xi, Y.; Saad, Y.

2017-12-01

We present a novel parallel computational approach to compute Earth's normal modes. We discretize Earth via an unstructured tetrahedral mesh and apply the continuous Galerkin finite element method to the elasto-gravitational system. To resolve the eigenvalue pollution issue, following the analysis separating the seismic point spectrum, we utilize explicitly a representation of the displacement for describing the oscillations of the non-seismic modes in the fluid outer core. Effectively, we separate out the essential spectrum which is naturally related to the Brunt-Väisälä frequency. We introduce two Lanczos approaches with polynomial and rational filtering for solving this generalized eigenvalue problem in prescribed intervals. The polynomial filtering technique only accesses the matrix pair through matrix-vector products and is an ideal candidate for solving three-dimensional large-scale eigenvalue problems. The matrix-free scheme allows us to deal with fluid separation and self-gravitation in an efficient way, while the standard shift-and-invert method typically needs an explicit shifted matrix and its factorization. The rational filtering method converges much faster than the standard shift-and-invert procedure when computing all the eigenvalues inside an interval. Both two Lanczos approaches solve for the internal eigenvalues extremely accurately, comparing with the standard eigensolver. In our computational experiments, we compare our results with the radial earth model benchmark, and visualize the normal modes using vector plots to illustrate the properties of the displacements in different modes.

Asymmetric correlation matrices: an analysis of financial data

NASA Astrophysics Data System (ADS)

Livan, G.; Rebecchi, L.

2012-06-01

We analyse the spectral properties of correlation matrices between distinct statistical systems. Such matrices are intrinsically non-symmetric, and lend themselves to extend the spectral analyses usually performed on standard Pearson correlation matrices to the realm of complex eigenvalues. We employ some recent random matrix theory results on the average eigenvalue density of this type of matrix to distinguish between noise and non-trivial correlation structures, and we focus on financial data as a case study. Namely, we employ daily prices of stocks belonging to the American and British stock exchanges, and look for the emergence of correlations between two such markets in the eigenvalue spectrum of their non-symmetric correlation matrix. We find several non trivial results when considering time-lagged correlations over short lags, and we corroborate our findings by additionally studying the asymmetric correlation matrix of the principal components of our datasets.

Almost analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2017-11-01

We present an almost analytical new approach to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of solving this matrix eigenvalue problem purely numerically, which may suffer from the computational inaccuracy for big data, first, we consider a pair of integral and differential equations, which are related to the so-called prolate spheroidal wave functions (PSWF). For the PSWF differential equation, the pair of the eigenvectors (PSWF) and eigenvalues can be obtained from a relatively small number of analytical Legendre functions. Then, the eigenvalues in the PSWF integral equation are expressed in terms of functional values of the PSWF and the eigenvalues of the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data; ordinary irregular waves and rogue waves. We found that the present almost analytical method is better than the conventional data-independent Fourier representation and, also, the conventional direct numerical K-L representation in terms of both accuracy and computational cost. This work was supported by the National Research Foundation of Korea (NRF). (NRF-2017R1D1A1B03028299).

Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments.

PubMed

Gasbarra, Dario; Pajevic, Sinisa; Basser, Peter J

2017-01-01

Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D , observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄ . When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model.

Eigenvalues of Random Matrices with Isotropic Gaussian Noise and the Design of Diffusion Tensor Imaging Experiments*

PubMed Central

Gasbarra, Dario; Pajevic, Sinisa; Basser, Peter J.

2017-01-01

Tensor-valued and matrix-valued measurements of different physical properties are increasingly available in material sciences and medical imaging applications. The eigenvalues and eigenvectors of such multivariate data provide novel and unique information, but at the cost of requiring a more complex statistical analysis. In this work we derive the distributions of eigenvalues and eigenvectors in the special but important case of m×m symmetric random matrices, D, observed with isotropic matrix-variate Gaussian noise. The properties of these distributions depend strongly on the symmetries of the mean tensor/matrix, D̄. When D̄ has repeated eigenvalues, the eigenvalues of D are not asymptotically Gaussian, and repulsion is observed between the eigenvalues corresponding to the same D̄ eigenspaces. We apply these results to diffusion tensor imaging (DTI), with m = 3, addressing an important problem of detecting the symmetries of the diffusion tensor, and seeking an experimental design that could potentially yield an isotropic Gaussian distribution. In the 3-dimensional case, when the mean tensor is spherically symmetric and the noise is Gaussian and isotropic, the asymptotic distribution of the first three eigenvalue central moment statistics is simple and can be used to test for isotropy. In order to apply such tests, we use quadrature rules of order t ≥ 4 with constant weights on the unit sphere to design a DTI-experiment with the property that isotropy of the underlying true tensor implies isotropy of the Fisher information. We also explain the potential implications of the methods using simulated DTI data with a Rician noise model. PMID:28989561

Stellarator Saddle Coils

NASA Astrophysics Data System (ADS)

Boozer, Allen H.

1999-11-01

Modern stellarators are designed using J. Nuehrenberg’s method of varying Fourier coefficients in the shape of the plasma boundary to maximize a target function. The matrix of second derivatives of the target function at the optimum determines a quality matrix. This matrix gives the degradation in the quality of the configuration as the normal magnetic field is varied on a control surface, which lies on or outside the plasma surface. The task is finding saddle coils that produce the desired configuration in the presence of a given toroidal field. An eigenvector of the quality matrix can be important for two reasons: (1) the normal field that must be produced by the saddles is large or (2) the eigenvalue is large (an island-causing resonant perturbation). The rank of the important part of the quality matrix is the number of important eigenvectors. The current in each saddle coil produces a normal field on the control surface, which can be described by an inductance matrix. The relevant part of the inductance matrix has large eigenvalues. The coils can produce the configuration if the rank of the important part of the quality matrix and its product with the relevant part of the inductance matrix are the same. Existing coil design codes, pioneered by P. Merkel, approximate the quality matrix by the unit matrix. Stellarator flexibility could be enhanced by using a more realistic quality matrix and by using trim coils to balance large eigenvalues.

Solving an inverse eigenvalue problem with triple constraints on eigenvalues, singular values, and diagonal elements

NASA Astrophysics Data System (ADS)

Wu, Sheng-Jhih; Chu, Moody T.

2017-08-01

An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.

Group identification in Indonesian stock market

NASA Astrophysics Data System (ADS)

Nurriyadi Suparno, Ervano; Jo, Sung Kyun; Lim, Kyuseong; Purqon, Acep; Kim, Soo Yong

2016-08-01

The characteristic of Indonesian stock market is interesting especially because it represents developing countries. We investigate the dynamics and structures by using Random Matrix Theory (RMT). Here, we analyze the cross-correlation of the fluctuations of the daily closing price of stocks from the Indonesian Stock Exchange (IDX) between January 1, 2007, and October 28, 2014. The eigenvalue distribution of the correlation matrix consists of noise which is filtered out using the random matrix as a control. The bulk of the eigenvalue distribution conforms to the random matrix, allowing the separation of random noise from original data which is the deviating eigenvalues. From the deviating eigenvalues and the corresponding eigenvectors, we identify the intrinsic normal modes of the system and interpret their meaning based on qualitative and quantitative approach. The results show that the largest eigenvector represents the market-wide effect which has a predominantly common influence toward all stocks. The other eigenvectors represent highly correlated groups within the system. Furthermore, identification of the largest components of the eigenvectors shows the sector or background of the correlated groups. Interestingly, the result shows that there are mainly two clusters within IDX, natural and non-natural resource companies. We then decompose the correlation matrix to investigate the contribution of the correlated groups to the total correlation, and we find that IDX is still driven mainly by the market-wide effect.

FastSKAT: Sequence kernel association tests for very large sets of markers.

PubMed

Lumley, Thomas; Brody, Jennifer; Peloso, Gina; Morrison, Alanna; Rice, Kenneth

2018-06-22

The sequence kernel association test (SKAT) is widely used to test for associations between a phenotype and a set of genetic variants that are usually rare. Evaluating tail probabilities or quantiles of the null distribution for SKAT requires computing the eigenvalues of a matrix related to the genotype covariance between markers. Extracting the full set of eigenvalues of this matrix (an n×n matrix, for n subjects) has computational complexity proportional to n 3 . As SKAT is often used when n>104, this step becomes a major bottleneck in its use in practice. We therefore propose fastSKAT, a new computationally inexpensive but accurate approximations to the tail probabilities, in which the k largest eigenvalues of a weighted genotype covariance matrix or the largest singular values of a weighted genotype matrix are extracted, and a single term based on the Satterthwaite approximation is used for the remaining eigenvalues. While the method is not particularly sensitive to the choice of k, we also describe how to choose its value, and show how fastSKAT can automatically alert users to the rare cases where the choice may affect results. As well as providing faster implementation of SKAT, the new method also enables entirely new applications of SKAT that were not possible before; we give examples grouping variants by topologically associating domains, and comparing chromosome-wide association by class of histone marker. © 2018 WILEY PERIODICALS, INC.

A differentiable reformulation for E-optimal design of experiments in nonlinear dynamic biosystems.

PubMed

Telen, Dries; Van Riet, Nick; Logist, Flip; Van Impe, Jan

2015-06-01

Informative experiments are highly valuable for estimating parameters in nonlinear dynamic bioprocesses. Techniques for optimal experiment design ensure the systematic design of such informative experiments. The E-criterion which can be used as objective function in optimal experiment design requires the maximization of the smallest eigenvalue of the Fisher information matrix. However, one problem with the minimal eigenvalue function is that it can be nondifferentiable. In addition, no closed form expression exists for the computation of eigenvalues of a matrix larger than a 4 by 4 one. As eigenvalues are normally computed with iterative methods, state-of-the-art optimal control solvers are not able to exploit automatic differentiation to compute the derivatives with respect to the decision variables. In the current paper a reformulation strategy from the field of convex optimization is suggested to circumvent these difficulties. This reformulation requires the inclusion of a matrix inequality constraint involving positive semidefiniteness. In this paper, this positive semidefiniteness constraint is imposed via Sylverster's criterion. As a result the maximization of the minimum eigenvalue function can be formulated in standard optimal control solvers through the addition of nonlinear constraints. The presented methodology is successfully illustrated with a case study from the field of predictive microbiology. Copyright © 2015. Published by Elsevier Inc.

A Novel Hyperbolization Procedure for The Two-Phase Six-Equation Flow Model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Samet Y. Kadioglu; Robert Nourgaliev; Nam Dinh

2011-10-01

We introduce a novel approach for the hyperbolization of the well-known two-phase six equation flow model. The six-equation model has been frequently used in many two-phase flow applications such as bubbly fluid flows in nuclear reactors. One major drawback of this model is that it can be arbitrarily non-hyperbolic resulting in difficulties such as numerical instability issues. Non-hyperbolic behavior can be associated with complex eigenvalues that correspond to characteristic matrix of the system. Complex eigenvalues are often due to certain flow parameter choices such as the definition of inter-facial pressure terms. In our method, we prevent the characteristic matrix receivingmore » complex eigenvalues by fine tuning the inter-facial pressure terms with an iterative procedure. In this way, the characteristic matrix possesses all real eigenvalues meaning that the characteristic wave speeds are all real therefore the overall two-phase flowmodel becomes hyperbolic. The main advantage of this is that one can apply less diffusive highly accurate high resolution numerical schemes that often rely on explicit calculations of real eigenvalues. We note that existing non-hyperbolic models are discretized mainly based on low order highly dissipative numerical techniques in order to avoid stability issues.« less

TIMEDELN: A programme for the detection and parametrization of overlapping resonances using the time-delay method

NASA Astrophysics Data System (ADS)

Little, Duncan A.; Tennyson, Jonathan; Plummer, Martin; Noble, Clifford J.; Sunderland, Andrew G.

2017-06-01

TIMEDELN implements the time-delay method of determining resonance parameters from the characteristic Lorentzian form displayed by the largest eigenvalues of the time-delay matrix. TIMEDELN constructs the time-delay matrix from input K-matrices and analyses its eigenvalues. This new version implements multi-resonance fitting and may be run serially or as a high performance parallel code with three levels of parallelism. TIMEDELN takes K-matrices from a scattering calculation, either read from a file or calculated on a dynamically adjusted grid, and calculates the time-delay matrix. This is then diagonalized, with the largest eigenvalue representing the longest time-delay experienced by the scattering particle. A resonance shows up as a characteristic Lorentzian form in the time-delay: the programme searches the time-delay eigenvalues for maxima and traces resonances when they pass through different eigenvalues, separating overlapping resonances. It also performs the fitting of the calculated data to the Lorentzian form and outputs resonance positions and widths. Any remaining overlapping resonances can be fitted jointly. The branching ratios of decay into the open channels can also be found. The programme may be run serially or in parallel with three levels of parallelism. The parallel code modules are abstracted from the main physics code and can be used independently.

The genealogical decomposition of a matrix population model with applications to the aggregation of stages.

PubMed

Bienvenu, François; Akçay, Erol; Legendre, Stéphane; McCandlish, David M

2017-06-01

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units. Copyright © 2017 Elsevier Inc. All rights reserved.

QCD dirac operator at nonzero chemical potential: lattice data and matrix model.

PubMed

Akemann, Gernot; Wettig, Tilo

2004-03-12

Recently, a non-Hermitian chiral random matrix model was proposed to describe the eigenvalues of the QCD Dirac operator at nonzero chemical potential. This matrix model can be constructed from QCD by mapping it to an equivalent matrix model which has the same symmetries as QCD with chemical potential. Its microscopic spectral correlations are conjectured to be identical to those of the QCD Dirac operator. We investigate this conjecture by comparing large ensembles of Dirac eigenvalues in quenched SU(3) lattice QCD at a nonzero chemical potential to the analytical predictions of the matrix model. Excellent agreement is found in the two regimes of weak and strong non-Hermiticity, for several different lattice volumes.

Free Fermions and the Classical Compact Groups

NASA Astrophysics Data System (ADS)

Cunden, Fabio Deelan; Mezzadri, Francesco; O'Connell, Neil

2018-06-01

There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.

Gaussian-reflectivity mirror resonator for a high-power transverse-flow CO2 laser.

PubMed

Ling, Dongxiong; Chen, Junruo; Li, Junchang

2006-05-01

A Gaussian-reflectivity mirror resonator is proposed to achieve high-quality laser beams. To analyze the laser fields in a Gaussian-reflectivity mirror resonator, the diffraction integral equations of a Gaussian-reflectivity mirror resonator are converted to the finite-sum matrix equations. Consequently, according to the Fox-Li laser self-reproducing principle, we describe the mode fields and their losses in the proposed resonator as eigenvectors and eigenvalues of a transfer matrix. The conclusion can be drawn from the numerical results that, if a Gaussian-reflectivity mirror is adopted for a plano-concave resonator, a fundamental mode can easily be obtained from a transverse-flow CO2 laser and high-quality laser beams can be expected.

Optimal port-based teleportation

NASA Astrophysics Data System (ADS)

Mozrzymas, Marek; Studziński, Michał; Strelchuk, Sergii; Horodecki, Michał

2018-05-01

Deterministic port-based teleportation (dPBT) protocol is a scheme where a quantum state is guaranteed to be transferred to another system without unitary correction. We characterise the best achievable performance of the dPBT when both the resource state and the measurement is optimised. Surprisingly, the best possible fidelity for an arbitrary number of ports and dimension of the teleported state is given by the largest eigenvalue of a particular matrix—Teleportation Matrix. It encodes the relationship between a certain set of Young diagrams and emerges as the optimal solution to the relevant semidefinite programme.

Q-operators for the open Heisenberg spin chain

NASA Astrophysics Data System (ADS)

Frassek, Rouven; Szécsényi, István M.

2015-12-01

We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang-Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.

Optimal trajectories for aeroassisted orbital transfer

NASA Technical Reports Server (NTRS)

Miele, A.; Venkataraman, P.

1983-01-01

Consideration is given to classical and minimax problems involved in aeroassisted transfer from high earth orbit (HEO) to low earth orbit (LEO). The transfer is restricted to coplanar operation, with trajectory control effected by means of lift modulation. The performance of the maneuver is indexed to the energy expenditure or, alternatively, the time integral of the heating rate. Firist-order optimality conditions are defined for the classical approach, as are a sequential gradient-restoration algorithm and a combined gradient-restoration algorithm. Minimization techniques are presented for the aeroassisted transfer energy consumption and time-delay integral of the heating rate, as well as minimization of the pressure. It is shown that the eigenvalues of the Jacobian matrix of the differential system is both stiff and unstable, implying that the sequential gradient restoration algorithm in its present version is unsuitable. A new method, involving a multipoint approach to the two-poing boundary value problem, is recommended.

Iterative Methods for Elliptic Problems and the Discovery of ’q’.

DTIC Science & Technology

1984-07-01

K = M’IlN LN 12 is a nonnegative irreducible matrix. Hence the Perron - Frobenius theory [19] tells us that there is exactly one eigenvalue A with W = p...earlier, the Perron - Frobenius theory implies that p is itself an eigenvalue. However, as we have said, in this instance the eigenvalue problem (l.12a

Eigenvalue statistics for the sum of two complex Wishart matrices

NASA Astrophysics Data System (ADS)

Kumar, Santosh

2014-09-01

The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement.

Structural robustness with suboptimal responses for linear state space model

NASA Technical Reports Server (NTRS)

Keel, L. H.; Lim, Kyong B.; Juang, Jer-Nan

1989-01-01

A relationship between the closed-loop eigenvalues and the amount of perturbations in the open-loop matrix is addressed in the context of performance robustness. If the allowable perturbation ranges of elements of the open-loop matrix A and the desired tolerance of the closed-loop eigenvalues are given such that max(j) of the absolute value of Delta-lambda(j) (A+BF) should be less than some prescribed value, what is a state feedback controller F which satisfies the closed-loop eigenvalue perturbation-tolerance requirement for a class of given perturbation in A? The paper gives an algorithm to design such a controller. Numerical examples are included for illustration.

A Thick-Restart Lanczos Algorithm with Polynomial Filtering for Hermitian Eigenvalue Problems

DOE PAGES

Li, Ruipeng; Xi, Yuanzhe; Vecharynski, Eugene; ...

2016-08-16

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric (or complex Hermitian) matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a thick-restart version of the Lanczos algorithm with deflation ("locking'') and a new type of polynomial filter obtained from a least-squares technique. Furthermore, the resulting algorithm can be utilized in a “spectrum-slicing” approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different subintervals independently from onemore » another.« less

Eigensolver for a Sparse, Large Hermitian Matrix

NASA Technical Reports Server (NTRS)

Tisdale, E. Robert; Oyafuso, Fabiano; Klimeck, Gerhard; Brown, R. Chris

2003-01-01

A parallel-processing computer program finds a few eigenvalues in a sparse Hermitian matrix that contains as many as 100 million diagonal elements. This program finds the eigenvalues faster, using less memory, than do other, comparable eigensolver programs. This program implements a Lanczos algorithm in the American National Standards Institute/ International Organization for Standardization (ANSI/ISO) C computing language, using the Message Passing Interface (MPI) standard to complement an eigensolver in PARPACK. [PARPACK (Parallel Arnoldi Package) is an extension, to parallel-processing computer architectures, of ARPACK (Arnoldi Package), which is a collection of Fortran 77 subroutines that solve large-scale eigenvalue problems.] The eigensolver runs on Beowulf clusters of computers at the Jet Propulsion Laboratory (JPL).

Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gurin, Péter; Varga, Szabolcs

2015-06-14

We extend the transfer matrix method of one-dimensional hard core fluids placed between confining walls for that case where the particles can pass each other and at most two layers can form. We derive an eigenvalue equation for a quasi-one-dimensional system of hard squares confined between two parallel walls, where the pore width is between σ and 3σ (σ is the side length of the square). The exact equation of state and the nearest neighbor distribution functions show three different structures: a fluid phase with one layer, a fluid phase with two layers, and a solid-like structure where the fluidmore » layers are strongly correlated. The structural transition between differently ordered fluids develops continuously with increasing density, i.e., no thermodynamic phase transition occurs. The high density structure of the system consists of clusters with two layers which are broken with particles staying in the middle of the pore.« less

Determination of eigenvalues of dynamical systems by symbolic computation

NASA Technical Reports Server (NTRS)

Howard, J. C.

1982-01-01

A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.

Random matrix theory and cross-correlations in global financial indices and local stock market indices

NASA Astrophysics Data System (ADS)

Nobi, Ashadun; Maeng, Seong Eun; Ha, Gyeong Gyun; Lee, Jae Woo

2013-02-01

We analyzed cross-correlations between price fluctuations of global financial indices (20 daily stock indices over the world) and local indices (daily indices of 200 companies in the Korean stock market) by using random matrix theory (RMT). We compared eigenvalues and components of the largest and the second largest eigenvectors of the cross-correlation matrix before, during, and after the global financial the crisis in the year 2008. We find that the majority of its eigenvalues fall within the RMT bounds [ λ -, λ +], where λ - and λ + are the lower and the upper bounds of the eigenvalues of random correlation matrices. The components of the eigenvectors for the largest positive eigenvalues indicate the identical financial market mode dominating the global and local indices. On the other hand, the components of the eigenvector corresponding to the second largest eigenvalue are positive and negative values alternatively. The components before the crisis change sign during the crisis, and those during the crisis change sign after the crisis. The largest inverse participation ratio (IPR) corresponding to the smallest eigenvector is higher after the crisis than during any other periods in the global and local indices. During the global financial the crisis, the correlations among the global indices and among the local stock indices are perturbed significantly. However, the correlations between indices quickly recover the trends before the crisis.

A divide and conquer approach to the nonsymmetric eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1991-01-01

Serial computation combined with high communication costs on distributed-memory multiprocessors make parallel implementations of the QR method for the nonsymmetric eigenvalue problem inefficient. This paper introduces an alternative algorithm for the nonsymmetric tridiagonal eigenvalue problem based on rank two tearing and updating of the matrix. The parallelism of this divide and conquer approach stems from independent solution of the updating problems. 11 refs.

Shape sensitivity analysis of flutter response of a laminated wing

NASA Technical Reports Server (NTRS)

Bergen, Fred D.; Kapania, Rakesh K.

1988-01-01

A method is presented for calculating the shape sensitivity of a wing aeroelastic response with respect to changes in geometric shape. Yates' modified strip method is used in conjunction with Giles' equivalent plate analysis to predict the flutter speed, frequency, and reduced frequency of the wing. Three methods are used to calculate the sensitivity of the eigenvalue. The first method is purely a finite difference calculation of the eigenvalue derivative directly from the solution of the flutter problem corresponding to the two different values of the shape parameters. The second method uses an analytic expression for the eigenvalue sensitivities of a general complex matrix, where the derivatives of the aerodynamic, mass, and stiffness matrices are computed using a finite difference approximation. The third method also uses an analytic expression for the eigenvalue sensitivities, but the aerodynamic matrix is computed analytically. All three methods are found to be in good agreement with each other. The sensitivities of the eigenvalues were used to predict the flutter speed, frequency, and reduced frequency. These approximations were found to be in good agreement with those obtained using a complete reanalysis.

Finite-difference solution of the compressible stability eigenvalue problem

NASA Technical Reports Server (NTRS)

Malik, M. R.

1982-01-01

A compressible stability analysis computer code is developed. The code uses a matrix finite difference method for local eigenvalue solution when a good guess for the eigenvalue is available and is significantly more computationally efficient than the commonly used initial value approach. The local eigenvalue search procedure also results in eigenfunctions and, at little extra work, group velocities. A globally convergent eigenvalue procedure is also developed which may be used when no guess for the eigenvalue is available. The global problem is formulated in such a way that no unstable spurious modes appear so that the method is suitable for use in a black box stability code. Sample stability calculations are presented for the boundary layer profiles of a Laminar Flow Control (LFC) swept wing.

Distribution of Schmidt-like eigenvalues for Gaussian ensembles of the random matrix theory

NASA Astrophysics Data System (ADS)

Pato, Mauricio P.; Oshanin, Gleb

2013-03-01

We study the probability distribution function P(β)n(w) of the Schmidt-like random variable w = x21/(∑j = 1nx2j/n), where xj, (j = 1, 2, …, n), are unordered eigenvalues of a given n × n β-Gaussian random matrix, β being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual (randomly chosen) eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such β-Gaussian random matrices. We show that in the asymptotic limit n → ∞ and for arbitrary β the distribution P(β)n(w) converges to the Marčenko-Pastur form, i.e. is defined as P_{n}^{( \\beta )}(w) \\sim \\sqrt{(4 - w)/w} for w ∈ [0, 4] and equals zero outside of the support, despite the fact that formally w is defined on the interval [0, n]. Furthermore, for Gaussian unitary ensembles (β = 2) we present exact explicit expressions for P(β = 2)n(w) which are valid for arbitrary n and analyse their behaviour.

Generalized Friedberg-Lee model for CP violation in neutrino physics

NASA Astrophysics Data System (ADS)

Razzaghi, N.; Gousheh, S. S.

2012-09-01

We propose a phenomenological model of Dirac neutrino mass operator based on the Friedberg-Lee neutrino mass model to include CP violation. By considering the most general set of complex coefficients, and imposing the condition that the mass eigenvalues are real, we find a neutrino mass matrix which is non-Hermitian, symmetric, and magic. In particular, we find that the requirement of obtaining real mass eigenvalues by transferring the residual phases to the mass eigenstates self-consistently dictates the following relationship between the imaginary part of the mass matrix elements B and the parameters of the Friedberg-Lee model: B=±(3)/(4)(a-br)2sin⁡22θ13cos⁡2θ12. We obtain inverted neutrino mass hierarchy m3=0. Making a correspondence between our model and the experimental data produces stringent conditions on the parameters as follows: 35.06°≲θ12≲36.27°, θ23=45°, 7.27°≲θ13≲11.09°, and 82.03°≲δ≲85.37°. We get mildly broken μ-τ symmetry, which reduces the resultant neutrino mixing pattern from tri-bimaximal to trimaximal. The CP violation as measured by the Jarlskog parameter is restricted by 0.027≲J≲0.044.

The feasibility and stability of large complex biological networks: a random matrix approach.

PubMed

Stone, Lewi

2018-05-29

In the 70's, Robert May demonstrated that complexity creates instability in generic models of ecological networks having random interaction matrices A. Similar random matrix models have since been applied in many disciplines. Central to assessing stability is the "circular law" since it describes the eigenvalue distribution for an important class of random matrices A. However, despite widespread adoption, the "circular law" does not apply for ecological systems in which density-dependence operates (i.e., where a species growth is determined by its density). Instead one needs to study the far more complicated eigenvalue distribution of the community matrix S = DA, where D is a diagonal matrix of population equilibrium values. Here we obtain this eigenvalue distribution. We show that if the random matrix A is locally stable, the community matrix S = DA will also be locally stable, providing the system is feasible (i.e., all species have positive equilibria D > 0). This helps explain why, unusually, nearly all feasible systems studied here are locally stable. Large complex systems may thus be even more fragile than May predicted, given the difficulty of assembling a feasible system. It was also found that the degree of stability, or resilience of a system, depended on the minimum equilibrium population.

Eigenvalue Attraction

NASA Astrophysics Data System (ADS)

Movassagh, Ramis

2016-02-01

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

NASA Astrophysics Data System (ADS)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; Govind, Niranjan; Yang, Chao

2017-12-01

We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.

Electron Raman scattering in a strained ZnO/MgZnO double quantum well

NASA Astrophysics Data System (ADS)

Mojab-abpardeh, M.; Karimi, M. J.

2018-02-01

In this work, the electron Raman scattering in a strained ZnO / MgZnO double quantum wells is studied. The energy eigenvalues and the wave functions are obtained using the transfer matrix method. The effects of Mg composition, well width and barrier width on the internal electric field in well and barrier layers are investigated. Then, the influences of these parameters on the differential cross-section of electron Raman scattering are studied. Results indicate that the position, magnitude and the number of the peaks depend on the Mg composition, well width and barrier width.

Random matrix approach to cross correlations in financial data

NASA Astrophysics Data System (ADS)

Plerou, Vasiliki; Gopikrishnan, Parameswaran; Rosenow, Bernd; Amaral, Luís A.; Guhr, Thomas; Stanley, H. Eugene

2002-06-01

We analyze cross correlations between price fluctuations of different stocks using methods of random matrix theory (RMT). Using two large databases, we calculate cross-correlation matrices C of returns constructed from (i) 30-min returns of 1000 US stocks for the 2-yr period 1994-1995, (ii) 30-min returns of 881 US stocks for the 2-yr period 1996-1997, and (iii) 1-day returns of 422 US stocks for the 35-yr period 1962-1996. We test the statistics of the eigenvalues λi of C against a ``null hypothesis'' - a random correlation matrix constructed from mutually uncorrelated time series. We find that a majority of the eigenvalues of C fall within the RMT bounds [λ-,λ+] for the eigenvalues of random correlation matrices. We test the eigenvalues of C within the RMT bound for universal properties of random matrices and find good agreement with the results for the Gaussian orthogonal ensemble of random matrices-implying a large degree of randomness in the measured cross-correlation coefficients. Further, we find that the distribution of eigenvector components for the eigenvectors corresponding to the eigenvalues outside the RMT bound display systematic deviations from the RMT prediction. In addition, we find that these ``deviating eigenvectors'' are stable in time. We analyze the components of the deviating eigenvectors and find that the largest eigenvalue corresponds to an influence common to all stocks. Our analysis of the remaining deviating eigenvectors shows distinct groups, whose identities correspond to conventionally identified business sectors. Finally, we discuss applications to the construction of portfolios of stocks that have a stable ratio of risk to return.

On complex matrices with simple spectrum that are unitarily similar to real matrices

NASA Astrophysics Data System (ADS)

Ikramov, Khakim D.

2011-04-01

Suppose that one should verify whether a given complex n × n matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues λ ,bar λ are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.

Density-matrix-based algorithm for solving eigenvalue problems

NASA Astrophysics Data System (ADS)

Polizzi, Eric

2009-03-01

A fast and stable numerical algorithm for solving the symmetric eigenvalue problem is presented. The technique deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques and takes its inspiration from the contour integration and density-matrix representation in quantum mechanics. It will be shown that this algorithm—named FEAST—exhibits high efficiency, robustness, accuracy, and scalability on parallel architectures. Examples from electronic structure calculations of carbon nanotubes are presented, and numerical performances and capabilities are discussed.

Quantitative fluorescence and elastic scattering tissue polarimetry using an Eigenvalue calibrated spectroscopic Mueller matrix system.

PubMed

Soni, Jalpa; Purwar, Harsh; Lakhotia, Harshit; Chandel, Shubham; Banerjee, Chitram; Kumar, Uday; Ghosh, Nirmalya

2013-07-01

A novel spectroscopic Mueller matrix system has been developed and explored for both fluorescence and elastic scattering polarimetric measurements from biological tissues. The 4 × 4 Mueller matrix measurement strategy is based on sixteen spectrally resolved (λ = 400 - 800 nm) measurements performed by sequentially generating and analyzing four elliptical polarization states. Eigenvalue calibration of the system ensured high accuracy of Mueller matrix measurement over a broad wavelength range, either for forward or backscattering geometry. The system was explored for quantitative fluorescence and elastic scattering spectroscopic polarimetric studies on normal and precancerous tissue sections from human uterine cervix. The fluorescence spectroscopic Mueller matrices yielded an interesting diattenuation parameter, exhibiting differences between normal and precancerous tissues.

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

NASA Technical Reports Server (NTRS)

Mielke, R. R.; Tung, L. J.; Carraway, P. I., III

1984-01-01

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The full state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system rmain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

Multitasking the Davidson algorithm for the large, sparse eigenvalue problem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Umar, V.M.; Fischer, C.F.

1989-01-01

The authors report how the Davidson algorithm, developed for handling the eigenvalue problem for large and sparse matrices arising in quantum chemistry, was modified for use in atomic structure calculations. To date these calculations have used traditional eigenvalue methods, which limit the range of feasible calculations because of their excessive memory requirements and unsatisfactory performance attributed to time-consuming and costly processing of zero valued elements. The replacement of a traditional matrix eigenvalue method by the Davidson algorithm reduced these limitations. Significant speedup was found, which varied with the size of the underlying problem and its sparsity. Furthermore, the range ofmore » matrix sizes that can be manipulated efficiently was expended by more than one order or magnitude. On the CRAY X-MP the code was vectorized and the importance of gather/scatter analyzed. A parallelized version of the algorithm obtained an additional 35% reduction in execution time. Speedup due to vectorization and concurrency was also measured on the Alliant FX/8.« less

Design of multivariable feedback control systems via spectral assignment using reduced-order models and reduced-order observers

NASA Technical Reports Server (NTRS)

Mielke, R. R.; Tung, L. J.; Carraway, P. I., III

1985-01-01

The feasibility of using reduced order models and reduced order observers with eigenvalue/eigenvector assignment procedures is investigated. A review of spectral assignment synthesis procedures is presented. Then, a reduced order model which retains essential system characteristics is formulated. A constant state feedback matrix which assigns desired closed loop eigenvalues and approximates specified closed loop eigenvectors is calculated for the reduced order model. It is shown that the eigenvalue and eigenvector assignments made in the reduced order system are retained when the feedback matrix is implemented about the full order system. In addition, those modes and associated eigenvectors which are not included in the reduced order model remain unchanged in the closed loop full order system. The fulll state feedback design is then implemented by using a reduced order observer. It is shown that the eigenvalue and eigenvector assignments of the closed loop full order system remain unchanged when a reduced order observer is used. The design procedure is illustrated by an actual design problem.

A new method for computation of eigenvector derivatives with distinct and repeated eigenvalues in structural dynamic analysis

NASA Astrophysics Data System (ADS)

Li, Zhengguang; Lai, Siu-Kai; Wu, Baisheng

2018-07-01

Determining eigenvector derivatives is a challenging task due to the singularity of the coefficient matrices of the governing equations, especially for those structural dynamic systems with repeated eigenvalues. An effective strategy is proposed to construct a non-singular coefficient matrix, which can be directly used to obtain the eigenvector derivatives with distinct and repeated eigenvalues. This approach also has an advantage that only requires eigenvalues and eigenvectors of interest, without solving the particular solutions of eigenvector derivatives. The Symmetric Quasi-Minimal Residual (SQMR) method is then adopted to solve the governing equations, only the existing factored (shifted) stiffness matrix from an iterative eigensolution such as the subspace iteration method or the Lanczos algorithm is utilized. The present method can deal with both cases of simple and repeated eigenvalues in a unified manner. Three numerical examples are given to illustrate the accuracy and validity of the proposed algorithm. Highly accurate approximations to the eigenvector derivatives are obtained within a few iteration steps, making a significant reduction of the computational effort. This method can be incorporated into a coupled eigensolver/derivative software module. In particular, it is applicable for finite element models with large sparse matrices.

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

NASA Astrophysics Data System (ADS)

Belinschi, Serban; Nowak, Maciej A.; Speicher, Roland; Tarnowski, Wojciech

2017-03-01

We extend the so-called ‘single ring theorem’ (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the Haagerup-Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.

Eigenvalue-eigenvector decomposition (EED) analysis of dissimilarity and covariance matrix obtained from total synchronous fluorescence spectral (TSFS) data sets of herbal preparations: Optimizing the classification approach

NASA Astrophysics Data System (ADS)

Tarai, Madhumita; Kumar, Keshav; Divya, O.; Bairi, Partha; Mishra, Kishor Kumar; Mishra, Ashok Kumar

2017-09-01

The present work compares the dissimilarity and covariance based unsupervised chemometric classification approaches by taking the total synchronous fluorescence spectroscopy data sets acquired for the cumin and non-cumin based herbal preparations. The conventional decomposition method involves eigenvalue-eigenvector analysis of the covariance of the data set and finds the factors that can explain the overall major sources of variation present in the data set. The conventional approach does this irrespective of the fact that the samples belong to intrinsically different groups and hence leads to poor class separation. The present work shows that classification of such samples can be optimized by performing the eigenvalue-eigenvector decomposition on the pair-wise dissimilarity matrix.

Eigentime identities for on weighted polymer networks

NASA Astrophysics Data System (ADS)

Dai, Meifeng; Tang, Hualong; Zou, Jiahui; He, Di; Sun, Yu; Su, Weiyi

2018-01-01

In this paper, we first analytically calculate the eigenvalues of the transition matrix of a structure with very complex architecture and their multiplicities. We call this structure polymer network. Based on the eigenvalues obtained in the iterative manner, we then calculate the eigentime identity. We highlight two scaling behaviors (logarithmic and linear) for this quantity, strongly depending on the value of the weight factor. Finally, by making use of the obtained eigenvalues, we determine the weighted counting of spanning trees.

CFD simulation of simultaneous monotonic cooling and surface heat transfer coefficient

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mihálka, Peter, E-mail: usarmipe@savba.sk; Matiašovský, Peter, E-mail: usarmat@savba.sk

The monotonic heating regime method for determination of thermal diffusivity is based on the analysis of an unsteady-state (stabilised) thermal process characterised by an independence of the space-time temperature distribution on initial conditions. At the first kind of the monotonic regime a sample of simple geometry is heated / cooled at constant ambient temperature. The determination of thermal diffusivity requires the determination rate of a temperature change and simultaneous determination of the first eigenvalue. According to a characteristic equation the first eigenvalue is a function of the Biot number defined by a surface heat transfer coefficient and thermal conductivity ofmore » an analysed material. Knowing the surface heat transfer coefficient and the first eigenvalue the thermal conductivity can be determined. The surface heat transport coefficient during the monotonic regime can be determined by the continuous measurement of long-wave radiation heat flow and the photoelectric measurement of the air refractive index gradient in a boundary layer. CFD simulation of the cooling process was carried out to analyse local convective and radiative heat transfer coefficients more in detail. Influence of ambient air flow was analysed. The obtained eigenvalues and corresponding surface heat transfer coefficient values enable to determine thermal conductivity of the analysed specimen together with its thermal diffusivity during a monotonic heating regime.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-12-01

In this article, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE PAGES

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue; ...

2017-08-24

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by amore » modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.« less

Eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems

NASA Astrophysics Data System (ADS)

Gorgizadeh, Shahnam; Flisgen, Thomas; van Rienen, Ursula

2018-07-01

Generalized eigenvalue problems are standard problems in computational sciences. They may arise in electromagnetic fields from the discretization of the Helmholtz equation by for example the finite element method (FEM). Geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices. Geometrical perturbations may arise by manufacturing tolerances, harsh operating conditions or during shape optimization. Directly solving the eigenvalue problem for each perturbation is computationally costly. The perturbed eigenpairs can be approximated using eigenpair derivatives. Two common approaches for the calculation of eigenpair derivatives, namely modal superposition method and direct algebraic methods, are discussed in this paper. Based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry.

Unreliable Retrial Queues in a Random Environment

DTIC Science & Technology

2007-09-01

equivalent to the stochasticity of the matrix Ĝ. It is generally known from Perron - Frobenius theory that a given square ma- trix M is stochastic if and...only if its maximum positive eigenvalue (i.e., its Perron eigenvalue) sp(M) is equal to unity. A simple analytical condition that guarantees the

Graph theory approach to the eigenvalue problem of large space structures

NASA Technical Reports Server (NTRS)

Reddy, A. S. S. R.; Bainum, P. M.

1981-01-01

Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.

Sparse Covariance Matrix Estimation With Eigenvalue Constraints

PubMed Central

LIU, Han; WANG, Lie; ZHAO, Tuo

2014-01-01

We propose a new approach for estimating high-dimensional, positive-definite covariance matrices. Our method extends the generalized thresholding operator by adding an explicit eigenvalue constraint. The estimated covariance matrix simultaneously achieves sparsity and positive definiteness. The estimator is rate optimal in the minimax sense and we develop an efficient iterative soft-thresholding and projection algorithm based on the alternating direction method of multipliers. Empirically, we conduct thorough numerical experiments on simulated datasets as well as real data examples to illustrate the usefulness of our method. Supplementary materials for the article are available online. PMID:25620866

Effects of sources on time-domain finite difference models.

PubMed

Botts, Jonathan; Savioja, Lauri

2014-07-01

Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed.

Properties of regular polygons of coupled microring resonators.

PubMed

Chremmos, Ioannis; Uzunoglu, Nikolaos

2007-11-01

The resonant properties of a closed and symmetric cyclic array of N coupled microring resonators (coupled-microring resonator regular N-gon) are for the first time determined analytically by applying the transfer matrix approach and Floquet theorem for periodic propagation in cylindrically symmetric structures. By solving the corresponding eigenvalue problem with the field amplitudes in the rings as eigenvectors, it is shown that, for even or odd N, this photonic molecule possesses 1 + N/2 or 1+N resonant frequencies, respectively. The condition for resonances is found to be identical to the familiar dispersion equation of the infinite coupled-microring resonator waveguide with a discrete wave vector. This result reveals the so far latent connection between the two optical structures and is based on the fact that, for a regular polygon, the field transfer matrix over two successive rings is independent of the polygon vertex angle. The properties of the resonant modes are discussed in detail using the illustration of Brillouin band diagrams. Finally, the practical application of a channel-dropping filter based on polygons with an even number of rings is also analyzed.

Comment on ‘Numerical estimates of the spectrum for anharmonic PT symmetric potentials’

NASA Astrophysics Data System (ADS)

Amore, Paolo; Fernández, Francisco M.

2013-04-01

We show that the authors of the commented paper (Bowen et al 2012 Phys. Scr. 85 065005) draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In some of the studied examples, the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operators and focused their attention on the complex ones that do not. We also show that the authors misread Bender's argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Bender 2007 Rep. Prog. Phys. 70 947).

Eigenvalue-eigenvector decomposition (EED) analysis of dissimilarity and covariance matrix obtained from total synchronous fluorescence spectral (TSFS) data sets of herbal preparations: Optimizing the classification approach.

PubMed

Tarai, Madhumita; Kumar, Keshav; Divya, O; Bairi, Partha; Mishra, Kishor Kumar; Mishra, Ashok Kumar

2017-09-05

The present work compares the dissimilarity and covariance based unsupervised chemometric classification approaches by taking the total synchronous fluorescence spectroscopy data sets acquired for the cumin and non-cumin based herbal preparations. The conventional decomposition method involves eigenvalue-eigenvector analysis of the covariance of the data set and finds the factors that can explain the overall major sources of variation present in the data set. The conventional approach does this irrespective of the fact that the samples belong to intrinsically different groups and hence leads to poor class separation. The present work shows that classification of such samples can be optimized by performing the eigenvalue-eigenvector decomposition on the pair-wise dissimilarity matrix. Copyright © 2017 Elsevier B.V. All rights reserved.

Sensitivity analysis and approximation methods for general eigenvalue problems

NASA Technical Reports Server (NTRS)

Murthy, D. V.; Haftka, R. T.

1986-01-01

Optimization of dynamic systems involving complex non-hermitian matrices is often computationally expensive. Major contributors to the computational expense are the sensitivity analysis and reanalysis of a modified design. The present work seeks to alleviate this computational burden by identifying efficient sensitivity analysis and approximate reanalysis methods. For the algebraic eigenvalue problem involving non-hermitian matrices, algorithms for sensitivity analysis and approximate reanalysis are classified, compared and evaluated for efficiency and accuracy. Proper eigenvector normalization is discussed. An improved method for calculating derivatives of eigenvectors is proposed based on a more rational normalization condition and taking advantage of matrix sparsity. Important numerical aspects of this method are also discussed. To alleviate the problem of reanalysis, various approximation methods for eigenvalues are proposed and evaluated. Linear and quadratic approximations are based directly on the Taylor series. Several approximation methods are developed based on the generalized Rayleigh quotient for the eigenvalue problem. Approximation methods based on trace theorem give high accuracy without needing any derivatives. Operation counts for the computation of the approximations are given. General recommendations are made for the selection of appropriate approximation technique as a function of the matrix size, number of design variables, number of eigenvalues of interest and the number of design points at which approximation is sought.

Emergent spectral properties of river network topology: an optimal channel network approach.

PubMed

Abed-Elmdoust, Armaghan; Singh, Arvind; Yang, Zong-Liang

2017-09-13

Characterization of river drainage networks has been a subject of research for many years. However, most previous studies have been limited to quantities which are loosely connected to the topological properties of these networks. In this work, through a graph-theoretic formulation of drainage river networks, we investigate the eigenvalue spectra of their adjacency matrix. First, we introduce a graph theory model for river networks and explore the properties of the network through its adjacency matrix. Next, we show that the eigenvalue spectra of such complex networks follow distinct patterns and exhibit striking features including a spectral gap in which no eigenvalue exists as well as a finite number of zero eigenvalues. We show that such spectral features are closely related to the branching topology of the associated river networks. In this regard, we find an empirical relation for the spectral gap and nullity in terms of the energy dissipation exponent of the drainage networks. In addition, the eigenvalue distribution is found to follow a finite-width probability density function with certain skewness which is related to the drainage pattern. Our results are based on optimal channel network simulations and validated through examples obtained from physical experiments on landscape evolution. These results suggest the potential of the spectral graph techniques in characterizing and modeling river networks.

An Application of the Vandermonde Determinant

ERIC Educational Resources Information Center

Xu, Junqin; Zhao, Likuan

2006-01-01

Eigenvalue is an important concept in Linear Algebra. It is well known that the eigenvectors corresponding to different eigenvalues of a square matrix are linear independent. In most of the existing textbooks, this result is proven using mathematical induction. In this note, a new proof using Vandermonde determinant is given. It is shown that this…

A multi-state fragment charge difference approach for diabatic states in electron transfer: Extension and automation

NASA Astrophysics Data System (ADS)

Yang, Chou-Hsun; Hsu, Chao-Ping

2013-10-01

The electron transfer (ET) rate prediction requires the electronic coupling values. The Generalized Mulliken-Hush (GMH) and Fragment Charge Difference (FCD) schemes have been useful approaches to calculate ET coupling from an excited state calculation. In their typical form, both methods use two eigenstates in forming the target charge-localized diabatic states. For problems involve three or four states, a direct generalization is possible, but it is necessary to pick and assign the locally excited or charge-transfer states involved. In this work, we generalize the 3-state scheme for a multi-state FCD without the need of manual pick or assignment for the states. In this scheme, the diabatic states are obtained separately in the charge-transfer or neutral excited subspaces, defined by their eigenvalues in the fragment charge-difference matrix. In each subspace, the Hamiltonians are diagonalized, and there exist off-diagonal Hamiltonian matrix elements between different subspaces, particularly the charge-transfer and neutral excited diabatic states. The ET coupling values are obtained as the corresponding off-diagonal Hamiltonian matrix elements. A similar multi-state GMH scheme can also be developed. We test the new multi-state schemes for the performance in systems that have been studied using more than two states with FCD or GMH. We found that the multi-state approach yields much better charge-localized states in these systems. We further test for the dependence on the number of state included in the calculation of ET couplings. The final coupling values are converged when the number of state included is increased. In one system where experimental value is available, the multi-state FCD coupling value agrees better with the previous experimental result. We found that the multi-state GMH and FCD are useful when the original two-state approach fails.

Frictionless Contact of Multilayered Composite Half Planes Containing Layers With Complex Eigenvalues

NASA Technical Reports Server (NTRS)

Zhang, Wang; Binienda, Wieslaw K.; Pindera, Marek-Jerzy

1997-01-01

A previously developed local-global stiffness matrix methodology for the response of a composite half plane, arbitrarily layered with isotropic, orthotropic or monoclinic plies, to indentation by a rigid parabolic punch is further extended to accommodate the presence of layers with complex eigenvalues (e.g., honeycomb or piezoelectric layers). First, a generalized plane deformation solution for the displacement field in an orthotropic layer or half plane characterized by complex eigenvalues is obtained using Fourier transforms. A local stiffness matrix in the transform domain is subsequently constructed for this class of layers and half planes, which is then assembled into a global stiffness matrix for the entire multilayered half plane by enforcing continuity conditions along the interfaces. Application of the mixed boundary condition on the top surface of the half plane indented by a rigid punch results in an integral equation for the unknown pressure in the contact region. The integral possesses a divergent kernel which is decomposed into Cauchy-type and regular parts using the asymptotic properties of the local stiffness matrix and a relationship between Fourier and finite Hilbert transform of the contact pressure. The solution of the resulting singular integral equation is obtained using a collocation technique based on the properties of orthogonal polynomials developed by Erdogan and Gupta. Examples are presented that illustrate the important influence of low transverse properties of layers with complex eigenvalues, such as those exhibited by honeycomb, on the load versus contact length response and contact pressure distributions for half planes containing typical composite materials.

Comparing the structure of an emerging market with a mature one under global perturbation

NASA Astrophysics Data System (ADS)

Namaki, A.; Jafari, G. R.; Raei, R.

2011-09-01

In this paper we investigate the Tehran stock exchange (TSE) and Dow Jones Industrial Average (DJIA) in terms of perturbed correlation matrices. To perturb a stock market, there are two methods, namely local and global perturbation. In the local method, we replace a correlation coefficient of the cross-correlation matrix with one calculated from two Gaussian-distributed time series, whereas in the global method, we reconstruct the correlation matrix after replacing the original return series with Gaussian-distributed time series. The local perturbation is just a technical study. We analyze these markets through two statistical approaches, random matrix theory (RMT) and the correlation coefficient distribution. By using RMT, we find that the largest eigenvalue is an influence that is common to all stocks and this eigenvalue has a peak during financial shocks. We find there are a few correlated stocks that make the essential robustness of the stock market but we see that by replacing these return time series with Gaussian-distributed time series, the mean values of correlation coefficients, the largest eigenvalues of the stock markets and the fraction of eigenvalues that deviate from the RMT prediction fall sharply in both markets. By comparing these two markets, we can see that the DJIA is more sensitive to global perturbations. These findings are crucial for risk management and portfolio selection.

A parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

NASA Technical Reports Server (NTRS)

Swarztrauber, Paul N.

1993-01-01

A parallel algorithm, called polysection, is presented for computing the eigenvalues of a symmetric tridiagonal matrix. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The signs of the polynomials at the interval endpoints are determined a priori and used to guarantee that all zeros are found. The use of finite-precision arithmetic may result in multiple zeros; however, in this case, the intervals coalesce and their number determines exactly the multiplicity of the zero. For an N x N matrix the eigenvalues can be determined in O(log-squared N) time with N-squared processors and O(N) time with N processors. The method is compared with a parallel variant of bisection that requires O(N-squared) time on a single processor, O(N) time with N processors, and O(log N) time with N-squared processors.

Comparison of two Galerkin quadrature methods

DOE PAGES

Morel, Jim E.; Warsa, James; Franke, Brian C.; ...

2017-02-21

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwisemore » sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

Comparison of two Galerkin quadrature methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Morel, Jim E.; Warsa, James; Franke, Brian C.

Here, we compare two methods for generating Galerkin quadratures. In method 1, the standard S N method is used to generate the moment-to-discrete matrix and the discrete-to-moment matrix is generated by inverting the moment-to-discrete matrix. This is a particular form of the original Galerkin quadrature method. In method 2, which we introduce here, the standard S N method is used to generate the discrete-to-moment matrix and the moment-to-discrete matrix is generated by inverting the discrete-to-moment matrix. With an N-point quadrature, method 1 has the advantage that it preserves N eigenvalues and N eigenvectors of the scattering operator in a pointwisemore » sense. With an N-point quadrature, method 2 has the advantage that it generates consistent angular moment equations from the corresponding S N equations while preserving N eigenvalues of the scattering operator. Our computational results indicate that these two methods are quite comparable for the test problem considered.« less

Solving large sparse eigenvalue problems on supercomputers

NASA Technical Reports Server (NTRS)

Philippe, Bernard; Saad, Youcef

1988-01-01

An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed.

A generalized Lyapunov theory for robust root clustering of linear state space models with real parameter uncertainty

NASA Technical Reports Server (NTRS)

Yedavalli, R. K.

1992-01-01

The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.

Survey of methods for calculating sensitivity of general eigenproblems

NASA Technical Reports Server (NTRS)

Murthy, Durbha V.; Haftka, Raphael T.

1987-01-01

A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.

Individual complex Dirac eigenvalue distributions from random matrix theory and comparison to quenched lattice QCD with a quark chemical potential.

PubMed

Akemann, G; Bloch, J; Shifrin, L; Wettig, T

2008-01-25

We analyze how individual eigenvalues of the QCD Dirac operator at nonzero quark chemical potential are distributed in the complex plane. Exact and approximate analytical results for both quenched and unquenched distributions are derived from non-Hermitian random matrix theory. When comparing these to quenched lattice QCD spectra close to the origin, excellent agreement is found for zero and nonzero topology at several values of the quark chemical potential. Our analytical results are also applicable to other physical systems in the same symmetry class.

Sensitivity analysis of hydrodynamic stability operators

NASA Technical Reports Server (NTRS)

Schmid, Peter J.; Henningson, Dan S.; Khorrami, Mehdi R.; Malik, Mujeeb R.

1992-01-01

The eigenvalue sensitivity for hydrodynamic stability operators is investigated. Classical matrix perturbation techniques as well as the concept of epsilon-pseudoeigenvalues are applied to show that parts of the spectrum are highly sensitive to small perturbations. Applications are drawn from incompressible plane Couette, trailing line vortex flow and compressible Blasius boundary layer flow. Parametric studies indicate a monotonically increasing effect of the Reynolds number on the sensitivity. The phenomenon of eigenvalue sensitivity is due to the non-normality of the operators and their discrete matrix analogs and may be associated with large transient growth of the corresponding initial value problem.

Uncovering the Internal Structure of the Indian Financial Market: Large Cross-correlation Behavior in the NSE

NASA Astrophysics Data System (ADS)

Sinha, Sitabhra; Pan, Raj Kumar

The cross-correlations between price fluctuations of 201 frequently traded stocks in the National Stock Exchange (NSE) of India are analyzed in this paper. We use daily closing prices for the period 1996-2006, which coincides with the period of rapid transformation of the market following liberalization. The eigenvalue distribution of the cross-correlation matrix, C, of NSE is found to be similar to that of developed markets, such as the New York Stock Exchange (NYSE): the majority of eigenvalues fall within the bounds expected for a random matrix constructed from mutually uncorrelated time series. Of the few largest eigenvalues that deviate from the bulk, the largest is identified with market-wide movements. The intermediate eigenvalues that occur between the largest and the bulk have been associated in NYSE with specific business sectors with strong intra-group interactions. However, in the Indian market, these deviating eigenvalues are comparatively very few and lie much closer to the bulk. We propose that this is because of the relative lack of distinct sector identity in the market, with the movement of stocks dominantly influenced by the overall market trend. This is shown by explicit construction of the interaction network in the market, first by generating the minimum spanning tree from the unfiltered correlation matrix, and later, using an improved method of generating the graph after filtering out the market mode and random effects from the data. Both methods show, compared to developed markets, the relative absence of clusters of co-moving stocks that belong to the same business sector. This is consistent with the general belief that emerging markets tend to be more correlated than developed markets.

Topology-driven phase transitions in the classical monomer-dimer-loop model.

PubMed

Li, Sazi; Li, Wei; Chen, Ziyu

2015-06-01

In this work, we investigate the classical loop models doped with monomers and dimers on a square lattice, whose partition function can be expressed as a tensor network (TN). In the thermodynamic limit, we use the boundary matrix product state technique to contract the partition function TN, and determine the thermodynamic properties with high accuracy. In this monomer-dimer-loop model, we find a second-order phase transition between a trivial monomer-condensation and a loop-condensation (LC) phase, which cannot be distinguished by any local order parameter, while nevertheless the two phases have distinct topological properties. In the LC phase, we find two degenerate dominating eigenvalues in the transfer-matrix spectrum, as well as a nonvanishing (nonlocal) string order parameter, both of which identify the topological ergodicity breaking in the LC phase and can serve as the order parameter for detecting the phase transitions.

Some Results on Mean Square Error for Factor Score Prediction

ERIC Educational Resources Information Center

Krijnen, Wim P.

2006-01-01

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix gamma[subscript rho] = theta[superscript 1/2] lambda[subscript rho] 'psi[subscript rho] [superscript…

Entropy-driven phase transitions of entanglement

NASA Astrophysics Data System (ADS)

Facchi, Paolo; Florio, Giuseppe; Parisi, Giorgio; Pascazio, Saverio; Yuasa, Kazuya

2013-05-01

We study the behavior of bipartite entanglement at fixed von Neumann entropy. We look at the distribution of the entanglement spectrum, that is, the eigenvalues of the reduced density matrix of a quantum system in a pure state. We report the presence of two continuous phase transitions, characterized by different entanglement spectra, which are deformations of classical eigenvalue distributions.

Dimension from covariance matrices.

PubMed

Carroll, T L; Byers, J M

2017-02-01

We describe a method to estimate embedding dimension from a time series. This method includes an estimate of the probability that the dimension estimate is valid. Such validity estimates are not common in algorithms for calculating the properties of dynamical systems. The algorithm described here compares the eigenvalues of covariance matrices created from an embedded signal to the eigenvalues for a covariance matrix of a Gaussian random process with the same dimension and number of points. A statistical test gives the probability that the eigenvalues for the embedded signal did not come from the Gaussian random process.

Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: Analytical results and applications

NASA Astrophysics Data System (ADS)

Julaiti, Alafate; Wu, Bin; Zhang, Zhongzhi

2013-05-01

The eigenvalues of the normalized Laplacian matrix of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian matrices of a family of fractal trees and dendrimers modeled by Cayley trees, both of which are built in an iterative way. For the fractal trees, we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities, with the eigenvalues provided by a recursive relation governing the eigenvalues of networks at two successive generations. For Cayley trees, we show that all their eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. By using the relation between normalized Laplacian spectra and eigentime identity, we derive the explicit solution to the eigentime identity for random walks on the two treelike networks, the leading scalings of which follow quite different behaviors. In addition, we corroborate the obtained eigenvalues and their degeneracies through the link between them and the number of spanning trees.

Periodic orbit spectrum in terms of Ruelle-Pollicott resonances

NASA Astrophysics Data System (ADS)

Leboeuf, P.

2004-02-01

Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g., a trajectory “p” returns to its initial conditions after some fixed time τp. Our aim is to investigate the spectrum {τ1,τ2,…} of periods of the periodic orbits. An explicit formula for the density ρ(τ)=∑pδ(τ-τp) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle-Pollicott resonances). For large periods, corrections to the well-known exponential growth of the smooth part of the density are obtained. An alternative formula for ρ(τ) in terms of the zeros and poles of the Ruelle ζ function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random-matrix theory, and discrete maps are also considered. In particular, a random-matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.

Closed-form eigensolutions of nonviscously, nonproportionally damped systems based on continuous damping sensitivity

NASA Astrophysics Data System (ADS)

Lázaro, Mario

2018-01-01

In this paper, nonviscous, nonproportional, vibrating structures are considered. Nonviscously damped systems are characterized by dissipative mechanisms which depend on the history of the response velocities via hereditary kernel functions. Solutions of the free motion equation lead to a nonlinear eigenvalue problem involving mass, stiffness and damping matrices. Viscoelasticity leads to a frequency dependence of this latter. In this work, a novel closed-form expression to estimate complex eigenvalues is derived. The key point is to consider the damping model as perturbed by a continuous fictitious parameter. Assuming then the eigensolutions as function of this parameter, the computation of the eigenvalues sensitivity leads to an ordinary differential equation, from whose solution arises the proposed analytical formula. The resulting expression explicitly depends on the viscoelasticity (frequency derivatives of the damping function), the nonproportionality (influence of the modal damping matrix off-diagonal terms). Eigenvectors are obtained using existing methods requiring only the corresponding eigenvalue. The method is validated using a numerical example which compares proposed with exact ones and with those determined from the linear first order approximation in terms of the damping matrix. Frequency response functions are also plotted showing that the proposed approach is valid even for moderately or highly damped systems.

Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices

NASA Astrophysics Data System (ADS)

Böttcher, A.; Bogoya, J. M.; Grudsky, S. M.; Maximenko, E. A.

2017-11-01

Analysis of the asymptotic behaviour of the spectral characteristics of Toeplitz matrices as the dimension of the matrix tends to infinity has a history of over 100 years. For instance, quite a number of versions of Szegő's theorem on the asymptotic behaviour of eigenvalues and of the so-called strong Szegő theorem on the asymptotic behaviour of the determinants of Toeplitz matrices are known. Starting in the 1950s, the asymptotics of the maximum and minimum eigenvalues were actively investigated. However, investigation of the individual asymptotics of all the eigenvalues and eigenvectors of Toeplitz matrices started only quite recently: the first papers on this subject were published in 2009-2010. A survey of this new field is presented here. Bibliography: 55 titles.

Derivation of an eigenvalue probability density function relating to the Poincaré disk

NASA Astrophysics Data System (ADS)

Forrester, Peter J.; Krishnapur, Manjunath

2009-09-01

A result of Zyczkowski and Sommers (2000 J. Phys. A: Math. Gen. 33 2045-57) gives the eigenvalue probability density function for the top N × N sub-block of a Haar distributed matrix from U(N + n). In the case n >= N, we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition and integrating over all variables except the eigenvalues. The integration is done by identifying a recursive structure which reduces the dimension. This approach is inspired by an analogous approach which has been recently applied to determine the eigenvalue probability density function for random matrices A-1B, where A and B are random matrices with entries standard complex normals. We relate the eigenvalue distribution of the sub-blocks to a many-body quantum state, and to the one-component plasma, on the pseudosphere.

Determining entire mean first-passage time for Cayley networks

NASA Astrophysics Data System (ADS)

Wang, Xiaoqian; Dai, Meifeng; Chen, Yufei; Zong, Yue; Sun, Yu; Su, Weiyi

In this paper, we consider the entire mean first-passage time (EMFPT) with random walks for Cayley networks. We use Laplacian spectra to calculate the EMFPT. Firstly, we calculate the constant term and monomial coefficient of characteristic polynomial. By using the Vieta theorem, we then obtain the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix. Finally, we obtain the scaling of the EMFPT for Cayley networks by using the relationship between the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix and the EMFPT. We expect that our method can be adapted to other types of self-similar networks, such as vicsek networks, polymer networks.

A parallel algorithm for the eigenvalues and eigenvectors for a general complex matrix

NASA Technical Reports Server (NTRS)

Shroff, Gautam

1989-01-01

A new parallel Jacobi-like algorithm is developed for computing the eigenvalues of a general complex matrix. Most parallel methods for this parallel typically display only linear convergence. Sequential norm-reducing algorithms also exit and they display quadratic convergence in most cases. The new algorithm is a parallel form of the norm-reducing algorithm due to Eberlein. It is proven that the asymptotic convergence rate of this algorithm is quadratic. Numerical experiments are presented which demonstrate the quadratic convergence of the algorithm and certain situations where the convergence is slow are also identified. The algorithm promises to be very competitive on a variety of parallel architectures.

Statistical properties of the stock and credit market: RMT and network topology

NASA Astrophysics Data System (ADS)

Lim, Kyuseong; Kim, Min Jae; Kim, Sehyun; Kim, Soo Yong

We analyzed the dependence structure of the credit and stock market using random matrix theory and network topology. The dynamics of both markets have been spotlighted throughout the subprime crisis. In this study, we compared these two markets in view of the market-wide effect from random matrix theory and eigenvalue analysis. We found that the largest eigenvalue of the credit market as a whole preceded that of the stock market in the beginning of the financial crisis and that of two markets tended to be synchronized after the crisis. The correlation between the companies of both markets became considerably stronger after the crisis as well.

Eigenvalues of the Laplacian of a graph

NASA Technical Reports Server (NTRS)

Anderson, W. N., Jr.; Morley, T. D.

1971-01-01

Let G be a finite undirected graph with no loops or multiple edges. The Laplacian matrix of G, Delta(G), is defined by Delta sub ii = degree of vertex i and Delta sub ij = -1 if there is an edge between vertex i and vertex j. The structure of the graph G is related to the eigenvalues of Delta(G); in particular, it is proved that all the eigenvalues of Delta(G) are nonnegative, less than or equal to the number of vertices, and less than or equal to twice the maximum vertex degree. Precise conditions for equality are given.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brabec, Jiri; Lin, Lin; Shao, Meiyue

We present two iterative algorithms for approximating the absorption spectrum of molecules within linear response of time-dependent density functional theory (TDDFT) framework. These methods do not attempt to compute eigenvalues or eigenvectors of the linear response matrix. They are designed to approximate the absorption spectrum as a function directly. They take advantage of the special structure of the linear response matrix. Neither method requires the linear response matrix to be constructed explicitly. They only require a procedure that performs the multiplication of the linear response matrix with a vector. These methods can also be easily modified to efficiently estimate themore » density of states (DOS) of the linear response matrix without computing the eigenvalues of this matrix. We show by computational experiments that the methods proposed in this paper can be much more efficient than methods that are based on the exact diagonalization of the linear response matrix. We show that they can also be more efficient than real-time TDDFT simulations. We compare the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost.« less

Calculation of normal modes of the closed waveguides in general vector case

NASA Astrophysics Data System (ADS)

Malykh, M. D.; Sevastianov, L. A.; Tiutiunnik, A. A.

2018-04-01

The article is devoted to the calculation of normal modes of the closed waveguides with an arbitrary filling ɛ, μ in the system of computer algebra Sage. Maxwell equations in the cylinder are reduced to the system of two bounded Helmholtz equations, the notion of weak solution of this system is given and then this system is investigated as a system of ordinary differential equations. The normal modes of this system are an eigenvectors of a matrix pencil. We suggest to calculate the matrix elements approximately and to truncate the matrix by usual way but further to solve the truncated eigenvalue problem exactly in the field of algebraic numbers. This approach allows to keep the symmetry of the initial problem and in particular the multiplicity of the eigenvalues. In the work would be presented some results of calculations.

Hessian eigenvalue distribution in a random Gaussian landscape

NASA Astrophysics Data System (ADS)

Yamada, Masaki; Vilenkin, Alexander

2018-03-01

The energy landscape of multiverse cosmology is often modeled by a multi-dimensional random Gaussian potential. The physical predictions of such models crucially depend on the eigenvalue distribution of the Hessian matrix at potential minima. In particular, the stability of vacua and the dynamics of slow-roll inflation are sensitive to the magnitude of the smallest eigenvalues. The Hessian eigenvalue distribution has been studied earlier, using the saddle point approximation, in the leading order of 1/ N expansion, where N is the dimensionality of the landscape. This approximation, however, is insufficient for the small eigenvalue end of the spectrum, where sub-leading terms play a significant role. We extend the saddle point method to account for the sub-leading contributions. We also develop a new approach, where the eigenvalue distribution is found as an equilibrium distribution at the endpoint of a stochastic process (Dyson Brownian motion). The results of the two approaches are consistent in cases where both methods are applicable. We discuss the implications of our results for vacuum stability and slow-roll inflation in the landscape.

Fingerprint recognition of alien invasive weeds based on the texture character and machine learning

NASA Astrophysics Data System (ADS)

Yu, Jia-Jia; Li, Xiao-Li; He, Yong; Xu, Zheng-Hao

2008-11-01

Multi-spectral imaging technique based on texture analysis and machine learning was proposed to discriminate alien invasive weeds with similar outline but different categories. The objectives of this study were to investigate the feasibility of using Multi-spectral imaging, especially the near-infrared (NIR) channel (800 nm+/-10 nm) to find the weeds' fingerprints, and validate the performance with specific eigenvalues by co-occurrence matrix. Veronica polita Pries, Veronica persica Poir, longtube ground ivy, Laminum amplexicaule Linn. were selected in this study, which perform different effect in field, and are alien invasive species in China. 307 weed leaves' images were randomly selected for the calibration set, while the remaining 207 samples for the prediction set. All images were pretreated by Wallis filter to adjust the noise by uneven lighting. Gray level co-occurrence matrix was applied to extract the texture character, which shows density, randomness correlation, contrast and homogeneity of texture with different algorithms. Three channels (green channel by 550 nm+/-10 nm, red channel by 650 nm+/-10 nm and NIR channel by 800 nm+/-10 nm) were respectively calculated to get the eigenvalues.Least-squares support vector machines (LS-SVM) was applied to discriminate the categories of weeds by the eigenvalues from co-occurrence matrix. Finally, recognition ratio of 83.35% by NIR channel was obtained, better than the results by green channel (76.67%) and red channel (69.46%). The prediction results of 81.35% indicated that the selected eigenvalues reflected the main characteristics of weeds' fingerprint based on multi-spectral (especially by NIR channel) and LS-SVM model.

Random pure states: Quantifying bipartite entanglement beyond the linear statistics.

PubMed

Vivo, Pierpaolo; Pato, Mauricio P; Oshanin, Gleb

2016-05-01

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions N and M. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary N≤M, a general relation between the n-point densities and the cross moments of the eigenvalues of the reduced density matrix, i.e., the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite N,M. Then, we focus on the moments E{K^{a}} of the Schmidt number K, the reciprocal of the purity. This is a random variable supported on [1,N], which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for E{K^{a}} for N=2 and 3 and arbitrary M, and also for square N=M systems by spotting for the latter a connection with the probability P(x_{min}^{GUE}≥sqrt[2N]ξ) that the smallest eigenvalue x_{min}^{GUE} of an N×N matrix belonging to the Gaussian unitary ensemble is larger than sqrt[2N]ξ. As a by-product, we present an exact asymptotic expansion for P(x_{min}^{GUE}≥sqrt[2N]ξ) for finite N as ξ→∞. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

Large computer simulations on elastic networks: Small eigenvalues and eigenvalue spectra of the Kirchhoff matrix

NASA Astrophysics Data System (ADS)

Shy, L. Y.; Eichinger, B. E.

1989-05-01

Computer simulations of the formation of trifunctional and tetrafunctional polydimethyl-siloxane networks that are crosslinked by condensation of telechelic chains with multifunctional crosslinking agents have been carried out on systems containing up to 1.05×106 chains. Eigenvalue spectra of Kirchhoff matrices for these networks have been evaluated at two levels of approximation: (1) inclusion of all midchain modes, and (2) suppression of midchain modes. By use of the recursion method of Haydock and Nex, we have been able to effectively diagonalize matrices with 730 498 rows and columns without actually constructing matrices of this size. The small eigenvalues have been computed by use of the Lanczos algorithm. We demonstrate the following results: (1) The smallest eigenvalues (with chain modes suppressed) vary as μ-2/3 for sufficiently large μ, where μ is the number of junctions in the network; (2) the eigenvalue spectra of the Kirchhoff matrices are well described by McKay's theory for random regular graphs in the range of the larger eigenvalues, but there are significant departures in the region of small eigenvalues where computed spectra have many more small eigenvalues than random regular graphs; (3) the smallest eigenvalues vary as n-1.78 where n is the number of Rouse beads in the chains that comprise the network. Computations are done for both monodisperse and polydisperse chain length distributions. Large eigenvalues associated with localized motion of the junctions are found as predicted by theory. The relationship between the small eigenvalues and the equilibrium modulus of elasticity is discussed, as is the relationship between viscoelasticity and the band edge of the spectrum.

An O(log sup 2 N) parallel algorithm for computing the eigenvalues of a symmetric tridiagonal matrix

NASA Technical Reports Server (NTRS)

Swarztrauber, Paul N.

1989-01-01

An O(log sup 2 N) parallel algorithm is presented for computing the eigenvalues of a symmetric tridiagonal matrix using a parallel algorithm for computing the zeros of the characteristic polynomial. The method is based on a quadratic recurrence in which the characteristic polynomial is constructed on a binary tree from polynomials whose degree doubles at each level. Intervals that contain exactly one zero are determined by the zeros of polynomials at the previous level which ensures that different processors compute different zeros. The exact behavior of the polynomials at the interval endpoints is used to eliminate the usual problems induced by finite precision arithmetic.

The discrete hungry Lotka Volterra system and a new algorithm for computing matrix eigenvalues

NASA Astrophysics Data System (ADS)

Fukuda, Akiko; Ishiwata, Emiko; Iwasaki, Masashi; Nakamura, Yoshimasa

2009-01-01

The discrete hungry Lotka-Volterra (dhLV) system is a generalization of the discrete Lotka-Volterra (dLV) system which stands for a prey-predator model in mathematical biology. In this paper, we show that (1) some invariants exist which are expressed by dhLV variables and are independent from the discrete time and (2) a dhLV variable converges to some positive constant or zero as the discrete time becomes sufficiently large. Some characteristic polynomial is then factorized with the help of the dhLV system. The asymptotic behaviour of the dhLV system enables us to design an algorithm for computing complex eigenvalues of a certain band matrix.

Efficient, massively parallel eigenvalue computation

NASA Technical Reports Server (NTRS)

Huo, Yan; Schreiber, Robert

1993-01-01

In numerical simulations of disordered electronic systems, one of the most common approaches is to diagonalize random Hamiltonian matrices and to study the eigenvalues and eigenfunctions of a single electron in the presence of a random potential. An effort to implement a matrix diagonalization routine for real symmetric dense matrices on massively parallel SIMD computers, the Maspar MP-1 and MP-2 systems, is described. Results of numerical tests and timings are also presented.

On functional determinants of matrix differential operators with multiple zero modes

NASA Astrophysics Data System (ADS)

Falco, G. M.; Fedorenko, Andrei A.; Gruzberg, Ilya A.

2017-12-01

We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of r× r matrix second order differential operators O with 0 < n ≤slant 2r linearly independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r-n or 2r-n (depending on the boundary conditions) solutions of the homogeneous equation O h=0 , in the spirit of Gel’fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. This cancellation drastically simplifies instanton calculations when one uses our formulas.

Robust cooperation of connected vehicle systems with eigenvalue-bounded interaction topologies in the presence of uncertain dynamics

NASA Astrophysics Data System (ADS)

Li, Keqiang; Gao, Feng; Li, Shengbo Eben; Zheng, Yang; Gao, Hongbo

2017-12-01

This study presents a distributed H-infinity control method for uncertain platoons with dimensionally and structurally unknown interaction topologies provided that the associated topological eigenvalues are bounded by a predesigned range.With an inverse model to compensate for nonlinear powertrain dynamics, vehicles in a platoon are modeled by third-order uncertain systems with bounded disturbances. On the basis of the eigenvalue decomposition of topological matrices, we convert the platoon system to a norm-bounded uncertain part and a diagonally structured certain part by applying linear transformation. We then use a common Lyapunov method to design a distributed H-infinity controller. Numerically, two linear matrix inequalities corresponding to the minimum and maximum eigenvalues should be solved. The resulting controller can tolerate interaction topologies with eigenvalues located in a certain range. The proposed method can also ensure robustness performance and disturbance attenuation ability for the closed-loop platoon system. Hardware-in-the-loop tests are performed to validate the effectiveness of our method.

Fine structure of spectral properties for random correlation matrices: An application to financial markets

NASA Astrophysics Data System (ADS)

Livan, Giacomo; Alfarano, Simone; Scalas, Enrico

2011-07-01

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross correlations between stocks. We interpret and corroborate these findings in terms of factor models, and we compare empirical spectra to those predicted by random matrix theory for such models.

a Unified Matrix Polynomial Approach to Modal Identification

NASA Astrophysics Data System (ADS)

Allemang, R. J.; Brown, D. L.

1998-04-01

One important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single, consistent mathematical formulation with a corresponding set of definitions and unifying concepts. Particularly, a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least-squares complex exponential (LSCE), the polyreference time domain (PTD), Ibrahim time domain (ITD), eigensystem realization algorithm (ERA), rational fraction polynomial (RFP), polyreference frequency domain (PFD) and the complex mode indication function (CMIF) methods. Using this unified matrix polynomial approach (UMPA) allows a discussion of the similarities and differences of the commonly used methods. the use of least squares (LS), total least squares (TLS), double least squares (DLS) and singular value decomposition (SVD) methods is discussed in order to take advantage of redundant measurement data. Eigenvalue and SVD transformation methods are utilized to reduce the effective size of the resulting eigenvalue-eigenvector problem as well.

Derivatives of random matrix characteristic polynomials with applications to elliptic curves

NASA Astrophysics Data System (ADS)

Snaith, N. C.

2005-12-01

The value distribution of derivatives of characteristic polynomials of matrices from SO(N) is calculated at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. We consider subsets of matrices from SO(N) that are constrained to have at least n eigenvalues equal to 1 and investigate the first non-zero derivative of the characteristic polynomial at that point. The connection between the values of random matrix characteristic polynomials and values of L-functions in families has been well established. The motivation for this work is the expectation that through this connection with L-functions derived from families of elliptic curves, and using the Birch and Swinnerton-Dyer conjecture to relate values of the L-functions to the rank of elliptic curves, random matrix theory will be useful in probing important questions concerning these ranks.

Computing resonance energies, widths, and wave functions using a Lanczos method in real arithmetic.

PubMed

Tremblay, Jean Christophe; Carrington, Tucker

2005-06-22

We introduce new ideas for calculating resonance energies and widths. It is shown that a non-Hermitian-Lanczos approach can be used to compute eigenvalues of H+W, where H is the Hamiltonian and W is a complex absorbing potential (CAP), without evaluating complex matrix-vector products. This is done by exploiting the link between a CAP-modified Hamiltonian matrix and a real but nonsymmetric matrix U suggested by Mandelshtam and Neumaier [J. Theor. Comput. Chem. 1, 1 (2002)] and using a coupled-two-term Lanczos procedure. We use approximate resonance eigenvectors obtained from the non-Hermitian-Lanczos algorithm and a very good CAP to obtain very accurate energies and widths without solving eigenvalue problems for many values of the CAP strength parameter and searching for cusps. The method is applied to the resonances of HCO. We compare properties of the method with those of established approaches.

Partial transpose of random quantum states: Exact formulas and meanders

NASA Astrophysics Data System (ADS)

Fukuda, Motohisa; Śniady, Piotr

2013-04-01

We investigate the asymptotic behavior of the empirical eigenvalues distribution of the partial transpose of a random quantum state. The limiting distribution was previously investigated via Wishart random matrices indirectly (by approximating the matrix of trace 1 by the Wishart matrix of random trace) and shown to be the semicircular distribution or the free difference of two free Poisson distributions, depending on how dimensions of the concerned spaces grow. Our use of Wishart matrices gives exact combinatorial formulas for the moments of the partial transpose of the random state. We find three natural asymptotic regimes in terms of geodesics on the permutation groups. Two of them correspond to the above two cases; the third one turns out to be a new matrix model for the meander polynomials. Moreover, we prove the convergence to the semicircular distribution together with its extreme eigenvalues under weaker assumptions, and show large deviation bound for the latter.

Approximate equiangular tight frames for compressed sensing and CDMA applications

NASA Astrophysics Data System (ADS)

Tsiligianni, Evaggelia; Kondi, Lisimachos P.; Katsaggelos, Aggelos K.

2017-12-01

Performance guarantees for recovery algorithms employed in sparse representations, and compressed sensing highlights the importance of incoherence. Optimal bounds of incoherence are attained by equiangular unit norm tight frames (ETFs). Although ETFs are important in many applications, they do not exist for all dimensions, while their construction has been proven extremely difficult. In this paper, we construct frames that are close to ETFs. According to results from frame and graph theory, the existence of an ETF depends on the existence of its signature matrix, that is, a symmetric matrix with certain structure and spectrum consisting of two distinct eigenvalues. We view the construction of a signature matrix as an inverse eigenvalue problem and propose a method that produces frames of any dimensions that are close to ETFs. Due to the achieved equiangularity property, the so obtained frames can be employed as spreading sequences in synchronous code-division multiple access (s-CDMA) systems, besides compressed sensing.

Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate

NASA Astrophysics Data System (ADS)

Tokar, V. I.; Dreyssé, H.

2005-03-01

We consider a one-dimensional lattice gas model of strained epitaxy with the elastic strain accounted for through a finite number of cluster interactions comprising contiguous atomic chains. Interactions of this type arise in the models of strained epitaxy based on the Frenkel-Kontorova model. Furthermore, the deposited atoms interact with the substrate via an arbitrary periodic potential of period L . This model is solved exactly with the use of an appropriately adopted technique developed recently in the theory of protein folding. The advantage of the proposed approach over the standard transfer-matrix method is that it reduces the problem to finding the largest eigenvalue of a matrix of size L instead of 2L-1 , which is vital in the case of nanostructures where L may measure in hundreds of interatomic distances. Our major conclusion is that the substrate modulation always facilitates the size calibration of self-assembled nanoparticles in one- and two-dimensional systems.

On Distributed Strategies in Defense of a High Value Unit (HVU) Against a Swarm Attack

DTIC Science & Technology

2012-09-01

function [ lam ,U] = tqr(a,b,U) % [ lam u] = tqr(a,b) or [ lam U] = tqr(a,b,U... lam u] = tqr(a,b): % % The column lam contains the eigenvalues of the Hermitian tridiagonal % matrix T = mxt(a,b) computed by one version of the...computed. The computed eigenvalues are real and are sorted to be % nonincreasing. % % [ lam U] = tqr(a,b,U): % % This replaces the input U

A Problem-Centered Approach to Canonical Matrix Forms

ERIC Educational Resources Information Center

Sylvestre, Jeremy

2014-01-01

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

On Graph Isomorphism and the PageRank Algorithm

DTIC Science & Technology

2008-09-01

specifies the probability of visiting each node from any other node. The perturbed matrix satisfies the Perron - Frobenius theorem’s conditions. Therefore... Frobenius and Perron theorems establishes the matrix must yield the dominant eigenvalue, one. Normalizing the unique and associated dominant eigenvector...is constructed such that none of its entries equal zero. An arbitrary PageRank matrix, S, is irreducible and satisfies the Perron - Frobenius

Workshop report on large-scale matrix diagonalization methods in chemistry theory institute

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bischof, C.H.; Shepard, R.L.; Huss-Lederman, S.

The Large-Scale Matrix Diagonalization Methods in Chemistry theory institute brought together 41 computational chemists and numerical analysts. The goal was to understand the needs of the computational chemistry community in problems that utilize matrix diagonalization techniques. This was accomplished by reviewing the current state of the art and looking toward future directions in matrix diagonalization techniques. This institute occurred about 20 years after a related meeting of similar size. During those 20 years the Davidson method continued to dominate the problem of finding a few extremal eigenvalues for many computational chemistry problems. Work on non-diagonally dominant and non-Hermitian problems asmore » well as parallel computing has also brought new methods to bear. The changes and similarities in problems and methods over the past two decades offered an interesting viewpoint for the success in this area. One important area covered by the talks was overviews of the source and nature of the chemistry problems. The numerical analysts were uniformly grateful for the efforts to convey a better understanding of the problems and issues faced in computational chemistry. An important outcome was an understanding of the wide range of eigenproblems encountered in computational chemistry. The workshop covered problems involving self- consistent-field (SCF), configuration interaction (CI), intramolecular vibrational relaxation (IVR), and scattering problems. In atomic structure calculations using the Hartree-Fock method (SCF), the symmetric matrices can range from order hundreds to thousands. These matrices often include large clusters of eigenvalues which can be as much as 25% of the spectrum. However, if Cl methods are also used, the matrix size can be between 10{sup 4} and 10{sup 9} where only one or a few extremal eigenvalues and eigenvectors are needed. Working with very large matrices has lead to the development of« less

Monitoring hydraulic stimulation using telluric sounding

NASA Astrophysics Data System (ADS)

Rees, Nigel; Heinson, Graham; Conway, Dennis

2018-01-01

The telluric sounding (TS) method is introduced as a potential tool for monitoring hydraulic fracturing at depth. The advantage of this technique is that it requires only the measurement of electric fields, which are cheap and easy when compared with magnetotelluric measurements. Additionally, the transfer function between electric fields from two locations is essentially the identity matrix for a 1D Earth no matter what the vertical structure. Therefore, changes in the earth resulting from the introduction of conductive bodies underneath one of these sites can be associated with deviations away from the identity matrix, with static shift appearing as a galvanic multiplier at all periods. Singular value decomposition and eigenvalue analysis can reduce the complexity of the resulting telluric distortion matrix to simpler parameters that can be visualised in the form of Mohr circles. This technique would be useful in constraining the lateral extent of resistivity changes. We test the viability of utilising the TS method for monitoring on both a synthetic dataset and for a hydraulic stimulation of an enhanced geothermal system case study conducted in Paralana, South Australia. The synthetic data example shows small but consistent changes in the transfer functions associated with hydraulic stimulation, with grids of Mohr circles introduced as a useful diagnostic tool for visualising the extent of fluid movement. The Paralana electric field data were relatively noisy and affected by the dead band making the analysis of transfer functions difficult. However, changes in the order of 5% were observed from 5 s to longer periods. We conclude that deep monitoring using the TS method is marginal at depths in the order of 4 km and that in order to have meaningful interpretations, electric field data need to be of a high quality with low levels of site noise.[Figure not available: see fulltext.

Fully Parallel MHD Stability Analysis Tool

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2014-10-01

Progress on full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. It is a powerful tool for studying MHD and MHD-kinetic instabilities and it is widely used by fusion community. Parallel version of MARS is intended for simulations on local parallel clusters. It will be an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, already implemented in MARS. Parallelization of the code includes parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the present MARS algorithm using parallel libraries and procedures. Initial results of the code parallelization will be reported. Work is supported by the U.S. DOE SBIR program.

A novel scatter-matrix eigenvalues-based total variation (SMETV) regularization for medical image restoration

NASA Astrophysics Data System (ADS)

Huang, Zhenghua; Zhang, Tianxu; Deng, Lihua; Fang, Hao; Li, Qian

2015-12-01

Total variation(TV) based on regularization has been proven as a popular and effective model for image restoration, because of its ability of edge preserved. However, as the TV favors a piece-wise constant solution, the processing results in the flat regions of the image are easily produced "staircase effects", and the amplitude of the edges will be underestimated; the underlying cause of the problem is that the regularization parameter can not be changeable with spatial local information of image. In this paper, we propose a novel Scatter-matrix eigenvalues-based TV(SMETV) regularization with image blind restoration algorithm for deblurring medical images. The spatial information in different image regions is incorporated into regularization by using the edge indicator called difference eigenvalue to distinguish edges from flat areas. The proposed algorithm can effectively reduce the noise in flat regions as well as preserve the edge and detailed information. Moreover, it becomes more robust with the change of the regularization parameter. Extensive experiments demonstrate that the proposed approach produces results superior to most methods in both visual image quality and quantitative measures.

A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus

NASA Astrophysics Data System (ADS)

Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir

2016-07-01

This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.

Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects

NASA Astrophysics Data System (ADS)

Castellano, Claudio; Pastor-Satorras, Romualdo

2017-10-01

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: the hub with its immediate neighbors and the densely connected set of nodes with maximum K -core index. We validate this formula by showing that it predicts, with good accuracy, the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a by-product, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.

Eigenvalue routines in NASTRAN: A comparison with the Block Lanczos method

NASA Technical Reports Server (NTRS)

Tischler, V. A.; Venkayya, Vipperla B.

1993-01-01

The NASA STRuctural ANalysis (NASTRAN) program is one of the most extensively used engineering applications software in the world. It contains a wealth of matrix operations and numerical solution techniques, and they were used to construct efficient eigenvalue routines. The purpose of this paper is to examine the current eigenvalue routines in NASTRAN and to make efficiency comparisons with a more recent implementation of the Block Lanczos algorithm by Boeing Computer Services (BCS). This eigenvalue routine is now available in the BCS mathematics library as well as in several commercial versions of NASTRAN. In addition, CRAY maintains a modified version of this routine on their network. Several example problems, with a varying number of degrees of freedom, were selected primarily for efficiency bench-marking. Accuracy is not an issue, because they all gave comparable results. The Block Lanczos algorithm was found to be extremely efficient, in particular, for very large size problems.

Spectral analysis for weighted tree-like fractals

NASA Astrophysics Data System (ADS)

Dai, Meifeng; Chen, Yufei; Wang, Xiaoqian; Sun, Yu; Su, Weiyi

2018-02-01

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a study on the spectra of the normalized Laplacian of weighted tree-like fractals. We analytically obtain the relationship between the eigenvalues and their multiplicities for two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index and Kemeny's constant.

Generalized Eigenvalues for pairs on heritian matrices

NASA Technical Reports Server (NTRS)

Rublein, George

1988-01-01

A study was made of certain special cases of a generalized eigenvalue problem. Let A and B be nxn matrics. One may construct a certain polynomial, P(A,B, lambda) which specializes to the characteristic polynomial of B when A equals I. In particular, when B is hermitian, that characteristic polynomial, P(I,B, lambda) has real roots, and one can ask: are the roots of P(A,B, lambda) real when B is hermitian. We consider the case where A is positive definite and show that when N equals 3, the roots are indeed real. The basic tools needed in the proof are Shur's theorem on majorization for eigenvalues of hermitian matrices and the interlacing theorem for the eigenvalues of a positive definite hermitian matrix and one of its principal (n-1)x(n-1) minors. The method of proof first reduces the general problem to one where the diagonal of B has a certain structure: either diag (B) = diag (1,1,1) or diag (1,1,-1), or else the 2 x 2 principal minors of B are all 1. According as B has one of these three structures, we use an appropriate method to replace A by a positive diagonal matrix. Since it can be easily verified that P(D,B, lambda) has real roots, the result follows. For other configurations of B, a scaling and a continuity argument are used to prove the result in general.

Sparse Regression as a Sparse Eigenvalue Problem

NASA Technical Reports Server (NTRS)

Moghaddam, Baback; Gruber, Amit; Weiss, Yair; Avidan, Shai

2008-01-01

We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization

Analyzing Small Signal Stability of Power System based on Online Data by Use of SMES

NASA Astrophysics Data System (ADS)

Ishikawa, Hiroyuki; Shirai, Yasuyuki; Nitta, Tanzo; Shibata, Katsuhiko

The purpose of this study is to estimate eigen-values and eigen-vectors of a power system from on-line data to evaluate the power system stability. Power system responses due to the small power modulation of known pattern from SMES (Superconducting Magnetic Energy Storage) were analyzed, and the transfer functions between the power modulation and power oscillations of generators were obtained. Eigen-values and eigen-vectors were estimated from the transfer functions. Experiments were carried out by use of a model SMES and Advanced Power System Analyzer (APSA), which is an analogue type power system simulator of Kansai Electric Power Company Inc., Japan. Changes in system condition were observed by the estimated eigen-values and eigen-vectors. Result agreed well with the resent report and digital simulation. This method gives a new application for SMES, which will be installed for improving electric power quality.

La structure de Jordan des matrices de transfert des modeles de boucles et la relation avec les hamiltoniens XXZ

NASA Astrophysics Data System (ADS)

Morin-Duchesne, Alexi

Lattice models such as percolation, the Ising model and the Potts model are useful for the description of phase transitions in two dimensions. Finding analytical solutions is done by calculating the partition function, which in turn requires finding eigenvalues of transfer matrices. At the critical point, the two dimensional statistical models are invariant under conformal transformations and the construction of rational conformal field theories, as the continuum limit of these lattice models, allows one to compute the partition function at the critical point. Many researchers think however that the paradigm of rational conformal conformal field theories can be extended to include models with non diagonalizable transfer matrices. These models would then be described, in the scaling limit, by logarithmic conformal field theories and the representations of the Virasoro algebra coming into play would be indecomposable. We recall the construction of the double-row transfer matrix DN (λ, u) of the Fortuin-Kasteleyn model, seen as an element of the Temperley-Lieb algebra. This transfer matrix comes into play in physical theories through its representation in link modules (or standard modules). The vector space on which this representation acts decomposes into sectors labelled by a physical parameter d, the number of defects, which remains constant or decreases in the link representations. This thesis is devoted to the identification of the Jordan structure of DN(λ, u) in the link representations. The parameter β = 2 cos λ = -(q + q-1) fixes the theory : for instance β = 1 for percolation and 2 for the Ising model. On the geometry of the strip with open boundary conditions, we show that DN(λ, u) has the same Jordan blocks as its highest Fourier coefficient, FN. We study the non-diagonalizability of FN through the divergences of some of the eigenstates of ρ(F N) that appear at the critical values of λ. The Jordan cells we find in ρ(DN(λ, u)) have rank 2 and couple sectors d and d' when specific constraints on λ, d, d' and N are satisfied. For the model of critical dense polymers (β = 0) on the strip, the eigenvalues of ρ(DN(λ, u)) were known, but their degeneracies only conjectured. By constructing an isomorphism between the link modules on the strip and a subspace of spin modules of the XXZ model at q = i, we prove this conjecture. We also show that the restriction of the Hamiltonian to any sector d is diagonalizable, and that the XX Hamiltonian has rank 2 Jordan cells when N is even. Finally, we study the Jordan structure of the transfer matrix T N(λ, ν) for periodic boundary conditions. When λ = πa/b and a, b ∈ Z× , the matrix TN(λ, ν) has Jordan blocks between sectors, but also within sectors. The approach using FN admits a generalization to the present case and allows us to probe the Jordan cells that tie different sectors. The rank of these cells exceeds 2 in some cases and can grow indefinitely with N. For the Jordan blocks within a sector, we show that the link modules on the cylinder and the XXZ spin modules are isomorphic except for specific curves in the (q, ν) plane. By using the behavior of the transformation ĩd N in a neighborhood of the critical values (qc, ν c), we explicitly build Jordan partners of rank 2 and discuss the existence of Jordan cells with higher rank. Keywords : phase transitions, Ising model, Potts model, Fortuin-Kasteleyn model, transfer matrix method, XXZ Hamiltonian, logarithmic conformal field theory, Jordan structure.

Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vecharynski, Eugene; Brabec, Jiri; Shao, Meiyue

We present two efficient iterative algorithms for solving the linear response eigen- value problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-innermore » product. The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples.« less

The coprime quantum chain

NASA Astrophysics Data System (ADS)

Mussardo, G.; Giudici, G.; Viti, J.

2017-03-01

In this paper we introduce and study the coprime quantum chain, i.e. a strongly correlated quantum system defined in terms of the integer eigenvalues n i of the occupation number operators at each site of a chain of length M. The n i ’s take value in the interval [2,q] and may be regarded as S z eigenvalues in the spin representation j  =  (q  -  2)/2. The distinctive interaction of the model is based on the coprimality matrix \\boldsymbolΦ : for the ferromagnetic case, this matrix assigns lower energy to configurations where occupation numbers n i and n i+1 of neighbouring sites share a common divisor, while for the anti-ferromagnetic case it assigns a lower energy to configurations where n i and n i+1 are coprime. The coprime chain, both in the ferro and anti-ferromagnetic cases, may present an exponential number of ground states whose values can be exactly computed by means of graph theoretical tools. In the ferromagnetic case there are generally also frustration phenomena. A fine tuning of local operators may lift the exponential ground state degeneracy and, according to which operators are switched on, the system may be driven into different classes of universality, among which the Ising or Potts universality class. The paper also contains an appendix by Don Zagier on the exact eigenvalues and eigenvectors of the coprimality matrix in the limit q\\to ∞ .

Comparison of eigensolvers for symmetric band matrices.

PubMed

Moldaschl, Michael; Gansterer, Wilfried N

2014-09-15

We compare different algorithms for computing eigenvalues and eigenvectors of a symmetric band matrix across a wide range of synthetic test problems. Of particular interest is a comparison of state-of-the-art tridiagonalization-based methods as implemented in Lapack or Plasma on the one hand, and the block divide-and-conquer (BD&C) algorithm as well as the block twisted factorization (BTF) method on the other hand. The BD&C algorithm does not require tridiagonalization of the original band matrix at all, and the current version of the BTF method tridiagonalizes the original band matrix only for computing the eigenvalues. Avoiding the tridiagonalization process sidesteps the cost of backtransformation of the eigenvectors. Beyond that, we discovered another disadvantage of the backtransformation process for band matrices: In several scenarios, a lot of gradual underflow is observed in the (optional) accumulation of the transformation matrix and in the (obligatory) backtransformation step. According to the IEEE 754 standard for floating-point arithmetic, this implies many operations with subnormal (denormalized) numbers, which causes severe slowdowns compared to the other algorithms without backtransformation of the eigenvectors. We illustrate that in these cases the performance of existing methods from Lapack and Plasma reaches a competitive level only if subnormal numbers are disabled (and thus the IEEE standard is violated). Overall, our performance studies illustrate that if the problem size is large enough relative to the bandwidth, BD&C tends to achieve the highest performance of all methods if the spectrum to be computed is clustered. For test problems with well separated eigenvalues, the BTF method tends to become the fastest algorithm with growing problem size.

Fission matrix-based Monte Carlo criticality analysis of fuel storage pools

DOE Office of Scientific and Technical Information (OSTI.GOV)

Farlotti, M.; Ecole Polytechnique, Palaiseau, F 91128; Larsen, E. W.

2013-07-01

Standard Monte Carlo transport procedures experience difficulties in solving criticality problems in fuel storage pools. Because of the strong neutron absorption between fuel assemblies, source convergence can be very slow, leading to incorrect estimates of the eigenvalue and the eigenfunction. This study examines an alternative fission matrix-based Monte Carlo transport method that takes advantage of the geometry of a storage pool to overcome this difficulty. The method uses Monte Carlo transport to build (essentially) a fission matrix, which is then used to calculate the criticality and the critical flux. This method was tested using a test code on a simplemore » problem containing 8 assemblies in a square pool. The standard Monte Carlo method gave the expected eigenfunction in 5 cases out of 10, while the fission matrix method gave the expected eigenfunction in all 10 cases. In addition, the fission matrix method provides an estimate of the error in the eigenvalue and the eigenfunction, and it allows the user to control this error by running an adequate number of cycles. Because of these advantages, the fission matrix method yields a higher confidence in the results than standard Monte Carlo. We also discuss potential improvements of the method, including the potential for variance reduction techniques. (authors)« less

A new approach to the method of source-sink potentials for molecular conduction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pickup, Barry T., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Fowler, Patrick W., E-mail: B.T.Pickup@sheffield.ac.uk, E-mail: P.W.Fowler@sheffield.ac.uk; Borg, Martha

2015-11-21

We re-derive the tight-binding source-sink potential (SSP) equations for ballistic conduction through conjugated molecular structures in a form that avoids singularities. This enables derivation of new results for families of molecular devices in terms of eigenvectors and eigenvalues of the adjacency matrix of the molecular graph. In particular, we define the transmission of electrons through individual molecular orbitals (MO) and through MO shells. We make explicit the behaviour of the total current and individual MO and shell currents at molecular eigenvalues. A rich variety of behaviour is found. A SSP device has specific insulation or conduction at an eigenvalue ofmore » the molecular graph (a root of the characteristic polynomial) according to the multiplicities of that value in the spectra of four defined device polynomials. Conduction near eigenvalues is dominated by the transmission curves of nearby shells. A shell may be inert or active. An inert shell does not conduct at any energy, not even at its own eigenvalue. Conduction may occur at the eigenvalue of an inert shell, but is then carried entirely by other shells. If a shell is active, it carries all conduction at its own eigenvalue. For bipartite molecular graphs (alternant molecules), orbital conduction properties are governed by a pairing theorem. Inertness of shells for families such as chains and rings is predicted by selection rules based on node counting and degeneracy.« less

Wigner surmises and the two-dimensional homogeneous Poisson point process.

PubMed

Sakhr, Jamal; Nieminen, John M

2006-04-01

We derive a set of identities that relate the higher-order interpoint spacing statistics of the two-dimensional homogeneous Poisson point process to the Wigner surmises for the higher-order spacing distributions of eigenvalues from the three classical random matrix ensembles. We also report a remarkable identity that equates the second-nearest-neighbor spacing statistics of the points of the Poisson process and the nearest-neighbor spacing statistics of complex eigenvalues from Ginibre's ensemble of 2 x 2 complex non-Hermitian random matrices.

New matrix bounds and iterative algorithms for the discrete coupled algebraic Riccati equation

NASA Astrophysics Data System (ADS)

Liu, Jianzhou; Wang, Li; Zhang, Juan

2017-11-01

The discrete coupled algebraic Riccati equation (DCARE) has wide applications in control theory and linear system. In general, for the DCARE, one discusses every term of the coupled term, respectively. In this paper, we consider the coupled term as a whole, which is different from the recent results. When applying eigenvalue inequalities to discuss the coupled term, our method has less error. In terms of the properties of special matrices and eigenvalue inequalities, we propose several upper and lower matrix bounds for the solution of DCARE. Further, we discuss the iterative algorithms for the solution of the DCARE. In the fixed point iterative algorithms, the scope of Lipschitz factor is wider than the recent results. Finally, we offer corresponding numerical examples to illustrate the effectiveness of the derived results.

Inflationary dynamics for matrix eigenvalue problems

PubMed Central

Heller, Eric J.; Kaplan, Lev; Pollmann, Frank

2008-01-01

Many fields of science and engineering require finding eigenvalues and eigenvectors of large matrices. The solutions can represent oscillatory modes of a bridge, a violin, the disposition of electrons around an atom or molecule, the acoustic modes of a concert hall, or hundreds of other physical quantities. Often only the few eigenpairs with the lowest or highest frequency (extremal solutions) are needed. Methods that have been developed over the past 60 years to solve such problems include the Lanczos algorithm, Jacobi–Davidson techniques, and the conjugate gradient method. Here, we present a way to solve the extremal eigenvalue/eigenvector problem, turning it into a nonlinear classical mechanical system with a modified Lagrangian constraint. The constraint induces exponential inflationary growth of the desired extremal solutions. PMID:18511564

The Performance Analysis Based on SAR Sample Covariance Matrix

PubMed Central

Erten, Esra

2012-01-01

Multi-channel systems appear in several fields of application in science. In the Synthetic Aperture Radar (SAR) context, multi-channel systems may refer to different domains, as multi-polarization, multi-interferometric or multi-temporal data, or even a combination of them. Due to the inherent speckle phenomenon present in SAR images, the statistical description of the data is almost mandatory for its utilization. The complex images acquired over natural media present in general zero-mean circular Gaussian characteristics. In this case, second order statistics as the multi-channel covariance matrix fully describe the data. For practical situations however, the covariance matrix has to be estimated using a limited number of samples, and this sample covariance matrix follow the complex Wishart distribution. In this context, the eigendecomposition of the multi-channel covariance matrix has been shown in different areas of high relevance regarding the physical properties of the imaged scene. Specifically, the maximum eigenvalue of the covariance matrix has been frequently used in different applications as target or change detection, estimation of the dominant scattering mechanism in polarimetric data, moving target indication, etc. In this paper, the statistical behavior of the maximum eigenvalue derived from the eigendecomposition of the sample multi-channel covariance matrix in terms of multi-channel SAR images is simplified for SAR community. Validation is performed against simulated data and examples of estimation and detection problems using the analytical expressions are as well given. PMID:22736976

Eigenvectors determination of the ribosome dynamics model during mRNA translation using the Kleene Star algorithm

NASA Astrophysics Data System (ADS)

Ernawati; Carnia, E.; Supriatna, A. K.

2018-03-01

Eigenvalues and eigenvectors in max-plus algebra have the same important role as eigenvalues and eigenvectors in conventional algebra. In max-plus algebra, eigenvalues and eigenvectors are useful for knowing dynamics of the system such as in train system scheduling, scheduling production systems and scheduling learning activities in moving classes. In the translation of proteins in which the ribosome move uni-directionally along the mRNA strand to recruit the amino acids that make up the protein, eigenvalues and eigenvectors are used to calculate protein production rates and density of ribosomes on the mRNA. Based on this, it is important to examine the eigenvalues and eigenvectors in the process of protein translation. In this paper an eigenvector formula is given for a ribosome dynamics during mRNA translation by using the Kleene star algorithm in which the resulting eigenvector formula is simpler and easier to apply to the system than that introduced elsewhere. This paper also discusses the properties of the matrix {B}λ \\otimes n of model. Among the important properties, it always has the same elements in the first column for n = 1, 2,… if the eigenvalue is the time of initiation, λ = τin , and the column is the eigenvector of the model corresponding to λ.

Staggered chiral random matrix theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Osborn, James C.

2011-02-01

We present a random matrix theory for the staggered lattice QCD Dirac operator. The staggered random matrix theory is equivalent to the zero-momentum limit of the staggered chiral Lagrangian and includes all taste breaking terms at their leading order. This is an extension of previous work which only included some of the taste breaking terms. We will also present some results for the taste breaking contributions to the partition function and the Dirac eigenvalues.

RANDOM MATRIX DIAGONALIZATION--A COMPUTER PROGRAM

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fuchel, K.; Greibach, R.J.; Porter, C.E.

A computer prograra is described which generates random matrices, diagonalizes them and sorts appropriately the resulting eigenvalues and eigenvector components. FAP and FORTRAN listings for the IBM 7090 computer are included. (auth)

Financial time series: A physics perspective

NASA Astrophysics Data System (ADS)

Gopikrishnan, Parameswaran; Plerou, Vasiliki; Amaral, Luis A. N.; Rosenow, Bernd; Stanley, H. Eugene

2000-06-01

Physicists in the last few years have started applying concepts and methods of statistical physics to understand economic phenomena. The word ``econophysics'' is sometimes used to refer to this work. One reason for this interest is the fact that Economic systems such as financial markets are examples of complex interacting systems for which a huge amount of data exist and it is possible that economic problems viewed from a different perspective might yield new results. This article reviews the results of a few recent phenomenological studies focused on understanding the distinctive statistical properties of financial time series. We discuss three recent results-(i) The probability distribution of stock price fluctuations: Stock price fluctuations occur in all magnitudes, in analogy to earthquakes-from tiny fluctuations to very drastic events, such as market crashes, eg., the crash of October 19th 1987, sometimes referred to as ``Black Monday''. The distribution of price fluctuations decays with a power-law tail well outside the Lévy stable regime and describes fluctuations that differ by as much as 8 orders of magnitude. In addition, this distribution preserves its functional form for fluctuations on time scales that differ by 3 orders of magnitude, from 1 min up to approximately 10 days. (ii) Correlations in financial time series: While price fluctuations themselves have rapidly decaying correlations, the magnitude of fluctuations measured by either the absolute value or the square of the price fluctuations has correlations that decay as a power-law and persist for several months. (iii) Correlations among different companies: The third result bears on the application of random matrix theory to understand the correlations among price fluctuations of any two different stocks. From a study of the eigenvalue statistics of the cross-correlation matrix constructed from price fluctuations of the leading 1000 stocks, we find that the largest 5-10% of the eigenvalues and the corresponding eigenvectors show systematic deviations from the predictions for a random matrix, whereas the rest of the eigenvalues conform to random matrix behavior-suggesting that these 5-10% of the eigenvalues contain system-specific information about correlated behavior. .

Properties of networks with partially structured and partially random connectivity

NASA Astrophysics Data System (ADS)

Ahmadian, Yashar; Fumarola, Francesco; Miller, Kenneth D.

2015-01-01

Networks studied in many disciplines, including neuroscience and mathematical biology, have connectivity that may be stochastic about some underlying mean connectivity represented by a non-normal matrix. Furthermore, the stochasticity may not be independent and identically distributed (iid) across elements of the connectivity matrix. More generally, the problem of understanding the behavior of stochastic matrices with nontrivial mean structure and correlations arises in many settings. We address this by characterizing large random N ×N matrices of the form A =M +L J R , where M ,L , and R are arbitrary deterministic matrices and J is a random matrix of zero-mean iid elements. M can be non-normal, and L and R allow correlations that have separable dependence on row and column indices. We first provide a general formula for the eigenvalue density of A . For A non-normal, the eigenvalues do not suffice to specify the dynamics induced by A , so we also provide general formulas for the transient evolution of the magnitude of activity and frequency power spectrum in an N -dimensional linear dynamical system with a coupling matrix given by A . These quantities can also be thought of as characterizing the stability and the magnitude of the linear response of a nonlinear network to small perturbations about a fixed point. We derive these formulas and work them out analytically for some examples of M ,L , and R motivated by neurobiological models. We also argue that the persistence as N →∞ of a finite number of randomly distributed outlying eigenvalues outside the support of the eigenvalue density of A , as previously observed, arises in regions of the complex plane Ω where there are nonzero singular values of L-1(z 1 -M ) R-1 (for z ∈Ω ) that vanish as N →∞ . When such singular values do not exist and L and R are equal to the identity, there is a correspondence in the normalized Frobenius norm (but not in the operator norm) between the support of the spectrum of A for J of norm σ and the σ pseudospectrum of M .

A new method for multi-bit and qudit transfer based on commensurate waveguide arrays

NASA Astrophysics Data System (ADS)

Petrovic, J.; Veerman, J. J. P.

2018-05-01

The faithful state transfer is an important requirement in the construction of classical and quantum computers. While the high-speed transfer is realized by optical-fibre interconnects, its implementation in integrated optical circuits is affected by cross-talk. The cross-talk between densely packed optical waveguides limits the transfer fidelity and distorts the signal in each channel, thus severely impeding the parallel transfer of states such as classical registers, multiple qubits and qudits. Here, we leverage on the suitably engineered cross-talk between waveguides to achieve the parallel transfer on optical chip. Waveguide coupling coefficients are designed to yield commensurate eigenvalues of the array and hence, periodic revivals of the input state. While, in general, polynomially complex, the inverse eigenvalue problem permits analytic solutions for small number of waveguides. We present exact solutions for arrays of up to nine waveguides and use them to design realistic buses for multi-(qu)bit and qudit transfer. Advantages and limitations of the proposed solution are discussed in the context of available fabrication techniques.

Capabilities of Fully Parallelized MHD Stability Code MARS

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2016-10-01

Results of full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. Parallel version of MARS, named PMARS, has been recently developed at FAR-TECH. Parallelized MARS is an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, implemented in MARS. Parallelization of the code included parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse vector iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the MARS algorithm using parallel libraries and procedures. Parallelized MARS is capable of calculating eigenmodes with significantly increased spatial resolution: up to 5,000 adapted radial grid points with up to 500 poloidal harmonics. Such resolution is sufficient for simulation of kink, tearing and peeling-ballooning instabilities with physically relevant parameters. Work is supported by the U.S. DOE SBIR program.

Fully Parallel MHD Stability Analysis Tool

NASA Astrophysics Data System (ADS)

Svidzinski, Vladimir; Galkin, Sergei; Kim, Jin-Soo; Liu, Yueqiang

2015-11-01

Progress on full parallelization of the plasma stability code MARS will be reported. MARS calculates eigenmodes in 2D axisymmetric toroidal equilibria in MHD-kinetic plasma models. It is a powerful tool for studying MHD and MHD-kinetic instabilities and it is widely used by fusion community. Parallel version of MARS is intended for simulations on local parallel clusters. It will be an efficient tool for simulation of MHD instabilities with low, intermediate and high toroidal mode numbers within both fluid and kinetic plasma models, already implemented in MARS. Parallelization of the code includes parallelization of the construction of the matrix for the eigenvalue problem and parallelization of the inverse iterations algorithm, implemented in MARS for the solution of the formulated eigenvalue problem. Construction of the matrix is parallelized by distributing the load among processors assigned to different magnetic surfaces. Parallelization of the solution of the eigenvalue problem is made by repeating steps of the present MARS algorithm using parallel libraries and procedures. Results of MARS parallelization and of the development of a new fix boundary equilibrium code adapted for MARS input will be reported. Work is supported by the U.S. DOE SBIR program.

ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL

PubMed Central

Huang, Jian; Sun, Tingni; Ying, Zhiliang; Yu, Yi; Zhang, Cun-Hui

2013-01-01

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities. PMID:24086091

ORACLE INEQUALITIES FOR THE LASSO IN THE COX MODEL.

PubMed

Huang, Jian; Sun, Tingni; Ying, Zhiliang; Yu, Yi; Zhang, Cun-Hui

2013-06-01

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities.

The behaviour of resonances in Hecke triangular billiards under deformation

NASA Astrophysics Data System (ADS)

Howard, P. J.; O'Mahony, P. F.

2007-08-01

The right-hand boundary of Artin's billiard on the Poincaré half-plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the Hecke groups Γ(2, n) at certain values of the deformation parameter. The quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated. The transitions to arithmetic chaos in the cases n ∈ {4, 6} are closely examined and the explicit analytic form for the scattering matrix is given together with the Fourier coefficients for the scattered wavefunction. The n = 4 and 6 cases have an additional set of regular equally spaced resonances compared to Artin's billiard (n = 3). For a general deformation, a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution. For deformations away from the non-generic arithmetic cases, including that of the tiling Hecke triangular billiard n = 5, the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory.

Solution of an eigenvalue problem for the Laplace operator on a spherical surface. M.S. Thesis - Maryland Univ.

NASA Technical Reports Server (NTRS)

Walden, H.

1974-01-01

Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (also referred to as the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two specific types of spherical surface domains are considered: (1) the interior of a spherical triangle, i.e., the region bounded by arcs of three great circles, and (2) the exterior of a great circle arc extending for less than pi radians on the sphere (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is derived. The symmetric (generally five-point) finite difference equations that develop are written in matrix form and then solved by the iterative method of point successive overrelaxation. Upon convergence of this iterative method, the fundamental eigenvalue is approximated by iteration utilizing the power method as applied to the finite Rayleigh quotient.

Spectral analysis of Chinese language: Co-occurrence networks from four literary genres

NASA Astrophysics Data System (ADS)

Liang, Wei; Chen, Guanrong

2016-05-01

The eigenvalues and eigenvectors of the adjacency matrix of a network contain essential information about its topology. For each of the Chinese language co-occurrence networks constructed from four literary genres, i.e., essay, popular science article, news report, and novel, it is found that the largest eigenvalue depends on the network size N, the number of edges, the average shortest path length, and the clustering coefficient. Moreover, it is found that their node-degree distributions all follow a power-law. The number of different eigenvalues, Nλ, is found numerically to increase in the manner of Nλ ∝ log N for novel and Nλ ∝ N for the other three literary genres. An ;M; shape or a triangle-like distribution appears in their spectral densities. The eigenvector corresponding to the largest eigenvalue is mostly localized to a node with the largest degree. For the above observed phenomena, mathematical analysis is provided with interpretation from a linguistic perspective.

On Fluctuations of Eigenvalues of Random Band Matrices

NASA Astrophysics Data System (ADS)

Shcherbina, M.

2015-10-01

We consider the fluctuations of linear eigenvalue statistics of random band matrices whose entries have the form with i.i.d. possessing the th moment, where the function u has a finite support , so that M has only nonzero diagonals. The parameter b (called the bandwidth) is assumed to grow with n in a way such that . Without any additional assumptions on the growth of b we prove CLT for linear eigenvalue statistics for a rather wide class of test functions. Thus we improve and generalize the results of the previous papers (Jana et al., arXiv:1412.2445; Li et al. Random Matrices 2:04, 2013), where CLT was proven under the assumption . Moreover, we develop a method which allows to prove automatically the CLT for linear eigenvalue statistics of the smooth test functions for almost all classical models of random matrix theory: deformed Wigner and sample covariance matrices, sparse matrices, diluted random matrices, matrices with heavy tales etc.

Spectral properties of the temporal evolution of brain network structure.

PubMed

Wang, Rong; Zhang, Zhen-Zhen; Ma, Jun; Yang, Yong; Lin, Pan; Wu, Ying

2015-12-01

The temporal evolution properties of the brain network are crucial for complex brain processes. In this paper, we investigate the differences in the dynamic brain network during resting and visual stimulation states in a task-positive subnetwork, task-negative subnetwork, and whole-brain network. The dynamic brain network is first constructed from human functional magnetic resonance imaging data based on the sliding window method, and then the eigenvalues corresponding to the network are calculated. We use eigenvalue analysis to analyze the global properties of eigenvalues and the random matrix theory (RMT) method to measure the local properties. For global properties, the shifting of the eigenvalue distribution and the decrease in the largest eigenvalue are linked to visual stimulation in all networks. For local properties, the short-range correlation in eigenvalues as measured by the nearest neighbor spacing distribution is not always sensitive to visual stimulation. However, the long-range correlation in eigenvalues as evaluated by spectral rigidity and number variance not only predicts the universal behavior of the dynamic brain network but also suggests non-consistent changes in different networks. These results demonstrate that the dynamic brain network is more random for the task-positive subnetwork and whole-brain network under visual stimulation but is more regular for the task-negative subnetwork. Our findings provide deeper insight into the importance of spectral properties in the functional brain network, especially the incomparable role of RMT in revealing the intrinsic properties of complex systems.

Spectral properties of the temporal evolution of brain network structure

NASA Astrophysics Data System (ADS)

Wang, Rong; Zhang, Zhen-Zhen; Ma, Jun; Yang, Yong; Lin, Pan; Wu, Ying

2015-12-01

The temporal evolution properties of the brain network are crucial for complex brain processes. In this paper, we investigate the differences in the dynamic brain network during resting and visual stimulation states in a task-positive subnetwork, task-negative subnetwork, and whole-brain network. The dynamic brain network is first constructed from human functional magnetic resonance imaging data based on the sliding window method, and then the eigenvalues corresponding to the network are calculated. We use eigenvalue analysis to analyze the global properties of eigenvalues and the random matrix theory (RMT) method to measure the local properties. For global properties, the shifting of the eigenvalue distribution and the decrease in the largest eigenvalue are linked to visual stimulation in all networks. For local properties, the short-range correlation in eigenvalues as measured by the nearest neighbor spacing distribution is not always sensitive to visual stimulation. However, the long-range correlation in eigenvalues as evaluated by spectral rigidity and number variance not only predicts the universal behavior of the dynamic brain network but also suggests non-consistent changes in different networks. These results demonstrate that the dynamic brain network is more random for the task-positive subnetwork and whole-brain network under visual stimulation but is more regular for the task-negative subnetwork. Our findings provide deeper insight into the importance of spectral properties in the functional brain network, especially the incomparable role of RMT in revealing the intrinsic properties of complex systems.

Products of random matrices from fixed trace and induced Ginibre ensembles

NASA Astrophysics Data System (ADS)

Akemann, Gernot; Cikovic, Milan

2018-05-01

We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main tool of investigation is the Laplace transform, that maps back the fixed trace to the induced Ginibre ensemble. Products of random matrices have been used to study the Lyapunov and stability exponents for chaotic dynamical systems, where the latter are based on the complex eigenvalues of the product matrix. Because little is known about the universality of the eigenvalue distribution of such product matrices, we then study the product of m induced Ginibre matrices with a fixed trace constraint—which are clearly non-Gaussian—and M  ‑  m such Ginibre matrices without constraint. Using an m-fold inverse Laplace transform, we obtain a concise result for the spectral density of such a mixed product matrix at finite matrix size, for arbitrary fixed m and M. Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble in the bulk of the spectrum. Here, we argue that the spectral density of mixed products is in the same universality class as the product of M independent induced Ginibre ensembles.

Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

NASA Astrophysics Data System (ADS)

Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.

2017-04-01

In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O (Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O (Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No

Targeting functional motifs of a protein family

NASA Astrophysics Data System (ADS)

Bhadola, Pradeep; Deo, Nivedita

2016-10-01

The structural organization of a protein family is investigated by devising a method based on the random matrix theory (RMT), which uses the physiochemical properties of the amino acid with multiple sequence alignment. A graphical method to represent protein sequences using physiochemical properties is devised that gives a fast, easy, and informative way of comparing the evolutionary distances between protein sequences. A correlation matrix associated with each property is calculated, where the noise reduction and information filtering is done using RMT involving an ensemble of Wishart matrices. The analysis of the eigenvalue statistics of the correlation matrix for the β -lactamase family shows the universal features as observed in the Gaussian orthogonal ensemble (GOE). The property-based approach captures the short- as well as the long-range correlation (approximately following GOE) between the eigenvalues, whereas the previous approach (treating amino acids as characters) gives the usual short-range correlations, while the long-range correlations are the same as that of an uncorrelated series. The distribution of the eigenvector components for the eigenvalues outside the bulk (RMT bound) deviates significantly from RMT observations and contains important information about the system. The information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate, which shows that for the β -lactamase family the smallest eigenvectors (low eigenmodes) are highly localized as well as informative. These small eigenvectors when processed gives clusters involving positions that have well-defined biological and structural importance matching with experiments. The approach is crucial for the recognition of structural motifs as shown in β -lactamase (and other families) and selectively identifies the important positions for targets to deactivate (activate) the enzymatic actions.

Subspace Iteration Method for Complex Eigenvalue Problems with Nonsymmetric Matrices in Aeroelastic System

NASA Technical Reports Server (NTRS)

Pak, Chan-gi; Lung, Shun-fat

2009-01-01

Modern airplane design is a multidisciplinary task which combines several disciplines such as structures, aerodynamics, flight controls, and sometimes heat transfer. Historically, analytical and experimental investigations concerning the interaction of the elastic airframe with aerodynamic and in retia loads have been conducted during the design phase to determine the existence of aeroelastic instabilities, so called flutter .With the advent and increased usage of flight control systems, there is also a likelihood of instabilities caused by the interaction of the flight control system and the aeroelastic response of the airplane, known as aeroservoelastic instabilities. An in -house code MPASES (Ref. 1), modified from PASES (Ref. 2), is a general purpose digital computer program for the analysis of the closed-loop stability problem. This program used subroutines given in the International Mathematical and Statistical Library (IMSL) (Ref. 3) to compute all of the real and/or complex conjugate pairs of eigenvalues of the Hessenberg matrix. For high fidelity configuration, these aeroelastic system matrices are large and compute all eigenvalues will be time consuming. A subspace iteration method (Ref. 4) for complex eigenvalues problems with nonsymmetric matrices has been formulated and incorporated into the modified program for aeroservoelastic stability (MPASES code). Subspace iteration method only solve for the lowest p eigenvalues and corresponding eigenvectors for aeroelastic and aeroservoelastic analysis. In general, the selection of p is ranging from 10 for wing flutter analysis to 50 for an entire aircraft flutter analysis. The application of this newly incorporated code is an experiment known as the Aerostructures Test Wing (ATW) which was designed by the National Aeronautic and Space Administration (NASA) Dryden Flight Research Center, Edwards, California to research aeroelastic instabilities. Specifically, this experiment was used to study an instability known as flutter. ATW was a small-scale airplane wing comprised of an airfoil and wing tip boom. This wing was formulated based on a NACA-65A004 airfoil shape with a 3.28 aspect ratio. The wing had a span of 18 inch with root chord length of 13.2 inch and tip chord length of 8.7 inch. The total area of this wing was 197 square inch. The wing tip boom was a 1 inch diameter hollow tube of length 21.5 inch. The total weight of the wing was 2.66 lbs.

Three-dimensional analytical solution for the instability of a parallel array of mutually attracting identical simply supported piezoelectric microplates

NASA Astrophysics Data System (ADS)

Liu, Lei; Wang, Xu

2017-12-01

Three-dimensional analytical solutions are derived for the structural instability of a parallel array of mutually attracting identical simply supported orthotropic piezoelectric rectangular microplates by means of a linear perturbation analysis. The two surfaces of each plate can be either insulating or conducting. By considering the fact that the shear stresses and the normal electric displacement (or electric potential) are zero on the two surfaces of each plate, a 2 × 2 transfer matrix for a plate can be obtained directly from the 8 × 8 fundamental piezoelectricity matrix without resolving the original Stroh eigenrelation. The critical interaction coefficient can be determined by solving the resulting generalized eigenvalue problem for the piezoelectric plate array. Also considered in our analysis is the in-plane uniform edge compression acting on the four sides of each piezoelectric plate. Our results indicate that the stabilizing influence of the piezoelectric effect on the structural instability is unignorable; the edge compression always plays a destabilizing role in the structural instability of the plate array with interactions.

Numerical method of applying shadow theory to all regions of multilayered dielectric gratings in conical mounting.

PubMed

Wakabayashi, Hideaki; Asai, Masamitsu; Matsumoto, Keiji; Yamakita, Jiro

2016-11-01

Nakayama's shadow theory first discussed the diffraction by a perfectly conducting grating in a planar mounting. In the theory, a new formulation by use of a scattering factor was proposed. This paper focuses on the middle regions of a multilayered dielectric grating placed in conical mounting. Applying the shadow theory to the matrix eigenvalues method, we compose new transformation and improved propagation matrices of the shadow theory for conical mounting. Using these matrices and scattering factors, being the basic quantity of diffraction amplitudes, we formulate a new description of three-dimensional scattering fields which is available even for cases where the eigenvalues are degenerate in any region. Some numerical examples are given for cases where the eigenvalues are degenerate in the middle regions.

Edge connectivity and the spectral gap of combinatorial and quantum graphs

NASA Astrophysics Data System (ADS)

Berkolaiko, Gregory; Kennedy, James B.; Kurasov, Pavel; Mugnolo, Delio

2017-09-01

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds corresponds to a well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers. Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are ‘asymptotically correct’, i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

The method of trend analysis of parameters time series of gas-turbine engine state

NASA Astrophysics Data System (ADS)

Hvozdeva, I.; Myrhorod, V.; Derenh, Y.

2017-10-01

This research substantiates an approach to interval estimation of time series trend component. The well-known methods of spectral and trend analysis are used for multidimensional data arrays. The interval estimation of trend component is proposed for the time series whose autocorrelation matrix possesses a prevailing eigenvalue. The properties of time series autocorrelation matrix are identified.

Direct structural parameter identification by modal test results

NASA Technical Reports Server (NTRS)

Chen, J.-C.; Kuo, C.-P.; Garba, J. A.

1983-01-01

A direct identification procedure is proposed to obtain the mass and stiffness matrices based on the test measured eigenvalues and eigenvectors. The method is based on the theory of matrix perturbation in which the correct mass and stiffness matrices are expanded in terms of analytical values plus a modification matrix. The simplicity of the procedure enables real time operation during the structural testing.

Correlation and volatility in an Indian stock market: A random matrix approach

NASA Astrophysics Data System (ADS)

Kulkarni, Varsha; Deo, Nivedita

2007-11-01

We examine the volatility of an Indian stock market in terms of correlation of stocks and quantify the volatility using the random matrix approach. First we discuss trends observed in the pattern of stock prices in the Bombay Stock Exchange for the three-year period 2000 2002. Random matrix analysis is then applied to study the relationship between the coupling of stocks and volatility. The study uses daily returns of 70 stocks for successive time windows of length 85 days for the year 2001. We compare the properties of matrix C of correlations between price fluctuations in time regimes characterized by different volatilities. Our analyses reveal that (i) the largest (deviating) eigenvalue of C correlates highly with the volatility of the index, (ii) there is a shift in the distribution of the components of the eigenvector corresponding to the largest eigenvalue across regimes of different volatilities, (iii) the inverse participation ratio for this eigenvector anti-correlates significantly with the market fluctuations and finally, (iv) this eigenvector of C can be used to set up a Correlation Index, CI whose temporal evolution is significantly correlated with the volatility of the overall market index.

Considering Horn's Parallel Analysis from a Random Matrix Theory Point of View.

PubMed

Saccenti, Edoardo; Timmerman, Marieke E

2017-03-01

Horn's parallel analysis is a widely used method for assessing the number of principal components and common factors. We discuss the theoretical foundations of parallel analysis for principal components based on a covariance matrix by making use of arguments from random matrix theory. In particular, we show that (i) for the first component, parallel analysis is an inferential method equivalent to the Tracy-Widom test, (ii) its use to test high-order eigenvalues is equivalent to the use of the joint distribution of the eigenvalues, and thus should be discouraged, and (iii) a formal test for higher-order components can be obtained based on a Tracy-Widom approximation. We illustrate the performance of the two testing procedures using simulated data generated under both a principal component model and a common factors model. For the principal component model, the Tracy-Widom test performs consistently in all conditions, while parallel analysis shows unpredictable behavior for higher-order components. For the common factor model, including major and minor factors, both procedures are heuristic approaches, with variable performance. We conclude that the Tracy-Widom procedure is preferred over parallel analysis for statistically testing the number of principal components based on a covariance matrix.

Research on the application of a decoupling algorithm for structure analysis

NASA Technical Reports Server (NTRS)

Denman, E. D.

1980-01-01

The mathematical theory for decoupling mth-order matrix differential equations is presented. It is shown that the decoupling precedure can be developed from the algebraic theory of matrix polynomials. The role of eigenprojectors and latent projectors in the decoupling process is discussed and the mathematical relationships between eigenvalues, eigenvectors, latent roots, and latent vectors are developed. It is shown that the eigenvectors of the companion form of a matrix contains the latent vectors as a subset. The spectral decomposition of a matrix and the application to differential equations is given.

nu-TRLan User Guide Version 1.0: A High-Performance Software Package for Large-Scale Harmitian Eigenvalue Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yamazaki, Ichitaro; Wu, Kesheng; Simon, Horst

2008-10-27

The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs,more » users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 [Installation], page 2, we provide an installation guide of the nu-TRLan software package. In Chapter 3 [Example], page 3, we present a simple nu-TRLan example program. In Chapter 4 [trl-info structure], page 7, and Chapter 5 [trlan subroutine], page 14, we describe the solver parameters and interfaces in detail. In Chapter 6 [Solver parameters], page 21, we discuss the selection of the user-specified parameters. In Chapter 7 [Contact information], page 22, we give the acknowledgements and contact information of the authors. In Chapter 8 [References], page 23, we list reference to related works.« less

Spectral Analysis for Weighted Iterated Triangulations of Graphs

NASA Astrophysics Data System (ADS)

Chen, Yufei; Dai, Meifeng; Wang, Xiaoqian; Sun, Yu; Su, Weiyi

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

Random Matrix Theory and Econophysics

NASA Astrophysics Data System (ADS)

Rosenow, Bernd

2000-03-01

Random Matrix Theory (RMT) [1] is used in many branches of physics as a ``zero information hypothesis''. It describes generic behavior of different classes of systems, while deviations from its universal predictions allow to identify system specific properties. We use methods of RMT to analyze the cross-correlation matrix C of stock price changes [2] of the largest 1000 US companies. In addition to its scientific interest, the study of correlations between the returns of different stocks is also of practical relevance in quantifying the risk of a given stock portfolio. We find [3,4] that the statistics of most of the eigenvalues of the spectrum of C agree with the predictions of RMT, while there are deviations for some of the largest eigenvalues. We interpret these deviations as a system specific property, e.g. containing genuine information about correlations in the stock market. We demonstrate that C shares universal properties with the Gaussian orthogonal ensemble of random matrices. Furthermore, we analyze the eigenvectors of C through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum - a situation reminiscent of localization theory results. This work was done in collaboration with V. Plerou, P. Gopikrishnan, T. Guhr, L.A.N. Amaral, and H.E Stanley and is related to recent work of Laloux et al.. 1. T. Guhr, A. Müller Groeling, and H.A. Weidenmüller, ``Random Matrix Theories in Quantum Physics: Common Concepts'', Phys. Rep. 299, 190 (1998). 2. See, e.g. R.N. Mantegna and H.E. Stanley, Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, England, 1999). 3. V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, and H.E. Stanley, ``Universal and Nonuniversal Properties of Cross Correlations in Financial Time Series'', Phys. Rev. Lett. 83, 1471 (1999). 4. V. Plerou, P. Gopikrishnan, T. Guhr, B. Rosenow, L.A.N. Amaral, and H.E. Stanley, ``Random Matrix Theory Analysis of Diffusion in Stock Price Dynamics, preprint

Expendable launch vehicle studies

NASA Technical Reports Server (NTRS)

Bainum, Peter M.; Reiss, Robert

1995-01-01

Analytical support studies of expendable launch vehicles concentrate on the stability of the dynamics during launch especially during or near the region of maximum dynamic pressure. The in-plane dynamic equations of a generic launch vehicle with multiple flexible bending and fuel sloshing modes are developed and linearized. The information from LeRC about the grids, masses, and modes is incorporated into the model. The eigenvalues of the plant are analyzed for several modeling factors: utilizing diagonal mass matrix, uniform beam assumption, inclusion of aerodynamics, and the interaction between the aerodynamics and the flexible bending motion. Preliminary PID, LQR, and LQG control designs with sensor and actuator dynamics for this system and simulations are also conducted. The initial analysis for comparison of PD (proportional-derivative) and full state feedback LQR Linear quadratic regulator) shows that the split weighted LQR controller has better performance than that of the PD. In order to meet both the performance and robustness requirements, the H(sub infinity) robust controller for the expendable launch vehicle is developed. The simulation indicates that both the performance and robustness of the H(sub infinity) controller are better than that for the PID and LQG controllers. The modelling and analysis support studies team has continued development of methodology, using eigensensitivity analysis, to solve three classes of discrete eigenvalue equations. In the first class, the matrix elements are non-linear functions of the eigenvector. All non-linear periodic motion can be cast in this form. Here the eigenvector is comprised of the coefficients of complete basis functions spanning the response space and the eigenvalue is the frequency. The second class of eigenvalue problems studied is the quadratic eigenvalue problem. Solutions for linear viscously damped structures or viscoelastic structures can be reduced to this form. Particular attention is paid to Maxwell and Kelvin models. The third class of problems consists of linear eigenvalue problems in which the elements of the mass and stiffness matrices are stochastic. dynamic structural response for which the parameters are given by probabilistic distribution functions, rather than deterministic values, can be cast in this form. Solutions for several problems in each class will be presented.

Modelling of Rigid-Body and Elastic Aircraft Dynamics for Flight Control Development.

DTIC Science & Technology

1986-06-01

AMAT MATSAV AUGMENT MI NV BMAT MMULT EVAL RLPLOT FASTCHG STABDER The subroutines are fairly well commented so that a person familiar with the theory...performed as in a typical flutter solution. C C Subroutine BMAT computes the B matrix from the forcing function C matrix Q. B is a function of dynamic...and BMAT multiplies matrices. C This is used to form the A and B matrices. C C Subroutine EVAL computes the eigenvalues of the A matrix C The

Computing the full spectrum of large sparse palindromic quadratic eigenvalue problems arising from surface Green's function calculations

NASA Astrophysics Data System (ADS)

Huang, Tsung-Ming; Lin, Wen-Wei; Tian, Heng; Chen, Guan-Hua

2018-03-01

Full spectrum of a large sparse ⊤-palindromic quadratic eigenvalue problem (⊤-PQEP) is considered arguably for the first time in this article. Such a problem is posed by calculation of surface Green's functions (SGFs) of mesoscopic transistors with a tremendous non-periodic cross-section. For this problem, general purpose eigensolvers are not efficient, nor is advisable to resort to the decimation method etc. to obtain the Wiener-Hopf factorization. After reviewing some rigorous understanding of SGF calculation from the perspective of ⊤-PQEP and nonlinear matrix equation, we present our new approach to this problem. In a nutshell, the unit disk where the spectrum of interest lies is broken down adaptively into pieces small enough that they each can be locally tackled by the generalized ⊤-skew-Hamiltonian implicitly restarted shift-and-invert Arnoldi (G⊤SHIRA) algorithm with suitable shifts and other parameters, and the eigenvalues missed by this divide-and-conquer strategy can be recovered thanks to the accurate estimation provided by our newly developed scheme. Notably the novel non-equivalence deflation is proposed to avoid as much as possible duplication of nearby known eigenvalues when a new shift of G⊤SHIRA is determined. We demonstrate our new approach by calculating the SGF of a realistic nanowire whose unit cell is described by a matrix of size 4000 × 4000 at the density functional tight binding level, corresponding to a 8 × 8nm2 cross-section. We believe that quantum transport simulation of realistic nano-devices in the mesoscopic regime will greatly benefit from this work.

Divergence and Necessary Conditions for Extremums

NASA Technical Reports Server (NTRS)

Quirein, J. A.

1973-01-01

The problem is considered of finding a dimension reducing transformation matrix B that maximizes the divergence in the reduced dimension for multi-class cases. A comparitively simple expression for the gradient of the average divergence with respect to B is developed. The developed expression for the gradient contains no eigenvectors or eigenvalues; also, all matrix inversions necessary to evaluate the gradient are available from computing the average divergence.

Material identification based on electrostatic sensing technology

NASA Astrophysics Data System (ADS)

Liu, Kai; Chen, Xi; Li, Jingnan

2018-04-01

When the robot travels on the surface of different media, the uncertainty of the medium will seriously affect the autonomous action of the robot. In this paper, the distribution characteristics of multiple electrostatic charges on the surface of materials are detected, so as to improve the accuracy of the existing electrostatic signal material identification methods, which is of great significance to help the robot optimize the control algorithm. In this paper, based on the electrostatic signal material identification method proposed by predecessors, the multi-channel detection circuit is used to obtain the electrostatic charge distribution at different positions of the material surface, the weights are introduced into the eigenvalue matrix, and the weight distribution is optimized by the evolutionary algorithm, which makes the eigenvalue matrix more accurately reflect the surface charge distribution characteristics of the material. The matrix is used as the input of the k-Nearest Neighbor (kNN)classification algorithm to classify the dielectric materials. The experimental results show that the proposed method can significantly improve the recognition rate of the existing electrostatic signal material recognition methods.

The difference between two random mixed quantum states: exact and asymptotic spectral analysis

NASA Astrophysics Data System (ADS)

Mejía, José; Zapata, Camilo; Botero, Alonso

2017-01-01

We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson’s theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.

Eigenvectors of optimal color spectra.

PubMed

Flinkman, Mika; Laamanen, Hannu; Tuomela, Jukka; Vahimaa, Pasi; Hauta-Kasari, Markku

2013-09-01

Principal component analysis (PCA) and weighted PCA were applied to spectra of optimal colors belonging to the outer surface of the object-color solid or to so-called MacAdam limits. The correlation matrix formed from this data is a circulant matrix whose biggest eigenvalue is simple and the corresponding eigenvector is constant. All other eigenvalues are double, and the eigenvectors can be expressed with trigonometric functions. Found trigonometric functions can be used as a general basis to reconstruct all possible smooth reflectance spectra. When the spectral data are weighted with an appropriate weight function, the essential part of the color information is compressed to the first three components and the shapes of the first three eigenvectors correspond to one achromatic response function and to two chromatic response functions, the latter corresponding approximately to Munsell opponent-hue directions 9YR-9B and 2BG-2R.

Inflation with a graceful exit in a random landscape

NASA Astrophysics Data System (ADS)

Pedro, F. G.; Westphal, A.

2017-03-01

We develop a stochastic description of small-field inflationary histories with a graceful exit in a random potential whose Hessian is a Gaussian random matrix as a model of the unstructured part of the string landscape. The dynamical evolution in such a random potential from a small-field inflation region towards a viable late-time de Sitter (dS) minimum maps to the dynamics of Dyson Brownian motion describing the relaxation of non-equilibrium eigenvalue spectra in random matrix theory. We analytically compute the relaxation probability in a saddle point approximation of the partition function of the eigenvalue distribution of the Wigner ensemble describing the mass matrices of the critical points. When applied to small-field inflation in the landscape, this leads to an exponentially strong bias against small-field ranges and an upper bound N ≪ 10 on the number of light fields N participating during inflation from the non-observation of negative spatial curvature.

Coherence analysis of a class of weighted networks

NASA Astrophysics Data System (ADS)

Dai, Meifeng; He, Jiaojiao; Zong, Yue; Ju, Tingting; Sun, Yu; Su, Weiyi

2018-04-01

This paper investigates consensus dynamics in a dynamical system with additive stochastic disturbances that is characterized as network coherence by using the Laplacian spectrum. We introduce a class of weighted networks based on a complete graph and investigate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. First, the recursive relationship of its eigenvalues at two successive generations of Laplacian matrix is deduced. Then, we compute the sum and square sum of reciprocal of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first- and second-order coherence with network size obey four and five laws, respectively, along with the range of the weight factor. Finally, it indicates that the scalings of our studied networks are smaller than other studied networks when 1/√{d }

On adaptive weighted polynomial preconditioning for Hermitian positive definite matrices

NASA Technical Reports Server (NTRS)

Fischer, Bernd; Freund, Roland W.

1992-01-01

The conjugate gradient algorithm for solving Hermitian positive definite linear systems is usually combined with preconditioning in order to speed up convergence. In recent years, there has been a revival of polynomial preconditioning, motivated by the attractive features of the method on modern architectures. Standard techniques for choosing the preconditioning polynomial are based only on bounds for the extreme eigenvalues. Here a different approach is proposed, which aims at adapting the preconditioner to the eigenvalue distribution of the coefficient matrix. The technique is based on the observation that good estimates for the eigenvalue distribution can be derived after only a few steps of the Lanczos process. This information is then used to construct a weight function for a suitable Chebyshev approximation problem. The solution of this problem yields the polynomial preconditioner. In particular, we investigate the use of Bernstein-Szego weights.

Preliminary demonstration of a robust controller design method

NASA Technical Reports Server (NTRS)

Anderson, L. R.

1980-01-01

Alternative computational procedures for obtaining a feedback control law which yields a control signal based on measurable quantitites are evaluated. The three methods evaluated are: (1) the standard linear quadratic regulator design model; (2) minimization of the norm of the feedback matrix, k via nonlinear programming subject to the constraint that the closed loop eigenvalues be in a specified domain in the complex plane; and (3) maximize the angles between the closed loop eigenvectors in combination with minimizing the norm of K also via the constrained nonlinear programming. The third or robust design method was chosen to yield a closed loop system whose eigenvalues are insensitive to small changes in the A and B matrices. The relationship between orthogonality of closed loop eigenvectors and the sensitivity of closed loop eigenvalues is described. Computer programs are described.

Analysis of spontaneous oscillations for a three-state power-stroke model.

PubMed

Washio, Takumi; Hisada, Toshiaki; Shintani, Seine A; Higuchi, Hideo

2017-02-01

Our study considers the mechanism of the spontaneous oscillations of molecular motors that are driven by the power stroke principle by applying linear stability analysis around the stationary solution. By representing the coupling equation of microscopic molecular motor dynamics and mesoscopic sarcomeric dynamics by a rank-1 updated matrix system, we derived the analytical representations of the eigenmodes of the Jacobian matrix that cause the oscillation. Based on these analytical representations, we successfully derived the essential conditions for the oscillation in terms of the rate constants of the power stroke and the reversal stroke transitions of the molecular motor. Unlike the two-state model, in which the dependence of the detachment rates on the motor coordinates or the applied forces on the motors plays a key role for the oscillation, our three-state power stroke model demonstrates that the dependence of the rate constants of the power and reversal strokes on the strains in the elastic elements in the motor molecules plays a key role, where these rate constants are rationally determined from the free energy available for the power stroke, the stiffness of the elastic element in the molecular motor, and the working stroke size. By applying the experimentally confirmed values to the free energy, the stiffness, and the working stroke size, our numerical model reproduces well the experimentally observed oscillatory behavior. Furthermore, our analysis shows that two eigenmodes with real positive eigenvalues characterize the oscillatory behavior, where the eigenmode with the larger eigenvalue indicates the transient of the system of the quick sarcomeric lengthening induced by the collective reversal strokes, and the smaller eigenvalue correlates with the speed of sarcomeric shortening, which is much slower than lengthening. Applying the perturbation analyses with primal physical parameters, we find that these two real eigenvalues occur on two branches derived from a merge point of a pair of complex-conjugate eigenvalues generated by Hopf bifurcation.

Parallel solution of the symmetric tridiagonal eigenproblem. Research report

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1989-10-01

This thesis discusses methods for computing all eigenvalues and eigenvectors of a symmetric tridiagonal matrix on a distributed-memory Multiple Instruction, Multiple Data multiprocessor. Only those techniques having the potential for both high numerical accuracy and significant large-grained parallelism are investigated. These include the QL method or Cuppen's divide and conquer method based on rank-one updating to compute both eigenvalues and eigenvectors, bisection to determine eigenvalues and inverse iteration to compute eigenvectors. To begin, the methods are compared with respect to computation time, communication time, parallel speed up, and accuracy. Experiments on an IPSC hypercube multiprocessor reveal that Cuppen's method ismore » the most accurate approach, but bisection with inverse iteration is the fastest and most parallel. Because the accuracy of the latter combination is determined by the quality of the computed eigenvectors, the factors influencing the accuracy of inverse iteration are examined. This includes, in part, statistical analysis of the effect of a starting vector with random components. These results are used to develop an implementation of inverse iteration producing eigenvectors with lower residual error and better orthogonality than those generated by the EISPACK routine TINVIT. This thesis concludes with adaptions of methods for the symmetric tridiagonal eigenproblem to the related problem of computing the singular value decomposition (SVD) of a bidiagonal matrix.« less

Parallel solution of the symmetric tridiagonal eigenproblem

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jessup, E.R.

1989-01-01

This thesis discusses methods for computing all eigenvalues and eigenvectors of a symmetric tridiagonal matrix on a distributed memory MIMD multiprocessor. Only those techniques having the potential for both high numerical accuracy and significant large-grained parallelism are investigated. These include the QL method or Cuppen's divide and conquer method based on rank-one updating to compute both eigenvalues and eigenvectors, bisection to determine eigenvalues, and inverse iteration to compute eigenvectors. To begin, the methods are compared with respect to computation time, communication time, parallel speedup, and accuracy. Experiments on an iPSC hyper-cube multiprocessor reveal that Cuppen's method is the most accuratemore » approach, but bisection with inverse iteration is the fastest and most parallel. Because the accuracy of the latter combination is determined by the quality of the computed eigenvectors, the factors influencing the accuracy of inverse iteration are examined. This includes, in part, statistical analysis of the effects of a starting vector with random components. These results are used to develop an implementation of inverse iteration producing eigenvectors with lower residual error and better orthogonality than those generated by the EISPACK routine TINVIT. This thesis concludes with adaptations of methods for the symmetric tridiagonal eigenproblem to the related problem of computing the singular value decomposition (SVD) of a bidiagonal matrix.« less

The symmetries of the system matrix and propagator matrix for anisotropic media and of the system matrix forperiodically layered media

NASA Astrophysics Data System (ADS)

Xu, Guo-Ming; Ni, Si-Dao

1998-11-01

The `auxiliary' symmetry properties of the system matrix (symmetry with respect to the trailing diagonal) for a general anisotropic dissipative medium and the special form for a monoclinic medium are revealed by rearranging the motion-stress vector. The propagator matrix of a single-layer general anisotropic dissipative medium is also shown to have auxiliary symmetry. For the multilayered case, a relatively simple matrix method is utilized to obtain the inverse of the propagator matrix. Further, Woodhouse's inverse of the propagator matrix for a transversely isotropic medium is extended in a clearer form to handle the monoclinic symmetric medium. The properties of a periodic layer system are studied through its system matrix Aly , which is computed from the propagator matrix P. The matrix Aly is then compared with Aeq , the system matrix for the long-wavelength equivalent medium of the periodic isotropic layers. Then we can find how the periodic layered medium departs from its long-wavelength equivalent medium when the wavelength decreases. In our numerical example, the results show that, when λ/D decreases to 6-8, the components of the two matrices will depart from each other. The component ratio of these two matrices increases to its maximum (more than 15 in our numerical test) when λ/D is reduced to 2.3, and then oscillates with λ/D when it is further reduced. The eigenvalues of the system matrix Aly show that the velocities of P and S waves decrease when λ/D is reduced from 6-8 and reach their minimum values when λ/D is reduced to 2.3 and then oscillate afterwards. We compute the time shifts between the peaks of the transmitted waves and the incident waves. The resulting velocity curves show a similar variation to those computed from the eigenvalues of the system matrix Aly , but on a smaller scale. This can be explained by the spectrum width of the incident waves.

Fourth-order convergence of a compact scheme for the one-dimensional biharmonic equation

NASA Astrophysics Data System (ADS)

Fishelov, D.; Ben-Artzi, M.; Croisille, J.-P.

2012-09-01

The convergence of a fourth-order compact scheme to the one-dimensional biharmonic problem is established in the case of general Dirichlet boundary conditions. The compact scheme invokes value of the unknown function as well as Pade approximations of its first-order derivative. Using the Pade approximation allows us to approximate the first-order derivative within fourth-order accuracy. However, although the truncation error of the discrete biharmonic scheme is of fourth-order at interior point, the truncation error drops to first-order at near-boundary points. Nonetheless, we prove that the scheme retains its fourth-order (optimal) accuracy. This is done by a careful inspection of the matrix elements of the discrete biharmonic operator. A number of numerical examples corroborate this effect. We also present a study of the eigenvalue problem uxxxx = νu. We compute and display the eigenvalues and the eigenfunctions related to the continuous and the discrete problems. By the positivity of the eigenvalues, one can deduce the stability of of the related time-dependent problem ut = -uxxxx. In addition, we study the eigenvalue problem uxxxx = νuxx. This is related to the stability of the linear time-dependent equation uxxt = νuxxxx. Its continuous and discrete eigenvalues and eigenfunction (or eigenvectors) are computed and displayed graphically.

Faces of matrix models

NASA Astrophysics Data System (ADS)

Morozov, A.

2012-08-01

Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and nonlinear equations, as τ-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of W-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.

Accurate Quasiparticle Spectra from the T-Matrix Self-Energy and the Particle-Particle Random Phase Approximation.

PubMed

Zhang, Du; Su, Neil Qiang; Yang, Weitao

2017-07-20

The GW self-energy, especially G 0 W 0 based on the particle-hole random phase approximation (phRPA), is widely used to study quasiparticle (QP) energies. Motivated by the desirable features of the particle-particle (pp) RPA compared to the conventional phRPA, we explore the pp counterpart of GW, that is, the T-matrix self-energy, formulated with the eigenvectors and eigenvalues of the ppRPA matrix. We demonstrate the accuracy of the T-matrix method for molecular QP energies, highlighting the importance of the pp channel for calculating QP spectra.

Typical entanglement

NASA Astrophysics Data System (ADS)

Deelan Cunden, Fabio; Facchi, Paolo; Florio, Giuseppe; Pascazio, Saverio

2013-05-01

Let a pure state | ψ> be chosen randomly in an NM-dimensional Hilbert space, and consider the reduced density matrix ρ A of an N-dimensional subsystem. The bipartite entanglement properties of | ψ> are encoded in the spectrum of ρ A . By means of a saddle point method and using a "Coulomb gas" model for the eigenvalues, we obtain the typical spectrum of reduced density matrices. We consider the cases of an unbiased ensemble of pure states and of a fixed value of the purity. We finally obtain the eigenvalue distribution by using a statistical mechanics approach based on the introduction of a partition function.

A Riemann solver for RANS

NASA Astrophysics Data System (ADS)

Chuvakhov, P. V.

2014-01-01

An exact expression for a system of both eigenvalues and right/left eigenvectors of a Jacobian matrix for a convective two-equation differential closure RANS operator split along a curvilinear coordinate is derived. It is shown by examples of numerical modeling of supersonic flows over a flat plate and a compression corner with separation that application of the exact system of eigenvalues and eigenvectors to the Roe approach for approximate solution of the Riemann problem gives rise to an increase in the convergence rate, better stability and higher accuracy of a steady-state solution in comparison with those in the case of an approximate system.

On the Wigner law in dilute random matrices

NASA Astrophysics Data System (ADS)

Khorunzhy, A.; Rodgers, G. J.

1998-12-01

We consider ensembles of N × N symmetric matrices whose entries are weakly dependent random variables. We show that random dilution can change the limiting eigenvalue distribution of such matrices. We prove that under general and natural conditions the normalised eigenvalue counting function coincides with the semicircle (Wigner) distribution in the limit N → ∞. This can be explained by the observation that dilution (or more generally, random modulation) eliminates the weak dependence (or correlations) between random matrix entries. It also supports our earlier conjecture that the Wigner distribution is stable to random dilution and modulation.

A three dimensional point cloud registration method based on rotation matrix eigenvalue

NASA Astrophysics Data System (ADS)

Wang, Chao; Zhou, Xiang; Fei, Zixuan; Gao, Xiaofei; Jin, Rui

2017-09-01

We usually need to measure an object at multiple angles in the traditional optical three-dimensional measurement method, due to the reasons for the block, and then use point cloud registration methods to obtain a complete threedimensional shape of the object. The point cloud registration based on a turntable is essential to calculate the coordinate transformation matrix between the camera coordinate system and the turntable coordinate system. We usually calculate the transformation matrix by fitting the rotation center and the rotation axis normal of the turntable in the traditional method, which is limited by measuring the field of view. The range of exact feature points used for fitting the rotation center and the rotation axis normal is approximately distributed within an arc less than 120 degrees, resulting in a low fit accuracy. In this paper, we proposes a better method, based on the invariant eigenvalue principle of rotation matrix in the turntable coordinate system and the coordinate transformation matrix of the corresponding coordinate points. First of all, we control the rotation angle of the calibration plate with the turntable to calibrate the coordinate transformation matrix of the corresponding coordinate points by using the least squares method. And then we use the feature decomposition to calculate the coordinate transformation matrix of the camera coordinate system and the turntable coordinate system. Compared with the traditional previous method, it has a higher accuracy, better robustness and it is not affected by the camera field of view. In this method, the coincidence error of the corresponding points on the calibration plate after registration is less than 0.1mm.

Statistical classification techniques for engineering and climatic data samples

NASA Technical Reports Server (NTRS)

Temple, E. C.; Shipman, J. R.

1981-01-01

Fisher's sample linear discriminant function is modified through an appropriate alteration of the common sample variance-covariance matrix. The alteration consists of adding nonnegative values to the eigenvalues of the sample variance covariance matrix. The desired results of this modification is to increase the number of correct classifications by the new linear discriminant function over Fisher's function. This study is limited to the two-group discriminant problem.

Sequential design of discrete linear quadratic regulators via optimal root-locus techniques

NASA Technical Reports Server (NTRS)

Shieh, Leang S.; Yates, Robert E.; Ganesan, Sekar

1989-01-01

A sequential method employing classical root-locus techniques has been developed in order to determine the quadratic weighting matrices and discrete linear quadratic regulators of multivariable control systems. At each recursive step, an intermediate unity rank state-weighting matrix that contains some invariant eigenvectors of that open-loop matrix is assigned, and an intermediate characteristic equation of the closed-loop system containing the invariant eigenvalues is created.

Asymptotic coincidence of the statistics for degenerate and non-degenerate correlated real Wishart ensembles

NASA Astrophysics Data System (ADS)

Wirtz, Tim; Kieburg, Mario; Guhr, Thomas

2017-06-01

The correlated Wishart model provides the standard benchmark when analyzing time series of any kind. Unfortunately, the real case, which is the most relevant one in applications, poses serious challenges for analytical calculations. Often these challenges are due to square root singularities which cannot be handled using common random matrix techniques. We present a new way to tackle this issue. Using supersymmetry, we carry out an anlaytical study which we support by numerical simulations. For large but finite matrix dimensions, we show that statistical properties of the fully correlated real Wishart model generically approach those of a correlated real Wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues. This holds for the local and global spectral statistics. With Monte Carlo simulations we show that this is even approximately true for small matrix dimensions. We explicitly investigate the k-point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case. Moreover we show that on the local scale the k-point correlation function exhibits the sine and the Airy kernel in the bulk and at the soft edges, respectively. We also address the positions and the fluctuations of the possible outliers in the data.

Detecting Seismic Activity with a Covariance Matrix Analysis of Data Recorded on Seismic Arrays

NASA Astrophysics Data System (ADS)

Seydoux, L.; Shapiro, N.; de Rosny, J.; Brenguier, F.

2014-12-01

Modern seismic networks are recording the ground motion continuously all around the word, with very broadband and high-sensitivity sensors. The aim of our study is to apply statistical array-based approaches to processing of these records. We use the methods mainly brought from the random matrix theory in order to give a statistical description of seismic wavefields recorded at the Earth's surface. We estimate the array covariance matrix and explore the distribution of its eigenvalues that contains information about the coherency of the sources that generated the studied wavefields. With this approach, we can make distinctions between the signals generated by isolated deterministic sources and the "random" ambient noise. We design an algorithm that uses the distribution of the array covariance matrix eigenvalues to detect signals corresponding to coherent seismic events. We investigate the detection capacity of our methods at different scales and in different frequency ranges by applying it to the records of two networks: (1) the seismic monitoring network operating on the Piton de la Fournaise volcano at La Réunion island composed of 21 receivers and with an aperture of ~15 km, and (2) the transportable component of the USArray composed of ~400 receivers with ~70 km inter-station spacing.

Adaptive Inverse Control for Rotorcraft Vibration Reduction

NASA Technical Reports Server (NTRS)

Jacklin, Stephen A.

1985-01-01

This thesis extends the Least Mean Square (LMS) algorithm to solve the mult!ple-input, multiple-output problem of alleviating N/Rev (revolutions per minute by number of blades) helicopter fuselage vibration by means of adaptive inverse control. A frequency domain locally linear model is used to represent the transfer matrix relating the higher harmonic pitch control inputs to the harmonic vibration outputs to be controlled. By using the inverse matrix as the controller gain matrix, an adaptive inverse regulator is formed to alleviate the N/Rev vibration. The stability and rate of convergence properties of the extended LMS algorithm are discussed. It is shown that the stability ranges for the elements of the stability gain matrix are directly related to the eigenvalues of the vibration signal information matrix for the learning phase, but not for the control phase. The overall conclusion is that the LMS adaptive inverse control method can form a robust vibration control system, but will require some tuning of the input sensor gains, the stability gain matrix, and the amount of control relaxation to be used. The learning curve of the controller during the learning phase is shown to be quantitatively close to that predicted by averaging the learning curves of the normal modes. For higher order transfer matrices, a rough estimate of the inverse is needed to start the algorithm efficiently. The simulation results indicate that the factor which most influences LMS adaptive inverse control is the product of the control relaxation and the the stability gain matrix. A small stability gain matrix makes the controller less sensitive to relaxation selection, and permits faster and more stable vibration reduction, than by choosing the stability gain matrix large and the control relaxation term small. It is shown that the best selections of the stability gain matrix elements and the amount of control relaxation is basically a compromise between slow, stable convergence and fast convergence with increased possibility of unstable identification. In the simulation studies, the LMS adaptive inverse control algorithm is shown to be capable of adapting the inverse (controller) matrix to track changes in the flight conditions. The algorithm converges quickly for moderate disturbances, while taking longer for larger disturbances. Perfect knowledge of the inverse matrix is not required for good control of the N/Rev vibration. However it is shown that measurement noise will prevent the LMS adaptive inverse control technique from controlling the vibration, unless the signal averaging method presented is incorporated into the algorithm.

A general solution strategy of modified power method for higher mode solutions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Peng; Lee, Hyunsuk; Lee, Deokjung, E-mail: deokjung@unist.ac.kr

2016-01-15

A general solution strategy of the modified power iteration method for calculating higher eigenmodes has been developed and applied in continuous energy Monte Carlo simulation. The new approach adopts four features: 1) the eigen decomposition of transfer matrix, 2) weight cancellation for higher modes, 3) population control with higher mode weights, and 4) stabilization technique of statistical fluctuations using multi-cycle accumulations. The numerical tests of neutron transport eigenvalue problems successfully demonstrate that the new strategy can significantly accelerate the fission source convergence with stable convergence behavior while obtaining multiple higher eigenmodes at the same time. The advantages of the newmore » strategy can be summarized as 1) the replacement of the cumbersome solution step of high order polynomial equations required by Booth's original method with the simple matrix eigen decomposition, 2) faster fission source convergence in inactive cycles, 3) more stable behaviors in both inactive and active cycles, and 4) smaller variances in active cycles. Advantages 3 and 4 can be attributed to the lower sensitivity of the new strategy to statistical fluctuations due to the multi-cycle accumulations. The application of the modified power method to continuous energy Monte Carlo simulation and the higher eigenmodes up to 4th order are reported for the first time in this paper. -- Graphical abstract: -- Highlights: •Modified power method is applied to continuous energy Monte Carlo simulation. •Transfer matrix is introduced to generalize the modified power method. •All mode based population control is applied to get the higher eigenmodes. •Statistic fluctuation can be greatly reduced using accumulated tally results. •Fission source convergence is accelerated with higher mode solutions.« less

Exact evaluation of the causal spectrum and localization properties of electronic states on a scale-free network

NASA Astrophysics Data System (ADS)

Xie, Pinchen; Yang, Bingjia; Zhang, Zhongzhi; Andrade, Roberto F. S.

2018-07-01

A deterministic network with tree structure is considered, for which the spectrum of its adjacency matrix can be exactly evaluated by a recursive renormalization approach. It amounts to successively increasing number of contributions at any finite step of construction of the tree, resulting in a causal chain. The resulting eigenvalues can be related the full energy spectrum of a nearest-neighbor tight-binding model defined on this structure. Given this association, it turns out that further properties of the eigenvectors can be evaluated, like the degree of quantum localization of the tight-binding eigenstates, expressed by the inverse participation ratio (IPR). It happens that, for the current model, the IPR's are also suitable to be analytically expressed in terms in corresponding eigenvalue chain. The resulting IPR scaling behavior is expressed by the tails of eigenvalue chains as well.

Unsteady combustion of solid propellants

NASA Astrophysics Data System (ADS)

Chung, T. J.; Kim, P. K.

The oscillatory motions of all field variables (pressure, temperature, velocity, density, and fuel fractions) in the flame zone of solid propellant rocket motors are calculated using the finite element method. The Arrhenius law with a single step forward chemical reaction is used. Effects of radiative heat transfer, impressed arbitrary acoustic wave incidence, and idealized mean flow velocities are also investigated. Boundary conditions are derived at the solid-gas interfaces and at the flame edges which are implemented via Lagrange multipliers. Perturbation expansions of all governing conservation equations up to and including the second order are carried out so that nonlinear oscillations may be accommodated. All excited frequencies are calculated by means of eigenvalue analyses, and the combustion response functions corresponding to these frequencies are determined. It is shown that the use of isoparametric finite elements, Gaussian quadrature integration, and the Lagrange multiplier boundary matrix scheme offers a convenient approach to two-dimensional calculations.

Effect of tumor resection on the characteristics of functional brain networks.

PubMed

Wang, H; Douw, L; Hernández, J M; Reijneveld, J C; Stam, C J; Van Mieghem, P

2010-08-01

Brain functioning such as cognitive performance depends on the functional interactions between brain areas, namely, the functional brain networks. The functional brain networks of a group of patients with brain tumors are measured before and after tumor resection. In this work, we perform a weighted network analysis to understand the effect of neurosurgery on the characteristics of functional brain networks. Statistically significant changes in network features have been discovered in the beta (13-30 Hz) band after neurosurgery: the link weight correlation around nodes and within triangles increases which implies improvement in local efficiency of information transfer and robustness; the clustering of high link weights in a subgraph becomes stronger, which enhances the global transport capability; and the decrease in the synchronization or virus spreading threshold, revealed by the increase in the largest eigenvalue of the adjacency matrix, which suggests again the improvement of information dissemination.

Matrix Perturbation Techniques in Structural Dynamics

NASA Technical Reports Server (NTRS)

Caughey, T. K.

1973-01-01

Matrix perturbation are developed techniques which can be used in the dynamical analysis of structures where the range of numerical values in the matrices extreme or where the nature of the damping matrix requires that complex valued eigenvalues and eigenvectors be used. The techniques can be advantageously used in a variety of fields such as earthquake engineering, ocean engineering, aerospace engineering and other fields concerned with the dynamical analysis of large complex structures or systems of second order differential equations. A number of simple examples are included to illustrate the techniques.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heifetz, Alexander; Vilim, Richard

Super-critical carbon dioxide (S-CO2) is a promising thermodynamic cycle for advanced nuclear reactors and solar energy conversion applications. Dynamic control of the proposed recompression S-CO2 cycle is accomplished with input from resistance temperature detector (RTD) measurements of the process fluid. One of the challenges in practical implementation of S-CO2 cycle is high corrosion rate of component and sensor materials. In this paper, we develop a mathematical model of RTD sensing using eigendecomposition model of radial heat transfer in a layered long cylinder. We show that the value of RTD time constant primarily depends on the rate of heat transfer frommore » the fluid to the outer wall of RTD. We also show that for typical material properties, RTD time constant can be calculated as the sum of reciprocal eigen-values of the heat transfer matrix. Using the computational model and a set of RTD and CO2 fluid thermophysical parameter values, we calculate the value of time constant of thermowell-mounted RTD sensor at the hot side of the precooler in the S-CO2 cycle. The eigendecomposition model of RTD will be used in future studies to model sensor degradation and its impact on control of S-CO2. (C) 2016 Elsevier B.V. All rights reserved.« less

Sensitivity of coronal loop sausage mode frequencies and decay rates to radial and longitudinal density inhomogeneities: a spectral approach

NASA Astrophysics Data System (ADS)

Cally, Paul S.; Xiong, Ming

2018-01-01

Fast sausage modes in solar magnetic coronal loops are only fully contained in unrealistically short dense loops. Otherwise they are leaky, losing energy to their surrounds as outgoing waves. This causes any oscillation to decay exponentially in time. Simultaneous observations of both period and decay rate therefore reveal the eigenfrequency of the observed mode, and potentially insight into the tubes’ nonuniform internal structure. In this article, a global spectral description of the oscillations is presented that results in an implicit matrix eigenvalue equation where the eigenvalues are associated predominantly with the diagonal terms of the matrix. The off-diagonal terms vanish identically if the tube is uniform. A linearized perturbation approach, applied with respect to a uniform reference model, is developed that makes the eigenvalues explicit. The implicit eigenvalue problem is easily solved numerically though, and it is shown that knowledge of the real and imaginary parts of the eigenfrequency is sufficient to determine the width and density contrast of a boundary layer over which the tubes’ enhanced internal densities drop to ambient values. Linearized density kernels are developed that show sensitivity only to the extreme outside of the loops for radial fundamental modes, especially for small density enhancements, with no sensitivity to the core. Higher radial harmonics do show some internal sensitivity, but these will be more difficult to observe. Only kink modes are sensitive to the tube centres. Variation in internal and external Alfvén speed along the loop is shown to have little effect on the fundamental dimensionless eigenfrequency, though the associated eigenfunction becomes more compact at the loop apex as stratification increases, or may even displace from the apex.

Intraday seasonalities and nonstationarity of trading volume in financial markets: Collective features.

PubMed

Graczyk, Michelle B; Duarte Queirós, Sílvio M

2017-01-01

Employing Random Matrix Theory and Principal Component Analysis techniques, we enlarge our work on the individual and cross-sectional intraday statistical properties of trading volume in financial markets to the study of collective intraday features of that financial observable. Our data consist of the trading volume of the Dow Jones Industrial Average Index components spanning the years between 2003 and 2014. Computing the intraday time dependent correlation matrices and their spectrum of eigenvalues, we show there is a mode ruling the collective behaviour of the trading volume of these stocks whereas the remaining eigenvalues are within the bounds established by random matrix theory, except the second largest eigenvalue which is robustly above the upper bound limit at the opening and slightly above it during the morning-afternoon transition. Taking into account that for price fluctuations it was reported the existence of at least seven significant eigenvalues-and that its autocorrelation function is close to white noise for highly liquid stocks whereas for the trading volume it lasts significantly for more than 2 hours -, our finding goes against any expectation based on those features, even when we take into account the Epps effect. In addition, the weight of the trading volume collective mode is intraday dependent; its value increases as the trading session advances with its eigenversor approaching the uniform vector as well, which corresponds to a soar in the behavioural homogeneity. With respect to the nonstationarity of the collective features of the trading volume we observe that after the financial crisis of 2008 the coherence function shows the emergence of an upset profile with large fluctuations from that year on, a property that concurs with the modification of the average trading volume profile we noted in our previous individual analysis.

The Hölder continuity of spectral measures of an extended CMV matrix

NASA Astrophysics Data System (ADS)

Munger, Paul E.; Ong, Darren C.

2014-09-01

We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip, and Lenz ["Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity," Commun. Math. Phys. 212, 191-204 (2000)] about the spectral measure of the discrete Schrödinger operator.

The Hölder continuity of spectral measures of an extended CMV matrix.

PubMed

Munger, Paul E; Ong, Darren C

2014-09-01

We prove results about the Hölder continuity of the spectral measures of the extended CMV matrix, given power law bounds of the solution of the eigenvalue equation. We thus arrive at a unitary analogue of the results of Damanik, Killip, and Lenz ["Uniform spectral properties of one-dimensional quasicrystals, III. α-continuity," Commun. Math. Phys.55, 191-204 (2000)] about the spectral measure of the discrete Schrödinger operator.

A generalization of random matrix theory and its application to statistical physics.

PubMed

Wang, Duan; Zhang, Xin; Horvatic, Davor; Podobnik, Boris; Eugene Stanley, H

2017-02-01

To study the statistical structure of crosscorrelations in empirical data, we generalize random matrix theory and propose a new method of cross-correlation analysis, known as autoregressive random matrix theory (ARRMT). ARRMT takes into account the influence of auto-correlations in the study of cross-correlations in multiple time series. We first analytically and numerically determine how auto-correlations affect the eigenvalue distribution of the correlation matrix. Then we introduce ARRMT with a detailed procedure of how to implement the method. Finally, we illustrate the method using two examples taken from inflation rates for air pressure data for 95 US cities.

Method of locating related items in a geometric space for data mining

DOEpatents

Hendrickson, B.A.

1999-07-27

A method for locating related items in a geometric space transforms relationships among items to geometric locations. The method locates items in the geometric space so that the distance between items corresponds to the degree of relatedness. The method facilitates communication of the structure of the relationships among the items. The method is especially beneficial for communicating databases with many items, and with non-regular relationship patterns. Examples of such databases include databases containing items such as scientific papers or patents, related by citations or keywords. A computer system adapted for practice of the present invention can include a processor, a storage subsystem, a display device, and computer software to direct the location and display of the entities. The method comprises assigning numeric values as a measure of similarity between each pairing of items. A matrix is constructed, based on the numeric values. The eigenvectors and eigenvalues of the matrix are determined. Each item is located in the geometric space at coordinates determined from the eigenvectors and eigenvalues. Proper construction of the matrix and proper determination of coordinates from eigenvectors can ensure that distance between items in the geometric space is representative of the numeric value measure of the items' similarity. 12 figs.

A contracting-interval program for the Danilewski method. Ph.D. Thesis - Va. Univ.

NASA Technical Reports Server (NTRS)

Harris, J. D.

1971-01-01

The concept of contracting-interval programs is applied to finding the eigenvalues of a matrix. The development is a three-step process in which (1) a program is developed for the reduction of a matrix to Hessenberg form, (2) a program is developed for the reduction of a Hessenberg matrix to colleague form, and (3) the characteristic polynomial with interval coefficients is readily obtained from the interval of colleague matrices. This interval polynomial is then factored into quadratic factors so that the eigenvalues may be obtained. To develop a contracting-interval program for factoring this polynomial with interval coefficients it is necessary to have an iteration method which converges even in the presence of controlled rounding errors. A theorem is stated giving sufficient conditions for the convergence of Newton's method when both the function and its Jacobian cannot be evaluated exactly but errors can be made proportional to the square of the norm of the difference between the previous two iterates. This theorem is applied to prove the convergence of the generalization of the Newton-Bairstow method that is used to obtain quadratic factors of the characteristic polynomial.

Method of locating related items in a geometric space for data mining

DOEpatents

Hendrickson, Bruce A.

1999-01-01

A method for locating related items in a geometric space transforms relationships among items to geometric locations. The method locates items in the geometric space so that the distance between items corresponds to the degree of relatedness. The method facilitates communication of the structure of the relationships among the items. The method is especially beneficial for communicating databases with many items, and with non-regular relationship patterns. Examples of such databases include databases containing items such as scientific papers or patents, related by citations or keywords. A computer system adapted for practice of the present invention can include a processor, a storage subsystem, a display device, and computer software to direct the location and display of the entities. The method comprises assigning numeric values as a measure of similarity between each pairing of items. A matrix is constructed, based on the numeric values. The eigenvectors and eigenvalues of the matrix are determined. Each item is located in the geometric space at coordinates determined from the eigenvectors and eigenvalues. Proper construction of the matrix and proper determination of coordinates from eigenvectors can ensure that distance between items in the geometric space is representative of the numeric value measure of the items' similarity.

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev model

NASA Astrophysics Data System (ADS)

García-García, Antonio M.; Jia, Yiyang; Verbaarschot, Jacobus J. M.

2018-05-01

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions with infinite range interactions in 0 +1 dimensions. We have found that, close to the ground state E ≈0 , discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite N , which we compute analytically and numerically, grows exponentially with N for E ≈0 . However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above E ≈0 , the spectral density grows exponentially with the energy. Deep in the quantum regime, corresponding to the first O (N ) eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for E ≈0 is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalue separations smaller than the Thouless energy, which seems to scale linearly with N . Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universally characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor g (t ), obtained from the connected two-level correlation function of the unfolded spectrum, decays as 1 /t2 for times shorter but comparable to the Thouless time with g (0 ) related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.

Level repulsion and band sorting in phononic crystals

NASA Astrophysics Data System (ADS)

Lu, Yan; Srivastava, Ankit

2018-02-01

In this paper we consider the problem of avoided crossings (level repulsion) in phononic crystals and suggest a computationally efficient strategy to distinguish them from normal cross points. This process is essential for the correct sorting of the phononic bands and, subsequently, for the accurate determination of mode continuation, group velocities, and emergent properties which depend on them such as thermal conductivity. Through explicit phononic calculations using generalized Rayleigh quotient, we identify exact locations of exceptional points in the complex wavenumber domain which results in level repulsion in the real domain. We show that in the vicinity of the exceptional point the relevant phononic eigenvalue surfaces resemble the surfaces of a 2 by 2 parameter-dependent matrix. Along a closed loop encircling the exceptional point we show that the phononic eigenvalues are exchanged, just as they are for the 2 by 2 matrix case. However, the behavior of the associated eigenvectors is shown to be more complex in the phononic case. Along a closed loop around an exceptional point, we show that the eigenvectors can flip signs multiple times unlike a 2 by 2 matrix where the flip of sign occurs only once. Finally, we exploit these eigenvector sign flips around exceptional points to propose a simple and efficient method of distinguishing them from normal crosses and of correctly sorting the band-structure. Our proposed method is roughly an order-of-magnitude faster than the zoom-in method and correctly identifies > 96% of the cases considered. Both its speed and accuracy can be further improved and we suggest some ways of achieving this. Our method is general and, as such, would be directly applicable to other eigenvalue problems where the eigenspectrum needs to be correctly sorted.

Network trending; leadership, followership and neutrality among companies: A random matrix approach

NASA Astrophysics Data System (ADS)

Mobarhan, N. S. Safavi; Saeedi, A.; Roodposhti, F. Rahnamay; Jafari, G. R.

2016-11-01

In this article, we analyze the cross-correlation between returns of different stocks to answer the following important questions. The first one is: If there exists collective behavior in a financial market, how could we detect it? And the second question is: Is there a particular company among the companies of a market as the leader of the collective behavior? Or is there no specified leadership governing the system similar to some complex systems? We use the method of random matrix theory to answer the mentioned questions. Cross-correlation matrix of index returns of four different markets is analyzed. The participation ratio quantity related to each matrices' eigenvectors and the eigenvalue spectrum is calculated. We introduce shuffled-matrix created of cross correlation matrix in such a way that the elements of the later one are displaced randomly. Comparing the participation ratio quantities obtained from a correlation matrix of a market and its related shuffled-one, on the bulk distribution region of the eigenvalues, we detect a meaningful deviation between the mentioned quantities indicating the collective behavior of the companies forming the market. By calculating the relative deviation of participation ratios, we obtain a measure to compare the markets according to their collective behavior. Answering the second question, we show there are three groups of companies: The first group having higher impact on the market trend called leaders, the second group is followers and the third one is the companies who have not a considerable role in the trend. The results can be utilized in portfolio construction.

Three-dimensional geometry of coronal loops inferred by the Principal Component Analysis

NASA Astrophysics Data System (ADS)

Nisticò, Giuseppe; Nakariakov, Valery

We propose a new method for the determination of the three dimensional (3D) shape of coronal loops from stereoscopy. The common approach requires to find a 1D geometric curve, as circumference or ellipse, that best-fits the 3D tie-points which sample the loop shape in a given coordinate system. This can be easily achieved by the Principal Component (PC) analysis. It mainly consists in calculating the eigenvalues and eigenvectors of the covariance matrix of the 3D tie-points: the eigenvalues give a measure of the variability of the distribution of the tie-points, and the corresponding eigenvectors define a new cartesian reference frame directly related to the loop. The eigenvector associated with the smallest eigenvalues defines the normal to the loop plane, while the other two determine the directions of the loop axes: the major axis is related to the largest eigenvalue, and the minor axis with the second one. The magnitude of the axes is directly proportional to the square roots of these eigenvalues. The technique is fast and easily implemented in some examples, returning best-fitting estimations of the loop parameters and 3D reconstruction with a reasonable small number of tie-points. The method is suitable for serial reconstruction of coronal loops in active regions, providing a useful tool for comparison between observations and theoretical magnetic field extrapolations from potential or force-free fields.

Detecting, anticipating, and predicting critical transitions in spatially extended systems.

PubMed

Kwasniok, Frank

2018-03-01

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Detecting, anticipating, and predicting critical transitions in spatially extended systems

NASA Astrophysics Data System (ADS)

Kwasniok, Frank

2018-03-01

A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system. As a further step, a genuinely nonstationary POP analysis is introduced. Here, the system matrix of the linear stochastic model is time-dependent, allowing for extrapolation and prediction of instabilities beyond the learning data window. The methods are demonstrated and explored using the one-dimensional Swift-Hohenberg equation as an example, focusing on the dynamics of stochastic fluctuations around the homogeneous stable state prior to the first bifurcation. The POP-based techniques are able to extract and track the least stable eigenvalues and eigenvectors of the system; the nonstationary POP analysis successfully predicts the timing of the first instability and the unstable mode well beyond the learning data window.

Coherent mode decomposition using mixed Wigner functions of Hermite-Gaussian beams.

PubMed

Tanaka, Takashi

2017-04-15

A new method of coherent mode decomposition (CMD) is proposed that is based on a Wigner-function representation of Hermite-Gaussian beams. In contrast to the well-known method using the cross spectral density (CSD), it directly determines the mode functions and their weights without solving the eigenvalue problem. This facilitates the CMD of partially coherent light whose Wigner functions (and thus CSDs) are not separable, in which case the conventional CMD requires solving an eigenvalue problem with a large matrix and thus is numerically formidable. An example is shown regarding the CMD of synchrotron radiation, one of the most important applications of the proposed method.

A nonperturbative light-front coupled-cluster method

NASA Astrophysics Data System (ADS)

Hiller, J. R.

2012-10-01

The nonperturbative Hamiltonian eigenvalue problem for bound states of a quantum field theory is formulated in terms of Dirac's light-front coordinates and then approximated by the exponential-operator technique of the many-body coupled-cluster method. This approximation eliminates any need for the usual approximation of Fock-space truncation. Instead, the exponentiated operator is truncated, and the terms retained are determined by a set of nonlinear integral equations. These equations are solved simultaneously with an effective eigenvalue problem in the valence sector, where the number of constituents is small. Matrix elements can be calculated, with extensions of techniques from standard coupled-cluster theory, to obtain form factors and other observables.

Linear quadratic regulators with eigenvalue placement in a specified region

NASA Technical Reports Server (NTRS)

Shieh, Leang S.; Dib, Hani M.; Ganesan, Sekar

1988-01-01

A linear optimal quadratic regulator is developed for optimally placing the closed-loop poles of multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at + or - pi/2k (k = 2 or 3) from the negative real axis with a sector angle of pi/2 or less, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane. The design method is mainly based on the solution of a linear matrix Liapunov equation, and the resultant closed-loop system with its eigenvalues in the desired region is optimal with respect to a quadratic performance index.

Wave Propagation and Localization via Quasi-Normal Modes and Transmission Eigenchannels

NASA Astrophysics Data System (ADS)

Wang, Jing; Shi, Zhou; Davy, Matthieu; Genack, Azriel Z.

2013-10-01

Field transmission coefficients for microwave radiation between arrays of points on the incident and output surfaces of random samples are analyzed to yield the underlying quasi-normal modes and transmission eigenchannels of each realization of the sample. The linewidths, central frequencies, and transmitted speckle patterns associated with each of the modes of the medium are found. Modal speckle patterns are found to be strongly correlated leading to destructive interference between modes. This explains distinctive features of transmission spectra and pulsed transmission. An alternate description of wave transport is obtained from the eigenchannels and eigenvalues of the transmission matrix. The maximum transmission eigenvalue, τ1 is near unity for diffusive waves even in turbid samples. For localized waves, τ1 is nearly equal to the dimensionless conductance, which is the sum of all transmission eigenvalues, g = Στn. The spacings between the ensemble averages of successive values of lnτn are constant and equal to the inverse of the bare conductance in accord with predictions by Dorokhov. The effective number of transmission eigenvalues Neff determines the contrast between the peak and background of radiation focused for maximum peak intensity. The connection between the mode and channel approaches is discussed.

Wave Propagation and Localization via Quasi-Normal Modes and Transmission Eigenchannels

NASA Astrophysics Data System (ADS)

Wang, Jing; Shi, Zhou; Davy, Matthieu; Genack, Azriel Z.

Field transmission coefficients for microwave radiation between arrays of points on the incident and output surfaces of random samples are analyzed to yield the underlying quasi-normal modes and transmission eigenchannels of each realization of the sample. The linewidths, central frequencies, and transmitted speckle patterns associated with each of the modes of the medium are found. Modal speckle patterns are found to be strongly correlated leading to destructive interference between modes. This explains distinctive features of transmission spectra and pulsed transmission. An alternate description of wave transport is obtained from the eigenchannels and eigenvalues of the transmission matrix. The maximum transmission eigenvalue, τ1 is near unity for diffusive waves even in turbid samples. For localized waves, τ1 is nearly equal to the dimensionless conductance, which is the sum of all transmission eigenvalues, g = Στn. The spacings between the ensemble averages of successive values of lnτn are constant and equal to the inverse of the bare conductance in accord with predictions by Dorokhov. The effective number of transmission eigenvalues Neff determines the contrast between the peak and background of radiation focused for maximum peak intensity. The connection between the mode and channel approaches is discussed.

Proper and improper zero energy modes in Hartree-Fock theory and their relevance for symmetry breaking and restoration.

PubMed

Cui, Yao; Bulik, Ireneusz W; Jiménez-Hoyos, Carlos A; Henderson, Thomas M; Scuseria, Gustavo E

2013-10-21

We study the spectra of the molecular orbital Hessian (stability matrix) and random-phase approximation (RPA) Hamiltonian of broken-symmetry Hartree-Fock solutions, focusing on zero eigenvalue modes. After all negative eigenvalues are removed from the Hessian by following their eigenvectors downhill, one is left with only positive and zero eigenvalues. Zero modes correspond to orbital rotations with no restoring force. These rotations determine states in the Goldstone manifold, which originates from a spontaneously broken continuous symmetry in the wave function. Zero modes can be classified as improper or proper according to their different mathematical and physical properties. Improper modes arise from symmetry breaking and their restoration always lowers the energy. Proper modes, on the other hand, correspond to degeneracies of the wave function, and their symmetry restoration does not necessarily lower the energy. We discuss how the RPA Hamiltonian distinguishes between proper and improper modes by doubling the number of zero eigenvalues associated with the latter. Proper modes in the Hessian always appear in pairs which do not double in RPA. We present several pedagogical cases exemplifying the above statements. The relevance of these results for projected Hartree-Fock methods is also addressed.

Matrix eigenvalue method for free-oscillations modelling of spherical elastic bodies

NASA Astrophysics Data System (ADS)

Zábranová, E.; Hanyk, L.; Matyska, C.

2017-11-01

Deformations and changes of the gravitational potential of pre-stressed self-gravitating elastic bodies caused by free oscillations are described by means of the momentum and Poisson equations and the constitutive relation. For spherically symmetric bodies, the equations and boundary conditions are transformed into ordinary differential equations of the second order by the spherical harmonic decomposition and further discretized by highly accurate pseudospectral difference schemes on Chebyshev grids; we pay special attention to the conditions at the centre of the models. We thus obtain a series of matrix eigenvalue problems for eigenfrequencies and eigenfunctions of the free oscillations. Accuracy of the presented numerical approach is tested by means of the Rayleigh quotients calculated for the eigenfrequencies up to 500 mHz. Both the modal frequencies and eigenfunctions are benchmarked against the output from the Mineos software package based on shooting methods. The presented technique is a promising alternative to widely used methods because it is stable and with a good capability up to high frequencies.

Time scales involved in emergent market coherence

NASA Astrophysics Data System (ADS)

Kwapień, J.; Drożdż, S.; Speth, J.

2004-06-01

In addressing the question of the time scales characteristic for the market formation, we analyze high-frequency tick-by-tick data from the NYSE and from the German market. By using returns on various time scales ranging from seconds or minutes up to 2 days, we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales. For various sets of stocks of different capitalization (and the average trading frequency), we observe a significant elevation of the largest eigenvalue with increasing time scale. Our results from the correlation matrix study can be considered as a manifestation of the so-called Epps effect. There is no unique explanation of this effect and it seems that many different factors play a role here. One of such factors is randomness in transaction moments for different stocks. Another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history.

Chaotic, informational and synchronous behaviour of multiplex networks

NASA Astrophysics Data System (ADS)

Baptista, M. S.; Szmoski, R. M.; Pereira, R. F.; Pinto, S. E. De Souza

2016-03-01

The understanding of the relationship between topology and behaviour in interconnected networks would allow to charac- terise and predict behaviour in many real complex networks since both are usually not simultaneously known. Most previous studies have focused on the relationship between topology and synchronisation. In this work, we provide analytical formulas that shows how topology drives complex behaviour: chaos, information, and weak or strong synchronisation; in multiplex net- works with constant Jacobian. We also study this relationship numerically in multiplex networks of Hindmarsh-Rose neurons. Whereas behaviour in the analytically tractable network is a direct but not trivial consequence of the spectra of eigenvalues of the Laplacian matrix, where behaviour may strongly depend on the break of symmetry in the topology of interconnections, in Hindmarsh-Rose neural networks the nonlinear nature of the chemical synapses breaks the elegant mathematical connec- tion between the spectra of eigenvalues of the Laplacian matrix and the behaviour of the network, creating networks whose behaviour strongly depends on the nature (chemical or electrical) of the inter synapses.

CCOMP: An efficient algorithm for complex roots computation of determinantal equations

NASA Astrophysics Data System (ADS)

Zouros, Grigorios P.

2018-01-01

In this paper a free Python algorithm, entitled CCOMP (Complex roots COMPutation), is developed for the efficient computation of complex roots of determinantal equations inside a prescribed complex domain. The key to the method presented is the efficient determination of the candidate points inside the domain which, in their close neighborhood, a complex root may lie. Once these points are detected, the algorithm proceeds to a two-dimensional minimization problem with respect to the minimum modulus eigenvalue of the system matrix. In the core of CCOMP exist three sub-algorithms whose tasks are the efficient estimation of the minimum modulus eigenvalues of the system matrix inside the prescribed domain, the efficient computation of candidate points which guarantee the existence of minima, and finally, the computation of minima via bound constrained minimization algorithms. Theoretical results and heuristics support the development and the performance of the algorithm, which is discussed in detail. CCOMP supports general complex matrices, and its efficiency, applicability and validity is demonstrated to a variety of microwave applications.

PCA meets RG

NASA Astrophysics Data System (ADS)

Bradde, Serena; Bialek, William

A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis (PCA) focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group (RG). Following this analogy, we can define relevant and irrelevant operators, where the role of dimensionality is played by properties of the eigenvalue density. These results also suggest an approach to the analysis of real data. As an example, we study neural activity in the vertebrate retina as it responds to naturalistic movies, and find evidence of behavior controlled by a nontrivial fixed point. Applied to financial data, our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non-Gaussian distribution.

HO2 rovibrational eigenvalue studies for nonzero angular momentum

NASA Astrophysics Data System (ADS)

Wu, Xudong T.; Hayes, Edward F.

1997-08-01

An efficient parallel algorithm is reported for determining all bound rovibrational energy levels for the HO2 molecule for nonzero angular momentum values, J=1, 2, and 3. Performance tests on the CRAY T3D indicate that the algorithm scales almost linearly when up to 128 processors are used. Sustained performance levels of up to 3.8 Gflops have been achieved using 128 processors for J=3. The algorithm uses a direct product discrete variable representation (DVR) basis and the implicitly restarted Lanczos method (IRLM) of Sorensen to compute the eigenvalues of the polyatomic Hamiltonian. Since the IRLM is an iterative method, it does not require storage of the full Hamiltonian matrix—it only requires the multiplication of the Hamiltonian matrix by a vector. When the IRLM is combined with a formulation such as DVR, which produces a very sparse matrix, both memory and computation times can be reduced dramatically. This algorithm has the potential to achieve even higher performance levels for larger values of the total angular momentum.

An O(N squared) method for computing the eigensystem of N by N symmetric tridiagonal matrices by the divide and conquer approach

NASA Technical Reports Server (NTRS)

Gill, Doron; Tadmor, Eitan

1988-01-01

An efficient method is proposed to solve the eigenproblem of N by N Symmetric Tridiagonal (ST) matrices. Unlike the standard eigensolvers which necessitate O(N cubed) operations to compute the eigenvectors of such ST matrices, the proposed method computes both the eigenvalues and eigenvectors with only O(N squared) operations. The method is based on serial implementation of the recently introduced Divide and Conquer (DC) algorithm. It exploits the fact that by O(N squared) of DC operations, one can compute the eigenvalues of N by N ST matrix and a finite number of pairs of successive rows of its eigenvector matrix. The rest of the eigenvectors--all of them or one at a time--are computed by linear three-term recurrence relations. Numerical examples are presented which demonstrate the superiority of the proposed method by saving an order of magnitude in execution time at the expense of sacrificing a few orders of accuracy.

Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional based tight-binding

DOE PAGES

Zhang, Hong; Zapol, Peter; Dixon, David A.; ...

2015-11-17

The Shift-and-invert parallel spectral transformations (SIPs), a computational approach to solve sparse eigenvalue problems, is developed for massively parallel architectures with exceptional parallel scalability and robustness. The capabilities of SIPs are demonstrated by diagonalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall metallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studied is a 128,000 (2000) atom nanotube for which ~330,000 (~5600) eigenvalues and eigenfunctions are obtained in ~190 (~5) seconds when parallelized over 266,144 (16,384) Blue Gene/Q cores. Weak scaling and strong scaling of SIPs are analyzed and the performance of SIPsmore » is compared with other novel methods. Different matrix ordering methods are investigated to reduce the cost of the factorization step, which dominates the time-to-solution at the strong scaling limit. As a result, a parallel implementation of assembling the density matrix from the distributed eigenvectors is demonstrated.« less

Shift-and-invert parallel spectral transformation eigensolver: Massively parallel performance for density-functional based tight-binding

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Hong; Zapol, Peter; Dixon, David A.

The Shift-and-invert parallel spectral transformations (SIPs), a computational approach to solve sparse eigenvalue problems, is developed for massively parallel architectures with exceptional parallel scalability and robustness. The capabilities of SIPs are demonstrated by diagonalization of density-functional based tight-binding (DFTB) Hamiltonian and overlap matrices for single-wall metallic carbon nanotubes, diamond nanowires, and bulk diamond crystals. The largest (smallest) example studied is a 128,000 (2000) atom nanotube for which ~330,000 (~5600) eigenvalues and eigenfunctions are obtained in ~190 (~5) seconds when parallelized over 266,144 (16,384) Blue Gene/Q cores. Weak scaling and strong scaling of SIPs are analyzed and the performance of SIPsmore » is compared with other novel methods. Different matrix ordering methods are investigated to reduce the cost of the factorization step, which dominates the time-to-solution at the strong scaling limit. As a result, a parallel implementation of assembling the density matrix from the distributed eigenvectors is demonstrated.« less

PCA Meets RG

NASA Astrophysics Data System (ADS)

Bradde, Serena; Bialek, William

2017-05-01

A system with many degrees of freedom can be characterized by a covariance matrix; principal components analysis focuses on the eigenvalues of this matrix, hoping to find a lower dimensional description. But when the spectrum is nearly continuous, any distinction between components that we keep and those that we ignore becomes arbitrary; it then is natural to ask what happens as we vary this arbitrary cutoff. We argue that this problem is analogous to the momentum shell renormalization group. Following this analogy, we can define relevant and irrelevant operators, where the role of dimensionality is played by properties of the eigenvalue density. These results also suggest an approach to the analysis of real data. As an example, we study neural activity in the vertebrate retina as it responds to naturalistic movies, and find evidence of behavior controlled by a nontrivial fixed point. Applied to financial data, our analysis separates modes dominated by sampling noise from a smaller but still macroscopic number of modes described by a non-Gaussian distribution.

Fast modal extraction in NASTRAN via the FEER computer program. [based on automatic matrix reduction method for lower modes of structures with many degrees of freedom

NASA Technical Reports Server (NTRS)

Newman, M. B.; Pipano, A.

1973-01-01

A new eigensolution routine, FEER (Fast Eigensolution Extraction Routine), used in conjunction with NASTRAN at Israel Aircraft Industries is described. The FEER program is based on an automatic matrix reduction scheme whereby the lower modes of structures with many degrees of freedom can be accurately extracted from a tridiagonal eigenvalue problem whose size is of the same order of magnitude as the number of required modes. The process is effected without arbitrary lumping of masses at selected node points or selection of nodes to be retained in the analysis set. The results of computational efficiency studies are presented, showing major arithmetic operation counts and actual computer run times of FEER as compared to other methods of eigenvalue extraction, including those available in the NASTRAN READ module. It is concluded that the tridiagonal reduction method used in FEER would serve as a valuable addition to NASTRAN for highly increased efficiency in obtaining structural vibration modes.

Thermal and Mechanical Buckling Analysis of Hypersonic Aircraft Hat-Stiffened Panels With Varying Face Sheet Geometry and Fiber Orientation

NASA Technical Reports Server (NTRS)

Ko, William L.

1996-01-01

Mechanical and thermal buckling behavior of monolithic and metal-matrix composite hat-stiffened panels were investigated. The panels have three types of face-sheet geometry: Flat face sheet, microdented face sheet, and microbulged face sheet. The metal-matrix composite panels have three types of face-sheet layups, each of which is combined with various types of hat composite layups. Finite-element method was used in the eigenvalue extractions for both mechanical and thermal buckling. The thermal buckling analysis required both eigenvalue and material property iterations. Graphical methods of the dual iterations are shown. The mechanical and thermal buckling strengths of the hat-stiffened panels with different face-sheet geometry are compared. It was found that by just microdenting or microbulging of the face sheet, the axial, shear, and thermal buckling strengths of both types of hat-stiffened panels could be enhanced considerably. This effect is more conspicuous for the monolithic panels. For the metal-matrix composite panels, the effect of fiber orientations on the panel buckling strengths was investigated in great detail, and various composite layup combinations offering, high panel buckling strengths are presented. The axial buckling strength of the metal-matrix panel was sensitive to the change of hat fiber orientation. However, the lateral, shear, and thermal buckling strengths were insensitive to the change of hat fiber orientation.

An analytical method for free vibration analysis of functionally graded beams with edge cracks

NASA Astrophysics Data System (ADS)

Wei, Dong; Liu, Yinghua; Xiang, Zhihai

2012-03-01

In this paper, an analytical method is proposed for solving the free vibration of cracked functionally graded material (FGM) beams with axial loading, rotary inertia and shear deformation. The governing differential equations of motion for an FGM beam are established and the corresponding solutions are found first. The discontinuity of rotation caused by the cracks is simulated by means of the rotational spring model. Based on the transfer matrix method, then the recurrence formula is developed to get the eigenvalue equations of free vibration of FGM beams. The main advantage of the proposed method is that the eigenvalue equation for vibrating beams with an arbitrary number of cracks can be conveniently determined from a third-order determinant. Due to the decrease in the determinant order as compared with previous methods, the developed method is simpler and more convenient to analytically solve the free vibration problem of cracked FGM beams. Moreover, free vibration analyses of the Euler-Bernoulli and Timoshenko beams with any number of cracks can be conducted using the unified procedure based on the developed method. These advantages of the proposed procedure would be more remarkable as the increase of the number of cracks. A comprehensive analysis is conducted to investigate the influences of the location and total number of cracks, material properties, axial load, inertia and end supports on the natural frequencies and vibration mode shapes of FGM beams. The present work may be useful for the design and control of damaged structures.

An experimental SMI adaptive antenna array simulator for weak interfering signals

NASA Technical Reports Server (NTRS)

Dilsavor, Ronald S.; Gupta, Inder J.

1991-01-01

An experimental sample matrix inversion (SMI) adaptive antenna array for suppressing weak interfering signals is described. The experimental adaptive array uses a modified SMI algorithm to increase the interference suppression. In the modified SMI algorithm, the sample covariance matrix is redefined to reduce the effect of thermal noise on the weights of an adaptive array. This is accomplished by subtracting a fraction of the smallest eigenvalue of the original covariance matrix from its diagonal entries. The test results obtained using the experimental system are compared with theoretical results. The two show a good agreement.

Simple derivation of the Lindblad equation

NASA Astrophysics Data System (ADS)

Pearle, Philip

2012-07-01

The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is ‘simple’ in that all it uses is the expression of a Hermitian matrix in terms of its orthonormal eigenvectors and real eigenvalues. Thus, it is appropriate for students who have learned the algebra of quantum theory. Where helpful, arguments are first given in a two-dimensional Hilbert space.

Random matrix theory and portfolio optimization in Moroccan stock exchange

NASA Astrophysics Data System (ADS)

El Alaoui, Marwane

2015-09-01

In this work, we use random matrix theory to analyze eigenvalues and see if there is a presence of pertinent information by using Marčenko-Pastur distribution. Thus, we study cross-correlation among stocks of Casablanca Stock Exchange. Moreover, we clean correlation matrix from noisy elements to see if the gap between predicted risk and realized risk would be reduced. We also analyze eigenvectors components distributions and their degree of deviations by computing the inverse participation ratio. This analysis is a way to understand the correlation structure among stocks of Casablanca Stock Exchange portfolio.

Introduction to the Application of the Dynalist Computer Program to the Analysis of Rail Systems Dynamics

DOT National Transportation Integrated Search

1974-08-01

DYNALIST, a computer program that extracts complex eigenvalues and eigenvectors for dynamic systems described in terms of matrix equations of motion, has been acquired and made operational at TSC. In this report, simple dynamic systems are used to de...

Biological Applications in the Mathematics Curriculum

ERIC Educational Resources Information Center

Marland, Eric; Palmer, Katrina M.; Salinas, Rene A.

2008-01-01

In this article we provide two detailed examples of how we incorporate biological examples into two mathematics courses: Linear Algebra and Ordinary Differential Equations. We use Leslie matrix models to demonstrate the biological properties of eigenvalues and eigenvectors. For Ordinary Differential Equations, we show how using a logistic growth…

Linear quadratic regulators with eigenvalue placement in a horizontal strip

NASA Technical Reports Server (NTRS)

Shieh, Leang S.; Dib, Hani M.; Ganesan, Sekar

1987-01-01

A method for optimally shifting the imaginary parts of the open-loop poles of a multivariable control system to the desirable closed-loop locations is presented. The optimal solution with respect to a quadratic performance index is obtained by solving a linear matrix Liapunov equation.

Sampled-Data Consensus of Linear Multi-agent Systems With Packet Losses.

PubMed

Zhang, Wenbing; Tang, Yang; Huang, Tingwen; Kurths, Jurgen

In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.In this paper, the consensus problem is studied for a class of multi-agent systems with sampled data and packet losses, where random and deterministic packet losses are considered, respectively. For random packet losses, a Bernoulli-distributed white sequence is used to describe packet dropouts among agents in a stochastic way. For deterministic packet losses, a switched system with stable and unstable subsystems is employed to model packet dropouts in a deterministic way. The purpose of this paper is to derive consensus criteria, such that linear multi-agent systems with sampled-data and packet losses can reach consensus. By means of the Lyapunov function approach and the decomposition method, the design problem of a distributed controller is solved in terms of convex optimization. The interplay among the allowable bound of the sampling interval, the probability of random packet losses, and the rate of deterministic packet losses are explicitly derived to characterize consensus conditions. The obtained criteria are closely related to the maximum eigenvalue of the Laplacian matrix versus the second minimum eigenvalue of the Laplacian matrix, which reveals the intrinsic effect of communication topologies on consensus performance. Finally, simulations are given to show the effectiveness of the proposed results.

Linking matrices in systems with periodic boundary conditions

NASA Astrophysics Data System (ADS)

Panagiotou, Eleni; Millett, Kenneth C.

2018-06-01

We study the linking matrix, a measure of entanglement for a collection of closed or open chains in 3-space based on the Gauss linking number. Periodic boundary conditions (PBC) are often used in the simulation of physical systems of filaments. To measure entanglement of closed or open chains in systems employing PBC we use the periodic linking matrix, based on the periodic linking number, defined in Panagiotou (2015 J. Comput. Phys. 300 533–73). We study the properties of the periodic linking matrix as a function of cell size. We provide analytical results concerning the eigenvalues of the periodic linking matrix and show that some of them are invariant of cell-size.

World currency exchange rate cross-correlations

NASA Astrophysics Data System (ADS)

Droå¼dż, S.; Górski, A. Z.; Kwapień, J.

2007-08-01

World currency network constitutes one of the most complex structures that is associated with the contemporary civilization. On a way towards quantifying its characteristics we study the cross correlations in changes of the daily foreign exchange rates within the basket of 60 currencies in the period December 1998 May 2005. Such a dynamics turns out to predominantly involve one outstanding eigenvalue of the correlation matrix. The magnitude of this eigenvalue depends however crucially on which currency is used as a base currency for the remaining ones. Most prominent it looks from the perspective of a peripheral currency. This largest eigenvalue is seen to systematically decrease and thus the structure of correlations becomes more heterogeneous, when more significant currencies are used as reference. An extreme case in this later respect is the USD in the period considered. Besides providing further insight into subtle nature of complexity, these observations point to a formal procedure that in general can be used for practical purposes of measuring the relative currencies significance on various time horizons.

Volatility and correlation-based systemic risk measures in the US market

NASA Astrophysics Data System (ADS)

Civitarese, Jamil

2016-10-01

This paper deals with the problem of how to use simple systemic risk measures to assess portfolio risk characteristics. Using three simple examples taken from previous literature, one based on raw and partial correlations, another based on the eigenvalue decomposition of the covariance matrix and the last one based on an eigenvalue entropy, a Granger-causation analysis revealed some of them are not always a good measure of risk in the S&P 500 and in the VIX. The measures selected do not Granger-cause the VIX index in all windows selected; therefore, in the sense of risk as volatility, the indicators are not always suitable. Nevertheless, their results towards returns are similar to previous works that accept them. A deeper analysis has shown that any symmetric measure based on eigenvalue decomposition of correlation matrices, however, is not useful as a measure of "correlation" risk. The empirical counterpart analysis of this proposition stated that negative correlations are usually small and, therefore, do not heavily distort the behavior of the indicator.

Some Results on Proper Eigenvalues and Eigenvectors with Applications to Scaling.

ERIC Educational Resources Information Center

McDonald, Roderick P.; And Others

1979-01-01

Problems in avoiding the singularity problem in analyzing matrices for optimal scaling are addressed. Conditions are given under which the stationary points and values of a ratio of quadratic forms in two singular matrices can be obtained by a series of simple matrix operations. (Author/JKS)

Quasinormal modes of Reissner-Nordstrom black holes

NASA Technical Reports Server (NTRS)

Leaver, Edward W.

1990-01-01

A matrix-eigenvalue algorithm is presented for accurately computing the quasi-normal frequencies and modes of charged static blackholes. The method is then refined through the introduction of a continued-fraction step. The approach should generalize to a variety of nonseparable wave equations, including the Kerr-Newman case of charged rotating blackholes.

Resonant-state expansion for open optical systems: generalization to magnetic, chiral, and bi-anisotropic materials

NASA Astrophysics Data System (ADS)

Muljarov, E. A.; Weiss, T.

2018-05-01

The resonant-state expansion, a recently developed powerful method in electrodynamics, is generalized here for open optical systems containing magnetic, chiral, or bi-anisotropic materials. It is shown that the key matrix eigenvalue equation of the method remains the same, but the matrix elements of the perturbation now contain variations of the permittivity, permeability, and bi-anisotropy tensors. A general normalization of resonant states in terms of the electric and magnetic fields is presented.

Generic Friedberg-Lee symmetry of Dirac neutrinos

DOE Office of Scientific and Technical Information (OSTI.GOV)

Luo Shu; Xing Zhizhong; Li Xin

2008-12-01

We write out the generic Dirac neutrino mass operator which possesses the Friedberg-Lee symmetry and find that its corresponding neutrino mass matrix is asymmetric. Following a simple way to break the Friedberg-Lee symmetry, we calculate the neutrino mass eigenvalues and show that the resultant neutrino mixing pattern is nearly tri-bimaximal. Imposing the Hermitian condition on the neutrino mass matrix, we also show that the simplified ansatz is consistent with current experimental data and favors the normal neutrino mass hierarchy.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smed, T.

Traditional eigenvalue sensitivity for power systems requires the formulation of the system matrix, which lacks sparsity. In this paper, a new sensitivity analysis, derived for a sparse formulation, is presented. Variables that are computed as intermediate results in established eigen value programs for power systems, but not used further, are given a new interpretation. The effect of virtually any control action can be assessed based on a single eigenvalue-eigenvector calculation. In particular, the effect of active and reactive power modulation can be found as a multiplication of two or three complex numbers. The method is illustrated in an example formore » a large power system when applied to the control design for an HVDC-link.« less

Instability of the cored barotropic disc: the linear eigenvalue formulation

NASA Astrophysics Data System (ADS)

Polyachenko, E. V.

2018-05-01

Gaseous rotating razor-thin discs are a testing ground for theories of spiral structure that try to explain appearance and diversity of disc galaxy patterns. These patterns are believed to arise spontaneously under the action of gravitational instability, but calculations of its characteristics in the gas are mostly obscured. The paper suggests a new method for finding the spiral patterns based on an expansion of small amplitude perturbations over Lagrange polynomials in small radial elements. The final matrix equation is extracted from the original hydrodynamical equations without the use of an approximate theory and has a form of the linear algebraic eigenvalue problem. The method is applied to a galactic model with the cored exponential density profile.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sun, Yongzheng, E-mail: yzsung@gmail.com; Li, Wang; Zhao, Donghua

In this paper, we propose a new consensus model in which the interactions among agents stochastically switch between attraction and repulsion. Such a positive-and-negative mechanism is described by the white-noise-based coupling. Analytic criteria for the consensus and non-consensus in terms of the eigenvalues of the noise intensity matrix are derived, which provide a better understanding of the constructive roles of random interactions. Specifically, we discover a positive role of noise coupling that noise can accelerate the emergence of consensus. We find that the converging speed of the multi-agent network depends on the square of the second smallest eigenvalue of itsmore » graph Laplacian. The influence of network topologies on the consensus time is also investigated.« less

Phase portraits of the full symmetric Toda systems on rank-2 groups

NASA Astrophysics Data System (ADS)

Sorin, A. S.; Chernyakov, Yu. B.; Sharygin, G. I.

2017-11-01

We continue investigations begun in our previous works where we proved that the phase diagram of the Toda system on special linear groups can be identified with the Bruhat order on the symmetric group if all eigenvalues of the Lax matrix are distinct or with the Bruhat order on permutations of a multiset if there are multiple eigenvalues. We show that the phase portrait of the Toda system and the Hasse diagram of the Bruhat order coincide in the case of an arbitrary simple Lie group of rank 2. For this, we verify this property for the two remaining rank-2 groups, Sp(4,ℝ) and the real form of G2.

Brain vascular image segmentation based on fuzzy local information C-means clustering

NASA Astrophysics Data System (ADS)

Hu, Chaoen; Liu, Xia; Liang, Xiao; Hui, Hui; Yang, Xin; Tian, Jie

2017-02-01

Light sheet fluorescence microscopy (LSFM) is a powerful optical resolution fluorescence microscopy technique which enables to observe the mouse brain vascular network in cellular resolution. However, micro-vessel structures are intensity inhomogeneity in LSFM images, which make an inconvenience for extracting line structures. In this work, we developed a vascular image segmentation method by enhancing vessel details which should be useful for estimating statistics like micro-vessel density. Since the eigenvalues of hessian matrix and its sign describes different geometric structure in images, which enable to construct vascular similarity function and enhance line signals, the main idea of our method is to cluster the pixel values of the enhanced image. Our method contained three steps: 1) calculate the multiscale gradients and the differences between eigenvalues of Hessian matrix. 2) In order to generate the enhanced microvessels structures, a feed forward neural network was trained by 2.26 million pixels for dealing with the correlations between multi-scale gradients and the differences between eigenvalues. 3) The fuzzy local information c-means clustering (FLICM) was used to cluster the pixel values in enhance line signals. To verify the feasibility and effectiveness of this method, mouse brain vascular images have been acquired by a commercial light-sheet microscope in our lab. The experiment of the segmentation method showed that dice similarity coefficient can reach up to 85%. The results illustrated that our approach extracting line structures of blood vessels dramatically improves the vascular image and enable to accurately extract blood vessels in LSFM images.

Preconditioning for the Navier-Stokes equations with finite-rate chemistry

NASA Technical Reports Server (NTRS)

Godfrey, Andrew G.

1993-01-01

The extension of Van Leer's preconditioning procedure to generalized finite-rate chemistry is discussed. Application to viscous flow is begun with the proper preconditioning matrix for the one-dimensional Navier-Stokes equations. Eigenvalue stiffness is resolved and convergence-rate acceleration is demonstrated over the entire Mach-number range from nearly stagnant flow to hypersonic. Specific benefits are realized at the low and transonic flow speeds typical of complete propulsion-system simulations. The extended preconditioning matrix necessarily accounts for both thermal and chemical nonequilibrium. Numerical analysis reveals the possible theoretical improvements from using a preconditioner for all Mach number regimes. Numerical results confirm the expectations from the numerical analysis. Representative test cases include flows with previously troublesome embedded high-condition-number areas. Van Leer, Lee, and Roe recently developed an optimal, analytic preconditioning technique to reduce eigenvalue stiffness over the full Mach-number range. By multiplying the flux-balance residual with the preconditioning matrix, the acoustic wave speeds are scaled so that all waves propagate at the same rate, an essential property to eliminate inherent eigenvalue stiffness. This session discusses a synthesis of the thermochemical nonequilibrium flux-splitting developed by Grossman and Cinnella and the characteristic wave preconditioning of Van Leer into a powerful tool for implicitly solving two and three-dimensional flows with generalized finite-rate chemistry. For finite-rate chemistry, the state vector of unknowns is variable in length. Therefore, the preconditioning matrix extended to generalized finite-rate chemistry must accommodate a flexible system of moving waves. Fortunately, no new kind of wave appears in the system. The only existing waves are entropy and vorticity waves, which move with the fluid, and acoustic waves, which propagate in Mach number dependent directions. The nonequilibrium vibrational energies and species densities in the unknown state vector act strictly as convective waves. The essential concept for extending the preconditioning to generalized chemistry models is determining the differential variables which symmetrize the flux Jacobians. The extension is then straight-forward. This algorithm research effort will be released in a future version of the production level computational code coined the General Aerodynamic Simulation Program (GASP), developed by Walters, Slack, and McGrory.

A comparison of linear approaches to filter out environmental effects in structural health monitoring

NASA Astrophysics Data System (ADS)

Deraemaeker, A.; Worden, K.

2018-05-01

This paper discusses the possibility of using the Mahalanobis squared-distance to perform robust novelty detection in the presence of important environmental variability in a multivariate feature vector. By performing an eigenvalue decomposition of the covariance matrix used to compute that distance, it is shown that the Mahalanobis squared-distance can be written as the sum of independent terms which result from a transformation from the feature vector space to a space of independent variables. In general, especially when the size of the features vector is large, there are dominant eigenvalues and eigenvectors associated with the covariance matrix, so that a set of principal components can be defined. Because the associated eigenvalues are high, their contribution to the Mahalanobis squared-distance is low, while the contribution of the other components is high due to the low value of the associated eigenvalues. This analysis shows that the Mahalanobis distance naturally filters out the variability in the training data. This property can be used to remove the effect of the environment in damage detection, in much the same way as two other established techniques, principal component analysis and factor analysis. The three techniques are compared here using real experimental data from a wooden bridge for which the feature vector consists in eigenfrequencies and modeshapes collected under changing environmental conditions, as well as damaged conditions simulated with an added mass. The results confirm the similarity between the three techniques and the ability to filter out environmental effects, while keeping a high sensitivity to structural changes. The results also show that even after filtering out the environmental effects, the normality assumption cannot be made for the residual feature vector. An alternative is demonstrated here based on extreme value statistics which results in a much better threshold which avoids false positives in the training data, while allowing detection of all damaged cases.

Matrix elements for type 1 unitary irreducible representations of the Lie superalgebra gl(m|n)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gould, Mark D.; Isaac, Phillip S.; Werry, Jason L.

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary generators, on finite dimensional type 1 unitary irreducible representations. We compare our results with existing works that deal with only subsets of the class of type 1 unitary representations, all of which only present explicit matrix elements for elementary generators. Our work therefore provides an important extension to existing methods, and thus highlights the strength of our techniques which exploit the characteristic identities.

Parallel-vector unsymmetric Eigen-Solver on high performance computers

NASA Technical Reports Server (NTRS)

Nguyen, Duc T.; Jiangning, Qin

1993-01-01

The popular QR algorithm for solving all eigenvalues of an unsymmetric matrix is reviewed. Among the basic components in the QR algorithm, it was concluded from this study, that the reduction of an unsymmetric matrix to a Hessenberg form (before applying the QR algorithm itself) can be done effectively by exploiting the vector speed and multiple processors offered by modern high-performance computers. Numerical examples of several test cases have indicated that the proposed parallel-vector algorithm for converting a given unsymmetric matrix to a Hessenberg form offers computational advantages over the existing algorithm. The time saving obtained by the proposed methods is increased as the problem size increased.

On-Line Identification of Simulation Examples for Forgetting Methods to Track Time Varying Parameters Using the Alternative Covariance Matrix in Matlab

NASA Astrophysics Data System (ADS)

Vachálek, Ján

2011-12-01

The paper compares the abilities of forgetting methods to track time varying parameters of two different simulated models with different types of excitation. The observed parameters in the simulations are the integral sum of the Euclidean norm, deviation of the parameter estimates from their true values and a selected band prediction error count. As supplementary information, we observe the eigenvalues of the covariance matrix. In the paper we used a modified method of Regularized Exponential Forgetting with Alternative Covariance Matrix (REFACM) along with Directional Forgetting (DF) and three standard regularized methods.

Robust Assignment Of Eigensystems For Flexible Structures

NASA Technical Reports Server (NTRS)

Juang, Jer-Nan; Lim, Kyong B.; Junkins, John L.

1992-01-01

Improved method for placement of eigenvalues and eigenvectors of closed-loop control system by use of either state or output feedback. Applied to reduced-order finite-element mathematical model of NASA's MAST truss beam structure. Model represents deployer/retractor assembly, inertial properties of Space Shuttle, and rigid platforms for allocation of sensors and actuators. Algorithm formulated in real arithmetic for efficient implementation. Choice of open-loop eigenvector matrix and its closest unitary matrix believed suitable for generating well-conditioned eigensystem with small control gains. Implication of this approach is that element of iterative search for "optimal" unitary matrix appears unnecessary in practice for many test problems.

Golden Matrix Families

ERIC Educational Resources Information Center

Fontaine, Anne; Hurley, Susan

2011-01-01

This student research project explores the properties of a family of matrices of zeros and ones that arises from the study of the diagonal lengths in a regular polygon. There is one family for each n greater than 2. A series of exercises guides the student to discover the eigenvalues and eigenvectors of the matrices, which leads in turn to…

Finite-range Coulomb gas models of banded random matrices and quantum kicked rotors

NASA Astrophysics Data System (ADS)

Pandey, Akhilesh; Kumar, Avanish; Puri, Sanjay

2017-11-01

Dyson demonstrated an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. We introduce finite-range Coulomb gas (FRCG) models via a Brownian matrix process, and study them analytically and by Monte Carlo simulations. These models yield new universality classes, and provide a theoretical framework for the study of banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth b and a QKR of chaos parameter α , the appropriate FRCG model has the effective range d =b2/N =α2/N , for large N matrix dimensionality. As d increases, there is a transition from Poisson to classical random matrix statistics.

Finite-range Coulomb gas models of banded random matrices and quantum kicked rotors.

PubMed

Pandey, Akhilesh; Kumar, Avanish; Puri, Sanjay

2017-11-01

Dyson demonstrated an equivalence between infinite-range Coulomb gas models and classical random matrix ensembles for the study of eigenvalue statistics. We introduce finite-range Coulomb gas (FRCG) models via a Brownian matrix process, and study them analytically and by Monte Carlo simulations. These models yield new universality classes, and provide a theoretical framework for the study of banded random matrices (BRMs) and quantum kicked rotors (QKRs). We demonstrate that, for a BRM of bandwidth b and a QKR of chaos parameter α, the appropriate FRCG model has the effective range d=b^{2}/N=α^{2}/N, for large N matrix dimensionality. As d increases, there is a transition from Poisson to classical random matrix statistics.

Symmetry Transition Preserving Chirality in QCD: A Versatile Random Matrix Model

NASA Astrophysics Data System (ADS)

Kanazawa, Takuya; Kieburg, Mario

2018-06-01

We consider a random matrix model which interpolates between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble while preserving chiral symmetry. This ensemble describes flavor symmetry breaking for staggered fermions in 3D QCD as well as in 4D QCD at high temperature or in 3D QCD at a finite isospin chemical potential. Our model is an Osborn-type two-matrix model which is equivalent to the elliptic ensemble but we consider the singular value statistics rather than the complex eigenvalue statistics. We report on exact results for the partition function and the microscopic level density of the Dirac operator in the ɛ regime of QCD. We compare these analytical results with Monte Carlo simulations of the matrix model.

Application of vector-valued rational approximations to the matrix eigenvalue problem and connections with Krylov subspace methods

NASA Technical Reports Server (NTRS)

Sidi, Avram

1992-01-01

Let F(z) be a vectored-valued function F: C approaches C sup N, which is analytic at z=0 and meromorphic in a neighborhood of z=0, and let its Maclaurin series be given. We use vector-valued rational approximation procedures for F(z) that are based on its Maclaurin series in conjunction with power iterations to develop bona fide generalizations of the power method for an arbitrary N X N matrix that may be diagonalizable or not. These generalizations can be used to obtain simultaneously several of the largest distinct eigenvalues and the corresponding invariant subspaces, and present a detailed convergence theory for them. In addition, it is shown that the generalized power methods of this work are equivalent to some Krylov subspace methods, among them the methods of Arnoldi and Lanczos. Thus, the theory provides a set of completely new results and constructions for these Krylov subspace methods. This theory suggests at the same time a new mode of usage for these Krylov subspace methods that were observed to possess computational advantages over their common mode of usage.

High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pieper, Andreas; Kreutzer, Moritz; Alvermann, Andreas, E-mail: alvermann@physik.uni-greifswald.de

2016-11-15

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need formore » matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 10{sup 2} innermost eigenpairs of a topological insulator matrix with dimension 10{sup 9} derived from quantum physics applications.« less

Analysis of cross-correlations between financial markets after the 2008 crisis

NASA Astrophysics Data System (ADS)

Sensoy, A.; Yuksel, S.; Erturk, M.

2013-10-01

We analyze the cross-correlation matrix C of the index returns of the main financial markets after the 2008 crisis using methods of random matrix theory. We test the eigenvalues of C for universal properties of random matrices and find that the majority of the cross-correlation coefficients arise from randomness. We show that the eigenvector of the largest deviating eigenvalue of C represents a global market itself. We reveal that high volatility of financial markets is observed at the same times with high correlations between them which lowers the risk diversification potential even if one constructs a widely internationally diversified portfolio of stocks. We identify and compare the connection and cluster structure of markets before and after the crisis using minimal spanning and ultrametric hierarchical trees. We find that after the crisis, the co-movement degree of the markets increases. We also highlight the key financial markets of pre and post crisis using main centrality measures and analyze the changes. We repeat the study using rank correlation and compare the differences. Further implications are discussed.

Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides

DOE Office of Scientific and Technical Information (OSTI.GOV)

Abdel-Rehim, A M; Stathopoulos, Andreas; Orginos, Kostas

2014-08-01

The technique that was used to build the EigCG algorithm for sparse symmetric linear systems is extended to the nonsymmetric case using the BiCG algorithm. We show that, similarly to the symmetric case, we can build an algorithm that is capable of computing a few smallest magnitude eigenvalues and their corresponding left and right eigenvectors of a nonsymmetric matrix using only a small window of the BiCG residuals while simultaneously solving a linear system with that matrix. For a system with multiple right-hand sides, we give an algorithm that computes incrementally more eigenvalues while solving the first few systems andmore » then uses the computed eigenvectors to deflate BiCGStab for the remaining systems. Our experiments on various test problems, including Lattice QCD, show the remarkable ability of EigBiCG to compute spectral approximations with accuracy comparable to that of the unrestarted, nonsymmetric Lanczos. Furthermore, our incremental EigBiCG followed by appropriately restarted and deflated BiCGStab provides a competitive method for systems with multiple right-hand sides.« less

Current density characteristics in the studies of electromagnetically induced transparency in a GaAs/GaAlAs quantum well

NASA Astrophysics Data System (ADS)

Jayarubi, J.; Peter, A. John

2017-05-01

Confinement potential profiles due to conduction and valence bands are obtained in a Ga0.7Al0.3As/ GaAs/ Ga0.7Al0.3As using variation formulism. The free electron distribution is carried out. The confined energy eigenvalue and its corresponding wavefunctions of charge carriers are found using self-consistent method. The confined energies with the geometrical confinement are computed. The potentials due to charges are done by Poisson equation. The effects of dielectric mismatch between the GaAs and GaAlAs semiconductors are introduced in the effective potential expressions. Transfer matrix method is employed to obtain the respective energies. The transmission probability is obtained for a constant well size. The high current density characteristics as a function of applied voltage is investigated. This investigation on the electromagnetically induced transparency in the photonic material will exploit in fabricating novel nonlinear optical devices in future.

Krylov subspace methods - Theory, algorithms, and applications

NASA Technical Reports Server (NTRS)

Sad, Youcef

1990-01-01

Projection methods based on Krylov subspaces for solving various types of scientific problems are reviewed. The main idea of this class of methods when applied to a linear system Ax = b, is to generate in some manner an approximate solution to the original problem from the so-called Krylov subspace span. Thus, the original problem of size N is approximated by one of dimension m, typically much smaller than N. Krylov subspace methods have been very successful in solving linear systems and eigenvalue problems and are now becoming popular for solving nonlinear equations. The main ideas in Krylov subspace methods are shown and their use in solving linear systems, eigenvalue problems, parabolic partial differential equations, Liapunov matrix equations, and nonlinear system of equations are discussed.

A parametric method for determining the number of signals in narrow-band direction finding

NASA Astrophysics Data System (ADS)

Wu, Qiang; Fuhrmann, Daniel R.

1991-08-01

A novel and more accurate method to determine the number of signals in the multisource direction finding problem is developed. The information-theoretic criteria of Yin and Krishnaiah (1988) are applied to a set of quantities which are evaluated from the log-likelihood function. Based on proven asymptotic properties of the maximum likelihood estimation, these quantities have the properties required by the criteria. Since the information-theoretic criteria use these quantities instead of the eigenvalues of the estimated correlation matrix, this approach possesses the advantage of not requiring a subjective threshold, and also provides higher performance than when eigenvalues are used. Simulation results are presented and compared to those obtained from the nonparametric method given by Wax and Kailath (1985).

Gaussian quadrature for multiple orthogonal polynomials

NASA Astrophysics Data System (ADS)

Coussement, Jonathan; van Assche, Walter

2005-06-01

We study multiple orthogonal polynomials of type I and type II, which have orthogonality conditions with respect to r measures. These polynomials are connected by their recurrence relation of order r+1. First we show a relation with the eigenvalue problem of a banded lower Hessenberg matrix Ln, containing the recurrence coefficients. As a consequence, we easily find that the multiple orthogonal polynomials of type I and type II satisfy a generalized Christoffel-Darboux identity. Furthermore, we explain the notion of multiple Gaussian quadrature (for proper multi-indices), which is an extension of the theory of Gaussian quadrature for orthogonal polynomials and was introduced by Borges. In particular, we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of Ln.

Quantization of an electromagnetic field in two-dimensional photonic structures based on the scattering matrix formalism ( S-quantization)

NASA Astrophysics Data System (ADS)

Ivanov, K. A.; Nikolaev, V. V.; Gubaydullin, A. R.; Kaliteevski, M. A.

2017-10-01

Based on the scattering matrix formalism, we have developed a method of quantization of an electromagnetic field in two-dimensional photonic nanostructures ( S-quantization in the two-dimensional case). In this method, the fields at the boundaries of the quantization box are expanded into a Fourier series and are related with each other by the scattering matrix of the system, which is the product of matrices describing the propagation of plane waves in empty regions of the quantization box and the scattering matrix of the photonic structure (or an arbitrary inhomogeneity). The quantization condition (similarly to the onedimensional case) is formulated as follows: the eigenvalues of the scattering matrix are equal to unity, which corresponds to the fact that the set of waves that are incident on the structure (components of the expansion into the Fourier series) is equal to the set of waves that travel away from the structure (outgoing waves). The coefficients of the matrix of scattering through the inhomogeneous structure have been calculated using the following procedure: the structure is divided into parallel layers such that the permittivity in each layer varies only along the axis that is perpendicular to the layers. Using the Fourier transform, the Maxwell equations have been written in the form of a matrix that relates the Fourier components of the electric field at the boundaries of neighboring layers. The product of these matrices is the transfer matrix in the basis of the Fourier components of the electric field. Represented in a block form, it is composed by matrices that contain the reflection and transmission coefficients for the Fourier components of the field, which, in turn, constitute the scattering matrix. The developed method considerably simplifies the calculation scheme for the analysis of the behavior of the electromagnetic field in structures with a two-dimensional inhomogeneity. In addition, this method makes it possible to obviate difficulties that arise in the analysis of the Purcell effect because of the divergence of the integral describing the effective volume of the mode in open systems.

Estimation and Control with Relative Measurements: Algorithms and Scaling Laws

DTIC Science & Technology

2007-09-01

eigenvector of L −1 corre- sponding to its largest eigenvalue. Since L−1 is a positive matrix, Perron - Frobenius theory tells us that |u1| := {|u11...the Frobenius norm of a matrix, and a linear vector space SV as the space of all bounded node-functions with respect to the above defined 144 norm...je‖2F where Eu is the set edges in E that are incident on u. It can be shown from the relationship between the Frobenius norm and the singular

Algorithms for the Equilibration of Matrices and Their Application to Limited-Memory Quasi-Newton Methods

DTIC Science & Technology

2010-05-01

irreducible, by the Perron - Frobenius theorem (see, for example, Theorem 8.4.4 in [28]), the eigenvalue 1 is simple. Next, the rank-one matrix Q has the...We refer to (2.1) as the scaling equation. Although algorithms must use A, existence and unique- ness theory need consider only the nonnegative matrix...B. If p = 1 and A is nonnegative , then A = B. We reserve the term binormalization for the case p = 2. We say A is scalable if there exists x > 0

Vibration control of large linear quadratic symmetric systems. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Jeon, G. J.

1983-01-01

Some unique properties on a class of the second order lambda matrices were found and applied to determine a damping matrix of the decoupled subsystem in such a way that the damped system would have preassigned eigenvalues without disturbing the stiffness matrix. The resulting system was realized as a time invariant velocity only feedback control system with desired poles. Another approach using optimal control theory was also applied to the decoupled system in such a way that the mode spillover problem could be eliminated. The procedures were tested successfully by numerical examples.

M-matrices with prescribed elementary divisors

NASA Astrophysics Data System (ADS)

Soto, Ricardo L.; Díaz, Roberto C.; Salas, Mario; Rojo, Oscar

2017-09-01

A real matrix A is said to be an M-matrix if it is of the form A=α I-B, where B is a nonnegative matrix with Perron eigenvalue ρ (B), and α ≥slant ρ (B) . This paper provides sufficient conditions for the existence and construction of an M-matrix A with prescribed elementary divisors, which are the characteristic polynomials of the Jordan blocks of the Jordan canonical form of A. This inverse problem on M-matrices has not been treated until now. We solve the inverse elementary divisors problem for diagonalizable M-matrices and the symmetric generalized doubly stochastic inverse M-matrix problem for lists of real numbers and for lists of complex numbers of the form Λ =\\{λ 1, a+/- bi, \\ldots, a+/- bi\\} . The constructive nature of our results allows for the computation of a solution matrix. The paper also discusses an application of M-matrices to a capacity problem in wireless communications.

Acoustooptic linear algebra processors - Architectures, algorithms, and applications

NASA Technical Reports Server (NTRS)

Casasent, D.

1984-01-01

Architectures, algorithms, and applications for systolic processors are described with attention to the realization of parallel algorithms on various optical systolic array processors. Systolic processors for matrices with special structure and matrices of general structure, and the realization of matrix-vector, matrix-matrix, and triple-matrix products and such architectures are described. Parallel algorithms for direct and indirect solutions to systems of linear algebraic equations and their implementation on optical systolic processors are detailed with attention to the pipelining and flow of data and operations. Parallel algorithms and their optical realization for LU and QR matrix decomposition are specifically detailed. These represent the fundamental operations necessary in the implementation of least squares, eigenvalue, and SVD solutions. Specific applications (e.g., the solution of partial differential equations, adaptive noise cancellation, and optimal control) are described to typify the use of matrix processors in modern advanced signal processing.

Contrasting eigenvalue and singular-value spectra for lasing and antilasing in a PT -symmetric periodic structure

NASA Astrophysics Data System (ADS)

Ge, Li; Feng, Liang

2017-01-01

It has been proposed and demonstrated that lasing and coherent perfect absorption (CPA or "antilasing") coexist in parity-time (PT ) symmetric photonic systems. In this work we show that the spectral signature of such a CPA laser displayed by the singular value spectrum of the scattering matrix (S ) can be orders of magnitude wider than that displayed by the eigenvalue spectrum of S . Since the former reflects how strongly light can be absorbed or amplified and the latter announces the spontaneous symmetry breaking of S , these contrasting spectral signatures indicate that near perfect absorption and extremely strong amplification can be achieved even in the PT -symmetric phase of S , which is known for and defined by its flux-conserving eigenstates. We also show that these contrasting spectral signatures are accompanied by strikingly different sensitivities to disorder and imperfection, suggesting that the eigenvalue spectrum is potentially suitable for sensing and the singular value spectrum for robust switching. A differential light amplifier may also be devised based on these two spectra.

Singular value decomposition utilizing parallel algorithms on graphical processors

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kotas, Charlotte W; Barhen, Jacob

2011-01-01

One of the current challenges in underwater acoustic array signal processing is the detection of quiet targets in the presence of noise. In order to enable robust detection, one of the key processing steps requires data and replica whitening. This, in turn, involves the eigen-decomposition of the sample spectral matrix, Cx = 1/K xKX(k)XH(k) where X(k) denotes a single frequency snapshot with an element for each element of the array. By employing the singular value decomposition (SVD) method, the eigenvectors and eigenvalues can be determined directly from the data without computing the sample covariance matrix, reducing the computational requirements formore » a given level of accuracy (van Trees, Optimum Array Processing). (Recall that the SVD of a complex matrix A involves determining V, , and U such that A = U VH where U and V are orthonormal and is a positive, real, diagonal matrix containing the singular values of A. U and V are the eigenvectors of AAH and AHA, respectively, while the singular values are the square roots of the eigenvalues of AAH.) Because it is desirable to be able to compute these quantities in real time, an efficient technique for computing the SVD is vital. In addition, emerging multicore processors like graphical processing units (GPUs) are bringing parallel processing capabilities to an ever increasing number of users. Since the computational tasks involved in array signal processing are well suited for parallelization, it is expected that these computations will be implemented using GPUs as soon as users have the necessary computational tools available to them. Thus, it is important to have an SVD algorithm that is suitable for these processors. This work explores the effectiveness of two different parallel SVD implementations on an NVIDIA Tesla C2050 GPU (14 multiprocessors, 32 cores per multiprocessor, 1.15 GHz clock - peed). The first algorithm is based on a two-step algorithm which bidiagonalizes the matrix using Householder transformations, and then diagonalizes the intermediate bidiagonal matrix through implicit QR shifts. This is similar to that implemented for real matrices by Lahabar and Narayanan ("Singular Value Decomposition on GPU using CUDA", IEEE International Parallel Distributed Processing Symposium 2009). The implementation is done in a hybrid manner, with the bidiagonalization stage done using the GPU while the diagonalization stage is done using the CPU, with the GPU used to update the U and V matrices. The second algorithm is based on a one-sided Jacobi scheme utilizing a sequence of pair-wise column orthogonalizations such that A is replaced by AV until the resulting matrix is sufficiently orthogonal (that is, equal to U ). V is obtained from the sequence of orthogonalizations, while can be found from the square root of the diagonal elements of AH A and, once is known, U can be found from column scaling the resulting matrix. These implementations utilize CUDA Fortran and NVIDIA's CUB LAS library. The primary goal of this study is to quantify the comparative performance of these two techniques against themselves and other standard implementations (for example, MATLAB). Considering that there is significant overhead associated with transferring data to the GPU and with synchronization between the GPU and the host CPU, it is also important to understand when it is worthwhile to use the GPU in terms of the matrix size and number of concurrent SVDs to be calculated.« less

The use of complete sets of orthogonal operators in spectroscopic studies

NASA Astrophysics Data System (ADS)

Raassen, A. J. J.; Uylings, P. H. M.

1996-01-01

Complete sets of orthogonal operators are used to calculate eigenvalues and eigenvector compositions in complex spectra. The latter are used to transform the LS-transition matrix into realistic intermediate coupling transition probabilities. Calculated transition probabilities for some close lying levels in Ni V and Fe III illustrate the power of the complete orthogonal operator approach.

Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

NASA Technical Reports Server (NTRS)

Lakin, W. D.

1981-01-01

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

Constraints and vibrations in static packings of ellipsoidal particles.

PubMed

Schreck, Carl F; Mailman, Mitch; Chakraborty, Bulbul; O'Hern, Corey S

2012-06-01

We numerically investigate the mechanical properties of static packings of frictionless ellipsoidal particles in two and three dimensions over a range of aspect ratio and compression Δφ. While amorphous packings of spherical particles at jamming onset (Δφ=0) are isostatic and possess the minimum contact number z_{iso} required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming (z

Dimensionality of genomic information and performance of the Algorithm for Proven and Young for different livestock species.

PubMed

Pocrnic, Ivan; Lourenco, Daniela A L; Masuda, Yutaka; Misztal, Ignacy

2016-10-31

A genomic relationship matrix (GRM) can be inverted efficiently with the Algorithm for Proven and Young (APY) through recursion on a small number of core animals. The number of core animals is theoretically linked to effective population size (N e ). In a simulation study, the optimal number of core animals was equal to the number of largest eigenvalues of GRM that explained 98% of its variation. The purpose of this study was to find the optimal number of core animals and estimate N e for different species. Datasets included phenotypes, pedigrees, and genotypes for populations of Holstein, Jersey, and Angus cattle, pigs, and broiler chickens. The number of genotyped animals varied from 15,000 for broiler chickens to 77,000 for Holsteins, and the number of single-nucleotide polymorphisms used for genomic prediction varied from 37,000 to 61,000. Eigenvalue decomposition of the GRM for each population determined numbers of largest eigenvalues corresponding to 90, 95, 98, and 99% of variation. The number of eigenvalues corresponding to 90% (98%) of variation was 4527 (14,026) for Holstein, 3325 (11,500) for Jersey, 3654 (10,605) for Angus, 1239 (4103) for pig, and 1655 (4171) for broiler chicken. Each trait in each species was analyzed using the APY inverse of the GRM with randomly selected core animals, and their number was equal to the number of largest eigenvalues. Realized accuracies peaked with the number of core animals corresponding to 98% of variation for Holstein and Jersey and closer to 99% for other breed/species. N e was estimated based on comparisons of eigenvalue decomposition in a simulation study. Assuming a genome length of 30 Morgan, N e was equal to 149 for Holsteins, 101 for Jerseys, 113 for Angus, 32 for pigs, and 44 for broilers. Eigenvalue profiles of GRM for common species are similar to those in simulation studies although they are affected by number of genotyped animals and genotyping quality. For all investigated species, the APY required less than 15,000 core animals. Realized accuracies were equal or greater with the APY inverse than with regular inversion. Eigenvalue analysis of GRM can provide a realistic estimate of N e .

Three flavor neutrino oscillations in matter: Flavor diagonal potentials, the adiabatic basis, and the CP phase

NASA Astrophysics Data System (ADS)

Kneller, James P.; McLaughlin, Gail C.

2009-09-01

We discuss the three neutrino flavor evolution problem with general, flavor-diagonal, matter potentials and a fully parametrized mixing matrix that includes CP violation, and derive expressions for the eigenvalues, mixing angles, and phases. We demonstrate that, in the limit that the mu and tau potentials are equal, the eigenvalues and matter mixing angles θ˜12 and θ˜13 are independent of the CP phase, although θ˜23 does have CP dependence. Since we are interested in developing a framework that can be used for S matrix calculations of neutrino flavor transformation, it is useful to work in a basis that contains only off-diagonal entries in the Hamiltonian. We derive the “nonadiabaticity” parameters that appear in the Hamiltonian in this basis. We then introduce the neutrino S matrix, derive its evolution equation and the integral solution. We find that this new Hamiltonian, and therefore the S matrix, in the limit that the μ and τ neutrino potentials are the same, is independent of both θ˜23 and the CP violating phase. In this limit, any CP violation in the flavor basis can only be introduced via the rotation matrices, and so effects which derive from the CP phase are then straightforward to determine. We then show explicitly that the electron neutrino and electron antineutrino survival probability is independent of the CP phase in this limit. Conversely, if the CP phase is nonzero and mu and tau matter potentials are not equal, then the electron neutrino survival probability cannot be independent of the CP phase.

Sloppy-model universality class and the Vandermonde matrix.

PubMed

Waterfall, Joshua J; Casey, Fergal P; Gutenkunst, Ryan N; Brown, Kevin S; Myers, Christopher R; Brouwer, Piet W; Elser, Veit; Sethna, James P

2006-10-13

In a variety of contexts, physicists study complex, nonlinear models with many unknown or tunable parameters to explain experimental data. We explain why such systems so often are sloppy: the system behavior depends only on a few "stiff" combinations of the parameters and is unchanged as other "sloppy" parameter combinations vary by orders of magnitude. We observe that the eigenvalue spectra for the sensitivity of sloppy models have a striking, characteristic form with a density of logarithms of eigenvalues which is roughly constant over a large range. We suggest that the common features of sloppy models indicate that they may belong to a common universality class. In particular, we motivate focusing on a Vandermonde ensemble of multiparameter nonlinear models and show in one limit that they exhibit the universal features of sloppy models.

Eigenvalue assignment by minimal state-feedback gain in LTI multivariable systems

NASA Astrophysics Data System (ADS)

Ataei, Mohammad; Enshaee, Ali

2011-12-01

In this article, an improved method for eigenvalue assignment via state feedback in the linear time-invariant multivariable systems is proposed. This method is based on elementary similarity operations, and involves mainly utilisation of vector companion forms, and thus is very simple and easy to implement on a digital computer. In addition to the controllable systems, the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs. Moreover, two types of state-feedback gain matrices can be achieved by this method: (1) the numerical one, which is unique, and (2) the parametric one, in which its parameters are determined in order to achieve a gain matrix with minimum Frobenius norm. The numerical examples are presented to demonstrate the advantages of the proposed method.

Resonance Extraction from the Finite Volume

DOE Office of Scientific and Technical Information (OSTI.GOV)

Doring, Michael; Molina Peralta, Raquel

2016-06-01

The spectrum of excited hadrons becomes accessible in simulations of Quantum Chromodynamics on the lattice. Extensions of Lüscher's method allow to address multi-channel scattering problems using moving frames or modified boundary conditions to obtain more eigenvalues in finite volume. As these are at different energies, interpolations are needed to relate different eigenvalues and to help determine the amplitude. Expanding the T- or the K-matrix locally provides a controlled scheme by removing the known non-analyticities of thresholds. This can be stabilized by using Chiral Perturbation Theory. Different examples to determine resonance pole parameters and to disentangle resonances from thresholds are dis-more » cussed, like the scalar meson f0(980) and the excited baryons N(1535)1/2^- and Lambda(1405)1/2^-.« less

Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems

NASA Astrophysics Data System (ADS)

Bäcker, A.

Summary: We give an introduction to some of the numerical aspects in quantum chaos. The classical dynamics of two-dimensional area-preserving maps on the torus is illustrated using the standard map and a perturbed cat map. The quantization of area-preserving maps given by their generating function is discussed and for the computation of the eigenvalues a computer program in Python is presented. We illustrate the eigenvalue distribution for two types of perturbed cat maps, one leading to COE and the other to CUE statistics. For the eigenfunctions of quantum maps we study the distribution of the eigenvectors and compare them with the corresponding random matrix distributions. The Husimi representation allows for a direct comparison of the localization of the eigenstates in phase space with the corresponding classical structures. Examples for a perturbed cat map and the standard map with different parameters are shown. Billiard systems and the corresponding quantum billiards are another important class of systems (which are also relevant to applications, for example in mesoscopic physics). We provide a detailed exposition of the boundary integral method, which is one important method to determine the eigenvalues and eigenfunctions of the Helmholtz equation. We discuss several methods to determine the eigenvalues from the Fredholm equation and illustrate them for the stadium billiard. The occurrence of spurious solutions is discussed in detail and illustrated for the circular billiard, the stadium billiard, and the annular sector billiard. We emphasize the role of the normal derivative function to compute the normalization of eigenfunctions, momentum representations or autocorrelation functions in a very efficient and direct way. Some examples for these quantities are given and discussed.

Data-Driven Model Reduction and Transfer Operator Approximation

NASA Astrophysics Data System (ADS)

Klus, Stefan; Nüske, Feliks; Koltai, Péter; Wu, Hao; Kevrekidis, Ioannis; Schütte, Christof; Noé, Frank

2018-06-01

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis, dynamic mode decomposition, and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods.

Spatial orientation of the vestibular system

NASA Technical Reports Server (NTRS)

Raphan, T.; Dai, M.; Cohen, B.

1992-01-01

1. A simplified three-dimensional state space model of visual vestibular interaction was formulated. Matrix and dynamical system operators representing coupling from the semicircular canals and the visual system to the velocity storage integrator were incorporated into the model. 2. It was postulated that the system matrix for a tilted position was a composition of two linear transformations of the system matrix for the upright position. One transformation modifies the eigenvalues of the system matrix while another rotates the pitch and roll eigenvectors with the head, while maintaining the yaw axis eigenvector approximately spatially invariant. Using this representation, the response characteristics of the pitch, roll, and yaw eye velocity were obtained in terms of the eigenvalues and associated eigenvectors. 3. Using OKAN data obtained from monkeys and comparing to the model predictions, the eigenvalues and eigenvectors of the system matrix were identified as a function of tilt to the side or of tilt to the prone positions, using a modification of the Marquardt algorithm. The yaw eigenvector for right-side-down tilt and for downward pitch cross-coupling was approximately 30 degrees from the spatial vertical. For the prone position, the eigenvector was computed to be approximately 20 degrees relative to the spatial vertical. For both side-down and prone positions, oblique OKN induced along eigenvector directions generated OKAN which decayed to zero along a straight line with approximately a single time constant. This was verified by a spectral analysis of the residual sequence about the straight line fit to the decaying data. The residual sequence was associated with a narrow autocorrelation function and a wide power spectrum. 4. Parameters found using the Marquardt algorithm were incorporated into the model. Diagonal matrices in a head coordinate frame were introduced to represent the direct pathway and the coupling of the visual system to the integrator. Model simulations predicted the behavior of yaw and pitch OKN and OKAN when the animal was upright, as well as the cross-coupling in the tilted position. The trajectories in velocity space were also accurately simulated. 5. There were similarities between the monkey eigenvectors and human perception of the spatial vertical. For side-down tilts and downward eye velocity cross-coupling, there was only an Aubert (A) effect. For upward eye velocity cross-coupling there were both Muller (E) and Aubert (A) effects. The mean of the eigenvectors for upward and downward eye velocities overlay human 1 x g perceptual data.(ABSTRACT TRUNCATED AT 400 WORDS).

The Combinatorial Trace Method in Action

ERIC Educational Resources Information Center

Krebs, Mike; Martinez, Natalie C.

2013-01-01

On any finite graph, the number of closed walks of length k is equal to the sum of the kth powers of the eigenvalues of any adjacency matrix. This simple observation is the basis for the combinatorial trace method, wherein we attempt to count (or bound) the number of closed walks of a given length so as to obtain information about the graph's…

Fourier method for modeling slanted lamellar gratings of arbitrary end-surface shapes in conical mounting.

PubMed

Li, Lifeng

2015-10-01

An efficient modal method for numerically modeling slanted lamellar gratings of isotropic dielectric or metallic media in conical mounting is presented. No restrictions are imposed on the slant angle and the length of the lamellae. The end surface of the lamellae can be arbitrary, subject to certain restrictions. An oblique coordinate system that is adapted to the slanted lamella sidewalls allows the most efficient way of representing and manipulating the electromagnetic fields. A translational coordinate system that is based on the oblique Cartesian coordinate system adapts to the end-surface profile of the lamellae, so that the latter can be handled simply and easily. Moreover, two matrix eigenvalue problems of size 2N × 2N, one for each fundamental polarization of the electromagnetic fields in the periodic lamellar structure, where N is the matrix truncation number, are derived to replace the 4N × 4N eigenvalue problem that has been used in the literature. The core idea leading to this success is the polarization decomposition of the electromagnetic fields inside the periodic lamellar region when the fields are expressed in the oblique translational coordinate system.

Nature of Driving Force for Protein Folding: A Result From Analyzing the Statistical Potential

NASA Astrophysics Data System (ADS)

Li, Hao; Tang, Chao; Wingreen, Ned S.

1997-07-01

In a statistical approach to protein structure analysis, Miyazawa and Jernigan derived a 20×20 matrix of inter-residue contact energies between different types of amino acids. Using the method of eigenvalue decomposition, we find that the Miyazawa-Jernigan matrix can be accurately reconstructed from its first two principal component vectors as Mij = C0+C1\\(qi+qj\\)+C2qiqj, with constant C's, and 20 q values associated with the 20 amino acids. This regularity is due to hydrophobic interactions and a force of demixing, the latter obeying Hildebrand's solubility theory of simple liquids.

Swimming of an assembly of rigid spheres at low Reynolds number.

PubMed

Felderhof, B U

2014-11-01

A matrix formulation is derived for the calculation of the swimming speed and the power required for swimming of an assembly of rigid spheres immersed in a viscous fluid of infinite extent. The spheres may have arbitrary radii and may interact with elastic forces. The analysis is based on the Stokes mobility matrix of the set of spheres, defined in low Reynolds number hydrodynamics. For small amplitude, swimming optimization of the swimming speed at given power leads to an eigenvalue problem. The method allows straightforward calculation of the swimming performance of structures modeled as assemblies of interacting rigid spheres.

Using an iterative eigensolver to compute vibrational energies with phase-spaced localized basis functions.

PubMed

Brown, James; Carrington, Tucker

2015-07-28

Although phase-space localized Gaussians are themselves poor basis functions, they can be used to effectively contract a discrete variable representation basis [A. Shimshovitz and D. J. Tannor, Phys. Rev. Lett. 109, 070402 (2012)]. This works despite the fact that elements of the Hamiltonian and overlap matrices labelled by discarded Gaussians are not small. By formulating the matrix problem as a regular (i.e., not a generalized) matrix eigenvalue problem, we show that it is possible to use an iterative eigensolver to compute vibrational energy levels in the Gaussian basis.

Eigenvalue computations with the QUAD4 consistent-mass matrix

NASA Technical Reports Server (NTRS)

Butler, Thomas A.

1990-01-01

The NASTRAN user has the option of using either a lumped-mass matrix or a consistent- (coupled-) mass matrix with the QUAD4 shell finite element. At the Sixteenth NASTRAN Users' Colloquium (1988), Melvyn Marcus and associates of the David Taylor Research Center summarized a study comparing the results of the QUAD4 element with results of other NASTRAN shell elements for a cylindrical-shell modal analysis. Results of this study, in which both the lumped-and consistent-mass matrix formulations were used, implied that the consistent-mass matrix yielded poor results. In an effort to further evaluate the consistent-mass matrix, a study was performed using both a cylindrical-shell geometry and a flat-plate geometry. Modal parameters were extracted for several modes for both geometries leading to some significant conclusions. First, there do not appear to be any fundamental errors associated with the consistent-mass matrix. However, its accuracy is quite different for the two different geometries studied. The consistent-mass matrix yields better results for the flat-plate geometry and the lumped-mass matrix seems to be the better choice for cylindrical-shell geometries.

The tunneling effect for a class of difference operators

NASA Astrophysics Data System (ADS)

Klein, Markus; Rosenberger, Elke

We analyze a general class of self-adjoint difference operators H𝜀 = T𝜀 + V𝜀 on ℓ2((𝜀ℤ)d), where V𝜀 is a multi-well potential and 𝜀 is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30-35]).Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs. The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian H𝜀 is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance d induced by H𝜀, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first n eigenvalues of H𝜀 converge to the first n eigenvalues of the direct sum of harmonic oscillators on ℝd located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of H𝜀. These are obtained from eigenfunctions or quasimodes for the operator H𝜀, acting on L2(ℝd), via restriction to the lattice (𝜀ℤ)d. Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrödinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted ℓ2-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two “wells” (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrödinger operator in [22].

Modelling and characteristic analysis of tri-axle trucks with hydraulically interconnected suspensions

NASA Astrophysics Data System (ADS)

Ding, Fei; Han, Xu; Luo, Zhen; Zhang, Nong

2012-12-01

In this paper, a new hydraulically interconnected suspension (HIS) system is proposed for the implementation of a resistance control for the pitch and bounce modes of tri-axle heavy trucks. A lumped-mass half-truck model is established using the free-body diagram method. The equations of motion of a mechanical and hydraulic coupled system are developed by incorporating the hydraulic strut forces into the mechanical subsystem as externally applied forces. The transfer matrix method (TMM) is used to evaluate the impedance matrix of the hydraulic subsystem consisting of models of fluid pipes, damper valves, accumulators, and three-way junctions. The TMM is further applied to find the quantitative relationships between the hydraulic strut forces and boundary flow of the mechanical-fluid interactive subsystem. The modal analysis method is employed to perform the vibration analysis between the trucks with the conventional suspension and the proposed HIS. Comparison analysis focuses on free vibration with identified eigenvalues and eigenvectors, isolation vibration capacity, and force vibration in terms of the power spectrum density responses. The obtained results show the effectiveness of the proposed HIS system in reducing the pitch motion of sprung mass and simultaneously maintaining the ride comfort. The pitch stiffness is increased while the bounce stiffness is slightly softened. The peak values of sprung mass and wheel hop motions are greatly reduced, and the vibration decay rate of sprung mass is also significantly increased.

Exact solution for four-order acousto-optic Bragg diffraction with arbitrary initial conditions.

PubMed

Pieper, Ron; Koslover, Deborah; Poon, Ting-Chung

2009-03-01

An exact solution to the four-order acousto-optic (AO) Bragg diffraction problem with arbitrary initial conditions compatible with exact Bragg angle incident light is developed. The solution, obtained by solving a 4th-order differential equation, is formalized into a transition matrix operator predicting diffracted light orders at the exit of the AO cell in terms of the same diffracted light orders at the entrance. It is shown that the transition matrix is unitary and that this unitary matrix condition is sufficient to guarantee energy conservation. A comparison of analytical solutions with numerical predictions validates the formalism. Although not directly related to the approach used to obtain the solution, it was discovered that all four generated eigenvalues from the four-order AO differential matrix operator are expressed simply in terms of Euclid's Divine Proportion.

The use of an analytic Hamiltonian matrix for solving the hydrogenic atom

NASA Astrophysics Data System (ADS)

Bhatti, Mohammad

2001-10-01

The non-relativistic Hamiltonian corresponding to the Shrodinger equation is converted into analytic Hamiltonian matrix using the kth order B-splines functions. The Galerkin method is applied to the solution of the Shrodinger equation for bound states of hydrogen-like systems. The program Mathematica is used to create analytic matrix elements and exact integration is performed over the knot-sequence of B-splines and the resulting generalized eigenvalue problem is solved on a specified numerical grid. The complete basis set and the energy spectrum is obtained for the coulomb potential for hydrogenic systems with Z less than 100 with B-splines of order eight. Another application is given to test the Thomas-Reiche-Kuhn sum rule for the hydrogenic systems.

Intraday seasonalities and nonstationarity of trading volume in financial markets: Collective features

PubMed Central

Graczyk, Michelle B.; Duarte Queirós, Sílvio M.

2017-01-01

Employing Random Matrix Theory and Principal Component Analysis techniques, we enlarge our work on the individual and cross-sectional intraday statistical properties of trading volume in financial markets to the study of collective intraday features of that financial observable. Our data consist of the trading volume of the Dow Jones Industrial Average Index components spanning the years between 2003 and 2014. Computing the intraday time dependent correlation matrices and their spectrum of eigenvalues, we show there is a mode ruling the collective behaviour of the trading volume of these stocks whereas the remaining eigenvalues are within the bounds established by random matrix theory, except the second largest eigenvalue which is robustly above the upper bound limit at the opening and slightly above it during the morning-afternoon transition. Taking into account that for price fluctuations it was reported the existence of at least seven significant eigenvalues—and that its autocorrelation function is close to white noise for highly liquid stocks whereas for the trading volume it lasts significantly for more than 2 hours —, our finding goes against any expectation based on those features, even when we take into account the Epps effect. In addition, the weight of the trading volume collective mode is intraday dependent; its value increases as the trading session advances with its eigenversor approaching the uniform vector as well, which corresponds to a soar in the behavioural homogeneity. With respect to the nonstationarity of the collective features of the trading volume we observe that after the financial crisis of 2008 the coherence function shows the emergence of an upset profile with large fluctuations from that year on, a property that concurs with the modification of the average trading volume profile we noted in our previous individual analysis. PMID:28753676

Free and forced vibrations of a tyre using a wave/finite element approach

NASA Astrophysics Data System (ADS)

Waki, Y.; Mace, B. R.; Brennan, M. J.

2009-06-01

Free and forced vibrations of a tyre are predicted using a wave/finite element (WFE) approach. A short circumferential segment of the tyre is modelled using conventional finite element (FE) methods, a periodicity condition applied and the mass and stiffness matrices post-processed to yield wave properties. Since conventional FE methods are used, commercial FE packages and existing element libraries can be utilised. An eigenvalue problem is formulated in terms of the transfer matrix of the segment. Zhong's method is used to improve numerical conditioning. The eigenvalues and eigenvectors give the wavenumbers and wave mode shapes, which in turn define transformations between the physical and wave domains. A method is described by which the frequency dependent material properties of the rubber components of the tyre can be included without the need to remesh the structure. Expressions for the forced response are developed which are numerically well-conditioned. Numerical results for a smooth tyre are presented. Dispersion curves for real, imaginary and complex wavenumbers are shown. The propagating waves are associated with various forms of motion of the tread supported by the stiffness of the side wall. Various dispersion phenomena are observed, including curve veering, non-zero cut-off and waves for which the phase velocity and the group velocity have opposite signs. Results for the forced response are compared with experimental measurements and good agreement is seen. The forced response is numerically determined for both finite area and point excitations. It is seen that the size of area of the excitation is particularly important at high frequencies. When the size of the excitation area is small enough compared to the tread thickness, the response at high frequencies becomes stiffness-like (reactive) and the effect of shear stiffness becomes important.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE Office of Scientific and Technical Information (OSTI.GOV)

Brabec, Jiri; Lin, Lin; Shao, Meiyue

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

A systematic linear space approach to solving partially described inverse eigenvalue problems

NASA Astrophysics Data System (ADS)

Hu, Sau-Lon James; Li, Haujun

2008-06-01

Most applications of the inverse eigenvalue problem (IEP), which concerns the reconstruction of a matrix from prescribed spectral data, are associated with special classes of structured matrices. Solving the IEP requires one to satisfy both the spectral constraint and the structural constraint. If the spectral constraint consists of only one or few prescribed eigenpairs, this kind of inverse problem has been referred to as the partially described inverse eigenvalue problem (PDIEP). This paper develops an efficient, general and systematic approach to solve the PDIEP. Basically, the approach, applicable to various structured matrices, converts the PDIEP into an ordinary inverse problem that is formulated as a set of simultaneous linear equations. While solving simultaneous linear equations for model parameters, the singular value decomposition method is applied. Because of the conversion to an ordinary inverse problem, other constraints associated with the model parameters can be easily incorporated into the solution procedure. The detailed derivation and numerical examples to implement the newly developed approach to symmetric Toeplitz and quadratic pencil (including mass, damping and stiffness matrices of a linear dynamic system) PDIEPs are presented. Excellent numerical results for both kinds of problem are achieved under the situations that have either unique or infinitely many solutions.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

DOE PAGES

Brabec, Jiri; Lin, Lin; Shao, Meiyue; ...

2015-10-06

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires halfmore » of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.« less

Random matrix theory analysis of cross-correlations in the US stock market: Evidence from Pearson’s correlation coefficient and detrended cross-correlation coefficient

NASA Astrophysics Data System (ADS)

Wang, Gang-Jin; Xie, Chi; Chen, Shou; Yang, Jiao-Jiao; Yang, Ming-Yan

2013-09-01

In this study, we first build two empirical cross-correlation matrices in the US stock market by two different methods, namely the Pearson’s correlation coefficient and the detrended cross-correlation coefficient (DCCA coefficient). Then, combining the two matrices with the method of random matrix theory (RMT), we mainly investigate the statistical properties of cross-correlations in the US stock market. We choose the daily closing prices of 462 constituent stocks of S&P 500 index as the research objects and select the sample data from January 3, 2005 to August 31, 2012. In the empirical analysis, we examine the statistical properties of cross-correlation coefficients, the distribution of eigenvalues, the distribution of eigenvector components, and the inverse participation ratio. From the two methods, we find some new results of the cross-correlations in the US stock market in our study, which are different from the conclusions reached by previous studies. The empirical cross-correlation matrices constructed by the DCCA coefficient show several interesting properties at different time scales in the US stock market, which are useful to the risk management and optimal portfolio selection, especially to the diversity of the asset portfolio. It will be an interesting and meaningful work to find the theoretical eigenvalue distribution of a completely random matrix R for the DCCA coefficient because it does not obey the Marčenko-Pastur distribution.

Evaluation of the Majorana phases of a general Majorana neutrino mass matrix: Testability of hierarchical flavour models

NASA Astrophysics Data System (ADS)

Samanta, Rome; Chakraborty, Mainak; Ghosal, Ambar

2016-03-01

We evaluate the Majorana phases for a general 3 × 3 complex symmetric neutrino mass matrix on the basis of Mohapatra-Rodejohann's phase convention using the three rephasing invariant quantities I12, I13 and I23 proposed by Sarkar and Singh. We find them interesting as they allow us to evaluate each Majorana phase in a model independent way even if one eigenvalue is zero. Utilizing the solution of a general complex symmetric mass matrix for eigenvalues and mixing angles we determine the Majorana phases for both the hierarchies, normal and inverted, taking into account the constraints from neutrino oscillation global fit data as well as bound on the sum of the three light neutrino masses (Σimi) and the neutrinoless double beta decay (ββ0ν) parameter |m11 |. This methodology of finding the Majorana phases is applied thereafter in some predictive models for both the hierarchical cases (normal and inverted) to evaluate the corresponding Majorana phases and it is shown that all the sub cases presented in inverted hierarchy section can be realized in a model with texture zeros and scaling ansatz within the framework of inverse seesaw although one of the sub cases following the normal hierarchy is yet to be established. Except the case of quasi degenerate neutrinos, the methodology obtained in this work is able to evaluate the corresponding Majorana phases, given any model of neutrino masses.

Stability analysis of an autocatalytic protein model

NASA Astrophysics Data System (ADS)

Lee, Julian

2016-05-01

A self-regulatory genetic circuit, where a protein acts as a positive regulator of its own production, is known to be the simplest biological network with a positive feedback loop. Although at least three components—DNA, RNA, and the protein—are required to form such a circuit, stability analysis of the fixed points of this self-regulatory circuit has been performed only after reducing the system to a two-component system, either by assuming a fast equilibration of the DNA component or by removing the RNA component. Here, stability of the fixed points of the three-component positive feedback loop is analyzed by obtaining eigenvalues of the full three-dimensional Hessian matrix. In addition to rigorously identifying the stable fixed points and saddle points, detailed information about the system can be obtained, such as the existence of complex eigenvalues near a fixed point.

Harmonic Bloch and dipole oscillations and their transition in elliptical optical waveguide arrays

NASA Astrophysics Data System (ADS)

Chan, Yun San; Zheng, Ming Jie; Yu, Kin Wah

2011-03-01

We have studied harmonic oscillations in an elliptical optical waveguide array in which the couplings between neighboring waveguides are varied in accord with a Kac matrix so that the propagation constant eigenvalues can take equally spaced values. As a result, the long-living optical Bloch oscillation (BO) and dipole oscillation (DO) are obtained. Moreover, when a linear gradient in the propagation constant is applied, we achieve a switching from DO to BO and vice versa by ramping up or down the gradient profile]. The various optical oscillations as well as their switching are investigated by field evolution analysis and confirmed by Hamiltonian optics. The equally spaced eigenvalues in the propagation constant allow viable applications in transmitting images, switching and routing of optical signals. Work supported by the General Research Fund of the Hong Kong SAR Government.

Eigenvalue approach to coupled thermoelasticity in a rotating isotropic medium

NASA Astrophysics Data System (ADS)

Bayones, F. S.; Abd-Alla, A. M.

2018-03-01

In this paper the linear theory of the thermoelasticity has been employed to study the effect of the rotation in a thermoelastic half-space containing heat source on the boundary of the half-space. It is assumed that the medium under consideration is traction free, homogeneous, isotropic, as well as without energy dissipation. The normal mode analysis has been applied in the basic equations of coupled thermoelasticity and finally the resulting equations are written in the form of a vector- matrix differential equation which is then solved by eigenvalue approach. Numerical results for the displacement components, stresses, and temperature are given and illustrated graphically. Comparison was made with the results obtained in the presence and absence of the rotation. The results indicate that the effect of rotation, non-dimensional thermal wave and time are very pronounced.

Individual eigenvalue distributions of crossover chiral random matrices and low-energy constants of SU(2) × U(1) lattice gauge theory

NASA Astrophysics Data System (ADS)

Yamamoto, Takuya; Nishigaki, Shinsuke M.

2018-02-01

We compute individual distributions of low-lying eigenvalues of a chiral random matrix ensemble interpolating symplectic and unitary symmetry classes by the Nyström-type method of evaluating the Fredholm Pfaffian and resolvents of the quaternion kernel. The one-parameter family of these distributions is shown to fit excellently the Dirac spectra of SU(2) lattice gauge theory with a constant U(1) background or dynamically fluctuating U(1) gauge field, which weakly breaks the pseudoreality of the unperturbed SU(2) Dirac operator. The observed linear dependence of the crossover parameter with the strength of the U(1) perturbations leads to precise determination of the pseudo-scalar decay constant, as well as the chiral condensate in the effective chiral Lagrangian of the AI class.

Photonic Breast Tomography and Tumor Aggressiveness Assessment

DTIC Science & Technology

2011-07-01

incorporates, in optical domain, the vector subspace classification method, Multiple Signal Classification ( MUSIC ). MUSIC was developed by Devaney...and co-workers for finding the location of scattering targets whose size is smaller than the wavelength of acoustic waves or electromagnetic waves...general area of array processing for acoustic and radar time-reversal imaging [12]. The eigenvalue equation of TR matrix is solved, and the signal and

Computation of free oscillations of the earth

USGS Publications Warehouse

Buland, Raymond P.; Gilbert, F.

1984-01-01

Although free oscillations of the Earth may be computed by many different methods, numerous practical considerations have led us to use a Rayleigh-Ritz formulation with piecewise cubic Hermite spline basis functions. By treating the resulting banded matrix equation as a generalized algebraic eigenvalue problem, we are able to achieve great accuracy and generality and a high degree of automation at a reasonable cost. ?? 1984.

Parallel Symmetric Eigenvalue Problem Solvers

DTIC Science & Technology

2015-05-01

tutoring, and mentoring experience as an undergraduate. Last but not least, I thank my family for their love and support. v TABLE OF CONTENTS Page LIST...34 4.6.2 Choice of the Ritz shifts . . . . . . . . . . . . . . . . . . . . 38 4.7 Relationship between TraceMin and...which are determined by the Ritz values of the matrix pencil. We conclude with a discussion of the relationship between TraceMin and simultaneous

Derivation of the Time-Reversal Anomaly for (2 +1 )-Dimensional Topological Phases

NASA Astrophysics Data System (ADS)

Tachikawa, Yuji; Yonekura, Kazuya

2017-09-01

We prove an explicit formula conjectured recently by Wang and Levin for the anomaly of time-reversal symmetry in (2 +1 )-dimensional fermionic topological quantum field theories. The crucial step is to determine the cross-cap state in terms of the modular S matrix and T2 eigenvalues, generalizing the recent analysis by Barkeshli et al. in the bosonic case.

Complex eigenvalue analysis of rotating structures

NASA Technical Reports Server (NTRS)

Patel, J. S.; Seltzer, S. M.

1972-01-01

A FORTRAN subroutine to NASTRAN which constructs coriolis and centripetal acceleration matrices, and a centrifugal load vector due to spin about a selected point or about the mass center of the structure is discussed. The rigid translational degrees of freedom can be removed by using a transformation matrix T and its explicitly given inverse. These matrices are generated in the subroutine and their explicit expressions are given.

A Multivariate Randomization Text of Association Applied to Cognitive Test Results

NASA Technical Reports Server (NTRS)

Ahumada, Albert; Beard, Bettina

2009-01-01

Randomization tests provide a conceptually simple, distribution-free way to implement significance testing. We have applied this method to the problem of evaluating the significance of the association among a number (k) of variables. The randomization method was the random re-ordering of k-1 of the variables. The criterion variable was the value of the largest eigenvalue of the correlation matrix.

Analysis of the Pre-stack Split-Step Migration Operator Using Ritz Values

NASA Astrophysics Data System (ADS)

Kaplan, S. T.; Sacchi, M. D.

2009-05-01

The Born approximation for the acoustic wave-field is often used as a basis for developing algorithms in seismic imaging (migration). The approximation is linear, and, as such, can be written as a matrix-vector multiplication (Am=d). In the seismic imaging problem, d is seismic data (the recorded wave-field), and we aim to find the seismic reflectivity m (a representation of earth structure and properties) so that Am=d is satisfied. This is the often studied inverse problem of seismic migration, where given A and d, we solve for m. This can be done in a least-squares sense, so that the equation of interest is, AHAm = AHd. Hence, the solution m is largely dependent on the properties of AHA. The imaging Jacobian J provides an approximation to AHA, so that J-1AHA is, in a broad sense, better behaved then AHA. We attempt to quantify this last statement by providing an analysis of AHA and J-1AHA using their Ritz values, and for the particular case where A is built using a pre-stack split-step migration algorithm. Typically, one might try to analyze the behaviour of these matrices using their eigenvalue spectra. The difficulty in the analysis of AHA and J-1AHA lie in their size. For example, a subset of the relatively small Marmousi data set makes AHA a complex valued matrix with, roughly, dimensions of 45 million by 45 million (requiring, in single-precision, about 16 Peta-bytes of computer memory). In short, the size of the matrix makes its eigenvalues difficult to compute. Instead, we compute the leading principal minors of similar tridiagonal matrices, Bk=Vk-1AHAVk and Ck = Uk-1 J-1 AHAUk. These can be constructed using, for example, the Lanczos decomposition. Up to some value of k it is feasible to compute the eigenvalues of Bk and Ck which, in turn, are the Ritz values of, respectively, AHA and J-1 AHA, and may allow us to make quantitative statements about their behaviours.

Minimizing the stochasticity of halos in large-scale structure surveys

NASA Astrophysics Data System (ADS)

Hamaus, Nico; Seljak, Uroš; Desjacques, Vincent; Smith, Robert E.; Baldauf, Tobias

2010-08-01

In recent work (Seljak, Hamaus, and Desjacques 2009) it was found that weighting central halo galaxies by halo mass can significantly suppress their stochasticity relative to the dark matter, well below the Poisson model expectation. This is useful for constraining relations between galaxies and the dark matter, such as the galaxy bias, especially in situations where sampling variance errors can be eliminated. In this paper we extend this study with the goal of finding the optimal mass-dependent halo weighting. We use N-body simulations to perform a general analysis of halo stochasticity and its dependence on halo mass. We investigate the stochasticity matrix, defined as Cij≡⟨(δi-biδm)(δj-bjδm)⟩, where δm is the dark matter overdensity in Fourier space, δi the halo overdensity of the i-th halo mass bin, and bi the corresponding halo bias. In contrast to the Poisson model predictions we detect nonvanishing correlations between different mass bins. We also find the diagonal terms to be sub-Poissonian for the highest-mass halos. The diagonalization of this matrix results in one large and one low eigenvalue, with the remaining eigenvalues close to the Poisson prediction 1/n¯, where n¯ is the mean halo number density. The eigenmode with the lowest eigenvalue contains most of the information and the corresponding eigenvector provides an optimal weighting function to minimize the stochasticity between halos and dark matter. We find this optimal weighting function to match linear mass weighting at high masses, while at the low-mass end the weights approach a constant whose value depends on the low-mass cut in the halo mass function. This weighting further suppresses the stochasticity as compared to the previously explored mass weighting. Finally, we employ the halo model to derive the stochasticity matrix and the scale-dependent bias from an analytical perspective. It is remarkably successful in reproducing our numerical results and predicts that the stochasticity between halos and the dark matter can be reduced further when going to halo masses lower than we can resolve in current simulations.

Liquid-gas phase transitions and C K symmetry in quantum field theories

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nishimura, Hiromichi; Ogilvie, Michael C.; Pangeni, Kamal

A general field-theoretic framework for the treatment of liquid-gas phase transitions is developed. Starting from a fundamental four-dimensional field theory at nonzero temperature and density, an effective three-dimensional field theory is derived. The effective field theory has a sign problem at finite density. Although finite density explicitly breaks charge conjugation C , there remains a symmetry under C K , where K is complex conjugation. Here, we consider four models: relativistic fermions, nonrelativistic fermions, static fermions and classical particles. The interactions are via an attractive potential due to scalar field exchange and a repulsive potential due to massive vector exchange.more » The field-theoretic representation of the partition function is closely related to the equivalence of the sine-Gordon field theory with a classical gas. The thermodynamic behavior is extracted from C K -symmetric complex saddle points of the effective field theory at tree level. In the cases of nonrelativistic fermions and classical particles, we find complex saddle point solutions but no first-order transitions, and neither model has a ground state at tree level. The relativistic and static fermions show a liquid-gas transition at tree level in the effective field theory. The liquid-gas transition, when it occurs, manifests as a first-order line at low temperature and high density, terminated by a critical end point. The mass matrix controlling the behavior of correlation functions is obtained from fluctuations around the saddle points. Due to the C K symmetry of the models, the eigenvalues of the mass matrix are not always real but can be complex. This then leads to the existence of disorder lines, which mark the boundaries where the eigenvalues go from purely real to complex. The regions where the mass matrix eigenvalues are complex are associated with the critical line. In the case of static fermions, a powerful duality between particles and holes allows for the analytic determination of both the critical line and the disorder lines. Depending on the values of the parameters, either zero, one, or two disorder lines are found. Our numerical results for relativistic fermions give a very similar picture.« less

Liquid-gas phase transitions and C K symmetry in quantum field theories

DOE PAGES

Nishimura, Hiromichi; Ogilvie, Michael C.; Pangeni, Kamal

2017-04-04

A general field-theoretic framework for the treatment of liquid-gas phase transitions is developed. Starting from a fundamental four-dimensional field theory at nonzero temperature and density, an effective three-dimensional field theory is derived. The effective field theory has a sign problem at finite density. Although finite density explicitly breaks charge conjugation C , there remains a symmetry under C K , where K is complex conjugation. Here, we consider four models: relativistic fermions, nonrelativistic fermions, static fermions and classical particles. The interactions are via an attractive potential due to scalar field exchange and a repulsive potential due to massive vector exchange.more » The field-theoretic representation of the partition function is closely related to the equivalence of the sine-Gordon field theory with a classical gas. The thermodynamic behavior is extracted from C K -symmetric complex saddle points of the effective field theory at tree level. In the cases of nonrelativistic fermions and classical particles, we find complex saddle point solutions but no first-order transitions, and neither model has a ground state at tree level. The relativistic and static fermions show a liquid-gas transition at tree level in the effective field theory. The liquid-gas transition, when it occurs, manifests as a first-order line at low temperature and high density, terminated by a critical end point. The mass matrix controlling the behavior of correlation functions is obtained from fluctuations around the saddle points. Due to the C K symmetry of the models, the eigenvalues of the mass matrix are not always real but can be complex. This then leads to the existence of disorder lines, which mark the boundaries where the eigenvalues go from purely real to complex. The regions where the mass matrix eigenvalues are complex are associated with the critical line. In the case of static fermions, a powerful duality between particles and holes allows for the analytic determination of both the critical line and the disorder lines. Depending on the values of the parameters, either zero, one, or two disorder lines are found. Our numerical results for relativistic fermions give a very similar picture.« less

Consensus Algorithms for Networks of Systems with Second- and Higher-Order Dynamics

NASA Astrophysics Data System (ADS)

Fruhnert, Michael

This thesis considers homogeneous networks of linear systems. We consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilizable. We show that, in continuous-time, consensus with a guaranteed rate of convergence can always be achieved using linear state feedback. For networks of continuous-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Hurwitz. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. Based on the conditions found, methods to compute feedback gains are proposed. We show that gains can be chosen such that consensus is achieved robustly over a variety of communication structures and system dynamics. We also consider the use of static output feedback. For networks of discrete-time second-order systems, we provide a new and simple derivation of the conditions for a second-order polynomials with complex coefficients to be Schur. We apply this result to obtain necessary and sufficient conditions to achieve consensus with networks whose graph Laplacian matrix may have complex eigenvalues. We show that consensus can always be achieved for marginally stable systems and discretized systems. Simple conditions for consensus achieving controllers are obtained when the Laplacian eigenvalues are all real. For networks of continuous-time time-variant higher-order systems, we show that uniform consensus can always be achieved if systems are quadratically stabilizable. In this case, we provide a simple condition to obtain a linear feedback control. For networks of discrete-time higher-order systems, we show that constant gains can be chosen such that consensus is achieved for a variety of network topologies. First, we develop simple results for networks of time-invariant systems and networks of time-variant systems that are given in controllable canonical form. Second, we formulate the problem in terms of Linear Matrix Inequalities (LMIs). The condition found simplifies the design process and avoids the parallel solution of multiple LMIs. The result yields a modified Algebraic Riccati Equation (ARE) for which we present an equivalent LMI condition.

Time-reversal optical tomography: detecting and locating extended targets in a turbid medium

NASA Astrophysics Data System (ADS)

Wu, Binlin; Cai, W.; Xu, M.; Gayen, S. K.

2012-03-01

Time Reversal Optical Tomography (TROT) is developed to locate extended target(s) in a highly scattering turbid medium, and estimate their optical strength and size. The approach uses Diffusion Approximation of Radiative Transfer Equation for light propagation along with Time Reversal (TR) Multiple Signal Classification (MUSIC) scheme for signal and noise subspaces for assessment of target location. A MUSIC pseudo spectrum is calculated using the eigenvectors of the TR matrix T, whose poles provide target locations. Based on the pseudo spectrum contours, retrieval of target size is modeled as an optimization problem, using a "local contour" method. The eigenvalues of T are related to optical strengths of targets. The efficacy of TROT to obtain location, size, and optical strength of one absorptive target, one scattering target, and two absorptive targets, all for different noise levels was tested using simulated data. Target locations were always accurately determined. Error in optical strength estimates was small even at 20% noise level. Target size and shape were more sensitive to noise. Results from simulated data demonstrate high potential for application of TROT in practical biomedical imaging applications.

Propagation of eigenmodes and transfer functions in waveguide WDM structures

NASA Astrophysics Data System (ADS)

Mashkov, Vladimir A.; Francoeur, S.; Geuss, U.; Neiser, K.; Temkin, Henryk

1998-02-01

A method of propagation functions and transfer amplitudes suitable for the design of integrated optical circuits is presented. The method is based on vectorial formulation of electrodynamics: the distributions and propagation of electromagnetic fields in optical circuits is described by equivalent surface sources. This approach permits a division of complex optical waveguide structures into sets of primitive blocks and to separately calculate the transfer function and the transfer amplitude for each block. The transfer amplitude of the entire optical system is represented by a convolution of transfer amplitudes of its primitive blocks. The eigenvalues and eigenfunctions of arbitrary waveguide structure are obtained in the WKB approximation and compared with other methods. The general approach is illustrated with the transfer amplitude calculations for Dragone's star coupler and router.

A fast, preconditioned conjugate gradient Toeplitz solver

NASA Technical Reports Server (NTRS)

Pan, Victor; Schrieber, Robert

1989-01-01

A simple factorization is given of an arbitrary hermitian, positive definite matrix in which the factors are well-conditioned, hermitian, and positive definite. In fact, given knowledge of the extreme eigenvalues of the original matrix A, an optimal improvement can be achieved, making the condition numbers of each of the two factors equal to the square root of the condition number of A. This technique is to applied to the solution of hermitian, positive definite Toeplitz systems. Large linear systems with hermitian, positive definite Toeplitz matrices arise in some signal processing applications. A stable fast algorithm is given for solving these systems that is based on the preconditioned conjugate gradient method. The algorithm exploits Toeplitz structure to reduce the cost of an iteration to O(n log n) by applying the fast Fourier Transform to compute matrix-vector products. Matrix factorization is used as a preconditioner.

Nature of Driving Force for Protein Folding-- A Result From Analyzing the Statistical Potential

NASA Astrophysics Data System (ADS)

Li, Hao; Tang, Chao; Wingreen, Ned S.

1998-03-01

In a statistical approach to protein structure analysis, Miyazawa and Jernigan (MJ) derived a 20× 20 matrix of inter-residue contact energies between different types of amino acids. Using the method of eigenvalue decomposition, we find that the MJ matrix can be accurately reconstructed from its first two principal component vectors as M_ij=C_0+C_1(q_i+q_j)+C2 qi q_j, with constant C's, and 20 q values associated with the 20 amino acids. This regularity is due to hydrophobic interactions and a force of demixing, the latter obeying Hildebrand's solubility theory of simple liquids.

Large planar maneuvers for articulated flexible manipulators

NASA Technical Reports Server (NTRS)

Huang, Jen-Kuang; Yang, Li-Farn

1988-01-01

An articulated flexible manipulator carried on a translational cart is maneuvered by an active controller to perform certain position control tasks. The nonlinear dynamics of the articulated flexible manipulator are derived and a transformation matrix is formulated to localize the nonlinearities within the inertia matrix. Then a feedback linearization scheme is introduced to linearize the dynamic equations for controller design. Through a pole placement technique, a robust controller design is obtained by properly assigning a set of closed-loop desired eigenvalues to meet performance requirements. Numerical simulations for the articulated flexible manipulators are given to demonstrate the feasibility and effectiveness of the proposed position control algorithms.

Factor Analytic Approach to Transitive Text Mining using Medline Descriptors

NASA Astrophysics Data System (ADS)

Stegmann, J.; Grohmann, G.

Matrix decomposition methods were applied to examples of noninteractive literature sets sharing implicit relations. Document-by-term matrices were created from downloaded PubMed literature sets, the terms being the Medical Subject Headings (MeSH descriptors) assigned to the documents. The loadings of the factors derived from singular value or eigenvalue matrix decomposition were sorted according to absolute values and subsequently inspected for positions of terms relevant to the discovery of hidden connections. It was found that only a small number of factors had to be screened to find key terms in close neighbourhood, being separated by a small number of terms only.

Trust Method for Multi-Agent Consensus

DTIC Science & Technology

2012-03-22

irreducible10 and by Lemma 1, is also stochastic. And according to the Perron - Frobenius theorem, the fact that has an eigenvalue of 1 with a positive...if the limit lim→∞ exists. According to the Perron - Frobenius theorem, this limit exists for primitive matrices and according to Lemma 2, ...and > 0. Here, is known as the Perron matrix of graph with parameter . If we substitute the normalized Laplacian for in

Minimal Krylov Subspaces for Dimension Reduction

DTIC Science & Technology

2013-01-01

these applications realized a maximal compute time improvement with minimal Krylov subspaces. More recently, Halko et . al . [36] have investigated... Halko et . al . proposed a variety of them in [36], but we focus on the “direct eigenvalue approximation for Hermitian matri- ces with random...result due to Halko et . al . Theorem 5 ( Halko , Martinsson and Tropp [36]). Let A ∈ Rn×m be the input matrix with partitioned singular value

Calculation of Radar Probability of Detection in K-Distributed Sea Clutter and Noise

DTIC Science & Technology

2011-04-01

Laguerre polynomials are generated from a recurrence relation, and the nodes and weights are calculated from the eigenvalues and eigenvectors of a...B.P. Flannery, Numerical Recipes in Fortran, Second Edition, Cambridge University Press (1992). 12. W. Gautschi, Orthogonal Polynomials (in Matlab...the integration, with the nodes and weights calculated using matrix methods, so that a general purpose numerical integration routine is not required

Incomplete augmented Lagrangian preconditioner for steady incompressible Navier-Stokes equations.

PubMed

Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun

2013-01-01

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids.

Column Subset Selection, Matrix Factorization, and Eigenvalue Optimization

DTIC Science & Technology

2008-07-01

Pietsch and Grothendieck, which are regarded as basic instruments in modern functional analysis [Pis86]. • The methods for computing these... Pietsch factorization and the maxcut semi- definite program [GW95]. 1.2. Overview. We focus on the algorithmic version of the Kashin–Tzafriri theorem...will see that the desired subset is exposed by factoring the random submatrix. This factorization, which was invented by Pietsch , is regarded as a basic

Control theory and splines, applied to signature storage

NASA Technical Reports Server (NTRS)

Enqvist, Per

1994-01-01

In this report the problem we are going to study is the interpolation of a set of points in the plane with the use of control theory. We will discover how different systems generate different kinds of splines, cubic and exponential, and investigate the effect that the different systems have on the tracking problems. Actually we will see that the important parameters will be the two eigenvalues of the control matrix.

Numerical methods in Markov chain modeling

NASA Technical Reports Server (NTRS)

Philippe, Bernard; Saad, Youcef; Stewart, William J.

1989-01-01

Several methods for computing stationary probability distributions of Markov chains are described and compared. The main linear algebra problem consists of computing an eigenvector of a sparse, usually nonsymmetric, matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous singular linear system. Several methods based on combinations of Krylov subspace techniques are presented. The performance of these methods on some realistic problems are compared.

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

NASA Technical Reports Server (NTRS)

Ceccobello, C.; Farinelli, R.; Titarchuk, L.

2014-01-01

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

Normal modes and frequencies from covariances in molecular dynamics or Monte Carlo simulations

NASA Astrophysics Data System (ADS)

Strachan, Alejandro

2004-01-01

We propose a simple method to obtain normal modes (NMs) and their characteristic frequencies from molecular dynamics or Monte Carlo simulations at any temperature. The resulting NM are consistent with the vibrational density of states (DOS) (every feature of the DOS can be attributed to one or few NMs). At low temperatures they coincide with the ones obtained from the Hessian matrix. We define the NMs (ρi) by imposing the condition that their velocities be uncorrelated to each other: <ρ˙i(t)ρ˙j(t)>∝δij, where < > denotes time average and δij is Kronecker's delta. With this definition the modes are the eigenvectors of the matrix Kijv=1/2<√mimj vivj> [i, j=1,…,3N (N being the number of atoms); m are masses and v atomic velocities]. The eigenvalues of Kijv, λiv, represent the kinetic energy in each NM. The ratio between the eigenvalues (λiv) and those obtained using positions (λir), accelerations (λia) in Kijv instead of velocities are a very good approximation to the mode frequencies: 2πνi˜(λiv/λix)(1/2)˜(λia/λix)(1/4). We demonstrate the new method using with two cases: an isolated water molecule and a crystalline polymer.

Polarization-resolved sensing with tilted fiber Bragg gratings: theory and limits of detection

NASA Astrophysics Data System (ADS)

Bialiayeu, Aliaksandr; Ianoul, Anatoli; Albert, Jacques

2015-08-01

Polarization based sensing with tilted fiber Bragg grating (TFBG) sensors is analysed theoretically by two alternative approaches. The first method is based on tracking the grating transmission for two orthogonal states of linear polarized light that are extracted from the measured Jones matrix or Stokes vectors of the TFBG transmission spectra. The second method is based on the measurements along the system principle axes and polarization dependent loss (PDL) parameter, also calculated from measured data. It is shown that the frequent crossing of the Jones matrix eigenvalues as a function of wavelength leads to a non-physical interchange of the calculated principal axes; a method to remove this unwanted mathematical artefact and to restore the order of the system eigenvalues and the corresponding principal axes is provided. A comparison of the two approaches reveals that the PDL method provides a smaller standard deviation and therefore lower limit of detection in refractometric sensing. Furthermore, the polarization analysis of the measured spectra allows for the identification of the principal states of polarization of the sensor system and consequentially for the calculation of the transmission spectrum for any incident polarization state. The stability of the orientation of the system principal axes is also investigated as a function of wavelength.

Computing singularities of perturbation series

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kvaal, Simen; Jarlebring, Elias; Michiels, Wim

2011-03-15

Many properties of current ab initio approaches to the quantum many-body problem, both perturbational and otherwise, are related to the singularity structure of the Rayleigh-Schroedinger perturbation series. A numerical procedure is presented that in principle computes the complete set of singularities, including the dominant singularity which limits the radius of convergence. The method approximates the singularities as eigenvalues of a certain generalized eigenvalue equation which is solved using iterative techniques. It relies on computation of the action of the Hamiltonian matrix on a vector and does not rely on the terms in the perturbation series. The method can be usefulmore » for studying perturbation series of typical systems of moderate size, for fundamental development of resummation schemes, and for understanding the structure of singularities for typical systems. Some illustrative model problems are studied, including a helium-like model with {delta}-function interactions for which Moeller-Plesset perturbation theory is considered and the radius of convergence found.« less

Operator mixing in the ɛ -expansion: Scheme and evanescent-operator independence

NASA Astrophysics Data System (ADS)

Di Pietro, Lorenzo; Stamou, Emmanuel

2018-03-01

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson-Fisher type in noninteger dimension d =4 -2 ɛ . Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in ɛ -expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in d =4 -2 ɛ at order O (ɛ2).

The Path Resistance Method for Bounding the Smallest Nontrivial Eigenvalue of a Laplacian

NASA Technical Reports Server (NTRS)

Guattery, Stephen; Leighton, Tom; Miller, Gary L.

1997-01-01

We introduce the path resistance method for lower bounds on the smallest nontrivial eigenvalue of the Laplacian matrix of a graph. The method is based on viewing the graph in terms of electrical circuits; it uses clique embeddings to produce lower bounds on lambda(sub 2) and star embeddings to produce lower bounds on the smallest Rayleigh quotient when there is a zero Dirichlet boundary condition. The method assigns priorities to the paths in the embedding; we show that, for an unweighted tree T, using uniform priorities for a clique embedding produces a lower bound on lambda(sub 2) that is off by at most an 0(log diameter(T)) factor. We show that the best bounds this method can produce for clique embeddings are the same as for a related method that uses clique embeddings and edge lengths to produce bounds.

Symmetric and Asymmetric Tendencies in Stable Complex Systems

PubMed Central

Tan, James P. L.

2016-01-01

A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by obtaining eigenvalue bounds of the Jacobian, we show that stable complex systems will favor mutualistic and competitive relationships that are asymmetrical (non-reciprocative) and trophic relationships that are symmetrical (reciprocative). Additionally, we define a measure called the interdependence diversity that quantifies how distributed the dependencies are between the dynamical variables in the system. We find that increasing interdependence diversity has a destabilizing effect on the equilibrium point, and the effect is greater for trophic relationships than for mutualistic and competitive relationships. These predictions are consistent with empirical observations in ecology. More importantly, our findings suggest stabilization algorithms that can apply very generally to a variety of complex systems. PMID:27545722

Symmetric and Asymmetric Tendencies in Stable Complex Systems.

PubMed

Tan, James P L

2016-08-22

A commonly used approach to study stability in a complex system is by analyzing the Jacobian matrix at an equilibrium point of a dynamical system. The equilibrium point is stable if all eigenvalues have negative real parts. Here, by obtaining eigenvalue bounds of the Jacobian, we show that stable complex systems will favor mutualistic and competitive relationships that are asymmetrical (non-reciprocative) and trophic relationships that are symmetrical (reciprocative). Additionally, we define a measure called the interdependence diversity that quantifies how distributed the dependencies are between the dynamical variables in the system. We find that increasing interdependence diversity has a destabilizing effect on the equilibrium point, and the effect is greater for trophic relationships than for mutualistic and competitive relationships. These predictions are consistent with empirical observations in ecology. More importantly, our findings suggest stabilization algorithms that can apply very generally to a variety of complex systems.

Tensor Sparse Coding for Positive Definite Matrices.

PubMed

Sivalingam, Ravishankar; Boley, Daniel; Morellas, Vassilios; Papanikolopoulos, Nikos

2013-08-02

In recent years, there has been extensive research on sparse representation of vector-valued signals. In the matrix case, the data points are merely vectorized and treated as vectors thereafter (for e.g., image patches). However, this approach cannot be used for all matrices, as it may destroy the inherent structure of the data. Symmetric positive definite (SPD) matrices constitute one such class of signals, where their implicit structure of positive eigenvalues is lost upon vectorization. This paper proposes a novel sparse coding technique for positive definite matrices, which respects the structure of the Riemannian manifold and preserves the positivity of their eigenvalues, without resorting to vectorization. Synthetic and real-world computer vision experiments with region covariance descriptors demonstrate the need for and the applicability of the new sparse coding model. This work serves to bridge the gap between the sparse modeling paradigm and the space of positive definite matrices.

Tensor sparse coding for positive definite matrices.

PubMed

Sivalingam, Ravishankar; Boley, Daniel; Morellas, Vassilios; Papanikolopoulos, Nikolaos

2014-03-01

In recent years, there has been extensive research on sparse representation of vector-valued signals. In the matrix case, the data points are merely vectorized and treated as vectors thereafter (for example, image patches). However, this approach cannot be used for all matrices, as it may destroy the inherent structure of the data. Symmetric positive definite (SPD) matrices constitute one such class of signals, where their implicit structure of positive eigenvalues is lost upon vectorization. This paper proposes a novel sparse coding technique for positive definite matrices, which respects the structure of the Riemannian manifold and preserves the positivity of their eigenvalues, without resorting to vectorization. Synthetic and real-world computer vision experiments with region covariance descriptors demonstrate the need for and the applicability of the new sparse coding model. This work serves to bridge the gap between the sparse modeling paradigm and the space of positive definite matrices.

Photonic band gap structure simulator

DOEpatents

Chen, Chiping; Shapiro, Michael A.; Smirnova, Evgenya I.; Temkin, Richard J.; Sirigiri, Jagadishwar R.

2006-10-03

A system and method for designing photonic band gap structures. The system and method provide a user with the capability to produce a model of a two-dimensional array of conductors corresponding to a unit cell. The model involves a linear equation. Boundary conditions representative of conditions at the boundary of the unit cell are applied to a solution of the Helmholtz equation defined for the unit cell. The linear equation can be approximated by a Hermitian matrix. An eigenvalue of the Helmholtz equation is calculated. One computation approach involves calculating finite differences. The model can include a symmetry element, such as a center of inversion, a rotation axis, and a mirror plane. A graphical user interface is provided for the user's convenience. A display is provided to display to a user the calculated eigenvalue, corresponding to a photonic energy level in the Brilloin zone of the unit cell.

Bifurcations of periodic motion in a three-degree-of-freedom vibro-impact system with clearance

NASA Astrophysics Data System (ADS)

Liu, Yongbao; Wang, Qiang; Xu, Huidong

2017-07-01

The smooth bifurcation and grazing non-smooth bifurcation of periodic motion of a three-degree-of-freedom vibro-impact system with clearance are studied in this paper. Firstly, a periodic solution of vibro-impact system is solved and a six-dimensional Poincaré map is established. Then, for the six-dimensional Poincaré map, the analytic expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that with application of the classical critical criterion described by the properties of eigenvalues, we have to numerically compute eigenvalues point by point and check their properties to search for the bifurcation points. Such the numerical calculation is a laborious job in the process of determining bifurcation points. To overcome the difficulty that originates from the classical bifurcation criteria, the explicit critical criteria without using eigenvalues calculation of high-dimensional map are applied to determine bifurcation points of Co-dimension-one period doubling bifurcation and Co-dimension-one Neimark-Sacker bifurcation and Co-dimension-two Flip-Neimark-Sacker bifurcation, and then local dynamical behaviors of these bifurcations are analyzed. Moreover, the directions of period doubling bifurcation and Neimark-Sacker bifurcation are analyzed by center manifold reduction theory and normal form approach. Finally, the existence of the grazing periodic motion of the vibro-impact system is analyzed and the grazing bifurcation point is obtained, the discontinuous grazing bifurcation behavior is studied based on the compound normal form map near the grazing point, the discontinuous jumping phenomenon and co-existing multiple solutions near the grazing bifurcation point are revealed.

Network-Based Detection and Classification of Seismovolcanic Tremors: Example From the Klyuchevskoy Volcanic Group in Kamchatka

NASA Astrophysics Data System (ADS)

Soubestre, Jean; Shapiro, Nikolai M.; Seydoux, Léonard; de Rosny, Julien; Droznin, Dmitry V.; Droznina, Svetlana Ya.; Senyukov, Sergey L.; Gordeev, Evgeniy I.

2018-01-01

We develop a network-based method for detecting and classifying seismovolcanic tremors. The proposed approach exploits the coherence of tremor signals across the network that is estimated from the array covariance matrix. The method is applied to four and a half years of continuous seismic data recorded by 19 permanent seismic stations in the vicinity of the Klyuchevskoy volcanic group in Kamchatka (Russia), where five volcanoes were erupting during the considered time period. We compute and analyze daily covariance matrices together with their eigenvalues and eigenvectors. As a first step, most coherent signals corresponding to dominating tremor sources are detected based on the width of the covariance matrix eigenvalues distribution. Thus, volcanic tremors of the two volcanoes known as most active during the considered period, Klyuchevskoy and Tolbachik, are efficiently detected. As a next step, we consider the daily array covariance matrix's first eigenvector. Our main hypothesis is that these eigenvectors represent the principal components of the daily seismic wavefield and, for days with tremor activity, characterize dominant tremor sources. Those daily first eigenvectors, which can be used as network-based fingerprints of tremor sources, are then grouped into clusters using correlation coefficient as a measure of the vector similarity. As a result, we identify seven clusters associated with different periods of activity of four volcanoes: Tolbachik, Klyuchevskoy, Shiveluch, and Kizimen. The developed method does not require a priori knowledge and is fully automatic; and the database of the network-based tremor fingerprints can be continuously enriched with newly available data.

Analytical development of disturbed matrix eigenvalue problem applied to mixed convection stability analysis in Darcy media

NASA Astrophysics Data System (ADS)

Hamed, Haikel Ben; Bennacer, Rachid

2008-08-01

This work consists in evaluating algebraically and numerically the influence of a disturbance on the spectral values of a diagonalizable matrix. Thus, two approaches will be possible; to use the theorem of disturbances of a matrix depending on a parameter, due to Lidskii and primarily based on the structure of Jordan of the no disturbed matrix. The second approach consists in factorizing the matrix system, and then carrying out a numerical calculation of the roots of the disturbances matrix characteristic polynomial. This problem can be a standard model in the equations of the continuous media mechanics. During this work, we chose to use the second approach and in order to illustrate the application, we choose the Rayleigh-Bénard problem in Darcy media, disturbed by a filtering through flow. The matrix form of the problem is calculated starting from a linear stability analysis by a finite elements method. We show that it is possible to break up the general phenomenon into other elementary ones described respectively by a disturbed matrix and a disturbance. A good agreement between the two methods was seen. To cite this article: H.B. Hamed, R. Bennacer, C. R. Mecanique 336 (2008).

From Principal Component to Direct Coupling Analysis of Coevolution in Proteins: Low-Eigenvalue Modes are Needed for Structure Prediction

PubMed Central

Cocco, Simona; Monasson, Remi; Weigt, Martin

2013-01-01

Various approaches have explored the covariation of residues in multiple-sequence alignments of homologous proteins to extract functional and structural information. Among those are principal component analysis (PCA), which identifies the most correlated groups of residues, and direct coupling analysis (DCA), a global inference method based on the maximum entropy principle, which aims at predicting residue-residue contacts. In this paper, inspired by the statistical physics of disordered systems, we introduce the Hopfield-Potts model to naturally interpolate between these two approaches. The Hopfield-Potts model allows us to identify relevant ‘patterns’ of residues from the knowledge of the eigenmodes and eigenvalues of the residue-residue correlation matrix. We show how the computation of such statistical patterns makes it possible to accurately predict residue-residue contacts with a much smaller number of parameters than DCA. This dimensional reduction allows us to avoid overfitting and to extract contact information from multiple-sequence alignments of reduced size. In addition, we show that low-eigenvalue correlation modes, discarded by PCA, are important to recover structural information: the corresponding patterns are highly localized, that is, they are concentrated in few sites, which we find to be in close contact in the three-dimensional protein fold. PMID:23990764

Computation of steady and unsteady quasi-one-dimensional viscous/inviscid interacting internal flows at subsonic, transonic, and supersonic Mach numbers

NASA Technical Reports Server (NTRS)

Swafford, Timothy W.; Huddleston, David H.; Busby, Judy A.; Chesser, B. Lawrence

1992-01-01

Computations of viscous-inviscid interacting internal flowfields are presented for steady and unsteady quasi-one-dimensional (Q1D) test cases. The unsteady Q1D Euler equations are coupled with integral boundary-layer equations for unsteady, two-dimensional (planar or axisymmetric), turbulent flow over impermeable, adiabatic walls. The coupling methodology differs from that used in most techniques reported previously in that the above mentioned equation sets are written as a complete system and solved simultaneously; that is, the coupling is carried out directly through the equations as opposed to coupling the solutions of the different equation sets. Solutions to the coupled system of equations are obtained using both explicit and implicit numerical schemes for steady subsonic, steady transonic, and both steady and unsteady supersonic internal flowfields. Computed solutions are compared with measurements as well as Navier-Stokes and inverse boundary-layer methods. An analysis of the eigenvalues of the coefficient matrix associated with the quasi-linear form of the coupled system of equations indicates the presence of complex eigenvalues for certain flow conditions. It is concluded that although reasonable solutions can be obtained numerically, these complex eigenvalues contribute to the overall difficulty in obtaining numerical solutions to the coupled system of equations.

Vibration properties of and power harvested by a system of electromagnetic vibration energy harvesters that have electrical dynamics

NASA Astrophysics Data System (ADS)

Cooley, Christopher G.

2017-09-01

This study investigates the vibration and dynamic response of a system of coupled electromagnetic vibration energy harvesting devices that each consist of a proof mass, elastic structure, electromagnetic generator, and energy harvesting circuit with inductance, resistance, and capacitance. The governing equations for the coupled electromechanical system are derived using Newtonian mechanics and Kirchhoff circuit laws for an arbitrary number of these subsystems. The equations are cast in matrix operator form to expose the device's vibration properties. The device's complex-valued eigenvalues and eigenvectors are related to physical characteristics of its vibration. Because the electrical circuit has dynamics, these devices have more natural frequencies than typical electromagnetic vibration energy harvesters that have purely resistive circuits. Closed-form expressions for the steady state dynamic response and average power harvested are derived for devices with a single subsystem. Example numerical results for single and double subsystem devices show that the natural frequencies and vibration modes obtained from the eigenvalue problem agree with the resonance locations and response amplitudes obtained independently from forced response calculations. This agreement demonstrates the usefulness of solving eigenvalue problems for these devices. The average power harvested by the device differs substantially at each resonance. Devices with multiple subsystems have multiple modes where large amounts of power are harvested.

Spherical space Bessel-Legendre-Fourier localized modes solver for electromagnetic waves.

PubMed

Alzahrani, Mohammed A; Gauthier, Robert C

2015-10-05

Maxwell's vector wave equations are solved for dielectric configurations that match the symmetry of a spherical computational domain. The electric or magnetic field components and the inverse of the dielectric profile are series expansion defined using basis functions composed of the lowest order spherical Bessel function, polar angle single index dependant Legendre polynomials and azimuthal complex exponential (BLF). The series expressions and non-traditional form of the basis functions result in an eigenvalue matrix formulation of Maxwell's equations that are relatively compact and accurately solvable on a desktop PC. The BLF matrix returns the frequencies and field profiles for steady states modes. The key steps leading to the matrix populating expressions are provided. The validity of the numerical technique is confirmed by comparing the results of computations to those published using complementary techniques.

Photonic Breast Tomography and Tumor Aggressiveness Assessment

DTIC Science & Technology

2010-07-01

removal of breast tumours (Specific Aim 4). While the TROT approach [7] has been introduced in other areas, such as, array processing for acoustic and...to the time-reversal matrix used in the general area of array processing for acoustic and radar time-reversal imaging [15]. The eigenvalue equation...spectrum [Eq.(1) in Ref. 8] is calculated directly for all voxels in the sample using the vector subspace method, Multiple Signal Classification ( MUSIC

On Dynamics of Spinning Structures

NASA Technical Reports Server (NTRS)

Gupta, K. K.; Ibrahim, A.

2012-01-01

This paper provides details of developments pertaining to vibration analysis of gyroscopic systems, that involves a finite element structural discretization followed by the solution of the resulting matrix eigenvalue problem by a progressive, accelerated simultaneous iteration technique. Thus Coriolis, centrifugal and geometrical stiffness matrices are derived for shell and line elements, followed by the eigensolution details as well as solution of representative problems that demonstrates the efficacy of the currently developed numerical procedures and tools.

Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations

PubMed Central

Tan, Ning-Bo; Huang, Ting-Zhu; Hu, Ze-Jun

2013-01-01

An incomplete augmented Lagrangian preconditioner, for the steady incompressible Navier-Stokes equations discretized by stable finite elements, is proposed. The eigenvalues of the preconditioned matrix are analyzed. Numerical experiments show that the incomplete augmented Lagrangian-based preconditioner proposed is very robust and performs quite well by the Picard linearization or the Newton linearization over a wide range of values of the viscosity on both uniform and stretched grids. PMID:24235888

A Note on Asymptotic Joint Distribution of the Eigenvalues of a Noncentral Multivariate F Matrix.

DTIC Science & Technology

1984-11-01

Krishnaiah (1982). Now, let us consider the samples drawn from the k multivariate normal popuiejons. Let (Xlt....Xpt) denote the mean vector of the t...to maltivariate problems. Sankh-ya, 4, 381-39(s. (71 KRISHNAIAH , P. R. (1982). Selection of variables in discrimlnant analysis. In Handbook of...Statistics, Volume 2 (P. R. Krishnaiah , editor), 805-820. North-Holland Publishing Company. 6. Unclassifie INSTRUCTIONS REPORT DOCUMENTATION PAGE

Classification Techniques for Multivariate Data Analysis.

DTIC Science & Technology

1980-03-28

analysis among biologists, botanists, and ecologists, while some social scientists may refer "typology". Other frequently encountered terms are pattern...the determinantal equation: lB -XW 0 (42) 49 The solutions X. are the eigenvalues of the matrix W-1 B 1 as in discriminant analysis. There are t non...Statistical Package for Social Sciences (SPSS) (14) subprogram FACTOR was used for the principal components analysis. It is designed both for the factor

Numerical linear algebra in data mining

NASA Astrophysics Data System (ADS)

Eldén, Lars

Ideas and algorithms from numerical linear algebra are important in several areas of data mining. We give an overview of linear algebra methods in text mining (information retrieval), pattern recognition (classification of handwritten digits), and PageRank computations for web search engines. The emphasis is on rank reduction as a method of extracting information from a data matrix, low-rank approximation of matrices using the singular value decomposition and clustering, and on eigenvalue methods for network analysis.

Closed form solution for a double quantum well using Gröbner basis

NASA Astrophysics Data System (ADS)

Acus, A.; Dargys, A.

2011-07-01

Analytical expressions for the spectrum, eigenfunctions and dipole matrix elements of a square double quantum well (DQW) are presented for a general case when the potential in different regions of the DQW has different heights and the effective masses are different. This was achieved by using a Gröbner basis algorithm that allowed us to disentangle the resulting coupled polynomials without explicitly solving the transcendental eigenvalue equation.

Electromagnetics. Volume 1, Number 4, October-December 1981.

DTIC Science & Technology

1981-01-01

terms. 1.6 Matrix and Operator Theory Integral equations have been cast in approximate numerical form by the moment method (MoM). In this numerical...introduced the eigenmode expansion method to find more properties of the SEM [3.4]. One defines eigenvalues and eigenmodes for the integral operator (kernel...exterior surface of the system. Mechanisms that play a role in the penetration are (1) diffusion through metal skins , (2) field leakage through

Singular-value demodulation of phase-shifted holograms.

PubMed

Lopes, Fernando; Atlan, Michael

2015-06-01

We report on phase-shifted holographic interferogram demodulation by singular-value decomposition. Numerical processing of optically acquired interferograms over several modulation periods was performed in two steps: (1) rendering of off-axis complex-valued holograms by Fresnel transformation of the interferograms; and (2) eigenvalue spectrum assessment of the lag-covariance matrix of hologram pixels. Experimental results in low-light recording conditions were compared with demodulation by Fourier analysis, in the presence of random phase drifts.

A new method to real-normalize measured complex modes

NASA Technical Reports Server (NTRS)

Wei, Max L.; Allemang, Randall J.; Zhang, Qiang; Brown, David L.

1987-01-01

A time domain subspace iteration technique is presented to compute a set of normal modes from the measured complex modes. By using the proposed method, a large number of physical coordinates are reduced to a smaller number of model or principal coordinates. Subspace free decay time responses are computed using properly scaled complex modal vectors. Companion matrix for the general case of nonproportional damping is then derived in the selected vector subspace. Subspace normal modes are obtained through eigenvalue solution of the (M sub N) sup -1 (K sub N) matrix and transformed back to the physical coordinates to get a set of normal modes. A numerical example is presented to demonstrate the outlined theory.

Matrix Sturm-Liouville equation with a Bessel-type singularity on a finite interval

NASA Astrophysics Data System (ADS)

Bondarenko, Natalia

2017-03-01

The matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval is studied. Special fundamental systems of solutions for this equation are constructed: analytic Bessel-type solutions with the prescribed behavior at the singular point and Birkhoff-type solutions with the known asymptotics for large values of the spectral parameter. The asymptotic formulas for Stokes multipliers, connecting these two fundamental systems of solutions, are derived. We also set boundary conditions and obtain asymptotic formulas for the spectral data (the eigenvalues and the weight matrices) of the boundary value problem. Our results will be useful in the theory of direct and inverse spectral problems.

Overview of Krylov subspace methods with applications to control problems

NASA Technical Reports Server (NTRS)

Saad, Youcef

1989-01-01

An overview of projection methods based on Krylov subspaces are given with emphasis on their application to solving matrix equations that arise in control problems. The main idea of Krylov subspace methods is to generate a basis of the Krylov subspace Span and seek an approximate solution the the original problem from this subspace. Thus, the original matrix problem of size N is approximated by one of dimension m typically much smaller than N. Krylov subspace methods have been very successful in solving linear systems and eigenvalue problems and are now just becoming popular for solving nonlinear equations. It is shown how they can be used to solve partial pole placement problems, Sylvester's equation, and Lyapunov's equation.

Approximation methods in relativistic eigenvalue perturbation theory

NASA Astrophysics Data System (ADS)

Noble, Jonathan Howard

In this dissertation, three questions, concerning approximation methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo--Hermitian Hamiltonian, and how can its eigenvalues be approximated via numerical calculations? This is a fairly broad topic, and the scope of the investigation is narrowed by focusing on a subgroup of pseudo--Hermitian operators, namely, PT--symmetric operators. Within a numerical approach, one projects a PT--symmetric Hamiltonian onto an appropriate basis, and uses a straightforward two--step algorithm to diagonalize the resulting matrix, leading to numerically approximated eigenvalues. (ii) Within an analytic ansatz, how can a relativistic Dirac Hamiltonian be decoupled into particle and antiparticle degrees of freedom, in appropriate kinematic limits? One possible answer is the Foldy--Wouthuysen transform; however, there are alter- native methods which seem to have some advantages over the time--tested approach. One such method is investigated by applying both the traditional Foldy--Wouthuysen transform and the "chiral" Foldy--Wouthuysen transform to a number of Dirac Hamiltonians, including the central-field Hamiltonian for a gravitationally bound system; namely, the Dirac-(Einstein-)Schwarzschild Hamiltonian, which requires the formal- ism of general relativity. (iii) Are there are pseudo--Hermitian variants of Dirac Hamiltonians that can be approximated using a decoupling transformation? The tachyonic Dirac Hamiltonian, which describes faster-than-light spin-1/2 particles, is gamma5--Hermitian, i.e., pseudo-Hermitian. Superluminal particles remain faster than light upon a Lorentz transformation, and hence, the Foldy--Wouthuysen program is unsuited for this case. Thus, inspired by the Foldy--Wouthuysen program, a decoupling transform in the ultrarelativistic limit is proposed, which is applicable to both sub- and superluminal particles.

Sensitivity analysis of eigenvalues for an electro-hydraulic servomechanism

NASA Astrophysics Data System (ADS)

Stoia-Djeska, M.; Safta, C. A.; Halanay, A.; Petrescu, C.

2012-11-01

Electro-hydraulic servomechanisms (EHSM) are important components of flight control systems and their role is to control the movement of the flying control surfaces in response to the movement of the cockpit controls. As flight-control systems, the EHSMs have a fast dynamic response, a high power to inertia ratio and high control accuracy. The paper is devoted to the study of the sensitivity for an electro-hydraulic servomechanism used for an aircraft aileron action. The mathematical model of the EHSM used in this paper includes a large number of parameters whose actual values may vary within some ranges of uncertainty. It consists in a nonlinear ordinary differential equation system composed by the mass and energy conservation equations, the actuator movement equations and the controller equation. In this work the focus is on the sensitivities of the eigenvalues of the linearized homogeneous system, which are the partial derivatives of the eigenvalues of the state-space system with respect the parameters. These are obtained using a modal approach based on the eigenvectors of the state-space direct and adjoint systems. To calculate the eigenvalues and their sensitivity the system's Jacobian and its partial derivatives with respect the parameters are determined. The calculation of the derivative of the Jacobian matrix with respect to the parameters is not a simple task and for many situations it must be done numerically. The system stability is studied in relation with three parameters: m, the equivalent inertial load of primary control surface reduced to the actuator rod; B, the bulk modulus of oil and p a pressure supply proportionality coefficient. All the sensitivities calculated in this work are in good agreement with those obtained through recalculations.

Semiblind channel estimation for MIMO-OFDM systems

NASA Astrophysics Data System (ADS)

Chen, Yi-Sheng; Song, Jyu-Han

2012-12-01

This article proposes a semiblind channel estimation method for multiple-input multiple-output orthogonal frequency-division multiplexing systems based on circular precoding. Relying on the precoding scheme at the transmitters, the autocorrelation matrix of the received data induces a structure relating the outer product of the channel frequency response matrix and precoding coefficients. This structure makes it possible to extract information about channel product matrices, which can be used to form a Hermitian matrix whose positive eigenvalues and corresponding eigenvectors yield the channel impulse response matrix. This article also tests the resistance of the precoding design to finite-sample estimation errors, and explores the effects of the precoding scheme on channel equalization by performing pairwise error probability analysis. The proposed method is immune to channel zero locations, and is reasonably robust to channel order overestimation. The proposed method is applicable to the scenarios in which the number of transmitters exceeds that of the receivers. Simulation results demonstrate the performance of the proposed method and compare it with some existing methods.

Mueller coherency matrix method for contrast image in tissue polarimetry

NASA Astrophysics Data System (ADS)

Arce-Diego, J. L.; Fanjul-Vélez, F.; Samperio-García, D.; Pereda-Cubián, D.

2007-07-01

In this work, we propose the use of the Mueller Coherency matrix of biological tissues in order to increase the information from tissue images and so their contrast. This method involves different Mueller Coherency matrix based parameters, like the eigenvalues analysis, the entropy factor calculation, polarization components crosstalks, linear and circular polarization degrees, hermiticity or the Quaternions analysis in case depolarisation properties of tissue are sufficiently low. All these parameters make information appear clearer and so increase image contrast, so pathologies like cancer could be detected in a sooner stage of development. The election will depend on the concrete pathological process under study. This Mueller Coherency matrix method can be applied to a single tissue point, or it can be combined with a tomographic technique, so as to obtain a 3D representation of polarization contrast parameters in pathological tissues. The application of this analysis to concrete diseases can lead to tissue burn depth estimation or cancer early detection.

Optimal Frequency-Domain System Realization with Weighting

NASA Technical Reports Server (NTRS)

Juang, Jer-Nan; Maghami, Peiman G.

1999-01-01

Several approaches are presented to identify an experimental system model directly from frequency response data. The formulation uses a matrix-fraction description as the model structure. Frequency weighting such as exponential weighting is introduced to solve a weighted least-squares problem to obtain the coefficient matrices for the matrix-fraction description. A multi-variable state-space model can then be formed using the coefficient matrices of the matrix-fraction description. Three different approaches are introduced to fine-tune the model using nonlinear programming methods to minimize the desired cost function. The first method uses an eigenvalue assignment technique to reassign a subset of system poles to improve the identified model. The second method deals with the model in the real Schur or modal form, reassigns a subset of system poles, and adjusts the columns (rows) of the input (output) influence matrix using a nonlinear optimizer. The third method also optimizes a subset of poles, but the input and output influence matrices are refined at every optimization step through least-squares procedures.

Effect of the interconnected network structure on the epidemic threshold.

PubMed

Wang, Huijuan; Li, Qian; D'Agostino, Gregorio; Havlin, Shlomo; Stanley, H Eugene; Van Mieghem, Piet

2013-08-01

Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a 2N×2N matrix that depicts the connectivity within each of two networks of size N, and B a 2N×2N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation, we determine that a critical susceptible-infected-susceptible (SIS) epidemic threshold in two interconnected networks is 1/λ(1)(A+αB), where the infection rate is β within each of the two individual networks and αβ in the interconnected links between the two networks and λ(1)(A+αB) is the largest eigenvalue of the matrix A+αB. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive λ(1)(A+αB) using a perturbation approximation for small and large α, the lower and upper bound for any α as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for λ(1)(A+αB) using numerical simulations, and determine how component network features affect λ(1)(A+αB). We note that, given two isolated networks G(1) and G(2) with principal eigenvectors x and y, respectively, λ(1)(A+αB) tends to be higher when nodes i and j with a higher eigenvector component product x(i)y(j) are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.

Fast Kalman Filter for Random Walk Forecast model

NASA Astrophysics Data System (ADS)

Saibaba, A.; Kitanidis, P. K.

2013-12-01

Kalman filtering is a fundamental tool in statistical time series analysis to understand the dynamics of large systems for which limited, noisy observations are available. However, standard implementations of the Kalman filter are prohibitive because they require O(N^2) in memory and O(N^3) in computational cost, where N is the dimension of the state variable. In this work, we focus our attention on the Random walk forecast model which assumes the state transition matrix to be the identity matrix. This model is frequently adopted when the data is acquired at a timescale that is faster than the dynamics of the state variables and there is considerable uncertainty as to the physics governing the state evolution. We derive an efficient representation for the a priori and a posteriori estimate covariance matrices as a weighted sum of two contributions - the process noise covariance matrix and a low rank term which contains eigenvectors from a generalized eigenvalue problem, which combines information from the noise covariance matrix and the data. We describe an efficient algorithm to update the weights of the above terms and the computation of eigenmodes of the generalized eigenvalue problem (GEP). The resulting algorithm for the Kalman filter with Random walk forecast model scales as O(N) or O(N log N), both in memory and computational cost. This opens up the possibility of real-time adaptive experimental design and optimal control in systems of much larger dimension than was previously feasible. For a small number of measurements (~ 300 - 400), this procedure can be made numerically exact. However, as the number of measurements increase, for several choices of measurement operators and noise covariance matrices, the spectrum of the (GEP) decays rapidly and we are justified in only retaining the dominant eigenmodes. We discuss tradeoffs between accuracy and computational cost. The resulting algorithms are applied to an example application from ray-based travel time tomography.

Effect of the interconnected network structure on the epidemic threshold

NASA Astrophysics Data System (ADS)

Wang, Huijuan; Li, Qian; D'Agostino, Gregorio; Havlin, Shlomo; Stanley, H. Eugene; Van Mieghem, Piet

2013-08-01

Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a 2N×2N matrix that depicts the connectivity within each of two networks of size N, and B a 2N×2N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation, we determine that a critical susceptible-infected-susceptible (SIS) epidemic threshold in two interconnected networks is 1/λ1(A+αB), where the infection rate is β within each of the two individual networks and αβ in the interconnected links between the two networks and λ1(A+αB) is the largest eigenvalue of the matrix A+αB. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive λ1(A+αB) using a perturbation approximation for small and large α, the lower and upper bound for any α as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for λ1(A+αB) using numerical simulations, and determine how component network features affect λ1(A+αB). We note that, given two isolated networks G1 and G2 with principal eigenvectors x and y, respectively, λ1(A+αB) tends to be higher when nodes i and j with a higher eigenvector component product xiyj are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.

A seismic coherency method using spectral amplitudes

NASA Astrophysics Data System (ADS)

Sui, Jing-Kun; Zheng, Xiao-Dong; Li, Yan-Dong

2015-09-01

Seismic coherence is used to detect discontinuities in underground media. However, strata with steeply dipping structures often produce false low coherence estimates and thus incorrect discontinuity characterization results. It is important to eliminate or reduce the effect of dipping on coherence estimates. To solve this problem, time-domain dip scanning is typically used to improve estimation of coherence in areas with steeply dipping structures. However, the accuracy of the time-domain estimation of dip is limited by the sampling interval. In contrast, the spectrum amplitude is not affected by the time delays in adjacent seismic traces caused by dipping structures. We propose a coherency algorithm that uses the spectral amplitudes of seismic traces within a predefined analysis window to construct the covariance matrix. The coherency estimates with the proposed algorithm is defined as the ratio between the dominant eigenvalue and the sum of all eigenvalues of the constructed covariance matrix. Thus, we eliminate the effect of dipping structures on coherency estimates. In addition, because different frequency bands of spectral amplitudes are used to estimate coherency, the proposed algorithm has multiscale features. Low frequencies are effective for characterizing large-scale faults, whereas high frequencies are better in characterizing small-scale faults. Application to synthetic and real seismic data show that the proposed algorithm can eliminate the effect of dip and produce better coherence estimates than conventional coherency algorithms in areas with steeply dipping structures.

Entanglement transitions induced by large deviations

NASA Astrophysics Data System (ADS)

Bhosale, Udaysinh T.

2017-12-01

The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as A and B , is computed analytically using a Coulomb gas method. It is shown that this probability, for large N , goes as exp[-β N2Φ (ζ ) ] , where the parameter β is the Dyson index of the ensemble, ζ is the large deviation parameter, while the rate function Φ (ζ ) is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems 1 and 2, obtained by further partitioning the subsystem A , using the properties of the density matrix's partial transpose ρ12Γ. The density of states of ρ12Γ is found to be close to the Wigner's semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter ζ . Log negativity is used to quantify the entanglement between subsystems 1 and 2. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.

Entanglement transitions induced by large deviations.

PubMed

Bhosale, Udaysinh T

2017-12-01

The probability of large deviations of the smallest Schmidt eigenvalue for random pure states of bipartite systems, denoted as A and B, is computed analytically using a Coulomb gas method. It is shown that this probability, for large N, goes as exp[-βN^{2}Φ(ζ)], where the parameter β is the Dyson index of the ensemble, ζ is the large deviation parameter, while the rate function Φ(ζ) is calculated exactly. Corresponding equilibrium Coulomb charge density is derived for its large deviations. Effects of the large deviations of the extreme (largest and smallest) Schmidt eigenvalues on the bipartite entanglement are studied using the von Neumann entropy. Effect of these deviations is also studied on the entanglement between subsystems 1 and 2, obtained by further partitioning the subsystem A, using the properties of the density matrix's partial transpose ρ_{12}^{Γ}. The density of states of ρ_{12}^{Γ} is found to be close to the Wigner's semicircle law with these large deviations. The entanglement properties are captured very well by a simple random matrix model for the partial transpose. The model predicts the entanglement transition across a critical large deviation parameter ζ. Log negativity is used to quantify the entanglement between subsystems 1 and 2. Analytical formulas for it are derived using the simple model. Numerical simulations are in excellent agreement with the analytical results.

Biophotonic applications of eigenchannels in a scattering medium (Conference Presentation)

NASA Astrophysics Data System (ADS)

Kim, Moonseok; Choi, Wonjun; Choi, Youngwoon; Yoon, Changhyeong; Choi, Wonshik

2016-03-01

When waves travel through disordered media such as ground glass and skin tissues, they are scattered multiple times. Most of the incoming energy bounces back at the superficial layers and only a small fraction can penetrate deep inside. This has been a limiting factor for the working depth of various optical techniques. We present a systematic method to enhance wave penetration to the scattering media. Specifically, we measured the reflection matrix of a disordered medium with wide angular coverage for each orthogonal polarization states. From the reflection matrix, we identified reflection eigenchannels of the medium, and shaped the incident wave into the reflection eigenchannel with smallest eigenvalue, which we call anti-reflection mode. This makes reflectance reduced and wave penetration increased as a result of the energy conservation. We demonstrated transmission enhancement by more than a factor of 3 by the coupling of the incident waves to the anti-reflection modes. Based on the uneven distribution of eigenvalues of reflection eigenchannels, we further developed an iterative feedback control method for finding and coupling light to anti-reflection modes. Since this adaptive control method can keep up with sample perturbation, it promotes the applicability of exploiting reflection eigenchannels. Our approach of delivering light deep into the scattering media will contribute to enhancing the sensitivity of detecting objects hidden under scattering layers, which is universal problem ranging from geology to life science.

Unifying model for random matrix theory in arbitrary space dimensions

NASA Astrophysics Data System (ADS)

Cicuta, Giovanni M.; Krausser, Johannes; Milkus, Rico; Zaccone, Alessio

2018-03-01

A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension d , and for arbitrary values of the lattice coordination number Z , are shown and discussed. As a function of these two parameters (and their ratio Z /d ), the most studied models in random matrix theory (Erdos-Renyi graphs, effective medium, and replicas) can be reproduced in the various limits of block dimensionality d . Remarkably, the Marchenko-Pastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or d →∞ , which clarifies the physical meaning of space dimension in these models. We feel that the approximate results for d =3 provided by our method may have many potential applications in the future, from the vibrational spectrum of glasses and elastic networks to wave localization, disordered conductors, random resistor networks, and random walks.

Random matrix theory and fund of funds portfolio optimisation

NASA Astrophysics Data System (ADS)

Conlon, T.; Ruskin, H. J.; Crane, M.

2007-08-01

The proprietary nature of Hedge Fund investing means that it is common practise for managers to release minimal information about their returns. The construction of a fund of hedge funds portfolio requires a correlation matrix which often has to be estimated using a relatively small sample of monthly returns data which induces noise. In this paper, random matrix theory (RMT) is applied to a cross-correlation matrix C, constructed using hedge fund returns data. The analysis reveals a number of eigenvalues that deviate from the spectrum suggested by RMT. The components of the deviating eigenvectors are found to correspond to distinct groups of strategies that are applied by hedge fund managers. The inverse participation ratio is used to quantify the number of components that participate in each eigenvector. Finally, the correlation matrix is cleaned by separating the noisy part from the non-noisy part of C. This technique is found to greatly reduce the difference between the predicted and realised risk of a portfolio, leading to an improved risk profile for a fund of hedge funds.

Periodic orbits around areostationary points in the Martian gravity field

NASA Astrophysics Data System (ADS)

Liu, Xiao-Dong; Baoyin, Hexi; Ma, Xing-Rui

2012-05-01

This study investigates the problem of areostationary orbits around Mars in three-dimensional space. Areostationary orbits are expected to be used to establish a future telecommunication network for the exploration of Mars. However, no artificial satellites have been placed in these orbits thus far. The characteristics of the Martian gravity field are presented, and areostationary points and their linear stability are calculated. By taking linearized solutions in the planar case as the initial guesses and utilizing the Levenberg-Marquardt method, families of periodic orbits around areostationary points are shown to exist. Short-period orbits and long-period orbits are found around linearly stable areostationary points, but only short-period orbits are found around unstable areostationary points. Vertical periodic orbits around both linearly stable and unstable areostationary points are also examined. Satellites in these periodic orbits could depart from areostationary points by a few degrees in longitude, which would facilitate observation of the Martian topography. Based on the eigenvalues of the monodromy matrix, the evolution of the stability index of periodic orbits is determined. Finally, heteroclinic orbits connecting the two unstable areostationary points are found, providing the possibility for orbital transfer with minimal energy consumption.

Field patterns: a new mathematical object

PubMed Central

Mattei, Ornella

2017-01-01

Field patterns occur in space–time microstructures such that a disturbance propagating along a characteristic line does not evolve into a cascade of disturbances, but rather concentrates on a pattern of characteristic lines. This pattern is the field pattern. In one spatial direction plus time, the field patterns occur when the slope of the characteristics is, in a sense, commensurate with the space–time microstructure. Field patterns with different spatial shifts do not generally interact, but rather evolve as if they live in separate dimensions, as many dimensions as the number of field patterns. Alternatively one can view a collection as a multi-component potential, with as many components as the number of field patterns. Presumably, if one added a tiny nonlinear term to the wave equation one would then see interactions between these field patterns in the multi-dimensional space that one can consider them to live, or between the different field components of the multi-component potential if one views them that way. As a result of PT-symmetry many of the complex eigenvalues of an appropriately defined transfer matrix have unit norm and hence the corresponding eigenvectors correspond to propagating modes. There are also modes that blow up exponentially with time. PMID:28293143

A simple physical model for forest fire spread

Treesearch

E. Koo; P. Pagni; J. Woycheese; S. Stephens; D. Weise; J. Huff

2005-01-01

Based on energy conservation and detailed heat transfer mechanisms, a simple physical model for fire spread is presented for the limit of one-dimensional steady-state contiguous spread of a line fire in a thermally-thin uniform porous fuel bed. The solution for the fire spread rate is found as an eigenvalue from this model with appropriate boundary conditions through a...

Spectral analysis of localized rotating waves in parabolic systems.

PubMed

Beyn, Wolf-Jürgen; Otten, Denny

2018-04-13

In this paper, we study the spectra and Fredholm properties of Ornstein-Uhlenbeck operators [Formula: see text]where [Formula: see text] is the profile of a rotating wave satisfying [Formula: see text] as [Formula: see text], the map [Formula: see text] is smooth and the matrix [Formula: see text] has eigenvalues with positive real parts and commutes with the limit matrix [Formula: see text] The matrix [Formula: see text] is assumed to be skew-symmetric with eigenvalues (λ 1 ,…,λ d )=(±i σ 1 ,…,±i σ k ,0,…,0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction-diffusion systems. We prove under appropriate conditions that every [Formula: see text] satisfying the dispersion relation [Formula: see text]belongs to the essential spectrum [Formula: see text] in L p For values Re λ to the right of the spectral bound for [Formula: see text], we show that the operator [Formula: see text] is Fredholm of index 0, solve the identification problem for the adjoint operator [Formula: see text] and formulate the Fredholm alternative. Moreover, we show that the set [Formula: see text]belongs to the point spectrum [Formula: see text] in L p We determine the associated eigenfunctions and show that they decay exponentially in space. As an application, we analyse spinning soliton solutions which occur in the Ginzburg-Landau equation and compute their numerical spectra as well as associated eigenfunctions. Our results form the basis for investigating the nonlinear stability of rotating waves in higher space dimensions and truncations to bounded domains. This article is part of the themed issue 'Stability of nonlinear waves and patterns and related topics'. © 2018 The Author(s).

A novel spatially-explicit condition for the onset of waterborne diseases in complex environments

NASA Astrophysics Data System (ADS)

Mari, L.; Gatto, M.; Bertuzzo, E.; Casagrandi, R.; Righetto, L.; Rodriguez-Iturbe, I.; Rinaldo, A.

2012-12-01

In spatial models of waterborne infections the condition that all the local reproduction numbers be larger than one is neither necessary nor sufficient for outbreaks to occur. Here, to properly determine epidemic onset conditions, we examine the transition from stable to unstable of the disease-free equilibrium of a system of nonlinear differential equations characterizing the evolution of susceptible and infected individuals within their respective settlements, and pathogen concentration in their accessible environment. Two different network connectivity layers are assumed to link human settlements: hydrologic pathways serve as ecological corridors for pathogens, while human mobility acts as disease vehicle through susceptibles contracting the disease and asymptomatic infectives shedding bacteria at their temporary destinations. We show that an epidemic outbreak can be triggered if the dominant eigenvalue of a generalized reproduction matrix G0, suitably accounting for spatial distribution of human settlements, hydrological pathways for pathogen dispersal and pathogen redistribution mechanisms due to human mobility, is larger than unity. Matrix G0 and its dominant eigenvalue thus replace the usual reproduction number whenever spatial effects on disease propagation cannot be ignored. Conversely, our novel criterion decays into the standard onset condition based on local reproduction numbers in nonspatial settings. By analyzing realistic test cases we show that within a connected network system the disease can start even if all the local reproduction numbers are smaller than unity, or might not start even if all the local reproduction numbers are larger than unity. We also show that onset geography in complex environments is linked to the dominant eigenvector of matrix G0. Applications to cholera outbreaks in developing countries demonstrate that our approach can be successfully used for disease prediction and emergency management.

Flutter analysis using transversality theory

NASA Technical Reports Server (NTRS)

Afolabi, D.

1993-01-01

A new method of calculating flutter boundaries of undamped aeronautical structures is presented. The method is an application of the weak transversality theorem used in catastrophe theory. In the first instance, the flutter problem is cast in matrix form using a frequency domain method, leading to an eigenvalue matrix. The characteristic polynomial resulting from this matrix usually has a smooth dependence on the system's parameters. As these parameters change with operating conditions, certain critical values are reached at which flutter sets in. Our approach is to use the transversality theorem in locating such flutter boundaries using this criterion: at a flutter boundary, the characteristic polynomial does not intersect the axis of the abscissa transversally. Formulas for computing the flutter boundaries and flutter frequencies of structures with two degrees of freedom are presented, and extension to multi-degree of freedom systems is indicated. The formulas have obvious applications in, for instance, problems of panel flutter at supersonic Mach numbers.

Complex Langevin simulation of a random matrix model at nonzero chemical potential

NASA Astrophysics Data System (ADS)

Bloch, J.; Glesaaen, J.; Verbaarschot, J. J. M.; Zafeiropoulos, S.

2018-03-01

In this paper we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass is inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.

PEM-PCA: a parallel expectation-maximization PCA face recognition architecture.

PubMed

Rujirakul, Kanokmon; So-In, Chakchai; Arnonkijpanich, Banchar

2014-01-01

Principal component analysis or PCA has been traditionally used as one of the feature extraction techniques in face recognition systems yielding high accuracy when requiring a small number of features. However, the covariance matrix and eigenvalue decomposition stages cause high computational complexity, especially for a large database. Thus, this research presents an alternative approach utilizing an Expectation-Maximization algorithm to reduce the determinant matrix manipulation resulting in the reduction of the stages' complexity. To improve the computational time, a novel parallel architecture was employed to utilize the benefits of parallelization of matrix computation during feature extraction and classification stages including parallel preprocessing, and their combinations, so-called a Parallel Expectation-Maximization PCA architecture. Comparing to a traditional PCA and its derivatives, the results indicate lower complexity with an insignificant difference in recognition precision leading to high speed face recognition systems, that is, the speed-up over nine and three times over PCA and Parallel PCA.

Hessian matrix approach for determining error field sensitivity to coil deviations

NASA Astrophysics Data System (ADS)

Zhu, Caoxiang; Hudson, Stuart R.; Lazerson, Samuel A.; Song, Yuntao; Wan, Yuanxi

2018-05-01

The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code (Zhu et al 2018 Nucl. Fusion 58 016008) is utilized to provide fast and accurate calculations of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.

Principal regression analysis and the index leverage effect

NASA Astrophysics Data System (ADS)

Reigneron, Pierre-Alain; Allez, Romain; Bouchaud, Jean-Philippe

2011-09-01

We revisit the index leverage effect, that can be decomposed into a volatility effect and a correlation effect. We investigate the latter using a matrix regression analysis, that we call ‘Principal Regression Analysis' (PRA) and for which we provide some analytical (using Random Matrix Theory) and numerical benchmarks. We find that downward index trends increase the average correlation between stocks (as measured by the most negative eigenvalue of the conditional correlation matrix), and makes the market mode more uniform. Upward trends, on the other hand, also increase the average correlation between stocks but rotates the corresponding market mode away from uniformity. There are two time scales associated to these effects, a short one on the order of a month (20 trading days), and a longer time scale on the order of a year. We also find indications of a leverage effect for sectorial correlations as well, which reveals itself in the second and third mode of the PRA.

Application of matrix singular value properties for evaluating gain and phase margins of multiloop systems. [stability margins for wing flutter suppression and drone lateral attitude control

NASA Technical Reports Server (NTRS)

Mukhopadhyay, V.; Newsom, J. R.

1982-01-01

A stability margin evaluation method in terms of simultaneous gain and phase changes in all loops of a multiloop system is presented. A universal gain-phase margin evaluation diagram is constructed by generalizing an existing method using matrix singular value properties. Using this diagram and computing the minimum singular value of the system return difference matrix over the operating frequency range, regions of guaranteed stability margins can be obtained. Singular values are computed for a wing flutter suppression and a drone lateral attitude control problem. The numerical results indicate that this method predicts quite conservative stability margins. In the second example if the eigenvalue magnitude is used instead of the singular value, as a measure of nearness to singularity, more realistic stability margins are obtained. However, this relaxed measure generally cannot guarantee global stability.

Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts

NASA Astrophysics Data System (ADS)

Noble, J. H.; Lubasch, M.; Stevens, J.; Jentschura, U. D.

2017-12-01

We describe a matrix diagonalization algorithm for complex symmetric (not Hermitian) matrices, A ̲ =A̲T, which is based on a two-step algorithm involving generalized Householder reflections based on the indefinite inner product 〈 u ̲ , v ̲ 〉 ∗ =∑iuivi. This inner product is linear in both arguments and avoids complex conjugation. The complex symmetric input matrix is transformed to tridiagonal form using generalized Householder transformations (first step). An iterative, generalized QL decomposition of the tridiagonal matrix employing an implicit shift converges toward diagonal form (second step). The QL algorithm employs iterative deflation techniques when a machine-precision zero is encountered "prematurely" on the super-/sub-diagonal. The algorithm allows for a reliable and computationally efficient computation of resonance and antiresonance energies which emerge from complex-scaled Hamiltonians, and for the numerical determination of the real energy eigenvalues of pseudo-Hermitian and PT-symmetric Hamilton matrices. Numerical reference values are provided.

Eigenspace techniques for active flutter suppression

NASA Technical Reports Server (NTRS)

Garrard, William L.; Liebst, Bradley S.; Farm, Jerome A.

1987-01-01

The use of eigenspace techniques for the design of an active flutter suppression system for a hypothetical research drone is discussed. One leading edge and two trailing edge aerodynamic control surfaces and four sensors (accelerometers) are available for each wing. Full state control laws are designed by selecting feedback gains which place closed loop eigenvalues and shape closed loop eigenvectors so as to stabilize wing flutter and reduce gust loads at the wing root while yielding accepatable robustness and satisfying constrains on rms control surface activity. These controllers are realized by state estimators designed using an eigenvalue placement/eigenvector shaping technique which results in recovery of the full state loop transfer characteristics. The resulting feedback compensators are shown to perform almost as well as the full state designs. They also exhibit acceptable performance in situations in which the failure of an actuator is simulated.

Effect of Fuel Temperature Profile on Eigenvalue Calculations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Greifenkamp, Tom E; Clarno, Kevin T; Gehin, Jess C

2008-01-01

Use of an average fuel temperature is a current practice when modeling fuel for eigenvalue (k-inf) calculations. This is an approximation, as it is known from Heat-transfer methods that a fuel pin having linear power q', will have a temperature that varies radially and has a maximum temperature at the center line [1]. This paper describes an investigation into the effects on k-inf and isotopic concentrations of modeling a fuel pin using a single average temperature versus a radially varying fuel temperature profile. The axial variation is not discussed in this paper. A single fuel pin was modeled having 1,more » 3, 5, 8, or 10 regions of equal volumes (areas). Fig. 1 shows a model of a 10-ring fuel pin surrounded by a gap and then cladding.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cameron, Maria K., E-mail: cameron@math.umd.edu

We develop computational tools for spectral analysis of stochastic networks representing energy landscapes of atomic and molecular clusters. Physical meaning and some properties of eigenvalues, left and right eigenvectors, and eigencurrents are discussed. We propose an approach to compute a collection of eigenpairs and corresponding eigencurrents describing the most important relaxation processes taking place in the system on its way to the equilibrium. It is suitable for large and complex stochastic networks where pairwise transition rates, given by the Arrhenius law, vary by orders of magnitude. The proposed methodology is applied to the network representing the Lennard-Jones-38 cluster created bymore » Wales's group. Its energy landscape has a double funnel structure with a deep and narrow face-centered cubic funnel and a shallower and wider icosahedral funnel. However, the complete spectrum of the generator matrix of the Lennard-Jones-38 network has no appreciable spectral gap separating the eigenvalue corresponding to the escape from the icosahedral funnel. We provide a detailed description of the escape process from the icosahedral funnel using the eigencurrent and demonstrate a superexponential growth of the corresponding eigenvalue. The proposed spectral approach is compared to the methodology of the Transition Path Theory. Finally, we discuss whether the Lennard-Jones-38 cluster is metastable from the points of view of a mathematician and a chemical physicist, and make a connection with experimental works.« less

Coupling coefficients for tensor product representations of quantum SU(2)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Groenevelt, Wolter, E-mail: w.g.m.groenevelt@tudelft.nl

2014-10-15

We study tensor products of infinite dimensional irreducible {sup *}-representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain self-adjoint elements using spectral analysis of Jacobi operators associated to well-known q-hypergeometric orthogonal polynomials. We also compute coupling coefficients between different eigenvectors corresponding to the same eigenvalue. Since the continuous spectrum has multiplicity two, the corresponding coupling coefficients can be considered as 2 × 2-matrix-valued orthogonal functions. We compute explicitly the matrix elements of these functions. The coupling coefficients can be considered as q-analogs of Bessel functions. As a results we obtain several q-integral identities involving q-hypergeometricmore » orthogonal polynomials and q-Bessel-type functions.« less

Matrix partitioning and EOF/principal component analysis of Antarctic Sea ice brightness temperatures

NASA Technical Reports Server (NTRS)

Murray, C. W., Jr.; Mueller, J. L.; Zwally, H. J.

1984-01-01

A field of measured anomalies of some physical variable relative to their time averages, is partitioned in either the space domain or the time domain. Eigenvectors and corresponding principal components of the smaller dimensioned covariance matrices associated with the partitioned data sets are calculated independently, then joined to approximate the eigenstructure of the larger covariance matrix associated with the unpartitioned data set. The accuracy of the approximation (fraction of the total variance in the field) and the magnitudes of the largest eigenvalues from the partitioned covariance matrices together determine the number of local EOF's and principal components to be joined by any particular level. The space-time distribution of Nimbus-5 ESMR sea ice measurement is analyzed.

Cause and cure of sloppiness in ordinary differential equation models.

PubMed

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

Inverse eigenproblem for R-symmetric matrices and their approximation

NASA Astrophysics Data System (ADS)

Yuan, Yongxin

2009-11-01

Let be a nontrivial involution, i.e., R=R-1[not equal to]±In. We say that is R-symmetric if RGR=G. The set of all -symmetric matrices is denoted by . In this paper, we first give the solvability condition for the following inverse eigenproblem (IEP): given a set of vectors in and a set of complex numbers , find a matrix such that and are, respectively, the eigenvalues and eigenvectors of A. We then consider the following approximation problem: Given an n×n matrix , find such that , where is the solution set of IEP and ||[dot operator]|| is the Frobenius norm. We provide an explicit formula for the best approximation solution by means of the canonical correlation decomposition.

Fast Multilevel Solvers for a Class of Discrete Fourth Order Parabolic Problems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zheng, Bin; Chen, Luoping; Hu, Xiaozhe

2016-03-05

In this paper, we study fast iterative solvers for the solution of fourth order parabolic equations discretized by mixed finite element methods. We propose to use consistent mass matrix in the discretization and use lumped mass matrix to construct efficient preconditioners. We provide eigenvalue analysis for the preconditioned system and estimate the convergence rate of the preconditioned GMRes method. Furthermore, we show that these preconditioners only need to be solved inexactly by optimal multigrid algorithms. Our numerical examples indicate that the proposed preconditioners are very efficient and robust with respect to both discretization parameters and diffusion coefficients. We also investigatemore » the performance of multigrid algorithms with either collective smoothers or distributive smoothers when solving the preconditioner systems.« less

Vertices cannot be hidden from quantum spatial search for almost all random graphs

NASA Astrophysics Data System (ADS)

Glos, Adam; Krawiec, Aleksandra; Kukulski, Ryszard; Puchała, Zbigniew

2018-04-01

In this paper, we show that all nodes can be found optimally for almost all random Erdős-Rényi G(n,p) graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires p=ω (log ^8(n)/n), while the second requires p≥ (1+ɛ )log (n)/n, where ɛ >0. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the \\Vert \\cdot \\Vert _∞ norm. At the same time for p<(1-ɛ )log (n)/n, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.

Unusual square roots in the ghost-free theory of massive gravity

NASA Astrophysics Data System (ADS)

Golovnev, Alexey; Smirnov, Fedor

2017-06-01

A crucial building block of the ghost free massive gravity is the square root function of a matrix. This is a problematic entity from the viewpoint of existence and uniqueness properties. We accurately describe the freedom of choosing a square root of a (non-degenerate) matrix. It has discrete and (in special cases) continuous parts. When continuous freedom is present, the usual perturbation theory in terms of matrices can be critically ill defined for some choices of the square root. We consider the new formulation of massive and bimetric gravity which deals directly with eigenvalues (in disguise of elementary symmetric polynomials) instead of matrices. It allows for a meaningful discussion of perturbation theory in such cases, even though certain non-analytic features arise.

Voxelwise Spectral Diffusional Connectivity and its Applications to Alzheimer’s Disease and Intelligence Prediction

PubMed Central

Li, Junning; Jin, Yan; Shi, Yonggang; Dinov, Ivo D.; Wang, Danny J.; Toga, Arthur W.; Thompson, Paul M.

2014-01-01

Human brain connectivity can be studied using graph theory. Many connectivity studies parcellate the brain into regions and count fibres extracted between them. The resulting network analyses require validation of the tractography, as well as region and parameter selection. Here we investigate whole brain connectivity from a different perspective. We propose a mathematical formulation based on studying the eigenvalues of the Laplacian matrix of the diffusion tensor field at the voxel level. This voxelwise matrix has over a million parameters, but we derive the Kirchhoff complexity and eigen-spectrum through elegant mathematical theorems, without heavy computation. We use these novel measures to accurately estimate the voxelwise connectivity in multiple biomedical applications such as Alzheimer’s disease and intelligence prediction. PMID:24505723

Implementing transmission eigenchannels of disordered media by a binary-control digital micromirror device

NASA Astrophysics Data System (ADS)

Kim, Donggyu; Choi, Wonjun; Kim, Moonseok; Moon, Jungho; Seo, Keumyoung; Ju, Sanghyun; Choi, Wonshik

2014-11-01

We report a method for measuring the transmission matrix of a disordered medium using a binary-control of a digital micromirror device (DMD). With knowledge of the measured transmission matrix, we identified the transmission eigenchannels of the medium. We then used binary control of the DMD to shape the wavefront of incident waves and to experimentally couple light to individual eigenchannels. When the wave was coupled to the eigenchannel with the largest eigenvalue, in particular, we were able to achieve about two times more energy transmission than the mean transmittance of the medium. Our study provides an elaborated use of the DMD as a high-speed wavefront shaping device for controlling the multiple scattering of waves in highly scattering media.

Cause and cure of sloppiness in ordinary differential equation models

NASA Astrophysics Data System (ADS)

Tönsing, Christian; Timmer, Jens; Kreutz, Clemens

2014-08-01

Data-based mathematical modeling of biochemical reaction networks, e.g., by nonlinear ordinary differential equation (ODE) models, has been successfully applied. In this context, parameter estimation and uncertainty analysis is a major task in order to assess the quality of the description of the system by the model. Recently, a broadened eigenvalue spectrum of the Hessian matrix of the objective function covering orders of magnitudes was observed and has been termed as sloppiness. In this work, we investigate the origin of sloppiness from structures in the sensitivity matrix arising from the properties of the model topology and the experimental design. Furthermore, we present strategies using optimal experimental design methods in order to circumvent the sloppiness issue and present nonsloppy designs for a benchmark model.

Effective Perron-Frobenius eigenvalue for a correlated random map

NASA Astrophysics Data System (ADS)

Pool, Roman R.; Cáceres, Manuel O.

2010-09-01

We investigate the evolution of random positive linear maps with various type of disorder by analytic perturbation and direct simulation. Our theoretical result indicates that the statistics of a random linear map can be successfully described for long time by the mean-value vector state. The growth rate can be characterized by an effective Perron-Frobenius eigenvalue that strongly depends on the type of correlation between the elements of the projection matrix. We apply this approach to an age-structured population dynamics model. We show that the asymptotic mean-value vector state characterizes the population growth rate when the age-structured model has random vital parameters. In this case our approach reveals the nontrivial dependence of the effective growth rate with cross correlations. The problem was reduced to the calculation of the smallest positive root of a secular polynomial, which can be obtained by perturbations in terms of Green’s function diagrammatic technique built with noncommutative cumulants for arbitrary n -point correlations.

Systemic risk and spatiotemporal dynamics of the US housing market.

PubMed

Meng, Hao; Xie, Wen-Jie; Jiang, Zhi-Qiang; Podobnik, Boris; Zhou, Wei-Xing; Stanley, H Eugene

2014-01-13

Housing markets play a crucial role in economies and the collapse of a real-estate bubble usually destabilizes the financial system and causes economic recessions. We investigate the systemic risk and spatiotemporal dynamics of the US housing market (1975-2011) at the state level based on the Random Matrix Theory (RMT). We identify richer economic information in the largest eigenvalues deviating from RMT predictions for the housing market than for stock markets and find that the component signs of the eigenvectors contain either geographical information or the extent of differences in house price growth rates or both. By looking at the evolution of different quantities such as eigenvalues and eigenvectors, we find that the US housing market experienced six different regimes, which is consistent with the evolution of state clusters identified by the box clustering algorithm and the consensus clustering algorithm on the partial correlation matrices. We find that dramatic increases in the systemic risk are usually accompanied by regime shifts, which provide a means of early detection of housing bubbles.

Social climber attachment in forming networks produces a phase transition in a measure of connectivity

NASA Astrophysics Data System (ADS)

Taylor, Dane; Larremore, Daniel B.

2012-09-01

The formation and fragmentation of networks are typically studied using percolation theory, but most previous research has been restricted to studying a phase transition in cluster size, examining the emergence of a giant component. This approach does not study the effects of evolving network structure on dynamics that occur at the nodes, such as the synchronization of oscillators and the spread of information, epidemics, and neuronal excitations. We introduce and analyze an alternative link-formation rule, called social climber (SC) attachment, that may be combined with arbitrary percolation models to produce a phase transition using the largest eigenvalue of the network adjacency matrix as the order parameter. This eigenvalue is significant in the analyses of many network-coupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important self-organized properties of SC attachment and discuss implications for controlling dynamics on networks.

Learning SVM in Kreĭn Spaces.

PubMed

Loosli, Gaelle; Canu, Stephane; Ong, Cheng Soon

2016-06-01

This paper presents a theoretical foundation for an SVM solver in Kreĭn spaces. Up to now, all methods are based either on the matrix correction, or on non-convex minimization, or on feature-space embedding. Here we justify and evaluate a solution that uses the original (indefinite) similarity measure, in the original Kreĭn space. This solution is the result of a stabilization procedure. We establish the correspondence between the stabilization problem (which has to be solved) and a classical SVM based on minimization (which is easy to solve). We provide simple equations to go from one to the other (in both directions). This link between stabilization and minimization problems is the key to obtain a solution in the original Kreĭn space. Using KSVM, one can solve SVM with usually troublesome kernels (large negative eigenvalues or large numbers of negative eigenvalues). We show experiments showing that our algorithm KSVM outperforms all previously proposed approaches to deal with indefinite matrices in SVM-like kernel methods.

Architecture of chaotic attractors for flows in the absence of any singular point

DOE Office of Scientific and Technical Information (OSTI.GOV)

Letellier, Christophe; Malasoma, Jean-Marc

2016-06-15

Some chaotic attractors produced by three-dimensional dynamical systems without any singular point have now been identified, but explaining how they are structured in the state space remains an open question. We here want to explain—in the particular case of the Wei system—such a structure, using one-dimensional sets obtained by vanishing two of the three derivatives of the flow. The neighborhoods of these sets are made of points which are characterized by the eigenvalues of a 2 × 2 matrix describing the stability of flow in a subspace transverse to it. We will show that the attractor is spiralling and twisted in themore » neighborhood of one-dimensional sets where points are characterized by a pair of complex conjugated eigenvalues. We then show that such one-dimensional sets are also useful in explaining the structure of attractors produced by systems with singular points, by considering the case of the Lorenz system.« less

An efficient solver for large structured eigenvalue problems in relativistic quantum chemistry

NASA Astrophysics Data System (ADS)

Shiozaki, Toru

2017-01-01

We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalisation of matrices of dimension N > 10, 000 is now routine on a single computer node. Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry. The implementation is based on a blocked version of the Paige-Van Loan algorithm, which allows us to use the Level 3 BLAS subroutines for most of the computations. Taking advantage of the symmetry, the program is faster by up to a factor of 2 than state-of-the-art implementations of complex Hermitian diagonalisation; diagonalising a 12, 800 × 12, 800 matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel Math Kernel Library's ZHEEV that is not structure-preserving, respectively. The source code is publicly available under the FreeBSD licence.

Linear state feedback, quadratic weights, and closed loop eigenstructures. M.S. Thesis

NASA Technical Reports Server (NTRS)

Thompson, P. M.

1979-01-01

Results are given on the relationships between closed loop eigenstructures, state feedback gain matrices of the linear state feedback problem, and quadratic weights of the linear quadratic regulator. Equations are derived for the angles of general multivariable root loci and linear quadratic optimal root loci, including angles of departure and approach. The generalized eigenvalue problem is used for the first time to compute angles of approach. Equations are also derived to find the sensitivity of closed loop eigenvalues and the directional derivatives of closed loop eigenvectors (with respect to a scalar multiplying the feedback gain matrix or the quadratic control weight). An equivalence class of quadratic weights that produce the same asymptotic eigenstructure is defined, sufficient conditions to be in it are given, a canonical element is defined, and an algorithm to find it is given. The behavior of the optimal root locus in the nonasymptotic region is shown to be different for quadratic weights with the same asymptotic properties.

Overview of the ArbiTER edge plasma eigenvalue code

NASA Astrophysics Data System (ADS)

Baver, Derek; Myra, James; Umansky, Maxim

2011-10-01

The Arbitrary Topology Equation Reader, or ArbiTER, is a flexible eigenvalue solver that is currently under development for plasma physics applications. The ArbiTER code builds on the equation parser framework of the existing 2DX code, extending it to include a topology parser. This will give the code the capability to model problems with complicated geometries (such as multiple X-points and scrape-off layers) or model equations with arbitrary numbers of dimensions (e.g. for kinetic analysis). In the equation parser framework, model equations are not included in the program's source code. Instead, an input file contains instructions for building a matrix from profile functions and elementary differential operators. The program then executes these instructions in a sequential manner. These instructions may also be translated into analytic form, thus giving the code transparency as well as flexibility. We will present an overview of how the ArbiTER code is to work, as well as preliminary results from early versions of this code. Work supported by the U.S. DOE.

Eigenvector decomposition of full-spectrum x-ray computed tomography.

PubMed

Gonzales, Brian J; Lalush, David S

2012-03-07

Energy-discriminated x-ray computed tomography (CT) data were projected onto a set of basis functions to suppress the noise in filtered back-projection (FBP) reconstructions. The x-ray CT data were acquired using a novel x-ray system which incorporated a single-pixel photon-counting x-ray detector to measure the x-ray spectrum for each projection ray. A matrix of the spectral response of different materials was decomposed using eigenvalue decomposition to form the basis functions. Projection of FBP onto basis functions created a de facto image segmentation of multiple contrast agents. Final reconstructions showed significant noise suppression while preserving important energy-axis data. The noise suppression was demonstrated by a marked improvement in the signal-to-noise ratio (SNR) along the energy axis for multiple regions of interest in the reconstructed images. Basis functions used on a more coarsely sampled energy axis still showed an improved SNR. We conclude that the noise-resolution trade off along the energy axis was significantly improved using the eigenvalue decomposition basis functions.

A fast signal subspace approach for the determination of absolute levels from phased microphone array measurements

NASA Astrophysics Data System (ADS)

Sarradj, Ennes

2010-04-01

Phased microphone arrays are used in a variety of applications for the estimation of acoustic source location and spectra. The popular conventional delay-and-sum beamforming methods used with such arrays suffer from inaccurate estimations of absolute source levels and in some cases also from low resolution. Deconvolution approaches such as DAMAS have better performance, but require high computational effort. A fast beamforming method is proposed that can be used in conjunction with a phased microphone array in applications with focus on the correct quantitative estimation of acoustic source spectra. This method bases on an eigenvalue decomposition of the cross spectral matrix of microphone signals and uses the eigenvalues from the signal subspace to estimate absolute source levels. The theoretical basis of the method is discussed together with an assessment of the quality of the estimation. Experimental tests using a loudspeaker setup and an airfoil trailing edge noise setup in an aeroacoustic wind tunnel show that the proposed method is robust and leads to reliable quantitative results.

Transfer matrix calculation for ion optical elements using real fields

NASA Astrophysics Data System (ADS)

Mishra, P. M.; Blaum, K.; George, S.; Grieser, M.; Wolf, A.

2018-03-01

With the increasing importance of ion storage rings and traps in low energy physics experiments, an efficient transport of ion species from the ion source area to the experimental setup becomes essential. Some available, powerful software packages rely on transfer matrix calculations in order to compute the ion trajectory through the ion-optical beamline systems of high complexity. With analytical approaches, so far the transfer matrices are documented only for a few ideal ion optical elements. Here we describe an approach (using beam tracking calculations) to determine the transfer matrix for any individual electrostatic or magnetostatic ion optical element. We verify the procedure by considering the well-known cases and then apply it to derive the transfer matrix of a 90-degree electrostatic quadrupole deflector including its realistic geometry and fringe fields. A transfer line consisting of a quadrupole deflector and a quadrupole doublet is considered, where the results from the standard first order transfer matrix based ion optical simulation program implementing the derived transfer matrix is compared with the real field beam tracking simulations.

Parallel algorithm for computation of second-order sequential best rotations

NASA Astrophysics Data System (ADS)

Redif, Soydan; Kasap, Server

2013-12-01

Algorithms for computing an approximate polynomial matrix eigenvalue decomposition of para-Hermitian systems have emerged as a powerful, generic signal processing tool. A technique that has shown much success in this regard is the sequential best rotation (SBR2) algorithm. Proposed is a scheme for parallelising SBR2 with a view to exploiting the modern architectural features and inherent parallelism of field-programmable gate array (FPGA) technology. Experiments show that the proposed scheme can achieve low execution times while requiring minimal FPGA resources.

Estimating the number of people in crowded scenes

NASA Astrophysics Data System (ADS)

Kim, Minjin; Kim, Wonjun; Kim, Changick

2011-01-01

This paper presents a method to estimate the number of people in crowded scenes without using explicit object segmentation or tracking. The proposed method consists of three steps as follows: (1) extracting space-time interest points using eigenvalues of the local spatio-temporal gradient matrix, (2) generating crowd regions based on space-time interest points, and (3) estimating the crowd density based on the multiple regression. In experimental results, the efficiency and robustness of our proposed method are demonstrated by using PETS 2009 dataset.

The influence of thermal and conductive temperatures in a nanoscale resonator

NASA Astrophysics Data System (ADS)

Hobiny, Aatef; Abbas, Ibrahim A.

2018-06-01

In this work, the thermoelastic interaction in a nano-scale resonator based on two-temperature Green-Naghdi model is established. The nanoscale resonator ends were simply supported. In the Laplace's domain, the analytical solution of conductivity temperature and thermodynamic temperature, the displacement and the stress components are obtained. The eigenvalue approach resorted to for solutions. In the vector-matrix differential equations form, the essential equations were written. The numerical results for all variables are presented and are illustrated graphically.

Sur les processus linéaires de naissance et de mort sous-critiques dans un environnement aléatoire.

PubMed

Bacaër, Nicolas

2017-07-01

An explicit formula is found for the rate of extinction of subcritical linear birth-and-death processes in a random environment. The formula is illustrated by numerical computations of the eigenvalue with largest real part of the truncated matrix for the master equation. The generating function of the corresponding eigenvector satisfies a Fuchsian system of singular differential equations. A particular attention is set on the case of two environments, which leads to Riemann's differential equation.

Koopman Mode Decomposition Methods in Dynamic Stall: Reduced Order Modeling and Control

DTIC Science & Technology

2015-11-10

the flow phenomena by separating them into individual modes. The technique of Proper Orthogonal Decomposition (POD), see [ Holmes : 1998] is a popular...sampled values h(k), k = 0,…,2M-1, of the exponential sum 1. Solve the following linear system where 2. Compute all zeros zj  D, j = 1,…,M...of the Prony polynomial i.e., calculate all eigenvalues of the associated companion matrix and form fj = log zj for j = 1,…,M, where log is the

Bounds for the Z-spectral radius of nonnegative tensors.

PubMed

He, Jun; Liu, Yan-Min; Ke, Hua; Tian, Jun-Kang; Li, Xiang

2016-01-01

In this paper, we have proposed some new upper bounds for the largest Z-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which improve the known upper bounds obtained in Chang et al. (Linear Algebra Appl 438:4166-4182, 2013), Song and Qi (SIAM J Matrix Anal Appl 34:1581-1595, 2013), He and Huang (Appl Math Lett 38:110-114, 2014), Li et al. (J Comput Anal Appl 483:182-199, 2015), He (J Comput Anal Appl 20:1290-1301, 2016).

Bunch-Kaufman factorization for real symmetric indefinite banded matrices

NASA Technical Reports Server (NTRS)

Jones, Mark T.; Patrick, Merrell L.

1989-01-01

The Bunch-Kaufman algorithm for factoring symmetric indefinite matrices was rejected for banded matrices because it destroys the banded structure of the matrix. Herein, it is shown that for a subclass of real symmetric matrices which arise in solving the generalized eigenvalue problem using Lanczos's method, the Bunch-Kaufman algorithm does not result in major destruction of the bandwidth. Space time complexities of the algorithm are given and used to show that the Bunch-Kaufman algorithm is a significant improvement over LU factorization.

Invariant criteria for bound states, degree of ionization, and plasma phase transition

NASA Technical Reports Server (NTRS)

Girardeau, M. D.

1990-01-01

Basis invariant characterizations of bound states and bound fraction of a partially ionized hydrogen plasma are given in terms of properties of the spectrum of eigenvalues and eigenfunctions of the equilibrium quantum statistical one-proton-one-electron reduced density matrix. It is suggested that these can be used to place theories of a proposed plasma-ionization phase transition on a firm foundation. This general approach may be relevant to cosmological questions such as the quark deconfinement-confinement transition.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kalugin, A. V., E-mail: Kalugin-AV@nrcki.ru; Tebin, V. V.

The specific features of calculation of the effective multiplication factor using the Monte Carlo method for weakly coupled and non-asymptotic multiplying systems are discussed. Particular examples are considered and practical recommendations on detection and Monte Carlo calculation of systems typical in numerical substantiation of nuclear safety for VVER fuel management problems are given. In particular, the problems of the choice of parameters for the batch mode and the method for normalization of the neutron batch, as well as finding and interpretation of the eigenvalue spectrum for the integral fission matrix, are discussed.

Low-dimensional Representation of Error Covariance

NASA Technical Reports Server (NTRS)

Tippett, Michael K.; Cohn, Stephen E.; Todling, Ricardo; Marchesin, Dan

2000-01-01

Ensemble and reduced-rank approaches to prediction and assimilation rely on low-dimensional approximations of the estimation error covariances. Here stability properties of the forecast/analysis cycle for linear, time-independent systems are used to identify factors that cause the steady-state analysis error covariance to admit a low-dimensional representation. A useful measure of forecast/analysis cycle stability is the bound matrix, a function of the dynamics, observation operator and assimilation method. Upper and lower estimates for the steady-state analysis error covariance matrix eigenvalues are derived from the bound matrix. The estimates generalize to time-dependent systems. If much of the steady-state analysis error variance is due to a few dominant modes, the leading eigenvectors of the bound matrix approximate those of the steady-state analysis error covariance matrix. The analytical results are illustrated in two numerical examples where the Kalman filter is carried to steady state. The first example uses the dynamics of a generalized advection equation exhibiting nonmodal transient growth. Failure to observe growing modes leads to increased steady-state analysis error variances. Leading eigenvectors of the steady-state analysis error covariance matrix are well approximated by leading eigenvectors of the bound matrix. The second example uses the dynamics of a damped baroclinic wave model. The leading eigenvectors of a lowest-order approximation of the bound matrix are shown to approximate well the leading eigenvectors of the steady-state analysis error covariance matrix.

Eigenvalue and eigenvector sensitivity and approximate analysis for repeated eigenvalue problems

NASA Technical Reports Server (NTRS)

Hou, Gene J. W.; Kenny, Sean P.

1991-01-01

A set of computationally efficient equations for eigenvalue and eigenvector sensitivity analysis are derived, and a method for eigenvalue and eigenvector approximate analysis in the presence of repeated eigenvalues is presented. The method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem. Examples are given to demonstrate the application of such equations for sensitivity and approximate analysis.

Perturbation solutions for the influence of forward flight on helicopter rotor flapping stability

NASA Technical Reports Server (NTRS)

Johnson, W.

1974-01-01

The stability of the flapping motion of a helicopter rotor blade in forward flight is investigated, using a perturbation technique which gives analytic expressions for the eigenvalues, including the influence of the periodic aerodynamic forces in forward flight. The perturbation solutions are based on small advance ratio (the ratio of the helicopter forward speed to the rotor tip speed). The rotor configurations considered are a single, independent blade; a teetering rotor; a gimballed rotor with three, four, and five or more blades; and a rotor with N independent blades. The constant coefficient approximation with the equations and degrees of freedom in the nonrotating frame represents the flap dynamic reasonably well for the lower frequency modes, although it cannot, of course, be completely correct. The transfer function of the rotor flap response to sinusoidal pitch input is examined, as an alternative to the eigenvalues as a representation of the dynamic characteristics of the flap motion.

Application of Variational Methods to the Thermal Entrance Region of Ducts

NASA Technical Reports Server (NTRS)

Sparrow, E. M.; Siegel. R.

1960-01-01

A variational method is presented for solving eigenvalue problems which arise in connection with the analysis of convective heat transfer in the thermal entrance region of ducts. Consideration is given, to both situations where the temperature profile depends upon one cross-sectional coordinate (e.g. circular tube) or upon two cross-sectional coordinates (e.g. rectangular duct). The variational method is illustrated and verified by application to laminar heat transfer in a circular tube and a parallel-plate channel, and good agreement with existing numerical solutions is attained. Then, application is made to laminar heat transfer in a square duct as a check, an alternate computation for the square duct is made using a method indicated by Misaps and Pohihausen. The variational method can, in principle, also be applied to problems in turbulent heat transfer.

Crossover ensembles of random matrices and skew-orthogonal polynomials

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kumar, Santosh, E-mail: skumar.physics@gmail.com; Pandey, Akhilesh, E-mail: ap0700@mail.jnu.ac.in

2011-08-15

Highlights: > We study crossover ensembles of Jacobi family of random matrices. > We consider correlations for orthogonal-unitary and symplectic-unitary crossovers. > We use the method of skew-orthogonal polynomials and quaternion determinants. > We prove universality of spectral correlations in crossover ensembles. > We discuss applications to quantum conductance and communication theory problems. - Abstract: In a recent paper (S. Kumar, A. Pandey, Phys. Rev. E, 79, 2009, p. 026211) we considered Jacobi family (including Laguerre and Gaussian cases) of random matrix ensembles and reported exact solutions of crossover problems involving time-reversal symmetry breaking. In the present paper we givemore » details of the work. We start with Dyson's Brownian motion description of random matrix ensembles and obtain universal hierarchic relations among the unfolded correlation functions. For arbitrary dimensions we derive the joint probability density (jpd) of eigenvalues for all transitions leading to unitary ensembles as equilibrium ensembles. We focus on the orthogonal-unitary and symplectic-unitary crossovers and give generic expressions for jpd of eigenvalues, two-point kernels and n-level correlation functions. This involves generalization of the theory of skew-orthogonal polynomials to crossover ensembles. We also consider crossovers in the circular ensembles to show the generality of our method. In the large dimensionality limit, correlations in spectra with arbitrary initial density are shown to be universal when expressed in terms of a rescaled symmetry breaking parameter. Applications of our crossover results to communication theory and quantum conductance problems are also briefly discussed.« less

Efficient Computation of Sparse Matrix Functions for Large-Scale Electronic Structure Calculations: The CheSS Library.

PubMed

Mohr, Stephan; Dawson, William; Wagner, Michael; Caliste, Damien; Nakajima, Takahito; Genovese, Luigi

2017-10-10

We present CheSS, the "Chebyshev Sparse Solvers" library, which has been designed to solve typical problems arising in large-scale electronic structure calculations using localized basis sets. The library is based on a flexible and efficient expansion in terms of Chebyshev polynomials and presently features the calculation of the density matrix, the calculation of matrix powers for arbitrary powers, and the extraction of eigenvalues in a selected interval. CheSS is able to exploit the sparsity of the matrices and scales linearly with respect to the number of nonzero entries, making it well-suited for large-scale calculations. The approach is particularly adapted for setups leading to small spectral widths of the involved matrices and outperforms alternative methods in this regime. By coupling CheSS to the DFT code BigDFT, we show that such a favorable setup is indeed possible in practice. In addition, the approach based on Chebyshev polynomials can be massively parallelized, and CheSS exhibits excellent scaling up to thousands of cores even for relatively small matrix sizes.

Computation of ancestry scores with mixed families and unrelated individuals.

PubMed

Zhou, Yi-Hui; Marron, James S; Wright, Fred A

2018-03-01

The issue of robustness to family relationships in computing genotype ancestry scores such as eigenvector projections has received increased attention in genetic association, and is particularly challenging when sets of both unrelated individuals and closely related family members are included. The current standard is to compute loadings (left singular vectors) using unrelated individuals and to compute projected scores for remaining family members. However, projected ancestry scores from this approach suffer from shrinkage toward zero. We consider two main novel strategies: (i) matrix substitution based on decomposition of a target family-orthogonalized covariance matrix, and (ii) using family-averaged data to obtain loadings. We illustrate the performance via simulations, including resampling from 1000 Genomes Project data, and analysis of a cystic fibrosis dataset. The matrix substitution approach has similar performance to the current standard, but is simple and uses only a genotype covariance matrix, while the family-average method shows superior performance. Our approaches are accompanied by novel ancillary approaches that provide considerable insight, including individual-specific eigenvalue scree plots. © 2017 The Authors. Biometrics published by Wiley Periodicals, Inc. on behalf of International Biometric Society.

Eigenvector dynamics: General theory and some applications

NASA Astrophysics Data System (ADS)

Allez, Romain; Bouchaud, Jean-Philippe

2012-10-01

We propose a general framework to study the stability of the subspace spanned by P consecutive eigenvectors of a generic symmetric matrix H0 when a small perturbation is added. This problem is relevant in various contexts, including quantum dissipation (H0 is then the Hamiltonian) and financial risk control (in which case H0 is the assets' return covariance matrix). We argue that the problem can be formulated in terms of the singular values of an overlap matrix, which allows one to define an overlap distance. We specialize our results for the case of a Gaussian orthogonal H0, for which the full spectrum of singular values can be explicitly computed. We also consider the case when H0 is a covariance matrix and illustrate the usefulness of our results using financial data. The special case where the top eigenvalue is much larger than all the other ones can be investigated in full detail. In particular, the dynamics of the angle made by the top eigenvector and its true direction defines an interesting class of random processes.

Complete polarization characterization of single plasmonic nanoparticle enabled by a novel Dark-field Mueller matrix spectroscopy system

PubMed Central

Chandel, Shubham; Soni, Jalpa; Ray, Subir kumar; Das, Anwesh; Ghosh, Anirudha; Raj, Satyabrata; Ghosh, Nirmalya

2016-01-01

Information on the polarization properties of scattered light from plasmonic systems are of paramount importance due to fundamental interest and potential applications. However, such studies are severely compromised due to the experimental difficulties in recording full polarization response of plasmonic nanostructures. Here, we report on a novel Mueller matrix spectroscopic system capable of acquiring complete polarization information from single isolated plasmonic nanoparticle/nanostructure. The outstanding issues pertaining to reliable measurements of full 4 × 4 spectroscopic scattering Mueller matrices from single nanoparticle/nanostructures are overcome by integrating an efficient Mueller matrix measurement scheme and a robust eigenvalue calibration method with a dark-field microscopic spectroscopy arrangement. Feasibility of quantitative Mueller matrix polarimetry and its potential utility is illustrated on a simple plasmonic system, that of gold nanorods. The demonstrated ability to record full polarization information over a broad wavelength range and to quantify the intrinsic plasmon polarimetry characteristics via Mueller matrix inverse analysis should lead to a novel route towards quantitative understanding, analysis/interpretation of a number of intricate plasmonic effects and may also prove useful towards development of polarization-controlled novel sensing schemes. PMID:27212687

Large Covariance Estimation by Thresholding Principal Orthogonal Complements

PubMed Central

Fan, Jianqing; Liao, Yuan; Mincheva, Martina

2012-01-01

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented. PMID:24348088

Large Covariance Estimation by Thresholding Principal Orthogonal Complements.

PubMed

Fan, Jianqing; Liao, Yuan; Mincheva, Martina

2013-09-01

This paper deals with the estimation of a high-dimensional covariance with a conditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

Spectrum of the Wilson Dirac operator at finite lattice spacings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Akemann, G.; Damgaard, P. H.; Splittorff, K.

2011-04-15

We consider the effect of discretization errors on the microscopic spectrum of the Wilson Dirac operator using both chiral perturbation theory and chiral random matrix theory. A graded chiral Lagrangian is used to evaluate the microscopic spectral density of the Hermitian Wilson Dirac operator as well as the distribution of the chirality over the real eigenvalues of the Wilson Dirac operator. It is shown that a chiral random matrix theory for the Wilson Dirac operator reproduces the leading zero-momentum terms of Wilson chiral perturbation theory. All results are obtained for a fixed index of the Wilson Dirac operator. The low-energymore » constants of Wilson chiral perturbation theory are shown to be constrained by the Hermiticity properties of the Wilson Dirac operator.« less

Anderson Localization in Quark-Gluon Plasma

NASA Astrophysics Data System (ADS)

Kovács, Tamás G.; Pittler, Ferenc

2010-11-01

At low temperature the low end of the QCD Dirac spectrum is well described by chiral random matrix theory. In contrast, at high temperature there is no similar statistical description of the spectrum. We show that at high temperature the lowest part of the spectrum consists of a band of statistically uncorrelated eigenvalues obeying essentially Poisson statistics and the corresponding eigenvectors are extremely localized. Going up in the spectrum the spectral density rapidly increases and the eigenvectors become more and more delocalized. At the same time the spectral statistics gradually crosses over to the bulk statistics expected from the corresponding random matrix ensemble. This phenomenon is reminiscent of Anderson localization in disordered conductors. Our findings are based on staggered Dirac spectra in quenched lattice simulations with the SU(2) gauge group.

State-independent uncertainty relations and entanglement detection

NASA Astrophysics Data System (ADS)

Qian, Chen; Li, Jun-Li; Qiao, Cong-Feng

2018-04-01

The uncertainty relation is one of the key ingredients of quantum theory. Despite the great efforts devoted to this subject, most of the variance-based uncertainty relations are state-dependent and suffering from the triviality problem of zero lower bounds. Here we develop a method to get uncertainty relations with state-independent lower bounds. The method works by exploring the eigenvalues of a Hermitian matrix composed by Bloch vectors of incompatible observables and is applicable for both pure and mixed states and for arbitrary number of N-dimensional observables. The uncertainty relation for the incompatible observables can be explained by geometric relations related to the parallel postulate and the inequalities in Horn's conjecture on Hermitian matrix sum. Practical entanglement criteria are also presented based on the derived uncertainty relations.

Scaling and correlations in foreign exchange market

NASA Astrophysics Data System (ADS)

Jiang, J.; Ma, K.; Cai, X.

2007-02-01

We observe that the distribution of the relative return, describing the variation of a certain currency, of 74 global currencies obeys a power-law. By using the random matrix theory we find that the distribution of eigenvalues of correlation matrix of relative return also follows a power-law. Using a scaled factorial moment we investigate the distribution of correlation coefficients of the relative return and observe intermittence phenomenon. Furthermore, we define the influence strength for a certain currency, which reflects the influence of its price change to the community interested. By doing that, we find that the distribution of influence strength is again a power-law. Beyond that, we compare the influence strength of Chinese Yuan (RMB) to those of other seven important currencies, which may have some interesting indications.

Localization in covariance matrices of coupled heterogenous Ornstein-Uhlenbeck processes

NASA Astrophysics Data System (ADS)

Barucca, Paolo

2014-12-01

We define a random-matrix ensemble given by the infinite-time covariance matrices of Ornstein-Uhlenbeck processes at different temperatures coupled by a Gaussian symmetric matrix. The spectral properties of this ensemble are shown to be in qualitative agreement with some stylized facts of financial markets. Through the presented model formulas are given for the analysis of heterogeneous time series. Furthermore evidence for a localization transition in eigenvectors related to small and large eigenvalues in cross-correlations analysis of this model is found, and a simple explanation of localization phenomena in financial time series is provided. Finally we identify both in our model and in real financial data an inverted-bell effect in correlation between localized components and their local temperature: high- and low-temperature components are the most localized ones.

Optical systolic solutions of linear algebraic equations

NASA Technical Reports Server (NTRS)

Neuman, C. P.; Casasent, D.

1984-01-01

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Oscillation of two-dimensional linear second-order differential systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kwong, M.K.; Kaper, H.G.

This article is concerned with the oscillatory behavior at infinity of the solution y: (a, infinity) ..-->.. R/sup 2/ of a system of two second-order differential equations, y''(t) + Q(t) y(t) = 0, t epsilon(a, infinity); Q is a continuous matrix-valued function on (a, infinity) whose values are real symmetric matrices of order 2. It is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix integral/sub a//sup t/ Q(s) ds tends to infinity as t ..-->.. infinity. This proves a conjecture of D. Hinton and R.T. Lewis for the two-dimensional case. Furthermore, it ismore » shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity. 7 references.« less

On the solution of two-point linear differential eigenvalue problems. [numerical technique with application to Orr-Sommerfeld equation

NASA Technical Reports Server (NTRS)

Antar, B. N.

1976-01-01

A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.

Accelerating molecular property calculations with nonorthonormal Krylov space methods

DOE Office of Scientific and Technical Information (OSTI.GOV)

Furche, Filipp; Krull, Brandon T.; Nguyen, Brian D.

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remainmore » small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.« less

Accelerating molecular property calculations with nonorthonormal Krylov space methods

DOE PAGES

Furche, Filipp; Krull, Brandon T.; Nguyen, Brian D.; ...

2016-05-03

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remainmore » small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.« less

IONIZATION EQUILIBRIUM TIMESCALES IN COLLISIONAL PLASMAS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smith, Randall K.; Hughes, John P., E-mail: rsmith@cfa.harvard.ed, E-mail: jph@physics.rutgers.ed

2010-07-20

Astrophysical shocks or bursts from a photoionizing source can disturb the typical collisional plasma found in galactic interstellar media or the intergalactic medium. The spectrum emitted by this plasma contains diagnostics that have been used to determine the time since the disturbing event, although this determination becomes uncertain as the elements in the plasma return to ionization equilibrium. A general solution for the equilibrium timescale for each element arises from the elegant eigenvector method of solution to the problem of a non-equilibrium plasma described by Masai and Hughes and Helfand. In general, the ionization evolution of an element Z inmore » a constant electron temperature plasma is given by a coupled set of Z + 1 first-order differential equations. However, they can be recast as Z uncoupled first-order differential equations using an eigenvector basis for the system. The solution is then Z separate exponential functions, with the time constants given by the eigenvalues of the rate matrix. The smallest of these eigenvalues gives the scale of the slowest return to equilibrium independent of the initial conditions, while conversely the largest eigenvalue is the scale of the fastest change in the ion population. These results hold for an ionizing plasma, a recombining plasma, or even a plasma with random initial conditions, and will allow users of these diagnostics to determine directly if their best-fit result significantly limits the timescale since a disturbance or is so close to equilibrium as to include an arbitrarily long time.« less

Non-Hermitian localization in biological networks.

PubMed

Amir, Ariel; Hatano, Naomichi; Nelson, David R

2016-04-01

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90^{∘} rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias-dependent shape of this hole tracks the bias-independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's law" (each site is purely excitatory or inhibitory) and conclude with perturbation theory results that describe the limit of large directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.

Non-Hermitian localization in biological networks

NASA Astrophysics Data System (ADS)

Amir, Ariel; Hatano, Naomichi; Nelson, David R.

2016-04-01

We explore the spectra and localization properties of the N -site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N , the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90∘ rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias-dependent shape of this hole tracks the bias-independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's law" (each site is purely excitatory or inhibitory) and conclude with perturbation theory results that describe the limit of large directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.

Effect of noise in principal component analysis with an application to ozone pollution

NASA Astrophysics Data System (ADS)

Tsakiri, Katerina G.

This thesis analyzes the effect of independent noise in principal components of k normally distributed random variables defined by a covariance matrix. We prove that the principal components as well as the canonical variate pairs determined from joint distribution of original sample affected by noise can be essentially different in comparison with those determined from the original sample. However when the differences between the eigenvalues of the original covariance matrix are sufficiently large compared to the level of the noise, the effect of noise in principal components and canonical variate pairs proved to be negligible. The theoretical results are supported by simulation study and examples. Moreover, we compare our results about the eigenvalues and eigenvectors in the two dimensional case with other models examined before. This theory can be applied in any field for the decomposition of the components in multivariate analysis. One application is the detection and prediction of the main atmospheric factor of ozone concentrations on the example of Albany, New York. Using daily ozone, solar radiation, temperature, wind speed and precipitation data, we determine the main atmospheric factor for the explanation and prediction of ozone concentrations. A methodology is described for the decomposition of the time series of ozone and other atmospheric variables into the global term component which describes the long term trend and the seasonal variations, and the synoptic scale component which describes the short term variations. By using the Canonical Correlation Analysis, we show that solar radiation is the only main factor between the atmospheric variables considered here for the explanation and prediction of the global and synoptic scale component of ozone. The global term components are modeled by a linear regression model, while the synoptic scale components by a vector autoregressive model and the Kalman filter. The coefficient of determination, R2, for the prediction of the synoptic scale ozone component was found to be the highest when we consider the synoptic scale component of the time series for solar radiation and temperature. KEY WORDS: multivariate analysis; principal component; canonical variate pairs; eigenvalue; eigenvector; ozone; solar radiation; spectral decomposition; Kalman filter; time series prediction

A FORTRAN program for the analysis of linear continuous and sample-data systems

NASA Technical Reports Server (NTRS)

Edwards, J. W.

1976-01-01

A FORTRAN digital computer program which performs the general analysis of linearized control systems is described. State variable techniques are used to analyze continuous, discrete, and sampled data systems. Analysis options include the calculation of system eigenvalues, transfer functions, root loci, root contours, frequency responses, power spectra, and transient responses for open- and closed-loop systems. A flexible data input format allows the user to define systems in a variety of representations. Data may be entered by inputing explicit data matrices or matrices constructed in user written subroutines, by specifying transfer function block diagrams, or by using a combination of these methods.

Krein signature for instability of PT-symmetric states

NASA Astrophysics Data System (ADS)

Chernyavsky, Alexander; Pelinovsky, Dmitry E.

2018-05-01

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT-symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

Employing Eigenvalue Ratios to Generate Prior Fracture-like Features for Stochastic Hydrogeophysical Characterization of a Fractured Aquifer System

NASA Astrophysics Data System (ADS)

Brewster, J.; Oware, E. K.

2017-12-01

Groundwater hosted in fractured rocks constitutes almost 65% of the principal aquifers in the US. The exploitation and contaminant management of fractured aquifers require fracture flow and transport modeling, which in turn requires a detailed understanding of the structure of the aquifer. The widely used equivalent porous medium approach to modeling fractured aquifer systems is inadequate to accurately predict fracture transport processes due to the averaging of the sharp lithological contrast between the matrix and the fractures. The potential of geophysical imaging (GI) to estimate spatially continuous subsurface profiles in a minimally invasive fashion is well proven. Conventional deterministic GI strategies, however, produce geologically unrealistic, smoothed-out results due to commonly enforced smoothing constraints. Stochastic GI of fractured aquifers is becoming increasing appealing due to its ability to recover realistic fracture features while providing multiple likely realizations that enable uncertainty assessment. Generating prior spatial features consistent with the expected target structures is crucial in stochastic imaging. We propose to utilize eigenvalue ratios to resolve the elongated fracture features expected in a fractured aquifer system. Eigenvalues capture the major and minor directions of variability in a region, which can be employed to evaluate shape descriptors, such as eccentricity (elongation) and orientation of features in the region. Eccentricity ranges from zero to one, representing a circularly sharped to a line feature, respectively. Here, we apply eigenvalue ratios to define a joint objective parameter consisting of eccentricity (shape) and direction terms to guide the generation of prior fracture-like features in some predefined principal directions for stochastic GI. Preliminary unconditional, synthetic experiments reveal the potential of the algorithm to simulate prior fracture-like features. We illustrate the strategy with a 2D, cross-borehole electrical resistivity tomography (ERT) in a fractured aquifer at the UB Environmental Geophysics Imaging Site, with tomograms validated with gamma and caliper logs obtained from the two ERT wells.

Treating Sample Covariances for Use in Strongly Coupled Atmosphere-Ocean Data Assimilation

NASA Astrophysics Data System (ADS)

Smith, Polly J.; Lawless, Amos S.; Nichols, Nancy K.

2018-01-01

Strongly coupled data assimilation requires cross-domain forecast error covariances; information from ensembles can be used, but limited sampling means that ensemble derived error covariances are routinely rank deficient and/or ill-conditioned and marred by noise. Thus, they require modification before they can be incorporated into a standard assimilation framework. Here we compare methods for improving the rank and conditioning of multivariate sample error covariance matrices for coupled atmosphere-ocean data assimilation. The first method, reconditioning, alters the matrix eigenvalues directly; this preserves the correlation structures but does not remove sampling noise. We show that it is better to recondition the correlation matrix rather than the covariance matrix as this prevents small but dynamically important modes from being lost. The second method, model state-space localization via the Schur product, effectively removes sample noise but can dampen small cross-correlation signals. A combination that exploits the merits of each is found to offer an effective alternative.

Complex Langevin simulation of a random matrix model at nonzero chemical potential

DOE PAGES

Bloch, Jacques; Glesaaen, Jonas; Verbaarschot, Jacobus J. M.; ...

2018-03-06

In this study we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass ismore » inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.« less

Hessian matrix approach for determining error field sensitivity to coil deviations.

DOE PAGES

Zhu, Caoxiang; Hudson, Stuart R.; Lazerson, Samuel A.; ...

2018-03-15

The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code [Zhu et al., Nucl. Fusion 58(1):016008 (2018)] is utilized to provide fast and accurate calculationsmore » of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.« less

Complex Langevin simulation of a random matrix model at nonzero chemical potential

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bloch, Jacques; Glesaaen, Jonas; Verbaarschot, Jacobus J. M.

In this study we test the complex Langevin algorithm for numerical simulations of a random matrix model of QCD with a first order phase transition to a phase of finite baryon density. We observe that a naive implementation of the algorithm leads to phase quenched results, which were also derived analytically in this article. We test several fixes for the convergence issues of the algorithm, in particular the method of gauge cooling, the shifted representation, the deformation technique and reweighted complex Langevin, but only the latter method reproduces the correct analytical results in the region where the quark mass ismore » inside the domain of the eigenvalues. In order to shed more light on the issues of the methods we also apply them to a similar random matrix model with a milder sign problem and no phase transition, and in that case gauge cooling solves the convergence problems as was shown before in the literature.« less

Hessian matrix approach for determining error field sensitivity to coil deviations.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhu, Caoxiang; Hudson, Stuart R.; Lazerson, Samuel A.

The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code [Zhu et al., Nucl. Fusion 58(1):016008 (2018)] is utilized to provide fast and accurate calculationsmore » of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.« less

Multilayer neural networks for reduced-rank approximation.

PubMed

Diamantaras, K I; Kung, S Y

1994-01-01

This paper is developed in two parts. First, the authors formulate the solution to the general reduced-rank linear approximation problem relaxing the invertibility assumption of the input autocorrelation matrix used by previous authors. The authors' treatment unifies linear regression, Wiener filtering, full rank approximation, auto-association networks, SVD and principal component analysis (PCA) as special cases. The authors' analysis also shows that two-layer linear neural networks with reduced number of hidden units, trained with the least-squares error criterion, produce weights that correspond to the generalized singular value decomposition of the input-teacher cross-correlation matrix and the input data matrix. As a corollary the linear two-layer backpropagation model with reduced hidden layer extracts an arbitrary linear combination of the generalized singular vector components. Second, the authors investigate artificial neural network models for the solution of the related generalized eigenvalue problem. By introducing and utilizing the extended concept of deflation (originally proposed for the standard eigenvalue problem) the authors are able to find that a sequential version of linear BP can extract the exact generalized eigenvector components. The advantage of this approach is that it's easier to update the model structure by adding one more unit or pruning one or more units when the application requires it. An alternative approach for extracting the exact components is to use a set of lateral connections among the hidden units trained in such a way as to enforce orthogonality among the upper- and lower-layer weights. The authors call this the lateral orthogonalization network (LON) and show via theoretical analysis-and verify via simulation-that the network extracts the desired components. The advantage of the LON-based model is that it can be applied in a parallel fashion so that the components are extracted concurrently. Finally, the authors show the application of their results to the solution of the identification problem of systems whose excitation has a non-invertible autocorrelation matrix. Previous identification methods usually rely on the invertibility assumption of the input autocorrelation, therefore they can not be applied to this case.

Graph Theory Meets Ab Initio Molecular Dynamics: Atomic Structures and Transformations at the Nanoscale

NASA Astrophysics Data System (ADS)

Pietrucci, Fabio; Andreoni, Wanda

2011-08-01

Social permutation invariant coordinates are introduced describing the bond network around a given atom. They originate from the largest eigenvalue and the corresponding eigenvector of the contact matrix, are invariant under permutation of identical atoms, and bear a clear signature of an order-disorder transition. Once combined with ab initio metadynamics, these coordinates are shown to be a powerful tool for the discovery of low-energy isomers of molecules and nanoclusters as well as for a blind exploration of isomerization, association, and dissociation reactions.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hayashi, A.; Hashimoto, T.; Horibe, M.

The quantum color coding scheme proposed by Korff and Kempe [e-print quant-ph/0405086] is easily extended so that the color coding quantum system is allowed to be entangled with an extra auxiliary quantum system. It is shown that in the extended scheme we need only {approx}2{radical}(N) quantum colors to order N objects in large N limit, whereas {approx}N/e quantum colors are required in the original nonextended version. The maximum success probability has asymptotics expressed by the Tracy-Widom distribution of the largest eigenvalue of a random Gaussian unitary ensemble (GUE) matrix.

An extension of the QZ algorithm for solving the generalized matrix eigenvalue problem

NASA Technical Reports Server (NTRS)

Ward, R. C.

1973-01-01

This algorithm is an extension of Moler and Stewart's QZ algorithm with some added features for saving time and operations. Also, some additional properties of the QR algorithm which were not practical to implement in the QZ algorithm can be generalized with the combination shift QZ algorithm. Numerous test cases are presented to give practical application tests for algorithm. Based on results, this algorithm should be preferred over existing algorithms which attempt to solve the class of generalized eigenproblems where both matrices are singular or nearly singular.

Automated Pole Placement Algorithm for Multivariable Optimal Control Synthesis.

DTIC Science & Technology

1985-09-01

set of Q and F The effective Qe and F, after n reassignments are given by .Q, Q Q. .. (eqn 4.11) and Fe =F, + Fa+... Fn (eqn 4.12) The above pole...Inverse transformation and determination of Q, and Fe are identical to the distinct eigenvalue case with M in equation 4.9 replaced by T. 3. System...and F and the augmented plant matrix become, Q -2.998 -149. 0.9994 -49.978 -149.9 7499 -0.00841 0.4211 The effective Q. and Fe required to move both

Generalized thermoelastic interaction in an isotropic solid cylinder without energy dissipation

NASA Astrophysics Data System (ADS)

Alshaikh, Fatimah

2018-04-01

In this paper, we constructed the generalized thermoelastic equations of an isotropic solid cylinder. The formulation is applied in the context of Green and Naghdi theory of types II (without energy dissipation). The material of the cylinder is supposed to be homogeneous isotropic both mechanically and thermally. The governing equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical results for the temperature distribution, displacement and radial stress are represented graphically.

Dynamical entropy via entropy of non-random matrices: application to stability and complexity in modelling ecosystems.

PubMed

Chakrabarti, C G; Ghosh, Koyel

2013-10-01

In the present paper we have first introduced a measure of dynamical entropy of an ecosystem on the basis of the dynamical model of the system. The dynamical entropy which depends on the eigenvalues of the community matrix of the system leads to a consistent measure of complexity of the ecosystem to characterize the dynamical behaviours such as the stability, instability and periodicity around the stationary states of the system. We have illustrated the theory with some model ecosystems. Copyright © 2013 Elsevier Inc. All rights reserved.