A new S-type eigenvalue inclusion set for tensors and its applications.

PubMed

Huang, Zheng-Ge; Wang, Li-Gong; Xu, Zhong; Cui, Jing-Jing

2016-01-01

In this paper, a new S -type eigenvalue localization set for a tensor is derived by dividing [Formula: see text] into disjoint subsets S and its complement. It is proved that this new set is sharper than those presented by Qi (J. Symb. Comput. 40:1302-1324, 2005), Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and Li et al. (Linear Algebra Appl. 481:36-53, 2015). As applications of the results, new bounds for the spectral radius of nonnegative tensors and the minimum H -eigenvalue of strong M -tensors are established, and we prove that these bounds are tighter than those obtained by Li et al. (Numer. Linear Algebra Appl. 21:39-50, 2014) and He and Huang (J. Inequal. Appl. 2014:114, 2014).

Statistical properties of cross-correlation in the Korean stock market

NASA Astrophysics Data System (ADS)

Oh, G.; Eom, C.; Wang, F.; Jung, W.-S.; Stanley, H. E.; Kim, S.

2011-01-01

We investigate the statistical properties of the cross-correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the cross-correlation matrix is positively skewed and changes over time. We find that the eigenvalue distribution of original cross-correlation matrix deviates from the eigenvalues predicted by the RMT, and the largest eigenvalue is 52 times larger than the maximum value among the eigenvalues predicted by the RMT. The β_{473} coefficient, which reflect the largest eigenvalue property, is 0.8, while one of the eigenvalues in the RMT is approximately zero. Notably, we show that the entropy function E(σ) with the portfolio risk σ for the original and filtered cross-correlation matrices are consistent with a power-law function, E( σ) σ^{-γ}, with the exponent γ 2.92 and those for Asian currency crisis decreases significantly.

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.

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.

A new localization set for generalized eigenvalues.

PubMed

Gao, Jing; Li, Chaoqian

2017-01-01

A new localization set for generalized eigenvalues is obtained. It is shown that the new set is tighter than that in (Numer. Linear Algebra Appl. 16:883-898, 2009). Numerical examples are given to verify the corresponding results.

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

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.

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.

Lattice topology dictates photon statistics.

PubMed

Kondakci, H Esat; Abouraddy, Ayman F; Saleh, Bahaa E A

2017-08-21

Propagation of coherent light through a disordered network is accompanied by randomization and possible conversion into thermal light. Here, we show that network topology plays a decisive role in determining the statistics of the emerging field if the underlying lattice is endowed with chiral symmetry. In such lattices, eigenmode pairs come in skew-symmetric pairs with oppositely signed eigenvalues. By examining one-dimensional arrays of randomly coupled waveguides arranged on linear and ring topologies, we are led to a remarkable prediction: the field circularity and the photon statistics in ring lattices are dictated by its parity while the same quantities are insensitive to the parity of a linear lattice. For a ring lattice, adding or subtracting a single lattice site can switch the photon statistics from super-thermal to sub-thermal, or vice versa. This behavior is understood by examining the real and imaginary fields on a lattice exhibiting chiral symmetry, which form two strands that interleave along the lattice sites. These strands can be fully braided around an even-sited ring lattice thereby producing super-thermal photon statistics, while an odd-sited lattice is incommensurate with such an arrangement and the statistics become sub-thermal.

Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra

DOE PAGES

Hatch, D. R.; Jenko, F.; Navarro, A. Banon; ...

2016-07-26

A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradient-driven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest inmore » the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeit in ways that cannot be fully explained by the linear eigenvalue approach; a non-modal treatment is necessary to understand key features of the turbulence.« less

Using Linear Regression To Determine the Number of Factors To Retain in Factor Analysis and the Number of Issues To Retain in Delphi Studies and Other Surveys.

ERIC Educational Resources Information Center

Jurs, Stephen; And Others

The scree test and its linear regression technique are reviewed, and results of its use in factor analysis and Delphi data sets are described. The scree test was originally a visual approach for making judgments about eigenvalues, which considered the relationships of the eigenvalues to one another as well as their actual values. The graph that is…

A method to stabilize linear systems using eigenvalue gradient information

NASA Technical Reports Server (NTRS)

Wieseman, C. D.

1985-01-01

Formal optimization methods and eigenvalue gradient information are used to develop a stabilizing control law for a closed loop linear system that is initially unstable. The method was originally formulated by using direct, constrained optimization methods with the constraints being the real parts of the eigenvalues. However, because of problems in trying to achieve stabilizing control laws, the problem was reformulated to be solved differently. The method described uses the Davidon-Fletcher-Powell minimization technique to solve an indirect, constrained minimization problem in which the performance index is the Kreisselmeier-Steinhauser function of the real parts of all the eigenvalues. The method is applied successfully to solve two different problems: the determination of a fourth-order control law stabilizes a single-input single-output active flutter suppression system and the determination of a second-order control law for a multi-input multi-output lateral-directional flight control system. Various sets of design variables and initial starting points were chosen to show the robustness of the method.

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.

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

Investigation, development and application of optimal output feedback theory. Vol. 4: Measures of eigenvalue/eigenvector sensitivity to system parameters and unmodeled dynamics

NASA Technical Reports Server (NTRS)

Halyo, Nesim

1987-01-01

Some measures of eigenvalue and eigenvector sensitivity applicable to both continuous and discrete linear systems are developed and investigated. An infinite series representation is developed for the eigenvalues and eigenvectors of a system. The coefficients of the series are coupled, but can be obtained recursively using a nonlinear coupled vector difference equation. A new sensitivity measure is developed by considering the effects of unmodeled dynamics. It is shown that the sensitivity is high when any unmodeled eigenvalue is near a modeled eigenvalue. Using a simple example where the sensor dynamics have been neglected, it is shown that high feedback gains produce high eigenvalue/eigenvector sensitivity. The smallest singular value of the return difference is shown not to reflect eigenvalue sensitivity since it increases with the feedback gains. Using an upper bound obtained from the infinite series, a procedure to evaluate whether the sensitivity to parameter variations is within given acceptable bounds is developed and demonstrated by an example.

XCOM intrinsic dimensionality for low-Z elements at diagnostic energies

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

Bornefalk, Hans

2012-02-15

Purpose: To determine the intrinsic dimensionality of linear attenuation coefficients (LACs) from XCOM for elements with low atomic number (Z = 1-20) at diagnostic x-ray energies (25-120 keV). H{sub 0}{sup q}, the hypothesis that the space of LACs is spanned by q bases, is tested for various q-values. Methods: Principal component analysis is first applied and the LACs are projected onto the first q principal component bases. The residuals of the model values vs XCOM data are determined for all energies and atomic numbers. Heteroscedasticity invalidates the prerequisite of i.i.d. errors necessary for bootstrapping residuals. Instead wild bootstrap is applied,more » which, by not mixing residuals, allows the effect of the non-i.i.d residuals to be reflected in the result. Credible regions for the eigenvalues of the correlation matrix for the bootstrapped LAC data are determined. If subsequent credible regions for the eigenvalues overlap, the corresponding principal component is not considered to represent true data structure but noise. If this happens for eigenvalues l and l + 1, for any l{<=}q, H{sub 0}{sup q} is rejected. Results: The largest value of q for which H{sub 0}{sup q} is nonrejectable at the 5%-level is q = 4. This indicates that the statistically significant intrinsic dimensionality of low-Z XCOM data at diagnostic energies is four. Conclusions: The method presented allows determination of the statistically significant dimensionality of any noisy linear subspace. Knowledge of such significant dimensionality is of interest for any method making assumptions on intrinsic dimensionality and evaluating results on noisy reference data. For LACs, knowledge of the low-Z dimensionality might be relevant when parametrization schemes are tuned to XCOM data. For x-ray imaging techniques based on the basis decomposition method (Alvarez and Macovski, Phys. Med. Biol. 21, 733-744, 1976), an underlying dimensionality of two is commonly assigned to the LAC of human tissue at diagnostic energies. The finding of a higher statistically significant dimensionality thus raises the question whether a higher assumed model dimensionality (now feasible with the advent of multibin x-ray systems) might also be practically relevant, i.e., if better tissue characterization results can be obtained.« less

An Eigenvalue Analysis of finite-difference approximations for hyperbolic IBVPs

NASA Technical Reports Server (NTRS)

Warming, Robert F.; Beam, Richard M.

1989-01-01

The eigenvalue spectrum associated with a linear finite-difference approximation plays a crucial role in the stability analysis and in the actual computational performance of the discrete approximation. The eigenvalue spectrum associated with the Lax-Wendroff scheme applied to a model hyperbolic equation was investigated. For an initial-boundary-value problem (IBVP) on a finite domain, the eigenvalue or normal mode analysis is analytically intractable. A study of auxiliary problems (Dirichlet and quarter-plane) leads to asymptotic estimates of the eigenvalue spectrum and to an identification of individual modes as either benign or unstable. The asymptotic analysis establishes an intuitive as well as quantitative connection between the algebraic tests in the theory of Gustafsson, Kreiss, and Sundstrom and Lax-Richtmyer L(sub 2) stability on a finite domain.

An eigenvalue localization set for tensors and its applications.

PubMed

Zhao, Jianxing; Sang, Caili

2017-01-01

A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al . (Linear Algebra Appl. 481:36-53, 2015) and Huang et al . (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of [Formula: see text]-tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al ., the advantage of our results is that, without considering the selection of nonempty proper subsets S of [Formula: see text], we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of [Formula: see text]-tensors. Finally, numerical examples are given to verify the theoretical results.

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.

An accurate method for solving a class of fractional Sturm-Liouville eigenvalue problems

NASA Astrophysics Data System (ADS)

Kashkari, Bothayna S. H.; Syam, Muhammed I.

2018-06-01

This article is devoted to both theoretical and numerical study of the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. In this paper, we implement a fractional-order Legendre Tau method to approximate the eigenvalues. This method transforms the Sturm-Liouville problem to a sparse nonsingular linear system which is solved using the continuation method. Theoretical results for the considered problem are provided and proved. Numerical results are presented to show the efficiency of the proposed method.

Matrix with Prescribed Eigenvectors

ERIC Educational Resources Information Center

Ahmad, Faiz

2011-01-01

It is a routine matter for undergraduates to find eigenvalues and eigenvectors of a given matrix. But the converse problem of finding a matrix with prescribed eigenvalues and eigenvectors is rarely discussed in elementary texts on linear algebra. This problem is related to the "spectral" decomposition of a matrix and has important technical…

An extended basis inexact shift-invert Lanczos for the efficient solution of large-scale generalized eigenproblems

NASA Astrophysics Data System (ADS)

Rewieński, M.; Lamecki, A.; Mrozowski, M.

2013-09-01

This paper proposes a technique, based on the Inexact Shift-Invert Lanczos (ISIL) method with Inexact Jacobi Orthogonal Component Correction (IJOCC) refinement, and a preconditioned conjugate-gradient (PCG) linear solver with multilevel preconditioner, for finding several eigenvalues for generalized symmetric eigenproblems. Several eigenvalues are found by constructing (with the ISIL process) an extended projection basis. Presented results of numerical experiments confirm the technique can be effectively applied to challenging, large-scale problems characterized by very dense spectra, such as resonant cavities with spatial dimensions which are large with respect to wavelengths of the resonating electromagnetic fields. It is also shown that the proposed scheme based on inexact linear solves delivers superior performance, as compared to methods which rely on exact linear solves, indicating tremendous potential of the 'inexact solve' concept. Finally, the scheme which generates an extended projection basis is found to provide a cost-efficient alternative to classical deflation schemes when several eigenvalues are computed.

Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials

NASA Astrophysics Data System (ADS)

Yang, Jianke; Nixon, Sean

2016-11-01

Stability of soliton families in one-dimensional nonlinear Schrödinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian-Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian-Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets (λ , - λ ,λ* , -λ*), similar to conservative systems and PT-symmetric systems. This quartet eigenvalue symmetry is very surprising for non- PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.

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.

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.

Rare-event statistics and modular invariance

NASA Astrophysics Data System (ADS)

Nechaev, S. K.; Polovnikov, K.

2018-01-01

Simple geometric arguments based on constructing the Euclid orchard are presented, which explain the equivalence of various types of distributions that result from rare-event statistics. In particular, the spectral density of the exponentially weighted ensemble of linear polymer chains is examined for its number-theoretic properties. It can be shown that the eigenvalue statistics of the corresponding adjacency matrices in the sparse regime show a peculiar hierarchical structure and are described by the popcorn (Thomae) function discontinuous in the dense set of rational numbers. Moreover, the spectral edge density distribution exhibits Lifshitz tails, reminiscent of 1D Anderson localization. Finally, a continuous approximation for the popcorn function is suggested based on the Dedekind η-function, and the hierarchical ultrametric structure of the popcorn-like distributions is demonstrated to be related to hidden SL(2,Z) modular symmetry.

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.

On the linear stability of blood flow through model capillary networks.

PubMed

Davis, Jeffrey M

2014-12-01

Under the approximation that blood behaves as a continuum, a numerical implementation is presented to analyze the linear stability of capillary blood flow through model tree and honeycomb networks that are based on the microvascular structures of biological tissues. The tree network is comprised of a cascade of diverging bifurcations, in which a parent vessel bifurcates into two descendent vessels, while the honeycomb network also contains converging bifurcations, in which two parent vessels merge into one descendent vessel. At diverging bifurcations, a cell partitioning law is required to account for the nonuniform distribution of red blood cells as a function of the flow rate of blood into each descendent vessel. A linearization of the governing equations produces a system of delay differential equations involving the discharge hematocrit entering each network vessel and leads to a nonlinear eigenvalue problem. All eigenvalues in a specified region of the complex plane are captured using a transformation based on contour integrals to construct a linear eigenvalue problem with identical eigenvalues, which are then determined using a standard QR algorithm. The predicted value of the dimensionless exponent in the cell partitioning law at the instability threshold corresponds to a supercritical Hopf bifurcation in numerical simulations of the equations governing unsteady blood flow. Excellent agreement is found between the predictions of the linear stability analysis and nonlinear simulations. The relaxation of the assumption of plug flow made in previous stability analyses typically has a small, quantitative effect on the stability results that depends on the specific network structure. This implementation of the stability analysis can be applied to large networks with arbitrary structure provided only that the connectivity among the network segments is known.

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.

On some stochastic formulations and related statistical moments of pharmacokinetic models.

PubMed

Matis, J H; Wehrly, T E; Metzler, C M

1983-02-01

This paper presents the deterministic and stochastic model for a linear compartment system with constant coefficients, and it develops expressions for the mean residence times (MRT) and the variances of the residence times (VRT) for the stochastic model. The expressions are relatively simple computationally, involving primarily matrix inversion, and they are elegant mathematically, in avoiding eigenvalue analysis and the complex domain. The MRT and VRT provide a set of new meaningful response measures for pharmacokinetic analysis and they give added insight into the system kinetics. The new analysis is illustrated with an example involving the cholesterol turnover in rats.

Chebyshev polynomials in the spectral Tau method and applications to Eigenvalue problems

NASA Technical Reports Server (NTRS)

Johnson, Duane

1996-01-01

Chebyshev Spectral methods have received much attention recently as a technique for the rapid solution of ordinary differential equations. This technique also works well for solving linear eigenvalue problems. Specific detail is given to the properties and algebra of chebyshev polynomials; the use of chebyshev polynomials in spectral methods; and the recurrence relationships that are developed. These formula and equations are then applied to several examples which are worked out in detail. The appendix contains an example FORTRAN program used in solving an eigenvalue problem.

Non-linear eigensolver-based alternative to traditional SCF methods

NASA Astrophysics Data System (ADS)

Gavin, B.; Polizzi, E.

2013-05-01

The self-consistent procedure in electronic structure calculations is revisited using a highly efficient and robust algorithm for solving the non-linear eigenvector problem, i.e., H({ψ})ψ = Eψ. This new scheme is derived from a generalization of the FEAST eigenvalue algorithm to account for the non-linearity of the Hamiltonian with the occupied eigenvectors. Using a series of numerical examples and the density functional theory-Kohn/Sham model, it will be shown that our approach can outperform the traditional SCF mixing-scheme techniques by providing a higher converge rate, convergence to the correct solution regardless of the choice of the initial guess, and a significant reduction of the eigenvalue solve time in simulations.

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

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 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.

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

NASA Technical Reports Server (NTRS)

Thurston, G. A.; Bains, N. J. C.

1986-01-01

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity.

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.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance.

PubMed

Wang, Weichen; Fan, Jianqing

2017-06-01

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies.

Asymptotics of empirical eigenstructure for high dimensional spiked covariance

PubMed Central

Wang, Weichen

2017-01-01

We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the magnitude of spiked eigenvalues, sample size, and dimensionality. This regime allows high dimensionality and diverging eigenvalues and provides new insights into the roles that the leading eigenvalues, sample size, and dimensionality play in principal component analysis. Our results are a natural extension of those in Paul (2007) to a more general setting and solve the rates of convergence problems in Shen et al. (2013). They also reveal the biases of estimating leading eigenvalues and eigenvectors by using principal component analysis, and lead to a new covariance estimator for the approximate factor model, called shrinkage principal orthogonal complement thresholding (S-POET), that corrects the biases. Our results are successfully applied to outstanding problems in estimation of risks of large portfolios and false discovery proportions for dependent test statistics and are illustrated by simulation studies. PMID:28835726

More insights into early brain development through statistical analyses of eigen-structural elements of diffusion tensor imaging using multivariate adaptive regression splines

PubMed Central

Chen, Yasheng; Zhu, Hongtu; An, Hongyu; Armao, Diane; Shen, Dinggang; Gilmore, John H.; Lin, Weili

2013-01-01

The aim of this study was to characterize the maturational changes of the three eigenvalues (λ1 ≥ λ2 ≥ λ3) of diffusion tensor imaging (DTI) during early postnatal life for more insights into early brain development. In order to overcome the limitations of using presumed growth trajectories for regression analysis, we employed Multivariate Adaptive Regression Splines (MARS) to derive data-driven growth trajectories for the three eigenvalues. We further employed Generalized Estimating Equations (GEE) to carry out statistical inferences on the growth trajectories obtained with MARS. With a total of 71 longitudinal datasets acquired from 29 healthy, full-term pediatric subjects, we found that the growth velocities of the three eigenvalues were highly correlated, but significantly different from each other. This paradox suggested the existence of mechanisms coordinating the maturations of the three eigenvalues even though different physiological origins may be responsible for their temporal evolutions. Furthermore, our results revealed the limitations of using the average of λ2 and λ3 as the radial diffusivity in interpreting DTI findings during early brain development because these two eigenvalues had significantly different growth velocities even in central white matter. In addition, based upon the three eigenvalues, we have documented the growth trajectory differences between central and peripheral white matter, between anterior and posterior limbs of internal capsule, and between inferior and superior longitudinal fasciculus. Taken together, we have demonstrated that more insights into early brain maturation can be gained through analyzing eigen-structural elements of DTI. PMID:23455648

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

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.

Intrinsic character of Stokes matrices

NASA Astrophysics Data System (ADS)

Gagnon, Jean-François; Rousseau, Christiane

2017-02-01

Two germs of linear analytic differential systems x k + 1Y‧ = A (x) Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

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.

Statistics of transmission eigenvalues in two-dimensional quantum cavities: Ballistic versus stochastic scattering

NASA Astrophysics Data System (ADS)

Rotter, Stefan; Aigner, Florian; Burgdörfer, Joachim

2007-03-01

We investigate the statistical distribution of transmission eigenvalues in phase-coherent transport through quantum dots. In two-dimensional ab initio simulations for both clean and disordered two-dimensional cavities, we find markedly different quantum-to-classical crossover scenarios for these two cases. In particular, we observe the emergence of “noiseless scattering states” in clean cavities, irrespective of sharp-edged entrance and exit lead mouths. We find the onset of these “classical” states to be largely independent of the cavity’s classical chaoticity, but very sensitive with respect to bulk disorder. Our results suggest that for weakly disordered cavities, the transmission eigenvalue distribution is determined both by scattering at the disorder potential and the cavity walls. To properly account for this intermediate parameter regime, we introduce a hybrid crossover scheme, which combines previous models that are valid in the ballistic and the stochastic limit, respectively.

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.

Initial values for the integration scheme to compute the eigenvalues for propagation in ducts

NASA Technical Reports Server (NTRS)

Eversman, W.

1977-01-01

A scheme for the calculation of eigenvalues in the problem of acoustic propagation in a two-dimensional duct is described. The computation method involves changing the coupled transcendental nonlinear algebraic equations into an initial value problem involving a nonlinear ordinary differential equation. The simplest approach is to use as initial values the hardwall eigenvalues and to integrate away from these values as the admittance varies from zero to its actual value with a linear variation. The approach leads to a powerful root finding routine capable of computing the transverse and axial wave numbers for two-dimensional ducts for any frequency, lining, admittance and Mach number without requiring initial guesses or starting points.

Design of a New Concentration Series for the Orthogonal Sample Design Approach and Estimation of the Number of Reactions in Chemical Systems.

PubMed

Shi, Jiajia; Liu, Yuhai; Guo, Ran; Li, Xiaopei; He, Anqi; Gao, Yunlong; Wei, Yongju; Liu, Cuige; Zhao, Ying; Xu, Yizhuang; Noda, Isao; Wu, Jinguang

2015-11-01

A new concentration series is proposed for the construction of a two-dimensional (2D) synchronous spectrum for orthogonal sample design analysis to probe intermolecular interaction between solutes dissolved in the same solutions. The obtained 2D synchronous spectrum possesses the following two properties: (1) cross peaks in the 2D synchronous spectra can be used to reflect intermolecular interaction reliably, since interference portions that have nothing to do with intermolecular interaction are completely removed, and (2) the two-dimensional synchronous spectrum produced can effectively avoid accidental collinearity. Hence, the correct number of nonzero eigenvalues can be obtained so that the number of chemical reactions can be estimated. In a real chemical system, noise present in one-dimensional spectra may also produce nonzero eigenvalues. To get the correct number of chemical reactions, we classified nonzero eigenvalues into significant nonzero eigenvalues and insignificant nonzero eigenvalues. Significant nonzero eigenvalues can be identified by inspecting the pattern of the corresponding eigenvector with help of the Durbin-Watson statistic. As a result, the correct number of chemical reactions can be obtained from significant nonzero eigenvalues. This approach provides a solid basis to obtain insight into subtle spectral variations caused by intermolecular interaction.

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.

Attractors in complex networks

NASA Astrophysics Data System (ADS)

Rodrigues, Alexandre A. P.

2017-10-01

In the framework of the generalized Lotka-Volterra model, solutions representing multispecies sequential competition can be predictable with high probability. In this paper, we show that it occurs because the corresponding "heteroclinic channel" forms part of an attractor. We prove that, generically, in an attracting heteroclinic network involving a finite number of hyperbolic and non-resonant saddle-equilibria whose linearization has only real eigenvalues, the connections corresponding to the most positive expanding eigenvalues form part of an attractor (observable in numerical simulations).

Attractors in complex networks.

PubMed

Rodrigues, Alexandre A P

2017-10-01

In the framework of the generalized Lotka-Volterra model, solutions representing multispecies sequential competition can be predictable with high probability. In this paper, we show that it occurs because the corresponding "heteroclinic channel" forms part of an attractor. We prove that, generically, in an attracting heteroclinic network involving a finite number of hyperbolic and non-resonant saddle-equilibria whose linearization has only real eigenvalues, the connections corresponding to the most positive expanding eigenvalues form part of an attractor (observable in numerical simulations).

Peculiar spectral statistics of ensembles of trees and star-like graphs

NASA Astrophysics Data System (ADS)

Kovaleva, V.; Maximov, Yu; Nechaev, S.; Valba, O.

2017-07-01

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and p-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the ‘Lifshitz singularity’ emerging in the one-dimensional localization, while the spectral statistics of p-branching star-like graphs is less universal, being strongly dependent on p. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, reflecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of an ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.

Peculiar spectral statistics of ensembles of trees and star-like graphs

DOE PAGES

Kovaleva, V.; Maximov, Yu; Nechaev, S.; ...

2017-07-11

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and p-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the \\Lifshitz singularity" emerging in the onedimensional localization, while the spectral statistics of p-branching star-like graphs is less universal, being strongly dependent on p. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However,more » the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, re ecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.« less

Peculiar spectral statistics of ensembles of trees and star-like graphs

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

Kovaleva, V.; Maximov, Yu; Nechaev, S.

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and p-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the \\Lifshitz singularity" emerging in the onedimensional localization, while the spectral statistics of p-branching star-like graphs is less universal, being strongly dependent on p. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However,more » the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, re ecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.« less

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.

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

Computing the Evans function via solving a linear boundary value ODE

NASA Astrophysics Data System (ADS)

Wahl, Colin; Nguyen, Rose; Ventura, Nathaniel; Barker, Blake; Sandstede, Bjorn

2015-11-01

Determining the stability of traveling wave solutions to partial differential equations can oftentimes be computationally intensive but of great importance to understanding the effects of perturbations on the physical systems (chemical reactions, hydrodynamics, etc.) they model. For waves in one spatial dimension, one may linearize around the wave and form an Evans function - an analytic Wronskian-like function which has zeros that correspond in multiplicity to the eigenvalues of the linearized system. If eigenvalues with a positive real part do not exist, the traveling wave will be stable. Two methods exist for calculating the Evans function numerically: the exterior-product method and the method of continuous orthogonalization. The first is numerically expensive, and the second reformulates the originally linear system as a nonlinear system. We develop a new algorithm for computing the Evans function through appropriate linear boundary-value problems. This algorithm is cheaper than the previous methods, and we prove that it preserves analyticity of the Evans function. We also provide error estimates and implement it on some classical one- and two-dimensional systems, one being the Swift-Hohenberg equation in a channel, to show the advantages.

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.

Evans function computation for the stability of travelling waves

NASA Astrophysics Data System (ADS)

Barker, B.; Humpherys, J.; Lyng, G.; Lytle, J.

2018-04-01

In recent years, the Evans function has become an important tool for the determination of stability of travelling waves. This function, a Wronskian of decaying solutions of the eigenvalue equation, is useful both analytically and computationally for the spectral analysis of the linearized operator about the wave. In particular, Evans-function computation allows one to locate any unstable eigenvalues of the linear operator (if they exist); this allows one to establish spectral stability of a given wave and identify bifurcation points (loss of stability) as model parameters vary. In this paper, we review computational aspects of the Evans function and apply it to multidimensional detonation waves. This article is part of the theme issue `Stability of nonlinear waves and patterns and related topics'.

Rapid solution of large-scale systems of equations

NASA Technical Reports Server (NTRS)

Storaasli, Olaf O.

1994-01-01

The analysis and design of complex aerospace structures requires the rapid solution of large systems of linear and nonlinear equations, eigenvalue extraction for buckling, vibration and flutter modes, structural optimization and design sensitivity calculation. Computers with multiple processors and vector capabilities can offer substantial computational advantages over traditional scalar computer for these analyses. These computers fall into two categories: shared memory computers and distributed memory computers. This presentation covers general-purpose, highly efficient algorithms for generation/assembly or element matrices, solution of systems of linear and nonlinear equations, eigenvalue and design sensitivity analysis and optimization. All algorithms are coded in FORTRAN for shared memory computers and many are adapted to distributed memory computers. The capability and numerical performance of these algorithms will be addressed.

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree-Fock.

PubMed

Hesselmann, Andreas; Görling, Andreas

2011-01-21

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.

Eigenvalue asymptotics for the damped wave equation on metric graphs

NASA Astrophysics Data System (ADS)

Freitas, Pedro; Lipovský, Jiří

2017-09-01

We consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues. In the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high-frequency abscissas, whose location is solely determined by the averages of the damping terms on each edge. We further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate.

Hydrodynamics of the Dirac spectrum

DOE PAGES

Liu, Yizhuang; Warchoł, Piotr; Zahed, Ismail

2015-12-15

We discuss a hydrodynamical description of the eigenvalues of the Dirac spectrum in even dimensions in the vacuum and in the large N (volume) limit. The linearized hydrodynamics supports sound waves. The hydrodynamical relaxation of the eigenvalues is captured by a hydrodynamical (tunneling) minimum configuration which follows from a pertinent form of Euler equation. As a result, the relaxation from a phase of unbroken chiral symmetry to a phase of broken chiral symmetry occurs over a time set by the speed of sound.

Local Bifurcation Control,

DTIC Science & Technology

1987-01-01

X, (0) in the open left half complex plane . (S) Eq. (1) has an equilibrium zo(p) when u = 0. Furthermore, the linearization of (1) near z0, p = 0...possesses a simple eigenvalue X(p) with XI(O) = 0, X; (0) 74 0, with the remaining eigenvalues X(0), . . . , X. (0) in the open left half complex plane ...Conference, Lausanne, June 1984. (11) "Chaos In dynamical systems by the Poincare -Melnikov-Arnold method" Proc. ARO Workshop, March 1984. %I 2.I JUAN C

Motivating the Concept of Eigenvectors via Cryptography

ERIC Educational Resources Information Center

Siap, Irfan

2008-01-01

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues,…

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.

A comparison of maximum likelihood and other estimators of eigenvalues from several correlated Monte Carlo samples

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

Beer, M.

1980-12-01

The maximum likelihood method for the multivariate normal distribution is applied to the case of several individual eigenvalues. Correlated Monte Carlo estimates of the eigenvalue are assumed to follow this prescription and aspects of the assumption are examined. Monte Carlo cell calculations using the SAM-CE and VIM codes for the TRX-1 and TRX-2 benchmark reactors, and SAM-CE full core results are analyzed with this method. Variance reductions of a few percent to a factor of 2 are obtained from maximum likelihood estimation as compared with the simple average and the minimum variance individual eigenvalue. The numerical results verify that themore » use of sample variances and correlation coefficients in place of the corresponding population statistics still leads to nearly minimum variance estimation for a sufficient number of histories and aggregates.« less

On Instability of Geostrophic Current with Linear Vertical Shear at Length Scales of Interleaving

NASA Astrophysics Data System (ADS)

Kuzmina, N. P.; Skorokhodov, S. L.; Zhurbas, N. V.; Lyzhkov, D. A.

2018-01-01

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability.

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.

Non-linear vibrations of sandwich viscoelastic shells

NASA Astrophysics Data System (ADS)

Benchouaf, Lahcen; Boutyour, El Hassan; Daya, El Mostafa; Potier-Ferry, Michel

2018-04-01

This paper deals with the non-linear vibration of sandwich viscoelastic shell structures. Coupling a harmonic balance method with the Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. This permits to get the non-linear frequency and the non-linear loss factor as functions of the displacement amplitude. To validate our approach, these relationships are illustrated in the case of a circular sandwich ring.

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

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

Probability distributions of linear statistics in chaotic cavities and associated phase transitions

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

Vivo, Pierpaolo; Majumdar, Satya N.; Bohigas, Oriol

2010-03-01

We establish large deviation formulas for linear statistics on the N transmission eigenvalues (T{sub i}) of a chaotic cavity, in the framework of random matrix theory. Given any linear statistics of interest A=SIGMA{sub i=1}{sup N}a(T{sub i}), the probability distribution P{sub A}(A,N) of A generically satisfies the large deviation formula lim{sub N-}>{sub i}nfinity[-2 log P{sub A}(Nx,N)/betaN{sup 2}]=PSI{sub A}(x), where PSI{sub A}(x) is a rate function that we compute explicitly in many cases (conductance, shot noise, and moments) and beta corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce newmore » formulas, such as a closed form expression for v(n)=lim{sub N-}>{sub i}nfinity var(T{sub n}) (where T{sub n}=SIGMA{sub i}T{sub i}{sup n}) for arbitrary integer n. The universal limit v*=lim{sub n-}>{sub i}nfinity v(n)=1/2pibeta is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly nonanalytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small N. Part of the results have been announced by Vivo et al. [Phys. Rev. Lett. 101, 216809 (2008)].« less

Electrical impedance tomography in anisotropic media with known eigenvectors

NASA Astrophysics Data System (ADS)

Abascal, Juan-Felipe P. J.; Lionheart, William R. B.; Arridge, Simon R.; Schweiger, Martin; Atkinson, David; Holder, David S.

2011-06-01

Electrical impedance tomography is an imaging method, with which volumetric images of conductivity are produced by injecting electrical current and measuring boundary voltages. It has the potential to become a portable non-invasive medical imaging technique. Until now, most implementations have neglected anisotropy even though human tissues like bone, muscle and brain white matter are markedly anisotropic. The recovery of an anisotropic conductivity tensor is uniquely determined by boundary measurements only up to a diffeomorphism that fixes the boundary. Nevertheless, uniqueness can be restored by providing information about the diffeomorphism. There are uniqueness results for two constraints: one eigenvalue and a multiple scalar of a general tensor. A useable constraint for medical applications is when the eigenvectors of the underlying tissue are known, which can be approximated from MRI or estimated from DT-MRI, although the eigenvalues are unknown. However there is no known theoretical result guaranteeing uniqueness for this constraint. In fact, only a few previous inversion studies have attempted to recover one or more eigenvalues assuming certain symmetries while ignoring nonuniqueness. In this work, the aim was to undertake a numerical study of the feasibility of the recovery of a piecewise linear finite element conductivity tensor in anisotropic media with known eigenvectors from the complete boundary data. The work suggests that uniqueness holds for this constraint, in addition to proposing a methodology for the incorporation of this prior for general conductivity tensors. This was carried out by performing an analysis of the Jacobian rank and by reconstructing four conductivity distributions: two diagonal tensors whose eigenvalues were linear and sinusoidal functions, and two general tensors whose eigenvectors resembled physiological tissue, one with eigenvectors spherically orientated like a spherical layered structure, and a sample of DT-MRI data of brain white matter. The Jacobian with respect to three eigenvalues was full-rank and it was possible to recover three eigenvalues for the four simulated distributions. This encourages further theoretical study of the uniqueness for this constraint and supports the use of this as a relevant usable method for medical applications.

A discourse on sensitivity analysis for discretely-modeled structures

NASA Technical Reports Server (NTRS)

Adelman, Howard M.; Haftka, Raphael T.

1991-01-01

A descriptive review is presented of the most recent methods for performing sensitivity analysis of the structural behavior of discretely-modeled systems. The methods are generally but not exclusively aimed at finite element modeled structures. Topics included are: selections of finite difference step sizes; special consideration for finite difference sensitivity of iteratively-solved response problems; first and second derivatives of static structural response; sensitivity of stresses; nonlinear static response sensitivity; eigenvalue and eigenvector sensitivities for both distinct and repeated eigenvalues; and sensitivity of transient response for both linear and nonlinear structural response.

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.

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.

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.

Physics, stability, and dynamics of supply networks

NASA Astrophysics Data System (ADS)

Helbing, Dirk; Lämmer, Stefan; Seidel, Thomas; Šeba, Pétr; Płatkowski, Tadeusz

2004-12-01

We show how to treat supply networks as physical transport problems governed by balance equations and equations for the adaptation of production speeds. Although the nonlinear behavior is different, the linearized set of coupled differential equations is formally related to those of mechanical or electrical oscillator networks. Supply networks possess interesting features due to their complex topology and directed links. We derive analytical conditions for absolute and convective instabilities. The empirically observed “bullwhip effect” in supply chains is explained as a form of convective instability based on resonance effects. Moreover, it is generalized to arbitrary supply networks. Their related eigenvalues are usually complex, depending on the network structure (even without loops). Therefore, their generic behavior is characterized by damped or growing oscillations. We also show that regular distribution networks possess two negative eigenvalues only, but perturbations generate a spectrum of complex eigenvalues.

Calculation of skin-stiffener interface stresses in stiffened composite panels

NASA Technical Reports Server (NTRS)

Cohen, David; Hyer, Michael W.

1987-01-01

A method for computing the skin-stiffener interface stresses in stiffened composite panels is developed. Both geometrically linear and nonlinear analyses are considered. Particular attention is given to the flange termination region where stresses are expected to exhibit unbounded characteristics. The method is based on a finite-element analysis and an elasticity solution. The finite-element analysis is standard, while the elasticity solution is based on an eigenvalue expansion of the stress functions. The eigenvalue expansion is assumed to be valid in the local flange termination region and is coupled with the finite-element analysis using collocation of stresses on the local region boundaries. Accuracy and convergence of the local elasticity solution are assessed using a geometrically linear analysis. Using this analysis procedure, the influence of geometric nonlinearities and stiffener parameters on the skin-stiffener interface stresses is evaluated.

Diffusive sensitivity to muscle architecture: a magnetic resonance diffusion tensor imaging study of the human calf.

PubMed

Galbán, Craig J; Maderwald, Stefan; Uffmann, Kai; de Greiff, Armin; Ladd, Mark E

2004-12-01

The aim of this study was to examine the diffusive properties of adjacent muscles at rest, and to determine the relationship between diffusive and architectural properties, which are task-specific to muscles. The principle, second, and third eigenvalues, trace of the diffusion tensor, and two anisotropic parameters, ellipsoid eccentricity (e) and fractional anisotropy (FA), of various muscles in the human calf were calculated by diffusion tensor imaging (DTI). Linear correlations of the calculated parameters to the muscle physiological cross-sectional area (PCSA), which is proportional to maximum muscle force, were performed to ascertain any linear relation between muscle architecture and diffusivity. Images of the left calf were acquired from six healthy male volunteers. Seven muscles were investigated in this study. These comprised the soleus, lateral gastrocnemius, medial gastrocnemius, posterior tibialis, anterior tibialis, extensor digitorum longus, and peroneus longus. All data were presented as the mean and standard error of the mean (SEM). In general, differences in diffusive parameter values occurred primarily between functionally different muscles. A strong correlation was also found between PCSA and the third eigenvalue, e, and FA. A mathematical derivation revealed a linear relationship between PCSA and the third eigenvalue as a result of their dependence on the average radius of all fibers within a single muscle. These findings demonstrated the ability of DTI to differentiate between functionally different muscles in the same region of the body on the basis of their diffusive properties.

Approximating natural connectivity of scale-free networks based on largest eigenvalue

NASA Astrophysics Data System (ADS)

Tan, S.-Y.; Wu, J.; Li, M.-J.; Lu, X.

2016-06-01

It has been recently proposed that natural connectivity can be used to efficiently characterize the robustness of complex networks. The natural connectivity has an intuitive physical meaning and a simple mathematical formulation, which corresponds to an average eigenvalue calculated from the graph spectrum. However, as a network model close to the real-world system that widely exists, the scale-free network is found difficult to obtain its spectrum analytically. In this article, we investigate the approximation of natural connectivity based on the largest eigenvalue in both random and correlated scale-free networks. It is demonstrated that the natural connectivity of scale-free networks can be dominated by the largest eigenvalue, which can be expressed asymptotically and analytically to approximate natural connectivity with small errors. Then we show that the natural connectivity of random scale-free networks increases linearly with the average degree given the scaling exponent and decreases monotonically with the scaling exponent given the average degree. Moreover, it is found that, given the degree distribution, the more assortative a scale-free network is, the more robust it is. Experiments in real networks validate our methods and results.

Convergence analysis of two-node CMFD method for two-group neutron diffusion eigenvalue problem

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

Jeong, Yongjin; Park, Jinsu; Lee, Hyun Chul

2015-12-01

In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also notedmore » that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations.« less

Transmission eigenvalues

NASA Astrophysics Data System (ADS)

Cakoni, Fioralba; Haddar, Houssem

2013-10-01

In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission eigenvalue problem. The need to answer these questions became important after a series of papers by Cakoni et al [5], and Cakoni et al [6] suggesting that these transmission eigenvalues could be used to obtain qualitative information about the material properties of the scattering object from far-field data. The first answer to the existence of transmission eigenvalues in the general case was given in 2008 when Päivärinta and Sylvester showed the existence of transmission eigenvalues for the index of refraction sufficiently large [7] followed in 2010 by the paper of Cakoni et al who removed the size restriction on the index of refraction [8]. More importantly, in the latter it was shown that transmission eigenvalues yielded qualitative information on the material properties of the scattering object and Cakoni et al established in [9] that transmission eigenvalues could be determined from the Tikhonov regularized solution of the far-field equation. Since the appearance of these papers there has been an explosion of interest in the transmission eigenvalue problem (we refer the reader to our recent survey paper [10] for a detailed account of the developments in this field up to 2012) and the papers in this special issue are representative of the myriad directions that this research has taken. Indeed, we are happy to see that many open theoretical and numerical questions raised in [10] have been answered (totally or partially) in the contributions of this special issue: the existence of transmission eigenvalues with minimal assumptions on the contrast, the numerical evaluation of transmission eigenvalues, the inverse spectral problem, applications to non-destructive testing, etc. In addition to these topics, many other new investigations and research directions have been proposed as we shall see in the brief content summary below. A number of papers in this special issue are concerned with the question of existence of transmission eigenvalues and the structure of the associated transmission eigenfunctions. The three papers by respectively Robbiano [11], Blasten and Päivärinta [12], and Lakshtanov and Vainberg [13] provide new complementary results on the existence of transmission eigenvalues for the scalar problem under weak assumptions on the (possibly complex valued) refractive index that mainly stipulates that the contrast does not change sign on the boundary. It is interesting here to see three different new methods to obtain these results. On the other hand, the paper by Bonnet-Ben Dhia and Chesnel [14] addresses the Fredholm properties of the interior transmission problem when the contrast changes sign on the boundary, exhibiting cases where this property fails. Using more standard approaches, the existence and structure of transmission eigenvalues are analyzed in the paper by Delbary [15] for the case of frequency dependent materials in the context of Maxwell's equations, whereas the paper by Vesalainen [16] initiates the study of the transmission eigenvalue problem in unbounded domains by considering the transmission eigenvalues for Schrödinger equation with non-compactly supported potential. The paper by Monk and Selgas [17] addresses the case where the dielectric is mounted on a perfect conductor and provides some numerical examples of the localization of associated eigenvalues using the linear sampling method. A series of papers then addresses the question of localization of transmission eigenvalues and the associated inverse spectral problem for spherically stratified media. More specifically, the paper by Colton and Leung [18] provides new results on complex transmission eigenvalues and a new proof for uniqueness of a solution to the inverse spectral problem, whereas the paper by Sylvester [19] provides sharp results on how to locate all the transmission eigenvalues associated with angular independent eigenfunctions when the index of refraction is constant. The paper by Gintides and Pallikarakis [20] investigates an iterative least square method to identify the spherically stratified index of refraction from transmission eigenvalues. On the characterization of transmission eigenvalues in terms of far-field measurements, a promising new result is obtained by Kirsch and Lechleiter [21] showing how one can identify the transmission eigenvalues using the eigenvalues of the scattering operator which are available in terms of measured scattering data. In the paper by Kleefeld [22], an accurate method for computing transmission eigenvalues based on a surface integral formulation of the interior transmission problem and numerical methods for nonlinear eigenvalue problems is proposed and numerically validated for the scalar problem in three dimensions. On the other hand, the paper by Sun and Xu [23] investigates the computation of transmission eigenvalues for Maxwell's equations using a standard iterative method associated with a variational formulation of the interior transmission problem with an emphasis on the effect of anisotropy on transmission eigenvalues. From the perspective of using transmission eigenvalues in non-destructive testing, the paper by Cakoni and Moskow [24] investigates the asymptotic behavior of transmission eigenvalues with respect to small inhomogeneities. The paper by Nakamura and Wang [25] investigates the linear sampling method for the time dependent heat equation and analyses the interior transmission problem associated with this equation. Finally, in the paper by Finch and Hickmann [26], the spectrum of the interior transmission problem is related to the unique determination of the acoustic properties of a body in thermoacoustic imaging. We hope that this collection of papers will stimulate further research in the rapidly growing area of transmission eigenvalues and inverse scattering theory.

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.

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.

Eigenenergies of a Relativistic Particle in an Infinite Range Linear Potential Using WKB Method

ERIC Educational Resources Information Center

Shivalingaswamy, T.; Kagali, B. A.

2011-01-01

Energy eigenvalues for a non-relativistic particle in a linear potential well are available. In this paper we obtain the eigenenergies for a relativistic spin less particle in a similar potential using an extension of the well-known WKB method treating the potential as the time component of a four-vector potential. Since genuine bound states do…

Eigenvalue assignment strategies in rotor systems

NASA Technical Reports Server (NTRS)

Youngblood, J. N.; Welzyn, K. J.

1986-01-01

The work done to establish the control and direction of effective eigenvalue excursions of lightly damped, speed dependent rotor systems using passive control is discussed. Both second order and sixth order bi-axis, quasi-linear, speed dependent generic models were investigated. In every case a single, bi-directional control bearing was used in a passive feedback stabilization loop to resist modal destabilization above the rotor critical speed. Assuming incomplete state measurement, sub-optimal control strategies were used to define the preferred location of the control bearing, the most effective measurement locations, and the best set of control gains to extend the speed range of stable operation. Speed dependent control gains were found by Powell's method to maximize the minimum modal damping ratio for the speed dependent linear model. An increase of 300 percent in stable speed operation was obtained for the sixth order linear system using passive control. Simulations were run to examine the effectiveness of the linear control law on nonlinear rotor models with bearing deadband. The maximum level of control effort (force) required by the control bearing to stabilize the rotor at speeds above the critical was determined for the models with bearing deadband.

Acoustic modes in fluid networks

NASA Technical Reports Server (NTRS)

Michalopoulos, C. D.; Clark, Robert W., Jr.; Doiron, Harold H.

1992-01-01

Pressure and flow rate eigenvalue problems for one-dimensional flow of a fluid in a network of pipes are derived from the familiar transmission line equations. These equations are linearized by assuming small velocity and pressure oscillations about mean flow conditions. It is shown that the flow rate eigenvalues are the same as the pressure eigenvalues and the relationship between line pressure modes and flow rate modes is established. A volume at the end of each branch is employed which allows any combination of boundary conditions, from open to closed, to be used. The Jacobi iterative method is used to compute undamped natural frequencies and associated pressure/flow modes. Several numerical examples are presented which include acoustic modes for the Helium Supply System of the Space Shuttle Orbiter Main Propulsion System. It should be noted that the method presented herein can be applied to any one-dimensional acoustic system involving an arbitrary number of branches.

The role of eigenvalues in linear feature selection theory

NASA Technical Reports Server (NTRS)

Brown, D. R.; Omalley, M. J.

1976-01-01

A particular measure of pattern class distinction called the average interclass divergence, or more simply, divergence, is considered. Here divergence will be the pairwise average of the expected interclass divergence derived from Hajek's two-class divergence.

FEAST fundamental framework for electronic structure calculations: Reformulation and solution of the muffin-tin problem

NASA Astrophysics Data System (ADS)

Levin, Alan R.; Zhang, Deyin; Polizzi, Eric

2012-11-01

In a recent article Polizzi (2009) [15], the FEAST algorithm has been presented as a general purpose eigenvalue solver which is ideally suited for addressing the numerical challenges in electronic structure calculations. Here, FEAST is presented beyond the “black-box” solver as a fundamental modeling framework which can naturally address the original numerical complexity of the electronic structure problem as formulated by Slater in 1937 [3]. The non-linear eigenvalue problem arising from the muffin-tin decomposition of the real-space domain is first derived and then reformulated to be solved exactly within the FEAST framework. This new framework is presented as a fundamental and practical solution for performing both accurate and scalable electronic structure calculations, bypassing the various issues of using traditional approaches such as linearization and pseudopotential techniques. A finite element implementation of this FEAST framework along with simulation results for various molecular systems is also presented and discussed.

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.

Linear state feedback, quadratic weights, and closed loop eigenstructures. M.S. Thesis. Final Report

NASA Technical Reports Server (NTRS)

Thompson, P. M.

1980-01-01

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 to compute angles of approach. Equations are also derived to find the sensitivity of closed loop eigenvalue and the directional derivatives of closed loop eigenvectors. An equivalence class of quadratic weights that produce the same asymptotic eigenstructure is defined, 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. An algorithm is presented that can be used to select a feedback gain matrix for the linear state feedback problem which produces a specified asymptotic eigenstructure. Another algorithm is given to compute the asymptotic eigenstructure properties inherent in a given set of quadratic weights. Finally, it is shown that optimal root loci for nongeneric problems can be approximated by generic ones in the nonasymptotic region.

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

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.

Lasing eigenvalue problems: the electromagnetic modelling of microlasers

NASA Astrophysics Data System (ADS)

Benson, Trevor; Nosich, Alexander; Smotrova, Elena; Balaban, Mikhail; Sewell, Phillip

2007-02-01

Comprehensive microcavity laser models should account for several physical mechanisms, e.g. carrier transport, heating and optical confinement, coupled by non-linear effects. Nevertheless, considerable useful information can still be obtained if all non-electromagnetic effects are neglected, often within an additional effective-index reduction to an equivalent 2D problem, and the optical modes viewed as solutions of Maxwell's equations. Integral equation (IE) formulations have many advantages over numerical techniques such as FDTD for the study of such microcavity laser problems. The most notable advantages of an IE approach are computational efficiency, the correct description of cavity boundaries without stair-step errors, and the direct solution of an eigenvalue problem rather than the spectral analysis of a transient signal. Boundary IE (BIE) formulations are more economic that volume IE (VIE) ones, because of their lower dimensionality, but they are only applicable to the constant cavity refractive index case. The Muller BIE, being free of 'defect' frequencies and having smooth or integrable kernels, provides a reliable tool for the modal analysis of microcavities. Whilst such an approach can readily identify complex-valued natural frequencies and Q-factors, the lasing condition is not addressed directly. We have thus suggested using a Muller BIE approach to solve a lasing eigenvalue problem (LEP), i.e. a linear eigenvalue solution in the form of two real-valued numbers (lasing wavelength and threshold information) when macroscopic gain is introduced into the cavity material within an active region. Such an approach yields clear insight into the lasing thresholds of individual cavities with uniform and non-uniform gain, cavities coupled as photonic molecules and cavities equipped with one or more quantum dots.

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.

Implicity restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations

NASA Technical Reports Server (NTRS)

Sorensen, Danny C.

1996-01-01

Eigenvalues and eigenfunctions of linear operators are important to many areas of applied mathematics. The ability to approximate these quantities numerically is becoming increasingly important in a wide variety of applications. This increasing demand has fueled interest in the development of new methods and software for the numerical solution of large-scale algebraic eigenvalue problems. In turn, the existence of these new methods and software, along with the dramatically increased computational capabilities now available, has enabled the solution of problems that would not even have been posed five or ten years ago. Until very recently, software for large-scale nonsymmetric problems was virtually non-existent. Fortunately, the situation is improving rapidly. The purpose of this article is to provide an overview of the numerical solution of large-scale algebraic eigenvalue problems. The focus will be on a class of methods called Krylov subspace projection methods. The well-known Lanczos method is the premier member of this class. The Arnoldi method generalizes the Lanczos method to the nonsymmetric case. A recently developed variant of the Arnoldi/Lanczos scheme called the Implicitly Restarted Arnoldi Method is presented here in some depth. This method is highlighted because of its suitability as a basis for software development.

Soliton's eigenvalue based analysis on the generation mechanism of rogue wave phenomenon in optical fibers exhibiting weak third order dispersion.

PubMed

Weerasekara, Gihan; Tokunaga, Akihiro; Terauchi, Hiroki; Eberhard, Marc; Maruta, Akihiro

2015-01-12

One of the extraordinary aspects of nonlinear wave evolution which has been observed as the spontaneous occurrence of astonishing and statistically extraordinary amplitude wave is called rogue wave. We show that the eigenvalues of the associated equation of nonlinear Schrödinger equation are almost constant in the vicinity of rogue wave and we validate that optical rogue waves are formed by the collision between quasi-solitons in anomalous dispersion fiber exhibiting weak third order dispersion.

Comparison of methods for calculating conditional expectations of sufficient statistics for continuous time Markov chains.

PubMed

Tataru, Paula; Hobolth, Asger

2011-12-05

Continuous time Markov chains (CTMCs) is a widely used model for describing the evolution of DNA sequences on the nucleotide, amino acid or codon level. The sufficient statistics for CTMCs are the time spent in a state and the number of changes between any two states. In applications past evolutionary events (exact times and types of changes) are unaccessible and the past must be inferred from DNA sequence data observed in the present. We describe and implement three algorithms for computing linear combinations of expected values of the sufficient statistics, conditioned on the end-points of the chain, and compare their performance with respect to accuracy and running time. The first algorithm is based on an eigenvalue decomposition of the rate matrix (EVD), the second on uniformization (UNI), and the third on integrals of matrix exponentials (EXPM). The implementation in R of the algorithms is available at http://www.birc.au.dk/~paula/. We use two different models to analyze the accuracy and eight experiments to investigate the speed of the three algorithms. We find that they have similar accuracy and that EXPM is the slowest method. Furthermore we find that UNI is usually faster than EVD.

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.

Veering and nonlinear interactions of a clamped beam in bending and torsion

NASA Astrophysics Data System (ADS)

Ehrhardt, David A.; Hill, Thomas L.; Neild, Simon A.; Cooper, Jonathan E.

2018-03-01

Understanding the linear and nonlinear dynamic behaviour of beams is critical for the design of many engineering structures such as spacecraft antennae, aircraft wings, and turbine blades. When the eigenvalues of such structures are closely-spaced, nonlinearity may lead to interactions between the underlying linear normal modes (LNMs). This work considers a clamped-clamped beam which exhibits nonlinear behaviour due to axial tension from large amplitudes of deformation. An additional cross-beam, mounted transversely and with a movable mass at each tip, allows tuning of the primary torsion LNM such that it is close to the primary bending LNM. Perturbing the location of one mass relative to that of the other leads to veering between the eigenvalues of the bending and torsion LNMs. For a number of selected geometries in the region of veering, a nonlinear reduced order model (NLROM) is created and the nonlinear normal modes (NNMs) are used to describe the underlying nonlinear behaviour of the structure. The relationship between the 'closeness' of the eigenvalues and the nonlinear dynamic behaviour is demonstrated in the NNM backbone curves, and veering-like behaviour is observed. Finally, the forced and damped dynamics of the structure are predicted using several analytical and numerical tools and are compared to experimental measurements. As well as showing a good agreement between the predicted and measured responses, phenomena such as a 1:1 internal resonance and quasi-periodic behaviour are identified.

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.

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.

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.

Wave Propagation inside Random Media

NASA Astrophysics Data System (ADS)

Cheng, Xiaojun

This thesis presents results of studies of wave scattering within and transmission through random and periodic systems. The main focus is on energy profiles inside quasi-1D and 1D random media. The connection between transport and the states of the medium is manifested in the equivalence of the dimensionless conductance, g, and the Thouless number which is the ratio of the average linewidth and spacing of energy levels. This equivalence and theories regarding the energy profiles inside random media are based on the assumption that LDOS is uniform throughout the samples. We have conducted microwave measurements of the longitudinal energy profiles within disordered samples contained in a copper tube supporting multiple waveguide channels with an antenna moving along a slit on the tube. These measurements allow us to determine the local density of states (LDOS) at a location which is the sum of energy from all incoming channels on both sides. For diffusive samples, the LDOS is uniform and the energy profile decays linearly as expected. However, for localized samples, we find that the LDOS drops sharply towards the middle of the sample and the energy profile does not follow the result of the local diffusion theory where the LDOS is assumed to be uniform. We analyze the field spectra into quasi-normal modes and found that the mode linewidth and the number of modes saturates as the sample length increases. Thus the Thouless number saturates while the dimensionless conductance g continues to fall with increasing length, indicating that the modes are localized near the boundaries. This is in contrast to the general believing that g and Thouless number follow the same scaling behavior. Previous measurements show that single parameter scaling (SPS) still holds in the same sample where the LDOS is suppressed te{shi2014microwave}. We explore the extension of SPS to the interior of the sample by analyzing statistics of the logrithm of the energy density ln W(x) and found that =-x/l where l is the transport mean free path. The result does not depend on the sample length, which is counterintuitive yet remarkably simple. More supprisingly, the linear fall-off of energy profile holds for totally disordered random 1D layered samples in simulations where the LDOS is uniform as well as for single mode random waveguide experiments and 1D nearly periodic samples where the LDOS is suppressed in the middle of the sample. The generalization of the transmission matrix to the interior of quasi-1D random samples, which is defined as the field matrix, and its eigenvalues statistics are also discussed. The maximum energy deposition at a location is not the intensity of the first transmission eigenchannel but the eigenvalue of the first energy density eigenchannels at that cross section, which can be much greater than the average value. The contrast, which is the ratio of the intensity at the focused point to the background intensity, in optimal focusing is determined by the participation number of the energy density eigenvalues and its inverse gives the variance of the energy density at that cross section in a single configuration. We have also studied topological states in photonic structures. We have demonstrated robust propagation of electromagnetic waves along reconfigurable pathways within a topological photonic metacrystal. Since the wave is confined within the domain wall, which is the boundary between two distinct topological insulating systems, we can freely steer the wave by reconstructing the photonic structure. Other topics, such as speckle pattern evolutions and the effects of boundary conditions on the statistics of transmission eigenvalues and energy profiles are also discussed.

Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept.

PubMed

Mazandarani, Mehran; Pariz, Naser

2018-05-01

This paper deals with sub-optimal control of a fuzzy linear dynamical system. The aim is to keep the state variables of the fuzzy linear dynamical system close to zero in an optimal manner. In the fuzzy dynamical system, the fuzzy derivative is considered as the granular derivative; and all the coefficients and initial conditions can be uncertain. The criterion for assessing the optimality is regarded as a granular integral whose integrand is a quadratic function of the state variables and control inputs. Using the relative-distance-measure (RDM) fuzzy interval arithmetic and calculus of variations, the optimal control law is presented as the fuzzy state variables feedback. Since the optimal feedback gains are obtained as fuzzy functions, they need to be defuzzified. This will result in the sub-optimal control law. This paper also sheds light on the restrictions imposed by the approaches which are based on fuzzy standard interval arithmetic (FSIA), and use strongly generalized Hukuhara and generalized Hukuhara differentiability concepts for obtaining the optimal control law. The granular eigenvalues notion is also defined. Using an RLC circuit mathematical model, it is shown that, due to their unnatural behavior in the modeling phenomenon, the FSIA-based approaches may obtain some eigenvalues sets that might be different from the inherent eigenvalues set of the fuzzy dynamical system. This is, however, not the case with the approach proposed in this study. The notions of granular controllability and granular stabilizability of the fuzzy linear dynamical system are also presented in this paper. Moreover, a sub-optimal control for regulating a Boeing 747 in longitudinal direction with uncertain initial conditions and parameters is gained. In addition, an uncertain suspension system of one of the four wheels of a bus is regulated using the sub-optimal control introduced in this paper. Copyright © 2018 ISA. Published by Elsevier Ltd. All rights reserved.

Parametric study of the dynamic JWL-EOS for detonation products

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

Urtiew, P.A.; Hayes, B.

1990-03-01

The JWL equation of state describing the adiabatic expansion of detonation products is revisited to complete the description of the principal eigenvalue, to reset the secondary eigenvalue to produce a well-behaved adiabatic gamma profile, and to normalize the characteristic equation of state in terms of conventional parameters having a clear experimental interpretation. This is accomplished by interjecting a dynamic flow condition concerning the value of the relative specific volume when the particle velocity of the detonation products is zero. In addition, a set of generic parameters based on the statistical distribution of the primary explosives making up the available datamore » base is presented. Unlike theoretical and statistical mechanical models, the adiabatic gamma function for these materials is seen to have a positive initial slope in accord with experimental findings. 10 refs., 4 figs.« less

Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions

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

Hintermueller, M., E-mail: hint@math.hu-berlin.de; Kao, C.-Y., E-mail: Ckao@claremontmckenna.edu; Laurain, A., E-mail: laurain@math.hu-berlin.de

2012-02-15

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this thresholdmore » value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.« less

Proposition for sensorless self-excitation by a piezoelectric device

NASA Astrophysics Data System (ADS)

Tanaka, Y.; Kokubun, Y.; Yabuno, H.

2018-04-01

In this paper, we propose a method to realize self-excitation in an oscillator actuated by a piezoelectric device without a sensor. In general, the positive feedback associated with the oscillator velocity causes the self-excitation. Instead of measuring the velocity with a sensor, we utilize the electro-mechanical coupling effect in the oscillator and piezoelectric device. We drive the piezoelectric device with a current proportional to the linear combination of the voltage across the terminals of the piezoelectric device and its differential voltage signal. Then, the oscillator with the piezoelectric device behaves like a third-order system, which has three eigenvalues. The self-excitation can be realized because appropriate feedback gains can set two of the eigenvalues to be conjugate complex roots with a positive real part and the other eigenvalue to be a negative real root. To confirm the validity of the proposed method, we experimentally demonstrated the sensorless self-excitation and, as an application example, carried out mass sensing in a sensorless self-excited macrocantilever.

The Cosmic V-Web

NASA Astrophysics Data System (ADS)

Pomarède, Daniel; Hoffman, Yehuda; Courtois, Hélène M.; Tully, R. Brent

2017-08-01

The network of filaments with embedded clusters surrounding voids, which has been seen in maps derived from redshift surveys and reproduced in simulations, has been referred to as the cosmic web. A complementary description is provided by considering the shear in the velocity field of galaxies. The eigenvalues of the shear provide information regarding whether or not a region is collapsing in three dimensions, which is the condition for a knot, expanding in three dimensions, which is the condition for a void, or in the intermediate condition of a filament or sheet. The structures that are quantitatively defined by the eigenvalues can be approximated by iso-contours that provide a visual representation of the cosmic velocity (V) web. The current application is based on radial peculiar velocities from the Cosmicflows-2 collection of distances. The three-dimensional velocity field is constructed using the Wiener filter methodology in the linear approximation. Eigenvalues of the velocity shear are calculated at each point on a grid. Here, knots and filaments are visualized across a local domain of diameter ˜ 0.1c.

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.

A Brief Historical Introduction to Determinants with Applications

ERIC Educational Resources Information Center

Debnath, L.

2013-01-01

This article deals with a short historical introduction to determinants with applications to the theory of equations, geometry, multiple integrals, differential equations and linear algebra. Included are some properties of determinants with proofs, eigenvalues, eigenvectors and characteristic equations with examples of applications to simple…

A framework for incorporating DTI Atlas Builder registration into Tract-Based Spatial Statistics and a simulated comparison to standard TBSS.

PubMed

Leming, Matthew; Steiner, Rachel; Styner, Martin

2016-02-27

Tract-based spatial statistics (TBSS) 6 is a software pipeline widely employed in comparative analysis of the white matter integrity from diffusion tensor imaging (DTI) datasets. In this study, we seek to evaluate the relationship between different methods of atlas registration for use with TBSS and different measurements of DTI (fractional anisotropy, FA, axial diffusivity, AD, radial diffusivity, RD, and medial diffusivity, MD). To do so, we have developed a novel tool that builds on existing diffusion atlas building software, integrating it into an adapted version of TBSS called DAB-TBSS (DTI Atlas Builder-Tract-Based Spatial Statistics) by using the advanced registration offered in DTI Atlas Builder 7 . To compare the effectiveness of these two versions of TBSS, we also propose a framework for simulating population differences for diffusion tensor imaging data, providing a more substantive means of empirically comparing DTI group analysis programs such as TBSS. In this study, we used 33 diffusion tensor imaging datasets and simulated group-wise changes in this data by increasing, in three different simulations, the principal eigenvalue (directly altering AD), the second and third eigenvalues (RD), and all three eigenvalues (MD) in the genu, the right uncinate fasciculus, and the left IFO. Additionally, we assessed the benefits of comparing the tensors directly using a functional analysis of diffusion tensor tract statistics (FADTTS 10 ). Our results indicate comparable levels of FA-based detection between DAB-TBSS and TBSS, with standard TBSS registration reporting a higher rate of false positives in other measurements of DTI. Within the simulated changes investigated here, this study suggests that the use of DTI Atlas Builder's registration enhances TBSS group-based studies.

Multiple eigenmodes of the Rayleigh-Taylor instability observed for a fluid interface with smoothly varying density

NASA Astrophysics Data System (ADS)

Yu, C. X.; Xue, C.; Liu, J.; Hu, X. Y.; Liu, Y. Y.; Ye, W. H.; Wang, L. F.; Wu, J. F.; Fan, Z. F.

2018-01-01

In this article, multiple eigen-systems including linear growth rates and eigen-functions have been discovered for the Rayleigh-Taylor instability (RTI) by numerically solving the Sturm-Liouville eigen-value problem in the case of two-dimensional plane geometry. The system called the first mode has the maximal linear growth rate and is just extensively studied in literature. Higher modes have smaller eigen-values, but possess multi-peak eigen-functions which bring on multiple pairs of vortices in the vorticity field. A general fitting expression for the first four eigen-modes is presented. Direct numerical simulations show that high modes lead to appearances of multi-layered spike-bubble pairs, and lots of secondary spikes and bubbles are also generated due to the interactions between internal spikes and bubbles. The present work has potential applications in many research and engineering areas, e.g., in reducing the RTI growth during capsule implosions in inertial confinement fusion.

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.

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.

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.

Linear instabilities near the DIII-D edge simulated in fluid models

NASA Astrophysics Data System (ADS)

Bass, Eric; Holland, Christopher

2017-10-01

The linear instability spectrum is reported near the DIII-D edge (within the separatrix) for L-mode and H-mode shots using the new eigenvalue solver FluTES (Fluid Toroidal Eigenvalue Solver). FluTES circumvents difficulties with convergence to clean linear eigenmodes (required for diagnosis of nonlinear simulations in codes such as BOUT++) often encountered with fluid initial-value solvers. FluTES is well-verified in analytic cases and against a BOUT++/ELITE benchmark toroidal case. We report results for both a 3-field, one-fluid model (the well-known ``elm-pb'' model) and a 5-field, two-fluid model. For the peeling-ballooning-dominated H-mode, the two solutions are qualitatively the same. In the driftwave-dominated L-mode edge, only the two-fluid solution gives robust instabilities which occur primarily at n > 50 . FluTES is optimized for this regime (near-flutelike limit, toroidally spectral). Cross-separatrix, coupled fluid and drift instabilities may play a role in explaining the gyrokinetic L-mode edge transport shortfall. Extension of FluTES into the open-field-line region is underway. Prepared by UCSD under Contract Number DE-FG02-06ER54871.

On the linear programming bound for linear Lee codes.

PubMed

Astola, Helena; Tabus, Ioan

2016-01-01

Based on an invariance-type property of the Lee-compositions of a linear Lee code, additional equality constraints can be introduced to the linear programming problem of linear Lee codes. In this paper, we formulate this property in terms of an action of the multiplicative group of the field [Formula: see text] on the set of Lee-compositions. We show some useful properties of certain sums of Lee-numbers, which are the eigenvalues of the Lee association scheme, appearing in the linear programming problem of linear Lee codes. Using the additional equality constraints, we formulate the linear programming problem of linear Lee codes in a very compact form, leading to a fast execution, which allows to efficiently compute the bounds for large parameter values of the linear codes.

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

Qualitative properties of large buckled states of spherical shells

NASA Technical Reports Server (NTRS)

Shih, K. G.; Antman, S. S.

1985-01-01

A system of 6th-order quasi-linear Ordinary Differential Equations is analyzed to show the global existence of axisymmetrically buckled states. A surprising nodal property is obtained which shows that everywhere along a branch of solutions that bifurcates from a simple eigenvalue of the linearized equation, the number of simultaneously vanishing points of both shear resultant and circumferential bending moment resultant remains invariant, provided that a certain auxiliary condition is satisfied.

The Shock and Vibration Digest, Volume 18, Number 3

DTIC Science & Technology

1986-03-01

Linear Distributed Parameter Des., Proc. Intl. Symp., 11th ONR Naval Struc. Systems by Shifted Legendre Polynomial Func- Mech. Symp., Tucson, AZ, pp...University, Atlanta, Georgia nonlinear problems with elementary algebra . It J. Sound Vib., 102 (2), pp 247-257 (Sept 22, uses i = -1, the Pascal’s...eigenvalues specified. The optimal avoid failure due to resonance under the action control problem of a linear distributed parameter 0School of Mechanical

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

Laplace-Beltrami Eigenvalues and Topological Features of Eigenfunctions for Statistical Shape Analysis

PubMed Central

Reuter, Martin; Wolter, Franz-Erich; Shenton, Martha; Niethammer, Marc

2009-01-01

This paper proposes the use of the surface based Laplace-Beltrami and the volumetric Laplace eigenvalues and -functions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and -functions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel. PMID:20161035

Experimental investigation of linear and nonlinear wave systems: A quantum chaos approach

NASA Astrophysics Data System (ADS)

Neicu, Toni

2002-09-01

An experimental and numerical study of linear and nonlinear wave systems using methods and ideas developed from quantum chaos is presented. We exploit the analogy of the wave equation for the flexural modes of a thin clover-shaped acoustic plate to the stationary solutions of the Schrodinger wave equation for a quantum clover-shaped billiard, a generic system that has regular and chaotic regions in its phase space. We observed periodic orbits in the spectral properties of the acoustic plate, the first such definitive acoustic experiment. We also solved numerically the linear wave equation of the acoustic plate for the first few hundred eigenmodes. The Fourier transform of the eigenvalues show peaks corresponding to the principal periodic orbits of the classical billiard. The signatures of the periodic orbits in the spectra were unambiguously verified by deforming one edge of the plate and observing that only the peaks corresponding to the orbits that hit this edge changed. The statistical measures of the eigenvalues are intermediate between universal forms for completely integrable and chaotic systems. The density distribution of the eigenfunctions agrees with the Porter-Thomas formula of chaotic systems. The viscosity dependence and effects of nonlinearity on the Faraday surface wave patterns in a stadium geometry were also investigated. The ray dynamics inside the stadium, a paradigm of quantum chaos, is completely chaotic. The majority of the observed patterns of the orbits resemble three eigenstates of the stadium: the bouncing ball, longitudinal, and bowtie patterns. We observed many disordered patterns that increase with the viscosity. The experimental results were analyzed using recent theoretical work that explains the suppression of certain modes. The theory also predicts that the perimeter dissipation is too strong for whispering gallery modes, which contradicts our observations of these modes for a fluid with low viscosity. Novel vortex patterns were observed in a strongly nonlinear, dissipative granular system of vertically vibrated rods. Above a critical packing fraction, moving domains of nearly vertical rods were seen to coexist with horizontal rods. The vertical domains coarsen to form several large vortices, which were driven by the anisotropy and inclination of the rods.

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.

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.

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.

Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble

NASA Astrophysics Data System (ADS)

Berggren, Tomas; Duits, Maurice

2017-09-01

In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.

The Modern Origin of Matrices and Their Applications

ERIC Educational Resources Information Center

Debnath, L.

2014-01-01

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show…

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…

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.

The Quasimonotonicity of Linear Differential Systems -The Complex Spectrum

DTIC Science & Technology

2001-09-12

proper, simplicial cone determined by the columns of B (see [10]) and that C is essentially nonnegative (see [11]). In [6], Heikkilä used Perron ...a B ≥ 0 such that Ae = B−1AB is essentially nonnegative and ir- reducible, then Perron - Frobenius theory tells us that Ae has a real eigenvalue λ1 with...systems requires that the comparison system be quasimonotone nondecreasing with respect to a cone contained in the nonnegative orthant. For linear

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1984-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, Kazufumi; Teglas, Russell

1987-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Investigation of cellular detonation structure formation via linear stability theory and 2D and 3D numerical simulations

NASA Astrophysics Data System (ADS)

Borisov, S. P.; Kudryavtsev, A. N.

2017-10-01

Linear and nonlinear stages of the instability of a plane detonation wave (DW) and the subsequent process of formation of cellular detonation structure are investigated. A simple model with one-step irreversible chemical reaction is used. The linear analysis is employed to predict the DW front structure at the early stages of its formation. An emerging eigenvalue problem is solved with a global method using a Chebyshev pseudospectral method and the LAPACK software library. A local iterative shooting procedure is used for eigenvalue refinement. Numerical simulations of a propagation of a DW in plane and rectangular channels are performed with a shock capturing WENO scheme of 5th order. A special method of a computational domain shift is implemented in order to maintain the DW in the domain. It is shown that the linear analysis gives certain predictions about the DW structure that are in agreement with the numerical simulations of early stages of DW propagation. However, at later stages, a merger of detonation cells occurs so that their number is approximately halved. Computations of DW propagation in a square channel reveal two different types of spatial structure of the DW front, "rectangular" and "diagonal" types. A spontaneous transition from the rectangular to diagonal type of structure is observed during propagation of the DW.

On the control canonical structure of a class of scalar input systems

NASA Technical Reports Server (NTRS)

Teglas, R.

1983-01-01

A discrete finite dimensional system, nonharmonic Fourier series and controllability, reduction to canonical form, and spectral synthesis are considered. The extent to which the eigenvalue associated with a controllable pair of a certain type may be modified via continuous linear state feedback is demonstrated.

Analysis techniques for multivariate root loci. [a tool in linear control systems

NASA Technical Reports Server (NTRS)

Thompson, P. M.; Stein, G.; Laub, A. J.

1980-01-01

Analysis and techniques are developed for the multivariable root locus and the multivariable optimal root locus. The generalized eigenvalue problem is used to compute angles and sensitivities for both types of loci, and an algorithm is presented that determines the asymptotic properties of the optimal root locus.

Sensitivity analysis for large-scale problems

NASA Technical Reports Server (NTRS)

Noor, Ahmed K.; Whitworth, Sandra L.

1987-01-01

The development of efficient techniques for calculating sensitivity derivatives is studied. The objective is to present a computational procedure for calculating sensitivity derivatives as part of performing structural reanalysis for large-scale problems. The scope is limited to framed type structures. Both linear static analysis and free-vibration eigenvalue problems are considered.

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…

Statistics of pressure fluctuations in decaying isotropic turbulence.

PubMed

Kalelkar, Chirag

2006-04-01

We present results from a systematic direct-numerical simulation study of pressure fluctuations in an unforced, incompressible, homogeneous, and isotropic three-dimensional turbulent fluid. At cascade completion, isosurfaces of low pressure are found to be organized as slender filaments, whereas the predominant isostructures appear sheetlike. We exhibit several results, including plots of probability distributions of the spatial pressure difference, the pressure-gradient norm, and the eigenvalues of the pressure-Hessian tensor. Plots of the temporal evolution of the mean pressure-gradient norm, and the mean eigenvalues of the pressure-Hessian tensor are also exhibited. We find the statistically preferred orientations between the eigenvectors of the pressure-Hessian tensor, the pressure gradient, the eigenvectors of the strain-rate tensor, the vorticity, and the velocity. Statistical properties of the nonlocal part of the pressure-Hessian tensor are also exhibited. We present numerical tests (in the viscous case) of some conjectures of Ohkitani [Phys. Fluids A 5, 2570 (1993)] and Ohkitani and Kishiba [Phys. Fluids 7, 411 (1995)] concerning the pressure-Hessian and the strain-rate tensors, for the unforced, incompressible, three-dimensional Euler equations.

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.

Folded concave penalized sparse linear regression: sparsity, statistical performance, and algorithmic theory for local solutions.

PubMed

Liu, Hongcheng; Yao, Tao; Li, Runze; Ye, Yinyu

2017-11-01

This paper concerns the folded concave penalized sparse linear regression (FCPSLR), a class of popular sparse recovery methods. Although FCPSLR yields desirable recovery performance when solved globally, computing a global solution is NP-complete. Despite some existing statistical performance analyses on local minimizers or on specific FCPSLR-based learning algorithms, it still remains open questions whether local solutions that are known to admit fully polynomial-time approximation schemes (FPTAS) may already be sufficient to ensure the statistical performance, and whether that statistical performance can be non-contingent on the specific designs of computing procedures. To address the questions, this paper presents the following threefold results: (i) Any local solution (stationary point) is a sparse estimator, under some conditions on the parameters of the folded concave penalties. (ii) Perhaps more importantly, any local solution satisfying a significant subspace second-order necessary condition (S 3 ONC), which is weaker than the second-order KKT condition, yields a bounded error in approximating the true parameter with high probability. In addition, if the minimal signal strength is sufficient, the S 3 ONC solution likely recovers the oracle solution. This result also explicates that the goal of improving the statistical performance is consistent with the optimization criteria of minimizing the suboptimality gap in solving the non-convex programming formulation of FCPSLR. (iii) We apply (ii) to the special case of FCPSLR with minimax concave penalty (MCP) and show that under the restricted eigenvalue condition, any S 3 ONC solution with a better objective value than the Lasso solution entails the strong oracle property. In addition, such a solution generates a model error (ME) comparable to the optimal but exponential-time sparse estimator given a sufficient sample size, while the worst-case ME is comparable to the Lasso in general. Furthermore, to guarantee the S 3 ONC admits FPTAS.

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

A spectral approach for the stability analysis of turbulent open-channel flows over granular beds

NASA Astrophysics Data System (ADS)

Camporeale, C.; Canuto, C.; Ridolfi, L.

2012-01-01

A novel Orr-Sommerfeld-like equation for gravity-driven turbulent open-channel flows over a granular erodible bed is here derived, and the linear stability analysis is developed. The whole spectrum of eigenvalues and eigenvectors of the complete generalized eigenvalue problem is computed and analyzed. The fourth-order eigenvalue problem presents singular non-polynomial coefficients with non-homogenous Robin-type boundary conditions that involve first and second derivatives. Furthermore, the Exner condition is imposed at an internal point. We propose a numerical discretization of spectral type based on a single-domain Galerkin scheme. In order to manage the presence of singular coefficients, some properties of Jacobi polynomials have been carefully blended with numerical integration of Gauss-Legendre type. The results show a positive agreement with the classical experimental data and allow one to relate the different types of instability to such parameters as the Froude number, wavenumber, and the roughness scale. The eigenfunctions allow two types of boundary layers to be distinguished, scaling, respectively, with the roughness height and the saltation layer for the bedload sediment transport.

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. .

Statistical properties of color-signal spaces.

PubMed

Lenz, Reiner; Bui, Thanh Hai

2005-05-01

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing.

Statistical properties of color-signal spaces

NASA Astrophysics Data System (ADS)

Lenz, Reiner; Hai Bui, Thanh

2005-05-01

In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing.

Ghost instabilities of cosmological models with vector fields nonminimally coupled to the curvature

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

Himmetoglu, Burak; Peloso, Marco; Contaldi, Carlo R.

2009-12-15

We prove that many cosmological models characterized by vectors nonminimally coupled to the curvature (such as the Turner-Widrow mechanism for the production of magnetic fields during inflation, and models of vector inflation or vector curvaton) contain ghosts. The ghosts are associated with the longitudinal vector polarization present in these models and are found from studying the sign of the eigenvalues of the kinetic matrix for the physical perturbations. Ghosts introduce two main problems: (1) they make the theories ill defined at the quantum level in the high energy/subhorizon regime (and create serious problems for finding a well-behaved UV completion), andmore » (2) they create an instability already at the linearized level. This happens because the eigenvalue corresponding to the ghost crosses zero during the cosmological evolution. At this point the linearized equations for the perturbations become singular (we show that this happens for all the models mentioned above). We explicitly solve the equations in the simplest cases of a vector without a vacuum expectation value in a Friedmann-Robertson-Walker geometry, and of a vector with a vacuum expectation value plus a cosmological constant, and we show that indeed the solutions of the linearized equations diverge when these equations become singular.« less

Development of a Three-Dimensional PSE Code for Compressible Flows: Stability of Three-Dimensional Compressible Boundary Layers

NASA Technical Reports Server (NTRS)

Balakumar, P.; Jeyasingham, Samarasingham

1999-01-01

A program is developed to investigate the linear stability of three-dimensional compressible boundary layer flows over bodies of revolutions. The problem is formulated as a two dimensional (2D) eigenvalue problem incorporating the meanflow variations in the normal and azimuthal directions. Normal mode solutions are sought in the whole plane rather than in a line normal to the wall as is done in the classical one dimensional (1D) stability theory. The stability characteristics of a supersonic boundary layer over a sharp cone with 50 half-angle at 2 degrees angle of attack is investigated. The 1D eigenvalue computations showed that the most amplified disturbances occur around x(sub 2) = 90 degrees and the azimuthal mode number for the most amplified disturbances range between m = -30 to -40. The frequencies of the most amplified waves are smaller in the middle region where the crossflow dominates the instability than the most amplified frequencies near the windward and leeward planes. The 2D eigenvalue computations showed that due to the variations in the azimuthal direction, the eigenmodes are clustered into isolated confined regions. For some eigenvalues, the eigenfunctions are clustered in two regions. Due to the nonparallel effect in the azimuthal direction, the eigenmodes are clustered into isolated confined regions. For some eigenvalues, the eigenfunctions are clustered in two regions. Due to the nonparallel effect in the azimuthal direction, the most amplified disturbances are shifted to 120 degrees compared to 90 degrees for the parallel theory. It is also observed that the nonparallel amplification rates are smaller than that is obtained from the parallel theory.

Aeroelastic analysis of a troposkien-type wind turbine blade

NASA Technical Reports Server (NTRS)

Nitzsche, F.

1981-01-01

The linear aeroelastic equations for one curved blade of a vertical axis wind turbine in state vector form are presented. The method is based on a simple integrating matrix scheme together with the transfer matrix idea. The method is proposed as a convenient way of solving the associated eigenvalue problem for general support conditions.

LINFLUX-AE: A Turbomachinery Aeroelastic Code Based on a 3-D Linearized Euler Solver

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, M. A.; Trudell, J. J.; Mehmed, O.; Stefko, G. L.

2004-01-01

This report describes the development and validation of LINFLUX-AE, a turbomachinery aeroelastic code based on the linearized unsteady 3-D Euler solver, LINFLUX. A helical fan with flat plate geometry is selected as the test case for numerical validation. The steady solution required by LINFLUX is obtained from the nonlinear Euler/Navier Stokes solver TURBO-AE. The report briefly describes the salient features of LINFLUX and the details of the aeroelastic extension. The aeroelastic formulation is based on a modal approach. An eigenvalue formulation is used for flutter analysis. The unsteady aerodynamic forces required for flutter are obtained by running LINFLUX for each mode, interblade phase angle and frequency of interest. The unsteady aerodynamic forces for forced response analysis are obtained from LINFLUX for the prescribed excitation, interblade phase angle, and frequency. The forced response amplitude is calculated from the modal summation of the generalized displacements. The unsteady pressures, work done per cycle, eigenvalues and forced response amplitudes obtained from LINFLUX are compared with those obtained from LINSUB, TURBO-AE, ASTROP2, and ANSYS.

Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

NASA Technical Reports Server (NTRS)

Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.

1986-01-01

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

Fluctuation-dissipation theory of input-output interindustrial relations

NASA Astrophysics Data System (ADS)

Iyetomi, Hiroshi; Nakayama, Yasuhiro; Aoyama, Hideaki; Fujiwara, Yoshi; Ikeda, Yuichi; Souma, Wataru

2011-01-01

In this study, the fluctuation-dissipation theory is invoked to shed light on input-output interindustrial relations at a macroscopic level by its application to indices of industrial production (IIP) data for Japan. Statistical noise arising from finiteness of the time series data is carefully removed by making use of the random matrix theory in an eigenvalue analysis of the correlation matrix; as a result, two dominant eigenmodes are detected. Our previous study successfully used these two modes to demonstrate the existence of intrinsic business cycles. Here a correlation matrix constructed from the two modes describes genuine interindustrial correlations in a statistically meaningful way. Furthermore, it enables us to quantitatively discuss the relationship between shipments of final demand goods and production of intermediate goods in a linear response framework. We also investigate distinctive external stimuli for the Japanese economy exerted by the current global economic crisis. These stimuli are derived from residuals of moving-average fluctuations of the IIP remaining after subtracting the long-period components arising from inherent business cycles. The observation reveals that the fluctuation-dissipation theory is applicable to an economic system that is supposed to be far from physical equilibrium.

Non-normality and classification of amplification mechanisms in stability and resolvent analysis

NASA Astrophysics Data System (ADS)

Symon, Sean; Rosenberg, Kevin; Dawson, Scott T. M.; McKeon, Beverley J.

2018-05-01

Eigenspectra and pseudospectra of the mean-linearized Navier-Stokes operator are used to characterize amplification mechanisms in laminar and turbulent flows in which linear mechanisms are important. Success of mean flow (linear) stability analysis for a particular frequency is shown to depend on whether two scalar measures of non-normality agree: (1) the product between the resolvent norm and the distance from the imaginary axis to the closest eigenvalue and (2) the inverse of the inner product between the most amplified resolvent forcing and response modes. If they agree, the resolvent operator can be rewritten in its dyadic representation to reveal that the adjoint and forward stability modes are proportional to the forcing and response resolvent modes at that frequency. Hence the real parts of the eigenvalues are important since they are responsible for resonant amplification and the resolvent operator is low rank when the eigenvalues are sufficiently separated in the spectrum. If the amplification is pseudoresonant, then resolvent analysis is more suitable to understand the origin of observed flow structures. Two test cases are studied: low Reynolds number cylinder flow and turbulent channel flow. The first deals mainly with resonant mechanisms, hence the success of both classical and mean stability analysis with respect to predicting the critical Reynolds number and global frequency of the saturated flow. Both scalar measures of non-normality agree for the base and mean flows, and the region where the forcing and response modes overlap scales with the length of the recirculation bubble. In the case of turbulent channel flow, structures result from both resonant and pseudoresonant mechanisms, suggesting that both are necessary elements to sustain turbulence. Mean shear is exploited most efficiently by stationary disturbances while bounds on the pseudospectra illustrate how pseudoresonance is responsible for the most amplified disturbances at spatial wavenumbers and temporal frequencies corresponding to well-known turbulent structures. Some implications for flow control are discussed.

Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model

NASA Astrophysics Data System (ADS)

Berglund, Nils; Landon, Damien

2012-08-01

We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of neuronal action potentials in parameter regimes characterized by mixed-mode oscillations. The interspike time interval is related to the random number of small-amplitude oscillations separating consecutive spikes. We prove that this number has an asymptotically geometric distribution, whose parameter is related to the principal eigenvalue of a substochastic Markov chain. We provide rigorous bounds on this eigenvalue in the small-noise regime and derive an approximation of its dependence on the system's parameters for a large range of noise intensities. This yields a precise description of the probability distribution of observed mixed-mode patterns and interspike intervals.

Computation of eigenpairs of Ax = lambda Bx for vibrations of spinning deformable bodies

NASA Technical Reports Server (NTRS)

Utku, S.; Clemente, J. L. M.

1984-01-01

It is shown that, when linear theory is used, the general eigenvalue problem related with the free vibrations of spinning deformable bodies is of the type AX = lambda Bx, where A is Hermitian, and B is real positive definite. Since the order n of the matrices may be large, and A and B are banded or block banded, due to the economics of the numerical solution, one is interested in obtaining only those eigenvalues which fall within the frequency band of interest of the problem. The paper extends the well known method of bisections and iteration of R to the n power to n dimensional complex spaces, i.e., to C to the n power, so that it can be applied to the present problem.

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.

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.

Nonlinear travelling waves in rotating Hagen–Poiseuille flow

NASA Astrophysics Data System (ADS)

Pier, Benoît; Govindarajan, Rama

2018-03-01

The dynamics of viscous flow through a rotating pipe is considered. Small-amplitude stability characteristics are obtained by linearizing the Navier–Stokes equations around the base flow and solving the resulting eigenvalue problems. For linearly unstable configurations, the dynamics leads to fully developed finite-amplitude perturbations that are computed by direct numerical simulations of the complete Navier–Stokes equations. By systematically investigating all linearly unstable combinations of streamwise wave number k and azimuthal mode number m, for streamwise Reynolds numbers {{Re}}z ≤slant 500 and rotational Reynolds numbers {{Re}}{{Ω }} ≤slant 500, the complete range of nonlinear travelling waves is obtained and the associated flow fields are characterized.

Design of linear quadratic regulators with eigenvalue placement in a specified region

NASA Technical Reports Server (NTRS)

Shieh, Leang-San; Zhen, Liu; Coleman, Norman P.

1990-01-01

Two linear quadratic regulators are developed for placing the closed-loop poles of linear multivariable continuous-time systems within the common region of an open sector, bounded by lines inclined at +/- pi/2k (for a specified integer k not less than 1) from the negative real axis, and the left-hand side of a line parallel to the imaginary axis in the complex s-plane, and simultaneously minimizing a quadratic performance index. The design procedure mainly involves the solution of either Liapunov equations or Riccati equations. The general expression for finding the lower bound of a constant gain gamma is also developed.

A sequential linear optimization approach for controller design

NASA Technical Reports Server (NTRS)

Horta, L. G.; Juang, J.-N.; Junkins, J. L.

1985-01-01

A linear optimization approach with a simple real arithmetic algorithm is presented for reliable controller design and vibration suppression of flexible structures. Using first order sensitivity of the system eigenvalues with respect to the design parameters in conjunction with a continuation procedure, the method converts a nonlinear optimization problem into a maximization problem with linear inequality constraints. The method of linear programming is then applied to solve the converted linear optimization problem. The general efficiency of the linear programming approach allows the method to handle structural optimization problems with a large number of inequality constraints on the design vector. The method is demonstrated using a truss beam finite element model for the optimal sizing and placement of active/passive-structural members for damping augmentation. Results using both the sequential linear optimization approach and nonlinear optimization are presented and compared. The insensitivity to initial conditions of the linear optimization approach is also demonstrated.

Parallel Symmetric Eigenvalue Problem Solvers

DTIC Science & Technology

2015-05-01

get research, tutoring, and mentoring experience as an undergraduate. Last but not least, I thank my family for their love and support. v TABLE OF...32 4.6.2 Choice of the Ritz shifts . . . . . . . . . . . . . . . . . . . . 37 4.7 Relationship between...pencil. I will conclude with a discussion of the relationship between Trace- Min and simultaneous iteration. If both methods solve the linear systems

Piecewise linear approximation for hereditary control problems

NASA Technical Reports Server (NTRS)

Propst, Georg

1987-01-01

Finite dimensional approximations are presented for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems when a quadratic cost integral has to be minimized subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in case the cost integral ranges over a finite time interval as well as in the case it ranges over an infinite time interval. The arguments in the latter case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space R(n) x L(2) and the favorable eigenvalue behavior of the piecewise linear approximations.

Statistical Aspects of Coherent States of the Higgs Algebra

NASA Astrophysics Data System (ADS)

Shreecharan, T.; Kumar, M. Naveen

2018-04-01

We construct and study various aspects of coherent states of a polynomial angular momentum algebra. The coherent states are constructed using a new unitary representation of the nonlinear algebra. The new representation involves a parameter γ that shifts the eigenvalues of the diagonal operator J 0.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Direct estimations of linear and nonlinear functionals of a quantum state.

PubMed

Ekert, Artur K; Alves, Carolina Moura; Oi, Daniel K L; Horodecki, Michał; Horodecki, Paweł; Kwek, L C

2002-05-27

We present a simple quantum network, based on the controlled-SWAP gate, that can extract certain properties of quantum states without recourse to quantum tomography. It can be used as a basic building block for direct quantum estimations of both linear and nonlinear functionals of any density operator. The network has many potential applications ranging from purity tests and eigenvalue estimations to direct characterization of some properties of quantum channels. Experimental realizations of the proposed network are within the reach of quantum technology that is currently being developed.

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.

On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry

NASA Astrophysics Data System (ADS)

Fyodorov, Yan V.

2018-06-01

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated `non-orthogonality overlap factor' (also known as the `eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size {N× N} . First we derive the general finite N expression for the JPD of a real eigenvalue {λ} and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its `bulk' and `edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue z and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its `bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (Phys Rev Lett 81(16):3367-3370, 1998), and we provide the `edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018. arXiv:1801.01219).

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 .

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.

Nonlinear Analysis of the Space Shuttle Superlightweight LO2 Tank. Part 2; Behavior Under 3g End-of-Flight Loads

NASA Technical Reports Server (NTRS)

Nemeth, Michael P.; Young, Richard D.; Collins, Timothy J.; Starnes, James H.,Jr.

1998-01-01

Results of linear bifurcation and nonlinear analyses of the Space Shuttle super lightweight (SLWT) external liquid-oxygen (LO2) tank are presented for an important end-of-flight loading condition. These results illustrate an important type of response mode for thin-walled shells, that are subjected to combined mechanical and thermal loads, that may be encountered in the design of other liquid-fuel launch vehicles. Linear bifurcation analyses are presented that predict several nearly equal eigenvalues that correspond to local buckling modes in the aft dome of the LO2 tank. In contrast, the nonlinear response phenomenon is shown to consist of a short-wavelength bending deformation in the aft elliptical dome of the LO2 tank that grows in amplitude in a stable manner with increasing load. Imperfection sensitivity analyses are presented that show that the presence of several nearly equal eigenvalues does not lead to a premature general instability mode for the aft dome. For the linear bifurcation and nonlinear analyses, the results show that accurate predictions of the response of the shell generally require a large-scale, high fidelity finite-element model. Results are also presented that show that the SLWT LO2 tank can support loads in excess of approximately 1.9 times the values of the operational loads considered.

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.

Numerical implementation of complex orthogonalization, parallel transport on Stiefel bundles, and analyticity

NASA Astrophysics Data System (ADS)

Avitabile, Daniele; Bridges, Thomas J.

2010-06-01

Numerical integration of complex linear systems of ODEs depending analytically on an eigenvalue parameter are considered. Complex orthogonalization, which is required to stabilize the numerical integration, results in non-analytic systems. It is shown that properties of eigenvalues are still efficiently recoverable by extracting information from a non-analytic characteristic function. The orthonormal systems are constructed using the geometry of Stiefel bundles. Different forms of continuous orthogonalization in the literature are shown to correspond to different choices of connection one-form on the Stiefel bundle. For the numerical integration, Gauss-Legendre Runge-Kutta algorithms are the principal choice for preserving orthogonality, and performance results are shown for a range of GLRK methods. The theory and methods are tested by application to example boundary value problems including the Orr-Sommerfeld equation in hydrodynamic stability.

Vibration analysis of rotor systems using reduced subsystem models

NASA Technical Reports Server (NTRS)

Fan, Uei-Jiun; Noah, Sherif T.

1989-01-01

A general impedance method using reduced submodels has been developed for the linear dynamic analysis of rotor systems. Formulated in terms of either modal or physical coordinates of the subsystems, the method enables imbalance responses at specific locations of the rotor systems to be efficiently determined from a small number of 'master' degrees of freedom. To demonstrate the capability of this impedance approach, the Space Shuttle Main Engine high-pressure oxygen turbopump has been investigated to determine the bearing loads due to imbalance. Based on the same formulation, an eigenvalue analysis has been performed to study the system stability. A small 5-DOF model has been utilized to illustrate the application of the method to eigenvalue analysis. Because of its inherent characteristics of allowing formulation of reduced submodels, the impedance method can significantly increase the computational speed.

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.

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

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.

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.

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.

Random discrete linear canonical transform.

PubMed

Wei, Deyun; Wang, Ruikui; Li, Yuan-Min

2016-12-01

Linear canonical transforms (LCTs) are a family of integral transforms with wide applications in optical, acoustical, electromagnetic, and other wave propagation problems. In this paper, we propose the random discrete linear canonical transform (RDLCT) by randomizing the kernel transform matrix of the discrete linear canonical transform (DLCT). The RDLCT inherits excellent mathematical properties from the DLCT along with some fantastic features of its own. It has a greater degree of randomness because of the randomization in terms of both eigenvectors and eigenvalues. Numerical simulations demonstrate that the RDLCT has an important feature that the magnitude and phase of its output are both random. As an important application of the RDLCT, it can be used for image encryption. The simulation results demonstrate that the proposed encryption method is a security-enhanced image encryption scheme.

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.

Predictability of a Coupled Model of ENSO Using Singular Vector Analysis: Optimal Growth and Forecast Skill.

NASA Astrophysics Data System (ADS)

Xue, Yan

The optimal growth and its relationship with the forecast skill of the Zebiak and Cane model are studied using a simple statistical model best fit to the original nonlinear model and local linear tangent models about idealized climatic states (the mean background and ENSO cycles in a long model run), and the actual forecast states, including two sets of runs using two different initialization procedures. The seasonally varying Markov model best fit to a suite of 3-year forecasts in a reduced EOF space (18 EOFs) fits the original nonlinear model reasonably well and has comparable or better forecast skill. The initial error growth in a linear evolution operator A is governed by the eigenvalues of A^{T}A, and the square roots of eigenvalues and eigenvectors of A^{T}A are named singular values and singular vectors. One dominant growing singular vector is found, and the optimal 6 month growth rate is largest for a (boreal) spring start and smallest for a fall start. Most of the variation in the optimal growth rate of the two forecasts is seasonal, attributable to the seasonal variations in the mean background, except that in the cold events it is substantially suppressed. It is found that the mean background (zero anomaly) is the most unstable state, and the "forecast IC states" are more unstable than the "coupled model states". One dominant growing singular vector is found, characterized by north-south and east -west dipoles, convergent winds on the equator in the eastern Pacific and a deepened thermocline in the whole equatorial belt. This singular vector is insensitive to initial time and optimization time, but its final pattern is a strong function of initial states. The ENSO system is inherently unpredictable for the dominant singular vector can amplify 5-fold to 24-fold in 6 months and evolve into the large scales characteristic of ENSO. However, the inherent ENSO predictability is only a secondary factor, while the mismatches between the model and data is a primary factor controlling the current forecast skill.

Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates

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

Perfetti, Christopher M; Rearden, Bradley T

2015-01-01

This study explored the impact of undersampling on the accuracy of tally estimates in Monte Carlo (MC) calculations. Steady-state MC simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity, and the impact of undersampling on eigenvalue and fuel pin flux/fission estimates was examined. This study observed biases in MC eigenvalue estimates as large as several percent and biases in fuel pin flux/fission tally estimates that exceeded tens, and in some cases hundreds, of percent. This study also investigated five statistical metrics for predicting the occurrence of undersampling biases in MC simulations. Threemore » of the metrics (the Heidelberger-Welch RHW, the Geweke Z-Score, and the Gelman-Rubin diagnostics) are commonly used for diagnosing the convergence of Markov chains, and two of the methods (the Contributing Particles per Generation and Tally Entropy) are new convergence metrics developed in the course of this study. These metrics were implemented in the KENO MC code within the SCALE code system and were evaluated for their reliability at predicting the onset and magnitude of undersampling biases in MC eigenvalue and flux tally estimates in two of the critical models. Of the five methods investigated, the Heidelberger-Welch RHW, the Gelman-Rubin diagnostics, and Tally Entropy produced test metrics that correlated strongly to the size of the observed undersampling biases, indicating their potential to effectively predict the size and prevalence of undersampling biases in MC simulations.« less

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.

A new method of imposing boundary conditions for hyperbolic equations

NASA Technical Reports Server (NTRS)

Funaro, D.; ative.

1987-01-01

A new method to impose boundary conditions for pseudospectral approximations to hyperbolic equations is suggested. This method involves the collocation of the equation at the boundary nodes as well as satisfying boundary conditions. Stability and convergence results are proven for the Chebyshev approximation of linear scalar hyperbolic equations. The eigenvalues of this method applied to parabolic equations are shown to be real and negative.

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.

Study of a mixed dispersal population dynamics model

DOE PAGES

Chugunova, Marina; Jadamba, Baasansuren; Kao, Chiu -Yen; ...

2016-08-27

In this study, we consider a mixed dispersal model with periodic and Dirichlet boundary conditions and its corresponding linear eigenvalue problem. This model describes the time evolution of a population which disperses both locally and non-locally. We investigate how long time dynamics depend on the parameter values. Furthermore, we study the minimization of the principal eigenvalue under the constraints that the resource function is bounded from above and below, and with a fixed total integral. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for the species to diemore » out more slowly or survive more easily. Our numerical simulations indicate that the optimal favorable region tends to be a simply-connected domain. Numerous results are shown to demonstrate various scenarios of optimal favorable regions for periodic and Dirichlet boundary conditions.« less

A new method for determining acoustic-liner admittance in a rectangular duct with grazing flow from experimental data

NASA Technical Reports Server (NTRS)

Watson, W. R.

1984-01-01

A method is developed for determining acoustic liner admittance in a rectangular duct with grazing flow. The axial propagation constant, cross mode order, and mean flow profile is measured. These measured data are then input into an analytical program which determines the unknown admittance value. The analytical program is based upon a finite element discretization of the acoustic field and a reposing of the unknown admittance value as a linear eigenvalue problem on the admittance value. Gaussian elimination is employed to solve this eigenvalue problem. The method used is extendable to grazing flows with boundary layers in both transverse directions of an impedance tube (or duct). Predicted admittance values are compared both with exact values that can be obtained for uniform mean flow profiles and with those from a Runge Kutta integration technique for cases involving a one dimensional boundary layer.

Pressure distribution under flexible polishing tools. I - Conventional aspheric optics

NASA Astrophysics Data System (ADS)

Mehta, Pravin K.; Hufnagel, Robert E.

1990-10-01

The paper presents a mathematical model, based on Kirchoff's thin flat plate theory, developed to determine polishing pressure distribution for a flexible polishing tool. A two-layered tool in which bending and compressive stiffnesses are equal is developed, which is formulated as a plate on a linearly elastic foundation. An equivalent eigenvalue problem and solution for a free-free plate are created from the plate formulation. For aspheric, anamorphic optical surfaces, the tool misfit is derived; it is defined as the result of movement from the initial perfect fit on the optic to any other position. The Polisher Design (POD) software for circular tools on aspheric optics is introduced. NASTRAN-based finite element analysis results are compared with the POD software, showing high correlation. By employing existing free-free eigenvalues and eigenfunctions, the work may be extended to rectangular polishing tools as well.

Chaotic Attractor Crisis and Climate Sensitivity: a Transfer Operator Approach

NASA Astrophysics Data System (ADS)

Tantet, A.; Lucarini, V.; Lunkeit, F.; Dijkstra, H. A.

2015-12-01

The rough response to a smooth parameter change of some non-chaotic climate models, such as the warm to snowball-Earth transition in energy balance models due to the ice-albedo feedback, can be studied in the framework of bifurcation theory, in particular by analysing the Lyapunov spectrum of fixed points or periodic orbits. However, bifurcation theory is of little help to study the destruction of a chaotic attractor which can occur in high-dimensional General Circulation Models (GCM). Yet, one would expect critical slowing down to occur before the crisis, since, as the system becomes susceptible to the physical instability mechanism responsible for the crisis, it turns out to be less and less resilient to exogenous perturbations and to spontaneous fluctuations due to other types of instabilities on the attractor. The statistical physics framework, extended to nonequilibrium systems, is particularly well suited for the study of global properties of chaotic and stochastic systems. In particular, the semigroup of transfer operators governs the evolution of distributions in phase space and its spectrum characterises both the relaxation rate of distributions to a statistical steady-state and the stability of this steady-state to perturbations. If critical slowing down indeed occurs in the approach to an attractor crisis, the gap in the spectrum of the semigroup of transfer operators is expected to shrink. We show that the chaotic attractor crisis due to the ice-albedo feedback and resulting in a transition from a warm to a snowball-Earth in the Planet Simulator (PlaSim), a GCM of intermediate complexity, is associated with critical slowing down, as observed by the slower decay of correlations before the crisis (cf. left panel). In addition, we demonstrate that this critical slowing down can be traced back to the shrinkage of the gap between the leading eigenvalues of coarse-grained approximations of the transfer operators and that these eigenvalues capture the fundamental features of the attractor crisis (cf. right panel). Finally, that the spectral gap is small close to the crisis suggests that the linear concept of Climate Sensitivity may be applied only far from an attractor crisis.

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.

Investigation of the flight mechanics simulation of a hovering helicopter

NASA Technical Reports Server (NTRS)

Chaimovich, M.; Rosen, A.; Rand, O.; Mansur, M. H.; Tischler, M. B.

1992-01-01

The flight mechanics simulation of a hovering helicopter is investigated by comparing the results of two different numerical models with flight test data for a hovering AH-64 Apache. The two models are the U.S. Army BEMAP and the Technion model. These nonlinear models are linearized by applying a numerical linearization procedure. The results of the linear models are compared with identification results in terms of eigenvalues, stability and control derivatives, and frequency responses. Detailed time histories of the responses of the complete nonlinear models, as a result of various pilots' inputs, are compared with flight test results. In addition the sensitivity of the models to various effects are also investigated. The results are discussed and problematic aspects of the simulation are identified.

A look-ahead variant of the Lanczos algorithm and its application to the quasi-minimal residual method for non-Hermitian linear systems. Ph.D. Thesis - Massachusetts Inst. of Technology, Aug. 1991

NASA Technical Reports Server (NTRS)

Nachtigal, Noel M.

1991-01-01

The Lanczos algorithm can be used both for eigenvalue problems and to solve linear systems. However, when applied to non-Hermitian matrices, the classical Lanczos algorithm is susceptible to breakdowns and potential instabilities. In addition, the biconjugate gradient (BCG) algorithm, which is the natural generalization of the conjugate gradient algorithm to non-Hermitian linear systems, has a second source of breakdowns, independent of the Lanczos breakdowns. Here, we present two new results. We propose an implementation of a look-ahead variant of the Lanczos algorithm which overcomes the breakdowns by skipping over those steps where a breakdown or a near-breakdown would occur. The new algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products per step as the classical Lanczos algorithm without look-ahead. Based on the proposed look-ahead Lanczos algorithm, we then present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which avoids the second source of breakdowns in the BCG algorithm. We present details of the new method and discuss some of its properties. In particular, we discuss the relationship between QMR and BCG, showing how one can recover the BCG iterates, when they exist, from the QMR iterates. We also present convergence results for QMR, showing the connection between QMR and the generalized minimal residual (GMRES) algorithm, the optimal method in this class of methods. Finally, we give some numerical examples, both for eigenvalue computations and for non-Hermitian linear systems.

Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals.

PubMed

Zuehlsdorff, T J; Hine, N D M; Payne, M C; Haynes, P D

2015-11-28

We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on a small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.

Monte Carlo Techniques for Nuclear Systems - Theory Lectures

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

Brown, Forrest B.

These are lecture notes for a Monte Carlo class given at the University of New Mexico. The following topics are covered: course information; nuclear eng. review & MC; random numbers and sampling; computational geometry; collision physics; tallies and statistics; eigenvalue calculations I; eigenvalue calculations II; eigenvalue calculations III; variance reduction; parallel Monte Carlo; parameter studies; fission matrix and higher eigenmodes; doppler broadening; Monte Carlo depletion; HTGR modeling; coupled MC and T/H calculations; fission energy deposition. Solving particle transport problems with the Monte Carlo method is simple - just simulate the particle behavior. The devil is in the details, however. Thesemore » lectures provide a balanced approach to the theory and practice of Monte Carlo simulation codes. The first lectures provide an overview of Monte Carlo simulation methods, covering the transport equation, random sampling, computational geometry, collision physics, and statistics. The next lectures focus on the state-of-the-art in Monte Carlo criticality simulations, covering the theory of eigenvalue calculations, convergence analysis, dominance ratio calculations, bias in Keff and tallies, bias in uncertainties, a case study of a realistic calculation, and Wielandt acceleration techniques. The remaining lectures cover advanced topics, including HTGR modeling and stochastic geometry, temperature dependence, fission energy deposition, depletion calculations, parallel calculations, and parameter studies. This portion of the class focuses on using MCNP to perform criticality calculations for reactor physics and criticality safety applications. It is an intermediate level class, intended for those with at least some familiarity with MCNP. Class examples provide hands-on experience at running the code, plotting both geometry and results, and understanding the code output. The class includes lectures & hands-on computer use for a variety of Monte Carlo calculations. Beginning MCNP users are encouraged to review LA-UR-09-00380, "Criticality Calculations with MCNP: A Primer (3nd Edition)" (available at http:// mcnp.lanl.gov under "Reference Collection") prior to the class. No Monte Carlo class can be complete without having students write their own simple Monte Carlo routines for basic random sampling, use of the random number generator, and simplified particle transport simulation.« less

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.

Chaotic dynamics and diffusion in a piecewise linear equation

NASA Astrophysics Data System (ADS)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-03-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

Self-diffusion in periodic porous media: a comparison of numerical simulation and eigenvalue methods.

PubMed

Schwartz, L M; Bergman, D J; Dunn, K J; Mitra, P P

1996-01-01

Random walk computer simulations are an important tool in understanding magnetic resonance measurements in porous media. In this paper we focus on the description of pulsed field gradient spin echo (PGSE) experiments that measure the probability, P(R,t), that a diffusing water molecule will travel a distance R in a time t. Because PGSE simulations are often limited by statistical considerations, we will see that valuable insight can be gained by working with simple periodic geometries and comparing simulation data to the results of exact eigenvalue expansions. In this connection, our attention will be focused on (1) the wavevector, k, and time dependent magnetization, M(k, t); and (2) the normalized probability, Ps(delta R, t), that a diffusing particle will return to within delta R of the origin after time t.

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

NASA Technical Reports Server (NTRS)

Kenny, Sean P.; Hou, Gene J. W.

1994-01-01

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

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.

Simultaneous structural and control optimization via linear quadratic regulator eigenstructure assignment

NASA Technical Reports Server (NTRS)

Becus, G. A.; Lui, C. Y.; Venkayya, V. B.; Tischler, V. A.

1987-01-01

A method for simultaneous structural and control design of large flexible space structures (LFSS) to reduce vibration generated by disturbances is presented. Desired natural frequencies and damping ratios for the closed loop system are achieved by using a combination of linear quadratic regulator (LQR) synthesis and numerical optimization techniques. The state and control weighing matrices (Q and R) are expressed in terms of structural parameters such as mass and stiffness. The design parameters are selected by numerical optimization so as to minimize the weight of the structure and to achieve the desired closed-loop eigenvalues. An illustrative example of the design of a two bar truss is presented.

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.

Compressed modes for variational problems in mathematics and physics

PubMed Central

Ozoliņš, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley

2013-01-01

This article describes a general formalism for obtaining spatially localized (“sparse”) solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger’s equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support (“compressed modes”). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size. PMID:24170861

Compressed modes for variational problems in mathematics and physics.

PubMed

Ozolins, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley

2013-11-12

This article describes a general formalism for obtaining spatially localized ("sparse") solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support ("compressed modes"). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size.

Heavy neutrino mixing and single production at linear collider

NASA Astrophysics Data System (ADS)

Gluza, J.; Maalampi, J.; Raidal, M.; Zrałek, M.

1997-02-01

We study the single production of heavy neutrinos via the processes e- e+ -> νN and e- γ -> W- N at future linear colliders. As a base of our considerations we take a wide class of models, both with vanishing and non-vanishing left-handed Majorana neutrino mass matrix mL. We perform a model independent analyses of the existing experimental data and find connections between the characteristic of heavy neutrinos (masses, mixings, CP eigenvalues) and the mL parameters. We show that with the present experimental constraints heavy neutrino masses almost up to the collision energy can be tested in the future experiments.

Modern digital flight control system design for VTOL aircraft

NASA Technical Reports Server (NTRS)

Broussard, J. R.; Berry, P. W.; Stengel, R. F.

1979-01-01

Methods for and results from the design and evaluation of a digital flight control system (DFCS) for a CH-47B helicopter are presented. The DFCS employed proportional-integral control logic to provide rapid, precise response to automatic or manual guidance commands while following conventional or spiral-descent approach paths. It contained altitude- and velocity-command modes, and it adapted to varying flight conditions through gain scheduling. Extensive use was made of linear systems analysis techniques. The DFCS was designed, using linear-optimal estimation and control theory, and the effects of gain scheduling are assessed by examination of closed-loop eigenvalues and time responses.

Statistical properties of MHD fluctuations associated with high speed streams from HELIOS 2 observations

NASA Technical Reports Server (NTRS)

Bavassano, B.; Dobrowolny, H.; Fanfoni, G.; Mariani, F.; Ness, N. F.

1981-01-01

Helios 2 magnetic data were used to obtain several statistical properties of MHD fluctuations associated with the trailing edge of a given stream served in different solar rotations. Eigenvalues and eigenvectors of the variance matrix, total power and degree of compressibility of the fluctuations were derived and discussed both as a function of distance from the Sun and as a function of the frequency range included in the sample. The results obtained add new information to the picture of MHD turbulence in the solar wind. In particular, a dependence from frequency range of the radial gradients of various statistical quantities is obtained.

Weak value controversy

NASA Astrophysics Data System (ADS)

Vaidman, L.

2017-10-01

Recent controversy regarding the meaning and usefulness of weak values is reviewed. It is argued that in spite of recent statistical arguments by Ferrie and Combes, experiments with anomalous weak values provide useful amplification techniques for precision measurements of small effects in many realistic situations. The statistical nature of weak values is questioned. Although measuring weak values requires an ensemble, it is argued that the weak value, similarly to an eigenvalue, is a property of a single pre- and post-selected quantum system. This article is part of the themed issue `Second quantum revolution: foundational questions'.

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

Cluster structure in the correlation coefficient matrix can be characterized by abnormal eigenvalues

NASA Astrophysics Data System (ADS)

Nie, Chun-Xiao

2018-02-01

In a large number of previous studies, the researchers found that some of the eigenvalues of the financial correlation matrix were greater than the predicted values of the random matrix theory (RMT). Here, we call these eigenvalues as abnormal eigenvalues. In order to reveal the hidden meaning of these abnormal eigenvalues, we study the toy model with cluster structure and find that these eigenvalues are related to the cluster structure of the correlation coefficient matrix. In this paper, model-based experiments show that in most cases, the number of abnormal eigenvalues of the correlation matrix is equal to the number of clusters. In addition, empirical studies show that the sum of the abnormal eigenvalues is related to the clarity of the cluster structure and is negatively correlated with the correlation dimension.

Structure preserving parallel algorithms for solving the Bethe–Salpeter eigenvalue problem

DOE PAGES

Shao, Meiyue; da Jornada, Felipe H.; Yang, Chao; ...

2015-10-02

The Bethe–Salpeter eigenvalue problem is a dense structured eigenvalue problem arising from discretized Bethe–Salpeter equation in the context of computing exciton energies and states. A computational challenge is that at least half of the eigenvalues and the associated eigenvectors are desired in practice. In this paper, we establish the equivalence between Bethe–Salpeter eigenvalue problems and real Hamiltonian eigenvalue problems. Based on theoretical analysis, structure preserving algorithms for a class of Bethe–Salpeter eigenvalue problems are proposed. We also show that for this class of problems all eigenvalues obtained from the Tamm–Dancoff approximation are overestimated. In order to solve large scale problemsmore » of practical interest, we discuss parallel implementations of our algorithms targeting distributed memory systems. Finally, several numerical examples are presented to demonstrate the efficiency and accuracy of our algorithms.« less

On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections

NASA Astrophysics Data System (ADS)

Ebrahimi, R.; Zohren, S.

2018-03-01

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar (J. Stat. Mech. P04001 arXiv:1102.0738), to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. More precisely, we show convergence of the rescaled eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential. Secondly, it is shown that this approach can be generalised to obtain convergence of the eigenvalue with smallest modulus and its universality for ring distributions. Most interestingly, the here presented techniques are used to compute all slowly varying finite N correction of the above distributions, which is important for practical applications, given the slow convergence. Another interesting aspect of this work is the fact that we can use standard techniques from Hermitian random matrices to obtain the extreme value statistics of non-Hermitian random matrices resembling the large N expansion used in context of the double scaling limit of Hermitian matrix models in string theory.

Program for solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

Sloate, H.

1973-01-01

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

Observability of discretized partial differential equations

NASA Technical Reports Server (NTRS)

Cohn, Stephen E.; Dee, Dick P.

1988-01-01

It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

A study of two cases of comma-cloud cyclogenesis using a semigeostrophic model

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

Craig, G.C.; Cho, Hanru

1992-12-01

The linear stability of two atmospheric flows is studied, with basic-state data taken from environments where comma clouds are observed to flow. Each basic state features a baroclinic zone associated with an upper-level jet, with conditional instability on the north side. The semigeostrophic approximation is utilized, along with a simple parameterization for cumulus heating, and the eigenvalue problem is solved employing a Chebyshev spectral technique. 47 refs.

Aircraft model prototypes which have specified handling-quality time histories

NASA Technical Reports Server (NTRS)

Johnson, S. H.

1976-01-01

Several techniques for obtaining linear constant-coefficient airplane models from specified handling-quality time histories are discussed. One technique, the pseudodata method, solves the basic problem, yields specified eigenvalues, and accommodates state-variable transfer-function zero suppression. The method is fully illustrated for a fourth-order stability-axis small-motion model with three lateral handling-quality time histories specified. The FORTRAN program which obtains and verifies the model is included and fully documented.

Robust L1-norm two-dimensional linear discriminant analysis.

PubMed

Li, Chun-Na; Shao, Yuan-Hai; Deng, Nai-Yang

2015-05-01

In this paper, we propose an L1-norm two-dimensional linear discriminant analysis (L1-2DLDA) with robust performance. Different from the conventional two-dimensional linear discriminant analysis with L2-norm (L2-2DLDA), where the optimization problem is transferred to a generalized eigenvalue problem, the optimization problem in our L1-2DLDA is solved by a simple justifiable iterative technique, and its convergence is guaranteed. Compared with L2-2DLDA, our L1-2DLDA is more robust to outliers and noises since the L1-norm is used. This is supported by our preliminary experiments on toy example and face datasets, which show the improvement of our L1-2DLDA over L2-2DLDA. Copyright © 2015 Elsevier Ltd. All rights reserved.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-10-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ω_i while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-09-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev

2017-11-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

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.

Linear and weakly nonlinear aspects of free shear layer instability, roll-up, subharmonic interaction and wall influence

NASA Technical Reports Server (NTRS)

Cain, A. B.; Thompson, M. W.

1986-01-01

The growth of the momentum thickness and the modal disturbance energies are examined to study the nature and onset of nonlinearity in a temporally growing free shear layer. A shooting technique is used to find solutions to the linearized eigenvalue problem, and pseudospectral weakly nonlinear simulations of this flow are obtained for comparison. The roll-up of a fundamental disturbance follows linear theory predictions even with a 20 percent disturbance amplitude. A weak nonlinear interaction of the disturbance creates a finite-amplitude mean shear stress which dominates the growth of the layer momentum thickness, and the disturbance growth rate changes until the fundamental disturbance dominates. The fundamental then becomes an energy source for the harmonic, resulting in an increase in the growth rate of the subharmonic over the linear prediction even when the fundamental has no energy to give. Also considered are phase relations and the wall influence.

LQR-Based Optimal Distributed Cooperative Design for Linear Discrete-Time Multiagent Systems.

PubMed

Zhang, Huaguang; Feng, Tao; Liang, Hongjing; Luo, Yanhong

2017-03-01

In this paper, a novel linear quadratic regulator (LQR)-based optimal distributed cooperative design method is developed for synchronization control of general linear discrete-time multiagent systems on a fixed, directed graph. Sufficient conditions are derived for synchronization, which restrict the graph eigenvalues into a bounded circular region in the complex plane. The synchronizing speed issue is also considered, and it turns out that the synchronizing region reduces as the synchronizing speed becomes faster. To obtain more desirable synchronizing capacity, the weighting matrices are selected by sufficiently utilizing the guaranteed gain margin of the optimal regulators. Based on the developed LQR-based cooperative design framework, an approximate dynamic programming technique is successfully introduced to overcome the (partially or completely) model-free cooperative design for linear multiagent systems. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design methods.

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

Study of T-wave morphology parameters based on Principal Components Analysis during acute myocardial ischemia

NASA Astrophysics Data System (ADS)

Baglivo, Fabricio Hugo; Arini, Pedro David

2011-12-01

Electrocardiographic repolarization abnormalities can be detected by Principal Components Analysis of the T-wave. In this work we studied the efect of signal averaging on the mean value and reproducibility of the ratio of the 2nd to the 1st eigenvalue of T-wave (T21W) and the absolute and relative T-wave residuum (TrelWR and TabsWR) in the ECG during ischemia induced by Percutaneous Coronary Intervention. Also, the intra-subject and inter-subject variability of T-wave parameters have been analyzed. Results showed that TrelWR and TabsWR evaluated from the average of 10 complexes had lower values and higher reproducibility than those obtained from 1 complex. On the other hand T21W calculated from 10 complexes did not show statistical diferences versus the T21W calculated on single beats. The results of this study corroborate that, with a signal averaging technique, the 2nd and the 1st eigenvalue are not afected by noise while the 4th to 8th eigenvalues are so much afected by this, suggesting the use of the signal averaged technique before calculation of absolute and relative T-wave residuum. Finally, we have shown that T-wave morphology parameters present high intra-subject stability.

Comparison of scalar measures used in magnetic resonance diffusion tensor imaging.

PubMed

Bahn, M M

1999-07-01

The tensors derived from diffusion tensor imaging describe complex diffusion in tissues. However, it is difficult to compare tensors directly or to produce images that contain all of the information of the tensor. Therefore, it is convenient to produce scalar measures that extract desired aspects of the tensor. These measures map the three-dimensional eigenvalues of the diffusion tensor into scalar values. The measures impose an order on eigenvalue space. Many invariant scalar measures have been introduced in the literature. In the present manuscript, a general approach for producing invariant scalar measures is introduced. Because it is often difficult to determine in clinical practice which of the many measures is best to apply to a given situation, two formalisms are introduced for the presentation, definition, and comparison of measures applied to eigenvalues: (1) normalized eigenvalue space, and (2) parametric eigenvalue transformation plots. All of the anisotropy information contained in the three eigenvalues can be retained and displayed in a two-dimensional plot, the normalized eigenvalue plot. An example is given of how to determine the best measure to use for a given situation by superimposing isometric contour lines from various anisotropy measures on plots of actual measured eigenvalue data points. Parametric eigenvalue transformation plots allow comparison of how different measures impose order on normalized eigenvalue space to determine whether the measures are equivalent and how the measures differ. These formalisms facilitate the comparison of scalar invariant measures for diffusion tensor imaging. Normalized eigenvalue space allows presentation of eigenvalue anisotropy information. Copyright 1999 Academic Press.

A Quantitative Structure-Property Relationship (QSPR) Study of Aliphatic Alcohols by the Method of Dividing the Molecular Structure into Substructure

PubMed Central

Liu, Fengping; Cao, Chenzhong; Cheng, Bin

2011-01-01

A quantitative structure–property relationship (QSPR) analysis of aliphatic alcohols is presented. Four physicochemical properties were studied: boiling point (BP), n-octanol–water partition coefficient (lg POW), water solubility (lg W) and the chromatographic retention indices (RI) on different polar stationary phases. In order to investigate the quantitative structure–property relationship of aliphatic alcohols, the molecular structure ROH is divided into two parts, R and OH to generate structural parameter. It was proposed that the property is affected by three main factors for aliphatic alcohols, alkyl group R, substituted group OH, and interaction between R and OH. On the basis of the polarizability effect index (PEI), previously developed by Cao, the novel molecular polarizability effect index (MPEI) combined with odd-even index (OEI), the sum eigenvalues of bond-connecting matrix (SX1CH) previously developed in our team, were used to predict the property of aliphatic alcohols. The sets of molecular descriptors were derived directly from the structure of the compounds based on graph theory. QSPR models were generated using only calculated descriptors and multiple linear regression techniques. These QSPR models showed high values of multiple correlation coefficient (R > 0.99) and Fisher-ratio statistics. The leave-one-out cross-validation demonstrated the final models to be statistically significant and reliable. PMID:21731451

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.

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.

The two-dimensional kinetic ballooning theory for ion temperature gradient mode in tokamak

NASA Astrophysics Data System (ADS)

Xie, T.; Zhang, Y. Z.; Mahajan, S. M.; Hu, S. L.; He, Hongda; Liu, Z. Y.

2017-10-01

The two-dimensional (2D) kinetic ballooning theory is developed for the ion temperature gradient mode in an up-down symmetric equilibrium (illustrated via concentric circular magnetic surfaces). The ballooning transform converts the basic 2D linear gyro-kinetic equation into two equations: (1) the lowest order equation (ballooning equation) is an integral equation essentially the same as that reported by Dong et al., [Phys. Fluids B 4, 1867 (1992)] but has an undetermined Floquet phase variable, (2) the higher order equation for the rapid phase envelope is an ordinary differential equation in the same form as the 2D ballooning theory in a fluid model [Xie et al., Phys. Plasmas 23, 042514 (2016)]. The system is numerically solved by an iterative approach to obtain the (phase independent) eigen-value. The new results are compared to the two earlier theories. We find a strongly modified up-down asymmetric mode structure, and non-trivial modifications to the eigen-value.

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

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.

Actinide electronic structure and atomic forces

NASA Astrophysics Data System (ADS)

Albers, R. C.; Rudin, Sven P.; Trinkle, Dallas R.; Jones, M. D.

2000-07-01

We have developed a new method[1] of fitting tight-binding parameterizations based on functional forms developed at the Naval Research Laboratory.[2] We have applied these methods to actinide metals and report our success using them (see below). The fitting procedure uses first-principles local-density-approximation (LDA) linear augmented plane-wave (LAPW) band structure techniques[3] to first calculate an electronic-structure band structure and total energy for fcc, bcc, and simple cubic crystal structures for the actinide of interest. The tight-binding parameterization is then chosen to fit the detailed energy eigenvalues of the bands along symmetry directions, and the symmetry of the parameterization is constrained to agree with the correct symmetry of the LDA band structure at each eigenvalue and k-vector that is fit to. By fitting to a range of different volumes and the three different crystal structures, we find that the resulting parameterization is robust and appears to accurately calculate other crystal structures and properties of interest.

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.

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.

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.

ASTROP2-LE: A Mistuned Aeroelastic Analysis System Based on a Two Dimensional Linearized Euler Solver

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Srivastava, R.; Mehmed, Oral

2002-01-01

An aeroelastic analysis system for flutter and forced response analysis of turbomachines based on a two-dimensional linearized unsteady Euler solver has been developed. The ASTROP2 code, an aeroelastic stability analysis program for turbomachinery, was used as a basis for this development. The ASTROP2 code uses strip theory to couple a two dimensional aerodynamic model with a three dimensional structural model. The code was modified to include forced response capability. The formulation was also modified to include aeroelastic analysis with mistuning. A linearized unsteady Euler solver, LINFLX2D is added to model the unsteady aerodynamics in ASTROP2. By calculating the unsteady aerodynamic loads using LINFLX2D, it is possible to include the effects of transonic flow on flutter and forced response in the analysis. The stability is inferred from an eigenvalue analysis. The revised code, ASTROP2-LE for ASTROP2 code using Linearized Euler aerodynamics, is validated by comparing the predictions with those obtained using linear unsteady aerodynamic solutions.

Methods, Software and Tools for Three Numerical Applications. Final report

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

E. R. Jessup

2000-03-01

This is a report of the results of the authors work supported by DOE contract DE-FG03-97ER25325. They proposed to study three numerical problems. They are: (1) the extension of the PMESC parallel programming library; (2) the development of algorithms and software for certain generalized eigenvalue and singular value (SVD) problems, and (3) the application of techniques of linear algebra to an information retrieval technique known as latent semantic indexing (LSI).

Acoustic-Liner Admittance in a Duct

NASA Technical Reports Server (NTRS)

Watson, W. R.

1986-01-01

Method calculates admittance from easily obtainable values. New method for calculating acoustic-liner admittance in rectangular duct with grazing flow based on finite-element discretization of acoustic field and reposing of unknown admittance value as linear eigenvalue problem on admittance value. Problem solved by Gaussian elimination. Unlike existing methods, present method extendable to mean flows with two-dimensional boundary layers as well. In presence of shear, results of method compared well with results of Runge-Kutta integration technique.

Probabilistic methods for rotordynamics analysis

NASA Technical Reports Server (NTRS)

Wu, Y.-T.; Torng, T. Y.; Millwater, H. R.; Fossum, A. F.; Rheinfurth, M. H.

1991-01-01

This paper summarizes the development of the methods and a computer program to compute the probability of instability of dynamic systems that can be represented by a system of second-order ordinary linear differential equations. Two instability criteria based upon the eigenvalues or Routh-Hurwitz test functions are investigated. Computational methods based on a fast probability integration concept and an efficient adaptive importance sampling method are proposed to perform efficient probabilistic analysis. A numerical example is provided to demonstrate the methods.

Entanglement entropy from tensor network states for stabilizer codes

NASA Astrophysics Data System (ADS)

He, Huan; Zheng, Yunqin; Bernevig, B. Andrei; Regnault, Nicolas

2018-03-01

In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of three-dimensional (3D) stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground states for some special cuts. In particular, we work out examples of the 3D toric code, the X-cube model, and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: For these, the constructed TNS is a singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground-state degeneracy. Moreover, the transfer matrices of these TNSs can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground-state degeneracy.

Effect of body aerodynamics on the dynamic flight stability of the hawkmoth Manduca sexta.

PubMed

Nguyen, Anh Tuan; Han, Jong-Seob; Han, Jae-Hung

2016-12-14

This study explores the effects of the body aerodynamics on the dynamic flight stability of an insect at various different forward flight speeds. The insect model, whose morphological parameters are based on measurement data from the hawkmoth Manduca sexta, is treated as an open-loop six-degree-of-freedom dynamic system. The aerodynamic forces and moments acting on the insect are computed by an aerodynamic model that combines the unsteady panel method and the extended unsteady vortex-lattice method. The aerodynamic model is then coupled to a multi-body dynamic code to solve the system of motion equations. First, the trimmed flight conditions of insect models with and without consideration of the body aerodynamics are obtained using a trim search algorithm. Subsequently, the effects of the body aerodynamics on the dynamic flight stability are analysed through modal structures, i.e., eigenvalues and eigenvectors in this case, which are based on linearized equations of motion. The solutions from the nonlinear and linearized equations of motion due to gust disturbances are obtained, and the effects of the body aerodynamics are also investigated through these solutions. The results showed the important effect of the body aerodynamics at high-speed forward flight (in this paper at 4.0 and 5.0 m s -1 ) and the movement trends of eigenvalues when the body aerodynamics is included.

Linear Temporal Stability Analysis of a Low-Density Round Gas Jet Injected into a High-Density Gas

NASA Technical Reports Server (NTRS)

Lawson, Anthony L.; Parthasarathy, Ramkumar N.

2002-01-01

It has been observed in previous experimental studies that round helium jets injected into air display a repetitive structure for a long distance, somewhat similar to the buoyancy-induced flickering observed in diffusion flames. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis of a round helium jet injected into air was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. The temporal growth rates of the disturbances increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity.

Technical notes and correspondence: Stochastic robustness of linear time-invariant control systems

NASA Technical Reports Server (NTRS)

Stengel, Robert F.; Ray, Laura R.

1991-01-01

A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluations of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but non-Gaussian cases, including uncertain-but-bounded variation. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation.

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

NASA Technical Reports Server (NTRS)

Freund, Roland W.; Nachtigal, Noel M.

1990-01-01

It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included.

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

Exact Solution of Klein-Gordon and Dirac Equations with Snyder-de Sitter Algebra

NASA Astrophysics Data System (ADS)

Merad, M.; Hadj Moussa, M.

2018-01-01

In this paper, we present the exact solution of the (1+1)-dimensional relativistic Klein-Gordon and Dirac equations with linear vector and scalar potentials in the framework of deformed Snyder-de Sitter model. We introduce some changes of variables, we show that a one-dimensional linear potential for the relativistic system in a space deformed can be equivalent to the trigonometric Rosen-Morse potential in a regular space. In both cases, we determine explicitly the energy eigenvalues and their corresponding eigenfunctions expressed in terms of Romonovski polynomials. The limiting cases are analyzed for α 1 and α 2 → 0 and are compared with those of literature.

A proposed method for enhanced eigen-pair extraction using finite element methods: Theory and application

NASA Technical Reports Server (NTRS)

Jara-Almonte, J.; Mitchell, L. D.

1988-01-01

The paper covers two distinct parts: theory and application. The goal of this work was the reduction of model size with an increase in eigenvalue/vector accuracy. This method is ideal for the condensation of large truss- or beam-type structures. The theoretical approach involves the conversion of a continuum transfer matrix beam element into an 'Exact' dynamic stiffness element. This formulation is implemented in a finite element environment. This results in the need to solve a transcendental eigenvalue problem. Once the eigenvalue is determined the eigenvectors can be reconstructed with any desired spatial precision. No discretization limitations are imposed on the reconstruction. The results of such a combined finite element and transfer matrix formulation is a much smaller FEM eigenvalue problem. This formulation has the ability to extract higher eigenvalues as easily and as accurately as lower eigenvalues. Moreover, one can extract many more eigenvalues/vectors from the model than the number of degrees of freedom in the FEM formulation. Typically, the number of eigenvalues accurately extractable via the 'Exact' element method are at least 8 times the number of degrees of freedom. In contrast, the FEM usually extracts one accurate (within 5 percent) eigenvalue for each 3-4 degrees of freedom. The 'Exact' element results in a 20-30 improvement in the number of accurately extractable eigenvalues and eigenvectors.

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.

Robustness of linear quadratic state feedback designs in the presence of system uncertainty. [application to Augmentor Wing Jet STOL Research Aircraft flare control autopilot design

NASA Technical Reports Server (NTRS)

Patel, R. V.; Toda, M.; Sridhar, B.

1977-01-01

The paper deals with the problem of expressing the robustness (stability) property of a linear quadratic state feedback (LQSF) design quantitatively in terms of bounds on the perturbations (modeling errors or parameter variations) in the system matrices so that the closed-loop system remains stable. Nonlinear time-varying and linear time-invariant perturbations are considered. The only computation required in obtaining a measure of the robustness of an LQSF design is to determine the eigenvalues of two symmetric matrices determined when solving the algebraic Riccati equation corresponding to the LQSF design problem. Results are applied to a complex dynamic system consisting of the flare control of a STOL aircraft. The design of the flare control is formulated as an LQSF tracking problem.

New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

NASA Astrophysics Data System (ADS)

Audibert, Lorenzo; Cakoni, Fioralba; Haddar, Houssem

2017-12-01

In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data.

Use of SCALE Continuous-Energy Monte Carlo Tools for Eigenvalue Sensitivity Coefficient Calculations

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

Perfetti, Christopher M; Rearden, Bradley T

2013-01-01

The TSUNAMI code within the SCALE code system makes use of eigenvalue sensitivity coefficients for an extensive number of criticality safety applications, such as quantifying the data-induced uncertainty in the eigenvalue of critical systems, assessing the neutronic similarity between different critical systems, and guiding nuclear data adjustment studies. The need to model geometrically complex systems with improved fidelity and the desire to extend TSUNAMI analysis to advanced applications has motivated the development of a methodology for calculating sensitivity coefficients in continuous-energy (CE) Monte Carlo applications. The CLUTCH and Iterated Fission Probability (IFP) eigenvalue sensitivity methods were recently implemented in themore » CE KENO framework to generate the capability for TSUNAMI-3D to perform eigenvalue sensitivity calculations in continuous-energy applications. This work explores the improvements in accuracy that can be gained in eigenvalue and eigenvalue sensitivity calculations through the use of the SCALE CE KENO and CE TSUNAMI continuous-energy Monte Carlo tools as compared to multigroup tools. The CE KENO and CE TSUNAMI tools were used to analyze two difficult models of critical benchmarks, and produced eigenvalue and eigenvalue sensitivity coefficient results that showed a marked improvement in accuracy. The CLUTCH sensitivity method in particular excelled in terms of efficiency and computational memory requirements.« less

Linear-scaling time-dependent density-functional theory beyond the Tamm-Dancoff approximation: Obtaining efficiency and accuracy with in situ optimised local orbitals

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

Zuehlsdorff, T. J., E-mail: tjz21@cam.ac.uk; Payne, M. C.; Hine, N. D. M.

2015-11-28

We present a solution of the full time-dependent density-functional theory (TDDFT) eigenvalue equation in the linear response formalism exhibiting a linear-scaling computational complexity with system size, without relying on the simplifying Tamm-Dancoff approximation (TDA). The implementation relies on representing the occupied and unoccupied subspaces with two different sets of in situ optimised localised functions, yielding a very compact and efficient representation of the transition density matrix of the excitation with the accuracy associated with a systematic basis set. The TDDFT eigenvalue equation is solved using a preconditioned conjugate gradient algorithm that is very memory-efficient. The algorithm is validated on amore » small test molecule and a good agreement with results obtained from standard quantum chemistry packages is found, with the preconditioner yielding a significant improvement in convergence rates. The method developed in this work is then used to reproduce experimental results of the absorption spectrum of bacteriochlorophyll in an organic solvent, where it is demonstrated that the TDA fails to reproduce the main features of the low energy spectrum, while the full TDDFT equation yields results in good qualitative agreement with experimental data. Furthermore, the need for explicitly including parts of the solvent into the TDDFT calculations is highlighted, making the treatment of large system sizes necessary that are well within reach of the capabilities of the algorithm introduced here. Finally, the linear-scaling properties of the algorithm are demonstrated by computing the lowest excitation energy of bacteriochlorophyll in solution. The largest systems considered in this work are of the same order of magnitude as a variety of widely studied pigment-protein complexes, opening up the possibility of studying their properties without having to resort to any semiclassical approximations to parts of the protein environment.« less

Normalized modes at selected points without normalization

NASA Astrophysics Data System (ADS)

Kausel, Eduardo

2018-04-01

As every textbook on linear algebra demonstrates, the eigenvectors for the general eigenvalue problem | K - λM | = 0 involving two real, symmetric, positive definite matrices K , M satisfy some well-defined orthogonality conditions. Equally well-known is the fact that those eigenvectors can be normalized so that their modal mass μ =ϕT Mϕ is unity: it suffices to divide each unscaled mode by the square root of the modal mass. Thus, the normalization is the result of an explicit calculation applied to the modes after they were obtained by some means. However, we show herein that the normalized modes are not merely convenient forms of scaling, but that they are actually intrinsic properties of the pair of matrices K , M, that is, the matrices already "know" about normalization even before the modes have been obtained. This means that we can obtain individual components of the normalized modes directly from the eigenvalue problem, and without needing to obtain either all of the modes or for that matter, any one complete mode. These results are achieved by means of the residue theorem of operational calculus, a finding that is rather remarkable inasmuch as the residues themselves do not make use of any orthogonality conditions or normalization in the first place. It appears that this obscure property connecting the general eigenvalue problem of modal analysis with the residue theorem of operational calculus may have been overlooked up until now, but which has in turn interesting theoretical implications.Á

Applying Rasch model analysis in the development of the cantonese tone identification test (CANTIT).

PubMed

Lee, Kathy Y S; Lam, Joffee H S; Chan, Kit T Y; van Hasselt, Charles Andrew; Tong, Michael C F

2017-01-01

Applying Rasch analysis to evaluate the internal structure of a lexical tone perception test known as the Cantonese Tone Identification Test (CANTIT). A 75-item pool (CANTIT-75) with pictures and sound tracks was developed. Respondents were required to make a four-alternative forced choice on each item. A short version of 30 items (CANTIT-30) was developed based on fit statistics, difficulty estimates, and content evaluation. Internal structure was evaluated by fit statistics and Rasch Factor Analysis (RFA). 200 children with normal hearing and 141 children with hearing impairment were recruited. For CANTIT-75, all infit and 97% of outfit values were < 2.0. RFA revealed 40.1% of total variance was explained by the Rasch measure. The first residual component explained 2.5% of total variance in an eigenvalue of 3.1. For CANTIT-30, all infit and outfit values were < 2.0. The Rasch measure explained 38.8% of total variance, the first residual component explained 3.9% of total variance in an eigenvalue of 1.9. The Rasch model provides excellent guidance for the development of short forms. Both CANTIT-75 and CANTIT-30 possess satisfactory internal structure as a construct validity evidence in measuring the lexical tone identification ability of the Cantonese speakers.

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.

Asymptotic approximations for pure bending of thin cylindrical shells

NASA Astrophysics Data System (ADS)

Coman, Ciprian D.

2017-08-01

A simplified partial wrinkling scenario for in-plane bending of thin cylindrical shells is explored by using several asymptotic strategies. The eighth-order boundary eigenvalue problem investigated here originates in the Donnel-Mushtari-Vlasov shallow shell theory coupled with a linear membrane pre-bifurcation state. It is shown that the corresponding neutral stability curve is amenable to a detailed asymptotic analysis based on the method of multiple scales. This is further complemented by an alternative WKB approximation that provides comparable information with significantly less effort.

Global Stability and Control Analysis of Aircraft at High Angles-of-Attack.

DTIC Science & Technology

1977-06-30

500 deg/sec to 500 deg/sec and com - puting the eigenvalues of the linearized system around different equilibrium points for 6e = 2°. It is noticed...SYSTEMS, INC. 0" -114- 2 Ä»" • + + + + + _+ + + + + + + + + M I o o o a i i 8 8 in 8 8 XNXX * x x» x x x< x...history com - parisons show significant improvement in dynamic response and performance using the BACTM ARI gain, as opposed to nulling the rudder

Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media.

PubMed

Nakkeeran, K

2001-10-01

We consider a family of N coupled nonlinear Schrödinger equations which govern the simultaneous propagation of N fields in the normal dispersion regime of an optical fiber with various important physical effects. The linear eigenvalue problem associated with the integrable form of all the equations is constructed with the help of the Ablowitz-Kaup-Newell-Segur method. Using the Hirota bilinear method, exact dark soliton solutions are explicitly derived.

Confining potential in momentum space

NASA Technical Reports Server (NTRS)

Norbury, John W.; Kahana, David E.; Maung, Khin Maung

1992-01-01

A method is presented for the solution in momentum space of the bound state problem with a linear potential in r space. The potential is unbounded at large r leading to a singularity at small q. The singularity is integrable, when regulated by exponentially screening the r-space potential, and is removed by a subtraction technique. The limit of zero screening is taken analytically, and the numerical solution of the subtracted integral equation gives eigenvalues and wave functions in good agreement with position space calculations.

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).

Eigenvalues of the Wentzell-Laplace operator and of the fourth order Steklov problems

NASA Astrophysics Data System (ADS)

Xia, Changyu; Wang, Qiaoling

2018-05-01

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell-Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.

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.

Direct Numerical Simulation of Transition in a Swept-Wing Boundary Layer

NASA Technical Reports Server (NTRS)

Duan, Lian; Choudhari, Meelan M.; Li, Fei

2013-01-01

Direct numerical simulation (DNS) is performed to examine laminar to turbulent transition due to high-frequency secondary instability of stationary crossflow vortices in a subsonic swept-wing boundary layer for a realistic natural-laminar-flow airfoil configuration. The secondary instability is introduced via inflow forcing derived from a two-dimensional, partial-differential-equation based eigenvalue computation; and the mode selected for forcing corresponds to the most amplified secondary instability mode which, in this case, derives a majority of its growth from energy production mechanisms associated with the wall-normal shear of the stationary basic state. Both the growth of the secondary instability wave and the resulting onset of laminar-turbulent transition are captured within the DNS computations. The growth of the secondary instability wave in the DNS solution compares well with linear secondary instability theory when the amplitude is small; the linear growth is followed by a region of reduced growth resulting from nonlinear effects before an explosive onset of laminar breakdown to turbulence. The peak fluctuations are concentrated near the boundary layer edge during the initial stage of transition, but rapidly propagates towards the surface during the process of laminar breakdown. Both time-averaged statistics and flow visualization based on the DNS reveal a sawtooth transition pattern that is analogous to previously documented surface flow visualizations of transition due to stationary crossflow instability. The memory of the stationary crossflow vortex is found to persist through the transition zone and well beyond the location of the maximum skin friction.

Beyond the spectral theorem: Spectrally decomposing arbitrary functions of nondiagonalizable operators

NASA Astrophysics Data System (ADS)

Riechers, Paul M.; Crutchfield, James P.

2018-06-01

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, the familiar linear operator techniques that one would then hope to use often fail since the operators cannot be diagonalized. The curse of nondiagonalizability also plays an important role even in finite-dimensional linear operators, leading to analytical impediments that occur across many scientific domains. We show how to circumvent it via two tracks. First, using the well-known holomorphic functional calculus, we develop new practical results about spectral projection operators and the relationship between left and right generalized eigenvectors. Second, we generalize the holomorphic calculus to a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. This simultaneously simplifies and generalizes functional calculus so that it is readily applicable to analyzing complex physical systems. Together, these results extend the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics arise, including memoryful stochastic processes, open nonunitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator, highlighting the special role of the zero eigenvalue. Furthermore, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a new general method to construct it. We provide new formulae for constructing spectral projection operators and delineate the relations among projection operators, eigenvectors, and left and right generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples. First, we analyze stochastic transition operators in discrete and continuous time. Second, we show that nondiagonalizability can be a robust feature of a stochastic process, induced even by simple counting. As a result, we directly derive distributions of the time-dependent Poisson process and point out that nondiagonalizability is intrinsic to it and the broad class of hidden semi-Markov processes. Third, we show that the Drazin inverse arises naturally in stochastic thermodynamics and that applying the meromorphic functional calculus provides closed-form solutions for the dynamics of key thermodynamic observables. Finally, we draw connections to the Ruelle-Frobenius-Perron and Koopman operators for chaotic dynamical systems and propose how to extract eigenvalues from a time-series.

Experimental Validation of Model Updating and Damage Detection via Eigenvalue Sensitivity Methods with Artificial Boundary Conditions

DTIC Science & Technology

2017-09-01

VALIDATION OF MODEL UPDATING AND DAMAGE DETECTION VIA EIGENVALUE SENSITIVITY METHODS WITH ARTIFICIAL BOUNDARY CONDITIONS by Matthew D. Bouwense...VALIDATION OF MODEL UPDATING AND DAMAGE DETECTION VIA EIGENVALUE SENSITIVITY METHODS WITH ARTIFICIAL BOUNDARY CONDITIONS 5. FUNDING NUMBERS 6. AUTHOR...unlimited. EXPERIMENTAL VALIDATION OF MODEL UPDATING AND DAMAGE DETECTION VIA EIGENVALUE SENSITIVITY METHODS WITH ARTIFICIAL BOUNDARY

Type I and Type II Error Rates and Overall Accuracy of the Revised Parallel Analysis Method for Determining the Number of Factors

ERIC Educational Resources Information Center

Green, Samuel B.; Thompson, Marilyn S.; Levy, Roy; Lo, Wen-Juo

2015-01-01

Traditional parallel analysis (T-PA) estimates the number of factors by sequentially comparing sample eigenvalues with eigenvalues for randomly generated data. Revised parallel analysis (R-PA) sequentially compares the "k"th eigenvalue for sample data to the "k"th eigenvalue for generated data sets, conditioned on"k"-…

Eigenvalue density of cross-correlations in Sri Lankan financial market

NASA Astrophysics Data System (ADS)

Nilantha, K. G. D. R.; Ranasinghe; Malmini, P. K. C.

2007-05-01

We apply the universal properties with Gaussian orthogonal ensemble (GOE) of random matrices namely spectral properties, distribution of eigenvalues, eigenvalue spacing predicted by random matrix theory (RMT) to compare cross-correlation matrix estimators from emerging market data. The daily stock prices of the Sri Lankan All share price index and Milanka price index from August 2004 to March 2005 were analyzed. Most eigenvalues in the spectrum of the cross-correlation matrix of stock price changes agree with the universal predictions of RMT. We find that the cross-correlation matrix satisfies the universal properties of the GOE of real symmetric random matrices. The eigen distribution follows the RMT predictions in the bulk but there are some deviations at the large eigenvalues. The nearest-neighbor spacing and the next nearest-neighbor spacing of the eigenvalues were examined and found that they follow the universality of GOE. RMT with deterministic correlations found that each eigenvalue from deterministic correlations is observed at values, which are repelled from the bulk distribution.

A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

NASA Astrophysics Data System (ADS)

Wang, Yan; Zhang, Yufeng; Zhang, Xiangzhi

2016-09-01

We first introduced a linear stationary equation with a quadratic operator in ∂x and ∂y, then a linear evolution equation is given by N-order polynomials of eigenfunctions. As applications, by taking N=2, we derived a (2+1)-dimensional generalized linear heat equation with two constant parameters associative with a symmetric space. When taking N=3, a pair of generalized Kadomtsev-Petviashvili equations with the same eigenvalues with the case of N=2 are generated. Similarly, a second-order flow associative with a homogeneous space is derived from the integrability condition of the two linear equations, which is a (2+1)-dimensional hyperbolic equation. When N=3, the third second flow associative with the homogeneous space is generated, which is a pair of new generalized Kadomtsev-Petviashvili equations. Finally, as an application of a Hermitian symmetric space, we established a pair of spectral problems to obtain a new (2+1)-dimensional generalized Schrödinger equation, which is expressed by the Riemann curvature tensors.

Dirac Theory on a Space with Linear Lie Type Fuzziness

NASA Astrophysics Data System (ADS)

Shariati, Ahmad; Khorrami, Mohammad; Fatollahi, Amir H.

2012-08-01

A spinor theory on a space with linear Lie type noncommutativity among spatial coordinates is presented. The model is based on the Fourier space corresponding to spatial coordinates, as this Fourier space is commutative. When the group is compact, the real space exhibits lattice characteristics (as the eigenvalues of space operators are discrete), and the similarity of such a lattice with ordinary lattices is manifested, among other things, in a phenomenon resembling the famous fermion doubling problem. A projection is introduced to make the dynamical number of spinors equal to that corresponding to the ordinary space. The actions for free and interacting spinors (with Fermi-like interactions) are presented. The Feynman rules are extracted and 1-loop corrections are investigated.

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.

Experimental and numerical investigation of development of disturbances in the boundary layer on sharp and blunted cone

NASA Astrophysics Data System (ADS)

Borisov, S. P.; Bountin, D. A.; Gromyko, Yu. V.; Khotyanovsky, D. V.; Kudryavtsev, A. N.

2016-10-01

Development of disturbances in the supersonic boundary layer on sharp and blunted cones is studied both experimentally and theoretically. The experiments were conducted at the Transit-M hypersonic wind tunnel of the Institute of Theoretical and Applied Mechanics. Linear stability calculations use the basic flow profiles provided by the numerical simulations performed by solving the Navier-Stokes equations with the ANSYS Fluent and the in-house CFS3D code. Both the global pseudospectral Chebyshev method and the local iteration procedure are employed to solve the eigenvalue problem and determine linear stability characteristics. The calculated amplification factors for disturbances of various frequencies are compared with the experimentally measured pressure fluctuation spectra at different streamwise positions. It is shown that the linear stability calculations predict quite accurately the frequency of the most amplified disturbances and enable us to estimate reasonably well their relative amplitudes.

Experimental and Numerical Study of the Buckling of Composite Profiles with Open Cross Section under Axial Compression

NASA Astrophysics Data System (ADS)

Rozylo, Patryk; Teter, Andrzej; Debski, Hubert; Wysmulski, Pawel; Falkowicz, Katarzyna

2017-10-01

The object of the research are short, thin-walled columns with an open top-hat cross section made of multilayer laminate. The walls of the investigated profiles are made of plate elements. The entire columns are subjected to uniform compression. A detailed analysis allowed us to determine critical forces and post-critical equilibrium paths. It is assumed that the columns are articulately supported on the edges forming their ends. The numerical investigation is performed by the finite element method. The study involves solving the problem of eigenvalue and the non-linear problem of stability of the structure. The numerical analysis is performed by the commercial simulation software ABAQUS®. The numerical results are then validated experimentally. In the discussed cases, it is assumed that the material operates within a linearly-elastic range, and the non-linearity of the FEM model is due to large displacements.

Robust consensus control with guaranteed rate of convergence using second-order Hurwitz polynomials

NASA Astrophysics Data System (ADS)

Fruhnert, Michael; Corless, Martin

2017-10-01

This paper considers homogeneous networks of general, linear time-invariant, second-order systems. We consider linear feedback controllers and require that the directed graph associated with the network contains a spanning tree and systems are stabilisable. We show that consensus with a guaranteed rate of convergence can always be achieved using linear state feedback. To achieve this, we provide a new and simple derivation of the conditions for a second-order polynomial 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.

Hierarchical Bayesian Markov switching models with application to predicting spawning success of shovelnose sturgeon

USGS Publications Warehouse

Holan, S.H.; Davis, G.M.; Wildhaber, M.L.; DeLonay, A.J.; Papoulias, D.M.

2009-01-01

The timing of spawning in fish is tightly linked to environmental factors; however, these factors are not very well understood for many species. Specifically, little information is available to guide recruitment efforts for endangered species such as the sturgeon. Therefore, we propose a Bayesian hierarchical model for predicting the success of spawning of the shovelnose sturgeon which uses both biological and behavioural (longitudinal) data. In particular, we use data that were produced from a tracking study that was conducted in the Lower Missouri River. The data that were produced from this study consist of biological variables associated with readiness to spawn along with longitudinal behavioural data collected by using telemetry and archival data storage tags. These high frequency data are complex both biologically and in the underlying behavioural process. To accommodate such complexity we developed a hierarchical linear regression model that uses an eigenvalue predictor, derived from the transition probability matrix of a two-state Markov switching model with generalized auto-regressive conditional heteroscedastic dynamics. Finally, to minimize the computational burden that is associated with estimation of this model, a parallel computing approach is proposed. ?? Journal compilation 2009 Royal Statistical Society.

How many atoms are required to characterize accurately trajectory fluctuations of a protein?

NASA Astrophysics Data System (ADS)

Cukier, Robert I.

2010-06-01

Large molecules, whose thermal fluctuations sample a complex energy landscape, exhibit motions on an extended range of space and time scales. Principal component analysis (PCA) is often used to extract dominant motions that in proteins are typically domain motions. These motions are captured in the large eigenvalue (leading) principal components. There is also information in the small eigenvalues, arising from approximate linear dependencies among the coordinates. These linear dependencies suggest that instead of using all the atom coordinates to represent a trajectory, it should be possible to use a reduced set of coordinates with little loss in the information captured by the large eigenvalue principal components. In this work, methods that can monitor the correlation (overlap) between a reduced set of atoms and any number of retained principal components are introduced. For application to trajectory data generated by simulations, where the overall translational and rotational motion needs to be eliminated before PCA is carried out, some difficulties with the overlap measures arise and methods are developed to overcome them. The overlap measures are evaluated for a trajectory generated by molecular dynamics for the protein adenylate kinase, which consists of a stable, core domain, and two more mobile domains, referred to as the LID domain and the AMP-binding domain. The use of reduced sets corresponding, for the smallest set, to one-eighth of the alpha carbon (CA) atoms relative to using all the CA atoms is shown to predict the dominant motions of adenylate kinase. The overlap between using all the CA atoms and all the backbone atoms is essentially unity for a sum over PCA modes that effectively capture the exact trajectory. A reduction to a few atoms (three in the LID and three in the AMP-binding domain) shows that at least the first principal component, characterizing a large part of the LID-binding and AMP-binding motion, is well described. Based on these results, the overlap criterion should be applicable as a guide to postulating and validating coarse-grained descriptions of generic biomolecular assemblies.

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

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.

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.

Weak value amplification considered harmful

NASA Astrophysics Data System (ADS)

Ferrie, Christopher; Combes, Joshua

2014-03-01

We show using statistically rigorous arguments that the technique of weak value amplification does not perform better than standard statistical techniques for the tasks of parameter estimation and signal detection. We show that using all data and considering the joint distribution of all measurement outcomes yields the optimal estimator. Moreover, we show estimation using the maximum likelihood technique with weak values as small as possible produces better performance for quantum metrology. In doing so, we identify the optimal experimental arrangement to be the one which reveals the maximal eigenvalue of the square of system observables. We also show these conclusions do not change in the presence of technical noise.

Noisy covariance matrices and portfolio optimization

NASA Astrophysics Data System (ADS)

Pafka, S.; Kondor, I.

2002-05-01

According to recent findings [#!bouchaud!#,#!stanley!#], empirical covariance matrices deduced from financial return series contain such a high amount of noise that, apart from a few large eigenvalues and the corresponding eigenvectors, their structure can essentially be regarded as random. In [#!bouchaud!#], e.g., it is reported that about 94% of the spectrum of these matrices can be fitted by that of a random matrix drawn from an appropriately chosen ensemble. In view of the fundamental role of covariance matrices in the theory of portfolio optimization as well as in industry-wide risk management practices, we analyze the possible implications of this effect. Simulation experiments with matrices having a structure such as described in [#!bouchaud!#,#!stanley!#] lead us to the conclusion that in the context of the classical portfolio problem (minimizing the portfolio variance under linear constraints) noise has relatively little effect. To leading order the solutions are determined by the stable, large eigenvalues, and the displacement of the solution (measured in variance) due to noise is rather small: depending on the size of the portfolio and on the length of the time series, it is of the order of 5 to 15%. The picture is completely different, however, if we attempt to minimize the variance under non-linear constraints, like those that arise e.g. in the problem of margin accounts or in international capital adequacy regulation. In these problems the presence of noise leads to a serious instability and a high degree of degeneracy of the solutions.

Global stability analysis of axisymmetric boundary layer over a circular cylinder

NASA Astrophysics Data System (ADS)

Bhoraniya, Ramesh; Vinod, Narayanan

2018-05-01

This paper presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at inflow boundary. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier-Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi's iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes is negative, and hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while they are moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat-plate boundary layer at low Reynolds number. However, at higher Reynolds number they approach 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.

A numerical projection technique for large-scale eigenvalue problems

NASA Astrophysics Data System (ADS)

Gamillscheg, Ralf; Haase, Gundolf; von der Linden, Wolfgang

2011-10-01

We present a new numerical technique to solve large-scale eigenvalue problems. It is based on the projection technique, used in strongly correlated quantum many-body systems, where first an effective approximate model of smaller complexity is constructed by projecting out high energy degrees of freedom and in turn solving the resulting model by some standard eigenvalue solver. Here we introduce a generalization of this idea, where both steps are performed numerically and which in contrast to the standard projection technique converges in principle to the exact eigenvalues. This approach is not just applicable to eigenvalue problems encountered in many-body systems but also in other areas of research that result in large-scale eigenvalue problems for matrices which have, roughly speaking, mostly a pronounced dominant diagonal part. We will present detailed studies of the approach guided by two many-body models.

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.

Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity

NASA Astrophysics Data System (ADS)

Gómez, D.; Nazarov, S. A.; Pérez, M. E.

2018-04-01

We consider a homogenization Winkler-Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler-Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order ɛ . For fixed ɛ , the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues {β _k^ɛ }_{k=1}^{∞} as ɛ → 0. We show that β _k^ɛ =O(ɛ ^{-1}) for each fixed k, and we observe a common limit point for all the rescaled eigenvalues ɛ β _k^ɛ while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain "groups" of eigenmodes.

The cosmological constant as an eigenvalue of a Sturm-Liouville problem

NASA Astrophysics Data System (ADS)

Astashenok, Artyom V.; Elizalde, Emilio; Yurov, Artyom V.

2014-01-01

It is observed that one of Einstein-Friedmann's equations has formally the aspect of a Sturm-Liouville problem, and that the cosmological constant, Λ, plays thereby the role of spectral parameter (what hints to its connection with the Casimir effect). The subsequent formulation of appropriate boundary conditions leads to a set of admissible values for Λ, considered as eigenvalues of the corresponding linear operator. Simplest boundary conditions are assumed, namely that the eigenfunctions belong to L 2 space, with the result that, when all energy conditions are satisfied, they yield a discrete spectrum for Λ>0 and a continuous one for Λ<0. A very interesting situation is seen to occur when the discrete spectrum contains only one point: then, there is the possibility to obtain appropriate cosmological conditions without invoking the anthropic principle. This possibility is shown to be realized in cyclic cosmological models, provided the potential of the matter field is similar to the potential of the scalar field. The dynamics of the universe in this case contains a sudden future singularity.

Cascade flutter analysis with transient response aerodynamics

NASA Technical Reports Server (NTRS)

Bakhle, Milind A.; Mahajan, Aparajit J.; Keith, Theo G., Jr.; Stefko, George L.

1991-01-01

Two methods for calculating linear frequency domain aerodynamic coefficients from a time marching Full Potential cascade solver are developed and verified. In the first method, the Influence Coefficient, solutions to elemental problems are superposed to obtain the solutions for a cascade in which all blades are vibrating with a constant interblade phase angle. The elemental problem consists of a single blade in the cascade oscillating while the other blades remain stationary. In the second method, the Pulse Response, the response to the transient motion of a blade is used to calculate influence coefficients. This is done by calculating the Fourier Transforms of the blade motion and the response. Both methods are validated by comparison with the Harmonic Oscillation method and give accurate results. The aerodynamic coefficients obtained from these methods are used for frequency domain flutter calculations involving a typical section blade structural model. An eigenvalue problem is solved for each interblade phase angle mode and the eigenvalues are used to determine aeroelastic stability. Flutter calculations are performed for two examples over a range of subsonic Mach numbers.

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

Eigenvalue sensitivity of sampled time systems operating in closed loop

NASA Astrophysics Data System (ADS)

Bernal, Dionisio

2018-05-01

The use of feedback to create closed-loop eigenstructures with high sensitivity has received some attention in the Structural Health Monitoring field. Although practical implementation is necessarily digital, and thus in sampled time, work thus far has center on the continuous time framework, both in design and in checking performance. It is shown in this paper that the performance in discrete time, at typical sampling rates, can differ notably from that anticipated in the continuous time formulation and that discrepancies can be particularly large on the real part of the eigenvalue sensitivities; a consequence being important error on the (linear estimate) of the level of damage at which closed-loop stability is lost. As one anticipates, explicit consideration of the sampling rate poses no special difficulties in the closed-loop eigenstructure design and the relevant expressions are developed in the paper, including a formula for the efficient evaluation of the derivative of the matrix exponential based on the theory of complex perturbations. The paper presents an easily reproduced numerical example showing the level of error that can result when the discrete time implementation of the controller is not considered.

INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY: Relaxation Property and Stability Analysis of the Quasispecies Models

NASA Astrophysics Data System (ADS)

Feng, Xiao-Li; Li, Yu-Xiao; Gu, Jian-Zhong; Zhuo, Yi-Zhong

2009-10-01

The relaxation property of both Eigen model and Crow-Kimura model with a single peak fitness landscape is studied from phase transition point of view. We first analyze the eigenvalue spectra of the replication mutation matrices. For sufficiently long sequences, the almost crossing point between the largest and second-largest eigenvalues locates the error threshold at which critical slowing down behavior appears. We calculate the critical exponent in the limit of infinite sequence lengths and compare it with the result from numerical curve fittings at sufficiently long sequences. We find that for both models the relaxation time diverges with exponent 1 at the error (mutation) threshold point. Results obtained from both methods agree quite well. From the unlimited correlation length feature, the first order phase transition is further confirmed. Finally with linear stability theory, we show that the two model systems are stable for all ranges of mutation rate. The Eigen model is asymptotically stable in terms of mutant classes, and the Crow-Kimura model is completely stable.

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.

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.

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.

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.

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

Linearized theory of inhomogeneous multiple 'water-bag' plasmas

NASA Technical Reports Server (NTRS)

Bloomberg, H. W.; Berk, H. L.

1973-01-01

Equations are derived for describing the inhomogeneous equilibrium and small deviations from the equilibrium, giving particular attention to systems with trapped particles. An investigation is conducted of periodic systems with a single trapped-particle water bag, taking into account the behavior of the perturbation equations at the turning points. An outline is provided concerning a procedure for obtaining the eigenvalues. The results of stability calculations connected with the sideband effects are considered along with questions regarding the general applicability of the multiple water-bag approach in stability calculations.

Study of electron-related intersubband optical properties in three coupled quantum wells wires with triangular transversal section

NASA Astrophysics Data System (ADS)

Tiutiunnyk, A.; Tulupenko, V.; Akimov, V.; Demediuk, R.; Morales, A. L.; Mora-Ramos, M. E.; Radu, A.; Duque, C. A.

2015-11-01

This work concerns theoretical study of confined electrons in a low-dimensional structure consisting of three coupled triangular GaAs/AlxGa1-xAs quantum wires. Calculations have been made in the effective mass and parabolic band approximations. In the calculations a diagonalization method to find the eigenfunctions and eigenvalues of the Hamiltonian was used. A comparative analysis of linear and nonlinear optical absorption coefficients and the relative change in the refractive index was made, which is tied to the intersubband electron transitions.

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.

Full characterization of modular values for finite-dimensional systems

NASA Astrophysics Data System (ADS)

Ho, Le Bin; Imoto, Nobuyuki

2016-06-01

Kedem and Vaidman obtained a relationship between the spin-operator modular value and its weak value for specific coupling strengths [14]. Here we give a general expression for the modular value in the n-dimensional Hilbert space using the weak values up to (n - 1)th order of an arbitrary observable for any coupling strength, assuming non-degenerated eigenvalues. For two-dimensional case, it shows a linear relationship between the weak value and the modular value. We also relate the modular value of the sum of observables to the weak value of their product.

Longitudinal Relaxation of Ferromagnetic Grains

NASA Astrophysics Data System (ADS)

Würger, Alois

1998-07-01

We study the activated longitudinal dynamics of a small single-domain magnet with uniaxial anisotropy, coupled to quantum noise. The smallest finite eigenvalue λ1 = γ0e-EB/kBT of the relaxation matrix is evaluated in a controlled approximation. For white noise we find γ0~T-1 at moderate temperatures and γ0 = const at very low T. Coupling to elastic waves leads to a prefactor that is linear in T or constant, depending on temperature. At very low T, the discreteness of the energy spectrum is crucial.

Multivariate Tensor-based Morphometry on Surfaces: Application to Mapping Ventricular Abnormalities in HIV/AIDS

PubMed Central

Wang, Yalin; Zhang, Jie; Gutman, Boris; Chan, Tony F.; Becker, James T.; Aizenstein, Howard J.; Lopez, Oscar L.; Tamburo, Robert J.; Toga, Arthur W.; Thompson, Paul M.

2010-01-01

Here we developed a new method, called multivariate tensor-based surface morphometry (TBM), and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using concepts from differential geometry and the theory of differential forms, we created mathematical structures known as holomorphic one-forms, to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also provides a natural way to register anatomical surfaces across subjects, and improves on prior methods as it handles surfaces that branch and join at complex 3D junctions. To analyze anatomical differences, we computed new statistics from the Riemannian surface metrics - these retain multivariate information on local surface geometry. We applied this framework to analyze lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8 healthy controls. Our method detected a 3D profile of surface abnormalities even in this small sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group differences, relative to other TBM-based methods including analysis of the Jacobian determinant, the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based morphometry studies of the brain. PMID:19900560

Conditions and Linear Stability Analysis at the Transition to Synchronization of Three Coupled Phase Oscillators in a Ring

NASA Astrophysics Data System (ADS)

El-Nashar, Hassan F.

2017-06-01

We consider a system of three nonidentical coupled phase oscillators in a ring topology. We explore the conditions that must be satisfied in order to obtain the phases at the transition to a synchrony state. These conditions lead to the correct mathematical expressions of phases that aid to find a simple analytic formula for critical coupling when the oscillators transit to a synchronization state having a common frequency value. The finding of a simple expression for the critical coupling allows us to perform a linear stability analysis at the transition to the synchronization stage. The obtained analytic forms of the eigenvalues show that the three coupled phase oscillators with periodic boundary conditions transit to a synchrony state when a saddle-node bifurcation occurs.

Stability analysis of the phytoplankton effect model on changes in nitrogen concentration on integrated multi-trophic aquaculture systems

NASA Astrophysics Data System (ADS)

Widowati; Putro, S. P.; Silfiana

2018-05-01

Integrated Multi-Trophic Aquaculture (IMTA) is a polyculture with several biotas maintained in it to optimize waste recycling as a food source. The interaction between phytoplankton and nitrogen as waste in fish cultivation including ammonia, nitrite, and nitrate studied in the form of mathematical models. The form model is non-linear systems of differential equations with the four variables. The analytical analysis was used to study the dynamic behavior of this model. Local stability analysis is performed at the equilibrium point with the first step linearized model by using Taylor series, then determined the Jacobian matrix. If all eigenvalues have negative real parts, then the equilibrium of the system is locally asymptotic stable. Some numerical simulations were also demonstrated to verify our analytical result.

A Linearized Prognostic Cloud Scheme in NASAs Goddard Earth Observing System Data Assimilation Tools

NASA Technical Reports Server (NTRS)

Holdaway, Daniel; Errico, Ronald M.; Gelaro, Ronald; Kim, Jong G.; Mahajan, Rahul

2015-01-01

A linearized prognostic cloud scheme has been developed to accompany the linearized convection scheme recently implemented in NASA's Goddard Earth Observing System data assimilation tools. The linearization, developed from the nonlinear cloud scheme, treats cloud variables prognostically so they are subject to linearized advection, diffusion, generation, and evaporation. Four linearized cloud variables are modeled, the ice and water phases of clouds generated by large-scale condensation and, separately, by detraining convection. For each species the scheme models their sources, sublimation, evaporation, and autoconversion. Large-scale, anvil and convective species of precipitation are modeled and evaporated. The cloud scheme exhibits linearity and realistic perturbation growth, except around the generation of clouds through large-scale condensation. Discontinuities and steep gradients are widely used here and severe problems occur in the calculation of cloud fraction. For data assimilation applications this poor behavior is controlled by replacing this part of the scheme with a perturbation model. For observation impacts, where efficiency is less of a concern, a filtering is developed that examines the Jacobian. The replacement scheme is only invoked if Jacobian elements or eigenvalues violate a series of tuned constants. The linearized prognostic cloud scheme is tested by comparing the linear and nonlinear perturbation trajectories for 6-, 12-, and 24-h forecast times. The tangent linear model performs well and perturbations of clouds are well captured for the lead times of interest.

S4 solution of the transport equation for eigenvalues using Legendre polynomials

NASA Astrophysics Data System (ADS)

Öztürk, Hakan; Bülbül, Ahmet

2017-09-01

Numerical solution of the transport equation for monoenergetic neutrons scattered isotropically through the medium of a finite homogeneous slab is studied for the determination of the eigenvalues. After obtaining the discrete ordinates form of the transport equation, separated homogeneous and particular solutions are formed and then the eigenvalues are calculated using the Gauss-Legendre quadrature set. Then, the calculated eigenvalues for various values of the c0, the mean number of secondary neutrons per collision, are given in the tables.

Bethe-Salpeter Eigenvalue Solver Package (BSEPACK) v0.1

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

SHAO, MEIYEU; YANG, CHAO

2017-04-25

The BSEPACK contains a set of subroutines for solving the Bethe-Salpeter Eigenvalue (BSE) problem. This type of problem arises in this study of optical excitation of nanoscale materials. The BSE problem is a structured non-Hermitian eigenvalue problem. The BSEPACK software can be used to compute all or subset of eigenpairs of a BSE Hamiltonian. It can also be used to compute the optical absorption spectrum without computing BSE eigenvalues and eigenvectors explicitly. The package makes use of the ScaLAPACK, LAPACK and BLAS.

Sturm-Liouville eigenproblems with an interior pole

NASA Technical Reports Server (NTRS)

Boyd, J. P.

1981-01-01

The eigenvalues and eigenfunctions of self-adjoint Sturm-Liouville problems with a simple pole on the interior of an interval are investigated. Three general theorems are proved, and it is shown that as n approaches infinity, the eigenfunctions more and more closely resemble those of an ordinary Sturm-Liouville problem. The low-order modes differ significantly from those of a nonsingular eigenproblem in that both eigenvalues and eigenfunctions are complex, and the eigenvalues for all small n may cluster about a common value in contrast to the widely separated eigenvalues of the corresponding nonsingular problem. In addition, the WKB is shown to be accurate for all n, and all eigenvalues of a normal one-dimensional Sturm-Liouville equation with nonperiodic boundary conditions are well separated.

Evaluation of the eigenvalue method in the solution of transient heat conduction problems

NASA Astrophysics Data System (ADS)

Landry, D. W.

1985-01-01

The eigenvalue method is evaluated to determine the advantages and disadvantages of the method as compared to fully explicit, fully implicit, and Crank-Nicolson methods. Time comparisons and accuracy comparisons are made in an effort to rank the eigenvalue method in relation to the comparison schemes. The eigenvalue method is used to solve the parabolic heat equation in multidimensions with transient temperatures. Extensions into three dimensions are made to determine the method's feasibility in handling large geometry problems requiring great numbers of internal mesh points. The eigenvalue method proves to be slightly better in accuracy than the comparison routines because of an exact treatment, as opposed to a numerical approximation, of the time derivative in the heat equation. It has the potential of being a very powerful routine in solving long transient type problems. The method is not well suited to finely meshed grid arrays or large regions because of the time and memory requirements necessary for calculating large sets of eigenvalues and eigenvectors.

A comparison of matrix methods for calculating eigenvalues in acoustically lined ducts

NASA Technical Reports Server (NTRS)

Watson, W.; Lansing, D. L.

1976-01-01

Three approximate methods - finite differences, weighted residuals, and finite elements - were used to solve the eigenvalue problem which arises in finding the acoustic modes and propagation constants in an absorptively lined two-dimensional duct without airflow. The matrix equations derived for each of these methods were solved for the eigenvalues corresponding to various values of wall impedance. Two matrix orders, 20 x 20 and 40 x 40, were used. The cases considered included values of wall admittance for which exact eigenvalues were known and for which several nearly equal roots were present. Ten of the lower order eigenvalues obtained from the three approximate methods were compared with solutions calculated from the exact characteristic equation in order to make an assessment of the relative accuracy and reliability of the three methods. The best results were given by the finite element method using a cubic polynomial. Excellent accuracy was consistently obtained, even for nearly equal eigenvalues, by using a 20 x 20 order matrix.

On the number of eigenvalues of the discrete one-dimensional Dirac operator with a complex potential

NASA Astrophysics Data System (ADS)

Hulko, Artem

2018-03-01

In this paper we define a one-dimensional discrete Dirac operator on Z . We study the eigenvalues of the Dirac operator with a complex potential. We obtain bounds on the total number of eigenvalues in the case where V decays exponentially at infinity. We also estimate the number of eigenvalues for the discrete Schrödinger operator with complex potential on Z . That is we extend the result obtained by Hulko (Bull Math Sci, to appear) to the whole Z.

SCALE Continuous-Energy Eigenvalue Sensitivity Coefficient Calculations

DOE PAGES

Perfetti, Christopher M.; Rearden, Bradley T.; Martin, William R.

2016-02-25

Sensitivity coefficients describe the fractional change in a system response that is induced by changes to system parameters and nuclear data. The Tools for Sensitivity and UNcertainty Analysis Methodology Implementation (TSUNAMI) code within the SCALE code system makes use of eigenvalue sensitivity coefficients for an extensive number of criticality safety applications, including quantifying the data-induced uncertainty in the eigenvalue of critical systems, assessing the neutronic similarity between different critical systems, and guiding nuclear data adjustment studies. The need to model geometrically complex systems with improved fidelity and the desire to extend TSUNAMI analysis to advanced applications has motivated the developmentmore » of a methodology for calculating sensitivity coefficients in continuous-energy (CE) Monte Carlo applications. The Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Tracklength importance CHaracterization (CLUTCH) and Iterated Fission Probability (IFP) eigenvalue sensitivity methods were recently implemented in the CE-KENO framework of the SCALE code system to enable TSUNAMI-3D to perform eigenvalue sensitivity calculations using continuous-energy Monte Carlo methods. This work provides a detailed description of the theory behind the CLUTCH method and describes in detail its implementation. This work explores the improvements in eigenvalue sensitivity coefficient accuracy that can be gained through the use of continuous-energy sensitivity methods and also compares several sensitivity methods in terms of computational efficiency and memory requirements.« less

A spectral analysis of the domain decomposed Monte Carlo method for linear systems

DOE PAGES

Slattery, Stuart R.; Evans, Thomas M.; Wilson, Paul P. H.

2015-09-08

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear oper- ator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approxi- mation and the mean chord approximation are applied to estimate the leakagemore » frac- tion of random walks from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem in numerical experiments to test the models for symmetric operators with spectral qualities similar to light water reactor problems. We find, in general, the derived approximations show good agreement with random walk lengths and leakage fractions computed by the numerical experiments.« less

Effect of size and indium-composition on linear and nonlinear optical absorption of InGaN/GaN lens-shaped quantum dot

NASA Astrophysics Data System (ADS)

Ahmed, S. Jbara; Zulkafli, Othaman; M, A. Saeed

2016-05-01

Based on the Schrödinger equation for envelope function in the effective mass approximation, linear and nonlinear optical absorption coefficients in a multi-subband lens quantum dot are investigated. The effects of quantum dot size on the interband and intraband transitions energy are also analyzed. The finite element method is used to calculate the eigenvalues and eigenfunctions. Strain and In-mole-fraction effects are also studied, and the results reveal that with the decrease of the In-mole fraction, the amplitudes of linear and nonlinear absorption coefficients increase. The present computed results show that the absorption coefficients of transitions between the first excited states are stronger than those of the ground states. In addition, it has been found that the quantum dot size affects the amplitudes and peak positions of linear and nonlinear absorption coefficients while the incident optical intensity strongly affects the nonlinear absorption coefficients. Project supported by the Ministry of Higher Education and Scientific Research in Iraq, Ibnu Sina Institute and Physics Department of Universiti Teknologi Malaysia (UTM RUG Vote No. 06-H14).

Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas

NASA Astrophysics Data System (ADS)

Schamel, Hans

2018-06-01

Two forms of the linearized eigenvalue problem with respect to linear perturbations of a privileged cnoidal electron hole as a structural nonlinear equilibrium element are established. Whereas its integral form involves integrations along the characteristics or unperturbed particle orbits, the differential form has to cope with a differential operator of infinite order. Both are hence faced with difficulties to obtain a solution. A first successful attempt is, however, made by addressing a single harmonic wave as a nonlinear equilibrium structure. By this microscopic nonlinear approach, its marginal stability against linear perturbations in both linear stability regimes, the sub- and super-critical one, is shown independent of the mobility of ions and in favor with recent observations. Responsible for vanishing damping (growth) is the microscopic distortion of the resonant distribution function. The macroscopic form of the trapping nonlinearity—the 3/2 power term of the electrostatic potential in the density—which disappears in the monochromatic harmonic wave limit is consequently necessary for the occurrence of a nonlinear plasma instability in the sub-critical regime.

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.

Irreversibility and entanglement spectrum statistics in quantum circuits

NASA Astrophysics Data System (ADS)

Shaffer, Daniel; Chamon, Claudio; Hamma, Alioscia; Mucciolo, Eduardo R.

2014-12-01

We show that in a quantum system evolving unitarily under a stochastic quantum circuit the notions of irreversibility, universality of computation, and entanglement are closely related. As the state evolves from an initial product state, it gets asymptotically maximally entangled. We define irreversibility as the failure of searching for a disentangling circuit using a Metropolis-like algorithm. We show that irreversibility corresponds to Wigner-Dyson statistics in the level spacing of the entanglement eigenvalues, and that this is obtained from a quantum circuit made from a set of universal gates for quantum computation. If, on the other hand, the system is evolved with a non-universal set of gates, the statistics of the entanglement level spacing deviates from Wigner-Dyson and the disentangling algorithm succeeds. These results open a new way to characterize irreversibility in quantum systems.

Towards random matrix model of breaking the time-reversal invariance of elastic waves in chaotic cavities by feedback

NASA Astrophysics Data System (ADS)

Antoniuk, Oleg; Sprik, Rudolf

2010-03-01

We developed a random matrix model to describe the statistics of resonances in an acoustic cavity with broken time-reversal invariance. Time-reversal invariance braking is achieved by connecting an amplified feedback loop between two transducers on the surface of the cavity. The model is based on approach [1] that describes time- reversal properties of the cavity without a feedback loop. Statistics of eigenvalues (nearest neighbor resonance spacing distributions and spectral rigidity) has been calculated and compared to the statistics obtained from our experimental data. Experiments have been performed on aluminum block of chaotic shape confining ultrasound waves. [1] Carsten Draeger and Mathias Fink, One-channel time- reversal in chaotic cavities: Theoretical limits, Journal of Acoustical Society of America, vol. 105, Nr. 2, pp. 611-617 (1999)

A time-domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems

NASA Astrophysics Data System (ADS)

Heinkenschloss, Matthias

2005-01-01

We study a class of time-domain decomposition-based methods for the numerical solution of large-scale linear quadratic optimal control problems. Our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete-time optimal control (DTOC) problem. The optimality conditions for this DTOC problem lead to a linear block tridiagonal system. The diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time-subintervals. This motivates the application of block Gauss-Seidel (GS)-type methods for the solution of the block tridiagonal systems. Numerical experiments show that the spectral radii of the block GS iteration matrices are larger than one for typical applications, but that the eigenvalues of the iteration matrices decay to zero fast. Hence, while the GS method is not expected to convergence for typical applications, it can be effective as a preconditioner for Krylov-subspace methods. This is confirmed by our numerical tests.A byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block GS method applied to the DTOC optimality system.

Accurate Valence Ionization Energies from Kohn-Sham Eigenvalues with the Help of Potential Adjustors.

PubMed

Thierbach, Adrian; Neiss, Christian; Gallandi, Lukas; Marom, Noa; Körzdörfer, Thomas; Görling, Andreas

2017-10-10

An accurate yet computationally very efficient and formally well justified approach to calculate molecular ionization potentials is presented and tested. The first as well as higher ionization potentials are obtained as the negatives of the Kohn-Sham eigenvalues of the neutral molecule after adjusting the eigenvalues by a recently [ Görling Phys. Rev. B 2015 , 91 , 245120 ] introduced potential adjustor for exchange-correlation potentials. Technically the method is very simple. Besides a Kohn-Sham calculation of the neutral molecule, only a second Kohn-Sham calculation of the cation is required. The eigenvalue spectrum of the neutral molecule is shifted such that the negative of the eigenvalue of the highest occupied molecular orbital equals the energy difference of the total electronic energies of the cation minus the neutral molecule. For the first ionization potential this simply amounts to a ΔSCF calculation. Then, the higher ionization potentials are obtained as the negatives of the correspondingly shifted Kohn-Sham eigenvalues. Importantly, this shift of the Kohn-Sham eigenvalue spectrum is not just ad hoc. In fact, it is formally necessary for the physically correct energetic adjustment of the eigenvalue spectrum as it results from ensemble density-functional theory. An analogous approach for electron affinities is equally well obtained and justified. To illustrate the practical benefits of the approach, we calculate the valence ionization energies of test sets of small- and medium-sized molecules and photoelectron spectra of medium-sized electron acceptor molecules using a typical semilocal (PBE) and two typical global hybrid functionals (B3LYP and PBE0). The potential adjusted B3LYP and PBE0 eigenvalues yield valence ionization potentials that are in very good agreement with experimental values, reaching an accuracy that is as good as the best G 0 W 0 methods, however, at much lower computational costs. The potential adjusted PBE eigenvalues result in somewhat less accurate ionization energies, which, however, are almost as accurate as those obtained from the most commonly used G 0 W 0 variants.

Cucheb: A GPU implementation of the filtered Lanczos procedure

NASA Astrophysics Data System (ADS)

Aurentz, Jared L.; Kalantzis, Vassilis; Saad, Yousef

2017-11-01

This paper describes the software package Cucheb, a GPU implementation of the filtered Lanczos procedure for the solution of large sparse symmetric eigenvalue problems. The filtered Lanczos procedure uses a carefully chosen polynomial spectral transformation to accelerate convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particularly effective for eigenvalue problems that arise in electronic structure calculations and density functional theory. We compare our implementation against an equivalent CPU implementation and show that using the GPU can reduce the computation time by more than a factor of 10. Program Summary Program title: Cucheb Program Files doi:http://dx.doi.org/10.17632/rjr9tzchmh.1 Licensing provisions: MIT Programming language: CUDA C/C++ Nature of problem: Electronic structure calculations require the computation of all eigenvalue-eigenvector pairs of a symmetric matrix that lie inside a user-defined real interval. Solution method: To compute all the eigenvalues within a given interval a polynomial spectral transformation is constructed that maps the desired eigenvalues of the original matrix to the exterior of the spectrum of the transformed matrix. The Lanczos method is then used to compute the desired eigenvectors of the transformed matrix, which are then used to recover the desired eigenvalues of the original matrix. The bulk of the operations are executed in parallel using a graphics processing unit (GPU). Runtime: Variable, depending on the number of eigenvalues sought and the size and sparsity of the matrix. Additional comments: Cucheb is compatible with CUDA Toolkit v7.0 or greater.

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

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

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2018-01-01

A new semi-analytical approach is presented 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 direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K-L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in 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 such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K-L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.

Design, Optimization and Evaluation of Integrally Stiffened Al 7050 Panel with Curved Stiffeners

NASA Technical Reports Server (NTRS)

Slemp, Wesley C. H.; Bird, R. Keith; Kapania, Rakesh K.; Havens, David; Norris, Ashley; Olliffe, Robert

2011-01-01

A curvilinear stiffened panel was designed, manufactured, and tested in the Combined Load Test Fixture at NASA Langley Research Center. The panel was optimized for minimum mass subjected to constraints on buckling load, yielding, and crippling or local stiffener failure using a new analysis tool named EBF3PanelOpt. The panel was designed for a combined compression-shear loading configuration that is a realistic load case for a typical aircraft wing panel. The panel was loaded beyond buckling and strains and out-of-plane displacements were measured. The experimental data were compared with the strains and out-of-plane deflections from a high fidelity nonlinear finite element analysis and linear elastic finite element analysis of the panel/test-fixture assembly. The numerical results indicated that the panel buckled at the linearly elastic buckling eigenvalue predicted for the panel/test-fixture assembly. The experimental strains prior to buckling compared well with both the linear and nonlinear finite element model.

Buckling and limit states of composite profiles with top-hat channel section subjected to axial compression

NASA Astrophysics Data System (ADS)

RóŻyło, Patryk; Debski, Hubert; Kral, Jan

2018-01-01

The subject of the research was a short thin-walled top-hat cross-section composite profile. The tested structure was subjected to axial compression. As part of the critical state research, critical load and the corresponding buckling mode was determined. Later in the study laminate damage areas were determined throughout numerical analysis. It was assumed that the profile is simply supported on the cross sections ends. Experimental tests were carried out on a universal testing machine Zwick Z100 and the results were compared with the results of numerical calculations. The eigenvalue problem and a non-linear problem of stability of thin-walled structures were carried out by the use of commercial software ABAQUS®. In the presented cases, it was assumed that the material is linear-elastic and non-linearity of the model results from the large displacements. Solution to the geometrically nonlinear problem was conducted by the use of the incremental-iterative Newton-Raphson method.

A study of the control problem of the shoot side environment delivery system of a closed crop growth research chamber

NASA Technical Reports Server (NTRS)

Blackwell, C. C.; Blackwell, A. L.

1992-01-01

The details of our initial study of the control problem of the crop shoot environment of a hypothetical closed crop growth research chamber (CGRC) are presented in this report. The configuration of the CGRC is hypothetical because neither a physical subject nor a design existed at the time the study began, a circumstance which is typical of large scale systems control studies. The basis of the control study is a mathematical model which was judged to adequately mimic the relevant dynamics of the system components considered necessary to provide acceptable realism in the representation. Control of pressure, temperature, and flow rate of the crop shoot environment, along with its oxygen, carbon dioxide, and water concentration is addressed. To account for mass exchange, the group of plants is represented in the model by a source of oxygen, a source of water vapor, and a sink for carbon dioxide. In terms of the thermal energy exchange, the group of plants is represented by a surface with an appropriate temperature. Most of the primitive equations about an experimental operating condition and a state variable representation which was extracted from the linearized equations are presented. Next, we present the results of a real Jordan decomposition and the repositioning of an undesirable eigenvalue via full state feedback. The state variable representation of the modeling system is of the nineteenth order and reflects the eleven control variables and eight system disturbances. Five real eigenvalues are very near zero, with one at zero, three having small magnitude positive values, and one having a small magnitude negative value. A Singular Value Decomposition analysis indicates that these non-zero eigenvalues are not results of numerical error.

An electron of helium atom under a high-intensity laser field

NASA Astrophysics Data System (ADS)

Falaye, Babatunde James; Sun, Guo-Hua; Adepoju, Adenike Grace; Liman, Muhammed S.; Oyewumi, K. J.; Dong, Shi-Hai

2017-02-01

We scrutinize the behavior of eigenvalues of an electron in a helium (He) atom as it interacts with electric field directed along the z-axis and is exposed to linearly polarized intense laser field radiation. To achieve this, we freeze one electron of the He atom at its ionic ground state and the motion of the second electron in the ion core is treated via a more general case of screened Coulomb potential model. Using the Kramers-Henneberger (KH) unitary transformation, which is the semiclassical counterpart of the Block-Nordsieck transformation in the quantized field formalism, the squared vector potential that appears in the equation of motion is eliminated and the resultant equation is expressed in the KH frame. Within this frame, the resulting potential and the corresponding wave function are expanded in Fourier series and using Ehlotzky’s approximation, we obtain a laser-dressed potential to simulate intense laser field. By fitting the more general case of screened Coulomb potential model into the laser-dressed potential, and then expanding it in Taylor series up to O≤ft({{r}4},α 09\\right) , we obtain the solution (eigenvalues and wave function) of an electron in a He atom under the influence of external electric field and high-intensity laser field, within the framework of perturbation theory formalism. We found that the variation in frequency of laser radiation has no effect on the eigenvalues of a He electron for a particular electric field intensity directed along z-axis. Also, for a very strong external electric field and an infinitesimal screening parameter, the system is strongly bound. This work has potential application in the areas of atomic and molecular processes in external fields including interactions with strong fields and short pulses.

Stability investigations of airfoil flow by global analysis

NASA Technical Reports Server (NTRS)

Morzynski, Marek; Thiele, Frank

1992-01-01

As the result of global, non-parallel flow stability analysis the single value of the disturbance growth-rate and respective frequency is obtained. This complex value characterizes the stability of the whole flow configuration and is not referred to any particular flow pattern. The global analysis assures that all the flow elements (wake, boundary and shear layer) are taken into account. The physical phenomena connected with the wake instability are properly reproduced by the global analysis. This enhances the investigations of instability of any 2-D flows, including ones in which the boundary layer instability effects are known to be of dominating importance. Assuming fully 2-D disturbance form, the global linear stability problem is formulated. The system of partial differential equations is solved for the eigenvalues and eigenvectors. The equations, written in the pure stream function formulation, are discretized via FDM using a curvilinear coordinate system. The complex eigenvalues and corresponding eigenvectors are evaluated by an iterative method. The investigations performed for various Reynolds numbers emphasize that the wake instability develops into the Karman vortex street. This phenomenon is shown to be connected with the first mode obtained from the non-parallel flow stability analysis. The higher modes are reflecting different physical phenomena as for example Tollmien-Schlichting waves, originating in the boundary layer and having the tendency to emerge as instabilities for the growing Reynolds number. The investigations are carried out for a circular cylinder, oblong ellipsis and airfoil. It is shown that the onset of the wake instability, the waves in the boundary layer, the shear layer instability are different solutions of the same eigenvalue problem, formulated using the non-parallel theory. The analysis offers large potential possibilities as the generalization of methods used till now for the stability analysis.

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.

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.

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.

Hybrid preconditioning for iterative diagonalization of ill-conditioned generalized eigenvalue problems in electronic structure calculations

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

Cai, Yunfeng, E-mail: yfcai@math.pku.edu.cn; Department of Computer Science, University of California, Davis 95616; Bai, Zhaojun, E-mail: bai@cs.ucdavis.edu

2013-12-15

The iterative diagonalization of a sequence of large ill-conditioned generalized eigenvalue problems is a computational bottleneck in quantum mechanical methods employing a nonorthogonal basis for ab initio electronic structure calculations. We propose a hybrid preconditioning scheme to effectively combine global and locally accelerated preconditioners for rapid iterative diagonalization of such eigenvalue problems. In partition-of-unity finite-element (PUFE) pseudopotential density-functional calculations, employing a nonorthogonal basis, we show that the hybrid preconditioned block steepest descent method is a cost-effective eigensolver, outperforming current state-of-the-art global preconditioning schemes, and comparably efficient for the ill-conditioned generalized eigenvalue problems produced by PUFE as the locally optimal blockmore » preconditioned conjugate-gradient method for the well-conditioned standard eigenvalue problems produced by planewave methods.« less

Analyzing Aeroelastic Stability of a Tilt-Rotor Aircraft

NASA Technical Reports Server (NTRS)

Kvaternil, Raymond G.

2006-01-01

Proprotor Aeroelastic Stability Analysis, now at version 4.5 (PASTA 4.5), is a FORTRAN computer program for analyzing the aeroelastic stability of a tiltrotor aircraft in the airplane mode of flight. The program employs a 10-degree- of-freedom (DOF), discrete-coordinate, linear mathematical model of a rotor with three or more blades and its drive system coupled to a 10-DOF modal model of an airframe. The user can select which DOFs are included in the analysis. Quasi-steady strip-theory aerodynamics is employed for the aerodynamic loads on the blades, a quasi-steady representation is employed for the aerodynamic loads acting on the vibrational modes of the airframe, and a stability-derivative approach is used for the aerodynamics associated with the rigid-body DOFs of the airframe. Blade parameters that vary with the blade collective pitch can be obtained by interpolation from a user-defined table. Stability is determined by examining the eigenvalues that are obtained by solving the coupled equations of motions as a matrix eigenvalue problem. Notwithstanding the relative simplicity of its mathematical foundation, PASTA 4.5 and its predecessors have played key roles in a number of engineering investigations over the years.

Oscilaciones estelares no-radiales: aplicación a configuraciones politrópicas y modelos de enanas blancas de He

NASA Astrophysics Data System (ADS)

Córsico, A. H.; Benvenuto, O. G.

Recently in our Observatory we have developed a new Stellar Pulsation Code, independently of other workers. Such program computes eigenvalues (eigenfrequencies) and eigenfunctions of non-radial modes in spherical non-perturbated stellar models. To accomplish this calculations, the four order eigenvalue problem (in the linear adiabatic approach) is solved by means of the well-know technique of Henyey on the finite differences scheme wich replace to the differential equations of the problem. In order to test the Code, we have computed numerous eigenmodes in polytropic configurations for several values of index n. In this comunication we show the excelent agreement of our results and that best available in the literature. Also, we present results of oscillations in models of white dwarf stars with homogeneus chemical composition (pure Helium). This models have been obtained with the Evolution Stellar Code of our Observatory. The calculations outlined above conform a first preliminary step in a major proyect whose main purpose is the study of pulsational properties of DA, DB and DO white dwarfs stars. Detailed investigations have demonstrated that such objets pulsates in non-radial g-modes with eigenperiods in the range 100-2000 sec.

On the behavior of the leading eigenvalue of Eigen's evolutionary matrices.

PubMed

Semenov, Yuri S; Bratus, Alexander S; Novozhilov, Artem S

2014-12-01

We study general properties of the leading eigenvalue w¯(q) of Eigen's evolutionary matrices depending on the replication fidelity q. This is a linear algebra problem that has various applications in theoretical biology, including such diverse fields as the origin of life, evolution of cancer progression, and virus evolution. We present the exact expressions for w¯(q),w¯(')(q),w¯('')(q) for q = 0, 0.5, 1 and prove that the absolute minimum of w¯(q), which always exists, belongs to the interval (0, 0.5]. For the specific case of a single peaked landscape we also find lower and upper bounds on w¯(q), which are used to estimate the critical mutation rate, after which the distribution of the types of individuals in the population becomes almost uniform. This estimate is used as a starting point to conjecture another estimate, valid for any fitness landscape, and which is checked by numerical calculations. The last estimate stresses the fact that the inverse dependence of the critical mutation rate on the sequence length is not a generally valid fact. Copyright © 2014 Elsevier Inc. All rights reserved.

Towards spectral geometric methods for Euclidean quantum gravity

NASA Astrophysics Data System (ADS)

Panine, Mikhail; Kempf, Achim

2016-04-01

The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis, respectively. Of particular interest in this regard is the field of spectral geometry, which studies to which extent the shape of a Riemannian manifold is describable in terms of the spectra of differential operators defined on the manifold. Spectral geometry is hard because it is highly nonlinear, but linearized spectral geometry, i.e., the task to determine small shape changes from small spectral changes, is much more tractable and may be iterated to approximate the full problem. Here, we generalize this approach, allowing, in particular, nonequal finite numbers of shape and spectral degrees of freedom. This allows us to study how well the shape degrees of freedom are encoded in the eigenvalues. We apply this strategy numerically to a class of planar domains and find that the reconstruction of small shape changes from small spectral changes is possible if enough eigenvalues are used. While isospectral nonisometric shapes are known to exist, we find evidence that generically shaped isospectral nonisometric shapes, if existing, are exceedingly rare.

Estimation of genetic parameters related to eggshell strength using random regression models.

PubMed

Guo, J; Ma, M; Qu, L; Shen, M; Dou, T; Wang, K

2015-01-01

This study examined the changes in eggshell strength and the genetic parameters related to this trait throughout a hen's laying life using random regression. The data were collected from a crossbred population between 2011 and 2014, where the eggshell strength was determined repeatedly for 2260 hens. Using random regression models (RRMs), several Legendre polynomials were employed to estimate the fixed, direct genetic and permanent environment effects. The residual effects were treated as independently distributed with heterogeneous variance for each test week. The direct genetic variance was included with second-order Legendre polynomials and the permanent environment with third-order Legendre polynomials. The heritability of eggshell strength ranged from 0.26 to 0.43, the repeatability ranged between 0.47 and 0.69, and the estimated genetic correlations between test weeks was high at > 0.67. The first eigenvalue of the genetic covariance matrix accounted for about 97% of the sum of all the eigenvalues. The flexibility and statistical power of RRM suggest that this model could be an effective method to improve eggshell quality and to reduce losses due to cracked eggs in a breeding plan.

The Effect of Stochastic Perturbation of Fuel Distribution on the Criticality of a One Speed Reactor and the Development of Multi-Material Multinomial Line Statistics

NASA Technical Reports Server (NTRS)

Jahshan, S. N.; Singleterry, R. C.

2001-01-01

The effect of random fuel redistribution on the eigenvalue of a one-speed reactor is investigated. An ensemble of such reactors that are identical to a homogeneous reference critical reactor except for the fissile isotope density distribution is constructed such that it meets a set of well-posed redistribution requirements. The average eigenvalue, , is evaluated when the total fissile loading per ensemble element, or realization, is conserved. The perturbation is proven to increase the reactor criticality on average when it is uniformly distributed. The various causes of the change in reactivity, and their relative effects are identified and ranked. From this, a path towards identifying the causes. and relative effects of reactivity fluctuations for the energy dependent problem is pointed to. The perturbation method of using multinomial distributions for representing the perturbed reactor is developed. This method has some advantages that can be of use in other stochastic problems. Finally, some of the features of this perturbation problem are related to other techniques that have been used for addressing similar problems.

Optimal linear-quadratic control of coupled parabolic-hyperbolic PDEs

NASA Astrophysics Data System (ADS)

Aksikas, I.; Moghadam, A. Alizadeh; Forbes, J. F.

2017-10-01

This paper focuses on the optimal control design for a system of coupled parabolic-hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear-quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

Effect of curvature on stationary crossflow instability of a three-dimensional boundary layer

NASA Technical Reports Server (NTRS)

Lin, Ray-Sing; Reed, Helen L.

1993-01-01

An incompressible three-dimensional laminar boundary-layer flow over a swept wing is used as a model to study both the wall-curvature and streamline-curvature effects on the stationary crossflow instability. The basic state is obtained by solving the full Navier-Stokes (N-S) equations numerically. The linear disturbance equations are cast on a fixed, body-intrinsic, curvilinear coordinate system. Those nonparallel terms which contribute mainly to the streamline-curvature effect are retained in the formulation of the disturbance equations and approximated by their local finite difference values. The resulting eigenvalue problem is solved by a Chebyshev collocation method. The present results indicate that the convex wall curvature has a stabilizing effect, whereas the streamline curvature has a destabilizing effect. A validation of these effects with an N-S solution for the linear disturbance flow is provided.

Using parallel banded linear system solvers in generalized eigenvalue problems

NASA Technical Reports Server (NTRS)

Zhang, Hong; Moss, William F.

1993-01-01

Subspace iteration is a reliable and cost effective method for solving positive definite banded symmetric generalized eigenproblems, especially in the case of large scale problems. This paper discusses an algorithm that makes use of two parallel banded solvers in subspace iteration. A shift is introduced to decompose the banded linear systems into relatively independent subsystems and to accelerate the iterations. With this shift, an eigenproblem is mapped efficiently into the memories of a multiprocessor and a high speed-up is obtained for parallel implementations. An optimal shift is a shift that balances total computation and communication costs. Under certain conditions, we show how to estimate an optimal shift analytically using the decay rate for the inverse of a banded matrix, and how to improve this estimate. Computational results on iPSC/2 and iPSC/860 multiprocessors are presented.

The Cr dependence problem of eigenvalues of the Laplace operator on domains in the plane

NASA Astrophysics Data System (ADS)

Haddad, Julian; Montenegro, Marcos

2018-03-01

The Cr dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding C r + 1-smooth one-parameter families of C1 perturbations of domains in Rn. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares in R2 and also for its second eigenvalue on balls in Rn for any n ≥ 3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces (Theorem 2) of independent interest.

Computing interior eigenvalues of nonsymmetric matrices: application to three-dimensional metamaterial composites.

PubMed

Terao, Takamichi

2010-08-01

We propose a numerical method to calculate interior eigenvalues and corresponding eigenvectors for nonsymmetric matrices. Based on the subspace projection technique onto expanded Ritz subspace, it becomes possible to obtain eigenvalues and eigenvectors with sufficiently high precision. This method overcomes the difficulties of the traditional nonsymmetric Lanczos algorithm, and improves the accuracy of the obtained interior eigenvalues and eigenvectors. Using this algorithm, we investigate three-dimensional metamaterial composites consisting of positive and negative refractive index materials, and it is demonstrated that the finite-difference frequency-domain algorithm is applicable to analyze these metamaterial composites.

The first eigenvalue of the p-Laplacian on quantum graphs

NASA Astrophysics Data System (ADS)

Del Pezzo, Leandro M.; Rossi, Julio D.

2016-12-01

We study the first eigenvalue of the p-Laplacian (with 1

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

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.

Viscoelastic stability in a single-screw channel flow

NASA Astrophysics Data System (ADS)

Agbessi, Y.; Bu, L. X.; Béreaux, Y.; Charmeau, J.-Y.

2018-05-01

In this work, we perform a linear stability analysis on pressure and drag flows of an Upper Convected Maxwell viscoelastic fluid. We use the well-recognised method of expanding the disturbances in Chebyschev polynomials and solve the resulting generalized eigenvalues problem with a collocation spectra method. Both the level of elasticity and the back-pressure vary. In a second stage, recent analytic solutions of viscoelastic fluid flows in slowly varying sections [1] are used to extend this stability analysis to flows in a compression or in a diverging section of a single screw channel, for example a wave mixing screw.

Stability analysis of ultrasound thick-shell contrast agents

PubMed Central

Lu, Xiaozhen; Chahine, Georges L.; Hsiao, Chao-Tsung

2012-01-01

The stability of thick shell encapsulated bubbles is studied analytically. 3-D small perturbations are introduced to the spherical oscillations of a contrast agent bubble in response to a sinusoidal acoustic field with different amplitudes of excitation. The equations of the perturbation amplitudes are derived using asymptotic expansions and linear stability analysis is then applied to the resulting differential equations. The stability of the encapsulated microbubbles to nonspherical small perturbations is examined by solving an eigenvalue problem. The approach then identifies the fastest growing perturbations which could lead to the breakup of the encapsulated microbubble or contrast agent. PMID:22280568

Identification of the structure parameters using short-time non-stationary stochastic excitation

NASA Astrophysics Data System (ADS)

Jarczewska, Kamila; Koszela, Piotr; Śniady, PaweŁ; Korzec, Aleksandra

2011-07-01

In this paper, we propose an approach to the flexural stiffness or eigenvalue frequency identification of a linear structure using a non-stationary stochastic excitation process. The idea of the proposed approach lies within time domain input-output methods. The proposed method is based on transforming the dynamical problem into a static one by integrating the input and the output signals. The output signal is the structure reaction, i.e. structure displacements due to the short-time, irregular load of random type. The systems with single and multiple degrees of freedom, as well as continuous systems are considered.

Aircraft model prototypes which have specified handling-quality time histories

NASA Technical Reports Server (NTRS)

Johnson, S. H.

1978-01-01

Several techniques for obtaining linear constant-coefficient airplane models from specified handling-quality time histories are discussed. The pseudodata method solves the basic problem, yields specified eigenvalues, and accommodates state-variable transfer-function zero suppression. The algebraic equations to be solved are bilinear, at worst. The disadvantages are reduced generality and no assurance that the resulting model will be airplane like in detail. The method is fully illustrated for a fourth-order stability-axis small motion model with three lateral handling quality time histories specified. The FORTRAN program which obtains and verifies the model is included and fully documented.

Pythagoras's theorem on a two-dimensional lattice from a `natural' Dirac operator and Connes's distance formula

NASA Astrophysics Data System (ADS)

Dai, Jian; Song, Xing-Chang

2001-07-01

One of the key ingredients of Connes's noncommutative geometry is a generalized Dirac operator which induces a metric (Connes's distance) on the pure state space. We generalize such a Dirac operator devised by Dimakis et al, whose Connes distance recovers the linear distance on an one-dimensional lattice, to the two-dimensional case. This Dirac operator has the local eigenvalue property and induces a Euclidean distance on this two-dimensional lattice, which is referred to as `natural'. This kind of Dirac operator can be easily generalized into any higher-dimensional lattices.

Wentzel-Kramers-Brillouin method in the Bargmann representation. [of quantum mechanics

NASA Technical Reports Server (NTRS)

Voros, A.

1989-01-01

It is demonstrated that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions.

Transactions of the Conference of Army Mathematicians (28th) Held at Bethesda, Maryland on 28-30 June 1982.

DTIC Science & Technology

1983-02-01

real part is the Hilbert transform of its imaginary part. Thus we have o) d4’ . (5.1)+," _ Here r(o) and 6(o) denote respectively T(4’,0-) and 0(o,0...linear operators A in a Hilbert space H, eigenuv.aueA are critical values of the Raqt.igh quo 5ient (5.1) R(y) = (Ay,y)/(y,y), y # 0. An eigenvalue X...Das Gupta Ballistic Research Laboratory Jim Greenberg National Science Foundation Charles Giardina Fairleigh-Dickinson University Frank P. Kuhl U. S

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.

Combustion-acoustic stability analysis for premixed gas turbine combustors

NASA Technical Reports Server (NTRS)

Darling, Douglas; Radhakrishnan, Krishnan; Oyediran, Ayo; Cowan, Lizabeth

1995-01-01

Lean, prevaporized, premixed combustors are susceptible to combustion-acoustic instabilities. A model was developed to predict eigenvalues of axial modes for combustion-acoustic interactions in a premixed combustor. This work extends previous work by including variable area and detailed chemical kinetics mechanisms, using the code LSENS. Thus the acoustic equations could be integrated through the flame zone. Linear perturbations were made of the continuity, momentum, energy, chemical species, and state equations. The qualitative accuracy of our approach was checked by examining its predictions for various unsteady heat release rate models. Perturbations in fuel flow rate are currently being added to the model.

Computing Evans functions numerically via boundary-value problems

NASA Astrophysics Data System (ADS)

Barker, Blake; Nguyen, Rose; Sandstede, Björn; Ventura, Nathaniel; Wahl, Colin

2018-03-01

The Evans function has been used extensively to study spectral stability of travelling-wave solutions in spatially extended partial differential equations. To compute Evans functions numerically, several shooting methods have been developed. In this paper, an alternative scheme for the numerical computation of Evans functions is presented that relies on an appropriate boundary-value problem formulation. Convergence of the algorithm is proved, and several examples, including the computation of eigenvalues for a multi-dimensional problem, are given. The main advantage of the scheme proposed here compared with earlier methods is that the scheme is linear and scalable to large problems.

The Use of Sphere Indentation Experiments to Characterize Ceramic Damage Models

DTIC Science & Technology

2011-09-01

state having two equal eigenvalues. For TXC, the axial stress (single eigenvalue) is more compressive than the lateral stresses (dual eigenvalues). For...parameters. These dynamic experiments supplement traditional characterization experiments such as tension, triaxial compression , Brazilian, and...These dynamic experiments supplement traditional characterization experiments such as tension, triaxial compression , Brazilian, and plate impact, which

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.

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.

The construction of partner potential from the general potential anharmonic in D-dimensional Schrodinger system

NASA Astrophysics Data System (ADS)

Suparmi; Cari, C.; Wea, K. N.; Wahyulianti

2018-03-01

The Schrodinger equation is the fundamental equation in quantum physics. The characteristic of the particle in physics potential field can be explained by using the Schrodinger equation. In this study, the solution of 4 dimensional Schrodinger equation for the anharmonic potential and the anharmonic partner potential have done. The method that used to solve the Schrodinger equation was the ansatz wave method, while to construction the partner potential was the supersymmetric method. The construction of partner potential used to explain the experiment result that cannot be explained by the original potential. The eigenvalue for anharmonic potential and the anharmonic partner potential have the same characteristic. Every increase of quantum orbital number the eigenvalue getting smaller. This result corresponds to Bohrn’s atomic theory that the eigenvalue is inversely proportional to the atomic shell. But the eigenvalue for the anharmonic partner potential higher than the eigenvalue for the anharmonic original potential.

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.

Probabilistic Component Mode Synthesis of Nondeterministic Substructures

NASA Technical Reports Server (NTRS)

Brown, Andrew M.; Ferri, Aldo A.

1996-01-01

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. We present a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Comparative analysis on the selection of number of clusters in community detection

NASA Astrophysics Data System (ADS)

Kawamoto, Tatsuro; Kabashima, Yoshiyuki

2018-02-01

We conduct a comparative analysis on various estimates of the number of clusters in community detection. An exhaustive comparison requires testing of all possible combinations of frameworks, algorithms, and assessment criteria. In this paper we focus on the framework based on a stochastic block model, and investigate the performance of greedy algorithms, statistical inference, and spectral methods. For the assessment criteria, we consider modularity, map equation, Bethe free energy, prediction errors, and isolated eigenvalues. From the analysis, the tendency of overfit and underfit that the assessment criteria and algorithms have becomes apparent. In addition, we propose that the alluvial diagram is a suitable tool to visualize statistical inference results and can be useful to determine the number of clusters.

Quantum statistics of Raman scattering model with Stokes mode generation

NASA Technical Reports Server (NTRS)

Tanatar, Bilal; Shumovsky, Alexander S.

1994-01-01

The model describing three coupled quantum oscillators with decay of Rayleigh mode into the Stokes and vibration (phonon) modes is examined. Due to the Manley-Rowe relations the problem of exact eigenvalues and eigenstates is reduced to the calculation of new orthogonal polynomials defined both by the difference and differential equations. The quantum statistical properties are examined in the case when initially: the Stokes mode is in the vacuum state; the Rayleigh mode is in the number state; and the vibration mode is in the number of or squeezed states. The collapses and revivals are obtained for different initial conditions as well as the change in time the sub-Poisson distribution by the super-Poisson distribution and vice versa.

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.

Fast Solution in Sparse LDA for Binary Classification

NASA Technical Reports Server (NTRS)

Moghaddam, Baback

2010-01-01

An algorithm that performs sparse linear discriminant analysis (Sparse-LDA) finds near-optimal solutions in far less time than the prior art when specialized to binary classification (of 2 classes). Sparse-LDA is a type of feature- or variable- selection problem with numerous applications in statistics, machine learning, computer vision, computational finance, operations research, and bio-informatics. Because of its combinatorial nature, feature- or variable-selection problems are NP-hard or computationally intractable in cases involving more than 30 variables or features. Therefore, one typically seeks approximate solutions by means of greedy search algorithms. The prior Sparse-LDA algorithm was a greedy algorithm that considered the best variable or feature to add/ delete to/ from its subsets in order to maximally discriminate between multiple classes of data. The present algorithm is designed for the special but prevalent case of 2-class or binary classification (e.g. 1 vs. 0, functioning vs. malfunctioning, or change versus no change). The present algorithm provides near-optimal solutions on large real-world datasets having hundreds or even thousands of variables or features (e.g. selecting the fewest wavelength bands in a hyperspectral sensor to do terrain classification) and does so in typical computation times of minutes as compared to days or weeks as taken by the prior art. Sparse LDA requires solving generalized eigenvalue problems for a large number of variable subsets (represented by the submatrices of the input within-class and between-class covariance matrices). In the general (fullrank) case, the amount of computation scales at least cubically with the number of variables and thus the size of the problems that can be solved is limited accordingly. However, in binary classification, the principal eigenvalues can be found using a special analytic formula, without resorting to costly iterative techniques. The present algorithm exploits this analytic form along with the inherent sequential nature of greedy search itself. Together this enables the use of highly-efficient partitioned-matrix-inverse techniques that result in large speedups of computation in both the forward-selection and backward-elimination stages of greedy algorithms in general.

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

Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian

NASA Astrophysics Data System (ADS)

Bender, Carl M.; Brody, Dorje C.

2018-04-01

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads to a challenging open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives the exact asymptotic behavior of the eigenfunction.

Applying nonlinear diffusion acceleration to the neutron transport k-Eigenvalue problem with anisotropic scattering

DOE PAGES

Willert, Jeffrey; Park, H.; Taitano, William

2015-11-01

High-order/low-order (or moment-based acceleration) algorithms have been used to significantly accelerate the solution to the neutron transport k-eigenvalue problem over the past several years. Recently, the nonlinear diffusion acceleration algorithm has been extended to solve fixed-source problems with anisotropic scattering sources. In this paper, we demonstrate that we can extend this algorithm to k-eigenvalue problems in which the scattering source is anisotropic and a significant acceleration can be achieved. Lastly, we demonstrate that the low-order, diffusion-like eigenvalue problem can be solved efficiently using a technique known as nonlinear elimination.

Solving complex band structure problems with the FEAST eigenvalue algorithm

NASA Astrophysics Data System (ADS)

Laux, S. E.

2012-08-01

With straightforward extension, the FEAST eigenvalue algorithm [Polizzi, Phys. Rev. B 79, 115112 (2009)] is capable of solving the generalized eigenvalue problems representing traveling-wave problems—as exemplified by the complex band-structure problem—even though the matrices involved are complex, non-Hermitian, and singular, and hence outside the originally stated range of applicability of the algorithm. The obtained eigenvalues/eigenvectors, however, contain spurious solutions which must be detected and removed. The efficiency and parallel structure of the original algorithm are unaltered. The complex band structures of Si layers of varying thicknesses and InAs nanowires of varying radii are computed as test problems.

Complex eigenvalue extraction in NASTRAN by the tridiagonal reduction (FEER) method

NASA Technical Reports Server (NTRS)

Newman, M.; Mann, F. I.

1977-01-01

An extension of the Tridiagonal Reduction (FEER) method to complex eigenvalue analysis in NASTRAN is described. As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum are extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order is much lower than that of the full size problem. The reduction process is effected automatically, and thus avoids the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admits mass, damping and stiffness matrices which are unrestricted in character, i.e., they may be real, complex, symmetric or unsymmetric, singular or non-singular.

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).

Statistical analysis of modeling error in structural dynamic systems

NASA Technical Reports Server (NTRS)

Hasselman, T. K.; Chrostowski, J. D.

1990-01-01

The paper presents a generic statistical model of the (total) modeling error for conventional space structures in their launch configuration. Modeling error is defined as the difference between analytical prediction and experimental measurement. It is represented by the differences between predicted and measured real eigenvalues and eigenvectors. Comparisons are made between pre-test and post-test models. Total modeling error is then subdivided into measurement error, experimental error and 'pure' modeling error, and comparisons made between measurement error and total modeling error. The generic statistical model presented in this paper is based on the first four global (primary structure) modes of four different structures belonging to the generic category of Conventional Space Structures (specifically excluding large truss-type space structures). As such, it may be used to evaluate the uncertainty of predicted mode shapes and frequencies, sinusoidal response, or the transient response of other structures belonging to the same generic category.

Experimental Study of Quantum Graphs with Microwave Networks

NASA Astrophysics Data System (ADS)

Fu, Ziyuan; Koch, Trystan; Antonsen, Thomas; Ott, Edward; Anlage, Steven; Wave Chaos Team

An experimental setup consisting of microwave networks is used to simulate quantum graphs. The networks are constructed from coaxial cables connected by T junctions. The networks are built for operation both at room temperature and superconducting versions that operate at cryogenic temperatures. In the experiments, a phase shifter is connected to one of the network bonds to generate an ensemble of quantum graphs by varying the phase delay. The eigenvalue spectrum is found from S-parameter measurements on one-port graphs. With the experimental data, the nearest-neighbor spacing statistics and the impedance statistics of the graphs are examined. It is also demonstrated that time-reversal invariance for microwave propagation in the graphs can be broken without increasing dissipation significantly by making nodes with circulators. Random matrix theory (RMT) successfully describes universal statistical properties of the system. We acknowledge support under contract AFOSR COE Grant FA9550-15-1-0171.

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.

Ultrarelativistic bound states in the spherical well

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

Żaba, Mariusz; Garbaczewski, Piotr

2016-07-15

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator (−Δ){sup 1/2}, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral data for lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into non-overlapping, orbitally labelled E{sub (k,l)} series. For each orbital label l = 0, 1, 2, …, the label k = 1, 2, … enumerates consecutive lth seriesmore » eigenvalues. Each of them is 2l + 1-degenerate. The l = 0 eigenvalues series E{sub (k,0)} are identical with the set of even labeled eigenvalues for the d = 1 Cauchy well: E{sub (k,0)}(d = 3) = E{sub 2k}(d = 1). Likewise, the eigenfunctions ψ{sub (k,0)}(d = 3) and ψ{sub 2k}(d = 1) show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for l ≤ 3).« less

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.

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

NASA Astrophysics Data System (ADS)

Kleefeld, Andreas; Pieronek, Lukas

2018-03-01

We analyze the method of fundamental solutions (MFS) in two different versions with focus on the computation of approximate acoustic interior transmission eigenvalues in 2D for homogeneous media. Our approach is mesh- and integration free, but suffers in general from the ill-conditioning effects of the discretized eigenoperator, which we could then successfully balance using an approved stabilization scheme. Our numerical examples cover many of the common scattering objects and prove to be very competitive in accuracy with the standard methods for PDE-related eigenvalue problems. We finally give an approximation analysis for our framework and provide error estimates, which bound interior transmission eigenvalue deviations in terms of some generalized MFS output.

Squeeze-film dampers for turbomachinery stabilization

NASA Technical Reports Server (NTRS)

Mclean, L. J.; Hahn, E. J.

1984-01-01

A technique for investigating the stability and damping present in centrally preloaded radially symmetric multi-mass flexible rotor bearing systems is presented. In general, one needs to find the eigenvalues of the linearized perturbation equations, though zero frequency stability maps may be found by solving as many simultaneous non-linear equations as there are dampers; and in the case of a single damper, such maps may be found directly, regardless of the number of degrees of freedom. The technique is illustrated for a simple symmetric four degree of freedom flexible rotor with an unpressurized damper. This example shows that whereas zero frequency stability maps are likely to prove to be a simple way to delineate multiple solution possibilities, they do not provide full stability information. Further, particularly for low bearing parameters, the introduction of an unpressurized squeeze film damper may promote instability in an otherwise stable system.

Probabilistic finite elements for transient analysis in nonlinear continua

NASA Technical Reports Server (NTRS)

Liu, W. K.; Belytschko, T.; Mani, A.

1985-01-01

The probabilistic finite element method (PFEM), which is a combination of finite element methods and second-moment analysis, is formulated for linear and nonlinear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in nonlinear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem. The moments calculated compare favorably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

HADY-I, a FORTRAN program for the compressible stability analysis of three-dimensional boundary layers. [on swept and tapered wings

NASA Technical Reports Server (NTRS)

El-Hady, N. M.

1981-01-01

A computer program HADY-I for calculating the linear incompressible or compressible stability characteristics of the laminar boundary layer on swept and tapered wings is described. The eigenvalue problem and its adjoint arising from the linearized disturbance equations with the appropriate boundary conditions are solved numerically using a combination of Newton-Raphson interative scheme and a variable step size integrator based on the Runge-Kutta-Fehlburh fifth-order formulas. The integrator is used in conjunction with a modified Gram-Schmidt orthonormalization procedure. The computer program HADY-I calculates the growth rates of crossflow or streamwise Tollmien-Schlichting instabilities. It also calculates the group velocities of these disturbances. It is restricted to parallel stability calculations, where the boundary layer (meanflow) is assumed to be parallel. The meanflow solution is an input to the program.

[Effects of different planting modes on the soil permeability of sloping farmlands in purple soil area].

PubMed

Li, Jian-Xing; He, Bing-Hui; Mei, Xue-Mei; Liang, Yan-Ling; Xiong, Jian

2013-03-01

Taking bare land as the control, this paper studied the effects of different planting modes on the soil permeability of sloping farmlands in purple soil area. For the test six planting modes, the soil permeability was in the order of Eriobotrya japonica > Citrus limon > Vetiveria zizanioides hedgerows +corn >Leucaena leucocephala hedgerows + corn> Hemerocallis fulva > corn> bare land, and decreased with increasing depth. The eigenvalues of soil infiltration were in the order of initial infiltration rate> average infiltration rate> stable infiltration rate. The soil permeability had significant positive linear correlations with soil total porosity, non-capillary porosity, initial moisture content, water holding capacity, and organic matter content, and significant negative linear correlation with soil bulk density. The common empirical infiltration model could well fit the soil moisture infiltration processes under the six planting modes, while the Kostiakov equation could not.

Some problems in applications of the linear variational method

NASA Astrophysics Data System (ADS)

Pupyshev, Vladimir I.; Montgomery, H. E.

2015-09-01

The linear variational method is a standard computational method in quantum mechanics and quantum chemistry. As taught in most classes, the general guidance is to include as many basis functions as practical in the variational wave function. However, if it is desired to study the patterns of energy change accompanying the change of system parameters such as the shape and strength of the potential energy, the problem becomes more complicated. We use one-dimensional systems with a particle in a rectangular or in a harmonic potential confined in an infinite rectangular box to illustrate situations where a variational calculation can give incorrect results. These situations result when the energy of the lowest eigenvalue is strongly dependent on the parameters that describe the shape and strength of the potential. The numerical examples described in this work are provided as cautionary notes for practitioners of numerical variational calculations.

Decentralized Observer with a Consensus Filter for Distributed Discrete-Time Linear Systems

NASA Technical Reports Server (NTRS)

Acikmese, Behcet; Mandic, Milan

2011-01-01

This paper presents a decentralized observer with a consensus filter for the state observation of a discrete-time linear distributed systems. In this setup, each agent in the distributed system has an observer with a model of the plant that utilizes the set of locally available measurements, which may not make the full plant state detectable. This lack of detectability is overcome by utilizing a consensus filter that blends the state estimate of each agent with its neighbors' estimates. We assume that the communication graph is connected for all times as well as the sensing graph. It is proven that the state estimates of the proposed observer asymptotically converge to the actual plant states under arbitrarily changing, but connected, communication and sensing topologies. As a byproduct of this research, we also obtained a result on the location of eigenvalues, the spectrum, of the Laplacian for a family of graphs with self-loops.

Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos.

PubMed

Basko, D M

2014-02-01

We study the discrete nonlinear Schröinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved but, on the other, strong enough for the dynamics of the normal mode amplitudes to be chaotic for almost all modes. We show that in this regime and in the limit of high temperature, the macroscopic density ρ satisfies the nonlinear diffusion equation with a density-dependent diffusion coefficient, D(ρ) = D(0)ρ(2). An explicit expression for D(0) is obtained in terms of the eigenfunctions and eigenvalues of the linear problem, which is then evaluated numerically. The role of the second conserved quantity (energy) in the transport is also quantitatively discussed.

Designing pinhole vacancies in graphene towards functionalization: Effects on critical buckling load

NASA Astrophysics Data System (ADS)

Georgantzinos, S. K.; Markolefas, S.; Giannopoulos, G. I.; Katsareas, D. E.; Anifantis, N. K.

2017-03-01

The effect of size and placement of pinhole-type atom vacancies on Euler's critical load on free-standing, monolayer graphene, is investigated. The graphene is modeled by a structural spring-based finite element approach, in which every interatomic interaction is approached as a linear spring. The geometry of graphene and the pinhole size lead to the assembly of the stiffness matrix of the nanostructure. Definition of the boundary conditions of the problem leads to the solution of the eigenvalue problem and consequently to the critical buckling load. Comparison to results found in the literature illustrates the validity and accuracy of the proposed method. Parametric analysis regarding the placement and size of the pinhole-type vacancy, as well as the graphene geometry, depicts the effects on critical buckling load. Non-linear regression analysis leads to empirical-analytical equations for predicting the buckling behavior of graphene, with engineered pinhole-type atom vacancies.

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.

Substructure Versus Property-Level Dispersed Modes Calculation

NASA Technical Reports Server (NTRS)

Stewart, Eric C.; Peck, Jeff A.; Bush, T. Jason; Fulcher, Clay W.

2016-01-01

This paper calculates the effect of perturbed finite element mass and stiffness values on the eigenvectors and eigenvalues of the finite element model. The structure is perturbed in two ways: at the "subelement" level and at the material property level. In the subelement eigenvalue uncertainty analysis the mass and stiffness of each subelement is perturbed by a factor before being assembled into the global matrices. In the property-level eigenvalue uncertainty analysis all material density and stiffness parameters of the structure are perturbed modified prior to the eigenvalue analysis. The eigenvalue and eigenvector dispersions of each analysis (subelement and property-level) are also calculated using an analytical sensitivity approximation. Two structural models are used to compare these methods: a cantilevered beam model, and a model of the Space Launch System. For each structural model it is shown how well the analytical sensitivity modes approximate the exact modes when the uncertainties are applied at the subelement level and at the property level.

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.

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.

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.

Non-linear eigensolver-based alternative to traditional SCF methods

NASA Astrophysics Data System (ADS)

Gavin, Brendan; Polizzi, Eric

2013-03-01

The self-consistent iterative procedure in Density Functional Theory calculations is revisited using a new, highly efficient and robust algorithm for solving the non-linear eigenvector problem (i.e. H(X)X = EX;) of the Kohn-Sham equations. This new scheme is derived from a generalization of the FEAST eigenvalue algorithm, and provides a fundamental and practical numerical solution for addressing the non-linearity of the Hamiltonian with the occupied eigenvectors. In contrast to SCF techniques, the traditional outer iterations are replaced by subspace iterations that are intrinsic to the FEAST algorithm, while the non-linearity is handled at the level of a projected reduced system which is orders of magnitude smaller than the original one. Using a series of numerical examples, it will be shown that our approach can outperform the traditional SCF mixing techniques such as Pulay-DIIS by providing a high converge rate and by converging to the correct solution regardless of the choice of the initial guess. We also discuss a practical implementation of the technique that can be achieved effectively using the FEAST solver package. This research is supported by NSF under Grant #ECCS-0846457 and Intel Corporation.

Social inequality, lifestyles and health - a non-linear canonical correlation analysis based on the approach of Pierre Bourdieu.

PubMed

Grosse Frie, Kirstin; Janssen, Christian

2009-01-01

Based on the theoretical and empirical approach of Pierre Bourdieu, a multivariate non-linear method is introduced as an alternative way to analyse the complex relationships between social determinants and health. The analysis is based on face-to-face interviews with 695 randomly selected respondents aged 30 to 59. Variables regarding socio-economic status, life circumstances, lifestyles, health-related behaviour and health were chosen for the analysis. In order to determine whether the respondents can be differentiated and described based on these variables, a non-linear canonical correlation analysis (OVERALS) was performed. The results can be described on three dimensions; Eigenvalues add up to the fit of 1.444, which can be interpreted as approximately 50 % of explained variance. The three-dimensional space illustrates correspondences between variables and provides a framework for interpretation based on latent dimensions, which can be described by age, education, income and gender. Using non-linear canonical correlation analysis, health characteristics can be analysed in conjunction with socio-economic conditions and lifestyles. Based on Bourdieus theoretical approach, the complex correlations between these variables can be more substantially interpreted and presented.

Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials

NASA Astrophysics Data System (ADS)

Volkmer, Hans

2008-04-01

Sequences of polynomials, orthogonal with respect to signed measures, are associated with a class of differential equations including the Mathieu, Lame and Whittaker-Hill equation. It is shown that the zeros of pn form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of pn are found. Applications to the numerical treatment of eigenvalue problems are given.

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.

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.

Market Correlation Structure Changes Around the Great Crash: A Random Matrix Theory Analysis of the Chinese Stock Market

NASA Astrophysics Data System (ADS)

Han, Rui-Qi; Xie, Wen-Jie; Xiong, Xiong; Zhang, Wei; Zhou, Wei-Xing

The correlation structure of a stock market contains important financial contents, which may change remarkably due to the occurrence of financial crisis. We perform a comparative analysis of the Chinese stock market around the occurrence of the 2008 crisis based on the random matrix analysis of high-frequency stock returns of 1228 Chinese stocks. Both raw correlation matrix and partial correlation matrix with respect to the market index in two time periods of one year are investigated. We find that the Chinese stocks have stronger average correlation and partial correlation in 2008 than in 2007 and the average partial correlation is significantly weaker than the average correlation in each period. Accordingly, the largest eigenvalue of the correlation matrix is remarkably greater than that of the partial correlation matrix in each period. Moreover, each largest eigenvalue and its eigenvector reflect an evident market effect, while other deviating eigenvalues do not. We find no evidence that deviating eigenvalues contain industrial sectorial information. Surprisingly, the eigenvectors of the second largest eigenvalues in 2007 and of the third largest eigenvalues in 2008 are able to distinguish the stocks from the two exchanges. We also find that the component magnitudes of the some largest eigenvectors are proportional to the stocks’ capitalizations.

A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem

NASA Astrophysics Data System (ADS)

Willert, Jeffrey; Park, H.; Knoll, D. A.

2014-10-01

Over the past several years a number of papers have been written describing modern techniques for numerically computing the dominant eigenvalue of the neutron transport criticality problem. These methods fall into two distinct categories. The first category of methods rewrite the multi-group k-eigenvalue problem as a nonlinear system of equations and solve the resulting system using either a Jacobian-Free Newton-Krylov (JFNK) method or Nonlinear Krylov Acceleration (NKA), a variant of Anderson Acceleration. These methods are generally successful in significantly reducing the number of transport sweeps required to compute the dominant eigenvalue. The second category of methods utilize Moment-Based Acceleration (or High-Order/Low-Order (HOLO) Acceleration). These methods solve a sequence of modified diffusion eigenvalue problems whose solutions converge to the solution of the original transport eigenvalue problem. This second class of methods is, in our experience, always superior to the first, as most of the computational work is eliminated by the acceleration from the LO diffusion system. In this paper, we review each of these methods. Our computational results support our claim that the choice of which nonlinear solver to use, JFNK or NKA, should be secondary. The primary computational savings result from the implementation of a HOLO algorithm. We display computational results for a series of challenging multi-dimensional test problems.

Changes in diffusion tensor imaging (DTI) eigenvalues of skeletal muscle due to hybrid exercise training.

PubMed

Okamoto, Yoshikazu; Kemp, Graham J; Isobe, Tomonori; Sato, Eisuke; Hirano, Yuji; Shoda, Junichi; Minami, Manabu

2014-12-01

Several studies have proposed the cell membrane as the main water diffusion restricting factor in the skeletal muscle cell. We sought to establish whether a particular form of exercise training (which is likely to affect only intracellular components) could affect water diffusion. The purpose of this study is to characterise prospectively the changes in diffusion tensor imaging (DTI) eigenvalues of thigh muscle resulting from hybrid training (HYBT) in patients with non-alcoholic fatty liver disease (NAFLD). Twenty-one NAFLD patients underwent HYBT for 30 minutes per day, twice a week for 6 months. Patients were scanned using DTI of the thigh pre- and post-HYBT. Fractional anisotropy (FA), apparent diffusion coefficient (ADC), the three eigenvalues lambda 1 (λ1), λ2, λ3, and the maximal cross sectional area (CSA) were measured in bilateral thigh muscles: knee flexors (biceps femoris (BF), semitendinosus (ST), semimembranous (SM)) and knee extensors (medial vastus (MV), intermediate vastus (IV), lateral vastus (LV), and rectus femoris (RF)), and compared pre- and post-HYBT by paired t-test. Muscle strength of extensors (P<0.01), but not flexors, increased significantly post-HYBT. For FA, ADC and eigenvalues, the overall picture was of increase. Some (P<0.05 in λ2 and P<0.01 in λ1) eigenvalues of flexors and all (λ1-λ3) eigenvalues of extensors increased significantly (P<0.01) post-HYBT. HYBT increased all 3 eigenvalues. We suggest this might be caused by enlargement of muscle intracellular space. Copyright © 2014 Elsevier Inc. All rights reserved.

Correlation, Cost Risk, and Geometry

NASA Technical Reports Server (NTRS)

Dean, Edwin B.

1992-01-01

The geometric viewpoint identifies the choice of a correlation matrix for the simulation of cost risk with the pairwise choice of data vectors corresponding to the parameters used to obtain cost risk. The correlation coefficient is the cosine of the angle between the data vectors after translation to an origin at the mean and normalization for magnitude. Thus correlation is equivalent to expressing the data in terms of a non orthogonal basis. To understand the many resulting phenomena requires the use of the tensor concept of raising the index to transform the measured and observed covariant components into contravariant components before vector addition can be applied. The geometric viewpoint also demonstrates that correlation and covariance are geometric properties, as opposed to purely statistical properties, of the variates. Thus, variates from different distributions may be correlated, as desired, after selection from independent distributions. By determining the principal components of the correlation matrix, variates with the desired mean, magnitude, and correlation can be generated through linear transforms which include the eigenvalues and the eigenvectors of the correlation matrix. The conversion of the data to a non orthogonal basis uses a compound linear transformation which distorts or stretches the data space. Hence, the correlated data does not have the same properties as the uncorrelated data used to generate it. This phenomena is responsible for seemingly strange observations such as the fact that the marginal distributions of the correlated data can be quite different from the distributions used to generate the data. The joint effect of statistical distributions and correlation remains a fertile area for further research. In terms of application to cost estimating, the geometric approach demonstrates that the estimator must have data and must understand that data in order to properly choose the correlation matrix appropriate for a given estimate. There is a general feeling by employers and managers that the field of cost requires little technical or mathematical background. Contrary to that opinion, this paper demonstrates that a background in mathematics equivalent to that needed for typical engineering and scientific disciplines at the masters or doctorate level is appropriate within the field of cost risk.

Assessment of psychometric properties of a modified PHEEM questionnaire.

PubMed

Gooneratne, I K; Munasinghe, S R; Siriwardena, C; Olupeliyawa, A M; Karunathilake, I

2008-12-01

An effective tool in analysing the learning environment, customised to the Sri Lankan setting, is vital for the assessment and delivery of quality healthcare training of preregistration house officers. Such a tool should be reliable and valid. We assessed psychometric properties such as internal reliability and construct validity of a modified version of the Postgraduate Hospital Educational Environment Measure (PHEEM). A modified PHEEM questionnaire customised to the Sri Lankan context was developed in accordance to the Sri Lanka Medical Council guidelines. The questionnaire was distributed to all interns at the National Hospital of Sri Lanka, Colombo North Teaching Hospital and Wathupitiwala Base Hospital during a calendar year (n = 100, response rate = 86%). Internal reliability and construct validity of the inventory were assessed by using Cronbach's alpha and exploratory factor analysis respectively as statistical methods. PHEEM consists of 3 subscales: perceptions of autonomy, social support and teaching, which are factors perceived to be influencing the educational environment. This administration demonstrated high internal reliability as reflected by a Cronbach's alpha value of 0.84. Exploratory factor analysis identified 12 factors with eigenvalue >1. However, the first factor had an eigenvalue of 6.7 (accounting for 19.7% of variance), while the rest had eigenvalues < 2.5. These results suggest a single predictive factor and thus a one-dimensional scale as opposed to the three-dimensional scale which is used in the current questionnaire. The psychometric properties of this tool reflect a high degree of internal reliability in assessing the educational environment of intern doctors in Sri Lanka. It is possible that the clinical educational environment is collectively represented as a single dimension. This may be due to the complex interplay between individual items in the questionnaire. Therefore the psychometric properties do not justify the interpretation of the educational environment through specified subscales.

Fast EEG spike detection via eigenvalue analysis and clustering of spatial amplitude distribution

NASA Astrophysics Data System (ADS)

Fukami, Tadanori; Shimada, Takamasa; Ishikawa, Bunnoshin

2018-06-01

Objective. In the current study, we tested a proposed method for fast spike detection in electroencephalography (EEG). Approach. We performed eigenvalue analysis in two-dimensional space spanned by gradients calculated from two neighboring samples to detect high-amplitude negative peaks. We extracted the spike candidates by imposing restrictions on parameters regarding spike shape and eigenvalues reflecting detection characteristics of individual medical doctors. We subsequently performed clustering, classifying detected peaks by considering the amplitude distribution at 19 scalp electrodes. Clusters with a small number of candidates were excluded. We then defined a score for eliminating spike candidates for which the pattern of detected electrodes differed from the overall pattern in a cluster. Spikes were detected by setting the score threshold. Main results. Based on visual inspection by a psychiatrist experienced in EEG, we evaluated the proposed method using two statistical measures of precision and recall with respect to detection performance. We found that precision and recall exhibited a trade-off relationship. The average recall value was 0.708 in eight subjects with the score threshold that maximized the F-measure, with 58.6  ±  36.2 spikes per subject. Under this condition, the average precision was 0.390, corresponding to a false positive rate 2.09 times higher than the true positive rate. Analysis of the required processing time revealed that, using a general-purpose computer, our method could be used to perform spike detection in 12.1% of the recording time. The process of narrowing down spike candidates based on shape occupied most of the processing time. Significance. Although the average recall value was comparable with that of other studies, the proposed method significantly shortened the processing time.

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

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.

Algorithm-Eigenvalue Estimation of Hyperspectral Wishart Covariance Matrices from a Limited Number of Samples

DTIC Science & Technology

2015-03-01

ALGORITHM—EIGENVALUE ESTIMATION OF HYPERSPECTRAL WISHART COVARIANCE MATRICES FROM A LIMITED NUMBER OF SAMPLES ECBC-TN-067 Avishai Ben- David ...NUMBER 6. AUTHOR(S) Ben- David , Avishai (ECBC) and Davidson, Charles E. (STC) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7...and published by Avishai Ben- David and Charles E. Davidson (Eigenvalue Estimation of Hyperspectral WishartCovariance Matrices from Limited Number of

Continuous-time quantum walks on multilayer dendrimer networks

NASA Astrophysics Data System (ADS)

Galiceanu, Mircea; Strunz, Walter T.

2016-08-01

We consider continuous-time quantum walks (CTQWs) on multilayer dendrimer networks (MDs) and their application to quantum transport. A detailed study of properties of CTQWs is presented and transport efficiency is determined in terms of the exact and average return probabilities. The latter depends only on the eigenvalues of the connectivity matrix, which even for very large structures allows a complete analytical solution for this particular choice of network. In the case of MDs we observe an interplay between strong localization effects, due to the dendrimer topology, and good efficiency from the linear segments. We show that quantum transport is enhanced by interconnecting more layers of dendrimers.

Stability analysis of ultrasound thick-shell contrast agents.

PubMed

Lu, Xiaozhen; Chahine, Georges L; Hsiao, Chao-Tsung

2012-01-01

The stability of thick shell encapsulated bubbles is studied analytically. 3-D small perturbations are introduced to the spherical oscillations of a contrast agent bubble in response to a sinusoidal acoustic field with different amplitudes of excitation. The equations of the perturbation amplitudes are derived using asymptotic expansions and linear stability analysis is then applied to the resulting differential equations. The stability of the encapsulated microbubbles to nonspherical small perturbations is examined by solving an eigenvalue problem. The approach then identifies the fastest growing perturbations which could lead to the breakup of the encapsulated microbubble or contrast agent. © 2012 Acoustical Society of America.

Nonlinear vibrations of thin arbitrarily laminated composite plates subjected to harmonic excitations using DKT elements

NASA Astrophysics Data System (ADS)

Chiang, C. K.; Xue, David Y.; Mei, Chuh

1993-04-01

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite thin plates using the Discrete Kirchhoff Theory (DKT) triangular elements. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite triangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, triangular plates, different boundary conditions, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Nonlinear vibrations of thin arbitrarily laminated composite plates subjected to harmonic excitations using DKT elements

NASA Technical Reports Server (NTRS)

Chiang, C. K.; Xue, David Y.; Mei, Chuh

1993-01-01

A finite element formulation is presented for determining the large-amplitude free and steady-state forced vibration response of arbitrarily laminated anisotropic composite thin plates using the Discrete Kirchhoff Theory (DKT) triangular elements. The nonlinear stiffness and harmonic force matrices of an arbitrarily laminated composite triangular plate element are developed for nonlinear free and forced vibration analyses. The linearized updated-mode method with nonlinear time function approximation is employed for the solution of the system nonlinear eigenvalue equations. The amplitude-frequency relations for convergence with gridwork refinement, triangular plates, different boundary conditions, lamination angles, number of plies, and uniform versus concentrated loads are presented.

Systematic study of baryons in a three-body quark model

NASA Astrophysics Data System (ADS)

Aslanzadeh, M.; Rajabi, A. A.

2016-09-01

We investigated the structure of baryons within a three-body quark model based on hypercentral approach. We considered an SU(6)-invariant potential consisting of the well-known "Coulomb-plus-linear" potential plus some multipole interactions as V ( x) ∝ x - n with n > 2. Then, through an analytical solution, we obtained the energy eigenvalues and eigenfunctions of the three-body problem and evaluated some observables such as the mass spectrum of light baryons and both the electromagnetic elastic form factors, and the charge radii of nucleons. We compared our results with the experimental data and showed that the present model provides a good description of the observed resonances.

Diagonalization of the symmetrized discrete i th right shift operator

NASA Astrophysics Data System (ADS)

Fuentes, Marc

2007-01-01

In this paper, we consider the symmetric part of the so-called ith right shift operator. We determine its eigenvalues as also the associated eigenvectors in a complete and closed form. The proposed proof is elementary, using only basical skills such as Trigonometry, Arithmetic and Linear algebra. The first section is devoted to the introduction of the tackled problem. Second and third parts contain almost all the ?technical? stuff of the proofE Afterwards, we continue with the end of the proof, provide a graphical illustration of the results, as well as an application on the polyhedral ?sandwiching? of a special compact of arising in Signal theory.

Integrability and Linearizability of the Lotka-Volterra System with a Saddle Point with Rational Hyperbolicity Ratio

NASA Astrophysics Data System (ADS)

Gravel, Simon; Thibault, Pierre

In this paper, we consider normalizability, integrability and linearizability properties of the Lotka-Volterra system in the neighborhood of a singular point with eigenvalues 1 and - λ. The results are obtained by generalizing and expanding two methods already known: the power expansion of the first integral or of the linearizing transformation and the transformation of the saddle into a node. With these methods we find conditions that are valid for λ∈ R+ or λ∈ Q. These conditions will allow us to find all the integrable and linearizable systems for λ= {p}/{2} and {2}/{p} with p∈ N+.

Multigrid techniques for nonlinear eigenvalue probems: Solutions of a nonlinear Schroedinger eigenvalue problem in 2D and 3D

NASA Technical Reports Server (NTRS)

Costiner, Sorin; Taasan, Shlomo

1994-01-01

This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.

The nonconforming virtual element method for eigenvalue problems

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

Gardini, Francesca; Manzini, Gianmarco; Vacca, Giuseppe

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L 2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problems. The proposed schemes provide a correct approximation of the spectrum and we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numericalmore » tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.« less

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.

The Impact of the Network Topology on the Viral Prevalence: A Node-Based Approach

PubMed Central

Yang, Lu-Xing; Draief, Moez; Yang, Xiaofan

2015-01-01

This paper addresses the impact of the structure of the viral propagation network on the viral prevalence. For that purpose, a new epidemic model of computer virus, known as the node-based SLBS model, is proposed. Our analysis shows that the maximum eigenvalue of the underlying network is a key factor determining the viral prevalence. Specifically, the value range of the maximum eigenvalue is partitioned into three subintervals: viruses tend to extinction very quickly or approach extinction or persist depending on into which subinterval the maximum eigenvalue of the propagation network falls. Consequently, computer virus can be contained by adjusting the propagation network so that its maximum eigenvalue falls into the desired subinterval. PMID:26222539

A modal aeroelastic analysis scheme for turbomachinery blading. M.S. Thesis - Case Western Reserve Univ. Final Report

NASA Technical Reports Server (NTRS)

Smith, Todd E.

1991-01-01

An aeroelastic analysis is developed which has general application to all types of axial-flow turbomachinery blades. The approach is based on linear modal analysis, where the blade's dynamic response is represented as a linear combination of contributions from each of its in-vacuum free vibrational modes. A compressible linearized unsteady potential theory is used to model the flow over the oscillating blades. The two-dimensional unsteady flow is evaluated along several stacked axisymmetric strips along the span of the airfoil. The unsteady pressures at the blade surface are integrated to result in the generalized force acting on the blade due to simple harmonic motions. The unsteady aerodynamic forces are coupled to the blade normal modes in the frequency domain using modal analysis. An iterative eigenvalue problem is solved to determine the stability of the blade when the unsteady aerodynamic forces are included in the analysis. The approach is demonstrated by applying it to a high-energy subsonic turbine blade from a rocket engine turbopump power turbine. The results indicate that this turbine could undergo flutter in an edgewise mode of vibration.

Numerical methods on some structured matrix algebra problems

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

Jessup, E.R.

1996-06-01

This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was tomore » translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.« less

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.

Extended reactance domain algorithms for DoA estimation onto an ESPAR antennas

NASA Astrophysics Data System (ADS)

Harabi, F.; Akkar, S.; Gharsallah, A.

2016-07-01

Based on an extended reactance domain (RD) covariance matrix, this article proposes new alternatives for directions of arrival (DoAs) estimation of narrowband sources through an electronically steerable parasitic array radiator (ESPAR) antennas. Because of the centro symmetry of the classic ESPAR antennas, an unitary transformation is applied to the collected data that allow an important reduction in both computational cost and processing time and, also, an enhancement of the resolution capabilities of the proposed algorithms. Moreover, this article proposes a new approach for eigenvalues estimation through only some linear operations. The developed DoAs estimation algorithms based on this new approach has illustrated a good behaviour with less calculation cost and processing time as compared to other schemes based on the classic eigenvalues approach. The conducted simulations demonstrate that high-precision and high-resolution DoAs estimation can be reached especially in very closely sources situation and low sources power as compared to the RD-MUSIC algorithm and the RD-PM algorithm. The asymptotic behaviours of the proposed DoAs estimators are analysed in various scenarios and compared with the Cramer-Rao bound (CRB). The conducted simulations testify the high-resolution of the developed algorithms and prove the efficiently of the proposed approach.

Guided waves dispersion equations for orthotropic multilayered pipes solved using standard finite elements code.

PubMed

Predoi, Mihai Valentin

2014-09-01

The dispersion curves for hollow multilayered cylinders are prerequisites in any practical guided waves application on such structures. The equations for homogeneous isotropic materials have been established more than 120 years ago. The difficulties in finding numerical solutions to analytic expressions remain considerable, especially if the materials are orthotropic visco-elastic as in the composites used for pipes in the last decades. Among other numerical techniques, the semi-analytical finite elements method has proven its capability of solving this problem. Two possibilities exist to model a finite elements eigenvalue problem: a two-dimensional cross-section model of the pipe or a radial segment model, intersecting the layers between the inner and the outer radius of the pipe. The last possibility is here adopted and distinct differential problems are deduced for longitudinal L(0,n), torsional T(0,n) and flexural F(m,n) modes. Eigenvalue problems are deduced for the three modes classes, offering explicit forms of each coefficient for the matrices used in an available general purpose finite elements code. Comparisons with existing solutions for pipes filled with non-linear viscoelastic fluid or visco-elastic coatings as well as for a fully orthotropic hollow cylinder are all proving the reliability and ease of use of this method. Copyright © 2014 Elsevier B.V. All rights reserved.

Eigenmodes of Ducted Flows With Radially-Dependent Axial and Swirl Velocity Components

NASA Technical Reports Server (NTRS)

Kousen, Kenneth A.

1999-01-01

This report characterizes the sets of small disturbances possible in cylindrical and annular ducts with mean flow whose axial and tangential components vary arbitrarily with radius. The linearized equations of motion are presented and discussed, and then exponential forms for the axial, circumferential, and time dependencies of any unsteady disturbances are assumed. The resultant equations form a generalized eigenvalue problem, the solution of which yields the axial wavenumbers and radial mode shapes of the unsteady disturbances. Two numerical discretizations are applied to the system of equations: (1) a spectral collocation technique based on Chebyshev polynomial expansions on the Gauss-Lobatto points, and (2) second and fourth order finite differences on uniform grids. The discretized equations are solved using a standard eigensystem package employing the QR algorithm. The eigenvalues fall into two primary categories: a discrete set (analogous to the acoustic modes found in uniform mean flows) and a continuous band (analogous to convected disturbances in uniform mean flows) where the phase velocities of the disturbances correspond to the local mean flow velocities. Sample mode shapes and eigensystem distributions are presented for both sheared axial and swirling flows. The physics of swirling flows is examined with reference to hydrodynamic stability and completeness of the eigensystem expansions. The effect of assuming exponential dependence in the axial direction is discussed.

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.

Polymer Crowding in Confined Polymer-Nanoparticle Mixtures

NASA Astrophysics Data System (ADS)

Davis, Wyatt J.; Denton, Alan R.

Crowding can influence the conformations and thus functionality of macromolecules in quasi-two-dimensional environments, such as DNA or proteins confined to a cell membrane. We explore such crowding within a model of polymers as penetrable ellipses, whose shapes are governed by the statistics of a 2D random walk. The principal radii of the polymers fluctuate according to probability distributions of the eigenvalues of the gyration tensor. Within this coarse-grained model, we perform Monte Carlo simulations of mixtures of polymers and hard nanodisks, including trial changes in polymer conformation (shape and orientation). Penetration of polymers by nanodisks is incorporated with a free energy cost predicted by polymer field theory. Over ranges of size ratio and nanodisk density, we analyze the influence of crowding on polymer shape by computing eigenvalue distributions, mean radius of gyration, and mean asphericity of the polymer. We compare results with predictions of free-volume theory and with corresponding results in three dimensions. Our approach may help to interpret recent (and motivate future) experimental studies of biopolymers interacting with cell membranes, with relevance for drug delivery and gene therapy. This work was supported by the National Science Foundation under Grant No. DMR-1106331.

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

NASA Astrophysics Data System (ADS)

Chekroun, Mickaël D.; Kondrashov, Dmitri

2017-09-01

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey, furthermore, a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled—provided the decay of temporal correlations is sufficiently well-resolved—within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.

Rasch Analysis of the General Self-Efficacy Scale in Workers with Traumatic Limb Injuries.

PubMed

Wu, Tzu-Yi; Yu, Wan-Hui; Huang, Chien-Yu; Hou, Wen-Hsuan; Hsieh, Ching-Lin

2016-09-01

Purpose The purpose of this study was to apply Rasch analysis to examine the unidimensionality and reliability of the General Self-Efficacy Scale (GSE) in workers with traumatic limb injuries. Furthermore, if the items of the GSE fitted the Rasch model's assumptions, we transformed the raw sum ordinal scores of the GSE into Rasch interval scores. Methods A total of 1076 participants completed the GSE at 1 month post injury. Rasch analysis was used to examine the unidimensionality and person reliability of the GSE. The unidimensionality of the GSE was verified by determining whether the items fit the Rasch model's assumptions: (1) item fit indices: infit and outfit mean square (MNSQ) ranged from 0.6 to 1.4; and (2) the eigenvalue of the first factor extracted from principal component analysis (PCA) for residuals was <2. Person reliability was calculated. Results The unidimensionality of the 10-item GSE was supported in terms of good item fit statistics (infit and outfit MNSQ ranging from 0.92 to 1.32) and acceptable eigenvalues (1.6) of the first factor of the PCA, with person reliability = 0.89. Consequently, the raw sum scores of the GSE were transformed into Rasch scores. Conclusions The results indicated that the items of GSE are unidimensional and have acceptable person reliability in workers with traumatic limb injuries. Additionally, the raw sum scores of the GSE can be transformed into Rasch interval scores for prospective users to quantify workers' levels of self-efficacy and to conduct further statistical analyses.

Optical coherence tomography angiography-based capillary velocimetry

NASA Astrophysics Data System (ADS)

Wang, Ruikang K.; Zhang, Qinqin; Li, Yuandong; Song, Shaozhen

2017-06-01

Challenge persists in the field of optical coherence tomography (OCT) when it is required to quantify capillary blood flow within tissue beds in vivo. We propose a useful approach to statistically estimate the mean capillary flow velocity using a model-based statistical method of eigendecomposition (ED) analysis of the complex OCT signals obtained with the OCT angiography (OCTA) scanning protocol. ED-based analysis is achieved by the covariance matrix of the ensemble complex OCT signals, upon which the eigenvalues and eigenvectors that represent the subsets of the signal makeup are calculated. From this analysis, the signals due to moving particles can be isolated by employing an adaptive regression filter to remove the eigencomponents that represent static tissue signals. The mean frequency (MF) of moving particles can be estimated by the first lag-one autocorrelation of the corresponding eigenvectors. Three important parameters are introduced, including the blood flow signal power representing the presence of blood flow (i.e., OCTA signals), the MF indicating the mean velocity of blood flow, and the frequency bandwidth describing the temporal flow heterogeneity within a scanned tissue volume. The proposed approach is tested using scattering phantoms, in which microfluidic channels are used to simulate the functional capillary vessels that are perfused with the scattering intralipid solution. The results indicate a linear relationship between the MF and mean flow velocity. In vivo animal experiments are also conducted by imaging mouse brain with distal middle cerebral artery ligation to test the capability of the method to image the changes in capillary flows in response to an ischemic insult, demonstrating the practical usefulness of the proposed method for providing important quantifiable information about capillary tissue beds in the investigations of neurological conditions in vivo.

An Independent Filter for Gene Set Testing Based on Spectral Enrichment.

PubMed

Frost, H Robert; Li, Zhigang; Asselbergs, Folkert W; Moore, Jason H

2015-01-01

Gene set testing has become an indispensable tool for the analysis of high-dimensional genomic data. An important motivation for testing gene sets, rather than individual genomic variables, is to improve statistical power by reducing the number of tested hypotheses. Given the dramatic growth in common gene set collections, however, testing is often performed with nearly as many gene sets as underlying genomic variables. To address the challenge to statistical power posed by large gene set collections, we have developed spectral gene set filtering (SGSF), a novel technique for independent filtering of gene set collections prior to gene set testing. The SGSF method uses as a filter statistic the p-value measuring the statistical significance of the association between each gene set and the sample principal components (PCs), taking into account the significance of the associated eigenvalues. Because this filter statistic is independent of standard gene set test statistics under the null hypothesis but dependent under the alternative, the proportion of enriched gene sets is increased without impacting the type I error rate. As shown using simulated and real gene expression data, the SGSF algorithm accurately filters gene sets unrelated to the experimental outcome resulting in significantly increased gene set testing power.

Correcting Too Much or Too Little? The Performance of Three Chi-Square Corrections.

PubMed

Foldnes, Njål; Olsson, Ulf Henning

2015-01-01

This simulation study investigates the performance of three test statistics, T1, T2, and T3, used to evaluate structural equation model fit under non normal data conditions. T1 is the well-known mean-adjusted statistic of Satorra and Bentler. T2 is the mean-and-variance adjusted statistic of Sattertwaithe type where the degrees of freedom is manipulated. T3 is a recently proposed version of T2 that does not manipulate degrees of freedom. Discrepancies between these statistics and their nominal chi-square distribution in terms of errors of Type I and Type II are investigated. All statistics are shown to be sensitive to increasing kurtosis in the data, with Type I error rates often far off the nominal level. Under excess kurtosis true models are generally over-rejected by T1 and under-rejected by T2 and T3, which have similar performance in all conditions. Under misspecification there is a loss of power with increasing kurtosis, especially for T2 and T3. The coefficient of variation of the nonzero eigenvalues of a certain matrix is shown to be a reliable indicator for the adequacy of these statistics.

Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

NASA Astrophysics Data System (ADS)

Costiner, Sorin; Ta'asan, Shlomo

1995-07-01

Algorithms for nonlinear eigenvalue problems (EP's) often require solving self-consistently a large number of EP's. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP's obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schrödinger Poisson EP's. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schrödinger-Poisson EP in two and three dimensions, presenting special computational difficulties that are due to the nonlinearity and to the equal and closely clustered eigenvalues are demonstrated. For these cases, the algorithm requires O(qN) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per fine level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained.

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

Asymptotic theory of a slender rotating beam with end masses.

NASA Technical Reports Server (NTRS)

Whitman, A. M.; Abel, J. M.

1972-01-01

The method of matched asymptotic expansions is employed to solve the singular perturbation problem of the vibrations of a rotating beam of small flexural rigidity with concentrated end masses. The problem is complicated by the appearance of the eigenvalue in the boundary conditions. Eigenfunctions and eigenvalues are developed as power series in the perturbation parameter beta to the 1/2 power, and results are given for mode shapes and eigenvalues through terms of the order of beta.

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).

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.

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.

Fast-Running Aeroelastic Code Based on Unsteady Linearized Aerodynamic Solver Developed

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.; Bakhle, Milind A.; Keith, T., Jr.

2003-01-01

The NASA Glenn Research Center has been developing aeroelastic analyses for turbomachines for use by NASA and industry. An aeroelastic analysis consists of a structural dynamic model, an unsteady aerodynamic model, and a procedure to couple the two models. The structural models are well developed. Hence, most of the development for the aeroelastic analysis of turbomachines has involved adapting and using unsteady aerodynamic models. Two methods are used in developing unsteady aerodynamic analysis procedures for the flutter and forced response of turbomachines: (1) the time domain method and (2) the frequency domain method. Codes based on time domain methods require considerable computational time and, hence, cannot be used during the design process. Frequency domain methods eliminate the time dependence by assuming harmonic motion and, hence, require less computational time. Early frequency domain analyses methods neglected the important physics of steady loading on the analyses for simplicity. A fast-running unsteady aerodynamic code, LINFLUX, which includes steady loading and is based on the frequency domain method, has been modified for flutter and response calculations. LINFLUX, solves unsteady linearized Euler equations for calculating the unsteady aerodynamic forces on the blades, starting from a steady nonlinear aerodynamic solution. First, we obtained a steady aerodynamic solution for a given flow condition using the nonlinear unsteady aerodynamic code TURBO. A blade vibration analysis was done to determine the frequencies and mode shapes of the vibrating blades, and an interface code was used to convert the steady aerodynamic solution to a form required by LINFLUX. A preprocessor was used to interpolate the mode shapes from the structural dynamic mesh onto the computational dynamics mesh. Then, we used LINFLUX to calculate the unsteady aerodynamic forces for a given mode, frequency, and phase angle. A postprocessor read these unsteady pressures and calculated the generalized aerodynamic forces, eigenvalues, and response amplitudes. The eigenvalues determine the flutter frequency and damping. As a test case, the flutter of a helical fan was calculated with LINFLUX and compared with calculations from TURBO-AE, a nonlinear time domain code, and from ASTROP2, a code based on linear unsteady aerodynamics.

The eigenvalue problem in phase space.

PubMed

Cohen, Leon

2018-06-30

We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

A multilevel finite element method for Fredholm integral eigenvalue problems

NASA Astrophysics Data System (ADS)

Xie, Hehu; Zhou, Tao

2015-12-01

In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen-Loève expansions of random fields, which play an important role in the applications of uncertainty quantification. In our MFE framework, solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh. Then, any existing efficient integration scheme can be used for the associated integration process. The error estimates are provided, and the computational complexity is analyzed. It is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh. Several numerical experiments are presented to validate the efficiency of the proposed numerical method.

Random matrix theory for analyzing the brain functional network in attention deficit hyperactivity disorder

NASA Astrophysics Data System (ADS)

Wang, Rong; Wang, Li; Yang, Yong; Li, Jiajia; Wu, Ying; Lin, Pan

2016-11-01

Attention deficit hyperactivity disorder (ADHD) is the most common childhood neuropsychiatric disorder and affects approximately 6 -7 % of children worldwide. Here, we investigate the statistical properties of undirected and directed brain functional networks in ADHD patients based on random matrix theory (RMT), in which the undirected functional connectivity is constructed based on correlation coefficient and the directed functional connectivity is measured based on cross-correlation coefficient and mutual information. We first analyze the functional connectivity and the eigenvalues of the brain functional network. We find that ADHD patients have increased undirected functional connectivity, reflecting a higher degree of linear dependence between regions, and increased directed functional connectivity, indicating stronger causality and more transmission of information among brain regions. More importantly, we explore the randomness of the undirected and directed functional networks using RMT. We find that for ADHD patients, the undirected functional network is more orderly than that for normal subjects, which indicates an abnormal increase in undirected functional connectivity. In addition, we find that the directed functional networks are more random, which reveals greater disorder in causality and more chaotic information flow among brain regions in ADHD patients. Our results not only further confirm the efficacy of RMT in characterizing the intrinsic properties of brain functional networks but also provide insights into the possibilities RMT offers for improving clinical diagnoses and treatment evaluations for ADHD patients.

Method for computing self-consistent solution in a gun code

DOEpatents

Nelson, Eric M

2014-09-23

Complex gun code computations can be made to converge more quickly based on a selection of one or more relaxation parameters. An eigenvalue analysis is applied to error residuals to identify two error eigenvalues that are associated with respective error residuals. Relaxation values can be selected based on these eigenvalues so that error residuals associated with each can be alternately reduced in successive iterations. In some examples, relaxation values that would be unstable if used alone can be used.

PT-symmetric eigenvalues for homogeneous potentials

NASA Astrophysics Data System (ADS)

Eremenko, Alexandre; Gabrielov, Andrei

2018-05-01

We consider one-dimensional Schrödinger equations with potential x2M(ix)ɛ, where M ≥ 1 is an integer and ɛ is real, under appropriate parity and time (PT)-symmetric boundary conditions. We prove the phenomenon which was discovered by Bender and Boettcher by numerical computation: as ɛ changes, the real spectrum suddenly becomes non-real in the sense that all but finitely many eigenvalues become non-real. We find the limit arguments of these non-real eigenvalues E as E → ∞.

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.

KARHUNEN-LOÈVE Basis Functions of Kolmogorov Turbulence in the Sphere

NASA Astrophysics Data System (ADS)

Mathar, Richard J.

In support of modeling atmospheric turbulence, the statistically independent Karhunen-Loève modes of refractive indices with isotropic Kolmogorov spectrum of the covariance are calculated inside a sphere of fixed radius, rendered as series of 3D Zernike functions. Many of the symmetry arguments of the well-known associated 2D problem for the circular input pupil remain valid. The technique of efficient diagonalization of the eigenvalue problem in wavenumber space is founded on the Fourier representation of the 3D Zernike basis, and extensible to the von-Kármán power spectrum.

A control system design approach for flexible spacecraft

NASA Technical Reports Server (NTRS)

Silverberg, L. M.

1985-01-01

A control system design approach for flexible spacecraft is presented. The control system design is carried out in two steps. The first step consists of determining the ideal control system in terms of a desirable dynamic performance. The second step consists of designing a control system using a limited number of actuators that possess a dynamic performance that is close to the ideal dynamic performance. The effects of using a limited number of actuators is that the actual closed-loop eigenvalues differ from the ideal closed-loop eigenvalues. A method is presented to approximate the actual closed-loop eigenvalues so that the calculation of the actual closed-loop eigenvalues can be avoided. Depending on the application, it also may be desirable to apply the control forces as impulses. The effect of digitizing the control to produce the appropriate impulses is also examined.

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.

The Theory of Quantized Fields. III

DOE R&D Accomplishments Database

Schwinger, J.

1953-05-01

In this paper we discuss the electromagnetic field, as perturbed by a prescribed current. All quantities of physical interest in various situations, eigenvalues, eigenfunctions, and transformation probabilities, are derived from a general transformation function which is expressed in a non-Hermitian representation. The problems treated are: the determination of the energy-momentum eigenvalues and eigenfunctions for the isolated electromagnetic field, and the energy eigenvalues and eigenfunctions for the field perturbed by a time-independent current that departs from zero only within a finite time interval, and for a time-dependent current that assumes non-vanishing time-independent values initially and finally. The results are applied in a discussion of the intra-red catastrophe and of the adiabatic theorem. It is shown how the latter can be exploited to give a uniform formulation for all problems requiring the evaluation of transition probabilities or eigenvalue displacements.

Multigrid method for stability problems

NASA Technical Reports Server (NTRS)

Ta'asan, Shlomo

1988-01-01

The problem of calculating the stability of steady state solutions of differential equations is addressed. Leading eigenvalues of large matrices that arise from discretization are calculated, and an efficient multigrid method for solving these problems is presented. The resulting grid functions are used as initial approximations for appropriate eigenvalue problems. The method employs local relaxation on all levels together with a global change on the coarsest level only, which is designed to separate the different eigenfunctions as well as to update their corresponding eigenvalues. Coarsening is done using the FAS formulation in a nonstandard way in which the right-hand side of the coarse grid equations involves unknown parameters to be solved on the coarse grid. This leads to a new multigrid method for calculating the eigenvalues of symmetric problems. Numerical experiments with a model problem are presented which demonstrate the effectiveness of the method.

Development of a Probabilistic Component Mode Synthesis Method for the Analysis of Non-Deterministic Substructures

NASA Technical Reports Server (NTRS)

Brown, Andrew M.; Ferri, Aldo A.

1995-01-01

Standard methods of structural dynamic analysis assume that the structural characteristics are deterministic. Recognizing that these characteristics are actually statistical in nature, researchers have recently developed a variety of methods that use this information to determine probabilities of a desired response characteristic, such as natural frequency, without using expensive Monte Carlo simulations. One of the problems in these methods is correctly identifying the statistical properties of primitive variables such as geometry, stiffness, and mass. This paper presents a method where the measured dynamic properties of substructures are used instead as the random variables. The residual flexibility method of component mode synthesis is combined with the probabilistic methods to determine the cumulative distribution function of the system eigenvalues. A simple cantilever beam test problem is presented that illustrates the theory.

Weak Value Amplification is Suboptimal for Estimation and Detection

NASA Astrophysics Data System (ADS)

Ferrie, Christopher; Combes, Joshua

2014-01-01

We show by using statistically rigorous arguments that the technique of weak value amplification does not perform better than standard statistical techniques for the tasks of single parameter estimation and signal detection. Specifically, we prove that postselection, a necessary ingredient for weak value amplification, decreases estimation accuracy and, moreover, arranging for anomalously large weak values is a suboptimal strategy. In doing so, we explicitly provide the optimal estimator, which in turn allows us to identify the optimal experimental arrangement to be the one in which all outcomes have equal weak values (all as small as possible) and the initial state of the meter is the maximal eigenvalue of the square of the system observable. Finally, we give precise quantitative conditions for when weak measurement (measurements without postselection or anomalously large weak values) can mitigate the effect of uncharacterized technical noise in estimation.

[The principal components analysis--method to classify the statistical variables with applications in medicine].

PubMed

Dascălu, Cristina Gena; Antohe, Magda Ecaterina

2009-01-01

Based on the eigenvalues and the eigenvectors analysis, the principal component analysis has the purpose to identify the subspace of the main components from a set of parameters, which are enough to characterize the whole set of parameters. Interpreting the data for analysis as a cloud of points, we find through geometrical transformations the directions where the cloud's dispersion is maximal--the lines that pass through the cloud's center of weight and have a maximal density of points around them (by defining an appropriate criteria function and its minimization. This method can be successfully used in order to simplify the statistical analysis on questionnaires--because it helps us to select from a set of items only the most relevant ones, which cover the variations of the whole set of data. For instance, in the presented sample we started from a questionnaire with 28 items and, applying the principal component analysis we identified 7 principal components--or main items--fact that simplifies significantly the further data statistical analysis.

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.

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.

Eigenvalue Detonation of Combined Effects Aluminized Explosives

NASA Astrophysics Data System (ADS)

Capellos, C.; Baker, E. L.; Nicolich, S.; Balas, W.; Pincay, J.; Stiel, L. I.

2007-12-01

Theory and performance for recently developed combined—effects aluminized explosives are presented. Our recently developed combined-effects aluminized explosives (PAX-29C, PAX-30, PAX-42) are capable of achieving excellent metal pushing, as well as high blast energies. Metal pushing capability refers to the early volume expansion work produced during the first few volume expansions associated with cylinder and wall velocities and Gurney energies. Eigenvalue detonation explains the observed detonation states achieved by these combined effects explosives. Cylinder expansion data and thermochemical calculations (JAGUAR and CHEETAH) verify the eigenvalue detonation behavior.

NASA Astrophysics Data System (ADS)

2018-05-01

Eigenvalues and eigenvectors, together, constitute the eigenstructure of the system. The design of vibrating systems aimed at satisfying specifications on eigenvalues and eigenvectors, which is commonly known as eigenstructure assignment, has drawn increasing interest over the recent years. The most natural mathematical framework for such problems is constituted by the inverse eigenproblems, which consist in the determination of the system model that features a desired set of eigenvalues and eigenvectors. Although such a problem is intrinsically challenging, several solutions have been proposed in the literature. The approaches to eigenstructure assignment can be basically divided into passive control and active control.

Extension of the tridiagonal reduction (FEER) method for complex eigenvalue problems in NASTRAN

NASA Technical Reports Server (NTRS)

Newman, M.; Mann, F. I.

1978-01-01

As in the case of real eigenvalue analysis, the eigensolutions closest to a selected point in the eigenspectrum were extracted from a reduced, symmetric, tridiagonal eigenmatrix whose order was much lower than that of the full size problem. The reduction process was effected automatically, and thus avoided the arbitrary lumping of masses and other physical quantities at selected grid points. The statement of the algebraic eigenvalue problem admitted mass, damping, and stiffness matrices which were unrestricted in character, i.e., they might be real, symmetric or nonsymmetric, singular or nonsingular.

Multigrid calculation of three-dimensional viscous cascade flows

NASA Technical Reports Server (NTRS)

Arnone, A.; Liou, M.-S.; Povinelli, L. A.

1991-01-01

A 3-D code for viscous cascade flow prediction was developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full multigrid method. The Baldwin-Lomax eddy viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

Quantization with maximally degenerate Poisson brackets: the harmonic oscillator!

NASA Astrophysics Data System (ADS)

Nutku, Yavuz

2003-07-01

Nambu's construction of multi-linear brackets for super-integrable systems can be thought of as degenerate Poisson brackets with a maximal set of Casimirs in their kernel. By introducing privileged coordinates in phase space these degenerate Poisson brackets are brought to the form of Heisenberg's equations. We propose a definition for constructing quantum operators for classical functions, which enables us to turn the maximally degenerate Poisson brackets into operators. They pose a set of eigenvalue problems for a new state vector. The requirement of the single-valuedness of this eigenfunction leads to quantization. The example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super-integrable systems.

A Relaxation Method for Nonlocal and Non-Hermitian Operators

NASA Astrophysics Data System (ADS)

Lagaris, I. E.; Papageorgiou, D. G.; Braun, M.; Sofianos, S. A.

1996-06-01

We present a grid method to solve the time dependent Schrödinger equation (TDSE). It uses the Crank-Nicholson scheme to propagate the wavefunction forward in time and finite differences to approximate the derivative operators. The resulting sparse linear system is solved by the symmetric successive overrelaxation iterative technique. The method handles local and nonlocal interactions and Hamiltonians that correspond to either Hermitian or to non-Hermitian matrices with real eigenvalues. We test the method by solving the TDSE in the imaginary time domain, thus converting the time propagation to asymptotic relaxation. Benchmark problems solved are both in one and two dimensions, with local, nonlocal, Hermitian and non-Hermitian Hamiltonians.

Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity.

PubMed

Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M

2011-01-01

We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.

Linear static structural and vibration analysis on high-performance computers

NASA Technical Reports Server (NTRS)

Baddourah, M. A.; Storaasli, O. O.; Bostic, S. W.

1993-01-01

Parallel computers offer the oppurtunity to significantly reduce the computation time necessary to analyze large-scale aerospace structures. This paper presents algorithms developed for and implemented on massively-parallel computers hereafter referred to as Scalable High-Performance Computers (SHPC), for the most computationally intensive tasks involved in structural analysis, namely, generation and assembly of system matrices, solution of systems of equations and calculation of the eigenvalues and eigenvectors. Results on SHPC are presented for large-scale structural problems (i.e. models for High-Speed Civil Transport). The goal of this research is to develop a new, efficient technique which extends structural analysis to SHPC and makes large-scale structural analyses tractable.

An "adiabatic-hindered-rotor" treatment allows para-H(2) to be treated as if it were spherical.

PubMed

Li, Hui; Roy, Pierre-Nicholas; Le Roy, Robert J

2010-09-14

In para-H(2)-{molecule} interactions, the common assumption that para-H(2) may be treated as a spherical particle is often substantially in error. For example, quantum mechanical eigenvalues on a full four-dimensional (4D) potential energy surface for para H(2)-{linear molecule} species often differ substantially from those calculated from the corresponding two-dimensional (2D) surface obtained by performing a simple spherical average over the relative orientations of the H(2) moiety. However, use of an "adiabatic-hindered-rotor" approximation can yield an effective 2D surface whose spectroscopic properties are an order of magnitude closer to those yielded by a full 4D treatment.

Multigrid calculation of three-dimensional viscous cascade flows

NASA Technical Reports Server (NTRS)

Arnone, A.; Liou, M.-S.; Povinelli, L. A.

1991-01-01

A three-dimensional code for viscous cascade flow prediction has been developed. The space discretization uses a cell-centered scheme with eigenvalue scaling to weigh the artificial dissipation terms. Computational efficiency of a four-stage Runge-Kutta scheme is enhanced by using variable coefficients, implicit residual smoothing, and a full-multigrid method. The Baldwin-Lomax eddy-viscosity model is used for turbulence closure. A zonal, nonperiodic grid is used to minimize mesh distortion in and downstream of the throat region. Applications are presented for an annular vane with and without end wall contouring, and for a large-scale linear cascade. The calculation is validated by comparing with experiments and by studying grid dependency.

The damped wave equation with unbounded damping

NASA Astrophysics Data System (ADS)

Freitas, Pedro; Siegl, Petr; Tretter, Christiane

2018-06-01

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices

NASA Technical Reports Server (NTRS)

Freund, Roland W.; Gutknecht, Martin H.; Nachtigal, Noel M.

1991-01-01

The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation is presented of a look-ahead version of the Lanczos algorithm that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead.