Eigenvalues Of Magnitude Research Articles

Floquet Theory for Linear Time-Periodic Delay Differential Equations Using Orthonormal History Functions

Abstract In the usual approach to determining the stability of a time-periodic delay differential equation (DDE), the DDE is converted into an approximate system of time-periodic ordinary differential equations (ODEs) using Galerkin approximations. Later, Floquet theory is applied to these ODEs. Alternatively, semidiscretization-like approaches can be used to construct an approximate Floquet transition matrix (FTM) for a DDE. In this paper, we develop a method to obtain the FTM directly. Our approach is analogous to the Floquet theory for ODEs: we consider one polynomial basis function at a time as the history function and stack the coefficients of the corresponding DDE solutions to construct the FTM. The largest magnitude eigenvalue of the FTM determines the stability of the DDE. Since the obtained FTM is an approximation of the actual infinite-dimensional FTM, the criterion developed for stability is approximate. We demonstrate the correctness, efficacy and convergence of our method by studying several candidate DDEs with time-periodic parameters and/or delays, and comparing the results with those obtained from the Galerkin approximations.

On a Maximum Eigenvalue of Third-Order Pairwise Comparison Matrix in Analytic Hierarchy Process and Convergence of Newton\u2019s Method

Nowadays, the analytic hierarchy process is an established method of multiple criteria decision making in the field of Operations Research. Pairwise comparison matrix plays a crucial role in the analytic hierarchy process. The principal (maximum magnitude) eigenvalue of the pairwise comparison matrix can be utilized for measuring the consistency of the decision maker’s judgment. The simple transformation of the maximum magnitude eigenvalue is known to be Saaty’s consistency index. In this short note, we shed light on the characteristic polynomial of a pairwise comparison matrix of third order. We will show that the only real-number root of the characteristic equation is the maximum magnitude eigenvalue of the third-order pairwise comparison matrix. The unique real-number root appears in the area where it is greater than 3, which is equal to the order of the matrix. By applying usual Newton’s method to the characteristic polynomial of the third-order pairwise comparison matrix, we see that the sequence generated from the initial value of 3 always converges to the maximum magnitude eigenvalue.

Non-contrast-enhanced ultrafast ultrasound Doppler imaging of spinal cord micro-vessels

With the advantages of non-radiation and low-cost, ultrasound imaging has been widely used in clinical diagnosis. However, due to the compromise between temporal and spatial resolution, the traditional ultrasound imaging method which collects images line by line fails to obtain the images at a high frame rate. Ultrafast ultrasound imaging method based on the plane-wave emission can achieve a high frame rate with the capability for instantaneous hemodynamic imaging of neurovascular response. Besides, by the coherent compounding of the echo signals received from emitting a set of tilting plane-waves, the image quality can be improved in terms of resolution, contrast and signal-to-noise ratio (SNR). Considering the fact that received signals are the mixture of echo signals from the low-speed soft tissue and high-speed blood flow, a clutter filtering method is used to remove the signals of soft tissue. In this study, the ultrafast ultrasound sequence of emission and reception is designed to image the spinal cord by using the groups of multiple steering-angle plane-waves. The so-called f-k migration algorithm based on an exploding reflector model (ERM) is used for coherent plane-wave compounding. Eigenvalue decomposition (EVD) is then applied to tissue and blood signal extraction. The static tissue signals correspond to eigenvectors with low Doppler frequency shift and large magnitude eigenvalue while the high-speed blood flow signals correspond to eigenvectors with high Doppler frequency shift and small eigenvalue. Therefore, frequency and amplitude thresholds can be applied to the accurate separation of the blood component and the tissue component. After the signal extraction, the hemodynamic imaging of blood vessels can thus be obtained from the power Doppler results. The experiments are carried out by using a programmable ultrasonic array system and a high-frequency linear array transducer L22-14vX with a central frequency of 15.625 MHz. The sample acquisition frequency is set to be 62.5 MHz. In rat experiments <i>in vivo</i>, 14040 angle-stilting images per second are compounded into 520 images, each of which is compounded from 27 tilting images (the tilting angles ranging from –10° to 10°). The experiments are conducted on the anesthetized rats with part of the vertebral plate removed. The ultrafast B-mode images are acquired from a 14.0 mm ×12.7 mm region-of-interest. The 520 frames acquired in one second are processed by the clutter filter based on eigenvalue decomposition and Doppler shift analysis. The eigenvectors and eigenvalues corresponding to the soft tissue are discarded. The power Doppler images of the spinal cord micro-vessels are obtained. The experimental results indicate that the ultrafast ultrasound Doppler imaging method is effective for monitoring the hemodynamic variation in spinal cord. The micro-hemorrhage can be identified from the power Doppler images. The quantitative results indicate that the SNR increases with the steering angles increasing. Compared with the results obtained by using 9 steering angles, 5 dB SNR enhancement can be obtained by using the 27 steering angles. In conclusion, the ultrafast ultrasound Doppler technology has the potential applications in spinal cord microvascular imaging and hemodynamic evaluation of neurovascular function of spinal cord.

Uniform exponential growth in twisted polynomial algebras

We consider the growth of a twisted polynomial algebra. We prove that the algebra has polynomially bounded growth if the twisting endomorphism has all eigenvalues of magnitude 0 or 1, and the algebra has uniform exponential growth otherwise.

Asymptotics of the eigenvalues for exponentially parameterized pentadiagonal matrices

SummaryLet P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit or two limits. Also, for n − 2 largest magnitude eigenvalues of P(t) we compute the asymptotics as n → ∞, in addition to t. When P(t) is also symmetric, these formulas allow us to compute the asymptotics of the 2‐norm condition number. The number of arithmetic operations involved, does not depend on n. We illustrate our results by some numerical tests.

Double, borderline, and extraordinary eigenvalues of Kac–Murdock–Szegő matrices with a complex parameter

For all sufficiently large complex ρ, and for arbitrary matrix dimension n, it is shown that the Kac–Murdock–Szegő matrix Kn(ρ)=[ρ|j−k|]j,k=1n possesses exactly two eigenvalues whose magnitude is larger than n. We discuss a number of properties of the two “extraordinary” eigenvalues. Conditions are developed that, given n, allow us—without actually computing eigenvalues—to find all values ρ that give rise to eigenvalues of magnitude n, termed “borderline” eigenvalues. The aforementioned values of ρ form two closed curves in the complex-ρ plane. We describe these curves, which are n-dependent, in detail. An interesting borderline case arises when an eigenvalue of Kn(ρ) equals −n: apart from certain exceptional cases, this occurs if and only if the eigenvalue is a double one; and if and only if the point ρ is a cusp-like singularity of one of the two closed curves.

Structural optimization considering smallest magnitude eigenvalues: a smooth approximation

An issue that frequently arises in structural optimization problems considering eigenvalues is the non differentiability of repeated eigenvalues. In order to overcome this difficulty, several schemes were already presented in the literature. However, these approaches generally have other disadvantages such as the inclusion of additional constraints, the inaccuracy of representation of smallest/largest eigenvalues, the significant increase in the computational effort required and incompatibility with finite differences schemes. In this paper a smooth p-norm approximation for the smallest magnitude eigenvalue is employed. The resulting approximation is differentiable, converges to the exact value as p is increased and is very simple to use (it is also compatible with finite difference schemes). Although the use of smooth approximations for maximum/minimum operators is a classical approach, for some reason it was not extensively studied in the context of structural optimization considering eigenvalues. Three examples concerning topology optimization for the maximization of the first natural vibration frequency of plane stress structures are presented in order to show the effectiveness of the proposed approach.

An iterative method for computing a symplectic SVD-like decomposition

In this paper, we present an effective iterative method for computing symplectic SVD-like decomposition for a 2n-by-m rectangular real matrix A. The main purpose here is a block-power iterative method with the ortho-symplectic SR decomposition in the normalization step. We compute an k-block symplectic SVD-like decomposition, namely \(A_k=S_{k} \Sigma _{k} V_{k}^{T}\) where \(S_{k}\in \mathbb {R}^{2n\times 2k}\) is symplectic and \(V_{k}\in \mathbb {R}^{m\times 2k}\) is orthogonal. For large matrices, we usually have to compute only part of eigenvalues. The main interest of the proposed method is to efficiently compute the desired number of ordered in magnitude eigenvalues of structured matrices.

A New Measure of Wireless Network Connectivity

Despite intensive research in the area of network connectivity, there is an important category of problems that remain unsolved: how to characterize and measure the quality of connectivity of a wireless network which has a realistic number of nodes, not necessarily large enough to warrant the use of asymptotic analysis, and which has unreliable connections, reflecting the inherent unreliability of wireless communications? The quality of connectivity measures how easily and reliably a packet sent by a node can reach another node. It complements the use of capacity to measure the quality of a network in saturated traffic scenarios and provides an intuitive measure of the quality of (end-to-end) network connections. In this paper, we introduce a probabilistic connectivity matrix as a tool to measure the quality of network connectivity. Some interesting properties of the probabilistic connectivity matrix and their connections to the quality of connectivity are demonstrated. We demonstrate that the largest magnitude eigenvalue of the probabilistic connectivity matrix, which is positive, can serve as a good measure of the quality of network connectivity. We provide a flooding algorithm whereby the nodes repeatedly flood the network with packets, and by measuring just the number of packets a given node receives, the node is able to asymptotically estimate this largest eigenvalue.

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

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 and 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.

Eigenvalue cluster traces for quantum graphs with equal edge lengths

On a finite metric graph with standard vertex conditions and equal edge lengths, the large magnitude eigenvalues of the Schrodinger op- erator ∆ + q cluster near the eigenvalues of the Laplace operator ∆. Based on the spectral 'periodicity' of ∆, the clusters can be parti- tioned into a finite collection of classes. There is a class dependent formula for the cluster trace (or average eigenvalue shift) in the large magnitude limit which expresses the trace as a function of the edge integrals e q and data from the underlying combinatorial graph.

State-space methods for calculating concentration dynamics in multizone buildings

This work explores analytic solutions to contaminant transport in multizone systems as a means of solving concentration dynamics more rapidly and providing insight into system behaviour. A general formulation is developed that is consistent with state-space theory. This is used to provide general and specific solutions for the concentration time series. In particular, an analytical expression is presented for the case of constant system parameters and constant input conditions using the eigenvalues and eigenvectors for the case of a diagonalisable state matrix for the multizone system. A method for using this expression to solve concentrations over a series of varying building and input states is developed, with potential for use as a rapid means of calculating concentrations at any time. In principle, this method could be used as a complementary method for solving concentration dynamics within multizone software. Analytical expressions for cumulative exposure for the same case are also presented. The characteristic behaviour of the solutions and their dependence on the eigenvalues and eigenvectors of the state matrix are explored for the steady state case and the decay solution. In particular, the behaviour in the late decay phase is shown to be characterised by a single decay rate, given by the smallest magnitude eigenvalue, and a fixed concentration ratio, given by the associated eigenvector. The independence of the concentration ratio in this phase to the initial conditions is also demonstrated. Two example cases are used to illustrate these and additional features.

Can there be a convergent perturbation theory of intermolecular energies?

We investigate the relationship between the convergence of Rayleigh-Schrödinger perturbation theory and the degree of correspondence between the eigenvalues and eigenfunctions of a Hamiltonian Ȟ and those of a zero-order Hamiltonian Ȟ0. We show that the radius of convergence of the perturbation expansion that transforms Ψ01, the lowest energy eigenfunction of Ȟ0, into Ψj, the jth lowest eigenfunction of Ȟ, is less than the magnitude of the reciprocal of the largest magnitude eigenvalue of a particular Hermitian operator. The operator depends on Ȟ and Ȟ0. This result is used to derive a sufficient condition for the perturbation expansion to have a radius of convergence less than one when Ψ01 is a good approximation to Ψj. When we apply the condition to our exchange perturbation theory calculations on HeH+, we find that the expansion must have a radius of convergence less than one. For this system, Ψ01 is a very good approximation to Ψ2. A radius of convergence less than one for the HeH+ calculations implies that the expansions diverge or that they are converging to a state other than Ψ2. We briefly consider the implications of our studies for the existence of a convergent perturbation theory of intermolecular energies.

A measure of centrality based on modularity matrix

In this paper, a kind of measure of structural centrality for networks, called modularity centrality, is introduced. This centrality index is based on the eigenvector belonging to the largest magnitude eigenvalue of modularity matrix. The measure is illustrated and compared with the standard centrality measures using a classic dataset. The statistical distribution of modularity centrality is investigated by considering large computer generated graphs and two networks from the real world.

Quasisaddles of liquids: Computational study of a bulk Lennard-Jones system

Inherent saddles of the potential energy surface, U, of a liquid are defined as configurations which correspond to the absolute minima of the pseudopotential surface, W=|∇U|2. Given finite numerical precision, multidimensional minimization procedures will sample both absolute and low-lying minima which are referred to collectively as quasisaddles. The sensitivity of statistical properties of these quasisaddles to the convergence criteria of the minimization procedure is investigated using, as a test system, a simple liquid bound by a quadratically shifted Lennard-Jones pair potential. The variation in statistical properties of quasisaddles is studied over a range of error tolerances spanning five orders of magnitude. Based on our results, it is clear that there are no qualitative changes in statistical properties of saddles over this range of error tolerances and even the quantitative changes are small. The results also show that it is not possible to set up an unambiguous numerical criterion to classify the quasisaddles into true saddles which contain no zero curvature, nontranslational normal modes, and inflexion points which have one or more zero-curvature normal mode directions.

Worst-case analysis and design of sampled-data control systems

Sampled-data systems with continuous-time inputs and outputs, where some state variables evolve in continuous time and others evolve in discrete time, are considered. Methods are presented for computing and optimizing the L/sub 2/-induced norm of such systems, considered as operators relating square integral signals. These worst-case analysis and design results are counterparts of the H/sub infinity /-norm analysis and synthesis for purely continuous- or discrete-time systems. The analysis shows that the L/sub 2/-induced norm of a sampled-data system is smaller than gamma if certain discrete-time descriptor systems have no eigenvalue of magnitude one, and if a certain discrete-time linear time-invariant plant depending on gamma has H/sub infinity / norm smaller than gamma . The synthesis shows that the optimal attenuation problem for a four-block continuous-time plant with a digital controller can be solved with a gamma -iteration on a certain discrete-time four-block plant which depends on gamma .< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Karhunen–Loeve method for data compression and speech synthesis

The use of the Karhunen–Loeve (KL) method in speech data compression and synthesis using the Fourier-Bessel (FB) expansion coefficient of speech signal is described. Bessel functions seem to make a natural basis for speech signal decomposition. Sinusoidal functions are the eigenfunctions of vibrating strings. Bessel functions are the eigenfunctions of vibrating pipes. The vocal tract resembles an excited pipe rather than a vibrating string. Good quality intelligible speech signal can be reconstructed using only a small portion of the FB expansion coefficient. Further data compression is possible through KL transformation of the speech signal FB expansion coefficient for efficient speech coding and synthesis. The transformation is implemented by first forming a covariance matrix of the FB coefficients. Eigenvalues and eigenvectors of the covariance matrix are computed and ranked according to the eigenvalue magnitude. Speech signal is then reconstructed using only the feature corresponding to the larger magnitude eigenvalues of the covariance matrix.

Improved Modelling of Plant Cell Suspension Cultures by Optimum Experiment Design

A simple mathematical model for the aerobic growth heterotrophic growth of cells in bioreactors is pointed out. The model is characterized by a switching function that separates between substrate limitation and oxygen limitation conditions.Simulations are shown using data from plant cell suspension cultures.During parameter estimation some problems arise: Some of the unknown parameters have an effect on the output variables only for a short fermentation period or are linearly dependent during certain fermentation states. In conjunction with the measurement errors this leads to considerable uncertainties in the parameter estimation. These uncertainties can be reduced by a suitable choice of the experimental conditions.An approach is used to calculate optimum input functions by minimizing the inverse of the minimum magnitude eigenvalue of the Fisher information matrix. A simulation with optimized input functions in this respect is shown.