Basis Of Eigenvectors Research Articles

Cardio-eigenoscopy: significance of this new method in prognosis of risks of fatal arrhythmia progression in AMI patients

The aim of the study was to evaluate the significance of cardio- eigenoscopy method for the prognosis of risks of fatal arrhythmia (the primary and secondary VF) progression in AMI individuals. The study was performed with the use of the PC-assisted hemodynamic analyzer Cardiocode. The cardio-eigenoscopy was developed as a methodology that is capable of presenting all ECG changes in the basis of eigenvectors of a covariance matrix of ECG amplitudes and providing an analysis of a spectrum of eigenvalues. The ECG analysis software is implemented as a real-time monitoring of a set of parameters that makes possible to evaluate the therapy effectiveness. The cardio-eigenoscopy method is capable of assessing the risk of progression of fatal arrhythmia events in MI patients. The major markers of the possible progression of fatal arrhythmia were identified. The cardio- eigenoscopy provided the maximum expressibility of an ECG curve at any specified EVE values.

Determination of spectral densities of dynamic characteristics of a nonlinear model of structure

A stochastic problem of the dynamics of a nonlinear model of structure is considered. The problem is solved with the use of an expansion of the solution in the truncated orthogonal basis of eigenvectors of the linear model. It is assumed that nonlinearities of the system do not result in a fundamental change in its dynamical behavior, but contribute a significant quantitative correction relative to the linear model. The nonlinear characteristics are statistically linearized. The coefficients of statistical linearization are formally interpreted as some variations in the corresponding elements of the stiffness matrix of the linear model. Spectral densities, displacement variances of the linearized system, as well as its eigenvalues and eigenvectors, are represented as a power series expansion in the above variations considering linear or square approximations. To find the coefficients in the expansion, the sensitivity theory is used. This allows one to form a set of algebraic transcendental equations in displacement variances, which is solved by the method of successive approximations. Then all the target spectral densities are found. The results are illustrated by examples.

Oscillations of networks: the role of soft nodes

To describe the flow of a miscible quantity on a network, we consider the graph wave equation where the standard continuous Laplacian is replaced by the graph Laplacian. The structure of the graph influences strongly the dynamics. Assuming the graph is forced and damped at specific nodes, we derive the amplitude equations using a basis of eigenvectors of the graph Laplacian. These lead us to introduce the notion of soft nodes. We give sufficient conditions for their existence in general graphs. They can cause several effects as we show on small graphs, for example the ineffectiveness of damping applied to them. Soft nodes may be of critical importance for complex physical networks and engineering networks like power grids.

Discrete Fourier transform of prime order: Eigenvectors with small support

We show how to construct an eigenvector basis of the discrete Fourier transform of odd prime order. The special feature of the new basis is that the basis vectors have small support.

Quantum correlations in different density-matrix representations of spin-1/2 open chain

We consider quantum correlations in a spin-1/2 open chain of $N$ nodes with the XY Hamiltonian using different bases for the density matrix representation and the initial state with a single polarized node. These bases of our choice are following: (i) the basis of eigenvectors of the fermion operators; this basis appears naturally through the Jordan-Wigner transformation (this representation of the density matrix is referred to as the $\beta$-representation), (ii) its Fourier representation ($c$-representation of the density matrix) and (iii) the basis of eigenvectors of the operators $I_{jz}$ (the $z$-projection of the $j$th spin, $j=1,...,N$). Although for the short chains (a few nodes) the qualitative behavior of the entanglement and the discord are very similar (the difference is quantitative), this is not valid for longer chains ($N\gtrsim 10$). In this case, there are qualitative and quantitative distinctions between the entanglement and the discord in all three cases. We underline three most important features: (i) the quantum discord is static in the $\beta$-representation, where the entanglement is identical to zero; (ii) in the $c$-representation, the concurrence may be non-zero only between the nearest neighbors (with a single exception), while the discord is nonzero between any two nodes; (iii) there is so-called "echo" in the evolution of the discord, which is not observed in the evolution of the concurrence. Using different bases, we may choose the preferable behavior of quantum correlations which allows a given quantum system to be more flexible in applications.

Real computation with least discrete advice: A complexity theory of nonuniform computability with applications to effective linear algebra

It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f:X∋x↦f(x)∈Y, additional structural information about the input x∈X (e.g. any kind of promise that x belongs to a certain subset X′⊆X, or does not) should be taken advantage of. In several examples from real number computation, such advice even makes the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable.Specifically, finding a nontrivial solution to a homogeneous linear equation A⋅x→=0 for a given singular real n×n-matrix A is possible when knowing rank(A)∈{0,1,…,n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors to) a given real symmetric n×n-matrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n-fold (i.e. roughly logn bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas for finding some single eigenvector of A, providing the truncated binary logarithm of the dimension of the least-dimensional eigenspace of A—i.e. ⌊1+log2n⌋-fold advice—is sufficient and optimal.Our proofs employ, in addition to topological considerations common in Recursive Analysis, also combinatorial arguments.

Marked drag reduction in non-affine viscoelastic turbulence in homogeneous isotropic and pipe flows

Effect of non-affinity of the molecular motions to the macroscopic deformation in the polymer-diluted flow on turbulent drag reduction (DR) is studied using the DNS data for homogeneous isotropic turbulence and pipe flow. The polymer stress is obtained by solving the non-affine Johnson-Segalman constitutive equation. In both flows, DR is maximal when non-affinity is either minimum or maximum, but the largest reduction is achieved when non-affinity is maximum. As an extreme case, in pipe flow, the mean velocity profile exceeds the Virk's maximum DR limit and almost complete relaminarization of turbulent state is achieved. The normal-stress difference (NSD) is obtained on the basis of new eigenvectors which span the isosurfaces of vortex tube and sheet. It is shown that the first NSD is predominantly positive, while the second NSD is negative along the sheets and tubes. Thus, an extra tension is exerted on the sheet and tube. With an increase of effective viscosity by an addition of elongation viscosity, resistance of the sheet and tube to their stretching is enhanced. The principal mechanism for DR when non-affinity is maximum is that the transformation of the sheet into the tube is restrained because the sheet tends to snap back to the original flat form. When non-affinity is minimum, the tubes are created but its stretching is suppressed by annihilation of lowering of the pressure in the tube-core region. In both cases, cascade of the energy into the small scales is diminished leading to the reduction of drag.

4D Cone‐beam CT reconstruction using a motion model based on principal component analysis

To provide a proof of concept validation of a novel 4D cone-beam CT (4DCBCT) reconstruction algorithm and to determine the best methods to train and optimize the algorithm. The algorithm animates a patient fan-beam CT (FBCT) with a patient specific parametric motion model in order to generate a time series of deformed CTs (the reconstructed 4DCBCT) that track the motion of the patient anatomy on a voxel by voxel scale. The motion model is constrained by requiring that projections cast through the deformed CT time series match the projections of the raw patient 4DCBCT. The motion model uses a basis of eigenvectors that are generated via principal component analysis (PCA) of a training set of displacement vector fields (DVFs) that approximate patient motion. The eigenvectors are weighted by a parameterized function of the patient breathing trace recorded during 4DCBCT. The algorithm is demonstrated and tested via numerical simulation. The algorithm is shown to produce accurate reconstruction results for the most complicated simulated motion, in which voxels move with a pseudo-periodic pattern and relative phase shifts exist between voxels. The tests show that principal component eigenvectors trained on DVFs from a novel 2D/3D registration method give substantially better results than eigenvectors trained on DVFs obtained by conventionally registering 4DCBCT phases reconstructed via filtered backprojection. Proof of concept testing has validated the 4DCBCT reconstruction approach for the types of simulated data considered. In addition, the authors found the 2D/3D registration approach to be our best choice for generating the DVF training set, and the Nelder-Mead simplex algorithm the most robust optimization routine.

Frame diagonalization of matrices

This paper introduces a concept of diagonalization that uses not a basis of eigenvectors, but a frame. A frame is a generalization of a basis which is used in a number of signal and image processing applications. We first investigate the properties of frame diagonalization, drawing parallels with those of basis diagonalization. We then describe several methods of constructing frames for frame diagonalization. In particular, we prove the existence of a universal diagonalizer for each n∈N that simultaneously diagonalizes all matrices in Mn(C), and create a method of frame diagonalization that works for any matrix in Mn(C), uses at most ⌊3n/2⌋ frame vectors and retains information about the eigenvalues of the matrix.

Technologies for Pattern Recognition of Late Potentials atria: classification approaches

In this paper the problem of atrial late potentials (ALP) recognition is solved by the formation of their signs in the basis of eigenvectors. Covariance matrix is formed for an ensemble of wavelet detail coefficients obtained by multi-level wavelet decomposition of electrocardiosignals (ECS). The results of cluster analysis in a model experiment on the classification of ECS with and without ALP on the background noise are presented.

Periodic Ising Correlations

In this paper, we first rework B. Kaufman's 1949 paper [Phys. Rev. 76, 1232 (1949)] by using representation theory. Our approach leads to a simpler and more direct way of deriving the spectrum of the transfer matrix for the finite periodic Ising model. We then determine formulas for the spin correlation functions that depend on the matrix elements of the induced rotation associated with the spin operator in a basis of eigenvectors for the transfer matrix. The representation of the spin matrix elements is obtained by considering the spin operator as an intertwining map. We exhibit the “new” elements \documentclass[12pt]{minimal}\begin{document}$V_{+}$\end{document}V+ and \documentclass[12pt]{minimal}\begin{document}$V_{-}$\end{document}V− in the Bugrij–Lisovyy formula [Phys. Lett. A 319, 390 (2003)] as part of a holomorphic factorization of the periodic and antiperiodic summability kernels on the spectral curve associated with the induced rotation for the transfer matrix.

IONIZATION EQUILIBRIUM TIMESCALES IN COLLISIONAL PLASMAS

Astrophysical shocks or bursts from a photoionizing source can disturb the typical collisional plasma found in galactic interstellar media or the intergalactic medium. The spectrum emitted by this plasma contains diagnostics that have been used to determine the time since the disturbing event, although this determination becomes uncertain as the elements in the plasma return to ionization equilibrium. A general solution for the equilibrium timescale for each element arises from the elegant eigenvector method of solution to the problem of a non-equilibrium plasma described by Masai (1984) and Hughes & Helfand (1985). In general the ionization evolution of an element Z in a constant electron temperature plasma is given by a coupled set of Z+1 first order differential equations. However, they can be recast as Z uncoupled first order differential equations using an eigenvector basis for the system. The solution is then Z separate exponential functions, with the time constants given by the eigenvalues of the rate matrix. The smallest of these eigenvalues gives the scale of slowest return to equilibrium independent of the initial conditions, while conversely the largest eigenvalue is the scale of the fastest change in the ion population. These results hold for an ionizing plasma, a recombining plasma, or even a plasma with random initial conditions, and will allow users of these diagnostics to determine directly if their best-fit result significantly limits the timescale since a disturbance or is so close to equilibrium as to include an arbitrarily-long time.

The Exact Controllability Property of Neutral Type Systems by the Moment Problem Approach Revisited

In a recent paper authors gave an analysis of the exact controllability problem via the moment problem approach. Namely, the steering conditions of controllable states are formulated as a vectorial moment problem using some Riesz basis. One of the main difficulties was the choice of the basis as, in general, a basis of eigenvectors does not exist. In this contribution we use a change of control by a feedback law and modify the structure of the system in such a way that there exists a basis of eigenvectors which allows a simpler expression of the moment problem. Hence, one obtains the result on exact controllability and on the time of exact controllability.

New enumeration of Walsh matrices

The discrete Walsh transform is a linear transform defined by a Walsh matrix. Three ways to construct Walsh matrices are known, which differ by the sequence order of rows and correspond to the Paley, Walsh, and Hadamard enumerations. We propose a new enumeration of Walsh matrices and study its properties. The new enumeration is constructed as a linear rearrangement; we obtain an eigenvector basis for it and propose a convenient-to-generate fast implementation algorithm; the new enumeration possesses certain symmetry properties, which make it similar to the discrete Fourier transform.

Admissibility and controllability of diagonal Volterra equations with scalar inputs

This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory.

A numerical method for a blood-artery interaction problem

We present a numerical method for the resolution of a bidimensional blood flow problem and more generally for a fluid flow surrounded by a time dependent domain. Our approach is based on an ALE formulation which is solved using a Galerkin method with an eigenvectors basis set on the initial fixed domain.

Direct evidence of overdamped Peierls-coupled modes in the temperature-induced phase transition in tetrathiafulvalene-chloranil

In this paper, we elucidate the optical response resulting from the interplay of charge distribution (ionicity) and Peierls instability (dimerization) in the neutral-ionic ferroelectric phase transition of tetrathiafulvalene-chloranil, a mixed-stack quasi-one-dimensional charge-transfer crystal. We present far-infrared reflectivity measurements down to $5\phantom{\rule{0.3em}{0ex}}{\text{cm}}^{\ensuremath{-}1}$ as a function of temperature above the phase transition (300--82 K). The coupling between electrons and lattice phonons in the pretransitional regime is analyzed on the basis of phonon eigenvectors obtained through quasiharmonic lattice dynamics, combined with the results of exact numerical diagonalization of the one-dimensional Peierls-Hubbard model. The resulting multiphonon Peierls coupling leads upon approaching the phase transition to a progressive shift of spectral weight and of the coupling strength toward the phonons at lower frequencies, ending in a soft-mode behavior only for the lowest-frequency phonon near the transition temperature. Moreover, in the proximity of the phase transition, the lowest-frequency phonon becomes overdamped due to anharmonicity induced by its coupling to electrons. The implications of these findings for the neutral-ionic transition mechanism are discussed.

On a special basis of approximate eigenvectors with local supports for an isolated narrow cluster of eigenvalues of a symmetric tridiagonal matrix

Let \( \tilde \lambda \) be an approximate eigenvalue of multiplicity mc = n − r of an n × n real symmetric tridiagonal matrix T having nonzero off-diagonal entries. A fast algorithm is proposed (and numerically tested) for deleting mc rows of T−\( \tilde \lambda \)I so that the condition number of the r × n matrix B formed of the remaining r rows is as small as possible. A special basis of mc vectors with local supports is constructed for the subspace kerB. These vectors are approximate eigenvectors of T corresponding to \( \tilde \lambda \). Another method for deleting mc rows of T−\( \tilde \lambda \)I is also proposed. This method uses a rank-revealing QR decomposition; however, it requires a considerably larger number of arithmetic operations. For the latter algorithm, the condition number of B is estimated, and orthogonality estimates for vectors of the special basis of kerB are derived.

2D, Quasi 3D and 3D Exact Solutions for Bending of Thick and Thin Sandwich Plates

A previous author's work related to the exact solution of a single layer isotropic plate is extended to the case of multilayered plate structures composed of isotropic layers. The method solves a first order linear system of differential equations in the unknown amplitudes of the displacements and stresses. An eigenvalue problem, in which analytical expressions for the basis of eigenvectors and generalized eigenvectors are available, is then formulated. This makes the coding of exact solution of multilayered structures very simple and effective compared with other exact methods present in the literature. The paper is mainly concerned about the analysis of sandwich structures made of isotropic layers. Challenging cases of thick and moderately thick plates (a/h = 1, 4, 10) and thin plates (a/h = 100) are analyzed in detail and the displacements and stresses are found and plotted through the thickness of the plate. An extensive study of the ratio between the elastic modulus of the skins and the elastic modulus of the core, as well as the change in the displacements and stresses with that parameter, is conducted. The exact three-dimensional results are also validated with quasi-3D results obtained by adopting a mixed assiomatic theory of order 9 presented here for the first time. Twenty three two-dimensional theories are assessed, five of which are presented here for the first time. The use of Murakami's zig—zag function and the consequent improvement of the ESL theories is also analyzed and discussed. All the results of this work are an useful tool to test and compare approximated methods, such as Finite Element Method.

On discrete Gauss–Hermite functions and eigenvectors of the discrete Fourier transform

The problem of furnishing an orthogonal basis of eigenvectors for the discrete Fourier transform (DFT) is fundamental to signal processing and also a key step in the recent development of discrete fractional Fourier transforms with projected applications in data multiplexing, compression, and hiding. Existing solutions toward furnishing this basis of DFT eigenvectors are based on the commuting matrix framework. However, none of the existing approaches are able to furnish a commuting matrix where both the eigenvalue spectrum and the eigenvectors are a close match to corresponding properties of the continuous differential Gauss–Hermite (G–H) operator. Furthermore, any linear combination of commuting matrices produced by existing approaches also commutes with the DFT, thereby bringing up issues of uniqueness. In this paper, inspired by concepts from quantum mechanics in finite dimensions, we present an approach that furnishes a basis of orthogonal eigenvectors for both versions of the DFT. This approach furnishes a commuting matrix whose eigenvalue spectrum is a very close approximation to that of the G–H differential operator and in the process furnishes two generators of the group of matrices that commute with the DFT.