Equal Eigenvalues Research Articles

Unified Krylov–Bogoliubov–Mitropolskii method under a critical condition

A modified and compact form of Krylov–Bogoliubov–Mitropolskii (KBM) unified method is extended to obtain approximate solution of an nth order, n=2,3,…, ordinary differential equation with small nonlinearities when unperturbed equation has some repeated real eigenvalues. The existing unified method is used when the eigenvalues are distinct whether they are purely imaginary or complex or real. The new form is presented generalizing all the previous formulae derived individually for second-, third- and fourth-order equations to obtain undamped, damped, over-damped and critically damped solutions. Therefore, all types of oscillatory and non-oscillatory solutions are determined by suitable substitution of the eigenvalues in a general result. The formulation of the method is very simple and the determination of the solution is easy. The method is illustrated by an example of a fourth-order equation when unperturbed equation has two real and equal eigenvalues. The solution agrees with a numerical solution nicely. Moreover, this solution is useful when the differences between conjugate eigenvalues (real or complex) are small. Thus the method is a complement of the existing modified and compact form of KBM method.

Computation of the exponential and other isotropic tensor functions and their derivatives

In the present paper we focus on numerical aspects of the computation of isotropic tensor functions and their derivative. In the general case of non-symmetric tensor arguments only two numerical algorithms appear to be appropriate. The first one represents a recurrent procedure resulting from the Taylor power series expansion of an isotropic tensor function. The second algorithm is based on a recently proposed closed-form representation which can be obtained from the definition of an isotropic tensor function either by the tensor power series or by the Dunford–Taylor integral. To improve the accuracy in the case of nearly equal eigenvalues a series expansion of this closed formula is proposed. Both algorithms are finally illustrated by an example of the exponential tensor function where emphasis is placed on the precision issue.

Detection-estimation of more uncorrelated gaussian sources than sensors in nonuniform linear antenna arrays-part II: partially augmentable arrays

Previous studies dealing with direction-of-arrival (DOA) estimation for uncorrelated planes waves incident on nonuniform M-sensor arrays assumed that the number of signal sources m was known or had already been estimated. In the case (m<M), traditional detection techniques such as Akaike's information criterion (AIC) and minimum description length (MDL) that are based on the equality of several smallest eigenvalues in the covariance matrix may be applied, although we demonstrate that these results can be misleading for nonuniform arrays. In the case (m/spl ges/M), these standard techniques are not applicable. We introduce a new approach to the detection problem for fully augmentable arrays (whose set of intersensor differences is complete). We show that the well-known direct augmentation approach applied to the sample covariance matrix is not a solution by itself since the resulting Toeplitz matrix is generally not positive definite for realistic sample volumes. We propose a transformation of this augmented matrix into a positive definite Toeplitz matrix T/sub /spl mu// with the proper number of equal minimum eigenvalues that are appropriate for the candidate number of sources /spl mu/. Comparison of the results of these best-fit transformations over the permissible range of candidates then allows us to select the most likely number of sources m/spl circ/ using traditional criteria and yields uniquely defined DOAs. Simulation results demonstrate the high performance of this method. Since detection techniques for superior scenarios have not been previously described in the literature, we compare our method with the standard AIC and MDL techniques in a conventional case with similar Cramer-Rao bound (CRB) and find that it has a similar detection performance.

Propagation of a relativistic particle in terms of the unitary irreducible representations of the Lorentz group

In a generalized Heisenberg/Schroedinger picture we use an invariant space-time transformation to describe the motion of a relativistic particle. We discuss the relation with the relativistic mechanics and find that the propagation of the particle may be defined as space-time transition between states with equal eigenvalues of the first and second Casimir operators of the Lorentz algebra. In addition we use a vector on the light-cone. A massive relativistic particle with spin 0 is considered. We also consider the nonrelativistic limit.

Detection-estimation of more uncorrelated Gaussian sources than sensors in nonuniform linear antenna arrays .I. Fully augmentable arrays

Previous studies dealing with direction-of-arrival (DOA) estimation for uncorrelated planes waves incident on nonuniform M-sensor arrays assumed that the number of signal sources m was known or had already been estimated. In the "conventional" case (m<M), traditional detection techniques such as Akaike's information criterion (AIC) and minimum description length (MDL) that are based on the equality of several smallest eigenvalues in the covariance matrix may be applied, although we demonstrate that these results can be misleading for nonuniform arrays. In the "superior" case (m/spl ges/M), these standard techniques are not applicable. We introduce a new approach to the detection problem for "fully augmentable" arrays (whose set of intersensor differences is complete). We show that the well-known direct augmentation approach applied to the sample covariance matrix is not a solution by itself since the resulting Toeplitz matrix is generally not positive definite for realistic sample volumes. We propose a transformation of this augmented matrix into a positive definite Toeplitz matrix T/sub /spl mu// with the proper number of equal minimum eigenvalues that are appropriate for the candidate number of sources /spl mu/. Comparison of the results of these best-fit transformations over the permissible range of candidates then allows us to select the most likely number of sources m/spl circ/ using traditional criteria and yields uniquely defined DOAs. Simulation results demonstrate the high performance of this method. Since detection techniques for superior scenarios have not been previously described in the literature, we compare our method with the standard AIC and MDL techniques in a conventional case with similar Cramer-Rao bound (CRB) and find that it has a similar detection performance.

Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill's family of generalized strain tensors

AbstractThe paper discusses algorithms for the computation of stresses and elasticity moduli for stored energy functions which are given in terms of a Seth–Hill‐type generalized strain measure. The key contribution is distinct computational representations of a chain rule representation for the stresses and moduli which may serve as a basic tool for an effective numerical implementation of complex elasticity models. The representations are formulated in the Lagrangian geometric setting based on a spectral decomposition of the right Cauchy–Green tensor and contain particular representations for the case of equal eigenvalues. Two algorithmic boxes are pointed out for three‐ and two‐dimensional problems, respectively. Furthermore, a distinct algorithmic setting is developed for applications of the Seth–Hill‐type measures to the isochoric part of the deformation based on a spectral decomposition of the unimodular part of the right Cauchy–Green tensor. The computation of the stress response in connection with generalized Hookean laws is considered as a simple application. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Determination of the optimal number of expansion terms in the HELS method

The Helmholtz equation least-square (HELS) method has been proven to be an efficient way of reconstructing acoustic radiation from a vibrating structure [Wu, J. Acoust. Soc. Am. 107, 2511–2522 (2000)]. In this method, the optimal number of expansion terms is determined by a heuristic method of searching the minimum–norm errors of the reconstructed acoustic pressures with respect to the measured data. While this approach works, it is time consuming. In this paper, a new method of a priori determination of the optimal number of expansions is developed based on the eigenvalue analysis of the measured information. Assuming a Gaussian white measurement noise, the hypothesis of equal noise eigenvalues is tested based on the statistical properties of the eigenvalues. This hypothesis test yields an upper bound of the optimal number of expansions in general. Also developed is a method for testing the correlation coefficients after eliminating some principal components, which gives a lower bound of the optimum number of expansions. Since the curve of minimum–norm errors in the previous work shows local minima when the number of terms is squares of the integer, the range produced by the present method leads to only a few candidates for the optimal number of expansions. Examples of reconstructing the radiated acoustic pressure field from a vehicle front end using the optimal number of expansion terms thus obtained are demonstrated in parallel with the results of the a posteriori method discussed by Wu. [Work supported by NSF.]

A Workstation Farm Optimized for Monte Carlo Shell Model Calculations : Alphleet

The Quantum Monte Carlo Diagonalization (QMCD) method which is heart of the MCSM, has a peculiar characteristic as converting the original large matrix into a small one giving nearly equal eigenvalues by means of transforming basis vectors, as mentioned by another paper on this proceedings. 1) In the QMCD, most important basis vectors of the many-body Hilbert space are searched in a stochastic way, and the Hamiltonian is diagonalized with respect to those basis vectors. Certainly, it takes almost all cpu time to calculate each Hamiltonian matrix element with many trial basis vectors to find out the (nearly) best transformation. This characteristic is suitable to implement on parallel calculation with using many CPUs, especially on loosely coupled system because of the calculation of each matrix element can be divided into individual segments not talking among themselves. We decided to make such a computer system as known as “cluster” or “farm” by ourselves because in absence of the computer system which realize such simple parallelism in the market. Every parallel computer in the market is so sophisticate, that is, too expensive for our purpose. Performance of floating point calculation is also so crucial. Vector CPUs, state of the art architecture for fastest floating point calculation, are however not suitable for our calculation which does not treat vector long enough for the CPUs. We chose the Alpha 21264 which has highest performance of the floating point calculation (58.7 SPECfp95 at 500 MHz) as scaler CPU about 4 times better than most popular CPU in the commodity market. The workstation farm that we have built is named “Alphleet”, which consists of many “Alpha” CPUs like as a “Fleet”. Figure 1 shows a schematic diagram and an actual photo (half nodes) of the Alphleet. It consists of a server and 70 clients. It is our goal of the system to maximize performance for the MCSM calculation and to minimize management task. The latter means that easy management against such a large scale system is another big issue for non-professionals of computer itself

Nonlinear Behavior of Space Shuttle Superlightweight Tank Under Booster Ascent Loads

Results of linear-bifurcation and nonlinear analyses of the Space Shuttle superlightweight (SLWT) external liquid-oxygen (LO2) tank for an important early booster ascent loading condition are presented. These results for thin-walled linear elastic shells that are subjected to combined mechanical and thermal loads illustrate an important type of response mode 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 forward ogive section of the LO 2 tank. In contrast, the nonlinear response phenomenon is shown to consist of short-wavelength bending deformations in the forward-ogive and barrel sections of the LO2 tank that grow 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 forward-ogive section. 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-e delity, e nite element model, and that a design based on a linear-bifurcation buckling analysis and a buckling-load knockdown factor is overly conservative. Results are presented that show that the SLWT LO 2 tank can support loads in excess of approximately 2.6 timesthevaluesof theoperationalloads considered. In addition, results are presented that show that local bending deformations may cause failure of the thermal protection system (TPS) at load levels less than the load level corresponding to structural collapse. Results are presented that can be used to estimate the load level at which TPS failure is likely to occur.

Nonlinear Behavior of Space Shuttle Superlightweight Tank Under End-of-Flight Loads

Results of elastic, linear-bifurcation buckling and nonlinear analyses of the new Space Shuttle superlightweight external liquid-oxygen tank are presented for an important end-of-e ight loading condition. These results illustrate an important type of response mode for thin-walled shells subjected to combined mechanical and thermal loads that may be encountered in the design of other liquid-fuel launch vehicles. Linear-bifurcation buckling analyses are presented that predict several nearly equal eigenvalues that correspond to local buckling modes in the aft dome of the liquid-oxygen tank. In contrast, the nonlinear response phenomenon is shown to consist of a shortwavelength bending deformation in the aft elliptical dome of the liquid-oxygen 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-e delity, e nite element model, and that a design based on a linear-bifurcation buckling analysis and a buckling-load knockdown factor is overly conservative. Results are also presented that showthatthesuperlightweightliquid-oxygentankcansupportloadsin excessofapproximately1.9 timesthevalues of the operational loads considered.

First- and Second-Order Epi-Differentiability in Eigenvalue Optimization

The subject of this work is the first- and second-order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi-differentiability. We show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. We also prove that the mth largest eigenvalue function is twice epi-differentiable if and only if it ranks first in a group of equal eigenvalues. Finally, we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems.

The Clarke and Michel-Penot Subdifferentials of the Eigenvalues of a Symmetric Matrix

We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classical directional derivative.

Effect of long range order on sheared liquid crystalline materials Part 1: compatibility between tumbling behavior and fixed anchoring

A viscoelastic constitutive equation for liquid crystalline flows is presented in terms of tensor order parameter dynamics. The presented equation takes into account short range order, long range order (Frank elasticity), and flow effects, and captures the spatio-temporal evolution of the tensor order parameter. The dimensionless parameters arising in the theory are the Deborah (De) number and Ericksen (Er) number, and their ratio is a new dimensionless number R which represent the ratio of short range elasticity to long range elasticity. At Er → ∞ the model converges to the Doi equation, and for De → 0 to the Leslie-Ericksen director equation; the consistency with the Doi theory is shown by the classical shear flow solutions: tumbling, oscillating, and steady state. The model is used to analyze rectilinear simple shear flow subjected to strong anchoring conditions. For the range of governing parameters studied in this paper it is found that for small Er, a spatially nonhomogenous steady state of the tensor order parameters exists. For De corresponding to the tumbling solutions of the Doi equation, the present theory predicts the presence of a composite three layer structure: (a) a center bulk region characterized by director tumbling; and (b) two director oscillating regions next to the bounding surfaces. The amplitude of the director oscillations decrease from a maximum at the boundaries between tumbling-oscillating modes to zero at the bounding surfaces. The boundary between the tumbling-oscillating mode is characterized by the time periodic emergence of an abnormal nematic state, which is characterized by a tensor order parameter with two equal eigenvalues on the shear plane. The emergence of the abnormal nematic state provides for a compatibility mechanism between oscillations and rotations. Thus, we have found through the use of computational modeling a new mechanism that explains the compatibility of Doi's tumbling solutions and fixed surface anchoring conditions. The new flow mechanism is independent of the type of strong anchoring conditions, and arises with planar, homeotropic, and any arbitrary director anchoring.

On curvature homogeneous three-dimensional Lorentzian manifolds

The aim of this paper is the study of three-dimensional Lorentzian manifolds whose Ricci tensor has three equal constant eigenvalues, whose associated eigenspace is two-dimensional. A complete local classification of this class of curvature homogeneous manifolds is presented. It turns out that, if the eigenvalue is zero, these are exactly the curvature homogeneous manifolds modelled on an indecomposable, non-irreducible Lorentzian symmetric space, which were first studied in Cahen etaal. (1990), and the techniques presented in this paper can therefore be applied to obtain a complete (local) classification of these manifolds, and to construct a number of new examples of such manifolds.

Nonlinear dynamics of an elliptic magnetic stagnation line

The nonlinear evolution of the kink instability of a plasma with an elliptic magnetic stagnation line is studied by means of an amplitude expansion of the ideal magnetohydrodynamic (MHD) equations. A cylindrically symmetric plasma with circular field lines is used to model the magnetic field geometry close to the stagnation line. Due to the symmetry with respect to ±z, the linear stability problem of such a system has a two-folded degeneracy, with equal eigenvalues for helical kink perturbations with positive and negative polarization. It is shown that, near marginal stability, the nonlinear evolution of the instability can be described in terms of a two-dimensional potential U(X,Y), where X and Y represent the amplitudes of the perturbations with positive and negative helical polarization. The potential U(X,Y) is found to be nonlinearly stabilizing for all values of the polarization. Furthermore, in addition to the equilibrium point (X,Y)=(0,0), the nonlinear potential has eight equilibrium points in the XY-plane, four corresponding to helical polarization (X or Y=0) and four to plane polarization (|X|=|Y|). The latter equilibria have the lowest energy, indicating that plane kinks preferably should be formed as the stagnation line instability evolves. The equilibria with helical polarization agree with the bifurcated Z-pinch equilibria obtained by means of a different method in a previous paper [Plasma Phys. Controlled Fusion 35, 551 (1993)].

Instability of ordination results under changes in input data order: explanations and remedies

Abstract.Correspondence analysis (CA) and its Detrended form (DCA) produced by the program CANOCO are unstable under reordering of the species and sites in the input data matrix. In CA, the main cause of the instability is the use of insufficiently stringent convergence criteria in the power algorithm used to estimate the eigenvalues. The use of stricter criteria gives results that are acceptably stable. The divisive classification program TWINSPAN uses CA based on a similar algorithm, but with extremely lax convergence criteria, and is thus susceptible to extreme instability. We detected an order‐dependent programming error in the non‐linear rescaling procedure that forms part of DCA. When this bug is corrected, much of the instability in DCA disappears. The stability of DCA solutions is further enhanced by the use of strict convergence criteria. In our trials, much of the instability occurred on axes 3 and 4, but one should not assume that published two‐dimensional ordinations are sufficiently accurate. Data sets which have pairs of almost equal eigenvalues among the first three axes could suffer from marked instability in the first two dimensions. We recommend that a debugged, strict version of CANOCO be released. Meanwhile, users can check the stability of their CA and DCA ordinations using the software that we have made available on the World Wide Web (http://www.helsinki.fi/jhoksane/). An accurate program for CA, a debugged, strict version of DECORANA (for DCA) and a strict version of TWINSPAN are also available at our site.

Tracking analysis of the sign-sign algorithm for nonstationary adaptive filtering with Gaussian data

This correspondence is concerned with the analysis of the sign-sign algorithm (SSA) when used to track a plant with randomly time-varying parameters. The input of the plant and the plant noise are assumed stationary and Gaussian. The paper derives expressions of the steady-state excess mean square error /spl xi/, the steady-state mean square weight misalignment /spl eta/, and the step sizes that minimize each one of them. The paper also presents a comparison among the tracking properties of the SSA, the sign algorithm (SA), and the signed regressor algorithm (SRA). It is found that the three algorithms share the features that (1) /spl xi/ does not depend on the eigenvalue spread of the input covariance matrix, (2) /spl eta/ depends on the eigenvalue spread, and (3) the step size that minimizes /spl xi/ is different from the one that minimizes /spl eta/. The minimum values of /spl xi/ attained by the SA and the SRA are equal to each other, and they are 1 dB less than the one attained by the SSA. The ratio of the minimum value of /spl eta/ of the SSA to the one of the SA is found to be dependent on the input eigenvalue spread; for equal eigenvalues, this ratio is equal to 1 dB. The minimum value of /spl eta/ of the SSA is found to be 1 dB higher than the one of the SRA independently of the input eigenvalue spread. It is found that an advantage of the SSA with respect to both the SA and the SRA is that the two optimum step sizes of the SSA are independent of the mean square plant input and the mean square plant noise.

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

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 Schr\odinger 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 Schr\odinger-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.

Multilevel methods applied to the design of resonant cavities

An application of multilevel (ML) methods to compute the modes and eigenvalues of resonant cavities is presented. The involved methods include an ML eigenvalue solver, an ML mode separation technique, a boundary treatment method, and a subspace continuation technique (SCT) for sequences of problems. In the presented numerical experiments, an asymptotic convergence factor of order 0.1 is obtained for ML cycles on all fine levels, while performing only a few relaxations per cycle. This factor is obtained for a rectangular cavity as well as for cavities having reentrant corners, holes and narrow regions, and presenting clusters of close and equal eigenvalues. A second order scheme is obtained for the computed eigenvalues and modes with an amount of work of order O(qN) for q modes of size N on the finest level. The SCT is illustrated on a moving boundary problem, where solutions change fast at a small boundary change. Such computations are applied to the design of new microwave selective devices. >

Benzenoid graphs with equal maximum eigenvalues

The authors stated that according to their calculation the maximum eigenvalues of G1 and G2 could be equal. Since we are in a position to carry out the diagonalization of the adjacency matrix to high accuracy, we found the answer: Benzenoid graphs G1 and G2 do not possess equal maximum eigenvalues. The calculated maximum eigenvalues are as follows: Xma x ( G 1 ) = 2.6369724379565793242087106873803083111541 and Xmax (G2) = 2.6369724369094896734421158581663075141815.