Unique Eigenvalue Research Articles

On existence of a bound state in an L-shaped waveguide

We investigate the Laplace operator in an L-shaped strip of a widthd with Dirichlet boundary conditions. It is shown that it has a unique eigenvalue corresponding to a square-integrable eigenfunction, namely λ = 0·93 (π/d)2. This result has implications for the theory of waveguides as well as for electron motion in some microscopic semiconductor devices.

31P magnetic shielding tensor components of a phosphirane. Angular dependence of the 31P magnetic shielding anisotropy

We report the measurement of the 31P magnetic shielding components of a phosphirane molecule. Its high field isotropic resonance is found to result from a unique eigenvalue at very high field. A comparison of this behaviour with that of cycloalkenes and of other phosphines is presented. A relationship between the magnetic shielding anisotropy and a geometrical parameter θ, characteristic of the strain at the nucleus under study, is emphasized.

Calculation of eigenvector derivatives for structures with repeated eigenvalues

In structural optimization and system identification, eigenvector derivatives provide important information for updating design/model parameters. When the current parameter values yield repeated eigenvalues, it has not been possible previously to calculate unique eigenvector derivatives. Recent work has provided a method for determining unique eigenvalue derivatives for this case, but methods for calculating the eigenvector sensitivities have been incomplete. In this work, a complete method for calculation of repeated-root eigenvector derivatives is shown for the real, symmetric structural eigenproblem. The derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities. As an example, the repeated-root eigenvector sensitivities are calculated for a simple three degree-of-freedom beam grillage. Comparisons of linear approximations (using these derivatives) to the calculated eigenvectors support the accuracy of the formulation.

3D transient eddy current fields using the u-v integral-eigenvalue formulation

The three-dimensional eddy current transient field problem is formulated using the u-v method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null field integral equations and the appropriate boundary conditions are used to set up an identification matrix which is independent of null field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. When the initial transient forcing function is Fourier decomposed into its spatial harmonics, each Fourier component can be associated with a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response, so obtained with the particular external field decay governing the problem at hand. The technique is applied to the FELIX (fusion electromagnetic induction experiments) medium cylinder experiment; computed results are compared with data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure.

On the use of the eigenvalue approach in the prediction of transient eddy current fields

Very complex time dependent three-dimensional eddy-current problems are considered. The use of global spatial basis functions with assumed eigenvalue (exp(- lambda t)) dependence, which is extremely efficient for many problems, is shown to have embedded in it some serious pitfalls which lead to error. Examination shows that extreme caution should be exercised when using an eigenvalue approach; the association of unique eigenvalues with what would appear to be unique eigenmodal shapes leads to errors that are more severe as the object shape gets more complex. An alternative approach to the problem using the standard space-dependent Green's function is presented. Both approaches above have been applied to the FELIX cylinder experiments, and the results are discussed. >

Stability of Singularly Perturbed Solutions to Systems of Reaction-Diffusion Equations

Stability theorem is presented for large amplitude singularly perturbed solutions (SPS) of reactiondiffusion systems on a finite interval. Spectral analysis shows that there exists a unique real critical eigenvalue $\lambda _c (\varepsilon )$ which behaves like $\lambda _c (\varepsilon ) \simeq \tau \varepsilon $ as $\varepsilon \downarrow 0$, where $\varepsilon $ is a small parameter contained in the system. All the other noncritical eigenvalues have strictly negative real parts independent of $\varepsilon $. The singular limit eigenvalue problem in §2 plays a key role to judge the sign of $\tau $ which determines the stability of SPS for small $\varepsilon $. Under a natural framework of nonlinearities, $\tau $ becomes negative, namely, SPS is asymptotically stable. Instability result is also shown in §4.

An integral eigenvalue approach to three-dimensional field problems--A low-frequency variation of SEM

The three-dimensional (3-D) eddy-current transient field problem is formulated first using the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u-v</tex> method. This method breaks the vector Helmholtz equation into two scalar Helmholtz equations. Null-field integral equations and the appropriate boundary conditions germane to the problem are used to set up an identification matrix which is independent of null-field point locations. Embedded in the identification matrix are the unknown eigenvalues of the problem representing its impulse response in time. These eigenvalues are found by equating the determinant of the identification matrix to zero. The eigenvalues, which can be equated with temporal response, are found to be intimately linked to the initial forcing function which triggers the transient in question. When this initial forcing function is Fourier decomposed into its respective spatial harmonics, it is possible to associate with each Fourier component a unique eigenvalue by this technique. The true transient solution comes through a convolution of the impulse response so obtained with the particular imposed external field governing the problem at hand. The technique is applied to the FELIX medium cylinder (a conducting cylinder placed in a collapsing external field) and compared to data. A pseudoanalytic confirmation of the eigenvalues so obtained is formulated to validate the procedure. The technique proposed is applied in the low-frequency regime where the near-field effects must be considered. Application of the technique to a high frequency follows directly if the Coulomb gauge is adopted to represent the vector potential.

The existence, uniqueness, and instability of spherically symmetric solutions of a system of reaction—Diffusion equations

The system ∂x ∂t = Δx + F(x,y), ∂y ∂t = G(x,y) is investigated, where x and y are scalar functions of time ( t ⩾ 0), and n space variables (ξ 1,…, ξ n), Δx ≡ ∑ i = 1 n ∂ 2x ∂ξ i 2 , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution (x(r),y(r)), where r = (ξ 1 2 + … + ξ n 2) 1 2 , which is bounded for r ⩾ 0 and satisfies x(0) > x 0, y(0) > y 0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ∀ r > 0. Thus, ( x( r), y( r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around ( x( r), y( r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique.

Exact differential renormalization group equations for Ising models on square lattices

The differential real space renormalization theory of Hilhorst et al. is applied to Ising models on square lattices with nearest-neighbour interactions only. The renormalization flow equations for these two interaction parameters contain two auxiliary parameters; these parameters have to be determined by solving two additional equations, one partial differential equation and one ordinary equation. The necessity of introducing these additional parameters is explained by arguing that for the present formulation of differential real space renormalization theory in d dimensions, at least d + 1 parameters are required. The concept of local fixed point is introduced; this fixed point can be determined by solving algebraic equations. The linearized flow around it describes local properties of the system and is therefore related to the critical properties of homogeneous Ising systems. We study temperature-like perturbations around the local fixed point and find a unique eigenvalue yT = 1, in agreement with the known exact result.

Positivity and monotonicity properties of transport equations with spatially dependent cross sections

Abstract We investigate the transport equation with suitable boundary conditions through an equivalent integral equation. Assuming the incoming fluxes, the internal source term f(x,μ), the cross section c(x) and the parameter ξ to be nonnegative, we prove the existence of a unique dominant eigenvalue ξ=ξ0(τ) for which the homogeneous problem has a positive solution (critical case), the existence of a unique positive solution for ξ ξ0(τ) (supercritical case). We show ξ0(τ) to decrease continuously from ∞ to some ξ0(∞)>0 whenever ξ increases from 0 to ∞ (monotonicity). The results are obtained by studying an operator that leaves invariant the cone of nonnegative functions in L∞(0,τ).

Exact differential renormalization group for the square Ising model

We present an exact differential renormalization-group method for the square Ising model. This method is analogous to that introduced by Hilhorst et al. for the triangular Ising model. Our approach is based upon the construction of such a renormalization transformation which remains confined to the space of four spatially dependent nearest-neighbor interactions. We derive the renormalization-group equations in the form of a set of four linear first-order partial differential equations for the interactions. We obtain a class of solutions of these equations and determine a nontrivial fixed point. The analysis of the flow around the fixed point shows that this fixed point is stable with respect to perturbations within the critical surface and is unstable in the temperaturelike direction. A unique eigenvalue, ${y}_{T}=1$, from the eigenvalue equation for the temperature direction is found, in accordance with the exact Onsager solution.

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)1/2, where D is the diffusivity of the mutant species and f is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lower-concentration uniform equilibrium state (UES). Depicting the persistence of a mutant-deficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points as t→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.

The universal metric properties of nonlinear transformations

The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx n+1 =λf(x n ) independent of a specificf is formally developed. A hierarchy of universal functionsg r (x) exists, each descriptive of the same local structure but at levels of a cluster of 2 r points. The hierarchy obeysg r−1 (x)=−αg r(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticg r ∼ g − δ−r h * where δ > 1 andh are determined as the associated eigenvalue and eigenvector of the operator ℒ: $$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} \right)} \right]$$ We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλ rather than just discreteλ r and bifurcation valuesΛ r and dynamics at suchλ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of theg r limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate tog r 's, thereby providing some elementary approximation schemes for obtainingλ r for a chosenf.

Certain infinite Markov chains and sequential decoding

It is shown that the sequential decoding of rate one-half convolutional codes leads to a special type of infinite Markov chain in which only one transition of one step toward the origin state is permitted but any number of transitions away from the origin state are permitted. It is shown that such chains have the singular property that the stationary state probabilities π 0,π 1,π 2,… can be calculated successively without summing any infinite series or solving for any eigenvalues. It is further shown that the analogous infinite Markov chains in which a single transition of one step away from the origin is permitted are also related to a problem in sequential decoding and also have a singular property, namely that π i = (1 − β) β i where β is the unique real eigenvalue of the characteristic polynomial such that 0 < β < 1.

Relativistic energies of excited states of atoms and ions of the second period

Relativistic energies are computed for a series of j states corresponding to 1s22sm2pn (0?m?2; 0?n?6) configurations by solution of the hartree-fock-dirac equations. The j states calculated are those states whose eigenfunctions within a configuration correspond to a unique and maximum eigenvalue of the operator j2. In addition, average relativistic energies are obtained for all configurations considered. The two sets of results are compared, and in certain cases (for the 1s22p, 1s22p5, 1s22s22p, 1s22s22p5 configurations) permit a determination of multiplet splittings (e(2p12/-2p32/)). The results of these splittings are in excellent agreement with available experimental data. The present results also tend to confirm the assumption that the relativistic energy contributions for the j averaged ls states are the same for all states arising from the same configuration. This makes it possible to evaluate the relativistic energies of all states belonging to the configurations of interest to this paper. These in turn serve to re-evaluate more correctly recently obtained correlation energies for the same states.

Model of Interacting Radiation and Matter. III. Multimode Gas Lasers

We extend the investigation of the long-time behavior of a model consisting of N two-level atoms interacting with a single electromagnetic cavity mode to include interaction with many cavity modes. We show that, as a consequence of the coupling between radiation modes produced by spatial density variations of the population inversion, there is no strictly stationary state possible for multimode behavior. However we obtain a stationary state by neglecting the rapidly oscillating terms. The steady-state population inversion is then a solution of an eigenvalue problem. The eigenvalue determinate is a function of the number of modes and the coupling between modes in addition to the usual dependence on frequencies and relaxation times. We explicitly solve for the unique eigenvalue in special cases. The corresponding eigenvector gives the steady-state mode intensity ratios. The absolute values of the steady-state intensities are determined by the energy conservation equation generalized to include pumping and dissipation. We also calculate the steady-state frequency shifts for each mode. The mode frequency shifts are practically independent of each other and have the same functional form as the single-mode frequency shifts.

III.—The Eigenvalue Problem for Boolean Matrices

SynopsisIn the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.