Sum Of Eigenvalues Research Articles

Schrödinger Operators with Rapidly Oscillating Potentials

Schrodinger operators \(\hat{H} = -\Delta + V\) with rapidly oscillating potentials V such as \(cos |x|^{2}\) are considered. Such potentials are not relatively compact with respect to the free Schrodinger operator −Δ. We show that the oscillating potential V do not change the essential spectrum of −Δ. Moreover we derive upper bounds for negative eigenvalue sums of Ĥ.

Entropies and entanglement for decoherence without energy relaxation in a two-level atom

In this paper, we investigate the temporal evolution of various measures of degree of entanglement and total correlation of a single-mode light field interacting with a two-level atom. The phase-damped cavity is taken into account. The effects of phase damping on a temporal of partial entropies for the atom and the field, total entropy, the entropy difference and the sum of negative eigenvalues of the partially transposed density matrix as a quantitative entanglement measure for the master equation in the dressed-state approximation are taken into account.

On the lifetime of a conditioned Brownian motion in the ball

Consider the Brownian motion conditioned to start in x, to converge to y, with x , y ∈ Ω ¯ , and to be killed at the boundary ∂ Ω. Here Ω is a bounded domain in R n . For which x and y is the lifetime of this Brownian motion maximal? One would guess for x and y being opposite boundary points and we will show that this holds true for balls in R n . As a consequence we find the best constant for the positivity preserving property of some elliptic systems and an identity between this constant and a sum of inverse Dirichlet eigenvalues.

Accurate Error Bounds for the Eigenvalues of the Kernel Matrix

The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to infinity. We derive probabilistic finite sample size bounds on the approximation error of individual eigenvalues which have the important property that the bounds scale with the eigenvalue under consideration, reflecting the actual behavior of the approximation errors as predicted by asymptotic results and observed in numerical simulations. Such scaling bounds have so far only been known for tail sums of eigenvalues. Asymptotically, the bounds presented here have a slower than stochastic rate, but the number of sample points necessary to make this disadvantage noticeable is often unrealistically large. Therefore, under practical conditions, and for all but the largest few eigenvalues, the bounds presented here form a significant improvement over existing non-scaling bounds.

Exploratory factor analysis: strategies used by patients to promote health

Strategies used by patients to promote health (SUPPH) was used to measure self-care self-efficacy in patients with cancer. The objectives of this study were (1) to determine the extent to which self-efficacy theory explained the factor structure of the SUPPH and (2) to determine the relationship of demographic data with factors of the SUPPH. Subjects were diagnosed with prostate cancer (PCa) and treated with either: (a) radical prostatectomy, (b) intensity modulated radiation therapy (IMRT) + radioactive seed implantation, or (c) IMRT + high dose rate. Subjects completed a demographic questionnaire and the SUPPH. Exploratory factor analysis of the SUPPH was performed using a varimax rotation. Subjects (n = 265) were predominately white and averaged 68 years of age. The model explained 81.3% of the total sum of eigenvalues. Two factors of the SUPPH were identified: physiological efficacy information and performance efficacy information. Younger subjects who were fully employed and earning more money had significantly higher performance self-efficacy than older subjects who were working part time and earning less money. Results are congruent with Bandura's (1997) description of self-efficacy. Use of the SUPPH may facilitate research validating Bandura's (1997) assertion that an individual's self-efficacy is related to quality of life (QOL) during chronic illness. Additional research focusing on self-efficacy and PCa patients' QOL may lead to efficacy enhancing interventions that will improve QOL of patients with PCa.

Linking Confinement to Spectral Properties of the Dirac Operator

We represent Polyakov loops and their correlators as spectral sums of eigenvalues and eigenmodes of the lattice Dirac operator. The deconfinement transition of pure gauge theory is characterized as a change in the response of moments of eigenvalues to varying the boundary conditions of the Dirac operator. We argue that the potential between static quarks is linked to spatial correlations of Dirac eigenvectors.

A new barrier for a class of semidefinite problems

We introduce a new barrier function to solve a class of Semidefinite Optimization Problems (SOP) with bounded variables. That class is motivated by some (SOP) as the minimization of the sum of the first few eigenvalues of symmetric matrices and graph partitioning problems. We study the primal-dual central path defined by the new barrier and we show that this path is analytic, bounded and that all cluster points are optimal solutions of the primal-dual pair of problems. Then, using some ideas from semi-analytic geometry we prove its full convergence. Finally, we introduce a new proximal point algorithm for that class of problems and prove its convergence.

Optimization of generalized mean square error in signal processing and communication

Two matrix optimization problems are analyzed. These problems arise in signal processing and communication. In the first problem, the trace of the mean square error matrix is minimized, subject to a power constraint. The solution is the training sequence, which yields the best estimate of a communication channel. The solution is expressed in terms of the eigenvalues and eigenvectors of correlation and covariance matrices describing the communication, and an unknown permutation. Our analysis exhibits the optimal permutation when the power is either very large or very small. Based on the structure of the optimal permutation in these limiting cases, we propose a small class of permutations to focus on when computing the optimal permutation for arbitrary power. In numerical experiments, with randomly generated matrices, the optimal solution is contained in the proposed permutation class with high probability. The second problem is connected with the optimization of the sum capacity of a communication channel. The second problem, which is obtained from the first by replacing the trace operator in the objective function by the determinant, minimizes the product of eigenvalues, while the first problem minimizes the sum of eigenvalues. For small values of the power, both problems have the same solution. As the power increases, the solutions are different, since the permutation matrix appearing in the solution of the trace problem is not present in the solution of the determinant problem. For large power, the ordering of the eigenvectors in the solution of the trace problem is the opposite of the ordering in the determinant problem.

J-pairing interaction, number of states, and nine-jsum rules of four identical particles

In this paper we study $J$-pairing Hamiltonian and find that the sum of eigenvalues of spin $I$ states equals sum of norm matrix elements within the pair basis for four identical particles such as four fermions in a single-$j$ shell or four bosons with spin $l$. We relate number of states to sum rules of nine-$j$ coefficients. We obtained sum rules for nine-$j$ coefficients $<(jj)J,(jj)K:I| (jj)J, (jj)K:I>$ and $<(ll)J,(ll)K:I| (ll)J, (ll)K:I>$ summing over (1) even $J$ and $K$, (2) even $J$ and odd $K$, (3) odd $J$ and odd $K$, and (4) both even and odd $J,K$, where $j$ is a half integer and $l$ is an integer.

Subadditivity of eigenvalue sums

Let f ( t ) f(t) be a nonnegative concave function on 0 ≤ t &gt; ∞ 0 \leq t &gt;\infty with f ( 0 ) = 0 f(0)=0 , and let X , Y X, Y be n × n n\times n matrices. Then it is known that ‖ f ( | X + Y | ) ‖ 1 ≤ ‖ f ( | X | ) ‖ 1 + ‖ f ( | Y | ) ‖ 1 \Vert f(|X+Y|)\Vert _1\leq \Vert f(|X|)\Vert _1 +\Vert f(|Y|)\Vert _1 , where ‖ ⋅ ‖ 1 \Vert \cdot \Vert _1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

Effect of Low-Lying Fermion Modes in the isin-Regime of QCD

We investigate the effects of low-lying fermion eigenmodes on the QCD partition function in the $\epsilon$-regime. The fermion determinant is approximated by a truncated product of low-lying eigenvalues of the overlap-Dirac operator. With two flavors of dynamical quarks, we observe that the lattice results for the lowest eigenvalue distribution, eigenvalue sum rules and partition function reproduce the analytic predictions made by Leutwyler and Smilga, which strongly depend on the topological charge of the background gauge configuration. The value of chiral condensate extracted from these measurements are consistent with each other. For one dynamical quark flavor, on the other hand, we find an apparent disagreement among different determinations of the chiral condensate, which may suggest the failure of the $\epsilon$-expansion in the absence of massless Nambu-Goldstone boson.

Proper orthogonal decomposition analysis of autoignition simulation data of nonhomogeneous hydrogen–air mixtures

The proper orthogonal decomposition (POD) method is implemented on unsteady 2D direct numerical simulation of autoignition in nonhomogeneous hydrogen–air mixtures. The analysis is implemented to evaluate requirements for the reproduction of transient, multidimensional and multiscalar processes in combustion. Data reduction is implemented on a set of 30 snapshots of 2D fields of a passive scalar, the mixture fraction, and a reactive scalar, the mass fraction of the intermediate species, HO 2. The snapshots cover the evolution of the hydrogen–air mixture from induction to the early stages of high-temperature combustion. The standard method by which the POD technique is measured, the cumulative energy criterion, based on the sum of the largest eigenvalues, suggests that the bulk of this energy may be represented by the first three to four modes for the reactive scalars. However, this criterion may not be sufficient to characterize the performance of the POD reduction approach. Therefore the number of required eigenmodes for each data set is tested. A number of preprocessing strategies of the scalar fields are explored to reduce the number of required eigenmodes. The strategies are designed to reduce the temporal and spatial spans of scalar values. The results show that different preprocessing strategies may yield different outcomes for the passive scalars, represented by the mixture fraction, and reactive scalars, represented by the intermediate species, HO 2 mass fraction. More importantly, there are different requirements to reproduce passive and reactive scalars during the autoignition process. The mixture fraction, which is affected by the mixing process only, requires the least number of eigenmodes, and yields a sufficient representation of the original data with only two to three eigenmodes. The reactive scalar reduction improves significantly with preprocessing, which reduces the required number of eigenmodes to approximately six.

Tiling a Plane in a Dynamical Process and its Applications to Arrays of Quantum Dots, Drums, and Heat Transfer

We present a reaction-diffusion system consisting of N components. The evolution of the system leads to the partition of the plane into cells, each occupied by only one component. For large N, the stationary state becomes a periodic array of hexagonal cells. We present a functional of the densities of the components, which decreases monotonically during the evolution and attains its minimal value in the stationary state. This value is equal to the sum of the first Laplacian eigenvalues for all cells. Thus, the resulting partition of the plane is determined by minimization of the sum of the eigenvalues, and not by the minimization of the total perimeter of the cells as in the famous honeycomb problem.

A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix

For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalization of their statistic for the test that the smallest eigenvalues of a covariance matrix are equal. Some inequalities are obtained for sums of eigenvalues and sums of squared eigenvalues. These bounds permit us to obtain the asymptotic null distribution of our statistic, as the dimensionality and sample size go to infinity together, by using distributional results obtained by Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102]. Some empirical results comparing our test with the likelihood ratio test are also given.

An interval estimation for the number of signals

We propose a multi-step procedure for constructing a confidence interval for the number of signals present. The proposed procedure uses the ratios of a sample eigen-value and the sum of different sample eigen-values sequentially to determine the upper and lower limits for the confidence interval. A preference zone in the parameter space of the population eigen-values is defined to separate the signals and the noise. We derive the probability of a correct estimation, P (CE), and the least favorable configuration (LFC) asymptotically under the preference zone. Some important procedure properties are shown. Under the asymptotic LFC, the P (CE) attains its minimum over the preference zone in the parameter space of all eigen-values. Therefore a minimum sample size can be determined in order to implement our procedure with a guaranteed probability requirement.

On a conditioned Brownian motion and a maximum principle on the disk

Bounds for the 3G-expression∫ΩG(x,z)G(z,y)d,z/G(x,y) play a fundamental role in potential theory. Here,G(x,y) is the Green function for the Laplace problem with zero dirichlet boundary conditions on Ω. The 3G-formula equals\({\mathbb{E}}_x^y (\tau _\Omega )\), the expected lifetime for a Brownian motion starting in\(x \in \bar \Omega \) that is killed on exiting ω and conditioned to converge to and to be stopped at\(y \in \bar \Omega \). Although it was shown by probabilistic methods for bounded (simply connected) 2d-domains that ifx e δΩ, then the supremum ofy \at Exy is assumed for somey at the boundary, the analogous question remained open forx in the interior. Here we are able to give an answer in the case thatB ⊂ ℝ is the unit disk. The dependence of this quantity on the positions ofx andy is investigated, and it is shown that indeed Exy(\gt\om) is maximized on\(\bar B^2 \) by opposite boundary points. The result also gives an answer to a number of questions related to the best constant for the positivity-preserving property of some elliptic systems. In particular, it confirms a, relationExy(\gt\om) with a ‘sum of inverse eigenvalues’ that was conjectured recently by Kawohl and Sweers.

Uniform Lieb-Thirring Inequality for the Three-Dimensional Pauli Operator with a Strong Non-Homogeneous Magnetic Field

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. A new Lieb-Thirring type inequality on the sum of the negative eigenvalues is presented. The main feature compared to earlier results is that in the large field regime the present estimate grows with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain an optimal upper bound on the pointwise density of zero energy eigenfunctions of the Dirac operator. The main technical tools are: (i) a new localization scheme for the square of the resolvent of a general class of second order elliptic operators; (ii) a geometric construction of a Dirac operator with a constant magnetic field that approximates the original Dirac operator in a tubular neighborhood of a fixed field line. The errors may depend on the regularity of the magnetic field but they are uniform in the field strength.

Magnetic Lieb–Thirring Inequalities with Optimal Dependence on the Field Strength

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb-Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality \cite{ES-IV} and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.

Determination of the regularization level of truncated singular‐value decomposition inversion: The case of 1D inversion of MT data

ABSTRACTThe total mean‐square error (MSE) of the estimated model, defined as the sum of the standard model variance and the bias variance, is used to define the truncation level of the singular‐value decomposition to give a reasonable balance between model resolution and model variance. This balance is determined largely by the data and no further assumptions are necessary except that the bias terms are estimated sufficiently well. This principle has been tested on the 1D magnetotelluric inverse problem with special emphasis on high‐frequency radio magnetotelluric (RMT) data. Simulations clearly demonstrate that the method provides a good balance between resolution and variance. Starting from a homogeneous half‐space, the best solution is sought for a fixed set of singular values. The model variance is estimated from the sum of the inverse eigenvalues squared, up to a certain threshold, and the bias variance is estimated from the model projections on the remaining eigenvectors. By varying the threshold, the minimum of the MSE is found for an increasing number of fixed singular values until the number of active singular values becomes greater than or equal to the estimated number. As a side‐effect, the depth of penetration of a given set of measurements can be estimated very efficiently by simply noting at which depth the final model deviates little from the starting homogeneous half‐space model. A suite of synthetic data is inverted and an example of inversion of one site is shown to illustrate how the truncation is carried out as the non‐linear inversion process proceeds. A field example with a profile across a plume of contaminated groundwater in the Netherlands shows good agreement with the electrical resistivity obtained in a nearby borehole.

Equilibrium geometries of low-lying isomers of some Li clusters, within Hartree-Fock theory plus bond order or MP2 correlation corrections.

In a recent study by Kornath et al. [J. Chem. Phys. 118, 6957 (2003)], the Li(n) clusters with n=2, 4, and 8 have been isolated in argon matrices at 15 K and characterized by Raman spectroscopy. This has prompted us to carry out a theoretical study on such clusters up to n=10, using Hartree-Fock theory, plus low-order Møller-Plesset perturbation corrections. To check against the above study of Kornath et al., as a by-product we have made the same approximations for n=6 and 8 as we have for n=10. This has led us to emphasize trends with n through the Li(n) clusters for (i) ground-state energy, (ii) HOMO-LUMO energy gap, (iii) dissociation energy, and (iv) Hartree-Fock eigenvalue sum. The role of electron correlation in distinguishing between low-lying isomers is plainly crucial, and will need a combination of experiment and theory to obtain decisive results such as that of Kornath et al. for Li(8). In particular, it is shown that Hartree-Fock theory plus bond order correlations does account for the experimentally observed symmetry T(d) symmetry for Li(8).