Sum Of Eigenvalues Research Articles

Frequency Domain Variation of Eigenvalues in Adaptive MIMO OFDM Systems

We consider the effects of the frequency variation of frequency selective fading channels on adaptive multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. The magnitude of this variation has important effects on system performance and design as well as applications to temporal variation, feedback delay and channel estimation.We first examine the variation of the eigenvalues and eigenvalue sums across the frequency bins of a MIMO OFDM system focusing on the changes in the ordered eigenvalues and the eigenvalue sum or link gain between two distinct frequency bins. Here we derive distributions and moments for changes in the link gain and maximum eigenvalue and autocorrelation functions for the link gain and maximum eigenvalue as well as simple approximate results for the ordered eigenvalue differences. We then consider variations in performance via the level crossing rates (LCRs) of the bit error rates (BERs) across the frequency bins of the MIMO OFDM system operating over a Rayleigh fading channel. Each eigenmode constitutes a random process in the frequency domain and we compute the LCR for the BER of transmission down the eigenchannels of the MIMO OFDM channel. We finally verify our analysis using Monte Carlo simulations.

DFT characterization of the first step of methyl acrylate polymerization: Performance of modern functionals in the complete basis limit

We obtain values of the reaction barrier for the reaction of methyl acrylate CH 2 CHCOOCH 3 (MA) with the radical CH 3CHCOOCH 3 (HMA ) by density functional theory (DFT) using a variety of functionals and basis sets. Structures for the reactants and the transition state are optimized in B3LYP/cc-pVTZ. We extrapolate energies for these structures to the complete basis set (CBS) limit for each of the functionals B3LYP, PBE, TPSS, BMK, HSE2PBE, mPW1PW91, B97-1, wB97-XD, and M06-2X. The extrapolation follows the energies obtained by the basis sets cc-pVnZ with n = 2, 3, and 4. The estimate of the barrier height is sensitive to the basis and the choice of functional. In order to recover the rate constant for the radical addition we require partition functions as well as the barrier height. To obtain the partition functions for internal rotation in MA, the radical HMA , and the transition state for their addition HMAMA (TS), we trace one-dimensional torsional potentials in B3LYP/cc-pVTZ. Using this data we employ a range of approximations to the partition function ranging from the harmonic oscillator limit, interpolation schemes linking the harmonic oscillator and free rotor limits, and semi-classical expressions. Comparison with the partition functions obtained by direct sum of Boltzmann factors with energy eigenvalues obtained by solution of the Schrödinger equations (total eigenvalue sum or TES) for the one-dimensional torsional potentials show that Mielke and Truhlar’s TDPPI-HS approximation is very accurate. Estimates of activation energies and rate constants for the addition reaction based on the modern functionals wB97-XD and M06-2X in the CBS limit and the TES partition functions reproduce the best experimental measurement.

A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem

Based on the well-known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem (SDP), we formulate the nearest low-rank correlation matrix problem as a nonconvex SDP and propose a numerical method that solves a sequence of least-square problems. Each of the least-square problems can be solved by a specifically designed semismooth Newton method, which is shown to be quadratically convergent. The sequential method is guaranteed to produce a stationary point of the nonconvex SDP. Our numerical results demonstrate the high efficiency of the proposed method on large scale problems.

Eigenvalue Bounds for Schrödinger Operators with a Homogeneous Magnetic Field

We prove Lieb-Thirring inequalities for Schr\"odinger operators with a homogeneous magnetic field in two and three space dimensions. The inequalities bound sums of eigenvalues by a semi-classical approximation which depends on the strength of the magnetic field, and hence quantifies the diamagnetic behavior of the system. For a harmonic oscillator in a homogenous magnetic field, we obtain the sharp constants in the inequalities.

A Multidomain Flexible Docking Approach to Deal with Large Conformational Changes in the Modeling of Biomolecular Complexes

Binding-induced backbone and large-scale conformational changes represent one of the major challenges in the modeling of biomolecular complexes by docking. To address this challenge, we have developed a flexible multidomain docking protocol that follows a "divide-and-conquer" approach to model both large-scale domain motions and small- to medium-scale interfacial rearrangements: the flexible binding partner is treated as an assembly of subparts/domains that are docked simultaneously making use of HADDOCK's multidomain docking ability. For this, the flexible molecules are cut at hinge regions predicted using an elastic network model. The performance of this approach is demonstrated on a benchmark covering an unprecedented range of conformational changes of 1.5 to 19.5 Å. We show from a statistical survey of known complexes that the cumulative sum of eigenvalues obtained from the elastic network has some predictive power to indicate the extent of the conformational change to be expected.

A nonsplit sum of coefficients of modular forms

We introduce and study certain truncated sums of Hecke eigenvalues of GL2-automorphic forms along quadratic polynomials. A power-saving estimate is established, and new applications to moments of critical L-values associated to quadratic fields are derived. An application to the asymptotic behavior of the height of Heegner points and singular moduli is discussed in detail.

Integral power sums of Hecke eigenvalues

In this paper, we investigate the $\ell$th power sum of Hecke eigenvalues of classical holomorphic cusp forms for $3\le \ell\le 8$ and improve the related results of L\u \cite{Lv09a, Lv09b, Lv10}. We also establish $\Omega$-estimates for $2\le \ell\le 6$ and affirm a conjecture of Ivi\'c \cite[(7.23)]{Ivic90}.

Sums of Laplace eigenvalues—rotationally symmetric maximizers in the plane

The sum of the first n ⩾ 1 eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio ( area ) 3 / ( moment of inertia ) for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schrödinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.

A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues

In this paper, we derive “universal” inequalities for the sums of eigenvalues of the Hodge de Rham Laplacian on Euclidean closed submanifolds and of eigenvalues of the Kohn Laplacian on the Heisenberg group. These inequalities generalize the Levitin–Parnovski inequality obtained for the sums of eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.

Influence of Atoms-in-Molecules Methods on Shared-Electron Distribution Indices and Domain-Averaged Fermi Holes†

The effect of using different methods to obtain atoms in molecules (AIMs) on the shared-electron distribution indices (SEDIs) and domain-averaged Fermi holes (DAFHs) is examined using a test set of diatomic molecules. Use of Bader's binary AIM model gives significantly different SEDIs as a function of internuclear distances than do self-consistent Hirshfeld-based AIM models. DAFH eigenvectors remain very similar for all AIM among the different methods. The corresponding eigenvalues are found to differ significantly, although the sums of complementary eigenvalues show only relatively small changes. The choice of a binary division of molecular space into AIM domains is found to lead to extra structure in SEDI curves, but this probably has limited chemical significance.

Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

We consider the Schrodinger operator H = −Δ − V(x), V > 0, acting in the space $ L^2 (\mathbb{R}^d) $ and study relations between the behavior of V at infinity and properties of the negative spectrum of H. Bibliography: 34 titles.

Distance spectral radius of trees with fixed maximum degree

C ‡ Abstract. Distance energy is a newly introduced molecular graph-based analog of the total �-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree �, we prove that the broom Bn,� (consisting of a star S�+1 and a path of length n � 1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.

Error bound for a perturbed minimization problem related with the sum of smallest eigenvalues

Let C be a n×n symmetric matrix. For each integer 1 < k < n we consider the minimization problem m(e): = minX{ Tr{CX} + eƒ(X)}. Here the variable X is an n×n symmetric matrix, whose eigenvalues satisfy the number e is a positive (perturbation) parameter and ƒ is a Lipchitz-continuous function (in general nonlinear). It is well known that when e = 0 the minimum value, m(0), is the sum of the smallest k eigenvalues of C. Assuming that the eigenvalues of C satisfy λ1(C) < ... < λk(C) < λk+1(C) < ∙∙∙ < λn(C), we establish the following upper and lower bounds for the minimum value m(e): where is the minimum value of ƒ over the solution set of unperturbed problem and L is the Lipschitz-constant of ƒ. The above inequality shows that the error by replacing the upper bound (or the lower bound) by the exact value is at least quadratic in the perturbation parameter. We also treat the case that λk+1(C) = λk(C). We compare the exact solution with the upper and lower bounds for some examples. Mathematical subject classification: 15A42, 15A18, 90C22.

A General Framework for Multivariate Analysis with Optimal Scaling: TheRPackageaspect

In a series of papers De Leeuw developed a general framework for multivariate analysis with optimal scaling. The basic idea of optimal scaling is to transform the observed variables (categories) in terms of quantifications. In the approach presented here the multivariate data are collected into a multivariable. An <em>aspect</em> of a multivariable is a function that is used to measure how well the multivariable satisfies some criterion. Basically we can think of two different families of aspects which unify many well-known multivariate methods: Correlational aspects based on sums of correlations, eigenvalues and determinants which unify multiple regression, path analysis, correspondence analysis, nonlinear PCA, etc. Non-correlational aspects which linearize bivariate regressions and can be used for SEM preprocessing with categorical data. Additionally, other aspects can be established that do not correspond to classical techniques at all. By means of the <b>R</b> package <b>aspect</b> we provide a unified majorization-based implementation of this methodology. Using various data examples we will show the flexibility of this approach and how the optimally scaled results can be represented using graphical tools provided by the package.

Optimization of Training Signal Transmission for Estimating MIMO Channel under Antenna Mutual Coupling Conditions

This paper reports investigations on the effect of antenna mutual coupling on performance of training-based Multiple-Input Multiple-Output (MIMO) channel estimation. The influence of mutual coupling is assessed for two training-based channel estimation methods, Scaled Least Square (SLS) and Minimum Mean Square Error (MMSE). It is shown that the accuracy of MIMO channel estimation is governed by the sum of eigenvalues of channel correlation matrix which in turn is influenced by the mutual coupling in transmitting and receiving array antennas. A water-filling-based procedure is proposed to optimize the training signal transmission to minimize the MIMO channel estimation errors.

Eigenvalues of sums of pseudo-Hermitian matrices

We study analogues of classical inequalities for the eigenvalues of sums of Hermitian matrices for the cone of admissible elements in the pseudo-Hermitian case. In particular, we obtain analogues of the Lidskii-Wielandt inequalities.

Effect of atomic spontaneous decay on entanglement in the generalized Jaynes–Cummings model

Some aspects of the irreversible dynamics of a generalized Jaynes–Cummings model are addressed. By working in the dressed-state representation, it is possible to split the dynamics of the entanglement and coherence. The exact solution of the master equation in the case of a high- Q cavity with atomic decay is found. Effects of the atomic spontaneous decay on the temporal evolution of partial entropies of the atom or the field and the total entropy as a quantitative measure entanglement are elucidated. The degree of entanglement, through the sum of the negative eigenvalues of the partially transposed density matrix and the negative mutual information has been studied and compared with other measures.

CP-MLR directed QSAR study of carbonic anhydrase inhibitors: sulfonamide and sulfamate inhibitors

Abstract The inhibition activities of sulfonamide and sulfamate derivatives for human carbonic anhydrases have been quantitatively analyzed using DRAGON descriptors. QSAR models have been obtained through combinatorial protocol-multiple linear regression (CP-MLR) computational procedure. For the hCA I inhibition activity, a higher value of information content index of the 1-order neighborhood symmetry (IC1) and a lower value of the Moran autocorrelations, MATS2v and MATS1p, along with a lower number of sulfur atoms in a molecular structure (nRSR) is beneficial to the activity. A higher number of 5-membered rings (nR05), a bigger distance between nitrogen and sulfur T(N..S), and a higher value of van der Waals volume weighted descriptor (GATS6v), are helpful to improve the hCA II inhibition activity. For the inhibition of hpCA, a lower value of the descriptors Jhetv and PW5, and higher values of the eigenvalue sum from Z weighted distance matrix, SEigZ, the Moran autocorrelation of lag 8 weighted by atomic van der Waals volumes, MATS8v and the Moran autocorrelation of lag 4 weighted by atomic Sanderson electronegativities, MATS4e are favorable. The derived significant models in such descriptors may further be used to synthesize new potential compounds and to decipher the mode of their actions at molecular level.

Dynamical entanglement for Fermi coupled stretching and bending modes

The dynamical entanglement for Fermi coupled C-H stretch and bend vibrations in molecule CHD3 is studied in terms of two negativities and the reduced von Neumann entropy, where initial states are taken to be direct products of photon-added coherent states on each mode. It is demonstrated that the negativity defined by the sum of negative eigenvalues of the partial transpose of density matrices is positively correlated with the von Neumann entropy. The entanglement difference between photon-added coherent states and usual coherent states is discussed as well.

Entanglement dynamics in Coriolis coupled rovibrational states of formaldehyde

The entanglement dynamics of two vibrational modes of a polyatomic molecule coupled by Coriolis interaction to overall molecular rotation is studied in terms of two negativities, N ( t ) and N s ( t ) , respectively, defined by the minimum of the eigenvalues and by the sum of the negative eigenvalues of the partial transpose of a density matrix. Various initial states are the products of Dicke states and the products of coherent states of vibrations and rotations. Formaldehyde is taken as an example, and the von Neumann entropy s ( t ) is simulated for the comparison with both negativities. It is shown that negativity N s ( t ) is positively correlated with entropy s ( t ) , and the correlated behavior between negativity N ( t ) and entropy s ( t ) strongly depends on initial states. However, these three indicators of entanglement display a dominantly positive correlation for the coherent states with small or large parameters. In addition, for the latter state two quantities N ( t ) and s ( t ) are nearly unchanged for a long time. This time can be further increased by the increasing of vibrational quantum number so that molecular information processing and quantum computing is allowed. These results are useful in quantum information theory.