Theoretical Eigenvalues Research Articles

Ultrahigh-Resolution Wireless Capacitance Readout Based on a Single Real Mode in a Perturbed PT -Symmetric Electronic Trimer Sandwich

High-performance interrogation of an inductor-capacitor microsensor has been a long-standing challenge due to the limited size of the sensor. Parity-time ($PT$) symmetry, an intriguing concept originated from quantum physics, has been utilized to improve the spectral resolution and sensitivity of the conventional readout circuit, while the $PT$-symmetry condition has to be satisfied. In this work, a sandwich-type wireless capacitance readout mechanism based on perturbed $PT$-symmetric electronic trimer without manual tuning of the reader circuit is proposed. Theoretical eigenvalue analysis by solving the system equations shows that the system exhibits single real mode in weak coupling regime whose eigenfrequency changes in response to the capacitance of the neutral resonator. Furthermore, the proposed readout system exhibits wider readout capacitance range compared to the standard $PT$-symmetry-based system while retaining higher $Q$ factor compared to the conventional readout method, which has been validated with the experimental prototype based on printed circuit board. Our work not only enriches the underlying theory of non-Hermitian physics, but also shows potential applications in scenarios where longer interrogation distance is required, such as implanted medical devices, parameter detection in harsh environment, etc.

Effects of Polarization Function on the Spin Contamination and Distribution in <i>β'</i>- Me<sub>4</sub>P[Pd(dmit)<sub>2</sub>]<sub>2</sub>

The effects of polarization function on the spin contamination and distribution in β'-Me4P[Pd(dmit)2]2 were studied using the DFT cluster method. Two basis sets, SV and SVP were considered in the calculations, where B3LYP functional was employed in the doublet state of the one-fragment and dimer clusters. The values of <S2> before annihilation for both SV and SVP basis sets are excellent and very close to the perfect theoretical eigenvalue of 0.75. The values of the spin densities at thiolate and thione calculated using SVP were found to be smaller than the ones using SV. The difference, however, is less than eight percent. In contrast, the difference in the spin density at Pd atoms in both monomers is significantly larger for the SVP, being about 21%. The inclusion of polarization function resulted in the shifting of electron density from the sulfur atoms to the central Pd atoms. The calculated spin densities revealed the inhomogeneous distribution of the electron spin in the dimer that leads to the existence of electron-rich and electron-poor regions.

Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity

We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer.

Convergence analysis of the modified frequency-domain block LMS algorithm with guaranteed optimal steady state performance

Although the bin-normalized frequency domain block LMS (NFBLMS) algorithm has theoretically very fast convergence speed, it suffers from convergence to a biased steady state solution when causality is not met or when the adaptive filter is of deficient length. A modified FBLMS (MFBLMS) algorithm with guaranteed optimal steady state performance has been proposed, but the theoretical analysis on its convergence properties has not been presented. This paper analyzes the convergence behavior of the algorithm by using the theory of asymptotically equivalent matrices. The eigenvalues of the matrix controlling the convergence behavior is proven to have the tendency to be equally distributed, and a theoretical eigenvalue spread is derived based on the first-order autoregressive (AR) signal model, which is significantly lower than that of the time domain LMS (TDLMS) algorithm. Therefore the convergence speed of the MFBLMS is significantly higher than that of the TDLMS algorithm for colored reference signal. Simulations are carried out to validate the convergence behavior predicted from the theoretical analysis.

Effects of Geometric Imperfections on the Bracing Performance of Cross Beams during Construction of Composite Bridges

Steel girders require stability controls for different construction stages before the lateral-torsional support from the concrete deck occurs. The load-carrying capacity of bridge girders and bracing forces generated in their bracings are very sensitive to the girders' initial imperfections in terms of both the magnitude and distribution along the span. Relatively little knowledge is available in this matter; however, decisions on the worst shape of imperfections that gives a conservative resistance and/or maximizes the bracings forces is often not an easy task in practice. The present paper reports the test results (in terms of the load-carrying capacities and bracing forces) of a large-scale bridge of twin I-girder type in which the location of the intermediate cross beam was varied across the depth of the main girders. Moreover, extensive numerical investigations were performed to study the effects of some relevant shape of imperfections on both the load-carrying capacities of the studied bridges and the magnitude of bracing forces generated in the cross beams that were involved. The test results showed that the load-carrying capacity of steel girders can exceed their lowest theoretical eigenvalue because of the presence of initial imperfections. In the finite-element analyses of the studied bridge cases, in some cases, the girders followed their shape of geometric imperfections and reached a load value that was greater than the lowest eigenvalue of the systems.

Block-diagonal preconditioning for spectral stochastic finite-element systems

Deterministic models of fluid flow and the transport of chemicals in flows in heterogeneous porous media incorporate partial differential equations (PDEs) whose material parameters are assumed to be known exactly. To tackle more realistic stochastic flow problems, it is fitting to represent the permeability coefficients as random fields with prescribed statistics. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions are averaged to obtain statistical properties of the solution variables. Alternatively, so-called stochastic finite-element methods (SFEMs) discretize the probabilistic dimension of the PDE directly leading to a single structured linear system. The latter approach is becoming extremely popular but its computational cost is still perceived to be problematic as this system is orders of magnitude larger than for the corresponding deterministic problem. A simple block-diagonal preconditioning strategy incorporating only the mean component of the random field coefficient and based on incomplete factorizations has been employed in the literature and observed to be robust, for problems of moderate variance, but without theoretical analysis. We solve the stochastic Darcy flow problem in primal formulation via the spectral SFEM and focus on its efficient iterative solution. To achieve optimal computational complexity, we base our block-diagonal preconditioner on algebraic multigrid. In addition, we provide new theoretical eigenvalue bounds for the preconditioned system matrix. By highlighting the dependence of these bounds on all the SFEM parameters, we illustrate, in particular, why enriching the stochastic approximation space leads to indefinite system matrices when unbounded random variables are employed.

Final states of decaying 2D turbulence in bounded domains: Influence of the geometry

Direct numerical simulations of two-dimensional decaying turbulence in wall bounded domains are presented. The Navier–Stokes equations are solved using a Fourier pseudo-spectral method with volume penalization. Starting from random initial conditions, we study the influence of the geometry of the domain on the flow dynamics, in particular on the long time behaviour. Circular, square, triangular and annular domains are considered and we show how the geometry plays a crucial role regarding the decay scenario towards final states. Three stages can be distinguished: formation of coherent vortices from random initial conditions, vortex wall interactions, and finally relaxation towards a quasi-steady structure. The eigenvalues estimated from the decay rate of both energy and enstrophy depend on the geometry and agree well with the theoretical eigenvalues based on the Stokes mode of the corresponding domain. For the final states we find a linear functional relation between vorticity and streamfunction.

Horizontal layered structure with heterogeneity beneath continents and island arcs from particle orbits of long-period P waves

We investigate the particle orbits of long-period (about 20 s) P waves observed with the global seismic network. By analysing 84 three-component seismograms recorded at 25 stations from 60 earthquakes occurring beneath 300 km, we quantitatively evaluate the orbits by three sets of eigenvalues and eigenvectors, using a covariance matrix method. The eigenvalues for P waves recorded at stations located on continents are explained by the standard horizontal layered structure model (iasp91). On the other hand, the orbits observed at stations close to island arcs are affected not only by the horizontal layered structure but also by heterogeneity due to subducting plates, mantle diapirs and so on. On the basis of a single-scattering model for a plane P wave, we quantify the heterogeneities by an isotropic scattering coefficient g0. Fitting the theoretical eigenvalues to the observed ones, we estimate g0 for the crust and upper mantle beneath continents to be less than 0.0005 km-1, and the mean g0 for the structure beneath island arcs to be about 0.0015 to 0.003 km-1.

The bending dynamics of acetylene

The dynamics and spectroscopy of (J=0) acetylene bending degrees of freedom are investigated using a reduced dimensional Hamiltonian. This Hamiltonian is obtained by applying an adiabatic approximation to average the vibrational Hamiltonian over the ground state in the three stretch coordinates. Within this approximation, an effective bend force field is obtained by adjusting force constants in the adiabatic potential to improve agreement between experimental and theoretical eigenvalues. With minor modification, a global bend force field is determined that qualitatively describes the vinylidene vibrations and quantitatively describes the acetylene vibrations. This surface is compared to the results of a recent ab initio calculation. A dispersed fluorescence spectrum out of the excited à state, calculated from this model, is found to agree well with results of a recent experimental study.

Estimation of the shape factor of a palaeostress ellipsoid by comparison with theoretical slickenline patterns and application of an eigenvalue method

Eigenvector analysis of the orientation matrix of slickenside striations in britteley deformed rocks is used to estimate the shape factor R = (σ 2 − σ 3) (σ 1 − σ 3) of a palaeostress ellipsoid. In a first step the displacement data are drawn with a computer program as fault slip lines following the method of Hoeppener (1955). Data which obviously do not fit into the general pattern can interactively be erased. This procedure is supported by superimposition of theoretical slickenlines calculated with the formula of Bott (1959) for a pattern of uniformly distributed theoretical fault planes. In a second step the eigenvalues of the orientation matrix of the measured striation data are calculated and compared with the eigenvalues of the calculated directions of the maximum resolved shear stress on the measured fault planes. In order to make a comparison between the eigenvalues of the measured striations and the theoretical maximum resolved shear stresses easier, all sets of eigenvalues are plotted as points in a triangle plot. The R-factor of the sample can be found by interpolation with the nearest theoretical eigenvalues. The arrangement of points which represent the theoretical eigenvalues and the position of the sample point can be used as criteria for the consistency of the data set. In a consistent data set all points, the sample point included, should lie on a straight line.

On the discretization of n-dimensional Laplace equation by finite differences and its theoretical eigenvalue analysis

On the discretization of n-dimensional Laplace equation by finite differences and its theoretical eigenvalue analysis

New f-values in C I and the CNO abundances in the sun

In an attempt to refine our knowledge of the solar abundances of the CNO elements, we discuss in this paper the implications of a new accurate set of oscillator strengths recently calculated in intermediate coupling for the 2p( 2 P o )nl-2p( 2 P o )n'l' E1 transitions of astrophysical interest in C I and also for the forbidden transitions within the 2p 2 ground configuration. As with previous analyses devoted to nitrogen and oxygen, configuration interaction effects have been considered in the calculations in a detailed way and empirical adjustments have been introduced in the diagonal Hamiltonian matrix elements in order to improve the agreement between theoretical eigenvalues and experimental energy differences

Eine Beschreibung von Hückel-Parametern mit Hilfe von Kennzahlen / A Description of Hüchel Parameters by Use of Characterizing Indices

Abstract In many cases simple graph theoretical eigenvalue methods suffice to explain properties of molecules. Introducing valuations or weights of graphs (e.g. Hückel-Coulomb-and Hückel-resonance integrals) one may ask how they are related to effective nuclear charges and bond distances. We are interested in whether such relations are bijective (injective and surjective) mappings. Injectivity and/or sujectivity can be characterized by indices, for instance “solution numbers” of non linear equations. Using as physically motivated starting points (I) a simple effective model Hamiltonian and (II) a semiempirical formula for β (resonance integral) and a general expression for α (Coulomb integral) we can show that the mappings resulting from (I) and (II) agree with respect to their indices. Therefore we conclude that omitted terms in (I) are immaterial with respect to the mapping properties.

Alternate interpretation of final state relaxation for chemisorbed molecules

Ab initio Xα electronic spectra have been calculated for CO before and after chemisorption onto Ni(001) in the observed c(2 × 2) symmetry. The electronic spectra of a monolayer of C 2H 2 has also been calculated and referenced to E F of Ni. The theoretical eigenvalue spectra of these two molecules agrees colosely with ionization potentials measured by UPS when these molecules are chemisorbed onto Ni. Charge density calculations for CO indicate that the molecular orbitais are delocalized by the COCO and CONi bonds. These results suggest that final state relaxation effects are small and possibly zero for the chemisorbed molecule and that the relaxation energy of chemisorbed molecules is approximately the correlation energy.

The shape of low-speed capillary jets of Newtonian liquids

The shape of a jet of Newtonian liquid issuing from a capillary needle into air is considered. The results of two theoretical approaches are presented. One approach is a perturbation analysis about the final state of the jet and the other is a boundary-layer analysis near the point of jet formation. Comparison of the predictions with experimental jet shapes shows them to be in semi-quantitative agreement. Especially interesting is the presence of a ‘discontinuity’ in the empirical exponential decay rate of the jet radius occurring at a Reynolds number somewhere between 14 and 20 and the correspondence of this discontinuity with the peculiar behaviour in this range of the Reynolds number of the theoretical eigenvalue.

The stability and transition of a two-dimensional jet

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.