Expansion Coefficients Of Functions Research Articles

Acceleration of Spectral Domain Approach for Generalized Multilayered Shielded Microstrip Interconnects Using Two Fast Convergent Series

A simple and versatile approach for the acceleration of the spectral domain approach for generalized shielded microstrip interconnects is presented. This approach uses asymptotic expansion for the Bessel's function and the Green's function followed by an approximation of infinite summation using two fast convergent sine cosine series. The asymptotic expansion coefficients of the Green's functions are found by a combination of analytical and numerical approaches. The infinite summation involved in the computation of the elements of the Galerkin matrix is accelerated using two fast convergent sine cosine series. The use of a few entire-domain basis functions gives very accurate results for the propagation constants for any mode in the general case of a multilayered shielded microstrip line. In addition to this, closed-form expressions are developed to choose the number of terms and some parameters to adaptively accelerate the convergence of the second type of fast convergent series for a given accuracy. This approach can be extended to handle multiple interconnects in the same layer and in different layers and also to include the effect of finite thickness and conductivity.

Green's function retrieval from the CCF of random waves and energy conservation for an obstacle of arbitrary shape: noise source distribution on a large surrounding shell

SUMMARY For imaging the earth structure, the cross-correlation function (CCF) of random waves as ambient noise or coda waves has been widely used for the estimation of the Green’s function. We precisely study the condition for the Green’s function retrieval in relation to the energy conservation for a single obstacle of arbitrary shape. When an obstacle is placed in a 2-D homogeneous medium, the Green’s function is written by a double series expansion using Hankel functions of the first kind which represent outgoing waves. When two receivers and the scattering obstacle are illuminated by uncorrelated noise sources randomly and uniformly distributed on a closed circle of a large radius surrounding them, the lag-time derivative of the CCF of random waves at the two receivers can be written by a convolution of the antisymmetrized Green’s function and the autocorrelation function of the noise source time function. We explicitly derive the constraint for the Hankel function expansion coefficients as the sufficient condition for the Green’s function retrieval. We show that the constraint is equal to the generalized optical theorem derived from the energy conservation principle. Physical meaningofthegeneralizedopticaltheorembecomesclearwhentheHankelfunctionexpansion coefficients are transformed into scattering amplitudes in the framework of the conventional scattering theory. In the 3-D case, the Green’s function is written by a double series expansion using spherical Hankel functions of the first kind and spherical harmonic functions. When two receivers and the scattering obstacle are illuminated by noise sources randomly and uniformly distributed on a closed spherical shell of a large radius surrounding them, we explicitly derive the constraint for the spherical Hankel function expansion coefficients for the Green’s function retrieval and the energy conservation. We note that the derivation of the constraint does not assume that two receivers are in the far field of the scattering obstacle.

A method of multivariable Hermite basis function approximation

A method of multivariable (multivariate) Hermite function based approximation is presented and discussed. The multivariable basis is constructed as a product of one-variable Hermite functions with adjustable scaling parameters. Thanks basis orthonormality, the approximated function expansion coefficients are calculated by using explicit, non-search formulae. The scaling parameters are determined via a search algorithm. Initially, an excessive number of functions in the basis is calculated, then a simple pruning method is applied. Only those are taken which contribute the most to error decrease, down to a desired level. The method ensures a very good generalization property. This claim is supported by both theoretical considerations and working examples.

Novel binding pocket descriptors based on DrugScore potential fields encoded by 3D Zernike descriptors

Proteins interact with other molecules, e.g. ligands or other proteins, in specific binding sites. Key factors for these interactions are the shape, size, and buriedness of the binding site, as well as its physicochemical composition. Since all these properties usually significantly vary among different proteins, up to now there is no standard definition what constitutes a binding site [1]. Thus, novel pocket descriptors allowing an in-depth characterization of binding sites are highly desired. Hence, we developed novel binding pocket descriptors based on 3D molecular interaction fields. The binding pocket of a protein is characterized using the distance dependent, knowledge-based pair potentials of the DrugScore scoring function [2] in combination with multiple ligand atom probes. To allow an efficient comparison of the resulting potential fields, the 3D grids are encoded using 3D Zernike descriptors. The 3D Zernike polynomials Znlm are orthonormal basis functions on the unit sphere. Thus, any 3D object can be represented as: using a 3D Zernike function expansion. We utilized the resulting function expansion coefficients cnlm, i.e. the 3D Zernike moments, to describe the 3D molecular potential fields characterizing a protein's binding pocket. The resulting descriptors are invariant under rotation, scaling, and translation and enable a fast comparison and an efficient characterization of protein binding pockets. Thus, these novel pocket descriptors can be used to predict target druggability or to calculate similarities between binding pockets, e.g. to predict potential off-targets or to perform protein function de-orphanization.

Computation of expansion coefficients of Melnikov functions near a nilpotent center

In this paper, we consider the computation problem of expansion coefficients of Melnikov functions for near-Hamiltonian systems with a nilpotent center. We develop a general method for finding the computation formulas for the expansion coefficients. Based on the new method, an efficient algorithm is established to systematically compute the coefficients. Moreover, as an application we consider a kind of Liénard systems with a nilpotent center and study the number of limit cycles near the center.

Stability of the Landau–Fermi Liquid Theory

The stability of the Landau–Fermi liquid theory is investigated. It has been shown that if the interaction function of the Fermi system is a finite function of the angle between the momenta of two particles at the Fermi surface, then the liquid can be stable. We have shown that the absolute value of the expansion coefficients of the interaction functions in Legendre polynomials are decreasing function of the coefficients indices. We solve the stability condition for one photon exchange (OPE) in an electron gas. The results show that we must use the massive boson propagator (higher order corrections to the photon propagator). Similar to previous works (Abrikosov et al. in Method of Quantum Field Theory in Statistical Physics, Pergamon, Elmsford, 1965), our result is proportional to g2. The density and temperature dependence of results is occulted in the effective mass of the system.

A recursion formula for k-Schur functions

The Bernstein operators allow one to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t = 1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t = 1 in terms of homogeneous symmetric functions.

A spherical harmonic expansion of the Boltzmann equation for nonhydrodynamic weakly ionized plasma in the presence of both electric and magnetic fields

The Boltzmann kinetic equation for nonhydrodynamic weakly ionized plasma in the presence of both electric and magnetic fields is considered. The charged particles distribution function is decomposed in terms of spherical harmonics in momentum space. After substituting the expansion into the Boltzmann equation an infinite hierarchy of differential equations for the distribution function expansion coefficients is derived. The cases of Cartesian, cylindrical and spherical coordinates in configuration space are studied. We applied obtained equations to the description of electron transport in nitrogen at high values of rf electric field intensity to number density ratio E/N.

The aerosol influence upon the polarization state of the atmosphere solar radiation

The development of aerosol radiation scattering models for the purposes of remote sensing is related to the essential account of the polarization properties (the most complete) of the scattered radiation. The smoothness of the spatial spectrum of the light field distribution vector function caused by mathematical singularities of the boundary conditions for the vectorial radiative transfer equation (vectorial radiative transfer equation) boundary problem allows us to restrict the light‐field general spherical functions expansion coefficients in a Taylor series with respect to a discrete zenith‐number (the order) of the general spherical function to two terms. This gives relatively simple expressions for the matrix general spherical function expansion coefficients of the vectorial radiative transfer equation solution – the vectorial small angle modification of the spherical harmonics method (vectorially modified spherical harmonics). The subsequent determination of the solution smooth non‐small angle part using the boundary problem for the vectorial radiative transfer equation with the vectorially modified spherical harmonics as the source function gives the complete solution to the vectorial radiative transfer equation and prevents some calculation problems caused by ill conditions of the matrices.

Study on effective condition number for collocation methods

In this paper, the stability study is further made for three kinds of collocation methods. (I) The general collocation Trefftz methods (GCTMs) using piecewise particular solutions, while in Li et al. [Stability analysis for the penalty plus hybrid and the direct Trefftz methods for singularity problems. Eng Anal Boundary Elem 2004;31:163–75] the collocation Trefftz methods (CTMs) using the uniform particular solutions in the entire domain. (II) The method of fundamental solutions (MFSs), i.e., the CTM using the fundamental solution (FS). (III) The collocation method using the radial basis functions (CM-RBFs). In this paper, the new bounds of the traditional condition number Cond and the effective condition number Cond_eff are derived, and comparisons are made to confirm that the effective condition number is a better criterion for stability analysis. The main results are as follows. (1) For Motz's problem by GCTM, Cond _ eff = O ( N ) and Cond = O ( a - N ) ( 0 < a < 1 ) are obtained by a suitable scaling, where N is the number of admissible functions used. (2) For MFS and CM-RBF, both Cond and Cond_eff are exponential with respect to N, but Cond_eff is much smaller than Cond. (3) For MFS and CM-RBF, the expansion coefficients of FS and radial basis functions may be oscillatingly large, to cause a severe subtraction cancellation in the final solutions. Different behaviours of Cond_eff compared to Cond are explored for different collocation methods in this paper, to provide a comprehensive and intrinsic nature of Cond_eff.

Coulomb Rydberg electron L-mixing in collisions with atomic ions

The cross sections of the Rydberg electron L-mixing in a hydrogen atom and a hydrogen-like ion are calculated for slow collisions with atomic ions H*(n, L) + A+ = H*(n, L′) + A+ without variation of the principal quantum number n. The probability of the L-mixing L → L′ is associated with the quantum interference of the wave functions of adiabatic states, i.e., with the mixing of the time phases of these functions exp(−i∫Ek (t)dt). The effective cross section of such L-mixing for the states with n = 28 are 4–5 orders of magnitude greater than the cross sections determined in previous investigations. The expansion coefficients of spherical Coulomb wave functions in terms of parabolic ones and vice versa, which are necessary for determining cross sections, are calculated on the basis of a comprehensive analysis of the spatial properties of these functions.

Renormalization group functions for two-dimensional phase transitions: To the problem of singular contributions

According to the available publications, the field theoretical renormalization group approach in the two-dimensional case gives the critical exponents that differ from the known exact values. This property is associated with the existence of nonanalytic contributions in the renormalization group functions. The situation is analyzed in this work using a new algorithm for summing divergent series that makes it possible to determine the dependence of the results for the critical exponents on the expansion coefficients for the renormalization group functions. It has been shown that the exact values of all the exponents can be obtained with a reasonable form of the coefficient functions. These functions have small nonmonotonic sections or inflections, which are poorly reproduced in natural interpolations. It is not necessary to assume the existence of singular contributions in the renormalization group functions.

Locality properties of radial basis function expansion coefficients for equispaced interpolation

Many types of radial basis functions (RBFs) are global in terms of having large magnitude across the entire domain. Yet, in contrast, e.g. with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centred in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well-conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1D and 2D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.

Semiempirical study of perturbations of the landé g factors of electronic-vibrational-rotational levels of hydrogen: IV. The I 1Π−g, R 1Π−g, J 1Δ−g, and S 1Δ−g states of the H2 and D2 molecules

Values of the Lande g factors are obtained for the first time for rovibronic levels of the (3d)J1Δg− state of the D2 molecule and of the (4d)S1Δg− state of the H2 and D2 molecules. Semiempirical values of these quantities are obtained for the first time for the (4d)R1Πg− state of the H2 molecule and the (3d)I1Πg− and (4d)R1Πg− states of the D2 molecule. These values are determined in terms of the nonadiabatic model, which takes into account the interaction between the ndπ1Πg and ndδ1Δg (n = 3 and 4) states in the approximation of pure precession, with the use of the semiempirical values obtained by us for the expansion coefficients of wave functions in the adiabatic basis functions and the results of our numerical calculation of overlap integrals of the vibrational wave functions of these states. The values thus obtained are in good agreement with the data available in the literature. It is found that the interference effects of coupling of the ndπ1Πg and ndδ1Δg states lead to significant perturbations of the dependences of the g factors of vibronic levels on the rotational quantum number as compared to the adiabatic results. These perturbations are regular (reaching a factor of nine) for the I1Πg− and R1Πg− states and irregular (reaching a factor of 50) for the J1Δg− and S1Δg− states. The nonadiabatic values of g factors obtained in our previous studies for the H2, HD, and D2 molecules in the i3Πg− and j3Δg− states are improved by an additional inclusion of adiabatic effects. It is established that the effect of the electron spin on the g factor values for rovibronic levels of the 3dΔg− states is significant and reaches 25% for H2 and 130% for D2.

Infinite-σ Limits For Tikhonov Regularization

We consider the problem of Tikhonov regularization with a general convex loss function: this formalism includes support vector machines and regularized least squares. For a family of kernels that includes the Gaussian, parameterized by a bandwidth parameter σ, we characterize the limiting solution as σ → ∞. In particular, we show that if we set the regularization parameter λ = ~λ σ-2p, the regularization term of the Tikhonov problem tends to an indicator function on polynomials of degree ⌊p⌋ (with residual regularization in the case where p ∈ Z). The proof rests on two key ideas: epi-convergence, a notion of functional convergence under which limits of minimizers converge to minimizers of limits, and a value-based formulation of learning, where we work with regularization on the function output values (y) as opposed to the function expansion coefficients in the RKHS. Our result generalizes and unifies previous results in this area.

Recent Results on Four-loop Tadpoles

The experimental measurable R -ratio can be used to perform a precise determination of the charm- and bottom-quark mass with the help of theoretical computable expansion coefficients of the vacuum polarization function. The four-loop contributions in perturbative QCD to the first two expansion coefficients of the vacuum polarization functions are presented.

Ratio control variate method for efficiently determining high‐dimensional model representations

The High-Dimensional Model Representation (HDMR) technique is a family of approaches to efficiently interpolate high-dimensional functions. RS(Random Sampling)-HDMR is a practical form of HDMR based on randomly sampling the overall function, and utilizing orthonormal polynomial expansions to approximate the RS-HDMR component functions. The determination of the expansion coefficients for the component functions employs Monte Carlo integration, which controls the accuracy of the RS-HDMR interpolation. The control variate method is an established approach to improve the accuracy of Monte Carlo integration. However, this method is often not practical for an arbitrary function f(x) because there is no general way to find the analytical control variate function h(x), which needs to be very similar to f(x). In this article, we show that truncated RS-HDMR expansions can be used as h(x) for arbitrary f(x), and a more stable iterative ratio control variate method was developed for the determination of the expansion coefficients for the RS-HDMR component functions. As the asymptotic error (standard deviation) of the estimator given by the ratio control variate method is proportional to 1/N(sample size), it is more efficient than the direct Monte Carlo integration, whose error is proportional to 1/square root(N). In an illustration of a four-dimensional atmospheric model a few hundred random samples are sufficient to construct an RS-HDMR expansion by the ratio control variate method with an accuracy comparable to that obtained by direct Monte Carlo integration with thousands of samples.

Generation of the basis sets for multi-Gaussian ultrasonic beam models—An overview

By using a small number of Gaussian basis functions, one can synthesize the wave fields radiated from planar and focused piston transducers in the form of a superposition of Gaussian beams. Since Gaussian beams can be transmitted through complex geometries and media, such multi-Gaussian beam models have become powerful simulation tools. In previous studies the basis function expansion coefficients of multi-Gaussian beam models have been obtained by both spatial domain and k-space domain methods. Here, we will give an overview of these two methods and relate their expansion coefficients. We will demonstrate that the expansion coefficients that have been optimized for circular piston transducers can also be used to generate improved field simulations for rectangular probes. It will also be shown that because Gaussian beams are only approximate (paraxial) solutions to the wave equation, a multi-Gaussian beam model is ultimately limited in the accuracy it can obtain in the very near field.

Right and wrong near critical endpoints

The implications of convexity (or thermodynamic stability) for isothermal density (or composition) phase diagrams near critical endpoints are investigated. In particular, Schreinemakers’ rules for the geometry of three coexisting phases in a space of densities are refined and extended physically to apply at and in the vicinity of a critical endpoint. Seemingly plausible phase diagrams are presented which violate the extended rules (and, in most cases, the Second Law of Thermodynamics). The requirements of convexity, supplemented for Ising-related systems by the Griffiths–Kelly–Sherman inequalities, also restrict the signs of important expansion coefficients of thermodynamic scaling functions for criticality and provide bounds on universal critical amplitude ratios.

Calculation of fine-structure parameters of the npn′s configurations by the semiempirical method

Fine-structure parameters, expansion coefficients of wave functions in the LS-coupling basis, and gyromagnetic ratios are obtained for zero residuals in energies of fine-structure levels (differences between calculated and experimental energies) for a greater number of elements of the npn′s configurations in the intermediate-coupling scheme. The results are compared with data available in the literature.