Series Expansion Of Functions Research Articles

Diffraction patterns of amorphous materials as a series expansion of neighbor distribution functions

An exact analytical expression for the static structure factor in disordered materials is derived from Fourier transformed neighbor distribution decompositions in real space, and permits to reconstruct the function in an iterative fashion. The result is successfully compared to experimental data of archetypal glasses or amorphous materials (GeS2, As2Se3, GeTe), and links quantitatively knowledge of structural information on short and intermediate -range order with the motifs found on the diffraction patterns in reciprocal space. The approach furthermore reveals that only a limited number of neighbor shells is sufficient to reasonably describe the structure factor for k > 2 . In the limit of the high momentum transfer, the oscillation characteristics of the interference function are related with new informations on the short-range order of disordered materials.

Electron states for gapped pseudospin-1 fermions in the field of a charged impurity

The electron states of gapped pseudospin-1 fermions of the $\alpha-{\cal T}_3$ lattice in the Coulomb field of a charged impurity are studied. The free $\alpha-{\cal T}_3$ model has three dispersive bands with two energy gaps between them depending on the parameter $\Theta$ which controls the coupling of atoms of honeycomb lattice with atoms in the center of each hexagon, thus, interpolating between graphene $\Theta=0$ and the dice model $\Theta=\pi/4$. The middle band becomes flat one with zero energy in the dice model. The bound electron states are found in the two cases: the centrally symmetric potential well and a regularized Coulomb potential of the charged impurity. As the charge of impurity increases, bound state energy levels descend from the upper and central continua and dive at certain critical charges into the central and lower continuum, respectively. In the dice model, it is found that the flat band survives in the presence of a potential well, however, is absent in the case of the Coulomb potential. The analytical results are presented for the energy levels near continuum boundaries in the potential well. For the genuine Coulomb potential, we present the recursion relations that determine the coefficients of the series expansion of wave functions of bound states. It is shown that the condition for the termination of the series expansion gives two equations relating energy and charge values. Hence, analytical solutions can exist for a countably infinite set of values of impurity charge at fixed $\Theta$.

Quasi Linkage Equilibrium under Weak Two-Locus Viability Selection: I. Haploid Population with Diallelic Loci

A model of weak viability selection at two diallelic loci with standardization of approaches through the use of perturbation theory is examined. The estimate of the quasi-equilibrium value for the linkage disequilibrium coefficient D is analyzed, and results in terms of average effects in quantitative genetics and in terms of the theory of singular perturbations in mathematics are obtained. The approximation of a discrete-time model of a random mating population with non-overlapping generations under weak selection by ordinary differential equations is considered. Weak selection is considered as a perturbation of the model without selection. The resulting model is singularly perturbed; that is, fast (D) and slow (allele frequencies) variables can be distinguished. The first approximation equation for quasi-equilibrium of D is obtained using the first terms of the Taylor series expansion of the model functions. It coincides with the corresponding part of the system of the first approximation of the asymptotic series for solving singularly perturbed equations. The first approximation for quasi-equilibrium of D is D* = e(p) $$\frac{\mu }{r}$$ x(1 – x) y(1 – y), e(p) ≡ v11(p) – v12(p) – v21(p) + v22(p), where μ is the intensity of selection, r is the recombination coefficient, e(p) is the index of epistasis nonadditivity for the viabilities $${{{v}}_{{{{i}_{{\text{1}}}}{{i}_{{\text{2}}}}}}}$$ , x and y are the allele frequencies of the first and second locus, respectively. The evolution of a nonequilibrium state of the population actually takes place at quasi-equilibrium (as the result of the stage of fast variable D dynamics) and current values of slow x, y.

Second-order variational equations for spatial point processes with a view to pair correlation function estimation

Second-order variational type equations for spatial point processes are established. In case of log linear parametric models for pair correlation functions, it is demonstrated that the variational equations can be applied to construct estimating equations with closed form solutions for the parameter estimates. This result is used to fit orthogonal series expansions of log pair correlation functions of general form.

Stress Analytical Solution for Shallow Buried Lined Circular Tunnel Under the Deformation of Surrounding Rock Inner Edge

In this article, an analytical solution is presented for an elastic shallow buried lined tunnel, which consider a certain surrounding rock deformation at the inner boundary. Concrete lining and the surrounding rock was assumed as linearly elastic materials. The solution uses Muskhelishvili complex potential functions combined with conformal mapping method to determine stress components within concrete lining and the surrounding rock mass. The coefficient of Laurent series expansion of the stress functions is determined by a combination of analytical and numerical computations. As an example, the case of a uniform radial displacement of surrounding rock inner edge is considered in some detail. The solution was verified by FEM through an example, very good agreement was demonstrated between analytical solution and numerical solution. Through numerical examples, the effect of elasticity modular and the ratio of the diameter to buried depth on the stresses component were assessed.

Bohr inequalities for free holomorphic functions on polyballs

Multivariable operator theory is used to provide Bohr inequalities for free holomorphic functions with operator coefficients on the regular polyball Bn, n=(n1,…,nk)∈Nk, which is a noncommutative analogue of the scalar polyball (Cn1)1×⋯×(Cnk)1. The Bohr radius Kmh(Bn) (resp. Kh(Bn)) associated with the multi-homogeneous (resp. homogeneous) power series expansions of the free holomorphic functions are the main objects of study in this paper. We extend a theorem of Bombieri and Bourgain for the disc D:={z∈C:|z|<1} to the polyball, and obtain the estimations 13k<Kmh(Bn)<2log⁡kk if k>1, extending Boas-Khavinson result for the scalar polydisc Dk.With respect to the homogeneous power series expansion, we prove that Kh(Bn)=1/3, extending the classical result, and obtain analogues of Carathéodory, Fejér, and Egerváry-Szász inequalities for free holomorphic functions with operator coefficients and positive real parts on the polyball. These results are used to provide multivariable analogues of Landau's inequality and Bohr's inequality when the norm is replaced by the numerical radius of an operator.When specialized to the regular polydisc Dk (which corresponds to the case n1=⋯=nk=1), we obtain new results concerning Bohr, Landau, Fejér, and Harnack inequalities for operator-valued holomorphic functions and k-pluriharmonic functions on the scalar polydisc Dk. The results of the paper can be used to obtain Bohr type inequalities for the noncommutative ball algebra An, the Hardy algebra Fn∞, and the C⁎-algebra C⁎(S), generated by the universal model S of the polyball Bn.

Radial: A Fortran subroutine package for the solution of the radial Schrödinger and Dirac wave equations

The Fortran subroutine package radial for the numerical solution of the Schrödinger and Dirac wave equations of electrons in central potentials V(r) is described. The considered potentials are such that the function V(r)≡rV(r) is finite for all r and tends to constant values when r→0 and r→∞. This includes finite-range potentials as well as combinations of Coulomb and finite-range potentials. The function V(r) used in the numerical calculation is the natural cubic spline that interpolates a table of values provided by the user. The radial wave equations are solved by using piecewise exact power series expansions of the radial functions, which are summed up to the prescribed accuracy so that truncation errors can be completely avoided. The radial subroutines compute radial wave functions, eigenvalues for bound states and phase shifts for free states. Specific subroutines are also provided for computing the radial functions and phase shifts for free states of complex optical potentials having a finite-range absorptive imaginary part. The solution subroutines are accompanied by example main programs, as well as with specific programs that perform calculations relevant in atomic, nuclear, and radiation physics (the self-consistent solution of the Dirac–Hartee–Fock–Slater equations for neutral atoms and positive ions, and the calculation of cross sections for elastic scattering of high-energy electrons and positrons by atoms and of nucleons by nuclei). The distribution package includes a detailed manual with a description of the basic physics and the mathematical formulas implemented in the subroutines. Program summaryProgram Title:radialProgram Files doi:http://dx.doi.org/10.17632/h6nbws5h6s.1Licensing provisions: CC by NC 3.0Programming language: Fortran 90/95Supplementary material: Manual in pdf format. Includes a description of the basic physics, and the complete set of formulas implemented in the program.Journal reference of previous version: Comput. Phys. Commun. 90 (1995) 151–168.Does the new version supersede the previous version?: YesReasons for the new version: The present subroutine package contains a revision and extension of the radial subroutines [1], which have been recoded. The new subroutines implement a more accurate normalisation of bound states; the phase convention of Dirac’s central-field orbitals has been replaced with that of Rose [2]. Subroutines for computing free states of complex optical-model potentials with an absorptive imaginary part have been added to the package. Subroutines giving the coefficients of asymptotic power-series expansions of radial functions for bound and free orbitals are also included. Finally, the present package contains complementary programs that use the radial subroutines for solving generic problems of interest in atomic and nuclear physics, namely, the self-consistent solution of the Dirac–Hartree–Fock–Slater equations for free neutral atoms and positive ions, and the calculation of differential cross sections for elastic collisions of electrons and positrons with atoms, and the scattering of nucleons by nuclei. The package also includes a manual in pdf format with a summary of the underlying physics and a detailed description of the formulas and numerical algorithms implemented in the programs, as well as derivations of the most relevant components.Summary of revisions: The original subroutines have been recoded and extended to include 1) a more accurate normalisation of bound states, 2) the calculation of free states of complex optical-model potentials, 3) asymptotic power-series expansions of radial functions, and 4) three practical applications.Nature of problem: The radial subroutines solve the radial Schrödinger and Dirac wave equations for a particle in a central potential V(r). They deliver the radial function(s), the energy eigenvalue for bound orbitals, and the phase shift for free orbitals.Solution method: The potential function V(r)≡rV(r) is defined by a table of values for a grid of radii that is dense enough to allow accurate natural cubic spline interpolation. The radial function is calculated for the potential defined by the interpolating cubic spline by means of exact power-series expansions, which are summed up to the desired accuracy.Additional comments including restrictions and unusual features: The subroutines are not capable of computing either bound or free states with energies very close to zero. Since the power-series expansions of the radial functions define exact piecewise solutions of the radial wave equations, they can be summed up to the accuracy required by the user. Truncation errors are controlled by the tolerance parameter ϵ, which determines the accuracy of the series summations. With double-precision arithmetic and ϵ∼10−15 the results are affected by only round-off errors. The subroutines deliver the values of the radial functions on a grid of radii defined by the user, which may be different from the grid used to specify the potential.[1] Salvat, F., J.M. Fernández-Varea, and W. Williamson, Jr. (1995), “Accurate numerical solution of the radial Schrödinger and Dirac wave equations.” Comput. Phys. Commun. 90, 151–168.[2] Rose, M.E. (1961), Relativistic Electron Theory (John Wiley and Sons, New York).

Плоский напружений стан для смуги, що ослаблена тріщиною

The plane stress elastic infinite strip problem of a finite longitudinal crack is investigated. The method that can be applied to calculate the stress state and the displacements for an infinite and semi-infinite strip with the longitudinal crack and arbitrary configuration of the boundary conditions is proposed. The main advantage of this method lies in the absence of necessity for use of the apparatus of the matrix differential calculus. Initial problem is reduced to the one-dimensional boundary value problem with the help of the generalized scheme of the integral transform method. By using the inverse integral Fourier transform, the one-dimensional problem is reduced to solving of the system of singular integral equations on a finite interval. The solution of this system was constructed with the help of the method of orthogonal polynomials by means of the second kind Chebyshev polynomials series expansion of the unknown functions. A graph of dependence of the stress intensity factor (SIF) on the geometric parameters of the problem is plotted. It is shown that the SIF for the case of the said strip tends to the SIF for the case of an infinite plane as the width of the strip approaches infinity.

Computing Arithmetic Functions Using Stochastic Logic by Series Expansion

Stochastic logic implementations of complex arithmetic functions, such as trigonometric, exponential, and sigmoid, are derived based on truncated versions of their Maclaurin series expansions. This paper makes three contributions. First, it is shown that a polynomial can be implemented using multiple levels of NAND gates based on Horner’s rule, if the coefficients are alternately positive and negative and their magnitudes are monotonically decreasing. Truncated Maclaurin series expansions of arithmetic functions are used to generate polynomials which satisfy these constraints. The input and output in these functions are represented by unipolar representation. Functions including sine, cosine, tangent hyperbolic, logarithm and exponential can be implemented using this method. Second, for a polynomial that does not satisfy these constraints, it still can be implemented based on Horner’s rule if each factor of the polynomial satisfies these constraints. It is shown that functions such as $\sin \pi x/\pi$ , $e^{-ax}$ , $\tanh ax$ and $\text{sigmoid}(ax3)$ (for values of $a>1$ ) can be implemented using stochastic logic using factorization in combination with Horner’s rule. Third, format conversion is proposed for arithmetic functions with input and output represented in different formats, such as $\text{cos}\,\pi x$ given $x\in [0,1]$ and $\text{sigmoid(x)}$ given $x\in [-1,1]$ . Polynomials are transformed to equivalent forms that naturally exploit format conversions. The proposed stochastic logic circuits outperform the well-known Bernstein polynomial based and finite-state-machine (FSM) based implementations. Furthermore, the hardware complexity and the critical path of the proposed implementations are less than the well-known Bernstein polynomial based and FSM based implementations for most cases.

Динамика зажатой капли в неоднородном электрическом поле

The forced oscillations of an incompressible liquid drop under the influence of an alternating electric field are considered. The drop is sandwiched between two homogeneous plates and surrounded by an incompressible fluid of a different density. In equilibrium, the drop has the shape of a circular cylinder which is bounded axially by parallel solid plates. These plates have different surface properties. An external electric field acts as an external force which is a reason of the motion of the contact line. A modified Hocking boundary condition is used to describe the motion of the contact line: the velocity of the contact line is proportional to the deviation of the contact angle and the rate of fast relaxation processes, the frequency of which is proportional to the doubled frequency of the electric field. Using this equation allows one to qualitatively describe the experimental dependence of the contact angle as a function of stress, in contrast to the Young – Lippmann equation. The solution to the problem is presented as a Fourier series expansion in eigen functions of the Laplace operator. Graphs of the amplitude-frequency characteristics and evolution of the drop shape are constructed for various values of the problem parameters. In the case of identical surfaces, only odd modes of droplet oscillations are excited, while for different surfaces, even harmonics additionally appear. It was found that in a wide range of parameters the shape of the side surface of the drop is close to the description by an odd function. The oscillation amplitude of even modes is significant only near resonances at the frequencies of these modes. In the case of an inhomogeneous field, azimuthal oscillations are excited, which leads to the appearance of additional resonance peaks. Travel waves propagate along the lateral surface of the drop, which are caused by vibrations of the contact line and the contact angle. The dependences of the values of the contact angle on the upper and lower surfaces on the square root of the amplitude for different values of the frequency of the electric field, which qualitatively coincide with similar graphs of experimental data, are constructed. It is possible to determine the parameters of the Hocking.

A controllable-accuracy stochastic time-domain technique for randomly varying microwave applications

PurposeThis paper aims to introduce an efficient time-domain formulation of adjustable accuracy for a consistent and trustworthy computation of electromagnetic field characteristics in randomly varying configurations. The developed methodology is carefully certified via comprehensive comparisons with the corresponding outcomes obtained by the Monte Carlo approach.Design/methodology/approachThe presented methodology uses higher-order approximations of Taylor series expansions of stochastic multivariable functions for the rapid estimation of the electromagnetic field component mean value and confidence intervals of their variance. Toward this objective, new time-update equations for the mean value and the variance of the involved electromagnetic field are elaborately derived.FindingsThe featured technique presents an efficient alternative to the excessively resource-consuming Monte Carlo finite-difference time-domain (MC–FDTD) implementation, which requires an unduly number of realizations to achieve a satisfying convergence. The higher-order stochastic algorithm retrieves accurately the statistical properties of all electromagnetic field in a single simulation, presenting promising accuracy, stability and convergence.Originality/valueThe adjustable-accuracy higher-order scheme introduces a new framework for the derivation of the stochastic explicit time-update equations and precisely computes the required confidence intervals for the electromagnetic field variance instead of the variance itself, which can be deemed a key advantage over existing schemes. This fully controllable formulation results in significantly more accurate calculations of the electromagnetic field variance, especially for larger fluctuations of the involved electromagnetic media parameters.

Sums of finite products of Chebyshev polynomials of the third and fourth kinds

In this paper, we study sums of finite products of Chebyshev polynomials of the third and fourth kinds and obtain Fourier series expansions of functions associated with them. Then from these Fourier series expansions we will be able to express those sums of finite products as linear combinations of Bernoulli polynomials and to have some identities from those expressions.

Sums of finite products of Legendre and Laguerre polynomials

In this paper, we study sums of finite products of Legendre and Laguerre polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we are going to express those sums of finite products as linear combinations of Bernoulli polynomials. Further, by using a method other than Fourier series expansions, we will be able to express those sums in terms of Euler polynomials.

Mean-field density functional theory of a nanoconfined classical, three-dimensional Heisenberg fluid. II. The interplay between molecular packing and orientational order.

As in Paper I of this series of papers [S. M. Cattes et al., J. Chem. Phys. 144, 194704 (2016)], we study a Heisenberg fluid confined to a nanoscopic slit pore with smooth walls. The pore walls can either energetically discriminate specific orientations of the molecules next to them or are indifferent to molecular orientations. Unlike in Paper I, we employ a version of classical density functional theory that allows us to explicitly account for the stratification of the fluid (i.e., the formation of molecular layers) as a consequence of the symmetry-breaking presence of the pore walls. We treat this stratification within the White Bear version (Mark I) of fundamental measure theory. Thus, in this work, we focus on the interplay between local packing of the molecules and orientational features. In particular, we demonstrate why a critical end point can only exist if the pore walls are not energetically discriminating specific molecular orientations. We analyze in detail the positional and orientational order of the confined fluid and show that reorienting molecules across the pore space can be a two-dimensional process. Last but not least, we propose an algorithm based upon a series expansion of Bessel functions of the first kind with which we can solve certain types of integrals in a very efficient manner.

Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models

We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^{2}$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parameterised in terms of a series expansion of basis functions (such as that given by a Karhunen–Loeve expansion). We introduce and analyse schemes for direct unbiased estimation along such an expansion. However, a substantial component of our paper is devoted to the study of MCMC schemes which, due to their infinite dimensionality, cannot be directly implemented, but which can be effectively estimated unbiasedly. For all our methods we give theory to justify the numerical stability for robust Monte Carlo implementation, and in some cases we illustrate using simulations. Interestingly the computational efficiency of our methods is usually comparable to simpler methods which are biased. Crucial to the effectiveness of our proposed methodology is the construction of appropriate couplings, many of which resonate strongly with the Monte Carlo constructions used in the coupling from the past algorithm.

Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials

In this paper, we consider sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we can express those sums of finite products in terms of Bernoulli polynomials and obtain some identities by using those expressions.

An improved method for Wiener–Hammerstein system identification based on the Fractional Approach

This paper develops and analyses a novel method for identifying Wiener–Hammerstein models, i.e. models consisting of two linear dynamic parts with a static non-linearity in between. Starting from the best linear model, which is a consistent estimate of the system dynamics for Gaussian excitation, the identification problem includes the partitioning of the poles and zeros between the two linear parts. This partitioning can be formulated as a discrete optimization problem. The fractional approach considers a relaxation of this problem into a continuous one, by parameterizing the partition of each pole and zero in a fractional way, and carrying out the computations in the frequency domain. In this paper it is shown that the fractional approach becomes ill-conditioned for some configurations of the poles and zeros of the linear dynamic parts, causing identifiability issues. A modification of the original fractional approach is then introduced, based on series expansion of the fractional transfer functions. This modification shares most of the properties of the fractional approach. However, it is shown that it provides an implicit regularization of the identification problem. It addresses the ill-conditioning problem while preserving meaningful statistical properties of the estimation. Furthermore, a lifted formulation of the estimation problem is proposed, which improves the algorithmic performance in the framework of Newton-based methods.

Magnetic properties of spinels GeNi2−xCoxO4 systems: Green's function and high-temperature series expansions

ABSTRACTThe Green's function theory and high-temperature series expansions technical have been developed for magnetic systems GeNi2−xCoxO4. We have applied the Green's function theory to evaluate thermal magnetization and magnetic susceptibility for different values of magnetic field and dilution x, considering all components of the magnetization when an external magnetic field is applied in (x,z)-plane. The second theory combined with the Padé approximants method for a randomly diluted Heisenberg magnet is used to deduce the magnetic phase diagram of GeNi2 − xCoxO4 systems. The critical exponents and associated with the magnetic susceptibility and the correlation length ξ, respectively, have been deduced. The theoretical results are compared with those given by magnetic measurements.

A novel single-phase-to-earth fault location method for distribution network based on zero-sequence components distribution characteristics

Based on distribution characteristics of the zero-sequence voltages and currents of the distributed lines, this paper proposes a novel single-phase-to-earth fault (SPTEF) location scheme for distribution network. Firstly, the distributed parameter model of distribution feeder lines is appropriately simplified according to Taylor series expansion of the hyperbolic functions in the model. The fault factor is constructed to select the faulted feeder. Then, the distribution features of the zero-sequence voltage (ZV) along the feeder line are analyzed. In each section of the faulted feeder, golden section search algorithm is adopted to distinguish the faulted section. Finally, based on the concept of maximum fault current, accurate fault location is realized by genetic algorithm (GA). A radial distribution network model containing different types of distribution lines is established in PSCAD/EMTDC. The calculation results of MATLAB prove the validity, reliability and accuracy of the proposed fault location scheme.

On the Andrews–Warnaar identities for partial theta functions

In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar but also unveils a new characteristic of such identities. As further applications, we establish a general form of Warnaar's identity and a general q-series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.