Expansion Coefficients Of Functions Research Articles

Dwork Crystals III: From Excellent Frobenius Lifts Towards Supercongruences

Abstract This paper is a continuation of our Dwork crystals series. Here we exploit the Cartier operation to prove supercongruences for expansion coefficients of rational functions. In the process, it appears that excellent Frobenius lifts are a driving force behind supercongruences. Originally introduced by Dwork, these excellent lifts have occurred rather infrequently in the literature, and only in the context of families of elliptic curves and abelian varieties. In the final sections of this paper, we present a list of examples that occur in the case of families of Calabi–Yau varieties.

Synthesis of Nonlinear Nonstationary Stochastic Systems by Wavelet Canonical Expansions

The article is devoted to Bayes optimization problems of nonlinear observable stochastic systems (NLOStSs) based on wavelet canonical expansions (WLCEs). Input stochastic processes (StPs) and output StPs of considered nonlinearly StSs depend on random parameters and additive independent Gaussian noises. For stochastic synthesis we use a Bayes approach with the given loss function and minimum risk condition. WLCEs are formed by covariance function expansion coefficients of two-dimensional orthonormal basis of wavelet with a compact carrier. New results: (i) a common Bayes’ criteria synthesis algorithm for NLOStSs by WLCE is presented; (ii) partial synthesis algorithms for three of Bayes’ criteria (minimum mean square error, damage accumulation and probability of error exit outside the limits) are given; (iii) an approximate algorithm based on statistical linearization; (iv) three test examples. Applications: wavelet optimization and parameter calibration in complex measurement and control systems. Some generalizations are formulated.

ARSENY: A program for computing inelastic transitions via hidden crossings in one-electron atomic ion–ion collisions with classical description of nuclear motion

The ARSENY program is intended to compute cross-sections of inelastic transitions via hidden crossings upon slow one-electron atomic ion–ion collisions using the approximation of classical description of nuclear motion in the collision energy region E<20 KeV/nucleon. Particular attention is paid to the calculation of the analytical properties of adiabatic potential energy curves of the two-center Coulomb problem for complex values of the internuclear distance. Complex branch points and hidden crossings are calculated and their significance for the dynamics of one-electron collisional systems is revealed. Benchmark calculations of cross sections for charge exchange, excitation, and ionization in slow He2++H(1s) collisions are presented and compared with experimental data. Program summaryProgram Title: ARSENYCPC Library link to program files:https://doi.org/10.17632/n43srxwdnm.1Licensing provisions: CC By 4.0Programming language: FORTRAN 90/95. Compilers: Intel(R) Visual Fortran Compiler 19.0.4.245 [IA-32]Nature of problem: The processes of charge exchange, excitation, and ionization in one-electron atomic ion–ion collisions in a low-temperature plasma are actively studied in astrophysics and tokamak Charge-eXchange Recombination Spectroscopy (CXRS) Edge diagnostics in ITER [1–7]. Numerical calculations of cross sections for charge exchange, excitation, and ionization for slow one-electron atomic ion–ion collisions within the approximation of classical description of nuclear motion have been conventionally implemented, in particular, by the electron-nuclear dynamics approach [8] and the close-coupled channel method using basis functions of the two-center Coulomb problem (see, e.g., [9–13]). However, with real-valued internuclear distance these approaches did not allow for the dynamical structure of discrete-discrete and discrete-continuous transitions in the two-center Coulomb problem. This disadvantage manifests itself in the disagreement between the theoretical and experimental cross-sections at small velocity of ions [13–15]. In a number of papers [1,14,16–20] the dynamical structure of the transitions is considered in terms of hidden crossing of potential curves, when using the analytic continuation to a complex plane of the internuclear distance. This approach is implemented in the ARSENY program aimed at computing cross sections of inelastic discrete-discrete and discrete-continuous transitions via hidden crossings for slow one-electron atomic ion–ion collisions within the approximation of classical description of nuclear motion announced in [1]. Benchmark calculations of cross sections for charge exchange, excitation, and ionization in slow He2++H(1s) collisions are presented and compared with experimental data.Solution method: The potential energy curves E=E(R) and the separation constant λ=λ(R) depending on the real-valued and complex-valued parameter R of the two-center Coulomb problem are calculated from the minimization condition for a quadratic functional corresponding to a pair of nonlinear equations with respect to a pair of unknowns E(R) and λ(R) by iterating three-term recurrent relations [21,22] using a mean least square method [23]. These equations are obtained from the known recurrent relations for the expansion coefficients of quasiradial and quasiangular spheroidal Coulomb functions. The series of branching points Rc sought for in the complex plane of internuclear distance R and the hidden crossings of complex potential energy curves E(R) are calculated by an iterative method using the known analytical behavior in the form of a square root of difference E(R)−E(Rc) in the vicinity of the unknown branch points Rc. Calculations of cross-sections of inelastic transitions via hidden crossings for the slow one-electron atomic ion–ion collisions within the approximation of classical description of nuclear motion are carried out by integration over the impact parameter of the nonadiabatic transition probability expressed in terms of the Stückelberg parameter determined integral of difference of imaginary parts of each pair of complex potential energy curves E(R) by a contour around the branching point Rc [18,19].Additional comments including restrictions and unusual features: In calculations of cross sections and required branch points, about 220 adiabatic states of the lower part of the discrete spectrum of the two-center problem are used.

On optimal convergence rates of Laguerre polynomial expansions for piecewise functions

By applying Hilb type formula and generalized van der Corput type Lemmas, this paper presents optimal decay rates on the expansion coefficients for given functions expanded in the form of Laguerre polynomial series, from which it leads to the convergence rates on the spectral orthogonal projections, and derives new estimates on the errors for Gauss–Laguerre quadrature. Ample numerical experiments are carried out to validate the optimality and accuracy of these theoretical results.

Reducing the computational load – atomic multiconfiguration calculations based on configuration state function generators

In configuration interaction (CI) calculations the atomic wave functions are given as expansions over configuration state functions (CSFs) built on relativistic one-electron orbitals. The expansion coefficients of the configuration state functions are obtained by constructing and diagonalizing the Hamiltonian matrix. Here we show how a regrouping of the configuration state functions and the introduction of configuration state function generators (CSFGs) allow for a substantial reduction of the computational load in relativistic CI calculations. The computational methodology based on configuration state function generators, recently implemented in the General Relativistic Atomic Structure package (Grasp2018, Froese Fischer et al. (2019) [16]), is applied to a number of atomic systems and correlation models with increasing sets of one-electron orbitals. We demonstrate a reduction of the CPU time with factors between 10 and 14 for the largest CI calculations. The inclusion of the Breit interaction into the calculations is time consuming. By applying restrictions on the Breit integrals we show that it is possible to further reduce the CPU times with factors between 2 and 3, with negligible changes to the computed excitation energies. We also demonstrate that the introduction of configuration state function generators allows for efficient a priori condensation techniques, with reductions of the expansions sizes with factors between 1.5 and 2.5 and the CPU time with factors between 2.5 and 4.5, again with negligible changes to the excitation energies. In total we demonstrate reductions of the CPU time with factors up to 68 for CI calculations based on configuration state function generators, restrictions on the Breit integrals and with a priori condensed expansions compared to ordinary CI calculations without restrictions on the Breit integrals and with full expansions. Further perspectives of the new methodology based on configuration state function generators are given.

Rapid Subentire-Domain Basis Functions Method Based on Adaptive Artificial Neural Networks

The subentire-domain (SED) basis functions method has shown its efficiency in solving the electromagnetic problem of large-scale finite periodic structures (LFPSs). However, calculating the expansion coefficients of SED basis functions is very time-consuming due to the consideration of the mutual coupling between all the elements in LFPSs, even after accelerated by conjugate-gradient fast Fourier transform (CG-FFT) and/or fast multipole method (FMM). In this article, based on the physics locations of observation cells, the adaptive artificial neural networks (AANNs) have been employed to rapidly predict the expansion coefficients of SED basis functions on interior cells, edge cells, and corner cells. By involving the AANNs, the mutual coupling between all the elements can be accounted into the neural networks without the construction of the mutual coupling matrix, and the expansion coefficients of SED basis functions can be obtained rapidly. Numerical experiments prove the accuracy and efficiency of the AANN-assisted SED basis functions method.

An Adaptive DGTD Algorithm Based on Hierarchical Vector Basis Function

To accurately obtain the radiation and propagation characteristics of the highly oscillating electromagnetic (EM) waves, the p-refinement method which expands the physical quantities by high-order basis functions has been proposed. Nevertheless, since the time-domain EM simulation is a dynamic process, a static refinement is not applicable. In this communication, an efficient dynamic p-adaptation method based on hierarchical vector basis functions having the property of nesting to facilitate the construction of basis functions has been developed for the discontinuous Galerkin time-domain (DGTD) method. In this approach, based on the relationship between the amplitude of the expansion coefficients of hierarchical vector basis functions and the corresponding expansion coefficients of the physical quantities captured by each element in the previous time step, an error estimator is proposed to determine the order of each element in the next time step. To reduce the computational overhead, the basis orders of each element can be dynamically adjusted when the pulse passes through or is dissipated. The numerical results demonstrate the accuracy and efficiency of the proposed approach.

The influence of a neighboring tunnel on the critical velocity of a three-dimensional tunnel-soil system

This paper proposes an analytical solution for the calculation of vibrations from a twin tunnel embedded in a half-space to assess the influence of a neighbouring tunnel on the ground-borne vibrations due to underground railway traffic. Using the wave function method, the total wave field in a half-space containing two cavities is expanded as the combination of down-going plane waves from the free surface and outgoing cylindrical waves from each cavity. The two tunnels are modelled as hollow cylinders, and the wave field inside them can be represented as the superposition of outgoing and regular cylindrical waves. To solve this multiple scattering problem, three sets of coordinate system defined at the scattering surfaces are used to describe the cylindrical waves in polar coordinates and the plane waves in rectangular coordinates. The application of transformation and translation properties between plane and cylindrical waves makes it possible to represent the waves emanating from one scattering surface in terms of the coordinates defined at the other two surfaces. The boundary conditions at the free surface and two tunnels can then be applied to solve for the expansion coefficients for the vector wave functions of plane and cylindrical waves. The present approach is also valid for a multi-layered ground, and in the layer with the tunnels, the field is composed of outgoing waves from the tunnels, down-going waves from the upper interface of the layer and up-going waves from the lower interface. Numerical results show that there is one critical velocity around shear wave speed of soils for a single/twin tunnel in a homogeneous half-space. For a multi-layered geometry, two global critical velocities may exist for the twin tunnel case, and they are determined by the properties of the soil layer where the twin tunnel is located in and the layer below the tunnels.

The causality from solar irradiation to ocean heat content detected via multi-scale Liang\u2013Kleeman information flow

Solar irradiation is the primary driving force for the Earth’s climate system. However, we are still short of powerful tools to study the variability of the Earth’s climate due to the solar activity. Here we apply the Liang–Kleeman information flow to quantify the causality from Total Solar Irradiance (TSI) to the global ocean heat content anomaly (OHCA). It reveals that the information flow from TSI to OHCA varies in both time and space. We adapt the method into a multi-scale version which describes the variation of information flow on different timescales. In different ocean basins, the significant information flow from TSI to OHCA varies on different timescales, which could be several decades, much longer than the timescale of the correlation revealed by wavelet coherence. Then we calculate the information flow from TSI to the first three expansion coefficients of the OHCA Empirical Orthogonal Functions. The results indicate that TSI is a part cause of the El Niño-Southern Oscillation (ENSO), especially in the 1970s. In the recent 40 years, the contribution of TSI to the variation of the OHCA becomes less significant probably due to the increasing influence of human activity on the climate system.

New Paley–Wiener Theorems

In this paper, for an arbitrary fixed compact set K, we find necessary and sufficient conditions on the Taylor expansion coefficients of entire functions of exponential type so that these functions are the Fourier image of distributions supported in K. In other words, we state the Paley–Wiener theorem in the language of Taylor expansion coefficients.

Relations between angular and Cartesian orientational expansions

Orientational expansions, which are widely used in natural sciences, exist in angular and Cartesian forms. Although these expansions are orderwise equivalent, it is difficult to relate them in practice. In this article, both types of expansions and their relations are explained in detail. We give explicit formulas for the conversion between angular and Cartesian expansion coefficients for functions depending on one, two, and three angles in two and three spatial dimensions. These formulas are useful, e.g., for comparing theoretical and experimental results in liquid crystal physics. The application of the expansions in the definition of orientational order parameters is also discussed.

An automatic and parameter-free information-based method for sparse representation in wavelet bases

In this article an information-based method for the selection of expansion coefficients of functions in a Hilbert basis is presented. An information-based measure, namely Entropic NID (ENID), is presented; the optimal separation point between more informative coefficients and less informative ones is selected by evaluating the information contribution of two competing sets of expansion coefficients. A consistent numerical scheme is given to approximate ENID and the numerical error is studied. Numerical simulations are provided to test the behaviour of ENID in different wavelet bases, as well as to perform comparative studies.

Seamless integration of elliptic Dirichlet-to-Neumann boundary condition and high order spectral element method for scattering problem

In this paper, we present a semi-analytic approach to enhance the integration of elliptic Dirichlet-to-Neumann (DtN) boundary condition and high order spectral element method in solving scattering problem with slender scatterer. By using appropriate elemental mapping in the spectral element discretization, semi-analytic formulas are obtained for the computation of Mathieu expansion coefficients involved in the global DtN operator. Further, a semi-analytic approach is proposed for the computation of global boundary integral terms in the spectral element discretization. The proposed semi-analytic formulas can also be used to calculate Mathieu expansion coefficients for functions given values on spectral element grids. Numerical examples show that spectral element method with the proposed semi-analytic approach can produce high order numerical solution for scattering problem with slender scatterer.

Determining the Thermo-Electro-Magneto-Elastic State of Multiply Connected Piecewise-Homogeneous Piezoelectric Plates

A method for studying the thermo-electro-magneto-elastic state of a multiply connected piecewise-homogeneous piezoelectric plate under the action of a linear heat flux is proposed. The solution of a problem using complex potentials and the generalized least squares method is reduced to solving a system of linear algebraic equations with respect to unknown expansion coefficients of functions into Laurent series and Faber polynomials. For the case of a plate with one inclusion, an exact analytical solution of the problem is obtained. The results of the numerical studies, which determine the effect of the electric and magnetic properties of the plate materials and inclusions and their location on the main characteristics of the thermo-electro-magneto-elastic state are described.

Zika virus: An emerging player in the global scenario

In this article an information-based method for the selection of expansion coefficients of functions in a Hilbert basis is presented. An information-based measure, namely Entropic NID (ENID), is presented; the optimal separation point between more informative coefficients and less informative ones is selected by evaluating the information contribution of two competing sets of expansion coefficients. A consistent numerical scheme is given to approximate ENID and the numerical error is studied. Numerical simulations are provided to test the behaviour of ENID in different wavelet bases, as well as to perform comparative studies.

Concentration dependence of the wings of a dipole-broadened magnetic resonance line in magnetically diluted lattices

The singularities of the time autocorrelation functions (ACFs) of magnetically diluted spin systems with dipole–dipole interaction (DDI), which determine the high-frequency asymptotics of autocorrelation functions and the wings of a magnetic resonance line, are studied. Using the self-consistent fluctuating local field approximation, nonlinear equations are derived for autocorrelation functions averaged over the independent random arrangement of spins (magnetic atoms) in a diamagnetic lattice with different spin concentrations. The equations take into account the specificity of the dipole–dipole interaction. First, due to its axial symmetry in a strong static magnetic field, the autocorrelation functions of longitudinal and transverse spin components are described by different equations. Second, the long-range type of the dipole–dipole interaction is taken into account by separating contributions into the local field from distant and near spins. The recurrent equations are obtained for the expansion coefficients of autocorrelation functions in power series in time. From them, the numerical value of the coordinate of the nearest singularity of the autocorrelation function is found on the imaginary time axis, which is equal to the radius of convergence of these expansions. It is shown that in the strong dilution case, the logarithmic concentration dependence of the coordinate of the singularity is observed, which is caused by the presence of a cluster of near spins whose fraction is small but contribution to the modulation frequency is large. As an example a silicon crystal with different 29Si concentrations in magnetic fields directed along three crystallographic axes is considered.

About One Method of On-Line Signature Verification Using Radial Basis Function

The article is devoted to the study of a possibility of on-line signature verification using the wavelet transform with the radial basis function. Representation of the signature in the form of function, invariant concerning a position, – the signature is replaced by a broken curve and is described using the angles between the adjacent links – is offered. Using the wavelet transforms for the signature description, comparison of expansion coefficients of functions of signatures by the radial basis function to verify signatures is described in detail. In the result of application of the offered method the magnitude of error of first kind has made 4.4%, the magnitude of error of second kind – 2.8%.

The Coefficient Problem and Multifractality of Whole-Plane SLE &amp; LLE

Karl Lowner (later known as Charles Loewner) introduced his famous differential equation in 1923 to solve the Bieberbach conjecture for series expansion coefficients of univalent analytic functions at level n = 3. His method was revived in 1999 by Oded Schramm when he introduced the Stochastic Loewner Evolution (SLE), a conformally invariant process which made it possible to prove many predictions from conformal field theory for critical planar models in statistical mechanics. The aim of this paper is to revisit the Bieberbach conjecture in the framework of SLE processes and, more generally, Levy processes. The study of their unbounded whole-plane versions leads to a discrete series of exact results for the expectations of coefficients and their variances, and, more generally, for the derivative moments of some prescribed order p. These results are generalized to the “oddified” or m-fold conformal maps of whole-plane SLEs or Levy–Loewner Evolutions. We also study the (average) integral means multifractal spectra of these unbounded whole-plane SLE curves. We prove the existence of a phase transition at a moment order p = p*(κ) > 0, at which one goes from the bulk SLEκ average integral means spectrum, as predicted by the first author (Duplantier Phys. Rev. Lett. 84:1363–1367, 2000) and established by Beliaev and Smirnov (Commun Math Phys 290:577–595, 2009) and valid for p ≤ p*(κ), to a new integral means spectrum for p ≥ p*(κ), as conjectured in part by Loutsenko (J Phys A Math Gen 45(26):265001, 2012). The latter spectrum is, furthermore, shown to be intimately related, via the associated packing spectrum, to the radial SLE derivative exponents obtained by Lawler, Schramm and Werner (Acta Math 187(2):237–273, 2001), and to the local SLE tip multifractal exponents obtained from quantum gravity by the first author (Duplantier Proc. Sympos. Pure Math. 72(2):365–482, 2004). This is generalized to the integral means spectrum of the m-fold transform of the unbounded whole-plane SLE map. A succinct, preliminary, version of this study first appeared in Duplantier et al. (Coefficient estimates for whole-plane SLE processes, Hal-00609774, 2011).

The multifacet graphically contracted function method. I. Formulation and implementation.

The basic formulation for the multifacet generalization of the graphically contracted function (MFGCF) electronic structure method is presented. The analysis includes the discussion of linear dependency and redundancy of the arc factor parameters, the computation of reduced density matrices, Hamiltonian matrix construction, spin-density matrix construction, the computation of optimization gradients for single-state and state-averaged calculations, graphical wave function analysis, and the efficient computation of configuration state function and Slater determinant expansion coefficients. Timings are given for Hamiltonian matrix element and analytic optimization gradient computations for a range of model problems for full-CI Shavitt graphs, and it is observed that both the energy and the gradient computation scale as O(N(2)n(4)) for N electrons and n orbitals. The important arithmetic operations are within dense matrix-matrix product computational kernels, resulting in a computationally efficient procedure. An initial implementation of the method is used to present applications to several challenging chemical systems, including N2 dissociation, cubic H8 dissociation, the symmetric dissociation of H2O, and the insertion of Be into H2. The results are compared to the exact full-CI values and also to those of the previous single-facet GCF expansion form.

A Gaussian graphical model approach to climate networks.

Distinguishing between direct and indirect connections is essential when interpreting network structures in terms of dynamical interactions and stability. When constructing networks from climate data the nodes are usually defined on a spatial grid. The edges are usually derived from a bivariate dependency measure, such as Pearson correlation coefficients or mutual information. Thus, the edges indistinguishably represent direct and indirect dependencies. Interpreting climate data fields as realizations of Gaussian Random Fields (GRFs), we have constructed networks according to the Gaussian Graphical Model (GGM) approach. In contrast to the widely used method, the edges of GGM networks are based on partial correlations denoting direct dependencies. Furthermore, GRFs can be represented not only on points in space, but also by expansion coefficients of orthogonal basis functions, such as spherical harmonics. This leads to a modified definition of network nodes and edges in spectral space, which is motivated from an atmospheric dynamics perspective. We construct and analyze networks from climate data in grid point space as well as in spectral space, and derive the edges from both Pearson and partial correlations. Network characteristics, such as mean degree, average shortest path length, and clustering coefficient, reveal that the networks posses an ordered and strongly locally interconnected structure rather than small-world properties. Despite this, the network structures differ strongly depending on the construction method. Straightforward approaches to infer networks from climate data while not regarding any physical processes may contain too strong simplifications to describe the dynamics of the climate system appropriately.