Interpolation Formula Research Articles

Active dynamics of colloidal particles in time-varying laser speckle patterns.

Colloidal particles immersed in a dynamic speckle pattern experience an optical force that fluctuates both in space and time. The resulting dynamics presents many interesting analogies with a broad class of non-equilibrium systems like: active colloids, self propelled microorganisms, transport in dynamical intracellular environments. Here we show that the use of a spatial light modulator allows to generate light fields that fluctuate with controllable space and time correlations and a prescribed average intensity profile. In particular we generate ring-shaped random patterns that can confine a colloidal particle over a quasi one-dimensional random energy landscape. We find a mean square displacement that is diffusive at both short and long times, while a superdiffusive or subdiffusive behavior is observed at intermediate times depending on the value of the speckles correlation time. We propose two alternative models for the mean square displacement in the two limiting cases of a short or long speckles correlation time. A simple interpolation formula is shown to account for the full phenomenology observed in the mean square displacement across the entire range from fast to slow fluctuating speckles.

Anti-Plane Interaction of a Coated Circular Inclusion with a Crack Located in Matrix

In this study, the interaction between a coated circular inclusion with an anti-plane crack located in matrix is considered. The solution procedures for solving this problem consist of two parts. In the first part, based on the method of analytical continuation in conjunction with the alternating technique, the complex potential functions of screw dislocation interacting with multi-layered composites are obtained. The second part consists of the derivation of logarithmic singular integral equations by introducing the complex potential functions of screw dislocation along the crack border together with superposition technique. The logarithmic singular integral equations is the solved numerically by modeling a crack in place of several segments. Linear interpolation formulae with undetermined coefficients are applied to approximate the dislocation distribution along the elements, except at vicinity of crack tip where the dislocation distribution preserves a square-root singularity. The stress intensity factors are then obtained numerically in terms of the values of the dislocation density functions of the logarithmic singular integral equations.

Fast and Simple Modular Interpolation Using Factorial Representation

We study a representation for polynomial functions over finite rings. This factorial representation is particularly useful for fast interpolation, and we show that it is computationally preferable to the Lagrange Interpolation Formula (LIF) and to Newton interpolation over finite fields and rings. Moreover, over arbitrary finite rings the calculation of the factorial representation aborts naturally when a given mapping does not arise as a polynomial function.

Interpolation of a function of two variables with large gradients in boundary layers

The paper is concerned with the interpolation of a function of two variables with large gradients in the boundary layers. An underlying function is assumed to be the sum of the regular componentwith derivatives bounded up to some order and two boundary layer components, the latter are known up to a multiplicative factor. Such a representation is typical for the solution of a singular perturbed elliptic problem. A two-dimensional interpolation formula exact on the boundary layer components is put forward. The formula has an arbitrary number of nodes in each direction. An error estimate is obtained which is uniform on the gradients of the underlying function in boundary layers. Results of numerical experiments are provided.

An approximating interpolation formula for bandlimited signals in the linear canonical transform domain associated with finite nonuniformly spaced samples

In the past literature, with the reproducing property of the reproducing kernel, Zhao et al. proposed an interpolation formula for bandlimited signals in the linear canonical transform (LCT) domain associated with finite set of irregularly spaced samples. Although the authors have discussed the reconstruction error of the formula, they did not investigate the conditions that lead to the error tends to zero when the number of samples runs to infinity, i.e., the approximating property of the formula. In this paper, we firstly present a sufficient condition achieving that property with respect to Zhao's formula. Then, under a similar condition we formulate an approximating interpolation formula for bandlimited signals in the LCT domain with parameter matrix A=(a, b;c, d) associated with finite nonuniform samples of the signals’ LCT with parameter matrix A¯=(−b,a;−d,c). We also provide some potential applications of the derived reconstruction formula to show the advantage of the theory.

Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term

ABSTRACTIn this paper, we propose an improved efficient difference method to solve the modified anomalous sub-diffusion equation with a nonlinear source term. The weighted and shifted Grünwald difference operator and compact difference are used to approximate the Riemann–Liouville fractional derivative and the space derivative, respectively. Since the semi-discrete system obtained by discretizing in space has some stiff properties when the space mesh is dense, the second-order backward differentiation formula (BDF) is then employed to solve the semi-discrete system and a second-order interpolation formula is used to deal with the nonlinear source term. The stability and convergence of the method are discussed and the convergence order is shown to be , where τ and h denote the time and space step size. The new method has higher accuracy compared with the existed methods and good stability in the numerical performance. Numerical examples are given to illustrate the theoretical results.

Discrete estimators of characteristics for periodically correlated time series

Results of an investigation of the characteristic estimator properties for periodically correlated time series obtained on the basis of finite data length are given. The formulae for the bias and variance of the estimators for mean and covariance function Fourier coefficients are found. The conditions for the choice of sampling interval value, for which aliasing effects do not appear, are obtained. The interpolation formulae for the mean and covariance function estimates are derived. The dependencies of the statistical characteristics of the estimators on sampling interval and sample size for modulated signals are analyzed.

Bulk and shear viscosities of the two-dimensional electron liquid in a doped graphene sheet

Hydrodynamic flow occurs in an electron liquid when the mean free path for electron-electron collisions is the shortest length scale in the problem. In this regime, transport is described by the Navier-Stokes equation, which contains two fundamental parameters, the bulk and shear viscosities. In this Article we present extensive results for these transport coefficients in the case of the two-dimensional massless Dirac fermion liquid in a doped graphene sheet. Our approach relies on microscopic calculations of the viscosities up to second order in the strength of electron-electron interactions and in the high-frequency limit, where perturbation theory is applicable. We then use simple interpolation formulae that allow to reach the low-frequency hydrodynamic regime where perturbation theory is no longer directly applicable. The key ingredient for the interpolation formulae is the "viscosity transport time" $\tau_{\rm v}$, which we calculate in this Article. The transverse nature of the excitations contributing to $\tau_{\rm v}$ leads to the suppression of scattering events with small momentum transfer, which are inherently longitudinal. Therefore, contrary to the quasiparticle lifetime, which goes as $-1/[T^2 \ln(T/T_{\rm F})]$, in the low temperature limit we find $\tau_{\rm v} \sim 1/T^2$.

Piecewise constructions of inverses of cyclotomic mapping permutation polynomials

Given a permutation polynomial of a large finite field, finding its inverse is usually a hard problem. Based on a piecewise interpolation formula, we construct the inverses of cyclotomic mapping permutation polynomials of arbitrary finite fields.

Interpolation formulas with derivatives in de Branges spaces

Interpolation formulas with derivatives in de Branges spaces

Simulation of the influence of the hot-pressing parameters and the mixture compositions in the ZrO2(m)–Al–C system on the hardness and the crack resistance of ceramics

The work is aimed at studying the change in physical and mechanical properties (Vickers hardness HV15 and the fracture toughness K1C) of ceramic tool material obtained by hot pressing of ZrO2–Al–C mixtures depending on the concentration of the initial components and the parameters of the synthesis process (temperature, pressure, and isothermal holding time). On the basis of the data obtained in the course of the five-factor experiment, the relationships that allow establishing the optimum level of baseline are derived for the synthesis of the material with an optimal combination of properties that predetermine the use of the ceramic as the basis of the blade tool (as a cutting insert). In the considered intervals of factors, the derived empirical models can also be used as an interpolation formula for determining the value of the Vickers hardness (HV15) and the fracture toughness (K1C) of materials synthesized in the ZrO2–Al–C system.

Approximation by amplitude and frequency operators

We study Padé interpolation at the node z=0 of functions f(z)=∑m=0∞fmzm, analytic in a neighbourhood of this node, by amplitude and frequency operators (sums) of the form ∑k=1nμkh(λkz),μk,λk∈C. Here h(z)=∑m=0∞hmzm, hm≠0, is a fixed (basis) function, analytic at the origin, and the interpolation is carried out by an appropriate choice of amplitudesμk and frequenciesλk. The solvability of the 2n-multiple interpolation problem is determined by the solvability of the associated moment problem ∑k=1nμkλkm=fm/hm,m=0,2n−1¯.In a number of cases, when the moment problem is consistent, it can be solved by the classical method due to Prony and Sylvester, moreover, one can easily construct the corresponding interpolating sum too. In the case of inconsistent moment problems, we propose a regularization method, which consists in adding a special binomial c1zn−1+c2z2n−1 to an amplitude and frequency sum so that the moment problem, associated with the sum obtained, can be already solved by the method of Prony and Sylvester. This approach enables us to obtain interpolation formulas with n nodes λkz, being exact for the polynomials of degree ⩽2n−1, whilst traditional formulas with the same number of nodes are usually exact only for the polynomials of degree ⩽n−1. The regularization method is applied to numerical differentiation and extrapolation.

Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

AbstractIn order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled withh-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

Dynamics of Dollard asymptotic variables. Asymptotic fields in Coulomb scattering

Generalizing Dollard’s strategy, we investigate the structure of the scattering theory associated to any large time reference dynamics [Formula: see text] allowing for the existence of Møller operators. We show that (for each scattering channel) [Formula: see text] uniquely identifies, for [Formula: see text], asymptotic dynamics [Formula: see text]; they are unitary groups acting on the scattering spaces, satisfy the Møller interpolation formulas and are interpolated by the [Formula: see text]-matrix. In view of the application to field theory models, we extend the result to the adiabatic procedure. In the Heisenberg picture, asymptotic variables are obtained as LSZ-like limits of Heisenberg variables; their time evolution is induced by [Formula: see text], which replace the usual free asymptotic dynamics. On the asymptotic states, (for each channel) the Hamiltonian can by written in terms of the asymptotic variables as [Formula: see text], [Formula: see text] the generator of the asymptotic dynamics. As an application, we obtain the asymptotic fields [Formula: see text] in repulsive Coulomb scattering by an LSZ modified formula; in this case, [Formula: see text], so that [Formula: see text] are free canonical fields and [Formula: see text].

A Lightweight Authenticated Communication Scheme for Smart Grid

In smart grid, various kinds of sensors have been equipped in smart meters to collect real-time consumption data and execute the instructions from the power management center, which leads to a more robust and customer-friendly grid system. However, a smart grid enables the two-way communication between the electricity supplier and users, which brings security challenges to the smart grid. In this paper, we propose a lightweight authenticated communication scheme for the smart grid, which ensures a secure two-way communication between the smart meters and the neighborhood gateway. Since in the proposed scheme, the smart meters only need to execute the bitwise exclusive-OR operations to encrypt collected usage data, and Lagrange interpolation formula is used to authenticate the sender; the transmitted messages can be real-timely authenticated. On the other side, the storage burden and communication cost of the smart meters can be significantly reduced. The security analysis shows that the messages transmission between the smart meters and the neighborhood gateway can be done in a confidential and authenticated manner. Furthermore, compared with other existing schemes, the performance analysis demonstrates that the proposed scheme has both lower storage and communication cost.

Special values of anticyclotomic L-functions for modular forms

Abstract In this article, we generalize some works of Bertolini–Darmon and Vatsal on anticyclotomic L-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p-adic L-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists.

High-order compact difference algorithm on half-staggered meshes for low Mach number flows

The paper presents a solution algorithm for variable density non-isothermal flows based on a high-order compact difference approximation combined with the projection method formulated on half-staggered meshes. In contrast to full-staggered grid arrangement, with the pressure and all scalar variables located in the cell centers, we shift only the pressure nodes while we keep the scalars and velocity components in the same nodes. This greatly simplifies the solution procedure as the interpolation between the nodes takes place only within the projection method. Derivative approximations and interpolation formulas are discretized using the high-order compact difference schemes up to 10th order in the central nodes and third/fourth order schemes near domain boundaries. The proposed algorithm is first verified in natural convection problems in square and rectangular wall bounded cavities for Rayleigh numbers Ra=3.4×105 and Ra=1.0×106 with maximal density ratio equals nine. The results are compared with published data for both steady and unsteady flow regimes. The performed simulations reveal that achievement of accurate results is conditioned by a mesh resolution in thermal boundary layers near the walls, and a way in which the governing equations are formulated, i.e., the conservative or non-conservative form. This turns out more important than a formal order of discretization method. Robustness of the proposed algorithm for the computations of turbulent flow is demonstrated based on the flow in a periodic channel at Reynolds number Reτ=200 with density ratio equal to two. In this case the profiles of mean velocity, temperature and their fluctuations are analyzed and they agree quite well with literature data. In all presented test cases the obtained pressure fields are smooth and without any oscillations observed when the collocated meshes are employed.

Numerical simulation of a compressible vortex–wall interaction

The wall interaction of isolated compressible vortices generated from a short driver section shock tube has been simulated numerically by solving the Navier–Stokes equations in axisymmetric form. The dynamics of shock-free (incident shock Mach number $$M = 1.36$$ ) and shock-embedded $$(M = 1.57)$$ compressible vortices near the wall has been studied in detail. The AUSM+ scheme with a fifth-order upwind interpolation formula is used for the convective fluxes. Time integration is performed using a low dissipative and dispersive fourth-order six-stage Runge–Kutta scheme. The evolution of primary and wall vortices has been shown using the velocity field, vorticity field, and numerical schlierens. The vortex impingement, shocklets, wall vortices, and their lift-off are clearly identified from the wall pressure time history. It has been observed that the maximum vorticity of the wall vortices reaches close to 30 % of the primary vortex for $$M = 1.36$$ and it reaches up to 60 % for $$M = 1.57$$ . The net pressure force on the wall due to incident shock impingement is dominant compared to the compressible vortex impingement and their evolution.

General Interpolation Formulae for Barycentric Blending Interpolation

General interpolation formulae for barycentric interpolation and barycentric rational Hermite interpolation are established by introducing multiple parameters,which include many kinds of barycentric interpolation and barycentric rational Hermite interpolation. We discussed the interpolation theorem, dual interpolation and special cases. Numerical example is given to show the effectiveness of the method.

On solving an n×n system of nonlinear Volterra integral equations by the Newton-Kantorovich method

We consider an nxn system of nonlinear integral equations of Volterra type (nonlinear VIEs) arising from an economic model. By applying the Newton-Kantorovich method to the nonlinear VIEs we linearize them into linear Volterra type integral equations (linear VIEs). Uniqueness of the solution of the system is shown. An idea has been proposed to find the approximate solution by transforming the system of linear VIEs into a system of linear Fredholm integral equations by using sub-collocation points. Then the backward Newton interpolation formula is used to find the approximate solution at the collocation points. Each iteration is solved by the Nystrom type Gauss-Legendre quadrature formula (QF). It is found that by increasing the number of collocation points of QF with fewer iterations, a high accurate approximate solution can be obtained. Finally, an illustrative example is demonstrated to validate the accuracy of the method.