Series Expansion Method Research Articles

Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions.

It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique has played an important role for designing equivariant convolutions, and has achieved success in making use of rotation symmetry of images. However, the current filter parametrization strategy still has its evident drawbacks, where the most critical one lies in the accuracy problem of filter representation. To address this issue, in this paper we explore an ameliorated Fourier series expansion for 2D filters, and propose a new filter parametrization method based on it. The proposed filter parametrization method not only finely represents 2D filters with zero error when the filter is not rotated (similar as the classical Fourier series expansion), but also substantially alleviates the aliasing-effect-caused quality degradation when the filter is rotated (which usually arises in classical Fourier series expansion method). Accordingly, we construct a new equivariant convolution method based on the proposed filter parametrization method, named F-Conv. We prove that the equivariance of the proposed F-Conv is exact in the continuous domain, which becomes approximate only after discretization. Moreover, we provide theoretical error analysis for the case when the equivariance is approximate, showing that the approximation error is related to the mesh size and filter size. Extensive experiments show the superiority of the proposed method. Particularly, we adopt rotation equivariant convolution methods to a typical low-level image processing task, image super-resolution. It can be substantiated that the proposed F-Conv based method evidently outperforms classical convolution based methods. Compared with pervious filter parametrization based methods, the F-Conv performs more accurately on this low-level image processing task, reflecting its intrinsic capability of faithfully preserving rotation symmetries in local image features.

A solution method of slowing-down distribution of energetic particles in tokamaks

The eigen equation of pitch-angle distribution derived from the slowing-down distribution equation with an energetic particle source term is solved by using the Legendre series expansion method. An iteration matrix is established when pitch-angle scattering terms become important. The whole pitch-angle region is separated into three parts, two passing regions, and one trapped area. The slowing-down distribution for each region is finally obtained. The method is applied to solve the slowing-down equations with source terms that the pitch-angle distribution is Maxwellian-like, neutral beam injection, and radial drifts. The distribution functions are convergent for each source with different pitch-angle distribution. The method is suitable for solving a kinetic equation that pitch-angle scattering collision is important.

Error-free strapdown inertial navigation system numerical updating algorithm in the launch-centered Earth-fixed frame for hypersonic vehicles

The two-sample numerical updating algorithm of strapdown inertial navigation systems (SINS) for hypersonic vehicles has a relatively large navigation error. To solve this problem, this paper proposes an error-free SINS numerical updating algorithm in the launch-centered Earth-fixed (LCEF) frame. First, the gyroscopes and accelerometers in SINS output angular increment and velocity increment respectively, while the proposed algorithm takes angular velocity and specific force as input, so the methods of angular increment fitting angular velocity and velocity increment fitting specific force are proposed. Then, according to the SINS differential equation in the LCEF frame, the error-free SINS numerical updating algorithm in the LCEF frame is designed by adopting the Taylor series expansion method. The derivation process of the algorithm shows that the proposed algorithm can achieve precise compensation for the non-exchangeable errors of attitude, velocity, and position, and selecting the LCEF frame as the navigation frame has greater advantages than the local-level frame. Finally, two-sample and error-free numerical updating algorithms are simulated and compared by taking the hypersonic boost-gliding vehicle as the object. Under the influence of a pure SINS algorithm, the velocity error of the two-sample algorithm with a simulation duration of 1100 s is in the order of 10−2 m/s, and the position error is in the order of 10 m. In contrast, the velocity error of the proposed algorithm is in the order of 10−7 m/s, and its position error is in the order of 10−4 m.

Numerical study on nonlinear vibration of FG-GNPRC circular membrane with dielectric properties

Dielectric elastomer membrane has been applied in a wide variety of engineering fields for their excellent dynamic performances. Functionally graded graphene nanoplatelet-reinforced composite (FG-GNPRC) shows great potential for developing high-performance and multifunctional structures. In this work, the damped nonlinear vibration of the FG-GNPRC dielectric membrane is investigated. The effects of damping and dielectric properties are considered in terms of energy while deriving governing equations. Taylor series expansion and differential quadrature together with direct iterative methods are used to discretize and numerically solve the obtained governing equations. The developed model and numerical solution are verified by comparing present results to existing studies. The influences of functionally graded distribution, damping, stretching ratio, dimensions of the membrane, and the attributes of the electrical field and GNP fillers on the nonlinear vibration of the structure are comprehensively investigated. It is found that compared to a uniform distribution, the membrane with functional distribution of GNP demonstrates exhibits more robust performances. The membrane with a smaller thickness-to-radius ratio is more sensitive to the external electrical field. In addition, the direct current voltage is evidenced to have a more significant effect on the nonlinear vibration when the membrane is subject to a relatively small stretching ratio. The present work suggests that the structural behavior of the FG-GNPRC membrane can be actively adjusted by varying the stretching ratio and the properties of the electrical field and the GNP filler. HIGHLIGHTS Effects of damping and dielectric properties are considered for structural behavior in terms of energy. Taylor series expansion and differential quadrature methods are combined to solve governing equations. Two transition regions in frequency ratio are identified due to AC frequency-facilitated polarization and electron tunneling. The effect of the electric field on nonlinear vibration can be regulated by varying the stretching ratio.

A fractional-order model to study the dynamics of the spread of crime

Numerous crucial factors and parameters influence the dynamic process of the spread of crime. Various integer-order differential models have been proposed to capture crime spread. Most of these introduced dynamic systems have not considered the history of the criminal and the impact of crime on society. To address these shortcomings, a fractional-order crime transmission model is proposed in this manuscript considering five different classes viz law-abiding citizens, non-incarcerated criminals, incarcerated criminals, prison-released and recidivists. The primary focus of the proposed model is to study the effect of recidivism in society and decide the adequate imprisonment for repeat offenders. The existence, uniqueness, non-negativity and boundedness of the solution of the proposed model are examined. The local stability of the equilibrium points is also analysed using Routh–Hurwitz Criteria with Matignon conditions. Further, the threshold condition for the uniform asymptotic stability of the system is evaluated by the Lyapunov stability method. Moreover, the long-term impact of the imprisonment of criminals on society is also examined in the current study. The numerical simulations of the model for a range of fractional orders are obtained using power series expansion method to strengthen the theoretical results.

Curved photonic nanojet generated by a rotating cylinder.

The curved photonic nanojet (CPNJ) produced due to the interaction between a dielectric circular cylinder rotating at a stable angular velocity and a plane wave is investigated. Based on this model, the optical Magnus effect of a dielectric circular cylinder is verified. And the analytical expression of both internal and external electric field are given based on the instantaneous rest-frame theory and the partial-wave series expansion method in cylindrical coordinates. The influence of the size parameter, the relative refractive index, and the rotating dimensionless parameter on the CPNJ are analyzed and discussed in numerical results. The "photonic nanojet curved" effect is highlighted, which can be used to generate the off-axis photonic nanojet (PNJ) controlling particles by adjusting the angular velocity of the dielectric cylinder. The results of this manuscript have promising application prospects in optical tweezers, particle manipulation, and optical trapping. Moreover, it also provides theoretical support for the particle spinning and generation of the off-axis CPNJ.

The Generalized Risk-Adjusted Cost-Effectiveness (GRACE) Model for Measuring the Value of Gains in Health: An Exact Formulation

AbstractThe generalized risk-adjusted cost-effectiveness (GRACE) analysis method modifies standard cost-effectiveness analysis (CEA), the primary method currently used worldwide to value health improvements arising from healthcare interventions. Generalizing standard CEA, GRACE allows for decreasing or even increasing returns to health. Previous presentations of GRACE have relied extensively on Taylor Series expansion methods to specify key model parameters, including those that properly adjust for illness severity and preexisting disability, consequences of uncertain treatment outcomes, and the marginal rate of substitution between life expectancy and health-related quality of life. Standard CEA cannot account for these sources of value or cost in its valuation of medical treatments. However, calculations of GRACE measures based on Taylor Series are approximations, which may be poorly behaved in some contexts. This paper provides a new approach for implementing GRACE, using exact utility functions instead of Taylor Series approximations. While any proper utility function will suffice, we illustrate with three well-known functions: constant relative risk aversion (CRRA) utility; hyperbolic absolute risk aversion (HARA) utility, of which CRRA is a special case; and expo-power (EP) utility, of which constant absolute risk aversion (CARA) is a special case. The analysis then extends from two-period to multiperiod models. We discuss methods to estimate parameters of HARA and EP functions using two different types of data, one from discrete choice experiments and the other from “happiness economics” methods. We conclude with some reflections on how this analysis might affect benefit-cost analysis studies of healthcare interventions.

Valuing equity-linked death benefits with a threshold expense structure under a regime-switching Lévy model

<p style='text-indent:20px;'>In this paper, we investigate the valuation problem of equity-linked death benefits with a threshold expense structure. Specifically, a regime-switching Lévy process is used to describe the underlying asset price process, which is monitored periodically. The fees are assumed to be continuously deducted at some constant rate from the policyholder's account between the current and next monitoring times, if the account value is smaller than a pre-specified level at the current observation time point. Under the modified threshold expense structure, some explicit valuation expressions for life-contingent call options are derived by the Fourier cosine series expansion method. Numerical results demonstrate the accuracy and efficiency of our method.</p>

Application of the Fourier Series Expansion Method for the Inversion of Gravity Gradients using Gravity Anomalies

Accurate and highly precise gravity gradient data are an important component of, for example, gravity field modeling, seabed topography inversion, and resource exploration. However, high-precision gravity gradient data are difficult to obtain. To address this difficulty, this work introduces the Fourier series expansion method to the modeling of gravity gradient fields. Based on gravity anomalies, the analytic expressions of the gravity gradient tensors have been deduced, which provides a new mathematical method for obtaining gravity gradient data. The expression’s derivation and verification processes are as follows. First, these analytic expressions for inverting the gravity gradient based on gravity anomaly data are derived according to the Laplace equation, the boundary value conditions of spherical approximation, and the Fourier series expansion method. Then, global 1’ × 1’ gravity field data provided by UCSD are used to verify the accuracy of these formulas. Finally, the results are analyzed. The experimental results show that the results obtained based on this inversion formula can sufficiently show the details of gravity gradient changes. The formulas derived in this paper have good computational efficiency in the inversion of regional gravity gradients and provide a new mathematical method for gravity gradient data acquisition.

Damped vibration analysis of graphene nanoplatelet reinforced dielectric membrane using Taylor series expansion and differential quadrature methods

Membrane has been attracting extensive attention due to their application in various engineering fields. Compared to traditional materials, graphene reinforced composites have demonstrated great potential in developing high-performance and multifunctional membrane structures. It is of great importance to investigate the structural behaviors of such composite membrane with considering damping and their significantly improved dielectric properties. This work studies the damped vibration of graphene nanoplatelets reinforced composite (GNPRC) dielectric membrane. Governing equations are established for the nonlinear damped vibration of the GNPRC dielectric membrane, in which the effects of dielectric properties and damping are incorporated in terms of energy. Taylor series expansion (TSE), differential quadrature (DQ) and direct iterative methods are used to numerically solve the governing equations. The model and the results are validated by comparing present results with previously reported ones. Parametric study is conducted to investigate the effects of the attributes of the electrical field and the GNP fillers, dielectric properties of the composites, damping and the stretching ratio on the nonlinear damped vibration. The result shows that the vibration frequency of the membrane can be actively tuned by changing the attributes of the applied electrical field. When the stretching ratio increases from 1.05 to 1.10, the nonlinear damped frequency can increase by 39.8%, indicating the significant effects of the stretching ratio.

Development of Converter Output Voltage Phase-Shift Keying Methods as Applied to a Seven-Phase Motor

Phase-shift keying (modulation) of the output voltage of a three-phase converter connected to a symmetrical three-phase winding is one of the stages of the motor vector control technology. This process has well been studied and lies in a practical application field. One of the supposed lines in the development of modern electric traction systems is to construct an electric drive based on a multiphase motor (with the number of phases more than three). In implementing a multiphase motor vector control technology, new opportunities appear, in particular, a new phase-shift keying method. The article considers the voltage phase-shift keying process as applied to a seven-phase winding with the aim to develop seven-phase converter control algorithms implementing the output voltage phase-shift keying, which is possible due to the features of a seven-phase symmetrical motor winding. In the study, the vector analysis techniques and Fourier series expansion methods were used. Experimental studies confirming the obtained theoretical results have been carried out. The scientific novelty of the work lies in that a new converter phase output voltage phase-shift keying method as applied to a seven-phase symmetrical winding and a new converter control algorithm implementing this method have been proposed. The results can be useful in implementing the vector control of a multiphase motor.

Analytical model to effective permeability loss monitoring in hydraulically fractured oil wells in pressure-sensitive reservoirs

This work develops a new analytical model to evaluate the effective permeability response during the oil’s production through a fractured vertical oil well fully penetrating a pressure-sensitive reservoir. The nonlinear hydraulic diffusivity equation (NHDE) is solved analytically through an integro-differential model coupled to Green’s function (GF) related to a semi-infinite oil source plane that represents a hydraulic fracture. The NHDE is perturbed using a first-order asymptotic series expansion method, and the solution is derived in terms of a new effective permeability pseudo-pressure function, considering the matrix and fracture permeability roles. A new hydraulic diffusivity deviation factor is also derived to represent the effective permeability loss as a function of the pore pressure throughout the well-reservoir life cycle. For this oil flow modeling, the general solution is expressed by the sum of the linear solution (constant permeability) plus a corrective term given by the combination of the exponential function and the complementary error function erfc(yD,tD). The model calibration is performed through a porous media numerical oil flow simulator, named IMEX®, widely used in the petroleum industry and scientific works. The results presented high accuracy when compared to the numerical simulator, thus showing that the developed model allows representing the permeability pressure-sensitive response.

Parametric testing for normality against bimodal and unimodal alternatives using higher moments

This study examines population and small sample properties of the standardized fifth and sixth moments – the “higher moments” – for assessing univariate normality against bimodal and selected unimodal alternatives. Population parameters and distributions for selected bimodal mixtures are calculated and contrasted with those for the normal distribution. Using Gram-Charlier series expansion methods, an omnibus goodness of fit test incorporating the higher moments is specified and Monte Carlo simulation used to compare test power with parametric tests based on the standardized third and fourth sample moments: the asymptotic and size corrected versions of the Jarque-Bera score test and the omnibus D’Agostino K2 test. The studentized range and directional tests using the third through sixth moments are also considered. The results demonstrate that incorporating the fifth and sixth moments can provide enhanced parametric normality test power for bimodal normal mixture alternatives but not for various unimodal alternatives.

A unified approach for dynamic characteristic analysis of a standing-wave thermoacoustic engine with general impedance ends

As promising devices for energy conversion, thermoacoustic engines (TAEs) are attracting more and more research attention with regard to renewable thermal energy application. For a TAE, the oscillation frequency is a critical parameter in its design and operation, which can only be estimated currently through an often tedious process using a shooting method or a numerical iteration method, and an analysis model with classical boundary conditions. In this context, this study seeks to establish a unified model for dynamic analysis of a one-dimensional (1D) standing-wave TAE with general impedance ends. First, the differential equations governing the thermoacoustic behavior of the TAE and its impedance boundary conditions are simultaneously solved using the Galerkin method and an improved Fourier series expansion method. Then, the characteristic distributions of oscillation frequency, sound pressure, and volume flow rate are obtained by solving a standard eigenvalue problem. Finally, the effects of engine length, stack length and position, and impedance ends on the dynamic behavior of the TAE’s thermoacoustic system are investigated and analyzed. The results show that engine length has a significantly negative correlation with the modal frequencies in different working gases, and stack position and stack length have a slight effect on the oscillation frequency of the TAE. The arbitrary variations in impedance boundary conditions can be simulated from the open end to the rigid wall and all intermediate cases. The proposed model can efficiently predict changes in dynamic parameters of 1-D TAEs with impedance ends in a unified pattern and can serve as a reliable analysis tool for further research on TAEs coupled with various energy harvesters.

High-order compact finite difference schemes for solving the regularized long-wave equation

In this paper, high-order compact finite difference schemes for the one-dimensional regularized long wave equation are proposed. Firstly, the original regularized long-wave equation is deformed in two ways. The fourth-order compact difference scheme and the fourth-order Padé scheme are used for discretization of the second and first derivatives of the two deformed equations in space direction, respectively, and the θ-weighted scheme is used to discretize the time derivative of the first deformed, the fourth-order backward difference formula is used for the competition of the time derivative of the second deformed equation. Secondly, the nonlinear terms in the two schemes are linearized by Taylor series expansion method. Thirdly, the existence, uniqueness and conservation of energy of numerical solutions are proved by discrete energy method and mathematical induction method. And the two new schemes are shown to be conserved and unconditionally stable. Since each time level involves three grid points, a tridiagonal linear system is formed, which can be solved directly using the Thomas algorithm. Finally, the accuracy and reliability of the present method are verified by numerical experiments.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Thermal analysis of multi-finned plate employing lattice Boltzmann method based on Taylor-series/least-squares

A comprehensive experimental/numerical investigation is carried for the purpose of heat transfer and hydrodynamic analysis of natural convection within a cavity equipped with active rectangular fins. Numbers of hot and cold fins are fitted at the surface of square cavity which have remarkable effect on the hydrothermal performance of cavity. In this work, some parameters with significant impact on hydrothermal performance are selected to be studied such as value of Rayleigh number (Ra), value of nanoparticle volume concentration (φV) and arrangement of active fins (Ther.Arr A, B, C and D). To reach this purpose, numerical approach and empirical observations are coupled to provide an accurate data. In order to make the standard Lattice Boltzmann Method (LBM) compatible with the non-uniform grid distribution, the Taylor series expansion and least square-based lattice Boltzmann method (TLLBM) is used. Furthermore, firstly, the SiC-EG nanofluid samples are prepared using modern experimental, and then its rheological and thermal properties are measured using modern devices. It is concluded that using of nanofluid and the thermal arrangement of fins, have significant impact on the thermal performance of cavity.

Simultaneous Localization of Scatterers and Target User Based on Indoor Prior Information in NLOS Environments

In complex indoor environments, there is often no Line of Sight (LOS) path between the transmitter and the receiver due to walls, moving people, and many other obstacles. Traditional localization method based on LOS measurement is no longer available which provides lower localization accuracy in Non-Line of Sight (NLOS) environments. In response to this problem, this paper proposes a joint localization algorithm for scatterers and target user based on indoor prior information in NLOS scenarios. The algorithm makes use of the prior localization information of the access point (AP) and the wall to determine the position of the scatterers and target user simultaneously. Since the established constrained optimization problem is non-convex which is a difference of convex (DC) problem, it is transformed into a convex one based on a Taylor series expansion (TSE) method before it is solved. At the same time, as only a single AP signal can be received normally, the proposed method is then extended to the scenario with only one AP. Simulation and real-world testing results both show that not only the proposed method based on multiple APs, but also the method based on a single AP achieve high positioning accuracy, which are better than the existing positioning methods mentioned in this article as well.

Three-Dimensional Fluctuating Flow of a Second-Grade Fluid along an Infinite Horizontal Plate with Periodic Suction

Laminar flow control plays a vital role in reducing drag resulting enhancement in the vehicle power by a significant amount. Theoretical and experimental studies have proven that the transition from laminar to turbulent flow (causing the drag coefficient to enhance) may be delayed/prevented by the suction of the fluid from the boundary layer to the wall. The purpose of this work is to study the effects of periodic suction on the unsteady 3-dimensional fluctuating flow of a second grade incompressible fluid flowing laminarly over a horizontal porous infinite plate. The plate is subjected to a sinusoidal transverse suction velocity while the free stream velocity oscillates in time about a constant mean. The flow turns out to be 3-dimensional because of transverse direction of the periodic suction velocity. By the series expansion method, analytic expressions for transient velocity, skin friction components, and pressure are attained. Impact of nondimensional parameters evolving in the mathematical model of the problem on these physical quantities are visualized graphically and discussed analytically. The velocity component u is found to be rising with a growth in suction parameter α. The pressure is noted to be growing with a growth in Reynolds number R e . Further, due to suction at the plate and transient free stream velocity, the pressure increases near the plate and then reaches to a steady value far-off the plate. The skin friction along the main flow is noticed to be decreasing with the rise in α for 0.5 ≤ R e ≤ 1.4 , however, an exponential rise is observed for R e > 1.4. The skin friction along the cross flow is noted to be enhancing for the rising values of suction parameter and elastic parameter. The present results have excellent agreement with previous published results in the limiting sense. This study is beneficial in designing and manufacturing laminar flow control system to enhance the vehicle power requirement.

Efficient valuation of guaranteed minimum maturity benefits in regime switching jump diffusion models with surrender risk

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.