Estimating the Gerber-Shiu Function in Lévy Insurance Risk Model by Fourier-Cosine Series Expansion

In this paper, we propose an estimator for the Gerber–Shiu function in a pure-jump Lévy risk model when the surplus process is observed at a high frequency. The estimator is constructed based on the Fourier–Cosine series expansion and its consistency property is thoroughly studied. Simulation examples reveal that our estimator performs better than the Fourier transform method estimator when the sample size is finite.

The vibration analysis of the elastically restrained functionally graded Timoshenko beam with arbitrary cross sections

In this study, an investigation on the free vibration of the beam with material properties and cross section varying arbitrarily along the axis direction is studied based on the so-called Spectro-Geometric Method. The cross-section area and second moment of area of the beam are both expanded into Fourier cosine series, which are mathematically capable of representing any variable cross section. The Young’s modulus, the mass density and the shear modulus varying along the lengthwise direction of the beam, are also expanded into Fourier cosine series. The translational displacement and rotation of cross section are expressed into the Fourier series by adding some polynomial functions which are used to handle the elastic boundary conditions with more accuracy and high convergence rate. According to Hamilton’s principle, the eigenvalues and the coefficients of the Fourier series can be obtained. Some examples are presented to validate the accuracy of this method and study the influence of the parameters on the vibration of the beam. The results show that the first four natural frequencies gradually decrease as the coefficient of the radius [Formula: see text] increases, and decreases as the gradient parameter n increases under clamped–clamped end supports. The stiffness of the functionally Timoshenko beam with arbitrary cross sections is variable compared with the uniform beam, which makes the vibration amplitude of the beam have different changes.

Reducing the near boundary errors of nonhomogeneous heat equations by boundary consistent methods

In the paper, we point out a drawback of the Fourier sine series method to represent a given odd function, where the boundary Gibbs phenomena would occur when the boundary values of the function are non-zero. We modify the Fourier sine series method by considering the consistent conditions on the boundaries, which can improve the accuracy near the boundaries. The modifications are extended to the Fourier cosine series and the Fourier series. Then, novel boundary consistent methods are developed to solve the 1D and 2D heat equations. Numerical examples confirm the accuracy of the boundary consistent methods, accounting for the non-zeros of the source terms and considering the consistency of heat equations on the boundaries, which can not only overcome the near boundary errors but also improve the accuracy of solution about four orders in the entire domain, upon comparing to the conventional Fourier sine series method and Duhamel’s principle.

Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

BRITE-Constellation photometry of π5 Orionis, an ellipsoidal SPB variable

ABSTRACT Results of an analysis of the BRITE-Constellation photometry of the SB1 system and ellipsoidal variable π5 Ori (B2 III) are presented. In addition to the orbital light-variation, which can be represented as a five-term Fourier cosine series with the frequencies forb, 2forb, 3forb, 4forb, and 6forb, where forb is the system’s orbital frequency, the star shows five low-amplitude but highly significant sinusoidal variations with frequencies fi (i = 2, .., 5, 7) in the range from 0.16 to 0.92 d−1. With an accuracy better than 1σ, the latter frequencies obey the following relations: f2 − f4 = 2forb, f7 − f3 = 2forb, f5 = f3 − f4 = f7 − f2. We interpret the first two relations as evidence that two high-order ℓ = 1, m = 0 gravity modes are self-excited in the system’s tidally distorted primary component. The star is thus an ellipsoidal SPB variable. The last relations arise from the existence of the first-order differential combination term between the two modes. Fundamental parameters, derived from photometric data in the literature and the Hipparcos parallax, indicate that the primary component is close to the terminal stages of its main-sequence (MS) evolution. Extensive Wilson–Devinney modelling leads to the conclusion that best fits of the theoretical to observed light curves are obtained for the effective temperature and mass consistent with the primary’s position in the HR diagram and suggests that the secondary is in an early MS evolutionary stage.

FOURIER SERIES REPRESENTATION OF FRACTAL INTERPOLATION FUNCTION

The purpose of this research is to show how the complicated and irregular fractal interpolation function is represented by Fourier series. First, on the closed interval [0,1], even prolongation is operated to the fractal interpolation function generated by iterated function system constituted by affine transform and Fourier cosine series representation of fractal interpolation function is proved. Second, for fractal interpolation function, odd prolongation is done and Fourier sine series formula of fractal interpolation function is proved. Final, Fourier series expansion of fractal interpolation function on the closed interval [Formula: see text] is proved. The result shows that complex fractal interpolation function can be represented by Fourier sine series and Fourier cosine series, so relatively simple Fourier series can be used to represent relatively complicated fractal interpolation function.

Dynamic Analysis of Rectangular Plate Stiffened by Any Number of Beams with Different Lengths and Orientations

The present work is concerned with dynamic characteristics of beam-stiffened rectangular plate by an improved Fourier series method (IFSM), including mobility characteristics, structural intensity, and transient response. The artificial coupling spring technology is introduced to establish the clamped or elastic connections at the interface between the plate and beams. According to IFSM, the displacement field of the plate and the stiffening beams are expressed as a combination of the Fourier cosine series and its auxiliary functions. Then, the Rayleigh–Ritz method is applied to solve the unknown Fourier coefficients, which determines the dynamic characteristics of the coupled structure. The Newmark method is adopted to obtain the transient response of the coupled structure, where the Rayleigh damping is taken into consideration. The rapid convergence of the current method is shown, and good agreement between the predicted results and FEM results is also revealed. On this basis, the effects of the factors related to the stiffening beam (including the length, orientations, and arrangement spacing of beams) and elastic parameters, as well as damping coefficients on the dynamic characteristics of the stiffened plate are investigated.

P2438Deep learning model for atrial fibrillation detection based on single beat morphology

Background The electrocardiogram (ECG) is commonly used, but most recent rhythm discrimination algorithms still lack both specificity and sensitivity. Deep learning techniques have shown promising results in the classification of physiological signals like ECGs. Purpose To develop and test a deep learning (DL) model to discriminate between atrial fibrillation (AF) and sinus rhythm (SR). Methods For the development of the DL model we used 1499 ECGs sampled at 500 Hz of patients diagnosed with AF. All ECGs were labeled by two experienced investigators. Only ECGs labeled as SR or AF were included in the dataset. To simplify the learning process, solely the first ECG channel was used. The ECG waveforms were preprocessed using the Fourier cosine series to correct for baseline wander. Input data was generated by normalizing and scaling all different heartbeats by centralizing the R peak, leading to 15744 single heart beat samples of 80 data points (figure A). Multiple feedforward architectures were tested with different numbers of layers, filters and activation functions. The models were trained by equally splitting the data (50%SR, 50%AF) in a training (65%), validation (25%) and test set (15%). The best performing model was chosen based on the accuracy. Results A total of 1469 ECGs (1061 (72%)SR, 408 (28%)AF) were included. The model with the best performance was a feedforward model consisting three dense layers with ReLU activation and four dense layers with Linear activation. Training of the model was performed in 32 epochs. Validation of the model resulted in an accuracy of 96% (figure B), precision of 95% and recall of 96%. Conclusions The morphology based deep learning model developed in this study was able to discriminate atrial fibrillation from sinus rhythm with a fairly high accuracy using a limited size dataset and only one lead.

Valuing Guaranteed Minimum Death Benefits by Cosine Series Expansion

Recently, the valuation of variable annuity products has become a hot topic in actuarial science. In this paper, we use the Fourier cosine series expansion (COS) method to value the guaranteed minimum death benefit (GMDB) products. We first express the value of GMDB by the discounted density function approach, then we use the COS method to approximate the valuation Equations. When the distribution of the time-until-death random variable is approximated by a combination of exponential distributions and the price of the fund is modeled by an exponential Lévy process, explicit equations for the cosine coefficients are given. Some numerical experiments are also made to illustrate the efficiency of our method.