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In this paper, an insurance company's surplus process is modeled by a time-changed Lévy process, where the continuous-time change is realized by integrating the Cox–Ingersoll–Ross (CIR) process. Assume that the surplus level is monitored periodically with constant monitoring frequency, and the insurer makes decisions on capital injections based on the observed surplus levels. Given a critical level b (b > 0), whenever the observed surplus level belongs to the interval [ 0 , b) , capitals are injected into the insurance company to bring the surplus level back to b ; whenever the observed surplus level is below level 0, ruin is declared, and the surplus process is stopped. The finite-time expected present value of operating costs until ruin is studied. Utilizing the Fourier series expansion in cosine form, we derive the closed-form approximation formula for this function and analyze the approximation error. Finally, many numerical examples are given to demonstrate the effectiveness of our method, and the impacts of some parameters are also studied. [ABSTRACT FROM AUTHOR]

This paper addresses the adaptive optimal containment control issue for non-affine nonlinear multi-agent systems in the presence of periodic disturbances. To deal with the disturbed internal dynamics, a fourier series expansion-neural networks-based adaptive identifier is designed for each follower, such that the restrictions posed on the system dynamics are released. Then,an adaptive dynamic programming technique is adopted to acquire the optimized virtual and actual controllers under a simplified actor-critic architecture, where the critic aims to appraise control performance and the actor aims to perform control task. Note that the above updating laws are constructed by the negative gradient of a designed function, which is constructed on the basis of the partial derivative of Hamilton-Jacobi-Bellman equation. Finally, simulation results are provided to show the applicability and effectiveness of the containment control scheme. [ABSTRACT FROM AUTHOR]

An analytical and meshless modeling for solving thickness shear vibration in an infinite piezoelectric quartz resonator with partial non-circular electrodes is proposed. Firstly, two-dimensional scalar differential equations derived by Tiersten and Smythe are adopted and transformed into the polar coordinate system. Secondly, displacement patterns for the electroded and unelectroded regions are assumed as a series of converging and outgoing cylindrical waves in the form of Bessel functions, where radiation conditions at infinity can be satisfied automatically. Finally, circumferential functions at interface are decomposed into Fourier series in order to deal with continuity conditions. It should be stressed that the general formulation proposed in this paper has a higher calculation accuracy and requires no division of the mesh compared to FEM/BEM, which can satisfy continuity conditions in an integrated manner over the whole interface or boundary. Resonance frequencies and mode shapes of different electrode shapes including circular, equilateral triangle, rectangular, elliptical, and pentagonal electrodes are numerically calculated and compared with FEM simulations, which efficiently validate high precision and wide applicability of this method. Utilizing this method, the influence of non-circular electrodes on the working performance of the quartz resonator is investigated systematically. The qualitative analysis and quantitative results obtained in this paper can provide the theoretical guidance for the design, measurement and manufacturing optimization of piezoelectric resonators. [ABSTRACT FROM AUTHOR]

This paper presents a pseudo-spectral method to solve PDEs in univocally describable irregular planar domains in polar coordinates, using a simple relationship that transforms the problem for the classic case in the unit circle, equipped with an orthogonal bivariate basis and complete in Sobolev sense. Given its generality, it applies to elliptical, parabolic or hyperbolic PDEs in Robin's condition. The choice of Radau points applied here allows the exclusion of the singularity in the origin and a posteriori error estimate also here developed provides us sharp error bounds of the numerical solution. • A method for solving 2nd order PDE in irregular domains with spectral precision is studied. • The domain transformation to the unit circle is presented. • The ideal ratio between expansion orders is derived. • The reversal verification allows to check solution quality by a posteriori analysis method. • The methodology can be extended to higher orders, even for non-linear problems. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the stabilization of a class of parabolic systems with time-varying delay, external disturbance and nonlinear periodic time-varying parameter (NPTVP). Firstly, combined with Fourier series expansion (FSE) method and neural networks (NNs) approximation technology, a NNs approximator is designed to approximately describe the uncertain dynamic term with NPTVP. Then, based on adaptive control theory, NNs approximation technology and reparameterization method, two robust adaptive neural network control (RANNC) algorithms are designed to make the parabolic systems with time-varying delay, external disturbance and NPTVP achieve asymptotic stability and meet a prescribed adaptive H ∞ performance of disturbance attenuation. Sufficient conditions of the asymptotical stability while satisfying the prescribed adaptive H ∞ performance index for the resulting closed-loop systems are also derived. Finally, several simulations are carried out to verify the effectiveness of the designed RANNC algorithms. [ABSTRACT FROM AUTHOR]

A recent study conducted by researchers at RUDN University explores the use of spectral analysis in diagnosing heart rhythm disorders. The study focuses on the Fourier transform and wavelet transform methods for analyzing electrocardiograms (ECGs). The researchers found that the accuracy of cardiac signal description is highest when the number of harmonics is half the number of sampling points during the heart's period. The study concludes that spectral analysis can be used to identify heart rhythm disorders and suggests further research in statistical analysis and the development of expert systems or technical devices. For more information, refer to the RUDN Journal of Engineering Research. [Extracted from the article]

This study presents a new meshfree approach for thermal vibration analysis of laminated composite cylindrical shell subject to arbitrary boundary conditions. The boundaries of laminated structures are constrained by artificial springs with variable stiffness. The spectro-geometric method (SGM) combined sine/consine functions containing circumferential wavenumber is employed to expand the displacement components of physical system. Following Reissner-Naghdi's thin shell theory, the Lagrangian energy functionals of the studied structure is derived, and the governing equation of laminated composite cylindrical shell under thermal environment is formulated with Rayleigh-Ritz method. After that, the convergence investigation and accuracy examination of the present method are both carried out by performing some numerical case studies. Finally, the effect of some crucial factors including temperature variable, boundary condition, length-radius ratio and laminated layers on the thermal vibration characteristics of laminated composite cylindrical shell are also studied. [ABSTRACT FROM AUTHOR]

In this paper, we study and solve linear neutral delay differential equations. We investigate the reliability and accuracy of applying the Laplace transform to obtain the solutions of linear neutral delay differential equations. These types of equations are more difficult to solve because the time delay appears in the derivative of the state variable. We rely on computer algebra and numerical methods to determine the poles which are required for computing the inverse Laplace transform. The form of the resulting solution is a non-harmonic Fourier series. A sufficient degree of accuracy can often be achieved by using a relatively small number of terms in the associated truncated series. We compare these solutions with the solutions obtained by the classical method of steps and numerical solutions obtained by the discretization of the linear delay differential equations. It is shown that the Laplace transform method provides very reliable and accurate solutions. [ABSTRACT FROM AUTHOR]

Purpose: The system of equations for fractional thermo-viscoelasticity is used to investigate two-dimensional bioheat transfer and heat-induced mechanical response in human skin tissue with rheological properties. Design/methodology/approach: Laplace and Fourier's transformations are used. The resulting formulation is applied to human skin tissue subjected to regional hyperthermia therapy for cancer treatment. The inversion process for Fourier and Laplace transforms is carried out using a numerical method based on Fourier series expansions. Findings: Comparisons are made with the results anticipated through the coupled and generalized theories. The influences of volume materials properties and fractional order parameters for all the regarded fields are examined. The results indicate that volume relaxation parameters, as well as fractional order parameters, play a major role in all considered distributions. Originality/value: Bio-thermo-mechanics includes bioheat transfer, biomechanics, burn injury and physiology. In clinical applications, knowledge of bio-thermo-mechanics in living tissues is very important. One can infer from the numerical results that, with a finite distance, the thermo-mechanical waves spread to skin tissue, removing the unrealistic predictions of the Pennes' model. [ABSTRACT FROM AUTHOR]

We present an efficient valuation approach for guaranteed minimum accumulation benefits (GMABs), guaranteed minimum death benefits (GMDBs), and surrender benefits (SBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We incorporate into the contract the risks of mortality and surrender, with these events generally monitored discretely over the life of the policy. Using a combination of the continuous-time Markov chain (CTMC) approximation and the Fourier cosine series expansion (COS) method, we determine that the valuation problem can be resolved within a regime-switching jump diffusion framework. Extensive numerical experiments showcase the efficiency of the proposed method, which proves to be more advantageous when compared to existing approaches like Monte Carlo (MC) simulation. The thorough analysis explores how model parameters affect the valuation outcomes. • We study the pricing of a complex insurance product. • We consider the effect of regime-switching on variable annuity price. • Using COS and CTMC methods, we obtain a semi-analytical approximation for VA pricing. • The computational efficiency of the existing method is greatly improved. [ABSTRACT FROM AUTHOR]

This paper explores an inverse source problem related to the heat equation, incorporating nonlocal boundary conditions and featuring two-term time-fractional derivatives. The task is to identify a source term that is independent of the spatial variable, as well as to define the temperature distribution based on energy measurements. Since the stated problem cannot be solved by direct use of the generalized Fourier method, we divide the problem into two sub-problems. The well-posedness of each problem is established through the application of the generalized Fourier method. [ABSTRACT FROM AUTHOR]

The classification of hazard is critical in industrial informatics, as it enhances early safety alerts, supports decision-making, and facilitates policy assessment. However, previous studies have generally neglected the temporal attributes of hazards, thereby constraining the effectiveness of models. This paper introduces a new model for hazard classification termed DLGM. DLGM represents a deep learning framework, with the structural parameters of grey models to encapsulate the hazard temporal attributes. To better accommodate the fluctuations of hazard series, a new grey model termed FSGM(1,1) equipped with Fourier series is proposed. Moreover, DLGM leverages a novel hierarchical feature fusion neural network (HFFNN) to optimize feature processing. Extensive experiments across three hazard themes involving 18 large-scale industrial processes have demonstrated the competitiveness of DLGM (e.g., it surpasses benchmark models such as Random Forest and BERT by approximately 2% in both accuracy and F1), the suitability of FSGM(1,1) (e.g., its mean absolute percentage error is less than 10% and mean squared error is below 0.02) and the effectiveness of HFFNN (e.g., it enhances the accuracy and F1 of DLGM by about 1%). [ABSTRACT FROM AUTHOR]

We investigate the steady-state solutions of the Whittaker–Hill equation, including a fractional derivative term. Using appropriate Fourier series, we characterize the behavior of the eigenvalue surfaces as a function of the differential equation parameters and describe the corresponding eigensolutions. We also examine the situation where the fractional derivative is zero to serve as a comparison for the fractional solutions. For some special combinations of the parameters, the fractional derivative term leads to the appearance of degenerate complex eigenvalues. • Characterization of the eigenvalue surfaces of the Whittaker–Hill equation in terms of the equation parameters. • Establishment of the recurrence relationships that connect the Fourier coefficients of the stable solutions of the fractional Whittaker–Hill equation. • Identification of the eigenvalue surfaces and determination of the factors leading to degenerate solutions of the fractional Whittaker–Hill equation. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to work out a method for calculating losses in induction motor steel taking into account its saturation. Design/methodology/approach: The theory of electric machines is applied during the analysis of induction motor equivalent circuits. The theory of Fourier series is used to determine the harmonic components of voltage, current and power. Instantaneous power theory and trigonometric transformations are used to solve algebraic and differential equations and their systems. The methods of approximation and interpolation are applied to obtain analytical expressions from the experimental data. Experimental research was carried out to verify the reliability of theoretical provisions and research results. Findings: A method for assessing an induction machine steel as a function of the generalized electromotive force has been proposed. It allows taking into account higher harmonics of the current, which are caused by the presence of nonlinearity of an induction motor magnetic circuit. Practical implications: The obtained results can be used in calculating the energy characteristics and operating modes of an induction motor, as well as in the construction of control systems. Originality/value: A method for determining the losses in the stator steel of an induction motor, using a generalized electromotive force, has been proposed for the first time. It enables taking into account the currents flowing both in the stator circuit and in the rotor circuit. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to present an analytical calculation for estimating the leakages field distribution in surface-mounted permanent magnet synchronous motors (SMPMSMs) according to a sub-domain field model for eccentricity fault detection. Design/methodology/approach: The magnetic field domain is classified into four sub-domains of PMs, air gap, stator core and outer region. In the proposed method, the governing equations taking the rotor eccentricity effect into account per region and the interface boundary conditions between sub-domains are formulated using the regular perturbation technique, Taylor series and Fourier series expansion. Maxwell's equations are solved in different regions in the polar coordinate system regarding the boundary conditions. Findings: The radial and tangential components of electromagnetic field distribution in all sub-domains of one SMPMSM are obtained using the proposed method analytically. Finite element analysis is used to validate the results of the proposed method; the results indicated that the analytical model matches the finite-element prediction up to 30% eccentricity, except for some peak values that depend on the harmonic order value. The results of this paper demonstrated that in the event of eccentricity, an asymmetric magnetic field is generated in the outer region of the machine. Although its amplitude is small, it can be an indicator for detecting eccentricity faults from the outside environment of the machine. Originality/value: The formulas presented in this paper can be applied as a new technique for detecting eccentricity faults in these motors from the outside environment. [ABSTRACT FROM AUTHOR]

This paper studies the consensus of nonlinear multi-agent systems with periodic disturbances and uncertain dynamics based on matrix theory, adaptive control, neural networks and fourier series expansion. Firstly, fourier series expansion and neural networks are used to describe the unknown periodic time-varying parameter and uncertain nonlinear dynamic, respectively. Secondly, based on adaptive control technology and reparameterization method, two new fully distributed control protocols are designed based on symbolic function and smooth hyperbolic tangent function, respectively, so that all agents can reach asymptotic consensus. Thirdly, a new positive integral bounded function is introduced to compensate for the approximation error caused by the smooth hyperbolic tangent function instead of the symbolic function, so that all network nodes achieve the same consensus effect. Finally, a simulation example is given to verify the effectiveness of the two algorithms and to illustrate their advantages and disadvantages. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid plane wave expansion/edge-based smoothed finite element method (PWE/ES-FEM) for band structures simulation of semi-infinite beam-like phononic crystals (PCs). The field variables are first approximated using linear shape functions in conjunction with the spatial Fourier series. Within the further formed edge-based smoothing domains, the gradient smoothing technique (GST) is employed to perform the strain smoothing operation. Based on the smoothed Galerkin weakform, the discretized system equations associated with the eigenvalue problem are finally obtained. Numerical examples demonstrate that the present method possesses higher computational accuracy in band structures simulation of semi-infinite beam-like PCs. [ABSTRACT FROM AUTHOR]

Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes. [ABSTRACT FROM AUTHOR]

The plane strain problem of an elastic matrix subjected to uniform far-field load and containing a Steigmann–Ogden material surface with circular arc cross-section is considered. The governing equations and the boundary conditions for the problem are reviewed. Exact complex integral representations for the elastic fields everywhere in the material are provided. The problem is further reduced to the system of real variables hypersingular boundary integral equations in terms of the first component of the surface stress tensor (surface stress) and the remaining component of that tensor and its second derivative, along with various problem parameters. The two unknowns are then approximated by the series of trigonometric functions that are multiplied by the square root weight functions to allow for automatic incorporation of the tip conditions. The unknown coefficients in series are found from the system of linear algebraic equations that is solved using standard collocation method. The numerical examples are presented to illustrate the influence of dimensionless parameters. The connection of the problem with that of rigid circular arc is discussed. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side Γ of the boundary of the computational domain Ω. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge–Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on ∂ Ω ∖ Γ. This new approach is called the Fourier-based dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

• A rigid-flexible nonlinear isolation system with an arbitrary boundary is modelled. • A novel method is proposed to analyze the arbitrary boundary nonlinear continuum. • The high-order energy Fourier series operate in temporal and spatial domains together. • An experiment rig is designed to prove the nonlinear isolation performance. The research on the nonlinear vibration isolation of continuum systems always focuses on the perfect boundary conditions. In this work, nonlinear isolation of transverse vibrations of the arbitrary boundary rectangular plate was investigated. A novel analytical approach was proposed. The rigid body dynamics and high-order energy Fourier series expansion method were utilized to establish a mathematical model of the arbitrary boundary nonlinear continuum system. This method involves the simultaneous Fourier series expansion in temporal and spatial domains. The dynamic behaviors of an arbitrary boundary rectangular plate with nonlinear vibration isolators can be described through a harmonic balance analysis together with arc-length continuation. The convergence and stability for the frequency response functions were checked via the Lyapunov stability theory. The numerical results supported the analytical solutions. Both the analytical and numerical results demonstrated that nonlinear broadband vibration isolation is sensitive to the elasticity of the arbitrary boundary flexible foundation. Furthermore, increasing damping can effectively suppress the transmission of low-order vibrations and control the divergence of higher-order vibrations. Finally, an experiment rig was designed to validate the nonlinear isolation of the transverse vibrations of an arbitrary boundary plate. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to analyze transient and steady state processes in series resistive capacitive-circuits when the supplied voltage is a sequence of periodic rectangular pulses. The purpose is to obtain an analytical formula for the capacitor voltage at any instant of time, both for the transient and the steady-state processes. Design/methodology/approach: The main approach is to use a combination of differential and recurrence equations. Findings: An analytical expression (formula) for obtaining the capacitor voltage at any instant of time has been found. Originality/value: The results obtained are new. They are much more convenient to use than the results obtained by the Fourier series. The exposed approach can also be used in other circuits and other forms of the supplied voltage. [ABSTRACT FROM AUTHOR]

Purpose: In the present paper, the new concept of "memory dependent derivative" in the Pennes' bioheat transfer and heat-induced mechanical response in human living tissue with variable thermal conductivity and rheological properties of the volume is considered. Design/methodology/approach: A problem of cancerous layered with arbitrary thickness is considered and solved analytically by Kirchhoff and Laplace transformation. The analytical expressions for temperature, displacement and stress are obtained in the Laplace transform domain. The inversion technique for Laplace transforms is carried out using a numerical technique based on Fourier series expansions. Findings: Comparisons are made with the results anticipated through the coupled and generalized theories. The influence of variable thermal, volume materials properties and time-delay parameters for all the regarded fields for different forms of kernel functions is examined. Originality/value: The results indicate that the thermal conductivity and volume relaxation parameters and MDD parameter play a major role in all considered distributions. This dissertation is an attempt to provide a theoretical thermo-viscoelastic structure to help researchers understand the complex thermo-mechanical processes present in thermal therapies. [ABSTRACT FROM AUTHOR]

A quiz with mathematical questions for class XI students on trigonometric functions and principle of mathematical induction is presented.

This paper concerns the fracture mechanics problem for elastic cracked materials under transient dynamic loading. The nonlinear contact problem for a linear crack under normal impact loading is solved by the boundary integral equations in the frequency domain and the components of the solution are presented by the Fourier exponential series. The algorithm convergence is proved and the solution is obtained for different friction coefficients. The dynamic stress intensity factors are computed for different stress pulses and compared with those obtained neglecting the crack closure and friction. [ABSTRACT FROM AUTHOR]

A new approximate approach to solve the two-dimensional boundary-value problems for the Stokes flows in a porous medium composed of a set of circular inclusions is proposed. The idea of the approach, based on the boundary element method (BEM), lies in Fourier series representations of unknown functions on the surfaces of inclusions. It is shown that for small volume concentrations of inclusions, a good accuracy is achieved by taking into account only the first three terms in the Fourier decomposition. Simultaneously, the total number of unknowns, which is equal to the product of the number of inclusions and that of unknowns on every inclusion, is reduced dramatically in comparison with the traditional BEM. The approach is tested by investigating the flow around a circular cylinder in a rectangular periodic cell, the flow in a porous medium with a large number of uniformly or randomly located inclusions. To demonstrate the capacity of the approach, we have calculated the Stokes flows in the porous medium which is modelled by locating 5000 inclusions of two different radii. It is shown that the method can also be used to estimate the properties of the fluid flows in highly porous media formed by non-circular inclusions. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to present the Galerkin method for analysis of steady-state processes in periodically time-varying circuits. Design/methodology/approach: A converter circuit working on a time-varying load is often controlled by different signals. In the case of incommensurable frequencies, one can find a steady-state process only via calculation of a transient process. As the obtained results will not be periodical, one must repeat this procedure to calculate the steady-state process on a different time interval. The proposed methodology is based on the expansion of ordinary differential equations with one time variable into a domain of two independent variables of time. In this case, the steady-state process will be periodical. This process is calculated by the use of the Galerkin method with bases and weight functions in the form of the double Fourier series. Findings: Expansion of differential equations and use of the Galerkin method enable discovery of the steady-state processes in converter circuits. Steady-state processes in the circuits of buck and boost converters are calculated and results are compared with numerical and generalized state-space averaging methods. Originality/value: The Galerkin method is used to find a steady-state process in a converter circuit with a time-varying load. Processes in such a load depend on two incommensurable signals. The state-space averaging method is generalized for extended differential equations. A balance of active power for extended equations is shown. [ABSTRACT FROM AUTHOR]

In this paper we present the inverse problem of determining a time dependent heat source in a two-dimensional heat equation accompanied with Dirichlet–Neumann–Wentzell boundary conditions. The model is of significant practical importance in applications where the time dependent internal source is to be controlled from total energy measurements in the case when the boundary is defined with partially absorbing (Dirichlet), partially reflecting (Neumann), and partially having the capacity for storing heat (Wentzell) conditions. The considered spatial domain is a rectangle which allows the application of a Fourier series analysis to the solution of the problem whose auxiliary spectral problem is a novel eigenproblem. This eigenvalue problem is defined using the two-dimensional Laplace operator where the spectral parameter resides in one of the accompanying boundary conditions. Under this setting we study the well-posedness of both the direct and inverse problems. In addition, we use the results obtained on the well-posedness to establish the numerical solvability of all the problems considered in the paper. In order to achieve this, we make use of a Chebyshev spectral collocation method for the spatial discretisation and a backward differentiation formula for the temporal discretisation. All the numerical results are shown to be consistent with the theory and in good agreement with the analytical solutions. • Inverse problem of finding a transient heat source in a 2D heat equation is analysed. • Partially absorbing, reflecting, and heat storing capacity boundaries are considered. • Fourier series analysis is employed for the well-posedness of the inverse problem. • The adjunct eigenproblem is defined by the Laplace operator with a Wentzell condition. • A numerical method is derived to solve all the direct, inverse, and eigenproblems. [ABSTRACT FROM AUTHOR]

A new expression for the fundamental solution is introduced, presenting three relevant characteristics: (i) it is explicit in terms of the Stroh's eigenvalues, (ii) it remains well-defined when some Stroh's eigenvalues are repeated, and (iii) it is exact. A fast and robust numerical scheme for the evaluation of the fundamental solution and its derivatives developed from double Fourier series representations is presented. The Fourier series representation is possible due to the periodic nature of the solution. The attractiveness of this series solution is that the information of the material properties is contained only in the Fourier coefficients, while the information of the dependence of the evaluation point is contained in simple trigonometric functions. This implies that any order derivatives can be determined by spatial differentiation of the trigonometric functions. Moreover, Fourier coefficients need to be obtained only once for a given material, leading to an efficient methodology. The robustness of the scheme arises from the properties (i) and (ii) of the new expression for the fundamental solution, which is used to compute the Fourier coefficients. The proposed approach combines the clean structure of the Stroh formalism with the simplicity of Fourier expansions, addressing the old drawbacks of anisotropic fundamental solutions. [ABSTRACT FROM AUTHOR]

This paper presents a multi-scale approach to analyze failure in laminated composites. First, at the continuum mesoscale, the boundary element method (BEM) is used with the anisotropic 3D fundamental solution based on double Fourier series. The dynamic effects in the continuum media are included owing to the application of high-rate boundary conditions. These effects induced domain integrals in the BEM formulation which are treated via the dual reciprocity method (DRM). After the evaluation of the mechanical behavior, the multi-scale cohesive zone model (MCZM) is employed. The highly fluctuant deformation confined in the matrix at the atomistic scale is homogenized through the Cauchy-Born rule which allows coupling of the coarse and fine scales at the continuum. The constitutive force field at the atomistic scale is evaluated using a specific potential for epoxy materials, taking the advantage offered by the coarse grained model. Finally, failure is deemed to occur when the criterion fitted at the atomistic scale is reached at the mesoscale. Numerical results are presented showing the crack propagation paths for different configurations. [ABSTRACT FROM AUTHOR]

This paper proposes a new methodology for subs-pace-based state-space identification for linear time-periodic (LTP) systems. Since LTP systems can be lifted to equivalent linear time-invariant (LTI) systems, we first lift input–output data from an unknown LTP system as if they were collected from an equivalent LTI system. Then, we use frequency-domain subspace identification methods to find the LTI system estimate. Subsequently, we propose a novel method to obtain a time-periodic realization for the estimated lifted LTI system by exploiting the specific parametric structure of Fourier series coefficients of the frequency-domain lifting method. Our method can be used to obtain state-space estimates for unknown LTP systems as well as to obtain Floquet transforms for known LTP systems. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study the statistical estimation of the discounted density of the deficit at ruin in the classical risk model. The estimator is constructed by the two-dimensional Fourier cosine series expansion. It is shown that the estimator is easily computed and has fast convergence rate. Some simulation results are presented to show that the estimator performs very well when the sample size is finite. [ABSTRACT FROM AUTHOR]

Abstract The main goal of this note is to provide new, mostly multidimensional densities, compactly supported and list many of its properties that enable effective calculations. The idea of obtaining such densities is firstly to build some one-dimensional densities depending on many parameters and then treat the constructed in this way distributions as conditional ones. Then of course by imposing certain distribution on the parameters and multiplying the two distributions we arrive at new multivariate distribution. To enable effective calculations, we utilize nice, simple and widely known properties of Chebyshev polynomials. Thus, in particular, the one-dimensional distribution mentioned above will have a form of arcsine distribution multiplied by some rational function. The fact that we use Chebyshev polynomials allows us to calculate all moments of this one-dimensional distribution as well as to find a family of polynomials that are orthogonal with respect to this distribution. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to describe a semi-analytical modeling technique to predict eddy currents in three-dimensional (3D) conducting structures with finite dimensions. Using the developed method, power losses and parasitic forces that result from eddy current distributions can be computed.Design/methodology/approach In conducting regions, the Fourier-based solutions are developed to include a spatially dependent conductivity in the expressions of electromagnetic quantities. To validate the method, it is applied to an electromagnetic configuration and the results are compared to finite element results.Findings The method shows good agreement with the finite element method for a large range of frequencies. The convergence of the presented model is analyzed.Research limitations/implications Because of the Fourier series basis of the solution, the results depend on the considered number of harmonics. When conducting structures are small with respect to the spatial period, the number of harmonics has to be relatively large.Practical implications Because of the general form of the solutions, the technique can be applied to a wide range of electromagnetic configurations to predict, e.g. eddy current losses in magnets or wireless energy transfer systems. By adaptation of the conductivity function in conducting regions, eddy current distributions in structures containing holes or slit patterns can be obtained.Originality/value With the presented technique, eddy currents in conducting structures of finite dimensions can be modeled. The semi-analytical model is for a relatively low number of harmonics computationally faster than 3D finite element methods. The method has been validated and shown to be computationally accurate. [ABSTRACT FROM AUTHOR]

Abstract Most high dimensional data visualization techniques embed or project the data onto a low dimensional space which is then used for viewing. Results are thus limited by how much of the information in the data can be conveyed in two or three dimensions. Methods 1 1 An R package implementing the proposed methodology is provided in the supplementary material. are described for a lossless functional representation of any real matrix that can capture key features of the data, such as distances and correlations. This approach can be used to visualize both subjects and variables as curves, allowing one to see patterns of subjects, patterns of variables, and how the subject and variable patterns relate to one another. A theoretical justification is provided for this approach, and various facets of the method’s usefulness are illustrated on both synthetic and real data sets. Highlights • This paper provides a novel visualization method for any arbitrary matrix. • The proposed approach can be used to inspect the mean and the covariance structure. • There is great flexibility for atypical data formats such as dissimilarity matrices. • An R package implementing this visualization methodology is provided. [ABSTRACT FROM AUTHOR]

Some methods are described for the generation of fractional and negative powers of any formal series, such as Poisson series or Chebyshev series. It is shown that, with the use of the three elementary operations of addition, subtraction, and multiplication, all rational (positive and negative) powers of a series con be constructed. There are basically two approaches: the binomial theorem and the iteration methods. Both methods are described here, and the relationship between them is pointed out. Some well-known classical formulas are obtained as particular cases, and it is shown how the convergence properties of these formulas can be improved with very little additional computations. Finally, at the end of the article, some numerical experiments are described with Chebyshev series and with Fourier series. [ABSTRACT FROM AUTHOR]

An algorithm for Filon quadrature is described. Considerable attention has been devoted to an analysis of the round-off and truncation errors. The algorithm includes an automatic error control feature. [ABSTRACT FROM AUTHOR]

This article discusses the ALGOL convolution procedure based on the fast Fourier transform. The operation of convolution is used in computing autocorrelation and cross-correlation functions, in digital filtering of time series, and many other applications. Procedure Convolution computes the convolution of two real vectors of dimension. The special features of this procedure are the usual reordering of the fast Fourier transform results is avoided, and the return from frequency to time is made with a transform of dimension n/2 instead of n. The two vectors A and B are first transformed with a single complex Fourier transform of dimension n. Procedure Convolution uses-procedure REALTRAN but repeated here with revisions to improve accuracy on computers using truncated floating-point arithmetic. Procedures FFT4 and REVFFT4 are also used and perform the same computation as procedures FFT2 and REVFFT2 given for use on a system with virtual memory. The transform procedures given are organized without regard to the problem of memory overlay.

The article presents the procedures for computing the finite Fourier transform of a complex data vector which is based on the Cooley-Tukey algorithm. An overview of the complextransform and realtransform's procedures are offered. Either the complex Fourier transform or its inverse are the procedure computed by complextransform, while realtransform computes either the Fourier coefficients of a sequence of real data points or evaluates a fourier series. The topic also includes orthogonal transformation, permutations, and spectral analysis.

• Nonlinear stability of saturated porous nanobeams embedded in an elastic foundation has been investigated. • Porous/monotonous, porous/nonlinear symmetric and porous/nonlinear non-symmetric distributions have consider for patterns for saturation. • The infinite terms of the power series are discretized thanks to Stokes' transformation and trigonometric series. • Effects of different material and geometrical parameters on nonlinear critical buckling loads are discussed. In this work, the nonlinear stability behaviors of the saturated porous nanobeams embedded in an elastic foundation are investigated. The restrained nanobeam is modeled using geometrical nonlinear equations in conjunction with the constitutive law of saturation. Three patterns for saturation along the thickness of the nanobeam are considered as porous/monotonous, porous/nonlinear symmetric, and porous/nonlinear non-symmetric distributions. The infinite terms of the power series are discretized thanks to Stokes' transformation and trigonometric series. Then, the effects of saturation and nonlinearity on buckling loads are studied by considering the restrained boundary conditions. Moreover, the nonlinear results are validated with the saturated nanobeam with the rigid supporting conditions. [ABSTRACT FROM AUTHOR]

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach. [Display omitted] • First ever lower theoretical bounds on the truncation error • Reveals novel relations between the genus and the truncation parameter • Proposes the use of hyperbolic crosses and demonstrates a new record of genus 60 • Tensor train approximations of the coefficient tensor do not seem to be practical [ABSTRACT FROM AUTHOR]

An instantaneous frequency estimation algorithm is developed for multi-component signals acquired through multiple sensors. The proposed method estimates the mixing matrix for the strongest source in the given multi-sensor recording. The information of the mixing matrix is used to extract the corresponding source through spatial filtering. The spatially filtered signal is then analyzed using adaptive directional time-frequency distribution. The instantaneous frequencies of the extracted sources are then estimated through the Random sample consensus (RANSAC) algorithm that involves the detection of peaks, fitting of the instantaneous frequency curve using Fourier series approximation, and evaluating the instantaneous frequency estimate using an objective function. Both the mixing matrix and instantaneous frequency information is used to remove the corresponding source from the mixture signal by employing time-frequency and spatial filtering. The process mentioned above is then repeated till the instantaneous frequencies of all the sources have been estimated. Once the instantaneous frequencies of all the sources have been estimated, each source's instantaneous frequency is re-estimated by removing all the other components using time-frequency and spatial filtering. Till convergence, the process is iterated. Through numerical results, it is shown that the proposed method performs better than the relevant methods in the state of the art. [ABSTRACT FROM AUTHOR]

The paper presents rhythm adaptive Fourier series decompositions of cyclic numerical functions and one-dimensional probabilistic characteristics of cyclic random processes, which are mathematical models of many deterministic and stochastic cyclic signals with a variable (irregular) rhythm. The adaptability of such decompositions to the change of rhythm of the investigated cyclic signals, on the one hand, ensures the elimination of the approximation error characteristic for classical Fourier series for periodic functions, and on the other hand, does not lead to an increase in the number of spectral components in the corresponding Fourier images, which occurs when Fourier transform is applied to cyclic deterministic and stochastic signals with variable rhythm. The work also presents the interconnected characteristics of the rhythm of cyclic deterministic and stochastic signals, and establishes their main properties for the base components of rhythm-adaptive Fourier series. The efficiency of rhythm adaptive Fourier series decompositions of cyclic numerical functions and one-dimensional probabilistic characteristics of cyclic random processes is confirmed by the results of a computer simulation. [ABSTRACT FROM AUTHOR]

The design and analysis of numerical methods are usually guided by the following: (a) von Neumann analysis using Fourier series expansion of unknowns, (b) the modified differential equation approach, and (c) a more generalized approach that analyzes numerical methods globally, using Fourier–Laplace transform to treat the total or disturbance quantities in terms of waves. This is termed as the global spectral analysis (GSA). GSA can easily handle non-periodic problems, by invoking wave properties of the field through the correct numerical dispersion relation, which is central to the design and analysis. This has transcended dimensionality of the problem, while incorporating various physical processes e.g. by studying convection, diffusion and reaction as the prototypical elements involved in defining the physics of the problem. Although this is used for fluid dynamical problems, it can also explain many multi-physics and multi-scale problems. This review describes this powerful tool of scientific computing, with new results originating from GSA: (i) providing a common framework to analyze both hyperbolic and dispersive wave problems; (ii) analyze numerical methods by comparing physical and numerical dispersion relation, which leads to the new class of dispersion relation preserving (DRP) schemes; (iii) developing error dynamics as a distinct tool, identifying sources of numerical errors involving both the truncation and round-off error. Such studies of error dynamics provide the epistemic tool of analysis rather than an aleatoric tool, which depends on uncertainty quantification for high performance computing (HPC). One of the central themes of GSA covers the recent advances in understanding numerical phenomenon like focusing, which defied analysis so far. An application of GSA shown here for the objective evaluation of the so-called DNS by pseudo-spectral method for spatial discretization along with time integration by two-stage Runge–Kutta method is performed. GSA clearly shows that this should not qualify as DNS for multiple reasons. A new design of HPC methods for peta- and exa-flop computing tools necessary for parallel computing by compact schemes are also described. [ABSTRACT FROM AUTHOR]

The Harmonic Balance method for simulating unsteady nonlinear periodic flows is presented in this paper. Its recent, yet rapid development was motivated by the possibility of significant CPU time consumption savings compared to conventional transient methods, while providing sufficient accuracy. Using the Harmonic Balance method, a single time-continuous primary variable is decomposed into a number of steady state snapshots throughout the representative period, depending on the number of resolved harmonics. Decomposition is done following the Fourier series, yielding the mutually coupled equations where time derivation term is replaced by a coupling source term. Therefore, the decomposition relates to temporal evolution and accounts for transient effects. The numerical method is based on the Finite Volume (FV) method for computational fluid dynamics and was implemented in the open-source software foam-extend . Along with the mathematical model, this work presents the additional derivation of temporal coupling over the interface for cases with more than one domain. With proper mathematical procedure, no additional time-interpolation between interfaces is needed, presenting a novelty in the field. The method is validated in two distinct fields: turbomachinery and naval hydrodynamics. Turbomachinery test cases consist of 2D ERCOFTAC centrifugal pump and a 3D Aachen turbine, for which the comparison with conventional transient and steady state results is performed. For the ERCOFTAC case, CPU time comparison is presented as well. Two phase variant of the Harmonic Balance method is validated on regular wave propagation test case, including the comparison with transient results, which is followed by a DTMB ship hull on waves test case. The two-phase variant of Harmonic Balance represents a novelty in the field of method application. The overall results show the approach is both efficient and accurate, while offering versatility in the area of application. [ABSTRACT FROM AUTHOR]

Purpose This paper aims to present a method for calculating electromagnetic fields, eddy currents and forces for a quasi-static two-dimensional (2-D) model of linear induction motors (LIMs) where the primary side is modeled as a collection of individual coils.Design/methodology/approach An analytical solution using Fourier series is derived for a general source with current excitations residing in an airgap and moving relative to a conducting plate and back iron. Ideal magnetic material with infinite permeability is used to model the primary iron above the primary source and the back iron below the conducting plate.Findings The analytical solution is compared to a commercial 2-D finite element analysis (FEA) simulation for validation and then compared to a 2-D FEA model with a more detailed geometry of the LIM. The analytical model accurately predicts LIM thrust even though the geometry of the primary core is simplified as an infinitely long flat slab. 2-D frequency FEA can be used successfully to predict in motion LIM performance.Originality/value The analytical solution presented here models the primary excitations as individual discrete coils instead of current sheets, which all existing models are based on. The discrete coils approach provides a more intuitive and realistic model of the LIM. [ABSTRACT FROM AUTHOR]

Purpose The purpose of this paper is to introduce a method for the analysis of steady-state processes in periodically time varying circuits. The method is based on a new definition of frequency responses for periodic time-varying circuits.Design/methodology/approach Processes in inverter circuits are often described by differential equations with periodically variable coefficients and forcing functions. To obtain a steady-state periodic solution, the expansion of differential equations into a domain of two independent variables of time is made. To obtain differential equations with constant coefficients the Lyapunov transformation is applied. The two-dimensional Laplace transform is used to find a steady-state solution. The steady-state solution is obtained in the form of the double Fourier series. The transfer function and frequency responses for the inverter circuit are introduced.Findings A set of frequency characteristics are defined. An example of a boost inverter is considered, and a set of frequency responses for voltage and current are presented. These responses show a resonance that is missed if the averaged state-space method is used.Originality/value A new definition of frequency responses is presented. On the basis of frequency responses, a modulation strategy and filters can be chosen to improve currents and voltages. [ABSTRACT FROM AUTHOR]

A low-power low-noise chopper amplifier for biosensor applications is proposed. To tackle the inherent ripple artifacts, it employs a simple ripple reduction method using a bandpass amplifier. The chopper amplifier is a linear periodic time-varying system. The method of harmonic transfer matrix is used to derive the signal and the noise harmonic transfer functions, and the theoretical results are confirmed by simulation and measurement results. Fabricated in a 130-nm standard CMOS process, the proof-of-concept prototype occupies a chip area of 0.28 mm2 and consumes 3.3~\mu \textA from a 1.2-V supply. The input referred noise is only 43 nV/ (\mathrm Hz)^1/2 . [ABSTRACT FROM AUTHOR]

In this paper, we essentially modernize the NAFASS (Non-orthogonal Amplitude Frequency Analysis of the Smoothed Signals) approach suggested earlier. Actually, we solved two important problems: (a) new and effective algorithm was proposed and (b) we proved that the segment of the Prony spectrum could be used as the fitting function for description of the desired frequency spectrum. These two basic elements open an alternative way for creation of the fluctuation spectroscopy when the segment of the Fourier series can fit any random signal with trend but the dispersion spectrum of the Fourier series ω 0 · k ( ω 0 ≡ 2 π / T ) ⇒ Ω k ( k = 0 , 1 , 2 , . . . , K − 1 ) is replaced by the specific dispersion law Ω k calculated with the help of original algorithm described below. It implies that any finite signal will have a compact amplitude-frequency response (AFR), where the number of the modes is much less in comparison with the number of data points ( K << N ). The NAFASS approach can be applicable for quantitative description of a wide set of random signals/fluctuations and allows one to compare them with each other based on one general platform. As the first example, we considered economic data and compare 30-years world prices for meat (beef, chicken, lamb and pork) entering as the basic components to every-day food consumption. These data were taken from the official site http://www.indexmundi.com/commodities/ . We fitted these random functions with the high accuracy and calculated the desired “amplitude-frequency” response for these random price fluctuations. The calculated distribution of the amplitudes ( Ac k , As k ) and frequency spectrum Ω k ( k = 0, 1,…, K −1) allows one to compress initial data ( K (number of modes) <<  N (number of data points), N / K ≅ 20–40) and receive an additional information for their comparison with each other. As the second example, we considered the transcendental/irrational numbers description in the frame of the proposed NAFASS approach, as well. This possibility was demonstrated on the quantitative description of the transcendental number π = 3.1415926535897932…, containing initially 6⋅10 4 digits. The results obtained for the second type of data can be useful for cryptography purposes. We do believe that the NAFASS approach can be widely used for creation of the new metrological standards based on comparison of different test fluctuations with the fluctuations registered from the pattern equipment. Apart from this obvious application, the NAFASS approach can be applicable for description of different nonlinear random signals containing the hidden beatings in radioelectronics and acoustics. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present dynamic responses of a multilayered magneto-electro-elastic half-space due to an external/internal time-harmonic shear mechanical loading. To do so, a concrete mathematical tool is applied to solve a set of complicated high-order coupled equations of motion for a layered magneto-electro-elastic structure. The mathematical tool is constructed based on a new infinite dimensional vector space utilizing multiplication of exponential functions as a base for Fourier series in the tangential direction and Bessel function as a base in the radial direction. A cylindrical coordinate system is attached to the multi-layered half-space with the depth axis oriented perpendicular to the surface. In this way, with the use of this vector space, the coupled governing partial differential equations are changed to two sets of ordinary first-order differential equations, where their unknown functions are the coefficients of base functions in the Hankel-Fourier domain. It should be noted that these two sets of equations are completely decoupled based on the decoupling of SH-wave from P- and SV-waves. The effect of the layering is considered with the use of the dual-variable-position method. A mechanically uniform surface/internal shear time-harmonic loading is applied as a common excitation in nature to pursue the pattern of energy transformation in multifunctional magneto-electro-elastic structures with different material properties. Also, the effects of loading frequency and resonance phenomena are investigated in this paper numerically. The results show clearly how the arrangement and the designing process of different layers with different material properties can change responses in smart devices. [ABSTRACT FROM AUTHOR]