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An intelligent lower-limb prosthesis can provide walking support and convenience for lower-limb amputees. Trajectory planning of prosthesis joints plays an important role in the intelligent prosthetic control system, which directly determines the performance and helps improve comfort when wearing the prosthesis. Due to the differences in physiology and walking habits, humans have their own walking mode that requires the prosthesis to consider the individual's demands when planning the prosthesis joint trajectories. The human is an integral part of the control loop, whose subjective feeling is important feedback information, as humans can evaluate many indicators that are difficult to quantify and model. In this study, trajectories were built using the phase variable method by normalizing the gait curve to a unified range. The deviations between the optimal trajectory and current were represented using Fourier series expansion. A gait dataset that contains multi-subject kinematics data is used in the experiments to prove the feasibility and effectiveness of this method. In the experiments, we optimized the subjects' gait trajectories from an average to an individual gait trajectory. By using the individual trajectory planning algorithm, the average gait trajectory can be effectively optimized into a personalized trajectory, which is beneficial for improving walking comfort and safety and bringing the prosthesis closer to intelligence. [ABSTRACT FROM AUTHOR]

Based on the concept of improving the convergence of Fourier series by elimination of the Gibbs phenomenon, this work proposes a rational approach to construct hierarchic sets of shape functions of any desired degree of continuity from suitably modified (enriched) trigonometric series. Hierarchic sets of either enriched cosines or enriched sines are then used in Ritz solutions of beam and plate bending problems to illustrate the benefits provided by the enrichment in the quality of approximations. For a given degree of continuity, it is observed that a set of enriched cosines yields, in general, more accurate results and faster convergence than a set of enriched sines. To the authors' knowledge, there is no published information yet on hierarchic sets of enriched cosines. Moreover, it is shown that some hierarchic sets, which are already well-established in the literature, may lead not only to spurious oscillations but also to erroneous results. [ABSTRACT FROM AUTHOR]

A theoretical model for the multi-span spinning beams with elastic constraints under an axial compressive force is proposed. The displacement and bending angle functions are represented through an improved Fourier series, which ensures the continuity of the derivative at the boundary and enhances the convergence. The exact characteristic equations of the multi-span spinning beams with elastic constraints under an axial compressive force are derived by the Lagrange equation. The efficiency and accuracy of the present method are validated in comparison with the finite element method (FEM) and other methods. The effects of the boundary spring stiffness, the number of spans, the spinning velocity, and the axial compressive force on the dynamic characteristics of the multi-span spinning beams are studied. The results show that the present method can freely simulate any boundary constraints without modifying the solution process. The elastic range of linear springs is larger than that of torsion springs, and it is not affected by the number of spans. With an increase in the axial compressive force, the attenuation rate of the natural frequency of a spinning beam with a large number of spans becomes larger, while the attenuation rate with an elastic boundary is lower than that under a classic simply supported boundary. [ABSTRACT FROM AUTHOR]

Recently, Devaiya and Srivastava [3] studied the T r -strong convergence of numerical sequences and Fourier series using a lower triangular matrix T = (b m , n) , and generalized the results of Kórus [8]. The main objective of this paper is to introduce [ T r , G , u , q ] -strongly convergent sequence spaces for r ∈ N , and defined by a sequence of modulus functions. We also provide a relationship between [ T , G , u , q ] and [ T r , G , u , q ] -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for T r -strong convergent sequences, statistical convergence and Fourier series using the idea of [ T r , G , u , q ] -strongly convergent sequence spaces. [ABSTRACT FROM AUTHOR]

We consider the corotational Maxwell model which is perhaps the simplest constitutive model with a nontrivial oscillatory shear response that can be solved analytically. The exact solution takes the form of an infinite series. Due to exponential convergence, accurate analytical approximations to the exact solution can be obtained by truncating the series after a modest number ( ≈ 10–20) of terms. We compare the speed and accuracy of this truncated analytical solution (AS) with a fast numerical method called harmonic balance (HB). HB represents the periodic steady-state solution using a Fourier series ansatz. Due to the linearity of the constitutive model, HB leads to a tridiagonal linear system of equations in the Fourier coefficients that can be solved very efficiently. Surprisingly, we find that the convergence properties of HB are superior to AS. In terms of computational cost, HB is about 200 times cheaper than AS. Thus, the answer to the question posed in the title is affirmative. [ABSTRACT FROM AUTHOR]

The Qilian Mountains (QM) possess a delicate vegetation ecosystem, amplifying the evident response of vegetation phenology to climate change. The relationship between changes in vegetation growth and climate remains complex. To this end, we used MODIS NDVI data to extract the phenological parameters of the vegetation including meadow (MDW), grassland (GSD), and alpine vegetation (ALV)) in the QM from 2002 to 2021. Then, we employed path analysis to reveal the direct and indirect impacts of seasonal climate change on vegetation phenology. Additionally, we decomposed the vegetation phenology in a time series using the trigonometric seasonality, Box-Cox transformation, ARMA errors, and Trend Seasonal components model (TBATS). The findings showed a distinct pattern in the vegetation phenology of the QM, characterized by a progressive shift towards an earlier start of the growing season (SOS), a delayed end of the growing season (EOS), and an extended length of the growing season (LOS). The growth cycle of MDW, GSD, and ALV in the QM species is clearly defined. The SOS for MDW and GSD occurred earlier, mainly between late April and August, while the SOS for ALVs occurred between mid-May and mid-August, a one-month delay compared to the other vegetation. The EOS in MDW and GSD were concentrated between late August and April and early September and early January, respectively. Vegetation phenology exhibits distinct responses to seasonal temperature and precipitation patterns. The advancement and delay of SOS were mainly influenced by the direct effect of spring temperatures and precipitation, which affected 19.59% and 22.17% of the study area, respectively. The advancement and delay of EOS were mainly influenced by the direct effect of fall temperatures and precipitation, which affected 30.18% and 21.17% of the area, respectively. On the contrary, the direct effects of temperature and precipitation in summer and winter on vegetation phenology seem less noticeable and were mainly influenced by indirect effects. The indirect effect of winter precipitation is the main factor affecting the advance or delay of SOS, and the area proportions were 16.29% and 23.42%, respectively. The indirect effects of fall temperatures and precipitation were the main factors affecting the delay and advancement of EOS, respectively, with an area share of 15.80% and 21.60%. This study provides valuable insight into the relationship between vegetation phenology and climate change, which can be of great practical value for the ecological protection of the Qinghai-Tibetan Plateau as well as for the development of GSD ecological animal husbandry in the QM alpine pastoral area. [ABSTRACT FROM AUTHOR]

Background: The estimation of the natural frequency is a key step in the design of an offshore wind turbine (OWT). In the literature, the natural frequency can be evaluated through two steps, the first one is the estimation of the fixed base natural frequency using formulas from the literature, and the second one is the estimation of the foundation system effect. These formulas are efficient for simple shapes of OWTs but become hard to use for recent OWTs. Method: The finite element method and the Fourier series have been used to develop a computer code under the name of "TurbiSoft" to compute the natural frequency of the OWT with the foundation system. Each of the linear, the quadratic volume elements, and the joint element was employed in TurbiSoft to model the whole system to examine the influence of the element type, the soil profile, and the interaction soil–pile state on the natural frequency. Results: The outcomes of TurbiSoft for several OWTs were compared with the measured frequencies to define the range error of each parameter. The results show that the quadratic elements are more suitable for the natural frequency calculation. Also, this study demonstrates that the natural frequency value is strongly related to the interaction soil–pile state, and is slightly affected by the soil profile variation. [ABSTRACT FROM AUTHOR]

The present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes' transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium. [ABSTRACT FROM AUTHOR]

In order to study vibro-acoustic characteristics between composite laminated rotationally stiffened plate and acoustic cavities in the coupled system, first-order shear deformation theory (FSDT) and modified Fourier series are used to construct a unified analysis model. The involved coupled systems primarily encompass three types: the coupled system between composite laminated rotationally stiffened plate and cylindrical-cylindrical cavities, spherical-cylindrical cavities, and conical-cylindrical cavities. First, the first-order shear deformation theory and the modified Fourier series are applied to construct the allowable displacement function of the composite laminated rotationally stiffened plate and the allowable sound pressure function of the acoustic cavities. Second, the energy functionals for the structural domain and the acoustic field domain are established, respectively. According to the continuity condition of the particle vibration velocity at the coupling boundary between the composite, laminated cylindrical shell and the enclosed cavity, the coupling potential energy between the stiffened plate and two acoustic cavities is introduced to obtain the energy functional of the coupled system. Third, the Rayleigh-Ritz method is utilized to solve the energy functional and, when combined with artificial virtual spring technology, the suggested theory may be used to study the vibro-acoustic characteristics of a coupled system with arbitrary elastic boundary conditions. Finally, based on validating the fast convergence and correctness of the model, this paper will analyze the impact of crucial parameters on vibro-acoustic characteristics. Furthermore, by incorporating internal point forces and point-sound source stimulation, a steady-state response analysis of the coupled system will be conducted. This research can give a theoretical foundation for the vibration and noise reduction of a vibro-acoustic coupling system. [ABSTRACT FROM AUTHOR]

In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution asymptotically approaches a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases become more accurate as the independent variable increases. [ABSTRACT FROM AUTHOR]

A permanent gravity wave propagating on deep water is a classic mathematical problem. However, the Fourier series approximation (FSA) based on the physical plane was examined to be valid for almost waves at all depths. The accuracy of the FSA for almost-limiting gravity waves remains unevaluated, which is the purpose of this study. We calculate some physical properties of almost-limiting waves on deep water using the FSA and compare them with other studies on the complex plane. The comparison results show that the closer the wave is, the greater the difference. We find that the main reason for this difference is that the wave profile in the FSA retains an original implicit form and is not represented by Fourier series. Therefore, the kinematic and dynamic conditions of the free surface around the wave crest cannot be satisfied at the same time. [ABSTRACT FROM AUTHOR]

This work investigates an initial-boundary value and an inverse coefficient problem of determining a space dependent coefficient in the fractional wave equation with the generalized Riemann–Liouville (Hilfer) time derivative. In the beginning, it is considered the initial boundary value problem (direct problem). By the Fourier method, this problem is reduced to equivalent integral equations, which contain Mittag-Leffler type functions in free terms and kernels. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we get a priori estimate for solution via unknown coefficient which will be used to study the inverse problem. The inverse problem is reduced to the equivalent integral equation of Volterra type. To show existence unique solution to this equation the Schauder principle is applied. The local existence and uniqueness results are obtained. [ABSTRACT FROM AUTHOR]

The GaoFen7 (GF7) optical satellite is the first Chinese civilian sub-meter high-resolution stereo mapping satellite and is equipped with a double linear array camera and laser altimeter to achieve large-scale topographic mapping. To improve the accuracy of attitude determination, an attitude determination system comprised of four star sensors is loaded. According to the measurement accuracy and steady performance, the star sensors 1a and 1b is usually used together for satellite attitude calculation, which is called the conventional mode of attitude determination. Then, the combination of star sensors 2a and 2b is called the unconventional mode of attitude determination. Affected by variations in the incident angle of sunlight and solar radiation, thermal deformation occurs in the body and installation structure of the star sensor, which causes Attitude Low-Frequency Error (ALFE) and seriously influences the consistency of attitude determination results of different combination modes for multiple star sensors system. This study proposes an ALFE analysis and calibration approach for the multiple star sensors system of GF7 satellite to ensure the consistency of attitude determination results of different combination modes. Based on the statistical characteristics of the angles of the three axes, the installation parameters of the four star sensors are first calibrated. After analyzing the characteristics of the optical axis angles within 1420 orbit periods over 135 days, the segmented ALFE compensation model between the unconventional and conventional modes is proposed based on the Fourier series model and input parameter of latitude. Based on the on-orbit installation parameters and the ALFE model, the precise attitude determination results of the unconventional mode are calculated. Experimental results show that the attitude determination consistency after compensation is better than 2". Moreover, the reliable application time range of the compensation model is 30 days to satisfy the requirements for high-precision attitude determination of GF7 satellite. [ABSTRACT FROM AUTHOR]

This work proposes a novel efficient method to track the evolution of amplitude extrema featured by frequency responses of nonlinear systems using the harmonic balance method. Means to compute the amplitude of a Fourier series are first outlined, and a set of equations characterizing a local extremum of a nonlinear frequency response amplitude curve is derived. Efficient numerical procedures are used to evaluate these equations and their derivatives (including second‐order ones) to embed them in a predictor‐corrector continuation framework. The proposed approach is illustrated on three examples of increasing complexity, namely a Helmholtz–Duffing oscillator, a two‐degree‐of‐freedom system with a modal interaction, and a doubly clamped von Kàrmàn beam with a nonlinear tuned vibration absorber. [ABSTRACT FROM AUTHOR]

Context. The YORP effect is the thermal torque generated by radiation from the surface of an asteroid. The effect is sensitive to surface topology, including small-scale roughness, boulders, and craters. Aims. The aim of this paper is to develop a computationally efficient semi-analytical model for the crater-induced YORP (CYORP) effect that can be used to investigate the functional dependence of this effect. Methods. This study linearizes the thermal radiation term as a function of the temperature in the boundary condition of the heat conductivity, and obtains the temperature field in a crater over a rotational period in the form of a Fourier series, accounting for the effects of self-sheltering, self-radiation, and self-scattering. By comparison with a numerical model, we find that this semi-analytical model for the CYORP effect works well for K > 0.1 Wm−1 K−1. This semi-analytical model is computationally three-orders-of-magnitude more efficient than the numerical approach. Results. We obtain the temperature field of a crater, accounting for the thermal inertia, crater shape, and crater location. We then find that the CYORP effect is negligible when the depth-to-diameter ratio is smaller than 0.05. In this case, it is reasonable to assume a convex shape for YORP calculations. Varying the thermal conductivity yields a consistent value of approximately 0.01 for the spin component of the CYORP coefficient, while the obliquity component is inversely related to thermal inertia, declining from 0.004 in basalt to 0.001 in metal. The CYORP spin component peaks at an obliquity of 0°, 90°, or 180°, while the obliquity component peaks at an obliquity of around 45° or 135°. For a z-axis symmetric shape, the CYORP spin component vanishes, while the obliquity component persists. Our model confirms that the total YORP torque is damped by a few tens of percent by uniformly distributed small-scale surface roughness. Furthermore, for the first time, we calculate the change in the YORP torque at each impact on the surface of an asteroid explicitly and compute the resulting stochastic spin evolution more precisely. Conclusions. This study shows that the CYORP effect due to small-scale surface roughness and impact craters is significant during the history of asteroids. The semi-analytical method that we developed, which benefits from fast computation, offers new perspectives for future investigations of the YORP modeling of real asteroids and for the complete rotational and orbital evolution of asteroids accounting for collisions. Future research employing our CYORP model may explore the implications of space-varying roughness distribution, roughness in binary systems, and the development of a comprehensive rotational evolution model for asteroid groups. [ABSTRACT FROM AUTHOR]

Several methods have been proposed to characterize the complex interactions in turbulent wakes, especially for flows with strong cyclic dynamics. This paper introduces the concept of Fourier-averaged Navier-Stokes (FANS) equations as a framework to obtain direct insights into the dynamics of complex coherent wake interactions. The method simplifies the interpretations of flow physics by identifying terms contributing to momentum transport at different time scales. The method also allows for direct interpretation of nonlinear interactions of the terms in the Navier-Stokes equations. By analysing well-known cases, the characteristics of FANS are evaluated. Particularly, we focus on physical interpretation of the terms as they relate to the interactions between modes at different time scales. Through comparison with established physics and other methods, FANS is shown to provide insight into the transfer of momentum between modes by extracting information about the contributing pressure, convective and diffusive forces. The FANS equations provide a simply calculated and more directly interpretable set of equations to analyse flow physics by leveraging momentum conservation principles and Fourier analysis. By representing the velocity as a Fourier series in time, for example, the triadic model interactions are apparent from the governing equations. The method is shown to be applicable to flows with complex cyclic waveforms, including broadband spectral energy distributions. [ABSTRACT FROM AUTHOR]

To address the drawbacks of the slow convergence speed and lack of individual information exchange in the cuckoo search (CS) algorithm, this study proposes an improved cuckoo search algorithm based on a sharing mechanism (ICSABOSM). The enhanced algorithm reinforces information sharing among individuals through the utilization of a sharing mechanism. Additionally, new search strategies are introduced in both the global and local searches of the CS. The results from numerical experiments on four standard test functions indicate that the improved algorithm outperforms the original CS in terms of search capability and performance. Building upon the improved algorithm, this paper introduces a numerical solution approach for differential equations involving the coupling of function approximation and intelligent algorithms. By constructing an approximate function using Fourier series to satisfy the conditions of the given differential equation and boundary conditions with minimal error, the proposed method minimizes errors while satisfying the differential equation and boundary conditions. The problem of solving the differential equation is then transformed into an optimization problem with the coefficients of the approximate function as variables. Furthermore, the improved cuckoo search algorithm is employed to solve this optimization problem. The specific steps of applying the improved algorithm to solve differential equations are illustrated through examples. The research outcomes broaden the application scope of the cuckoo optimization algorithm and provide a new perspective for solving differential equations. [ABSTRACT FROM AUTHOR]

It is well known that Luzin's conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin's conjecture has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved. [ABSTRACT FROM AUTHOR]

A new flow fluctuation abatement method based on noncircular gear variable speed drive is proposed and the mechanism principle of variable speed lobe pump is expounded. Considering the instantaneous flow rate of lobe pump and the closed constraint condition of noncircular gear, the design method of noncircular gear for flow pulsation suppression based on Fourier series is established. Based on the transient numerical simulation, the effects of design parameter of the noncircular gear, angular deviation, rotor-rotor gap and input velocity on fluid pulsation were investigated. The result shows that the outlet flow rate fluctuation of lobe pump decreases by 84.1 % after installing a Fourier noncircular gear, while the increase of angular deviation deteriorates the flow characteristics. With the increase of the number of terms in the Fourier series, the intensity of the fluid pulsation decreases gradually. Decreasing of the rotor-rotor gap from 0.4 mm down to 0.2 mm produces about 54.1 % reduction of flow rate fluctuation. As the input velocity is increased from 200 (r/min) to 600 (r/min), the flow pulsation is reduced by more than 60.0 %. A pump prototype was made to verify that the flow pulsation of the lobe pump can be optimized effectively by the external Fourier noncircular gears drive, which provides a beneficial support for the design of lobe pump with low flow pulsation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Background: Running means for total calcium (Ca) results at our laboratory exhibit a stable time-of-day (TOD) periodic pattern. We examined use of TOD-dependent targets for running means in patient-based quality control (PBQC) for Ca. Methods: Primary data were Ca results over a 3 month interval, restricted to weekday data within the Ca reference interval (8.5–10.3 mg/dL; 2.12–2.57 mmol/L). Running means were evaluated as sliding averages of 20 samples (20-mers). Results: Data comprised 39,629 consecutive Ca measurements (75.3% inpatient (IP)) for which Ca was 9.29±0.47 mg/dL. The all data average for 20-mers was 9.29 ± 0.18 mg/dL. When parsed in 1 h TOD intervals, however, averages among 20-mers ranged from 9.1 to 9.5 mg/dL, with blocs of contiguous results above (0800–2300 h; 53.3% of results; IP = 75.3%) and below (2300–0800 h; 46.7% of results; IP = 99.9%) the all-data mean. There was thus an inherent TOD-dependent pattern of deviation of means from target when using a fixed PBQC target. Using Fourier series analysis as an example approach, characterization of the pattern to produce TOD-dependent PBQC targets eliminated this inherent inaccuracy. Conclusions: In circumstances of periodic variation in running means, simple characterization of that variation can reduce the probability of both false positive and false negative flags in PBQC. [ABSTRACT FROM AUTHOR]

Applying r-repeated de la Vallée Poussin sums, we have proved four theorems which show the upper bound of the r-repeated de la Vall'ee Poussin kernel, their convergence at a point, the deviation between a continuous function and the r-repeated de la Vallée Poussin sums of partial sums of its Fourier series, and finally we determine the degree of approximation of functions belonging to ordinary Lipschitz class. [ABSTRACT FROM AUTHOR]

Atmospheric drag stands out as the predominant non-gravitational force acting on satellites in Low Earth Orbit (LEO), with altitudes below 2000 km. This drag exhibits a strong dependence on the thermospheric mass density (TMD), a parameter of vital significance in the realms of orbit determination, prediction, collision avoidance, and re-entry forecasting. A multitude of empirical TMD models have been developed, incorporating contemporary data sources, including TMD measurements obtained through onboard accelerometers on LEO satellites. This paper delves into three different TMD modelling techniques, specifically, Fourier series, spherical harmonics, and artificial neural networks (ANNs), during periods of geomagnetic quiescence. The TMD data utilised for modelling and evaluation are derived from three distinct LEO satellites: GOCE (at an altitude of approximately 250 km), CHAMP (around 400 km), and GRACE (around 500 km), spanning the years 2002 to 2013. The consistent utilisation of these TMD data sets allows for a clear performance assessment of the different modelling approaches. Subsequent research will shift its focus to TMD modelling during geomagnetic disturbances, while the present work can serve as a foundation for disentangling TMD variations stemming from geomagnetic activity. Furthermore, this study undertakes precise TMD modelling during geomagnetic quiescence using data obtained from the GRACE (at an altitude of approximately 500 km), CHAMP (around 400 km), and GOCE (roughly 250 km) satellites, covering the period from 2002 to 2013. It employs three distinct methods, namely Fourier analysis, spherical harmonics (SH) analysis, and the artificial neural network (ANN) technique, which are subsequently compared to identify the most suitable methodology for TMD modelling. Additionally, various combinations of time and coordinate representations are scrutinised within the context of TMD modelling. Our results show that the precision of low-order Fourier-based models can be enhanced by up to 10 % through the utilisation of geocentric solar magnetic coordinates. Both the Fourier- and SH-based models exhibit limitations in approximating the vertical gradient of TMD. Conversely, the ANN-based model possesses the capacity to capture vertical TMD variability without manifesting sensitivity to variations in time and coordinate inputs. [ABSTRACT FROM AUTHOR]

Considering the approximation of a function from a Sobolev space by the partial sums of Fourier series in a system of Sobolev orthogonal polynomials generated by classical Laguerre polynomials, we obtain an estimate for the convergence rate of the partial sums to . [ABSTRACT FROM AUTHOR]

In this paper, we consider an inhomogeneous hyperbolic type partial integro-differential equation with degenerate kernel, two redefinition functions and involution. Intermediate data are used to find these redefinition functions. Dirichlet boundary conditions with respect to spatial variable are used. The Fourier method of separation of variables is applied. The countable system of functional-integral equations is obtained. Theorem on a unique solvability of countable system of functional-integral equations is proved. The method of successive approximations is used in combination with the method of contraction mappings. The triple of solutions of the inverse problem is obtained in the form of Fourier series. Absolute convergence of Fourier series is proved. [ABSTRACT FROM AUTHOR]

Using the method of expanding unknown functions into Fourier series of Legendre polynomials, we have obtained a system of equations for the equilibrium state of a transversally isotropic plate that has a constant thickness under combined conditions on flat faces. The method for representing the general analytical solution of the system for symmetrical deformation of the plate with respect to the mid-plane has been described. A solution to the problem of the stress state of a plate with a circular hole to which a known splitting force is applied has been found. [ABSTRACT FROM AUTHOR]

In this paper, we develop a general mathematical framework for perfect and approximate hydrodynamic cloaking and shielding of electro-osmotic flow, which is governed by a coupled PDE system via the field-effect electro-osmosis. We first establish the representation formula of the solution of the coupled system using the layer potential techniques. Based on the Fourier series, the perfect hydrodynamic cloaking and shielding conditions are derived for the control region with the cross-sectional shape being an annulus or a confocal ellipses. Then we further propose an optimization scheme for the design of approximate cloaks and shields within general geometries. The well-posedness of the optimization problem is proved. In particular, the conditions that can ensure the occurrence of approximate cloaks and shields for general geometries are also established. Our theoretical findings are validated and supplemented by a variety of numerical results. The results in this paper also provide a mathematical foundation for more complex hydrodynamic cloaking and shielding. [ABSTRACT FROM AUTHOR]

In this paper, the influence of gravitational modulation on weakly nonlinear biothermal convection in a porous rotating layer is investigated. We consider a layer of porous medium saturated with Newtonian fluid, containing gyrotactic microorganisms, and subject to gravitational modulation, rotation, and internal heating. To analyze linear stability, it is sufficient to represent disturbances in the form of normal modes, while nonlinear analysis includes a truncated Fourier series containing a harmonic of the nonlinear interaction. A six-dimensional nonlinear Lorentz-type model is constructed, exhibiting both reflection symmetry and dissipation. We determined heat and mass transfer using a weakly nonlinear theory based on the representation of a truncated Fourier series. Additionally, the behavior of nonstationary Nusselt and Sherwood numbers was investigated by numerically solving finite amplitude equations. Applying the expansion of regular perturbations in a small parameter to a six-dimensional model of Lorentz equations with periodic coefficients, we obtained the Ginzburg-Landau (GL) equation. This equation describes the evolution of the finite amplitude of the onset of convection. The amplitude of convection in the unmodulated case is determined analytically and serves as a standard for comparison. The study examines the effect of various parameters on the system, including the Vadasz number, modified Rayleigh-Darcy number, Taylor number, cell eccentricity, and modulation parameters such as amplitude and frequency. By varying these parameters, in different cases, we analyzed heat and mass transfer, quantitatively expressed by the Nusselt and Sherwood numbers. It has been established that the modulation amplitude has a significant effect on the enhancement of heat and mass transfer, while the modulation frequency has a decreasing effect. [ABSTRACT FROM AUTHOR]

This paper is devoted to solving one of the extremal problems in the theory of approximation of functional classes by linear methods of summation of the Fourier series in the integral metric, namely, approximation of classes Lβ,1ψ by biharmonic Poisson integrals. As a result of the research, we have found the asymptotic equalities for the approximation values of classes of (ψ, β)-differentiable functions by biharmonic Poisson integrals, that is, have found solutions of the Kolmogorov-Nikol’skii problem for biharmonic Poisson integrals on classes Lβ,1ψ in the integral metric. [ABSTRACT FROM AUTHOR]

In this paper we summarize the most relevant results with respect to the convergence of Fourier series based on representative product systems. We focus on monomial systems defined on the complete product of quaternion groups. The results clearly indicate that these systems can be considered as a natural extension of the Vilenkin systems to the nonabelian harmonic analysis. [ABSTRACT FROM AUTHOR]

In this paper we present and prove some new results concerning the rate of approximation of T means of functions in Lp(G) and in lip (α, p), for 1 ≤ p < ∞ and α > 0. As a corollary, we obtain some new as well as known approximation inequalities. [ABSTRACT FROM AUTHOR]

This work aims to determine the effect of stresses caused by dislocation networks placed at the interface of a semiconductor heterostructure of the thin GaAs/Si system. In this case, we use a mathematical modeling by Fourier series expansion to numerically simulate the stresses for the two cases of isotropic and anisotropic elasticity in order to predict the mechanical behavior of the heterostructure in the presence of interfacial dislocations while respecting well-defined stress boundary conditions. The elastic stress relaxation is reached for a layer thickness threshold of the GaAs deposit on the Si substrate not exceeding 5 nm. [ABSTRACT FROM AUTHOR]

This paper aims to show the scale effects of vertical maneuvering derivatives for an underwater vehicle using computational fluid dynamics (CFD). The model scale and full-scale benchmark DARPA suboff hull forms were used in numerical vertical planar motion mechanism simulations to achieve this aim. The heave forces, pitch moments, and Euler coefficients are calculated together with the linear vertical maneuvering derivatives by implementing the Fourier series expansion approach. The results show that the coupled added mass terms are significantly influenced by the scale effects, whereas the impact of scale effects on the decoupled added mass and coupled and decoupled damping terms is negligible. Considering that the effects of the coupled added mass on the vertical maneuvering characteristics are almost negligible, the vertical maneuvering of a submarine is slightly influenced by the scale effects. [ABSTRACT FROM AUTHOR]

We obtain muti-dimensional generalizations of the N. Wiener and P. Lévy theorems. We also give two applications of the real analytic functional calculus for several variables in p-Banach algebras (0 < p ≤ 1). The first one consists of giving the real-analytic versions of P. Lévy theorem for several functions of a single variable. The second consists of giving a generalization of N. Wiener and P. Lévy theorems for functions of several variables. [ABSTRACT FROM AUTHOR]

In end milling, chatter vibration generated during processing becomes a problem in processing accuracy. Chatter vibration is classified into forced chatter and self-excited chatter depending on the cause of the vibration. Especially, stability of self-excited chatter is very important, because it becomes large amplitude vibration if it occurs. Generally, the chatter vibration phenomenon in end milling is modeled as a periodic system with time delay. However, all of the existing stability analysis methods of self-excited chatter are approximate and an accurate analysis method has not been established. In this study, a numerical method to calculate characteristic exponents of a periodic system with time delay equal to the fundamental period accurately using Fourier series is proposed. The proposed method is applied to the stability analysis of the self-excited chatter in end milling. The validity and accuracy of the proposed method is confirmed by comparing the results of the proposed method to well-known real phenomena and the numerical simulation results. Furthermore, the analysis method is applied to the case that the support of the tool is anisotropic, and influence of the anisotropic support for the self-excited vibration is studied.

We consider a mathematical model of two-dimensional electrically driven laminar free shear flows in a straight duct under action of an applied uniform homogeneous magnetic field. The mathematical approach is based on studies by J.C.R. Hunt and W.E. Williams [10], Yu. Kolesnikov and H. Kalis [22,23]. We solve the system of stationary partial differential equations (PDEs) with two unknown functions of velocity U and induced magnetic field H. The flows are generated as a result of the interaction of injected electric current in liquid and the applied field using one or two couples of linear electrodes located on duct walls: three cases are considered. In dependence on direction of current injection and uniform magnetic field, the flows between the end walls are realized. Distributions of velocities and induced magnetic fields, electric current density in dependence on the Hartmann number Ha are studied. The solution of this problem is obtained analytically and numerically, using the Fourier series method and Matlab.

Purpose. Correction of the mathematical model of electromagnetic processes in a two-pole induction motor with a short-circuited rotor, taking into account static rotor eccentricity to identify diagnostic correlations. Methodology. Analytical modeling using the method of specific magnetic conductivity, mathematical modeling of electromagnetic fields in a three-phase induction motor with a short-circuited rotor using methods of electromagnetic field theory and finite element methods. Obtained results. The necessity of improving mathematical models for induction motors with short-circuited rotors to establish new or refine connections between diagnostic features and diagnosed defects has been demonstrated. Refined mathematical expressions for calculating the specific conductivity of non-uniform air gaps in induction motors with static eccentricity are provided. Modeling was performed using the FEMM environment for a statically eccentric two-pole induction motor with a short-circuited rotor. It has been proven that the harmonic order values obtained using the numerical-field method are consistent with those obtained analytically. Findings. Based on the field approach and using the finite elements method, an analysis of the distribution of magnetic field in a two-pole induction motor with a short-circuited rotor was conducted. Harmonic analysis of the magnetic field in the air gap was performed to identify the fundamental harmonic and higher and lower-order harmonics when eccentricity occurs. The influence of static rotor eccentricity on the electromagnetic processes of the induction motor was analyzed. Practical value. The results of the study can be utilized for functional diagnosis of the rotor winding of induction motors based on the radial component of the magnetic field. This will contribute to enhancing the reliability of induction motors and enable the prevention of failure in induction motors with short-circuited rotors.

Abstract In this paper, fast algorithms for the extrapolation of band-limited signals are presented by the sampling theorem and Fourier series in the case of over sampling. Assume the band-limited signal is known in a finite interval. We update the signal outside the interval by the Shannon sampling theorem in the case of over sampling. Then we obtain a fast algorithm in the form of Fourier series instead of the Fourier transform in the Papoulis–Gerchberg algorithm. Gibbs phenomena is analyzed in the method. An algorithm is presented to control the Gibbs phenomena, and some examples are given in the experimental results.

The problem of high-frequency diffraction of a dipole field by a perfectly conducting strongly elongated spheroid is considered in parabolic equation approximation. The leading order term is represented in the form of Fourier series with each harmonics expressed by an integral involving the Whittaker functions. The amplitudes under the sign of integration are obtained as solutions of integral equations and are expressed explicitly in terms of the Whittaker functions. [ABSTRACT FROM AUTHOR]

This paper proposes an analytical design procedure for 2D FIR circular filter banks and also a novel, computationally efficient implementation of the designed filter bank based on a polyphase structure and a block filtering approach. The component filters of the bank are designed in the frequency domain using a specific frequency transformation applied to a low-pass, band-pass and high-pass 1D prototype with a specified Gaussian shape and imposed specifications (peak frequency, bandwidth). The 1D prototype filter frequency response is derived in a closed form as a trigonometric polynomial with a specified order using Fourier series, and then it is factored. Since the design starts from a 1D prototype with a factored transfer function, the frequency response of the designed 2D filter bank components also results directly in a factored form. The designed filters have an accurate shape, with negligible distortions at a relatively low order. We present the design of two types of circular filter banks: uniform and non-uniform (dyadic). An example of image analysis with the uniform filter bank is also provided, showing that the original image can be accurately reconstructed from the sub-band images. The proposed implementation is presented for a simpler case, namely for a smaller size of the filter kernel and of the input image. Using the polyphase and block filtering approach, a convenient implementation at the system level is obtained for the designed 2D FIR filter, with a relatively low computational complexity. [ABSTRACT FROM AUTHOR]