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This paper establishes a robust stability criterion in the radially symmetric heat equation that admits heat sources belonging to a set of bounded functions. The robust stability criterion is determined by extending a definition of stability under constant-acting perturbations that was originally established for systems of ordinary differential equations. It is assumed that heat sources admit a Fourier series representation whose coefficients are bounded and piecewise continuous functions. The robust stability criterion obtained is useful to conclude that the solution of the heat equation, as well as its first partial derivatives with respect to the radial axis and with respect to time, are bounded by a constant whose value is initially established. The results obtained are illustrated numerically. [ABSTRACT FROM AUTHOR]

We revisit Fourier’s approach to solve the heat equation on the circle in the context of (twisted) reduced group C*-algebras, convergence of Fourier series and semigroups associated to negative definite functions. We introduce some heat properties for countably infinite groups and investigate when they are satisfied. Kazhdan’s property (T) is an obstruction to the weakest property, and our findings leave open the possibility that this might be the only one. On the other hand, many groups with the Haagerup property satisfy the strongest version. We show that this heat property implies that the associated heat problem has a unique solution regardless of the choice of the initial datum. [ABSTRACT FROM AUTHOR]

Considering the approximation of , with , by the partial sums of the Fourier series in a system of polynomials orthogonal in the sense of Sobolev and generated by a system of classical Laguerre polynomials, we obtain some estimates for the convergence rate of to . [ABSTRACT FROM AUTHOR]

We study the triangular θ -mean of the partial sums of f ∈ L p (T q 2) and prove the following noncommutative weak and strong type maximal inequalities: ‖ (σ n Δ , θ (f)) n ≥ 1 ‖ Λ 1 , ∞ (T q 2 , ℓ ∞) ≤ c θ ‖ f ‖ L 1 (T q 2) , p = 1 and σ n Δ , θ (f) n ≥ 1 L p (T q 2 , ℓ ∞) ≤ c p , θ ‖ f ‖ L p (T q 2) , 1 < p < ∞ , where T q 2 is a 2-dimensional quantum torus. As a consequence, we obtain the bilateral almost uniform convergence of σ n Δ , θ (f) provided f ∈ L p (T q 2). [ABSTRACT FROM AUTHOR]

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of some two-dimensional rectangle (decreasing diagonal) Walsh-Fourier series in L p (G 2) space ( 1 ≤ p < ∞ ) and in C (G 2) . It implies in some special case norm convergence. We also show an application of our results for Lipschitz functions. At the end of the paper we show the most important result, the almost everywhere convergence theorem. We note that T summation is a common generalization of the following known summation methods Cesàro, Weierstrass, Riesz and Picar and Bessel methods. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of recovering the source function with pointwise overdetermination conditions. The source is represented as a finite segment of a series with Fourier coefficients depending on time. We prove that the solution is uniquely determined and continuously depends on the data. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

We consider nonharmonic Fourier series defined in terms of arbitrarily close exponentials. Our aim is to use the positivity of quadratic polynomials in order to get new Ingham-type weighted inequalities. The proof relies on an Ingham proof technique inspired by Jaffard et al. (J Fourier Anal Appl 3:577–582, 1997). As applications, we consider families of frequencies with relevance in control approximation theory, for which we can prove the uniform (with respect to the mesh-size) controllability property of the semi-discrete model, when the spurious frequencies (the gap between them tends to zero when the mesh size goes to zero) are eliminated. [ABSTRACT FROM AUTHOR]

Determining the stress state in a circular ring has been a classical topic in the stress analysis literature. Based on the principle of superposition, the results may be obtained by adding known solutions to an associated ring problem, where the boundary stresses on the inner and outer walls of the ring are represented in Fourier series. In this work, the coefficients of the Fourier series are generated through an algorithm based on the fast Fourier transform (FFT). In the case of concentrated loading, the required additional fundamental solutions are derived in closed-form. The presented numerical method allows for accurate and efficient computations of the stress distributions in a circular ring in static equilibrium under arbitrary in-plane loading; and generally, the FFT-based algorithm provides a convenient and versatile tool for handing some two-dimensional problems involving circular boundaries. [ABSTRACT FROM AUTHOR]

This article, titled "Functions Almost Universal in the Sense of Signs with Respect to the Trigonometric System and the Walsh System," discusses the existence and structure of functions whose Fourier series are almost universal in the sense of signs with respect to the trigonometric system and the Walsh system. The paper explores various definitions and theorems related to universal functions and poses unanswered questions about the existence of certain types of universal functions. The authors provide examples and results from previous studies in the field of complex and real analysis. The article concludes with acknowledgments and funding information. [Extracted from the article]

According to the Wik theorem, there exist massive Helson sets on the circle. In particular, they can be of Hausdorff dimension one. We extend Wik's result to the multidimensional case. [ABSTRACT FROM AUTHOR]

The mathematical formulations for transverse compression of a thin elastic disc are considered, including various boundary conditions along the faces of the disc. The mixed boundary conditions corresponding to the loading by normal stresses in absence of sliding are studied in detail. These conditions support an explicit solution in a Fourier series for the boundary layers localised near the edge of the disc and also do not assume making use of the Saint-Venant principle underlying the traditional asymptotic theory for thin elastic structures. As an example, an axisymmetric problem is studied. Along with the leading order solution for a plane boundary layer, a two-term outer expansion is derived. The latter is expressed through the derivatives of the prescribed stresses. Generalisations of the developed approach are addressed. [ABSTRACT FROM AUTHOR]

This paper values fixed-income (discrete- and continuous-time) European Asian and Australian options. We assume that the term structure of interest rates is modelled by the specification proposed in Moreno et al. (Econ Model 72:140–150, 2018, https://doi.org/10.1016/j.econmod.2018.01.015). We obtain closed-form expressions for the premiums of geometric average options and, for arithmetic average options, premiums are computed by numerical methods. We also perform a sensitivity analysis with respect to different parameters for both (geometric and arithmetic) options. [ABSTRACT FROM AUTHOR]

We consider the Lipschitz class functions on [0, 1] and special series of their Fourier coefficients with respect to general orthonormal systems (ONS). The convergence of classical Fourier series (trigonometric, Haar, Walsh systems) of Lip 1 class functions is a trivial problem and is well known. But general Fourier series, as it is known, even for the function f (x) = 1 does not converge. On the other hand, we show that such series do not converge with respect to general ONSs. In the paper we find the special conditions on the functions φ n of the system (φ n) such that the above-mentioned series are convergent for any Lipschitz class function. The obtained result is the best possible. [ABSTRACT FROM AUTHOR]

The degree of approximation (in Hölder metric) of a periodic function belonging to a specific Hölder class by its Abel–Poisson means of Fourier series and of the conjugate function by the conjugate Abel–Poisson means of the conjugate series of the Fourier series is obtained. [ABSTRACT FROM AUTHOR]

We justify an analytical method for constructing periodic solutions of nonlinear systems of ordinary differential equations of polynomial type. Periodic solutions are constructed in the form of Fourier series in which the coefficients are polynomials depending on a parameter, which is not assumed to be small. Two examples are considered: the van der Pol equation and the Lorenz system. [ABSTRACT FROM AUTHOR]

This paper presents a novel approach for assessing the uncertainty in vibration and static responses of laminated composite beams resulting from uncertainty in material properties and distributed loads. The proposed method utilizes surrogate models developed using polynomial chaos expansion (PCE) based on a relatively small sample size. These training samples are computed using a high-order shear beam model in which the governing equations are derived using Hamilton's principle, and solved by Ritz's approach using a trigonometric series approximation. The proposed PCE model's coefficients are estimated using the spectral projection and linear regression techniques. The first four statistical moments and probability distributions of the mid-span displacement and the fundamental frequency of laminated composite beams are predicted. Global sensitivity analysis is also conducted to assess how material property variation and stochastic loads affect the beam's deflection and the fundamental frequency. The accuracy and efficiency of the proposed PCE models are compared with those from Monte Carlo simulation (MCS). A remarkable reduction in the computational cost of PCE models compared to MCS is observed without compromising the predictions' accuracy. As most real-world systems are subjected to multiple sources of uncertainty, this study provides a state-of-the-art method to quantify such uncertain parameters more efficiently and allow for a better reliability assessment in composite beam design. [ABSTRACT FROM AUTHOR]

In this paper, the problem of -convergence of Fourier series with quasi-monotone coefficients is handled by using the -mean. Also, an example is given about the Fourier series of a signal (function) and its mean. [ABSTRACT FROM AUTHOR]

We study the convergence of Fourier series in the polynomial system orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of in this system converges to pointwise on the grid as . In addition, we study the approximation properties of partial sums of Fourier series in the system . [ABSTRACT FROM AUTHOR]

A generalized solution of a mixed problem for the wave equation is constructed under minimal conditions on the right side of the equation. The solution is represented as a series from the Fourier method, and its sum is found. The form of a generalized solution of a mixed problem for an inhomogeneous telegraph equation is given. [ABSTRACT FROM AUTHOR]

In this study, the space-time fractional generalised reaction duffing model is investigated analytically, which is a generalization for a collection of prominent fractional models describing various key phenomenon in science and engineering. The governing equation is converted to a nonlinear ODE by the compatible travelling wave transform. The investigation established that for analysing nonlinear evolution equations of fractional order, the recommended approach is more effective and realistic. The findings are given extensively in rational forms of trigonometric function series or clearly in powers of specific trigonometric functions. A collection of bright, dark, periodic, and optical solitons is discovered. Mathematica is used to flourish the presence of some obtained solutions in 3D graphs with different fractional orders. The results show that the recommended methods are more practical and effective ways to illustrate the dynamics of several complex wave structures in modern science and technology. [ABSTRACT FROM AUTHOR]

In this paper, a novel back-propagation error technique is presented. This neural network structure allows for two fundamental basic modes: (1) To decompose the neurones by transforming their variables, weights, and scalar functions into vectors. This conveys for the decomposition of the transfer function of every neurone (where the output variables are the components of the decomposition) and, consequently, to be written as the invariant sum of orthogonal functions, with the safeguard of preserving information This orthogonality is proven using Fourier theory. (2) In a second mode, a tuned neural network that occupies one of the channels of the neural network can see the weights of its supplementary channels adjusted to retain additional information. Only the decomposition algorithm of the network is presented here—Multi-back-propagation algorithm. The adopted methodology is validated step-by-step with some representative examples. Namely, to assess the performance of the splitting method, two different examples have been constructed from scratch: (1) a 2D classification problem and (2) a 3D surface. In both problems, the signal and transfer functions of the neural network are successfully decomposed without information losses. Therefore, since the main contribution of this work is to allow for the organisation of the information stored in neural network structure, through a split process, this promising method shows potential use in various areas—e.g. classification and/or pattern recognition problems, data analysis, modelling and so on. In the future, we expect to work further in the method computational aspects to render it more efficient, versatile and robust. [ABSTRACT FROM AUTHOR]

The paper investigates sufficient conditions for the absolute convergence of trigonometric Fourier series of almost-periodic functions in the sense of Besikovitch in the case when the Fourier exponents have a single limiting point at infinity. A higher-order modulus of continuity is used as a structural characteristic of the function under consideration. [ABSTRACT FROM AUTHOR]

Objective: The purpose of this work is to investigate the size-dependent free vibrations of carbon nanotube-reinforced nanowires/nanobeams with movable ends. For this purpose, two movable end conditions are considered for carbon nanotube-reinforced nanowires/nanobeams with different carbon nanotube distributions. The size effect is addressed with the modified couple stress theory, which includes a material length scale parameter. Methods: In this study, a solution approach based on the Fourier sine series and Stokes' transformation is used. With the help of this approach, both rigid and movable end conditions can be investigated. Firstly, equation sets consisting of infinite series and movable end parameters are derived. Then, eigenvalue problems are obtained for the free vibration of carbon nanotube-reinforced nanowires/nanobeams. The eigenvalues of these problems give the frequencies of the carbon nanotube-reinforced nanowires/nanobeams with movable ends. Conclusions: The eigenvalue problems obtained in this study can be used to examine both rigid and movable end conditions. The accuracy of the problems obtained for solution is proven by various comparison studies. Then, a series of analyses are carried out for nanowire/nanobeam reinforced with carbon nanotubes, including both the size effect and the rotary inertia effect. When the frequencies of nanowire/nanobeams reinforced with carbon nanotubes are examined, it is understood that the material length scale parameter, carbon nanotube distribution, carbon nanotube volume fraction, rotational and lateral spring parameters can cause significant changes in free vibration. [ABSTRACT FROM AUTHOR]

This article studies a inverse problem for diffusion equation on the rectangular domain. First we introduce a definition of a classical solution, and then the original problem investigate to use successive approximation. Second, the given problem is reduced to an equivalent problem. Existence and uniqueness of the solution of the equivalent problem is proved using a contraction mapping principle. Finally, using the equivalency, the existence and uniqueness of classical solution is obtained. [ABSTRACT FROM AUTHOR]

This work presents novel three-dimensional solutions for the multiphysics response of magneto-electro-elastic multilayered plates with interfacial imperfections in a thermal environment. The Stroh formalism is employed, incorporating thermal coupling with the Eringen nonlocal theory to capture small-scale effects. The laminated structures are simply supported and subjected to time-harmonic temperature distributions and extended tractions represented using Fourier series expansions. The dual variable and position technique is used to address the challenges posed by non-ideal thermal and mechanical bonded contacts between constituents, ensuring the consistency and stability of the recursive field relations. A wide range of application examples are analyzed, including the influence of material arrangements, aspect ratio and nonlocal length characteristics, elastically compliant and thermally/dielectrically weakly conducting interfaces, as well as forced vibrations in combined thermo-mechanical environments. The comprehensive results shed light on the intricate multiphysics response of multilayered structures and provide valuable insights into practical engineering implications for advanced materials and structures. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of almost periodic response solutions for the superlinear Duffing's equation with an almost periodic external force. Assume that the system is reversible, and if the almost periodic forcing term admits a rapidly converging Fourier series, moreover the Diophantine condition for the frequencies is satisfied, the existence of response solutions will be proved. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for reversible systems. [ABSTRACT FROM AUTHOR]

We propose a photonics scheme to generate rectified cosinusoidal shaped (RCS) and rectified sinusoidal shaped (RSS) microwave waveforms and the duty cycle of the generated waveforms can be freely tunable. In this proposed method, two cascaded Mach-Zehnder modulators (MZMs) with polarization sensitive characteristics are employed. By setting the rotation angle of polarization controller, phase shift of the phase shifters and modulation index of the front MZM's (modulation index of the rear MZM is unfixed) properly, the obtained RCS and RSS microwave waveforms with arbitrary duty cycle by the superposition of beating signals, can be seen as an approximation of the first three terms of the Fourier series expansion of the ideal waveforms. The detailed theoretical analysis and simulations are given. In the simulations, the duty cycle of 100%, 80% and 50% of the RCS and RSS microwave waveforms are successfully obtained. Also, fitting errors are introduced to measure the similarity between the generated waveforms and the ideal waveforms. Different from previous works, the RCS and RSS microwave waveforms can be both generated by our scheme and the duty cycle is freely tunable. Moreover, the modulation index of the rear MZM is unfixed, thus increasing the flexibility of the scheme. [ABSTRACT FROM AUTHOR]

Polar orthotropic materials as wood and filament-wound composites are ubiquitous in nature and technological applications, respectively. With annular domains rendering themselves naturally for polar orthotropy, applications featuring annular domains have received considerable attention; however, arbitrarily-shaped domains are less explored. In an attempt to address this gap, the present work explores elastostatics of arbitrarily-shaped layered 2D domains. After identifying the refinement required in an earlier work to yield a physically and mathematically consistent solution, the work considers analytically homogenous and layered polar orthotropic annular domains and corroborates the results with FEA. Subsequently, the arbitrarily-shaped layered domains—including octagonal-shaped, cruciform-shaped and annular domains—are analyzed by employing a coupled FE-analytical technique combining coarse-mesh boundary displacement-data from FEA and the refined solution. Results for the illustrative cases under symmetric and anti-symmetric loadings are depicted as contours over the domains for the resultant displacements and von Mises stresses. A comparison with converged FE results and L 2 -error norms indicate a good correspondence between the two—demonstrating the efficacy of the coupled technique. The potential extension of the technique to hybrid-experimental strategies, asymmetric cases, composite cylindrical FGMs and a complex-variable based formulation for the coupled technique is briefly discussed. [ABSTRACT FROM AUTHOR]

The article studies the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces. This theorem generalizes some known results of Hardy and Littlewood for number series. We find sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle. These conditions are expressed in terms of the Fourier coefficients and are weaker than Hardy's condition. We give a description of all Banach function spaces given over the circle and endowed with a norm been equivalent to a norm in a homogeneous Banach space. We study interpolation properties of such spaces and give new examples of them. We extend the classical Fejér theorem on the uniform Cesàro summability of the Fourier series on sets by means of a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces. We also generalize the classical Hardy theorem on the uniform convergence of the Fourier series on sets. [ABSTRACT FROM AUTHOR]

A moving object is often excited by time-dependent factors. If the equation of motion and time-dependent factor can be distilled from data, a foundation for further analysis can be established. This paper proposes a method to discover the equation of motion and time-dependent factors without prior information and customized preparation. We first construct statistics to determine whether the system is subject to time-dependent factors. For the motivated system, we introduce the time-dependent Fourier series to approximate the excitation. Finally, we achieve the identification by simple steps. The new method does not require the prior knowledge of the system. Since the new method determines whether the time-dependent factor is present and which state variable is motivated, it avoids the confusion caused by inappropriately adding time-dependent terms to the identification. As the designed Fourier series can approximate the unknown model accurately, the new method does not require developing customized identification for each problem. We apply the new method to discover dynamical systems from data collected from Liénard equation and SD oscillator driven by various time-dependent factors. The new method can accurately discover dynamical systems from the collected short-time data and predict many dynamical behaviors, including the long-time evolution of the system, the multi-stable dynamics, and qualitative changes in the dynamics as the time-dependent factor varies. The results show that the new identification method can discover nonautonomous dynamical systems effectively. [ABSTRACT FROM AUTHOR]

Objective: The aim of this study is to solve the free vibrations of embedded functionally graded porous and non-porous nanobeams with different material distributions (power-law and sigmoid) under general elastic boundary conditions to better understand the dynamic properties. This form of model has the benefit of allowing one to handle the rigid or restrained supporting conditions. Method: A solution method using the Fourier sine series and Stokes' transform together is adopted. This method is used to study the effects of deformable boundary conditions as well as rigid boundary conditions, which are common in the literature. In the current study, two sets of equations for both elastic support conditions consisting of infinite series are derived. Then, eigenvalue problems are set up for the analytical solution. The eigenvalues of the established problems give the vibration frequencies of the embedded functionally graded porous/non-porous nanobeams. Conclusions: The proposed models are effective for studying arbitrary boundary conditions. The accuracy of the model is compared with some results from the literature for rigid boundary conditions. Looking at the frequencies of functionally graded porous/non-porous nanobeams, it is seen that the studied parameters such as foundation parameters, nonlocal parameter, grading index, elastic spring stiffness produce changes that cannot be ignored. [ABSTRACT FROM AUTHOR]

We study the ℓ 1 -summability of functions in the d-dimensional torus T d and so-called ℓ 1 -invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the ℓ 1 -norm of their indices. Such functions are characterized as divided differences that have cos θ 1 , … , cos θ d as knots for (θ 1 … , θ d) ∈ T d . It leads us to consider the d-dimensional Fourier series of univariate B-splines with respect to its knots, which turns out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal series of the B-spline function. [ABSTRACT FROM AUTHOR]

We obtain an estimate for the convergence rate of the Fourier series of a continuous periodic function in terms of the modulus of continuity of the function and the value of its -variation. We prove that the leading term of the estimate is sharp. [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]

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]

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]

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]

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]

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]

Boundary-value problems are considered for harmonic and biharmonic equations, as well as the general polyharmonic equation for multiply connected domains on a plane. The problems are reduced to solving linear integral equations at boundary contours, which are assumed to be planar. An algorithm for deriving an approximation of integral equations by a linear system is presented, taking into account the logarithmic singularities of the kernels of integral operators, through which the integral equations are expressed. The algorithm uses the periodicity of functions defined for closed boundary contours. As the number of grid points increases, the approximation error decreases faster than the grid spacing to any fixed power. Applications to solving problems of hydrodynamics, filtration, and other problems of theoretical physics are considered. [ABSTRACT FROM AUTHOR]

This work is executed to predict the variation in global temperature and greenhouse gas (GHG) emissions resulting from climate change and global warming, taking into consideration the natural climate cycle. A mathematical model was developed using a Recurrent Neural Network (RNN) with Long–Short-Term Memory (LSTM) model. Data sets of global temperature were collected from 800,000 BC to 1950 AD from the National Oceanic and Atmospheric Administration (NOAA). Furthermore, another data set was obtained from The National Aeronautics and Space Administration (NASA) climate website. This contained records from 1880 to 2019 of global temperature and carbon dioxide levels. Curve fitting techniques, employing Sin, Exponential, and Fourier Series functions, were utilized to reconstruct both NOAA and NASA data sets, unifying them on a consistent time scale and expanding data size by representing the same information over smaller periods. The fitting quality, assessed using the R-squared measure, ensured a thorough process enhancing the model's accuracy and providing a more precise representation of historical climate data. Subsequently, the time-series data were converted into a supervised format for effective use with the LSTM model for prediction purposes. Augmented by the Mean Squared Error (MSE) as the analyzed loss function, normalization techniques, and refined data representation from curve fitting the LSTM model revealed a sharp increase in global temperature, reaching a temperature rise of 4.8 °C by 2100. Moreover, carbon dioxide concentrations will continue to boom, attaining a value of 713 ppm in 2100. In addition, the findings indicated that the RNN algorithm (LSTM model) provided higher accuracy and reliable forecasting results as the prediction outputs were closer to the international climate models and were found to be in good agreement. This study contributes valuable insights into the trajectory of global temperature and GHG emissions, emphasizing the potential of LSTM models in climate prediction. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new interference-suppression beamformer design and a Direction-of-Arrival (DoA) estimation method for Line-of-Sight (LoS) multi-user communication systems with circular Large Intelligent Surfaces (LISs). This symmetrical circular-shaped LIS allows signals defined on it to be formed by combining Fourier and Bessel series. The Fourier harmonics are orthogonal along the rotational direction and the Bessel functions constituting the Bessel series are orthogonal along the radial direction. Synthesizing basis functions from these two series results in the proposed interference-suppression beamformer, with their coefficients designed to cancel inter-user interference. Also by exploiting signal structure of a circular LIS communicating with Mobile Stations (MSs), we devise a DoA estimation method to obtain directional information of incoming signals. Under LoS channel, the DoA corresponds to elevation and Azimuth angles of MSs relative to the LIS. This method is numerically demonstrated to have promising performance with high-frequency carriers. [ABSTRACT FROM AUTHOR]

The subject of the research is the control algorithms for a seven-phase converter that implement space-vector voltage modulation of a seven-phase motor as an alternative to a three-phase engine in modern electric traction. The study used elements of set theory, combinatorics, Fourier series expansion and vector analysis. Checking research results was implemented on a special stand for experimental studies of spatial vector voltage modulation of a seven-phase motor. [ABSTRACT FROM AUTHOR]