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Kernel methods such as kernel ridge regression and Gaussian process regression with Matern-type kernels have been increasingly used, in particular, to fit potential energy surfaces (PES) and density functionals, and for materials informatics. When the dimensionality of the feature space is high, these methods are used with necessarily sparse data. In this regime, the optimal length parameter of a Matern-type kernel may become so large that the method effectively degenerates into a low-order polynomial regression and, therefore, loses any advantage over such regression. This is demonstrated theoretically as well as numerically in the examples of six- and fifteen-dimensional molecular PES using squared exponential and simple exponential kernels. The results shed additional light on the success of polynomial approximations such as PIP for medium-size molecules and on the importance of orders-of-coupling-based models for preserving the advantages of kernel methods with Matern-type kernels of on the use of physically motivated (reproducing) kernels. [ABSTRACT FROM AUTHOR]

In this work, the dynamic responses of an uncertain flexible shaft-disk-drum rotor under elastic support and connection are investigated, where uncertain parameters are considered as unknown but bounded interval variables. The energy method is used to derive deterministic analytical model for the flexible shaft-disk-drum rotor, which is validated by modal tests. A non-intrusive interval method based on Chebyshev polynomial approximation is introduced to evaluate the uncertain dynamic response of the flexible shaft-disk-drum rotor system. Comparative studies of Monte Carlo simulation, scanning method and the interval method are conducted to illustrate the validity and accuracy of the approach proposed here. [ABSTRACT FROM AUTHOR]

This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let r, n be positive integers with n ≥ 6r. Let Pn ∩ Mr be the space of polynomials of degree at most n on [0, 1] with integer coefficients such that P(k)(0)/k! and P(k)(1)/k! are integers for k = 0,. . ., r - 1 and let PZn ∩ Mr be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of PZn ∩ Mr from Pn ∩ Mr in L2(0, 1). We give rather precise quantitative estimations for successive minima of PZn in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in Pn ∩ Mr. [ABSTRACT FROM AUTHOR]

In approximation theory, the "direct theorem" is a fundamental concept that relates to how well more straightforward functions or signals can approximate a function or signal from a chosen approximation space. The direct theorem typically provides bounds on the error or the quality of the approximation in terms of various parameters, such as the degree of approximation, the dimension of the approximation space, and the properties of the approximated functions. The specific form of the direct theorem can vary depending on the context and the type of approximation being considered (e.g., polynomial approximation, Fourier series approximation, spline approximation, etc.). The direct theorem aims to establish the mathematical framework for comprehending how well elements from a given space can approximate a function, which is crucial in various fields like numerical analysis, signal processing, and scientific computing. Finally, the direct theorem is a crucial result in approximation theory. It describes how elements from a given approximation space get closer and better at approximating a given function as the number of terms in the approximation sequence grows. It provides essential insights into the behavior of approximations and plays a central role in various mathematical and computational disciplines. In this article, we aim to prove the direct theorem by defining a new appropriate operator for functions Lp ([[0, 2π]d) that uses our new k-mixed modulus of smoothness to estimate the best approximation degree and the new k-mixed difference. Also, to prove the direct theorem, we introduce the angular approximation in Lp ([0, 2π]d to reach the most accurate estimate. [ABSTRACT FROM AUTHOR]

Summary To address the grid stability problem caused by random changes in cable impedance when an offshore wind power generation system is sent out via a power cable, this study proposes a stochastic stability analysis method for an offshore doubly‐fed wind power generation system considering the frequency variation of the power cable parameters based on the polynomial approximation theory, which can be analyzed by equating the problem of the random change in the impedance of the offshore wind power public coupling point as a deterministic problem. First, a stochastic dynamics model of doubly‐fed wind farm is established by considering variation impedance variation. Second, using the polynomial approximation theory, the wind farm model containing stochastic parameters is approximated for analysis as an equivalent deterministic model. Finally, based on the approximated deterministic model, the grid‐connected stability of wind farms with different impedance fluctuation strengths is investigated, and the stable operation interval of the wind farms affected by the impedance is determined. The simulation results show that when the stochastic intensity exceeds a certain threshold, Hopf bifurcation occurs during the voltage and turndown, and the system is destabilized. This study provides a theoretical basis for the stability of offshore wind power. [ABSTRACT FROM AUTHOR]

The first part of this work provides explicit solutions for two integral equations; both are solved by means of Fourier transform. In the second part of this paper, sufficient conditions for the existence and uniqueness of the solutions satisfying sandwich constraints for two types of full moment problems are provided. The only given data are the moments of all positive integer orders of the solution and two other linear, not necessarily positive, constraints on it. Under natural assumptions, all the linear solutions are continuous. With their value in the subspace of polynomials being given by the moment conditions, the uniqueness follows. When the involved linear solutions and constraints are positive, the sufficient conditions mentioned above are also necessary. This is achieved in the third part of the paper. All these conditions are written in terms of quadratic expressions. [ABSTRACT FROM AUTHOR]

We present numerical algorithms for efficient and highly accurate computations of Fresnel's sine and cosine integrals S(z) and C(z) for real (z = x) and complex (z = x + iy) arguments. The algorithms are based on series expansion for small values of |z|, expansion in Chebyshev subinterval polynomial approximations for intermediate real values (y = 0), toggled with sum of half-integer order Bessel functions for complex arguments, together with asymptotic series expansion for large values of |z|. The present algorithms, implemented in a Fortran elemental module, can be run using any of the single, double, or quadruple precision arithmetic. Results from the present code have been benchmarked versus comprehensive data tables, generated using Maple, Mathematica, and Matlab symbolic toolbox. Compared to a maximum of 16 significant digits in the literature, the present algorithms can calculate S(x) and C(x) up to 28 significant digits in the range xϵ[0, 106] and S(z) and C(z) to the same accuracy in the domain where |S(z)| and |C(z)| each is less than the largest finite floating point number in the precision under consideration with ∣ arg z ∣ < π 2 or more specifically 3 × 10 - 4 < y x < 2 × 10 3 for S(z) and 4 × 10 - 4 < y x < 2.2 × 10 3 for C(z). [ABSTRACT FROM AUTHOR]

Accurate segmentation of thalamic nuclei, crucial for understanding their role in healthy cognition and in pathologies, is challenging to achieve on standard T1-weighted (T1w) magnetic resonance imaging (MRI) due to poor image contrast. White-matter-nulled (WMn) MRI sequences improve intrathalamic contrast but are not part of clinical protocols or extant databases. In this study, we introduce histogram-based polynomial synthesis (HIPS), a fast preprocessing transform step that synthesizes WMn-like image contrast from standard T1w MRI using a polynomial approximation for intensity transformation. HIPS was incorporated into THalamus Optimized Multi-Atlas Segmentation (THOMAS) pipeline, a method developed and optimized for WMn MRI. HIPS-THOMAS was compared to a convolutional neural network (CNN)-based segmentation method and THOMAS modified for the use of T1w images (T1w-THOMAS). The robustness and accuracy of the three methods were tested across different image contrasts (MPRAGE, SPGR, and MP2RAGE), scanner manufacturers (PHILIPS, GE, and Siemens), and field strengths (3 T and 7 T). HIPS-transformed images improved intra-thalamic contrast and thalamic boundaries, and HIPS-THOMAS yielded significantly higher mean Dice coefficients and reduced volume errors compared to both the CNN method and T1w-THOMAS. Finally, all three methods were compared using the frequently travelling human phantom MRI dataset for inter- and intra-scanner variability, with HIPS displaying the least inter-scanner variability and performing comparably with T1w-THOMAS for intra-scanner variability. In conclusion, our findings highlight the efficacy and robustness of HIPS in enhancing thalamic nuclei segmentation from standard T1w MRI. [ABSTRACT FROM AUTHOR]

The aim of this study was to design physics-preserving and precise surrogate models of the nonlinear elastic behaviour of an intervertebral disc (IVD). Based on artificial force-displacement data sets from detailed finite element (FE) disc models, we used greedy kernel and polynomial approximations of second, third and fourth order to train surrogate models for the scalar force-torque -potential. Doing so, the resulting models of the elastic IVD responses ensured the conservation of mechanical energy through their structure. At the same time, they were capable of predicting disc forces in a physiological range of motion and for the coupling of all six degrees of freedom of an intervertebral joint. The performance of all surrogate models for a subject-specific L4 | 5 disc geometry was evaluated both on training and test data obtained from uncoupled (one-dimensional), weakly coupled (two-dimensional), and random movement trajectories in the entire six-dimensional (6d) physiological displacement range, as well as on synthetic kinematic data. We observed highest precisions for the kernel surrogate followed by the fourth-order polynomial model. Both clearly outperformed the second-order polynomial model which is equivalent to the commonly used stiffness matrix in neuro-musculoskeletal simulations. Hence, the proposed model architectures have the potential to improve the accuracy and, therewith, validity of load predictions in neuro-musculoskeletal spine models. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical solution of Cauchy Singular Integral Equations based on some non standard polynomial quasi–projection of de la Vallée Poussin type. Such kind of approximation presents several advantages over classical Lagrange interpolation such as the uniform boundedness of the Lebesgue constants, the near–best order of uniform convergence to any continuous function, and a strong reduction of Gibbs phenomenon. These features will be inherited by the proposed numerical method which is stable and convergent, and provides a near-best polynomial approximation of the sought solution by solving a well conditioned linear system. The numerical tests confirm the theoretical error estimates and, in case of functions subject to Gibbs phenomenon, they show a better local approximation compared with analogous Lagrange projection methods. [ABSTRACT FROM AUTHOR]

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96 % compared to the algorithm from Gorodetsky et al. (Comput. Methods Appl. Mech. Eng. 347, 59–84 2019). [ABSTRACT FROM AUTHOR]

Background: Previously, there has been some work in the field of optical imaging on phase compensation by employing Legendre polynomials as an expansion of the phase function, and this seems to be an appealing unstudied area in the field of ultrasound imaging. Purpose: The paper is devoted to solving one of the problems of enhancing the authenticity of diagnostics data obtained in ultrasound visualization systems and presents a novel approach to aberration correction, which ensures a reliable eliminating of phase distortions. The novelty of the proposed approach consists in the use of the decomposition of the wave front by Legendre polynomials to approximate aberrations of the phase front of ultrasonic waves propagating through distorting layers. Methods: The phase aberrations are corrected using a single sector probe that captures echoes in synthetic aperture mode. The proposed method approximates the changes in the wave front by means of the Legendre polynomials. The performance was investigated by placing a 13‐mm‐wide ultrasonic probe with 64 elements operating at 2 MHz on the surface of a commercial quality control phantom ATS Model 539 through a distorting layer. The following metrics were measured: peak value, root mean square (RMS), full width at half maximum (FWHM), contrast‐to‐noise ratio (CNR). During the experiments, three different aberrators were used interchangeably as distorting layers, two of which were made of photopolymer resin with RMS values of 39 and 97 ns and a speed of sound close to that of a human temporal bone tissue, and the third was an ex vivo human temporal bone with the RMS value of 44 ns. To test the correction, three regions inside the phantom were examined. Two‐sided nonparametric Wilcoxon rank‐sum test was used for assessing the statistical significance. The significance level for this study was set at α ≤ 0.05. To counteract the problem of multiple comparisons, the p‐values were adjusted by the Holm–Bonferroni correction method. Results: The statistically significant results have demonstrated the possibility of increasing peak intensity value up to 3.08 times, reducing the RMS width and FWHM of the intensity angular distribution down to 60% and 82%, respectively, and increasing CNR up to 2.12 times by using the proposed phase distortion correction method, compared with the case without correction. The results show that the method can be used as an effective aberration correction technique for all tested regions inside the phantom and distorting layers. Its usage allowed correcting up to 97% of the aberrations caused by the ex vivo temporal bone model. Conclusion: The results show that the usage of the proposed method for aberration correction can successfully increase the intensity and reduce the angular width of the ultrasound wave scattered by the point targets inside the phantoms. The method works with different distorting layers and is capable of correcting phase aberrations in multiple sections of the sonogram. The limitation of the proposed method consists in the fact that it requires the use of aperture synthesis and access to raw radiofrequency data, which restricts its application in common scanners. [ABSTRACT FROM AUTHOR]

In this paper, we give direct theorems on approximation by Haar and Walsh polynomials in variable exponent grand Lebesgue space. Also the degree of approximation by Borel, Abel-Poisson, Riesz-Zygmund and Euler linear means of Haar-Fourier series are treated in the above-cited space. [ABSTRACT FROM AUTHOR]

Enhancing the through-flow capability of the turbine facilitates the potential to decrease the external dimension of the engine, leading to a reduction in weight and an increase in the thrust-to-weight ratio. The optimization of blade profile in terms of aerodynamic design is an essential strategy to improve the overall performance of the high through-flow design turbine. This study aims to develop an automated optimization technique, utilizing the response surface method (RSM), to enable reliable prediction and rapid optimization of aerodynamic performance. A parametric modeling method is devised to generate blade profiles with continuous curvature throughout the automated optimization process. The Box–Behnken experimental design, in combination with Reynolds-averaged Navier–Stokes numerical calculation, is employed to construct a second-order polynomial RSM approximation model. The optimization process comprises two levels: improving the through-flow capability and reducing blade profile loss. The optimized profile, Opt1, achieves a substantial 13.19% improvement in through-flow capability at the cost of a 12.27% increase in blade profile loss. Thus, further optimization is performed to minimize blade profile loss based on the Opt1 scheme. Geometric constraints are applied to the most influential parameters affecting through-flow capability to mitigate their impacts. Compared to the Opt1, the final optimized profile, Opt2, achieves a substantial 31.83% decrease in blade profile loss with a negligible sacrifice of 0.23% in through-flow capability. [ABSTRACT FROM AUTHOR]

Tensor robust principal component analysis (TRPCA) is a useful approach for obtaining low-rank data corrupted by noise or outliers. However, existing TRPCA methods face certain challenges when it comes to estimating the tensor rank and the sparsity accurately. The commonly used tensor nuclear norm (TNN) may lead to sub-optimal solutions due to the gap between TNN and the tensor rank. Additionally, the ℓ 1 -norm is not an ideal estimation of the ℓ 0 -norm, and solving TNN minimization can be computationally intensive because of the tensor singular value thresholding (t-SVT) scheme. To address these issues, a method called fast correction TNN (FC-TNN) is proposed for TRPCA. In contrast to existing methods, FC-TNN introduces a correction term to bridge the gap between TNN and the tensor rank. Furthermore, a new correction term is employed for the ℓ 1 -norm to achieve the desired solution. To improve computational efficiency, the Chebyshev polynomial approximation (CPA) technique is presented for computing t-SVT without requiring tensor singular value decomposition (t-SVD). The CPA technique is incorporated into the alternating direction method of multipliers (ADMM) algorithm to solve the proposed model effectively. Theoretical analysis demonstrates that FC-TNN offers a lower error bound compared to TNN under certain conditions. Extensive experiments conducted on various tensor-based datasets illustrate that the proposed method outperforms several state-of-the-art methods. • We consider the corrected tensor nuclear norm and l1 norm to solve the tensor recovery problem. • The proposed model is a more general framework, including several current methods as special cases. • A fast ADMM algorithm with chebyshev polynomial approximation technique is proposed. • Numerical experiments conducted on real-word datasets demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Suppose A = { a 1 , ... , a n + 2 } ⊂ Z n has cardinality n + 2 , with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = (f 1 , ... , f n) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixedn, counts exactly the number of real roots of F in time polynomial in log (d H) . We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well. [ABSTRACT FROM AUTHOR]

In this work, we construct a fifth‐order weighted essentially non‐oscillatory (WENO) scheme with exponential approximation space for solving dispersive equations. A conservative third‐order derivative formulation is developed directly using WENO spatial reconstruction procedure, and third‐order TVD Runge–Kutta scheme is used for the evaluation of time derivative. This exponential approximation space consists a tension parameter that may be optimized to fit the specific feature of the characteristic data, yielding better results without spurious oscillations compared to the polynomial approximation space. A detailed formulation is presented for the construction of conservative flux approximation, smoothness indicators, nonlinear weights, and verified that the proposed scheme provides the required fifth convergence order. One‐ and two‐dimensional numerical examples are presented to support the theoretical claims. [ABSTRACT FROM AUTHOR]

This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre's K -functional. Then, we show that, under special choices of A (t) , the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically. [ABSTRACT FROM AUTHOR]

We present algorithms to approximate the scaled complementary error function, exp x 2 e r f c (x) , and the Dawson integral, e - x 2 ∫ x 0 e t 2 d t , to the best accuracy in the standard single, double, and quadruple precision arithmetic. The algorithms are based on expansion in Chebyshev subinterval polynomial approximations together with expansion in terms of Taylor series and/or Laplace continued fraction. The present algorithms, implemented as Fortran elemental modules, have been benchmarked versus competitive algorithms available in the literature and versus functions built-in in modern Fortran compilers, in addition to comprehensive tables generated with variable precision computations using the Matlab™ symbolic toolbox. The present algorithm for calculating the scaled complementary error function showed an overall significant efficiency improvement (factors between 1.3 and 20 depending on the compiler and tested dataset) compared to the built-in function "Erfc_Scaled" in modern Fortran compilers, whereas the algorithm for calculating the Dawson integral is exceptional in calculating the function to 32 significant digits (compared to 19 significant digits reported in the literature) while being more efficient than competitive algorithms as well. [ABSTRACT FROM AUTHOR]

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider f(x_1, \dots, x_N), where x_i \in \mathbb {R}^d, and f is invariant under permutations of its N arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function f, and in particular study the dependence of that ratio on d, N and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where N becomes a parameter of the input. [ABSTRACT FROM AUTHOR]

In this work, we show that solvers of elliptic boundary value problems in d dimensions can be approximated to accuracy ε from only O(log(N) logd (N/ε)) matrix-vector products with carefully chosen vectors (right-hand sides). The solver is only accessed as a black box, and the under-lying operator may be unknown and of an arbitrarily high order. Our algorithm (1) has complexity O(N log²(N) log2d (N/ε)) and represents the solution operator as a sparse Cholesky factorization with O(N log (N) logd (N/ε)) nonzero entries, (2) allows for embarrassingly parallel evaluation of the solution operator and the computation of its log-determinant, (3) allows for O(log (N) logd (N/ε)) complexity computation of individual entries of the matrix representation of the solver that, in turn, enables its recompression to an O(N logd (N/ε)) complexity representation. As a byproduct, our compression scheme produces a homogenized solution operator with near-optimal approximation accuracy. By polynomial approximation, we can also approximate the continuous Green's function (in operator and Hilbert-Schmidt norm) to accuracy ε from O(log1+d(ε-1)) solutions of the PDE. We include rigorous proofs of these results. To the best of our knowledge, our algorithm achieves the best known trade-off between accuracy ε and the number of required matrix-vector products. [ABSTRACT FROM AUTHOR]

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α ∈ (1 , 2). The error has the asymptotic expansion (d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯) + (d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯) at any fixed time t N = T , N ∈ Z + , where d i , i = 3 , 4 , ... and d i ∗ , i = 2 , 3 , ... denote some suitable constants and τ = T / N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α ∈ (1 , 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings. [ABSTRACT FROM AUTHOR]

In this work we review and derive some elementary properties of the discrete renewal sequences based on a positive, finite and integer-valued random variable. Our results consider these sequences as dependent on the probability masses of the underlying random variable. In particular we study the minima and the maxima of these sequences and prove that they are attained for indices of the sequences smaller or equal than the support of the underlying random variable. Noting that the minimum itself is a minimum of multi-variate polynomials we conjecture that one universal polynomial envelopes the minimum from below and that it is maximal in some sense and largest in another. We prove this conjecture in a special case. [ABSTRACT FROM AUTHOR]

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods. [ABSTRACT FROM AUTHOR]

A method for radiometer-noise generation and its software implementation in the form of a Python library module is proposed. The method can be used to simulate radiometer noise with variable component properties: Gaussian white noise and flicker noise (random drift) with a power spectrum linearly increasing with decreasing frequency. The method is based on the transformation of spectral properties of Gaussian noise via sequential filtering by a differentiator, nonlinear threshold switch, and subsequent integration. By changing the integrator accumulation time constant and the threshold level relative to noise intensity one can change the resulting form of the spectrum and the corresponding statistical parameters of the output noise over a wide range: from stationary white noise to noise dominated by a -type spectrum component. Empirical polynomial approximation formulas for the dependence of the spectral index of the resulting noise records on the nonlinear threshold switch level in the 0.0– interval are derived based on numerical simulations using a software implementation of the model. Examples of noise-record implementations, their spectra, and autocorrelation functions are demonstrated. The generated noise is shown to be similar to the noise observed on radio telescopes of the Institute of Applied Astronomy of the Russian Academy of Sciences. [ABSTRACT FROM AUTHOR]

The exact value was calculated and the polynomial for the best weighted polynomial approximation was found for the kernels of the form A + B x (x 2 + λ 2) 2 , where A,B ∈ R and λ > 0, in the root-mean-square metric. [ABSTRACT FROM AUTHOR]

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data. [ABSTRACT FROM AUTHOR]

The Bessel function of the first kind J N k x is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind I N k x . The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for J N k x useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving   1 F 2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1 / 2 i 3 j 5 k 7 l 11 m 13 n 17 o 19 p multiplying powers of the coefficient k. [ABSTRACT FROM AUTHOR]

This paper constructs an interval partition linearization algorithm for solving minimax linear fractional programming (MLFP) problem. In this algorithm, MLFP is converted and decomposed into a series of linear programs by dividing the outer 1-dimensional space of the equivalent problem (EP) into polynomially countable intervals. To improve the computational efficiency of the algorithm, two new acceleration techniques are introduced, in which the regions in outer space where the optimal solution of EP does not exist are largely deleted. In addition, the global convergence of the proposed algorithm is summarized, and its computational complexity is illustrated to reveal that it is a fully polynomial time approximation scheme. Finally, the numerical results demonstrate that the proposed algorithm is feasible and effective. [ABSTRACT FROM AUTHOR]

This paper proposes Brenke-type polynomials for uncertainty approximation in adaptive control of cooperative electrically driven manipulators. The contribution of this paper is reducing the number of signals needed in the construction of the regressor vector for uncertainty approximation due to the simplicity of Brenke-type polynomials in comparison with other alternatives. It is verified that uniformly bounded stability of the tracking/approximation errors can be assured if the Brenke-type polynomials are utilised as regressors. Moreover, it has been assumed that velocity signals are not available for feedback. Different trajectories, including simple and highly complicated and fast paths, have been tested in simulation results. The results are also compared with four state-of-the-art powerful approximation methods to show the accuracy and efficiency of the proposed controller structure. [ABSTRACT FROM AUTHOR]

A new approximation method for the plasma dispersion function Z(ζ) is presented. Multipoint quasi-rational approximation technique is used to find a bridge function that connects the power series and the asymptotic expansion of the function Z(ζ) using rational functions combined with exponential functions. An approximation with a polynomial of degree 10 is performed for the function Z(ζ), and the results obtained are compared with those of previous approximations from the literature. The results of this approximation were a relative error of ɛ = 0.0035 for R e [ Z ̃ (ζ) ] and a relative error of ɛ = 0.0011 for I m [ Z ̃ (ζ) ] , which are lower than those of the other existing approximations. [ABSTRACT FROM AUTHOR]

The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions. [ABSTRACT FROM AUTHOR]

This paper proposes several methodologies whose objective consists of securing copula density estimates. More specifically, this aim will be achieved by differentiating bivariate least-squares polynomials fitted to Deheuvels' empirical copulas, by making use of Bernstein's approximating polynomials of appropriately selected orders; by differentiating linearized distribution functions evaluated at optimally spaced grid points; and by implementing the kernel density estimation technique in conjunction with a repositioning of the pseudo-observations and a certain criterion for determining suitable bandwidths. Smoother representations of such density estimates can further be secured by approximating them by means of moment-based bivariate polynomials. The various copula density estimation techniques being advocated herein are successfully applied to an actual dataset as well as a random sample generated from a known distribution. [ABSTRACT FROM AUTHOR]

Longitudinal data arise frequently in many economic studies and epidemiological research. In this paper, we investigate the partially linear additive model for longitudinal data in the framework of quantile regression. To incorporate the within-subject correlation, we develop an estimation procedure using quadratic inference function (QIF) and polynomial spline approximation for unknown nonparametric functions. The theoretical properties of the resulting estimators are established, where the nonparametric functions achieve the optimal convergence rate and the parametric components are asymptotically normal even when the number of parameters in the linear part is diverging. We also propose a variable selection procedure based on penalization. Since the objective function is discontinuous, a practical estimation procedure is proposed using induced smoothing and we prove that the smoothed estimator is asymptotically equivalent to the original estimator. The proposed methods are evaluated via simulation studies and a real data application. [ABSTRACT FROM AUTHOR]

In this work, the novel quadrilateral overlapping element incorporated with the harmonic trigonometric functions is presented to reduce the efforts in mesh division for modal analysis. The new partition of unity functions in the overlapping element are cubic polynomials. Therefore, low-order polynomials can be utilized to augment local approximation space without the linear dependence problem. Moreover, the original polynomial approximation spaces are further enriched by the harmonic trigonometric functions without introducing additional rank deficiencies in the system matrices. Thus, the presented enriched overlapping finite element method (EOFEM) provides very precise eigenfrequencies and eigenmodes by using a very coarse mesh. However, the shape functions in the presented EOFEM do not possess the Kronecker delta function property. Hence, a conforming method that modifies the shape functions rather than the variational principle is proposed to impose the essential boundary conditions. Several numerical examples in two dimensions are then investigated to elaborate on the performance of the presented EOFEM in modal analyses. Besides, a block-diagonal lumped mass matrix is presented and further extended to the structure vibration analysis by incorporating the explicit time integration method. It is proven that the enriched overlapping element is computationally more efficient than the quadratic finite element and the original overlapping element. [ABSTRACT FROM AUTHOR]

Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in H r (u) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L 2 space associated with u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of E n (f (r)) , the error of best approximation of f (r) in L 2 (u) . [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves two critical physical constraints: a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition in each cell, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for the high-order finite volume methods that return only the point values (no explicit approximation polynomials) in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, we adopt the recently developed geometric quasilinearization approach [K. Wu and C.-W. Shu, SIAM Rev., 65 (2023), pp. 1031--1073], which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques. [ABSTRACT FROM AUTHOR]

The mixed projection-mesh scheme of the finite element method was used to set a mixed approximation for three finite elements in the form of a hexagonal eight-node prism with the substitution of displacements after the polylinear law and stress and strain approximation with polynomial interpolation functions, defined for the nodes at the vertices, in the prism center, in the face centers, inner points, and a tetrahedron-shaped element in which displacements are sought in the nodes at the vertices, and stresses and strains are in the nodes at the vertices and in the center. The expressions for the coefficients of the decision matrices are obtained. Several test problems with analytical or known numerical solutions in the elastic statement are examined. The results obtained with the mixed method were compared with those obtained from the finite element method. The relative errors in determining examined characteristics were assessed in the elastic statement. The Irwin plastic zone correction was evaluated in the elastoplastic statement. One of the proposed elements would be appropriate for further application from the solution results and the estimation of time taken for solving specific problems and data body processing. [ABSTRACT FROM AUTHOR]

In this article, we design and analyze an arbitrary-order stabilized finite element method to approximate the unique continuation problem for laminar steady flow described by the linearized incompressible Navier–Stokes equation. We derive quantitative local error estimates for the velocity, which account for noise level and polynomial degree, using the stability of the continuous problem in the form of a conditional stability estimate. Numerical examples illustrate the performances of the method with respect to the polynomial order and perturbations in the data. We observe that the higher order polynomials may be efficient for ill-posed problems, but are also more sensitive for problems with poor stability due to the ill-conditioning of the system. [ABSTRACT FROM AUTHOR]

We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights. [ABSTRACT FROM AUTHOR]

The research of traffic congestion control has been developed rapidly in recent years. Researching the bifurcation characteristics of traffic flow and designing control schemes for unstable bifurcation points can provide some new methods for relieving the traffic congestion. In this paper, we study the bifurcation characteristics and nonlinear control of traffic flow based on the continuum model considering the new energy vehicles effect. First, a macroscopic traffic flow model considering the number of new energy vehicles and their range is proposed. Then, we transform it into a traffic flow stability model suitable for branching analysis using the traveling wave transformation. Therefore, we can transform the traffic flow problem into the stability analysis problem of systems and reflect unstable traffic characteristics such as congestion. Based on the stability model, the existence conditions of the Hopf bifurcation are proved and some bifurcation points of the traffic system are found. Second, we study the congestion and stability abrupt change behaviors near the equilibrium point and branching point to seek for the formation mechanism of traffic congestion. Finally, for the unstable bifurcation point, the control scheme is designed by Chebyshev polynomial approximation, stochastic feedback control, and other methods to delay or disappear the unstable bifurcation point and alleviate traffic congestion. It has important theoretical significance and application value. [ABSTRACT FROM AUTHOR]

This research is the first to deal with high stochastic solitons of the eighth-order NLSE that have higher polynomial nonlinearity and use Itô multiplicative white noise. We use each of an extended simplest approach, the addendum Kudryashov's technique to achieve this. The examination of this study provides bright, singular, bright-singular and combined bright-dark solitons. These findings contribute to the advancement of our understanding of high stochastic solitons and their dynamics, which is crucial for real-world applications, particularly in optical communication systems. By comprehending the effects of stochasticity on dispersive solitons, we can enhance the design and performance of optical networks, leading to more efficient and reliable communication over long distances. Representations of the bright and combined bright-dark solutions are available in 2D and 3D. The system being studied is new and special, as are the effects that are discussed in the paper. [ABSTRACT FROM AUTHOR]

Palatini f (R) gravity is probably the simplest extension of general relativity (GR) and the simplest realization of a metric-affine theory. It has the same number of degrees of freedom as GR and, in vacuum, it is straightforwardly mapped into GR with a cosmological constant. The mapping between GR and Palatini f (R) inside matter is possible but at the expense of reinterpreting the meaning of the matter fields. The physical meaning and consequences of such mapping will depend on the physical context. Here, we consider three such cases within the weak field limit: solar system dynamics, planetary internal dynamics (seismology), and galaxies. After revising our previous results on the solar system and Earth's seismology, we consider here the possibility of f (R) Palatini as a dark matter candidate. For any f (R) that admits a polynomial approximation in the weak field limit, we show here, using SPARC data and a recent method that we proposed, that the theory cannot be used to replace dark matter in galaxies. We also show that the same result applies to the Eddington-inspired Born–Infeld gravity. Differently from the metric f (R) case, the rotation curve data are sufficient for this conclusion. [ABSTRACT FROM AUTHOR]

Traffic congestion not only has a great impact on people's travel, but also increases energy consumption and air pollution. The control analysis of the macroscopic traffic flow model considering the vehicle braking effect is particularly important, reflecting the impact on the actual traffic flow density wave, so as to better solve the actual traffic problems. In this paper, based on a speed difference optimization speed model, the micro–macro-variables are transformed into a high-order continuous traffic flow model. Then, a random function considering the physical correlation of random components is added to the high-order continuous traffic flow model to establish a random traffic flow model that can reflect the uncertain behavior of traffic flow acceleration or deceleration. Based on this stochastic traffic model, the existence of Hopf bifurcation and bifurcation control of the traffic flow system model considering stochastic characteristics are derived by using Hopf bifurcation theorem. By Chebyshev polynomial approximation method, the stochastic problem of the system is transformed into the bifurcation control problem of its equivalent deterministic system. A feedback controller is designed to delay the occurrence of Hopf bifurcation and control the amplitude of the limit cycle. Without changing the equilibrium point of the system, the complete elimination of Hopf bifurcation can be achieved by controlling the amplitude of the limit cycle. That is, the feedback controller is used to modify the bifurcation characteristics of the system, such as the bifurcation appearing at the equilibrium point in the control system moves forward, moves backward or disappears, so as to achieve the effect of preventing or alleviating traffic congestion. [ABSTRACT FROM AUTHOR]

This article concerns a question asked by M. V. Nori on homotopy of sections of projective modules defined on the polynomial algebra over a smooth affine domain R. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) dim (R) = 2 ; or (2) d ≥ 3 but R is not smooth. We first prove that an affirmative answer can be given for dim (R) = 2 when R is an F ‾ p -algebra. Next, for d ≥ 3 we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring A is an affine F ‾ p -algebra of dimension d. We apply this improvement to define the n -th Euler class group E n (A) , where 2 n ≥ d + 2. Moreover, if A is smooth, we associate to a unimodular row v of length n + 1 its Euler class e (v) ∈ E n (A) and show that the corresponding stably free module, say, P (v) has a unimodular element if and only if e (v) vanishes in E n (A). [ABSTRACT FROM AUTHOR]