Spherical Polynomials Research Articles

Learning on manifolds without manifold learning

Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional Euclidean space. A great deal of research deals with obtaining information about this manifold, such as the eigendecomposition of the Laplace–Beltrami operator or coordinate charts, and using this information for function approximation. This two-step approach implies some extra errors in the approximation stemming from estimating the basic quantities of the data manifold in addition to the errors inherent in function approximation. In this paper, we project the unknown manifold as a submanifold of an ambient hypersphere and study the question of constructing a one-shot approximation using a specially designed sequence of localized spherical polynomial kernels on the hypersphere. Our approach does not require preprocessing of the data to obtain information about the manifold other than its dimension. We give optimal rates of approximation for relatively “rough” functions.

Image analysis by fractional-order weighted spherical Bessel-Fourier moments

Moment and moment invariants as effective feature descriptors have been widely applied in image analysis, pattern recognition and computer vision applications. Scholars have demonstrated that fractional-order orthogonal moments outperform their corresponding moments in representing the fine details of a given image. To this end, we propose a new fractional-order orthogonal moments based on fractional-order weighted spherical Bessel polynomial in this paper, which is named as Fractional-order weighted Spherical Bessel-Fourier Moments (FrSBFMs). Moreover, to address the issue of color image analysis, FrSBFMs are integrated with quaternion theory to develop Quaternion FrSBFMs (QFrSBFMs). In addition, the rotation invariance and parameter α analysis of FrSBFMs and QFrSBFMs are discussed in detail. The experimental results show the effectiveness of the proposed FrSBFMs and QFrSBFMs in terms of image reconstruction capability, pattern recognition accuracy and zero-watermark verification.

Extensions of polynomial plank covering theorems

AbstractWe prove a complex polynomial plank covering theorem for not necessarily homogeneous polynomials. As the consequence of this result, we extend the complex plank theorem of Ball to the case of planks that are not necessarily centrally symmetric and not necessarily round. We also prove a weaker version of the spherical polynomial plank covering conjecture for planks of different widths.

Random and mean Lyapunov exponents for

AbstractWe consider orthogonally invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $\operatorname {\mathrm {GL}}_n(\mathbb {C})$ . Astérisque287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

Comparison of polynomial and rational function cornea models for effective dimensionality reduction

This study aims to examine geometric models of the corneal surface that can be used to reduce in reasonable time the dimensionality of datasets of normal anterior corneas. Polynomial models (P) like Zernike polynomials (ZP) and spherical harmonic polynomials (SHP) were obvious candidates along with their rational function (R) counterparts, namely Zernike rational functions (ZR) and spherical harmonic rational functions (SHR, new model). Knowing that both SHP and ZR were more accurate than ZP for the modeling of normal and keratoconus corneas, it was expected that both spherical harmonic (SH) models (SHP and SHR) would be more accurate than their Zernike (Z) counterparts (ZP and ZR, respectively), and both rational (R) models (SHR and ZR) more accurate than their polynomial counterparts (SHP and ZP, respectively) for a low dimensional space (coefficient number J < 30). This was the case. The SH factor contributed more to accuracy than the R factor. Considering the corneal processing time as a function of J, P models were processed in quasi-linear time with a quasi-null slope and rational models in polynomial time. Z models were faster than SH models, and increasingly so in their R version. In sum, for corneal dimensionality reduction, SHR is the most accurate model, but its processing time is increasingly prohibitive unless the best coefficient combination is identified beforehand. ZP is the fastest model and is reasonably accurate with normal corneas for exploratory tasks. SHP is the best compromise between accuracy and speed.

Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L2-orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.

Characteristics of Low-Latitude Ionosphere Activity and Deterioration of TEC Model during the 7–9 September 2017 Magnetic Storm

Under the influence of space weather, abnormal disturbances in the ionosphere will distort the ionosphere model seriously and affect the global navigation satellite system negatively. This study analyzes the ionospheric activity characteristics and the ionospheric model performance in low latitude during a strong geomagnetic storm from 7 to 9 September 2017. The research goals are to determine the abnormal behavior of the ionosphere during the geomagnetic storm and to refine the ionosphere model in the low latitude. In the experiment, the vertical total electron content (VTEC) peak value at low latitudes caused by this geomagnetic storm was significantly higher than that on the geomagnetic quiet day, and the VTEC peak value increased by approximately 75%. In the main phase of the geomagnetic storm, the degree of VTEC variation with longitude is significantly higher than that of the geomagnetic quiet day. The VTEC variation trend in the northern hemisphere is more severe than that in the southern hemisphere. In the region where VTEC decreases with longitude, the VTEC in the northern hemisphere is higher than that in the southern hemisphere on the same longitude at low latitudes, and this phenomenon is not significantly affected by the geomagnetic disturbance of the recovery phase. During the geomagnetic storm, the daily minimum value of VTEC at different latitudes was basically the same, approximately 5 TECU, indicating that the nighttime VTEC of the ionosphere in low latitudes was weakly affected by latitude and geomagnetic storms. Geomagnetic disturbances during geomagnetic storms will lead to anomalous features of the “Fountain effect” in the ionosphere at low latitudes. In addition, this geomagnetic storm event caused the accuracy of spherical harmonics (SH), polynomial, and ICE models to decrease by 7.12%, 27.87%, and 48.56%, respectively, and caused serious distortion, which is negative VTEC values fitted by the polynomial model.

Neural network-based ionospheric modeling and predicting—To enhance high accuracy GNSS positioning and navigation

As a highly variable space plasma, the sophisticated monitoring and expression of the ionosphere are critically important for the performance of Global Navigation Satellite System (GNSS). To satisfy the requirement of more precise positioning and navigation, it is necessary to model and predict the ionosphere in a more accurate way. In this paper, the potential advantages of neural networks in ionosphere modeling and predicting are explored. Radial Basis Function Neural Network (RBF-NN) is used for modeling and the Long- and Short-Term Memory Neural Network (LSTM-NN) is used for predicting. Results validate that performance of the RBF-NN is superior to the traditional spherical harmonic and polynomial model. The average root mean square error of one-day modeling residuals by RBF-NN is 1.2951 Total Electron Content Unit (TECU, 1 TECU = 1016 electron/m2), which is reduced by 0.4172 and 0.2710 TECU respectively, compared with the traditional spherical harmonic and polynomial model. The predicting results show that the LSTM-NN has better short-time performance, compared with the finite Fourier series model and the autoregressive integrated moving average model. Finally, one-day predicted products are used as priori constraints for single-frequency static and dynamic precise point positioning (PPP). Positioning results show that constraint effect of the predicted products is equivalent to that of the International GNSS Services (IGS) final products. Compared with the results without ionospheric constraint, the PPP convergence time is obviously reduced, and the positioning performance in the horizontal direction is significantly improved.

The accuracy of geostatistics for regional geomagnetic modeling in an archipelago setting.

Indonesia as an archipelago country relies on a limited number and clustered distributed repeat station networks. This paper explores the use of geostatistical modeling to overcome this data limitation. The model data set consisted of repeat station data from 1985 to 2015 epoch. The geostatistical methods utilized included ordinary kriging (OK), collocated cokriging (CC), and kriging with external drift (KED). The model generated using these geostatistical methods was then compared to spherical cap harmonic analyses (SCHA) and polynomial models. The geostatistical model was shown to perform better, with greater accuracy in declination, inclination, and total intensity, as indicated by the root mean square error (RMSE). We have demonstrated that the geostatistical method is a promising approach in the modeling of regional geomagnetic field, especially in areas with limited and clustered distributed data.

Formula omitted] error estimates for polynomial discrete penalized least-squares approximation on the sphere from noisy data

We consider discrete penalized least-squares approximation on the unit sphere Sd, where the minimizer is sought in the space PL(Sd) of spherical polynomials of degree ≤L. The penalized least-squares functional is the sum of two terms both involving the weights and the nodes of a positive weight quadrature rule with polynomial degree of exactness at least 2L. The first one is a discrete least-squares functional measuring the squared weighted ℓ2 discrepancy between the noisy data and the approximation. The second term is the product of a regularization parameter λ≥0 times a penalization term that can be interpreted as an approximation of a squared semi-norm in the Sobolev (Hilbert) space Hs(Sd). The approximation can be computed directly via a summation process and does not require the solving of a linear system. For λ=0 (which is only appropriate if there is almost no noise), our approximation becomes a case of hyperinterpolation. As λ>0 increases, less weight is given to data fitting and more weight is given to keeping the approximation smooth. We derive L2(Sd) error estimates for the approximation of functions from the Sobolev Hilbert space Hs(Sd), where s>d/2, from noisy data for the regularization parameter λ chosen (i) as λ=0 (hyperinterpolation), (ii) for general λ>0, and (iii) with Morozov’s discrepancy principle (an a posteriori parameter choice strategy). The L2(Sd) error estimates in case (i) and (iii) are in a sense order-optimal. Numerical experiments explore the approximation method and illustrate the theoretical results.

Theta series for quadratic forms of signature (n − 1,1) with (spherical) polynomials

We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature [Formula: see text]. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a construction of the second author) to obtain holomorphic, almost holomorphic and modular theta series. We give a criterion for these series to coincide, enabling us to construct almost holomorphic and holomorphic cusp forms on congruence subgroups of the modular group. Further, we provide numerous explicit examples.

Non‐Singular Calculation of Geomagnetic Vectors and Geomagnetic Gradient Tensors

AbstractSome spherical harmonic expressions of gravitational and geomagnetic field elements will become infinite when computation point approaches polar regions, as the sine function of the geocentric co‐latitude contained in the denominator tends to be zero. Currently, this singularity problem has been solved for gravitational field case, however, it remains unsolved for geomagnetic vectors (GVs) and geomagnetic gradient tensors (GGTs). Because the latter use Schmidt semi‐normalized associated Legendre function (SNALF), which is different from fully‐normalized associated Legendre function used in the former. To overcome this singularity problem, we derive new non‐singular expressions of the first‐ and second‐order derivatives of Schmidt SNALF (order m equals to 0 or 1), and the corresponding two kinds of spherical harmonic polynomials. When the non‐singular expressions are applied to the traditional formulae of GVs and GGTs, more practical expressions of GVs and GGTs with non‐singularity are formulated by refining the cases that the order m equals to 0, 1, 2 and other values. Furthermore, to provide flexible calculation strategies for Schmidt SNALF, we derive four kinds of recursive formulae, including the standard forward row recursion, the standard forward column recursion, the cross degree and order recursion, and the Belikov recursion. The standard formula to check the calculation accuracy of various recursive formulae of SNALF are also given. Besides, we demonstrate the effectiveness and reliability of the new derived non‐singular expressions of GVs and GGTs and analyze the computation speed and stability of the four recursive formulae by extensive numerical experiments.

Analysis of disturbance observer-based control systems via spherical polynomials

Robustness analysis of disturbance observer (DOB)-based control systems under parametric uncertainty is addressed in this study. A spherical polynomial family-based approach is adopted to analyse how much uncertainty can be tolerated, and to capture a unified framework for different cases including non-minimum phase plant and different nominal and perturbed plant structures. Results are validated using the value set concept for spherical polynomial families. The study has shown that if the relative degrees of the perturbed plant and the nominal plant are equal, then robustness is achievable even if the plant model is low order. Although non-minimum phase zeros limit the selection of DOB bandwidth, the robustness margin can be exactly determined for a given DOB bandwidth. Furthermore, it is shown that when the plant numerator and denominator have uncertain parameters, the robustness margin is not increased as with the DOB filter bandwidth in general.

A mesh-free treatment for even parity neutron transport equation

Common family of numerical methods such as finite element and finite difference involve the discretization of domain into regular grids or meshes. Mesh generation and re-meshing process, however, for obtaining required accuracy are complicated and time-consuming. Recently, Mesh Free (MFree) methods have been developed to overcome the drawbacks of such conventional numerical methods. In the present work, a novel approach is developed for solving multi-group neutron transport equation in two-dimensional X-Y geometry based on a mesh free scheme using the Radial Point Interpolation Method (RPIM). The K+ principle which is an extremum variational principle for solving even parity neutron transport equation is our choice to implement RPIM. The directional dependence of even-parity angular flux is expanded by the spherical harmonic polynomials which leads to PN method. The multi-quadrics radial basis function is used to construct the RPIM shape functions for spatial approximation of angular flux to obtain the discretized MFree weak-form of neutron transport equation. As the RPIM shape functions possess the Kronecker delta function property, essential boundary conditions is enforced as efficiently as in the finite element method (FEM). The obtained results are compared with some mesh-based methods such as conventional finite element method to evaluate the performance of the presented approach. It is demonstrated that the proposed method is robust, stable, reliable and efficient for treatment of neutron transport.

Estimates of the asymptotic Nikolskii constants for spherical polynomials

Let Πnd denote the space of spherical polynomials of degree at most n on the unit sphere Sd⊂Rd+1 that is equipped with the surface Lebesgue measure dσ normalized by ∫Sddσ(x)=1. This paper establishes a close connection between the asymptotic Nikolskii constant, L∗(d)≔limn→∞1dimΠndsupf∈Πnd‖f‖L∞(Sd)‖f‖L1(Sd), and the following extremal problem: Iα≔infak‖jα+1(t)−∑k=1∞akjα(qα+1,kt∕qα+1,1)‖L∞(R+) with the infimum being taken over all sequences {ak}k=1∞⊂R such that the infinite series converges absolutely a.e. on R+. Here jα denotes the Bessel function of the first kind normalized so that jα(0)=1, and {qα+1,k}k=1∞ denotes the strict increasing sequence of all positive zeros of jα+1. We prove that for α≥−0.272, Iα=∫0qα+1,1jα+1(t)t2α+1dt∫0qα+1,1t2α+1dt=1F2(α+1;α+2,α+2;−qα+1,124). As a result, we deduce that the constant L∗(d) goes to zero exponentially fast as d→∞: 0.5d≤L∗(d)≤(0.857⋯)d(1+εd)with εd=O(d−2∕3).

Approximation by spherical polynomials in 𝐿𝑝 for 𝑝 &lt; 1

Based on recently proved estimates for the 𝐿1-Nikolskii constants for S𝑑 and R𝑑, effective bounds for the constant 𝐾 are given in the following inequality of the type Brown–Lucier for functions 𝑓 ∈ 𝐿𝑝(S𝑑), 0 < 𝑝 < 1: ‖𝑓 − 𝐸1𝑓‖𝑝 6 (1 + 2𝐾)1/𝑝 inf 𝑢∈Π𝑑 𝑛 ‖𝑓 − 𝑢‖𝑝, where Π𝑑 𝑛 is the subspace of spherical polynomials, 𝐸1𝑓 is a best approximant of 𝑓 from Π𝑑𝑛 in the metric 𝐿1(S𝑑). The results are generalized to the case of the Dunkl weight.

Константы Никольского — Бернштейна в $L^p$ на сфере с весом Данкля

Изучаются точные константы Никольского--Бернштейна для сферических полиномов впространстве $L^{p}(\mathbb{S}^{d})$ с весом Данкля. Устанавливаетсявзаимосвязь с одномерными константами для алгебраических полиномов впространстве $L^{p}[-1,1]$ с весом Гегенбауэра.

Tensor Expansions of the Angular Particle Distribution

We establish a relation between the class of symmetric spherical tensors and even/odd polynomials. The expansions of the scattering operator of photons or neutrons in a series of symmetric spherical tensors are obtained. Among them are expansions that have a higher rate of uniform convergence in comparison with expansions in spherical functions and Legendre polynomials. It is shown that, in problems of radiation transport in a substance with predominant forward or backward scattering, it is advisable to use expansions in the system of Chebyshev polynomials and tensors.

Scattering of elastic wave from poroelastic inclusions located in a fluid

The problem of scattering of a plane compressional acoustic wave in a fluid from a spherical poroelastic inclusion is considered. The elastic wave propagation in the inclusion is described by the equations of the Biot theory. The wave field in the inclusion consists of fast and slow compressional and shear waves. Outside the inclusion, a scattered spherical compressional wave is formed. The solution for an isolated inclusion is obtained in terms of series of spherical Bessel functions and Legendre polynomials. This solution is used for the calculation of effective wave number of compressional wave propagating in the fluid containing a set of poroelastic inclusions (suspension). For deriving the effective wave number, the theory of multiple scattering is used. It is shown that the effective wave number depends strongly on hydrodynamic permeability of inclusions and fluid properties in the inclusion pore space.

Quadrature Formulas of Gauss Type for a Sphere with Nodes Characterized by Regular Prism Symmetry

When the transport equation is solved by the discrete ordinate method, the problem arises of constructing quadrature formulas on a sphere characterized by the required accuracy and making it possible to use the quadrature nodes to approximate the transport equation in $$r,\;\vartheta ,\;z$$ geometry, in which quadrature nodes are simultaneously used to approximate the derivative with respect to the azimuth angle $$\varphi $$ of the transport equation, that is, must be located in levels on the sphere with the same values of the polar angle $$\theta $$ . An algorithm is considered to construct quadrature formulas of the needed form that are characterized by regular prism (dihedron) symmetry and exact for all spherical polynomials of degree not exceeding some maximal value $$L$$ . This study is a development of the work of A.N. Kazakov and V.I. Lebedev (1994). The constructed family of quadratures, unlike that in the above work, does not contain nodes with $$\varphi = 0,{\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2},\pi ,{{3\pi } \mathord{\left/ {\vphantom {{3\pi } 2}} \right. \kern-0em} 2}$$ , at the poles $$\theta = \pm {\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0em} 2}$$ , and on the equator $$\theta = 0$$ of the sphere. It is shown that this family ensures a significant computational gain when radiation transport problems are solved in three-dimensional geometry.