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The rotational excitation rates of NCCN species are studied for its collision with hydrogen (H2) in temperatures ranging from 1 to 100 K. Such collisions can occur in the interstellar medium with H2 in either para (p-) or ortho (o-) state, of which the p-H2 state can be approximated via its collision with He (using a scaling factor) or with a reduced rigid rotor-H2 surface (by averaging over various orientations of H2). In the current work, a four-dimensional (4D) ab initio potential energy surface (PES) is considered to study the collision dynamics of H2 in both p- and o-states and the results are compared with previous approximations. The 4D surface is constructed using the explicitly correlated coupled-cluster method CCSD(T)-F12b with the augmented triple zeta basis AVTZ and then fitted into an artificial neural networks (NN) model to augment the surface and account for missing data points. The radial coefficients are obtained from this NN fitted 4D PES via a least square fit over two spherical harmonics functions. The cross sections (σ) are computed using the close-coupling (CC) method (until 230 cm−1) for both p- and o-H2 collisions, and the rates are obtained by Boltzmann distribution over the translational energy of H2 until 100 K. The o-H2 rates are found to be higher by 25%–30% and 10%–20% compared to the p-H2 rates for Δj = 2 and higher order transitions, respectively. The coupled-state/centrifugal sudden approximated rates are also computed and found to have deviations as large as 40% when compared to CC rates, thus making quantitative descriptions unreliable. [ABSTRACT FROM AUTHOR]

Spherical harmonics provide a smooth, orthogonal, and symmetry-adapted basis to expand functions on a sphere, and they are used routinely in physical and theoretical chemistry as well as in different fields of science and technology, from geology and atmospheric sciences to signal processing and computer graphics. More recently, they have become a key component of rotationally equivariant models in geometric machine learning, including applications to atomic-scale modeling of molecules and materials. We present an elegant and efficient algorithm for the evaluation of the real-valued spherical harmonics. Our construction features many of the desirable properties of existing schemes and allows us to compute Cartesian derivatives in a numerically stable and computationally efficient manner. To facilitate usage, we implement this algorithm in sphericart, a fast C++ library that also provides C bindings, a Python API, and a PyTorch implementation that includes a GPU kernel. [ABSTRACT FROM AUTHOR]

Processing of single‐crystal X‐ray diffraction data from area detectors can be separated into two steps. First, raw intensities are obtained by integration of the diffraction images, and then data correction and reduction are performed to determine structure‐factor amplitudes and their uncertainties. The second step considers the diffraction geometry, sample illumination, decay, absorption and other effects. While absorption is only a minor effect in standard macromolecular crystallography (MX), it can become the largest source of uncertainty for experiments performed at long wavelengths. Current software packages for MX typically employ empirical models to correct for the effects of absorption, with the corrections determined through the procedure of minimizing the differences in intensities between symmetry‐equivalent reflections; these models are well suited to capturing smoothly varying experimental effects. However, for very long wavelengths, empirical methods become an unreliable approach to model strong absorption effects with high fidelity. This problem is particularly acute when data multiplicity is low. This paper presents an analytical absorption correction strategy (implemented in new software AnACor) based on a volumetric model of the sample derived from X‐ray tomography. Individual path lengths through the different sample materials for all reflections are determined by a ray‐tracing method. Several approaches for absorption corrections (spherical harmonics correction, analytical absorption correction and a combination of the two) are compared for two samples, the membrane protein OmpK36 GD, measured at a wavelength of λ = 3.54 Å, and chlorite dismutase, measured at λ = 4.13 Å. Data set statistics, the peak heights in the anomalous difference Fourier maps and the success of experimental phasing are used to compare the results from the different absorption correction approaches. The strategies using the new analytical absorption correction are shown to be superior to the standard spherical harmonics corrections. While the improvements are modest in the 3.54 Å data, the analytical absorption correction outperforms spherical harmonics in the longer‐wavelength data (λ = 4.13 Å), which is also reflected in the reduced amount of data being required for successful experimental phasing. [ABSTRACT FROM AUTHOR]

Positive semidefinite kernels on spheres cross time are on demand in spatial statistics, machine learning and numerical analysis, motivated by applications in the climate, environmental and earth sciences. Most of the recent literature has focused on kernels that are isotropic on the sphere and stationary in time, but this assumption is often overly limited in quantifying spatial dependence properly. A broader class of kernels can be obtained by trading isotropy for axial symmetry, i.e., stationarity with respect to longitudes only. This paper challenges theoretical aspects by providing a spectral representation that uniquely characterizes space-time positive semidefinite kernels that are axially symmetric on the sphere and stationary in time. We also address the problem of partial Fourier inversion and show that it is feasible under mild technical conditions. The second part of the paper focuses on constructive methods to build nonseparable kernels having axial symmetry on the sphere and stationarity over time. [ABSTRACT FROM AUTHOR]

Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine's, Poisson's, and Stokes's integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations. [ABSTRACT FROM AUTHOR]

The remove–compute–restore (RCR) approach is widely used in local quasi-geoid modeling. However, the classical RCR approach usually does not take into account the noise of the satellite-only global gravity field model (GGM), which may lead to a suboptimal result. This paper presents an approach for local quasi-geoid modeling based on spherical radial basis functions that combines local noisy datasets and a noisy satellite-only GGM. This approach includes an RCR procedure using a satellite-only GGM. This is a direct approach that takes the spherical harmonic coefficients of satellite-only GGM as a noisy dataset and includes the corresponding full-noise covariance matrix in the least-squares estimation, aiming to obtain a statistically optimal local quasi-geoid model. The direct approach goes beyond the indirect approach, which treats the height anomalies generated from the satellite-only GGM as a noisy dataset. However, the generated GGM height anomaly dataset is not an equivalent representation of the satellite-only GGM, which may result in the loss of information from the satellite-only GGM. Through mathematical deduction, we demonstrate the theoretical consistency between the direct approach and the indirect approach. The direct approach also has an advantage over the indirect approach in terms of computational complexity due to the simpler algorithm. We conducted a synthetic closed-loop test with a real data distribution in Colorado, and numerical results demonstrated the advantage of the direct approach in local quasi-geoid modeling. In terms of the root mean square of the differences between the predicted values and the true reference values, the direct approach provided an improvement of approximately 14% compared to the indirect approach. [ABSTRACT FROM AUTHOR]

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, ψℓm(τ)$$ {\psi}_{\mathit{\ell m}}^{\left(\tau \right)} $$, and vector, Mℓm(τ),Nℓm(τ)$$ {\mathbf{M}}_{\mathit{\ell m}}^{\left(\tau \right)},{\mathbf{N}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, Lℓm(τ)$$ {\mathbf{L}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed. [ABSTRACT FROM AUTHOR]

Traditionally, direction-of-arrival (DOA) estimations under near- and far-field scenarios are treated as independent tasks based on the corresponding acoustic model, hence necessitating a proper soundfield detector as an upstream processing tool, whereas there may not be a distinct boundary between different soundfield types, especially the mixed-field scenarios where both near- and far-field sources coexist simultaneously. To handle this issue, this article investigates a multisource DOA estimator that equally localizes multiple near-, far-, and mixed-field sources, not requiring any specialized adjustments. We (i) define a signal-invariant multichannel feature denoted generalized relative harmonic coefficients in the spherical harmonics domain; (ii) derive the analytical expression of this feature and summarize its unique properties, exhibiting consistence for both near- and far-field sources; (iii) estimate source elevation and azimuth using the magnitude and phase parts of this feature, respectively; (iv) detect single-source dominated periods from the mixed measurements based on an investigated distance measure; and (v) count the number of sources and localize their DOAs by clustering the single-source dominated estimates. Extensive experimental results, in both simulated and real-life environments, finally confirm the effectiveness of the proposed algorithm under diverse acoustic scenarios, and a superiority over baseline approaches in localizing mixed-field sources. [ABSTRACT FROM AUTHOR]

A new method for the fully generalized reconstruction of three-dimensional (3D) photoproduct distributions from velocity-map imaging (VMI) projection data is presented. This approach, dubbed Fourier–Hankel–Abel Nyquist-limited TOMography (FHANTOM), builds on recent previous work in tomographic image reconstruction [C. Sparling and D. Townsend, J. Chem. Phys. 157, 114201 (2022)] and takes advantage of the fact that the distributions produced in typical VMI experiments can be simply described as a sum over a small number of spherical harmonic functions. Knowing the solution is constrained in this way dramatically simplifies the reconstruction process and leads to a considerable reduction in the number of projections required for robust tomographic analysis. Our new method significantly extends basis set expansion approaches previously developed for the reconstruction of photoproduct distributions possessing an axis of cylindrical symmetry. FHANTOM, however, can be applied generally to any distribution—cylindrically symmetric or otherwise—that can be suitably described by an expansion in spherical harmonics. Using both simulated and real experimental data, this new approach is tested and benchmarked against other tomographic reconstruction strategies. In particular, the reconstruction of photoelectron angular distributions recorded in a strong-field ionization regime—marked by their extensive expansion in terms of spherical harmonics—serves as a key test of the FHANTOM methodology. With the increasing use of exotic optical polarization geometries in photoionization experiments, it is anticipated that FHANTOM and related reconstruction techniques will provide an easily accessible and relatively low-cost alternative to more advanced 3D-VMI spectrometers. [ABSTRACT FROM AUTHOR]

The transport of charged particles or photons in a scattering medium can be modelled with a Boltzmann equation. The mathematical treatment for scattering in such scenarios is often simplified if evaluated in a frame where the scattering centres are, on average, at rest. It is common therefore to use a mixed coordinate system, wherein space and time are measured in a fixed inertial frame, while momenta are measured in a 'co-moving' frame. To facilitate analytical and numerical solutions, the momentum dependence of the phase-space density may be expanded as a series of spherical harmonics, typically truncated at low order. A method for deriving the system of equations for the expansion coefficients of the spherical harmonics to arbitrary order is presented in the limit of isotropic, small-angle scattering. The method of derivation takes advantage of operators acting on the space of spherical harmonics. The matrix representations of these operators are employed to compute the system of equations. The computation of matrix representations is detailed and subsequently simplified with the aid of rotations of the coordinate system. The eigenvalues and eigenvectors of the matrix representations are investigated to prepare the application of standard numerical techniques, e.g. the finite volume method or the discontinuous Galerkin method, to solve the system. [ABSTRACT FROM AUTHOR]

An innovative method for synthesizing optimum difference patterns of the spherical sensor array is introduced, along with a sidelobe tapering technique. Firstly, we suggest employing the spherical harmonics of degree ±1 to synthesize the spherical array difference pattern; secondly, we study the mapping relationship between the difference pattern of the spherical sensor array and the difference pattern of the uniformly spaced linear array (ULA) with odd-numbered elements; finally, we enhance the Zolotarev difference pattern, which is a counterpart to the Dolph–Chebyshev sum pattern that traditionally allows synthesis only for ULA with even-numbered elements. Our modification extends its applicability to synthesize difference patterns for ULA with odd-numbered elements. Leveraging the optimal difference pattern, a generalized Bayliss difference pattern synthesis method designed for the ULA with odd-numbered elements is further proposed. To illustrate the effectiveness of our approach, we present several design examples through experimental simulation. [ABSTRACT FROM AUTHOR]

Two computational strategies for the evaluation of the spherical harmonic coefficients of the gravitational potential due to a generally shaped homogeneous polyhedral source are examined in detail. The techniques are implemented numerically for the known asteroid shape models of Eros and Didymos. The aim of the investigation is to quantify specific numerical aspects of the two algorithms, such as the accuracy of the techniques compared to a closed analytical solution for varying distance between source and computation point, the band-limited spectral analysis of the obtained spherical harmonic models and the convergence behavior of the corresponding series expansion in the vicinity of the characteristic Brillouin sphere. From a computational point of view, the line integral approach demands approximately three times the CPU time of Werner's method. The two sets of spherical harmonic coefficients are 100% correlated up to degree 45 for Eros and up to degree 49 for Didymos. Approaching degree 100, the correlation by degree decreases by 0.0004% for Eros and by 0.004% for Didymos, the corresponding values for the correlation by order being 0.0002% and 0.304%. Inside the Brillouin sphere and approaching its boundary, the numerical agreement of the gravitational potential between the line integral method and the analytical solution is at the 1E-4 level, while with Werner's approach at the 1E-7 level. At a distance of 33.5 km outside the Brillouin sphere for Eros and 2.2 km for Didymos, both methods are identical, reaching an agreement level with the analytical solution of 1E-11 level for Eros and 1E-14 for Didymos. In terms of spherical harmonic representation, the series defined by the line integral approach converges faster to the analytical value for the gravitational potential by 4 degrees. [ABSTRACT FROM AUTHOR]

Data harmonization is necessary for removing confounding effects in multi-site diffusion image analysis. One such harmonization method, LinearRISH, scales rotationally invariant spherical harmonic (RISH) features from one site ("target") to the second ("reference") to reduce confounding scanner effects. However, reference and target site designations are not arbitrary and resultant diffusion metrics (fractional anisotropy, mean diffusivity) are biased by this choice. In this work we propose MidRISH: rather than scaling reference RISH features to target RISH features, we project both sites to a mid-space. We validate MidRISH with the following experiments: harmonizing scanner differences from 37 matched patients free of cognitive impairment, and harmonizing acquisition and study differences on 117 matched patients free of cognitive impairment. We find that MidRISH reduces bias of reference selection while preserving harmonization efficacy of LinearRISH. Users should be cautious when performing LinearRISH harmonization. To select a reference site is to choose diffusion metric effect-size. Our proposed method eliminates the bias-inducing site selection step. [ABSTRACT FROM AUTHOR]

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a K -functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness. [ABSTRACT FROM AUTHOR]

Any complex-valued polynomial on (R n) k decomposes into an algebraic combination of O (n) -invariant polynomials and harmonic polynomials. This decomposition, separation of variables, is granted to be unique if n ≥ 2 k − 1. We prove that the condition n ≥ 2 k − 1 is not only sufficient, but also necessary for uniqueness of the separation. Moreover, we describe the structure of non-uniqueness of the separation in the boundary cases when n = 2 k − 2 and n = 2 k − 3. Formally, we study the kernel of a multiplication map ϕ carrying out separation of variables. We devise a general algorithmic procedure for describing Ker ϕ in the restricted non-stable range k ≤ n < 2 k − 1. In the full non-stable range n < 2 k − 1 , we give formulas for highest weights of generators of the kernel as well as formulas for its Hilbert series. Using the developed methods, we obtain a list of highest weight vectors generating Ker ϕ. [ABSTRACT FROM AUTHOR]

The paper studies discrete structural properties of polynomials that play an important role in the theory of spherical harmonics in any dimensions. These polynomials have their origin in the research on problems of harmonic analysis by means of generalized holomorphic (monogenic) functions of hypercomplex analysis. The Sturm‐Liouville equation that occurs in this context supplements the knowledge about generalized Vietoris number sequences Vn, first encountered as a special sequence (corresponding to n=2) by Vietoris in connection with positivity of trigonometric sums. Using methods of the calculus of holonomic differential equations, we obtain a general recurrence relation for Vn, and we derive an exponential generating function of Vn expressed by Kummer's confluent hypergeometric function. [ABSTRACT FROM AUTHOR]

The Weinstein equation tΔu+k∂u∂t=0, with k∈ℤ, considered in ℝ3=(x,y,t), is a modification of the classical Laplace equation Δu=0. Its solutions are called k‐modified harmonic functions. Whereas for positive integers k the Weinstein equation is relatively well understood, little is known if the parameter k is negative. The main result of this article is the statement that in case the negative integers are even, i.e., k=−2ℓ,ℓ∈ℕ, we still have a Fischer‐type decomposition. For k=0, the classical harmonic functions, this decomposition is well known. But also in case k∈ℕ, a Fischer‐type decomposition holds true, a Fischer‐type decomposition holds true. Surprisingly in case k=−3,k=−5, or k=−7 and probably in all higher negative odd cases, the decomposition doesn't hold. In case k=−1, we give a complete description of the vector space Hnk(ℝ3) of homogeneous k‐modified harmonic polynomials of degree n in ℝ3. Such a result is also at hand in case k∈ℕ. Finally, in case k=0 of the classical harmonic functions, we give a description of the vector space Hn(ℝ3)=Hn0(ℝ3). [ABSTRACT FROM AUTHOR]

Efficient computation of the geopotential gradient is essential for numerical propagators, particularly in scenarios involving low Earth orbits. Conventional geopotential calculations are based on spherical harmonics series, which become computationally demanding as the degree/order increases. This computational burden can be mitigated by means of parallelized algorithms. Additionally, certain situations lend themselves to high parallelization, such as the propagation of space debris catalogs, satellite mega-constellations, or the dispersion of particles resulting from a space collision event. This paper introduces an optimized Graphics Processing Unit (GPU) implementation designed to facilitate extensive parallelization in the geopotential gradient calculation. The formulation developed in this study is not specific to any GPU. However, to illustrate the low-level optimizations necessary for an efficient implementation, we selected the Compute Unified Device Architecture (CUDA) as the dominant and de facto standard in parallel computing. Nevertheless, most of the concepts and optimizations presented in this paper are also valid for other GPU architectures. Built upon the spherical harmonic expansion using the Cunningham formulation, which is well-suited for GPU computations, our implementation offers several variants with different tradeoffs between speed and accuracy. Besides GPU double precision, we introduced a mixed precision arithmetic –a hybrid between single and double precision– that exploits GPU capabilities with a low penalty in accuracy. The proposed algorithm was implemented as a software reusable module, and its performance was evaluated against GMAT, GODOT, and Orekit astrodynamic codes. The algorithm's accuracy in double precision is comparable to such codes. The mixed precision version showed enough accuracy for LEO satellite propagation, with around 1 m difference in four days. Testing across different CUDA architectures revealed very high speed-up factors compared to a single CPU, reaching a speed-up of 645 for the mixed precision variant and 450 for the double precision one in the propagation of about 3200 objects with a geopotential of degree/order 126 × 126 using an A100 GPU device. [ABSTRACT FROM AUTHOR]

In this paper we study critical sets of solutions uε$u_\varepsilon$ of second‐order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first‐order correctors, we show that the (d−2)$(d-2)$‐dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε$\varepsilon$, provided that doubling indices for solutions are bounded. The key step is an estimate of "turning" of an approximate tangent map, the projection of a non‐constant solution uε$u_\varepsilon$ onto the subspace of spherical harmonics of order ℓ$\ell$, when the doubling index for uε$u_\varepsilon$ on a sphere ∂B(0,r)$\partial B(0, r)$ is trapped between ℓ−δ$\ell -\delta$ and ℓ+δ$\ell +\delta$, for r$r$ between 1 and a minimal radius r∗≥C0ε$r^*\ge C_0\varepsilon$. This estimate is proved by using harmonic approximation successively. With a suitable L2$L^2$ renormalization as well as rescaling we are able to control the accumulated errors introduced by homogenization and projection. Our proof also gives uniform bounds for Minkowski contents of the critical sets. [ABSTRACT FROM AUTHOR]

Machine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighborhood of each atom in the system. Symmetry considerations support the use of spherical harmonics to expand the angular dependence of this density, but there is, as of yet, no clear rationale to choose one radial basis over another. Here, we investigate the basis that results from the solution of the Laplacian eigenvalue problem within a sphere around the atom of interest. We show that this generates a basis of controllable smoothness within the sphere (in the same sense as plane waves provide a basis with controllable smoothness for a problem with periodic boundaries) and that a tensor product of Laplacian eigenstates also provides a smooth basis for expanding any higher-order correlation of the atomic density within the appropriate hypersphere. We consider several unsupervised metrics of the quality of a basis for a given dataset and show that the Laplacian eigenstate basis has a performance that is much better than some widely used basis sets and competitive with data-driven bases that numerically optimize each metric. Finally, we investigate the role of the basis in building models of the potential energy. In these tests, we find that a combination of the Laplacian eigenstate basis and target-oriented heuristics leads to equal or improved regression performance when compared to both heuristic and data-driven bases in the literature. We conclude that the smoothness of the basis functions is a key aspect of successful atomic density representations. [ABSTRACT FROM AUTHOR]

Three- and four-center Coulomb integrals in the solid spherical harmonic Gaussian basis are solved by expansion in terms of two-center integrals. The two-electron Gaussian product rule, coupled with the addition theorem for solid spherical harmonics, reduces four-center Coulomb integrals into a linear combination of two-center Coulomb integrals and one-center overlap integrals. With this approach, three- and four-center Coulomb integrals can be reduced to the same form of two-center integrals. Resulting two-center Coulomb integrals can be further simplified into a simpler form, which can be related to the Boys function. Multi-center Coulomb integrals are solved hierarchically: simple two-center Coulomb integrals are used for calculation of more complicated two-center Coulomb integrals, which are used in the calculation of multicenter integrals. [ABSTRACT FROM AUTHOR]

It turns out that harmonic analysis on the superspace R m | 2 n is quite parallel to the classical theory on the Euclidean space R m unless the superdimension M := m − 2 n is even and non-positive. The underlying symmetry is given by the orthosymplectic superalgebra o s p (m | 2 n) . In this paper, when the symmetry is reduced to o s p (m − 1 | 2 n) we describe explicitly the corresponding branching laws for spherical harmonics on R m | 2 n also in exceptional cases, i.e, when M − 1 ∈ − 2 N 0 . In unexceptional cases, these branching laws are well-known and quite analogous as in the Euclidean framework. [ABSTRACT FROM AUTHOR]

In this paper, we consider a kind of singular integrals which appear in the generalized 2D dissipative quasi-geostrophic (QG) equation ∂ t ⁡ θ + u ⋅ ∇ ⁡ θ + κ ⁢ Λ 2 ⁢ β ⁢ θ = 0 , (x , t) ∈ ℝ 2 × ℝ + , κ > 0 , where u = - ∇ ⊥ ⁡ Λ - 2 + 2 ⁢ α ⁢ θ , α ∈ [ 0 , 1 2 ] and β ∈ (0 , 1 ] . First, we give a relationship between this kind of singular integrals and Calderón–Zygmund singular integral operators and obtain a uniform Besov estimates. As an application, we give the well-posedness of the generalized 2D dissipative quasi-geostrophic (QG) in the critical Besov space. [ABSTRACT FROM AUTHOR]

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier's separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions. [ABSTRACT FROM AUTHOR]

We present a novel method for estimating the magnetic field that can generate the desired magnetic force to move magnetic nanoparticles in a region of interest. The magnetic field is expanded using vector spherical harmonics, and the optimal multipole moments are estimated. The multipole moments are included in quadratic form because the magnetic force is defined as the product of the magnetic field and its gradient. We transform the system of quadratic equations into a rank-one-matrix estimation problem to construct a convex numerical algorithm based on a well-established proximal operation. The proposed method was validated using numerical simulations and can be used to design magnetic force patterns that cannot be generated using a simple coil pair. [ABSTRACT FROM AUTHOR]

Particle morphology at different length scales is important in understanding the mechanical behaviour of granular materials. In this sense, it is crucial to accurately describe and measure the size and shape of the grains using suitable definitions of morphological descriptors. Most of the research up until this point has analyzed particle shape in a two-dimensional framework, and sieving has typically been used to determine size. This paper describes the use of x-ray micro-computed tomography (µCT) which enables the visualization and quantification of three-dimensional particle morphology. Spherical harmonic analysis was used to reconstruct the three-dimensional (3D) realistic surface of the granular particles. 3D morphological descriptors were then introduced and computed to obtain the overall form, local features, and surface textures of the particle morphology based on the spherical harmonic reconstructed surface. To describe the fractal nature of the surfaces of natural sand particle morphology, the 3D fractal dimension was quantified using spherical harmonic-based fractal analysis. Complete volume-based distributions of particle morphological descriptors were presented and compared for four different sand samples with different grain size and shape characteristics. According to the statistical analysis, there is a clear correlation between the shape parameters at various characteristic scales, indicating that they are not independent measures. The correlation between any two parameters was observed to rely on the distance between the characteristic scales of the morphological parameters. [ABSTRACT FROM AUTHOR]

• A particle model of interaction of weakly non-spherical gas bubbles is developed. • Compared with available analogs, the model allows smaller inter-bubble distances. • The model shows good agreement with the results by boundary element method. • The model demonstrates good agreement with available experimental data. • Bubble dynamics in rectilinear, planar and spatial clusters is compared. A mathematical model of interaction of weakly non-spherical gas bubbles in liquid is proposed. It is a system оf the second order ODEs in the radii of bubbles, the position vectors of their centers and the amplitudes of their small deformations in the form of spherical harmonics. Compared to the available analogs, the proposed model equations are more accurate in terms of the ratio of the radii of bubbles to the distance between their centers. This allows one to study bubble-bubble interaction at smaller distances between the bubbles. The model equations are quite compact, which simplifies their theoretical analysis and construction of algorithms of their solution. Five problems (dynamics of a single bubble under harmonic excitation, bubble collapse near a plane rigid wall, interaction of three synchronized and nonsynchronized bubbles located in line, and interaction between sixteen bubbles with the centers located on a plane surface) are considered for validation. It is shown that in all cases the proposed model results well agree with the corresponding available experimental data and numerical solutions. For demonstration of the applicability, the proposed model is used to analyze radial oscillations, spatial movements and surface deformations of the bubbles in rectilinear, planar and spatial clusters under harmonic forcing. Initially 125, 121 and 125 bubbles of the clusters are located at the nodes of 1D, 2D and 3D uniform grids, respectively. It is found that the monotonically decaying radial oscillations of the 1D cluster bubbles during one period of forcing are quite similar to those of a single bubble. Unlike this, the attenuation of the radial oscillations of the central bubbles in the 2D and 3D clusters is essentially non-monotonic. The maximum pressures in the central bubbles of the 2D and 3D clusters are significantly higher than in the single and 1D cluster bubbles. In all clusters the maximum deformations of bubbles are in the form of the second spherical harmonics, except for the central bubble of the 3D cluster, the maximum deformations of which are in the form of the fourth harmonics. [ABSTRACT FROM AUTHOR]

The broadband constant beam pattern (CBP) spherical array synthesis theory is applied to suppress or to cancel wideband three-dimensional interfering noise sources by implanting zeros in the array's original synthesized shading function. This modified array angular shading function can be expanded by a series of spherical harmonics that will converge to the beam pattern in the far-field such that the created nulls are in the wideband interfering noise source directions per the CBP theory where the ratio of the spherical array radius to the operating frequency wavelength is large. The simulated numerical examples given for this wideband noise source suppression method demonstrate that a broadband CBP performance is maintained for the spherical array with a Legendre polynomial shading function, the classic Dolph–Chebyshev shading function, or a combination. With the CBP design, one set of the real shading functions works for all frequencies in the array's operating band to cancel or to suppress three-dimensional wideband interfering noise sources. [ABSTRACT FROM AUTHOR]

The Dawn spacecraft approached the asteroid Vesta and descended from a high-altitude mission orbit to a low-altitude mission orbit using low-thrust propulsion. During this descent, the spacecraft crossed the 2:3 and 1:1 ground-track resonances with Vesta, which posed a risk of capture that might strongly perturb the spacecraft's orbit. This study analyzes the effects of these resonances on the spacecraft's orbital elements and estimates the probability of capture into it through Monte Carlo simulations. Specifically, a comprehensive investigation is performed to assess the effects of 1:1 and 2:3 ground-track resonances on the semimajor axis, eccentricity, and inclination. The dynamical model includes the gravitational field of Vesta using a spherical harmonics approximation up to the fourth degree and order and the low-thrust acceleration that is assumed to be opposite to the spacecraft's velocity vector direction. It is observed that the eccentricity evolution is mostly influenced by the 2:3 ground-track resonance which results in a large variation when the spacecraft crosses that ground-track resonance, while the semimajor axis and inclination are mostly influenced by the 1:1 ground-track resonance. Then, the probability of capture into 1:1 ground-track resonance is found to have a negative correlation with the spacecraft's thrust magnitude and the probability of capture into 2:3 ground-track resonance is found to arise as the spacecraft's mass increases. It is found that for circular orbits below a certain inclination value the spacecraft's trajectory is subject to the "automatic entry into libration" phenomenon, due to the singularity in the Hamiltonian function. This research contributes to the design of successful transfer strategies when crossing resonance for future missions. [ABSTRACT FROM AUTHOR]

Cell shape is a powerful readout of cell state, fate and function. We describe a custom workflow to perform semi-automated, 3D cell and nucleus segmentation, and spherical harmonics and principal components analysis to distill cell and nuclear shape variation into discrete biologically meaningful parameters. We apply these methods to analyze shape in the neuromast cells of the zebrafish lateral line system, finding that shapes vary with cell location and identity. The distinction between hair cells and support cells accounted for much of the variation, which allowed us to train classifiers to predict cell identity from shape features. Using transgenic markers for support cell subpopulations, we found that subtypes had different shapes from each other. To investigate how loss of a neuromast cell type altered cell shape distributions, we examined atoh1a mutants that lack hair cells. We found that mutant neuromasts lacked the cell shape phenotype associated with hair cells, but did not exhibit a mutant-specific cell shape. Our results demonstrate the utility of using 3D cell shape features to characterize, compare and classify cells in a living developing organism. [ABSTRACT FROM AUTHOR]

Our work utilizes the quantum model of the hydrogen atom which is based on the Schrödinger equation with Coulomb potential. Specifically, we concentrate on the angular components of the wave eigenfunctions derived from this model. We consider the quantum states with n = 4. In order to visualize the orbital shapes of these states, we built in the spherical coordinates system their 3D geometric representations. Furthermore, we use the corresponding spherical harmonics, to calculate the - nodal values that describe the configurations of these orbital states. [ABSTRACT FROM AUTHOR]

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

We discuss strong local and global well–posedness for the three–dimensional NLS equation with nonlinearity concentrated on S 2 . Precisely, local well–posedness is proved for any C 2 power–nonlinearity, while global well–posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point–concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas. [ABSTRACT FROM AUTHOR]

Monitoring urban heat island (UHI) effect is critical because it causes health problems and excessive energy consumption more energy when cooling buildings. In this study, we propose an approach for UHI monitoring by fusing data from ground-based global navigation satellite system (GNSS), space-based GNSS radio occultation (RO), and radiosonde. The idea of the approach is as follows: First, the first and second grid tops are defined based on historical RO and radiosonde observations. Next, the wet refractivities between the first and second grid tops are fitted to higher-order spherical harmonics and they are used as the inputs of GNSS tomography. Then, the temperature and water vapor partial pressure are estimated by using best search method based on the tomography-derived wet refractivity. In the end, the UHI intensity is evaluated by calculating the temperature difference between the urban regions and nearby rural regions. Feasibility of the UHI intensity monitoring approach was evaluated with GNSS RO and radiosonde data in 2010–2019, as well as ground-based GNSS data in 2020 in Hong Kong, China, by taking synoptic temperature data as reference. The result shows that the proposed approach achieved an accuracy of 1.2 K at a 95% confidence level. [ABSTRACT FROM AUTHOR]

We analyze the well-posedness of certain field-only boundary integral equations (BIEs) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x) and (2) scalar quantity E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x)\cdot x are radiative solutions of the Helmholtz equation, we see that novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIEs. In this paper we use the combined field methodology of Burton and Miller within the fieldonly BIE approach, and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are nonzero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one--a property similar to second-kind integral equations. [ABSTRACT FROM AUTHOR]

Electron Probe Microanalysis (EPMA) is a nondestructive technique to determine the chemical composition of material samples in the micro- to nanometer range. Based on intensity measurements of x-radiation, the reconstruction of the material composition and structure poses an inverse problem. The reconstruction methods currently applied are based on models that assume a homogeneous or a layered structure of the studied material. To increase the spatial resolution of reconstruction in EPMA the combination of a more sophisticated reconstruction method, which is based on a model that allows complex material structure, together with multiple measurements with varying beam configurations is required. We present a deterministic k-ratio model that is based on the PN model, an approximation of the radiative transfer equation for electron transport. Our goal is to approximate a maximum likelihood solution of the inverse problem using gradient-based optimization. We detail the application of the model in the context of algorithmic differentiation, in particular by deriving its continuous adjoint formulation. Algorithmic differentiation provides the flexibility to adapt the reconstruction method to various material parametrizations and thus to regularize and take into account prior knowledge. Through examples, we verify our implementation and demonstrate the flexibility of the reconstruction/differentiation method. [ABSTRACT FROM AUTHOR]

This paper proposes the first linear scaling implementation for the domain decomposition approach of the polarizable continuum model (ddPCM) for the computation of the solvation energy and forces. The ddPCM-equation consists of a (non-local) integral equation on the van der Waals or solvent accessible surface of the solute's cavity resulting in a dense solution matrix, and, in turn, one matrix–vector multiplication has a quadratic arithmetic complexity with respect to the number of atoms of the solute molecule. The use of spherical harmonics as basis functions makes it natural to employ the fast multipole method (FMM) in order to provide an asymptotically linear scaling method. In this paper, we employ the FMM in a non-uniform manner with a clusterization based on a recursive inertial bisection. We present some numerical tests illustrating the accuracy and scaling of our implementation. [ABSTRACT FROM AUTHOR]

A solution to the electrostatic problem for the spheroidal particle with a spherical core is constructed. To involve the problem geometry in a full manner, in a vicinity of the particle surface the potentials of the fields are represented by their expansions in terms of spheroidal harmonics of Laplace equation, while in a vicinity of the core surface by the expansions in terms of spherical harmonics. Matching of the fields inside the particle shell is made by using the relations between the spheroidal and spherical harmonics. The T-matrix relates the coefficients of expansions of the incident and "scattered" fields. In the present paper, both the particle polarizability related to the main matrix element T11, and the whole T-matrix are considered. The symmetry of the matrix as well as its dependence on the size of the layered particle are shown. A relationship between the T-matrices in the spherical and spheroidal systems was also found. Numerical calculations made for particles of small and large aspect ratios (a/b = 1.5 − 5.0) confirmed high efficiency of the suggested solution in contrast to the methods that use a single spherical basis. [ABSTRACT FROM AUTHOR]

We analyze the directional dependence of the gravitational wave (GW) emission from 15 3D neutrino radiation hydrodynamic simulations of core-collapse supernovae (CCSNe). Using spin weighted spherical harmonics, we develop a new analytic technique to quantify the evolution of the distribution of GW emission over all angles. We construct a physics-informed toy model that can be used to approximate GW distributions for general ellipsoid-like systems, and use it to provide closed form expressions for the distribution of GWs for different CCSN phases. Using these toy models, we approximate the protoneutron star (PNS) dynamics during multiple CCSN stages and obtain similar GW distributions to simulation outputs. When considering all viewing angles, we apply this new technique to quantify the evolution of preferred directions of GW emission. For nonrotating cases, this dominant viewing angle drifts isotropically throughout the supernova, set by the dynamical timescale of the PNS. For rotating cases, during core bounce and the following tens of milliseconds, the strongest GW signal is observed along the equator. During the accretion phase, comparable—if not stronger—GW amplitudes are generated along the axis of rotation, which can be enhanced by the low T /∣ W ∣ instability. We show two dominant factors influencing the directionality of GW emission are the degree of initial rotation and explosion morphology. Lastly, looking forward, we note the sensitive interplay between GW detector site and supernova orientation, along with its effect on detecting individual polarization modes. [ABSTRACT FROM AUTHOR]

In this paper we study the large time asymptotic behaviour of the heat equation with Hardy inverse-square potential on corner spaces R N − k × (0 , ∞) k , k ≥ 0. We first show a new improved Hardy-Poincaré inequality for the quantum harmonic oscillator with Hardy potential. In view of that, we construct the appropriate functional setting in order to pose the Cauchy problem. Then we obtain optimal polynomial large time decay rates and subsequently the first term in the asymptotic expansion of the solutions in L 2 (R N − k × (0 , ∞) k). Particularly, we extend and improve the results obtained by Vázquez and Zuazua (J. Funct. Anal. 2000), which correspond to the case k = 0 , to any k ≥ 0. We emphasize that the higher the value of k the better time decay rates are. We employ a different and simplified approach than Vázquez and Zuazua, managing to remove the usage of spherical harmonics decomposition in our analysis. [ABSTRACT FROM AUTHOR]

In this article, we present Kinetic Transport Solver for Radiation Therapy (KiT-RT), an open source C++-based framework for solving kinetic equations in therapy applications available at https://github.com/CSMMLab/KiT-RT. This software framework aims to provide a collection of classical deterministic solvers for unstructured meshes that allow for easy extendability. Therefore, KiT-RT is a convenient base to test new numerical methods in various applications and compare them against conventional solvers. The implementation includes spherical harmonics, minimal entropy, neural minimal entropy, and discrete ordinates methods. Solution characteristics and efficiency are presented through several test cases ranging from radiation transport to electron radiation therapy. Due to the variety of included numerical methods and easy extendability, the presented open source code is attractive for both developers, who want a basis to build their numerical solvers, and users or application engineers, who want to gain experimental insights without directly interfering with the codebase. [ABSTRACT FROM AUTHOR]

Acoustic tweezers offer a contactless, three-dimensional, and selective approach to trapping objects by harnessing the acoustic radiation force. Precise control of this technique requires accurate calibration of the force, which depends on the object's properties and the spherical harmonics expansion of the incident field through the beam shape coefficients. Previous studies showed that these coefficients can be determined using either the Lebedev quadrature or the angular spectrum methods. However, the former is highly susceptible to noise, while the latter demands extensive implementation time due to the number of required measurement points. A filtered method with a reduced number of points is introduced to address these limitations. Initially, we emphasize the implicit filtering in the angular spectrum method, allowing relative noise insensitivity. Subsequently, we present its unfiltered version, enabling force estimation of a standing field. Finally, we develop a filtered method based on the Lebedev quadrature, requiring fewer points, and apply it to focused vortex beams. Numerical evaluation of the radiation force demonstrates the method's resilience to noise and a reduced need for points compared to previous methods. The filtered Lebedev method paves the way for characterizing high-frequency acoustic tweezers, where measurement constraints necessitate rapid and robust beam shape coefficient estimation techniques. [ABSTRACT FROM AUTHOR]

In this work, we extend the analysis of the relativistic Dirac–Rosen–Morse problem in curved space–time. For that, we consider the Dirac equation in curved space–time with line element ds2 = (1 + α2U(r))2(dt2 − dr2) − r2dθ2 − r2sin 2θdϕ2, where α is fine structural constant, U(r) is a scalar potential, and in the presence of the electromagnetic field Aμ = (V(r), cA(r), 0, 0). Because of the spherical symmetry, the angular spinor is given in terms of the spherical harmonics. For the radial spinor, we apply a unitary transformation and define the vector component of the electromagnetic field A(r) written as a function of V(r) and U(r) so as to solve the radial spinor for Dirac–Rosen–Morse problem. Graphical analyses were performed comparing the eigenenergies and the probability densities in curved and flat space–time to visualize the influence of curvature in space–time on the two-component radial spinor, with the upper and lower components representing the particle and antiparticle, respectively. [ABSTRACT FROM AUTHOR]

Although observation of gravity perturbations induced by earthquakes is possible, simulation of seismic wave propagation in a self-gravitating, rotating Earth model with 3-D heterogeneity is challenging due to the numerical complexities associated with the unbounded Poisson/Laplace equation that governs gravity perturbations. Therefore, gravity perturbations are generally omitted, and only the background gravity is taken into account using the so-called Cowling approximation. However, gravity perturbations may be significant for large earthquakes (M w ≥ 6.0) and long-period responses. In this study, we develop a time-domain solver based on the spectral-infinite-element approach, which combines the spectral element method inside the Earth domain with a mapped-infinite-element method in the infinite space outside. This combination allows us to solve the complete, coupled momentum-gravitational equations in a fully discretized domain while accommodating complex 3-D Earth models. We compute displacement and gravity perturbations considering various Earth models, including Preliminary Reference Earth Model and S40RTS and conduct comprehensive benchmarks of our method against the spherical harmonics normal-mode approach and the direct radial integration method. Our 3-D simulations accommodate topography, bathymetry, rotation, ellipticity and oceans. Results show that our technique is accurate and stable for long simulations. Our method provides a new scope for incorporating earthquake-induced gravity perturbations into source and adjoint tomographic inversions. [ABSTRACT FROM AUTHOR]

We study gravitational absorption effects using effective on-shell scattering amplitudes. We develop an in-in probability-based framework involving plane- and partial-wave coherent states for the incoming wave to describe the interaction of the wave with a black hole or another compact object. We connect this framework to a simplified single-quantum analysis. The basic ingredients are mass-changing three-point amplitudes, which model the leading absorption effects and a spectral-density function of the black hole. As an application, we consider a non-spinning black hole that may start spinning as a consequence of the dynamics. The corresponding amplitudes are found to correspond to covariant spin-weighted spherical harmonics, the properties of which we formulate and make use of. We perform a matching calculation to general-relativity results at the cross-section level and derive the effective absorptive three-point couplings. They are found to behave as O G Newton s + 1 , where s is the spin of the outgoing massive state. [ABSTRACT FROM AUTHOR]

A function f$$ f $$ from a domain in ℝ3$$ {\mathrm{\mathbb{R}}}&amp;#x0005E;3 $$ to the quaternions is said to be inframonogenic if ∂‾f∂‾=0$$ \overline{\partial}f\overline{\partial}&amp;#x0003D;0 $$, where ∂‾=∂/∂x0+(∂/∂x1)e1+(∂/∂x2)e2$$ \overline{\partial}&amp;#x0003D;\partial /\partial {x}_0&amp;#x0002B;\left(\partial /\partial {x}_1\right){e}_1&amp;#x0002B;\left(\partial /\partial {x}_2\right){e}_2 $$. All inframonogenic functions are biharmonic. In the context of functions f=f0+f1e1+f2e2$$ f&amp;#x0003D;{f}_0&amp;#x0002B;{f}_1{e}_1&amp;#x0002B;{f}_2{e}_2 $$ taking values in the reduced quaternions, we show that the inframonogenic homogeneous polynomials of degree n$$ n $$ form a subspace of dimension 6n+3$$ 6n&amp;#x0002B;3 $$. We use the homogeneous polynomials to construct an explicit, computable orthogonal basis for the Hilbert space of square‐integrable inframonogenic functions defined in the ball in ℝ3$$ {\mathrm{\mathbb{R}}}&amp;#x0005E;3 $$. [ABSTRACT FROM AUTHOR]

We extend the modal decomposition method, previously applied to compress the information in the real-space bispectrum to the anisotropic redshift-space galaxy bispectrum. In the modal method approach, the bispectrum is expanded on a basis of smooth functions of triangles and their orientations, such that a set of modal expansion coefficients can capture the information in the bispectrum. We assume a reference survey and compute Fisher forecasts for the compressed modal bispectrum and two other basis decompositions of the redshift-space bispectrum in the literature, one based on (single) spherical harmonics and another based on tripolar spherical harmonics. In each case, we compare the forecasted constraints from the compressed statistic with forecasted constraints from the full uncompressed bispectrum which includes all triangles and orientations. Our main result is that all three compression methods achieve good recovery of the full information content of the bispectrum, but the modal decomposition approach achieves this the most efficiently: only 14 (42) modal expansion coefficients are necessary to obtain constraints that are within 10 (2) per cent of the full bispectrum result. The next most efficient decomposition is the one based on tripolar spherical harmonics, while the spherical harmonic multipoles are the least efficient. [ABSTRACT FROM AUTHOR]

We present a high‐accuracy numerical method based on a decoupled dimensionality reduction scheme for Maxwell eigenvalue problem in spherical domains. Using the orthogonality of vector spherical harmonics and the variable separation approach, we decompose the original problem into two classes of decoupled one‐dimensional TE mode and TM mode. For the TE mode, we establish a variational formulation and its discrete scheme and give the error estimations of the approximate eigenvalues and eigenfunctions. For the TM mode, it is different from TE mode which naturally meets the divergence‐free condition and will not generate some spurious eigenvalues. We design a numerical algorithm based on a parameterized method to filter out the spurious eigenvalues. Finally, some numerical results are presented to confirm the theoretical results and validate the algorithms. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and ℓ p estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on ℓ p spaces. [ABSTRACT FROM AUTHOR]