Spherical Basis Research Articles

Implementation of the BIRCH algorithm to construct a data-adaptive network design for regional gravity field modeling via SRBF

This study presents a new methodology to establish an optimal network for the spherical radial basis functions (SRBFs) by employing the Balanced Iterative Reducing and Clustering using Hierarchies (BIRCH) algorithm. In the proposed methodology, sub-cluster centers obtained by the BIRCH algorithm are replaced with the center of the SRBFs. Since the horizontal positions of the observations are utilized in the clustering, the SRBFs are distributed adaptively to the data. The algorithm’s performance and the effects of the BIRCH parameters are investigated in detail with real and simulated data sets in the Auvergne and Colorado areas, respectively. The bandwidth of each SRBF is determined by the generalized cross-validation (GCV) technique. The turning point algorithm is employed to reduce long-wavelength errors that occur due to the always positivity of the selected Legendre coefficients in the spatial domain. The outcomes of the numerical tests show that only one parameter (threshold) is enough to construct a proper data-adaptive network design for SRBFs. Compared to existing algorithms, fewer SRBFs are required to achieve the same accuracy on the control points while saving more than 95% of the time in the network design. Furthermore, the proposed methodology improves the condition number of the normal equation matrix. That makes it possible to estimate unknown coefficients without regularization in the least-square procedure depending on the selected threshold parameter. Therefore, the BIRCH algorithm is very effective and suitable to establish an optimal data-adaptive network design, especially in large data sets.

Microscopic description of the pygmy and giant dipole resonances in even–even nuclei. Application to the isotopes 150,152,154,156,158,160Gd

This paper deals with a microscopic approach for the description of the pygmy dipole resonance (PDR) as well as of the giant dipole resonance (GDR). The formalism is based on a Hamiltonian which is summing a mean field term, the pairing for alike nucleons and the dipole–dipole interaction. Two features are specific for our description: (a) The use of a projected spherical single particle basis and (b) the involved dipole operator is a Schif-like operator including a corrective term which is cubic in the radial coordinate. The dipole strength distribution with energy is plotted for the PDR region, i.e., [0,10] MeV. The dominant transitions have an isovector character for three isotopes and isoscalar for another three. The PDR states carry only 0.4–1.8[Formula: see text] of the total EWSR. The dependence of the dipole strength on nuclear deformation is evidenced. For the whole interval of [0,20] MeV the dipole strength was first folded by Lorentzian of 1[Formula: see text]MeV width and then plotted as function of the QRPA energies. The peaks belonging to each of the two ranges are analyzed in detail and their nature, isovector/isoscalar, was pointed out. The r-cubic term and the nuclear deformation have opposite effects on the dipole strength. The famous Thomas–Reiche–Kuhn sum rule formula is generalized to the case of the Schiff dipole momentum. The new EWSR is very well satisfied. The photoabsorbtion cross-section is calculated and compared with experimental data in a figure showing its dependence on energy. The total cross-section, [Formula: see text], the moments of the integrated cross-sections, [Formula: see text], [Formula: see text], the dipole polarizability and the thickness of the neutron crust are also calculated. The main features of PDR and GDR are realistically described.

Spherical basis functions in Hardy spaces with localization constraints

Subspaces obtained by the orthogonal projection of locally supported square-integrable vector fields onto the Hardy spaces H+(S) and H−(S), respectively, play a role in various inverse potential field problems since they characterize the uniquely recoverable components of the underlying sources. Here, we consider approximation in these subspaces by a particular set of spherical basis functions. Error bounds are provided along with further considerations on norm-minimizing vector fields that satisfy the underlying localization constraint. The new aspect here is that the used spherical basis functions are themselves members of the subspaces under consideration.

Asymmetry of the Frontal Aslant Tract and Development of Supplementary Motor Area Syndrome.

Background/Objectives: The purpose of this study was to investigate preoperative interhemispheric differences of the FAT in relation to the onset of postoperative SMA syndrome. Methods: This was a single-center retrospective analysis of patients who underwent surgical resection of diffuse gliomas involving the SMA between 2018 and 2022. Inclusion criteria were availability of preoperative and postoperative Magnetic Resonance Imaging, no previous surgery, and no neurological deficits at presentation. Diffusion-weighted data were processed by spherical deconvolution (SD) and diffusion tensor imaging tractography algorithms, and TrackVis was used to dissect the FAT of both hemispheres. The FAT data were analyzed for correlation with postoperative SMA syndrome onset. Results:N = 25 cases were included in the study, among which n = 23 had preoperative bilaterally identifiable FAT by SD. N = 12 developed an SMA syndrome, 6 demonstrated a motor-only syndrome, 4 had a verbal-only syndrome, and 2 had mixed verbal and motor features. The SMA syndrome incidence was significantly more frequent in lower-grade gliomas (p = 0.005). On the tumor side, the FAT identified by SD was smaller than the contralateral (mean volume 6.53 cm3 and 13.33 cm3, respectively, p < 0.001). In the 6 cases that developed a verbal SMA syndrome, a normalized FAT volume asymmetry (FAT-VA) demonstrated an asymmetry shifted towards the non-dominant side (mean FAT-VA = -0.68), while the cases with no postoperative verbal impairment had opposite asymmetry towards the dominant side (mean FAT-VA = 0.42, p = 0.010). Conclusions: Preoperative interhemispheric FAT volume asymmetry estimated according to functional dominance can predict postoperative onset of verbal SMA syndrome, with proportionally smaller FAT on the affected dominant hemisphere.

Exact modeling of power spectrum multipole through spherical Fourier-Bessel basis

Exact modeling of power spectrum multipole through spherical Fourier-Bessel basis

Soft x-ray tomography on the high field spherical tokamak ST40.

As part of its roadmap to developing commercial fusion plants, Tokamak Energy Ltd. operates the high field spherical tokamak ST40. Studies on this device will help to expand the high field spherical tokamak physics basis by characterizing confinement and the fusion triple product. In support of this, bolometers and broadband and x-ray sensitive diodes can provide information on key energy loss mechanisms of the plasma. These mechanisms include core magnetohydrodynamic activity that deteriorates confinement, such as sawtooth crashes that can be used to characterize relaxations in the q-profile. In addition, combinations of these diagnostics can be used to infer the total radiated power losses and plasma composition. Here, we present results from a new, midplane, tangential, Be-filtered diode with 16 channels spanning the radial extent of the plasma. The system is shown to resolve magnetohydrodynamic instabilities (up to 100kHz) and be able to provide radiation profiles through tomography. The tomographic inversion routine is compared against other diagnostics on ST40 and provides emissivity measurements across a variety of operating scenarios. Finally, we look ahead to implementing multiple soft x-ray cameras on ST40 and the improvements this will have on the diagnostic capabilities.

Reduced form of statistical equilibrium equations and radiative transfer equations

Abstract An extension of the vector algebra for irreducible spherical tensor operators (ITOs) is proposed, which involves coupling two ITOs into a third one (V (kv)= T (kt)× U (ku)*), with the additional condition that one of the operators is a complex conjugate and therefore is not an ITO. The correctness condition is considered for this extension. The new coupling rule treats the density matrices ρ(Kl) and all tensors related to light polarization (J (Kr) or Τ (K)) equally.&amp;#xD;This approach has been applied to rewrite the fundamental statistical equilibrium equations and radiation transfer equations relevant to astrophysical polarimetry. The revised equations are more concise and eliminate the need for the nj symbols. Instead, they utilize vector and scalar products of ITOs. All rate coefficients in the equations consist of two independent factors, the reduced matrix element from dipole moments and the product (vector or scalar) of the tensors ρ(Kl) and J (Kr) (or Τ (K)).

Nonlinear Optics Through the Field Tensor Formalism

AbstractThe “field tensor” is the tensor product of the electric fields of the interacting waves during a sum‐ or difference‐frequency generation nonlinear optical interaction. It is therefore a tensor describing light interacting with matter, the latter being characterized by the “electric susceptibility tensor.” The contracted product of these two tensors of equal rank gives the light‐matter interaction energy, whether or not propagation occurs. This notion having been explicitly or implicitly present from the early pioneering studies in nonlinear optics, its practical use has led to original developments in many highly topical theoretical or experimental situations, at the microscopic as well macroscopic level throughout a variety of coherent or non‐coherent processes. The aim of this review article is to rigorously explain the field tensor formalism in the context of tensor algebra and nonlinear optics in terms of a general time‐space multi‐convolutional development, using spherical tensors, with components expressed in the frame of a common basis set of irreducible tensors, or Cartesian tensors. A wide variety of media are considered, including biological tissues and their imaging, artificially engineered by various combinations of optical and static electric fields, with the two extremes of all‐optical and purely electric poling, and also bulk single crystals.

To justification of concrete strength criterion at biaxial compression

Introduction. Numerous experimental data from Russian and foreign researchers indicate that the classical plasticity hypotheses do not take into account the different resistance to uniaxial tension and compression, the influence of the spherical tensor. At the same time, experiments show that the ultimate resistance depends on the type of stress state, and hydrostatic pressure increases the strength and plasticity of solids. Aim. To establish the dependence of the influence of the second component of stresses during biaxial compression of concrete on the parameters of the complete diagrams of deformation of the material σbR and εbR it is necessary to describe these diagrams, and to construct a closed curve on the plane of the main stresses (concrete strength criterion). Materials and methods. Based on the experimental materials of foreign and domestic researchers, including the experiments of the authors of the article, methods of mechanics of deformed solids, limit curves and a closed curve on the plane of the main stresses in the form of a chain line forming the smallest area strength surface in the form of a catenoid are proposed. Results. The article provides an analysis of the known strength criteria from the point of view of their geometric interpretation in the stress space. It is shown that these studies relate mainly to metals and metal structures, and for the design of reinforced concrete and steel-concrete structures in a complex stress state, it is necessary to develop an appropriate criterion for the strength of concrete. Conclusions. As a result, the proposed limit curves and the surface (material strength criterion) on the plane of main stresses in the form of a catenoid accurately reflect the behavior of concrete under conditions of uniform and uneven flat stress state, and the equation of the surface in the form of a catenoid is a generalization of the equations of limit curves for each of the three types of flat stress state. At the same time, there is no overestimation of strength in the "compression – compression" area in this case.

Fault-Tolerant Quantum Computation Using Large Spin-Cat Codes

We construct a fault-tolerant quantum error-correcting protocol based on a qubit encoded in a large spin qudit using a spin-cat code, analogous to the continuous-variable cat encoding. With this, we can correct the dominant error sources, namely processes that can be expressed as error operators that are linear or quadratic in the components of angular momentum. Such codes tailored to dominant error sources can exhibit superior thresholds and lower resource overheads when compared to those designed for unstructured noise models. A key component is the gate that preserves the rank of spherical tensor operators. Categorizing the dominant errors as phase and amplitude errors, we demonstrate how phase errors, analogous to phase-flip errors for qubits, can be effectively corrected. Furthermore, we propose a measurement-free error-correction scheme to address amplitude errors without relying on syndrome measurements. Through an in-depth analysis of logical gate errors, we establish that the fault-tolerant threshold for error correction in the spin-cat encoding surpasses that of standard qubit-based encodings. We consider a specific implementation based on neutral-atom quantum computing, with qudits encoded in the nuclear spin of 87Sr, and show how to generate the universal gate set, including the rank-preserving gate, using quantum control and the Rydberg blockade. These findings pave the way for encoding a qubit in a large spin with the potential to achieve fault tolerance, high threshold, and reduced resource overhead in quantum information processing. Published by the American Physical Society 2024

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

A Direct Approach for Local Quasi-Geoid Modeling Based on Spherical Radial Basis Functions Using a Noisy Satellite-Only Global Gravity Field Model

Differentiable design of a double-freeform lens with multi-level radial basis functions for extended source irradiance tailoring

Freeform optics are key for generating prescribed illumination patterns from given sources, which are crucial for solid-state lighting and machine vision illumination. There is an increasing demand for compact freeform optics, which presents a substantial challenge for current design methods since the source dimensions must be considered. Most current extended-source design methods, although requiring profound knowledge of optics and mathematics, focus on the modest goal of obtaining uniform irradiance distributions. We address a more challenging design problem of generating an irradiance distribution of arbitrary shape through a double-freeform lens that can fully encompass the extended source. We propose a differentiable design method whose uniqueness lies in the representation of the double-freeform surfaces using multi-level spherical radial basis functions, which has a natural link to a multi-scale optimization technique. In addition, we employ a sequential unconstrained minimization technology complemented with Lagrange multipliers that add key feasibility constraints on lens shape and size. The proposed method is flexible, general, and efficient in designing highly compact freeform lenses for generating both simple and complex irradiance distributions, as demonstrated through the design examples. This could enable a universal solution to the extended-source design problem.

A data-adaptive network design for the regional gravity field modelling using spherical radial basis functions

A high-precision regional gravity field model is significant in various geodesy applications. In the field of modelling regional gravity fields, the spherical radial basis functions (SRBFs) approach has recently gained widespread attention, while the modelling precision is primarily influenced by the base function network. In this study, we propose a method for constructing a data-adaptive network of SRBFs using a modified Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm, and the performance of the algorithm is verified by the observed gravity data in the Auvergne area. Furthermore, the turning point method is used to optimize the bandwidth of the basis function spectrum, which satisfies the demand for both high-precision gravity field and quasi-geoid modelling simultaneously. Numerical experimental results indicate that our algorithm has an accuracy of about 1.58 mGal in constructing the gravity field model and about 0.03 m in the regional quasi-geoid model. Compared to the existing methods, the number of SRBFs used for modelling has been reduced by 15.8%, and the time cost to determine the centre positions of SRBFs has been saved by 12.5%. Hence, the modified HDBSCAN algorithm presented here is a suitable design method for constructing the SRBF data adaptive network.

Comparative validation of automated presurgical tractography based on constrained spherical deconvolution and diffusion tensor imaging with direct electrical stimulation.

Accurate presurgical brain mapping enables preoperative risk assessment and intraoperative guidance. This cross-sectional study investigated whether constrained spherical deconvolution (CSD) methods were more accurate than diffusion tensor imaging (DTI)-based methods for presurgical white matter mapping using intraoperative direct electrical stimulation (DES) as the ground truth. Five different tractography methods were compared (three DTI-based and two CSD-based) in 22 preoperative neurosurgical patients undergoing surgery with DES mapping. The corticospinal tract (CST, N = 20) and arcuate fasciculus (AF, N = 7) bundles were reconstructed, then minimum distances between tractograms and DES coordinates were compared between tractography methods. Receiver-operating characteristic (ROC) curves were used for both bundles. For the CST, binary agreement, linear modeling, and posthoc testing were used to compare tractography methods while correcting for relative lesion and bundle volumes. Distance measures between 154 positive (functional response, pDES) and negative (no response, nDES) coordinates, and 134 tractograms resulted in 860 data points. Higher agreement was found between pDES coordinates and CSD-based compared to DTI-based tractograms. ROC curves showed overall higher sensitivity at shorter distance cutoffs for CSD (8.5 mm) compared to DTI (14.5 mm). CSD-based CST tractograms showed significantly higher agreement with pDES, which was confirmed by linear modeling and posthoc tests (PFWE < .05). CSD-based CST tractograms were more accurate than DTI-based ones when validated using DES-based assessment of motor and sensory function. This demonstrates the potential benefits of structural mapping using CSD in clinical practice.

Toward a new approach for the pygmy dipole resonance in even–even nuclei. Application to isotopes 144,148,150,152,154Sm

A many-body Hamiltonian consisting of a spherical shell model mean-field term, a pairing interaction for alike nucleons and a dipole–dipole interaction, with the dipole operator involving a cubic term in the radial coordinate, was studied within a quasiparticle random-phase approximation and applied numerically to five even–even isotopes of Sm. The resulting wavefunctions were further used to calculate the B(E1) values, which at their turn were employed to calculate the photoabsorption cross section, integrated moments of the cross section and energy weighted sum rule (EWSR). The calculated cross section and its integrated moments were compared with the available data, and a good agreement was observed. Two regions were distinguished: one corresponding to the pygmy dipole resonance (PDR) (1–10 MeV) and the other to the giant dipole resonance (GDR) (10–20 MeV), which were studied separately. The peaks belonging to each of the two ranges were analyzed in detail. The PDR states were located around the neutron separation energy and were mainly formed by the collective isoscalar and neutron collective states. The PDR states describe oscillations of the neutron excess against protons from the isospin-saturated core. The character of the states from the GDR region, isoscalar or isovector, is also pointed out. The PDR states carry only 0.8%–2.7% of the total EWSR and 0.4%–5.9% of the total E1 strength. The dependence of the dipole strength on nuclear deformation is evidenced. A comment on the cross section splitting into two branches for deformed isotopes is included. The r-cubic term and nuclear deformation have opposite effects on the dipole strength. In addition, it diminishes the effect of nonconservation of the center of mass momentum. The famous Thomas–Reiche–Kuhn sum rule formula is generalized to the case of the Schiff dipole momentum. The new sum rule is well satisfied. The projected spherical single-particle basis used in our formalism allows for a unified description of spherical transitional and deformed isotopes.

On the conjugate symmetry and sparsity of the harmonic decomposition of parametric surfaces with the randomised Kaczmarz method

The downside of increasing the resolution of surface scanning devices is that the amount of acquired data makes the morphological analysis of the scanned surfaces computationally challenging. This limitation is circumvented by using scalable and non–memory–intensive harmonic expansions. In this paper, the projection of parametric surfaces onto disk and spherical harmonics bases is investigated, and three novel computationally efficient algorithms are proposed based on the randomised Kaczmarz (RK). To boost the computational performance and convergence of the root mean square error (RMSE) of the reconstructed surfaces we exploited the conjugate symmetry property of the harmonic basis functions. Further, the sparsity of the signals is used for estimating the projection coefficients from an undersampled surface. The first algorithm only takes into consideration the conjugate symmetry property for enhancing the convergence of the RMSE. The second algorithm endows the sparse version of the RK algorithm with the conjugate symmetry property. The third algorithm combines the previous two to further accelerate the convergence of the RMSE. The performance of the developed algorithms is tested on three surfaces where we demonstrate that they outperform conventional reconstruction techniques in terms of processing time with comparable precision.

High-resolution regional gravity field modeling in data-challenging regions for the realization of geopotential-based height systems

Modern height systems are based on the combination of satellite positioning and gravity field models of high resolution. However, in many regions, especially developing or newly industrializing countries, there is no (reliable) regional gravity model at all, due to challenges such as limited data availability, unknown/low data quality, and missing metadata. This paper addresses this issue in a case study of Colombia, where eight decades of historical terrestrial and airborne gravity measurements are available but widely contain systematic errors, outliers, and biases. Correspondingly, processing strategies and structures are proposed and applied to validate and improve the quality of old gravity datasets. A novel method is developed based on spherical radial basis functions (SRBFs) for estimating biases, which are found in different airborne surveys with values exceeding 40 mGal. The validity of this bias estimation method is demonstrated both by a simulation test and by the evaluation of the airborne data in comparison to the SATOP (SAtellite-TOPography) model, which merges the satellite-only global gravity model GOCO06s with the Earth2014 topography model. The terrestrial and airborne data are then combined with a global gravity model (GGM), ultra-high-resolution topography models, as well as altimetry-derived gravity anomalies from DTU21GRA for the offshore areas. The results are presented in terms of height anomalies (QGeoidCOL2023), and they are thoroughly validated using GPS/leveling data both in the absolute and relative manner. The standard deviation in comparison to the GPS/leveling data after applying a correction surface to account for the datum inconsistencies amounts to 15.76 cm, which is 27% smaller compared to the mean standard deviation value given by five recent high-resolution GGMs, and 36% smaller than the one delivered by the latest South American quasi-geoid model QGEOID2021. The relative validation results show that QGeoidCOL2023 performs better, i.e., delivers lower RMS errors than the GGMs and QGEOID2021 in all the baseline length groups. These results indicate the validity and benefits of the developed methods and procedures, which can be used for other data-challenging areas to facilitate the realization of geopotential-based height systems.Graphical

Seafloor topography modeling by fusing ICESat-2 lidar, echo sounding, and airborne and altimetric gravity data from spherical radial basis functions

Seafloor topography modeling by fusing ICESat-2 lidar, echo sounding, and airborne and altimetric gravity data from spherical radial basis functions

Trapped atoms in spatially-structured vector light fields

Spatially-structured laser beams, eventually carrying orbital angular momentum, affect electronic transitions of atoms and their motional states in a complex way. We present a general framework, based on the spherical tensor decomposition of the interaction Hamiltonian, for computing atomic transition matrix elements for light fields of arbitrary spatial mode and polarization structures. We study both the bare electronic matrix elements, corresponding to transitions with no coupling to the atomic center-of-mass motion, as well as the matrix elements describing the coupling to the quantized atomic motion in the resolved side-band regime. We calculate the spatial dependence of electronic and motional matrix elements for tightly focused Hermite–Gaussian, Laguerre–Gaussian and for radially and azimuthally polarized beams. We show that near the diffraction limit, all these beams exhibit longitudinal fields and field gradients, which strongly affect the selection rules and could be used to tailor the light-matter interaction. The presented framework is useful for describing trapped atoms or ions in spatially-structured light fields and therefore for designing new protocols and setups in quantum optics, -sensing and -information processing. We provide open code to reproduce our results or to evaluate interaction matrix elements for different transition types, beam structures and interaction geometries.

Applying New Algorithms for Numerical Integration on the Sphere in the Far Field of Sound Pressure

For some sound sources, the function of the square of sound pressure amplitudes on the sphere in the far field is an integrable function or can be integrated with geometrical simplifications, so an exact or approximated analytical expression for the sound power can be calculated. However, often the sound pressure on the sphere in the far field can only be defined in discrete points, for which a numerical integration is required for the calculation of the sound power. In this paper, two new algorithms, Anchored Radially Projected Integration on Spherical Triangles (ARPIST) and Spherical Quadrature Radial Basis Function (SQRBF), for surface numerical integration are used to calculate the sound power from the sound pressures on the sphere surface in the far field, and their solutions are compared with the analytical and the finite element method solution. If function values are available at any location on a sphere, ARPIST has a greater accuracy and stability than SQRBF while being faster and easier to implement. If function values are available only at user-prescribed locations, SQRBF can directly calculate weights while ARPIST needs data interpolation to obtain function values at predefined node locations, which reduces the accuracy and increases the calculation time.