Search Results (271)

Addition theorems have been indispensable tools for the reduction of quantum transition amplitudes. They are normally utilized at the start of the process to move the angular dependence within plane waves, Coulomb potentials, and the like, into a sum over spherical harmonics that allows the angular integration to be carried out. These have historically been “two-range” addition theorems, characterized by the two-fold notation $r_{>}=Max[r_1,r_2]$ and $r_{<}=Min[r_1,r_2]$ and comprising a single infinite series. More recently, “one-range” addition theorems have been created that have no such piecewise notation, but at the cost of the introduction of another infinite series. We use a very different approach to derive an infinite set of addition theorems for Slater orbitals, hydrogenic and Hylleraas wave functions, and so on, that retain the one-range variable dependence but have, at worst, a finite second series rather than an infinite one. In addition, unlike previous addition theorems, they are applicable to more than one coordinate system. One of these addition theorems may also be used for Yukawa-like functions that may appear late in the reduction of amplitude integrals, and we show its utility for an integral that has stubbornly defied reduction to analytic form for nearly sixty years. Finally, we craft indefinite integrals of 15 half-integer Macdonald functions multiplied by (inverse) powers and negative exponentials containing squares of the integration variable. Full article

Height anomaly, defined as the separation between the quasi-geoid and the reference ellipsoid, is fundamental to quasi-geoid refinement. While the Goddard Mars Model-3 (GMM-3) developed by NASA’s Goddard Space Flight Center (GSFC) and the JPL Mars gravity field MRO120D (JGMRO_120D) model developed by NASA’s Jet Propulsion Laboratory (JPL) stand as two representative Martian gravity field models, the systematic differences between them and their associated physical implications remain insufficiently quantified. This study establishes a validated computational framework for Martian height anomaly determination using updated physical parameters and spherical harmonic expansions. Validation against terrestrial datasets confirms high reliability (standard deviation: 0.0695 m relative to International Centre for Global Earth Models (ICGEM)), ensuring confidence in subsequent analysis. Our analysis reveals three critical findings: (1) Systematic latitudinal biases between GMM-3 and JGMRO_120D exhibit a monotonic gradient from −1.3 m near the equator to +3.9 m at the North Pole, suggesting differential parameterization of polar mass loading or tidal models between the two centers. (2) Polar clustering of uncertainties and outliers exceeding the 95th percentile (>7 m) concentrate non-randomly at latitudes >60°, which is attributed to sparse satellite tracking and seasonal ice cap modeling limitations. (3) There is error amplification in lowland terrains, where relative errors exceed 60% in flat regions (near-zero anomalies), posing critical risks for precision landing missions. While global consistency between models is high (R² = 0.9999), the identified discrepancies provide new constraints on Mars’s geophysical models and essential guidance for future gravity field improvements and mission planning. Full article

KANECS is a 3D multigroup neutronics code based on the Simplified Spherical Harmonics (SP_N) approximation and the Continuous Galerkin Finite Element Method (CGFEM). In this work, the code is extended to solve the time-dependent neutron kinetics by implementing a fully implicit backward Euler scheme for the neutron transport equation and an implicit exponential integration for delayed neutron precursors. These schemes ensure unconditional stability and minimize temporal discretization errors, making the method suitable for fast transients. The new formulation transforms each time step into a transient fixed-source problem, which is solved efficiently using the GMRES solver with ILU preconditioning. The kinetics module is validated against established benchmark problems, including TWIGL, the C5G2 MOX benchmark, and both 2D and 3D mini-core rod-ejection transients. KANECS shows close agreement with the reference solutions from well-known neutron transport codes, with consistent accuracy in normalized power evolution, spatial power distributions, and steady-state eigenvalues. The results confirm that KANECS provides a reliable and accurate framework for solving neutron kinetics problems. Full article

In this paper, using harmonic analysis tools−including spherical harmonic decomposition of kernels, sharp maximal function estimates, and variable exponent space theory—we investigate the boundedness of the commutator $[b,{\mathcal{M}}_{\Omega ,\beta }]$ on variable exponent central Morrey spaces, under suitable regularity conditions on the variable exponents. Here, $\Omega \in L^l\left({\mathbf{S}}^{n-1}\right)$ ($l\ge 1$) denotes a zero-degree homogeneous function on the unit sphere ${\mathbf{S}}^{n-1}$, $\beta$ satisfies $0\le \beta <n$, and $b\in CBMO\left({\mathbf{R}}^n\right)$. Full article

Global Gravitational Models (GGMs) describe the Earth’s external gravitational field by a set of spherical harmonic (Stokes) coefficients. These coefficients are routinely used to compute the geoid model, while disregarding the upper continental crustal (i.e., topographic) masses above the geoid. Strictly speaking, however, these coefficients can describe only gravity field quantities at (or above) the Earth’s surface to satisfy Laplace’s equation. Consequently, the GGM coefficients cannot be used to define the geoid surface rigorously without accounting for the internal convergence domain and the gravitational effect of topographic masses. In most technical and scientific applications, the computation of the geoid model directly from the GGM coefficients has been accepted under the assumption that errors due to disregarding the internal convergence domain (inside the topographic masses) are typically less than a few centimeters (i.e., at the level of global geoid model uncertainties). In this study, we demonstrate that these errors reach several decimeters and even meters, with maxima in Tibet and Himalayas exceeding ~4 m. Moreover, relatively large errors, reaching decimeters, are already detected in regions with a moderately elevated topography. In scientific applications requiring a high accuracy, such errors cannot be ignored. Instead, GGM coefficients describing the Earth’s external gravitational field have to be corrected for the effect of (topographic) masses distributed above the geoid surface to obtain spherical harmonic coefficients that explicitly define the geoid globally. The explicit definition of the global geoid model in the spectral domain is derived in this study and used to compile spherical harmonic coefficients of the geoid up to degree/order 2160 from the EIGEN-6C4 global gravitational model. Full article

The description of the Earth’s gravitational field, governed by the fundamental potential equation (the Laplace equation), is conventionally expressed using spherical harmonics, yet the Cartesian formulation, using a Taylor series representation, offers significant algebraic advantages. This paper proposes a novel Direct Cartesian Method for generating spherical basis functions and coefficients directly within the Cartesian coordinate system, utilising the partial derivatives of the inverse distance ($1/R$) function. The present study investigates the structural correspondence between the Cartesian form of spherical basis functions and the high-order partial derivatives of $1/R$. The study reveals that spherical basis functions can be categorised into four distinct groups based on the parity of the degree n and order m. It is demonstrated that each spherical basis function is equivalent to a weighted summation of the partial derivatives of the inverse distance ($1/R$) with respect to Cartesian coordinates. Specifically, the basis functions are combined with those derivatives that share the same order of Z-differentiation and possess matching parities in their orders of differentiation with respect to X and Y. In order to facilitate the practical calculation of these high-degree derivatives, a recursive numerical algorithm has been developed. The method generates the polynomial coefficients for the numerator of the $1/R$ derivatives. A pivotal innovation is the implementation of a step-wise normalization scheme within the recursive relations. The integration of the recursive ratios of global normalization factors (including full Schmidt normalization) into each step of the algorithm effectively neutralises factorial growth, rendering the process immune to numerical overflow. The validity and numerical stability of the proposed method are demonstrated through a detailed step-by-step derivation of a sectorial basis function ($n=8,m=2$). Full article

Ionospheric delay errors significantly reduce the positioning accuracy of global navigation satellite systems (GNSSs), whereas precise ionospheric modeling can effectively mitigate this issue. The ionosphere exhibits large-scale symmetry, and spherical harmonics (SHs) can effectively describe this property due to their rotational symmetry on the sphere. However, mathematical fitting models such as spherical harmonic functions and polynomial models encounter boundary inaccuracies caused by edge effects. To address this problem, we developed a spherical harmonic–radial basis function (SH-RBF) hybrid method based on the integration of spherical harmonics and radial basis function interpolation techniques. This method leverages the global symmetry of spherical harmonics and utilizes the local adaptability of radial basis functions to correct regional distortions. Validation using European GNSS data during both geomagnetically quiet and active periods, in comparison with the CODE global ionospheric map (GIM), demonstrates that the modeling accuracy of spherical harmonics surpasses that of POLY during geomagnetically quiet periods. Compared to spherical harmonics, SH-RBF improves overall modeling accuracy by 8.87–27.27% and enhances accuracy in edge regions by 34.16–83.91%. During geomagnetically active periods, the SH-RBF method also achieves notable improvements. This study confirms that SH-RBF is a reliable technique for significantly reducing edge effects in regional ionospheric modeling. Full article

Introduction: In recent decades, the burden of mental disorders has become a major determinant of population health in the European Union, generating profound clinical, socioeconomic, and institutional consequences. Despite political recognition of this silent crisis, substantial methodological challenges persist in the transnational monitoring of mental health and in linking disease burden with the resources allocated to address it. The present analysis develops a multivariate taxonomy of EU Member States from a psychosocial perspective, using an integrative quantitative approach. Methods: This cross-sectional, comparative study follows international standards for transparent and reproducible quantitative reporting and is based on 18 harmonized clinical, epidemiological, and institutional indicators collected for 27 EU Member States over the period 2014–2023. The indicators used in this study were grouped according to their position along the care continuum. Hospital-based indicators refer to inpatient activity and institutional capacity, including total hospital discharges, psychiatric admissions (affective disorders, schizophrenia, dementia, alcohol- and drug-related disorders), and hospital bed availability. Outpatient and community-level indicators reflect the capacity of systems to provide non-hospital psychiatric care and consist primarily of psychiatrist density and total specialist medical workforce. Finally, subjective perception indicators capture population-level self-assessed health status, complementing clinical and institutional measures by integrating a psychosocial perspective. After harmonization and standardization, Principal Component Analysis (PCA) with Varimax rotation was applied to identify latent dimensions of mental health. Model adequacy was confirmed using the Kaiser–Meyer–Olkin coefficient (0.747) and Bartlett’s test of sphericity (p < 0.001). Results: Three latent dimensions explaining 77.7% of the total variance were identified: (1) institutionalized psychiatric burden, (2) functional capacity of the health care system, and (3) suicidal vulnerability associated with problematic substance use. Standardized factor scores allowed for the classification of Member States, revealing distinct patterns of psychosocial risk. For example, Germany and France display profiles marked by high levels of institutionalized psychiatric activity, while the Baltic and Southeast European countries exhibit elevated suicidal vulnerability in the context of limited medical resources. These results highlight the deep heterogeneity of psychiatric configurations in Europe and reveal persistent gaps between population needs and institutional response capacity. Conclusions: The analysis provides an empirical foundation for differentiated public policies aimed at prevention, early intervention, and stigma reduction. It also supports the case for institutionalizing a European mental health monitoring system based on harmonized indicators and common assessment standards. Overall, the findings clarify the underlying structure of mental health across the European Union and underscore the need for coherent, evidence-based strategies to reduce inequalities and strengthen system performance at the continental level. Full article

To mitigate the impact of north–south strip errors inherent in Gravity Recovery and Climate Experiment (GRACE) spherical harmonic coefficient solutions, this research develops a state-space model to generate a more robust solution. The efficacy of the state-space model is demonstrated by comparing its performance with that of conventional filtering methods and hydrological modeling schemes. The method is subsequently applied to estimate the GRACE Groundwater Drought Index in the California Central Valley basin, a region significantly affected by drought during the GRACE observation period. This analysis quantifies the severity of droughts and floods while investigating the direct influences of precipitation, runoff, evaporation, and anthropogenic activities. By incorporating the El Niño–Southern Oscillation (ENSO) and the Pacific Decadal Oscillation, the study offers a detailed causal analysis and proposes a novel methodology for water resource management and disaster early warning. The results indicate that a moderate-duration flood event in 2006 resulted in a recharge of 19.81 km³ of water resources in the California Central Valley basin, whereas prolonged droughts in 2008 and 2013, lasting over 15 months, led to groundwater depletion of 41.53 km³ and 91.45 km³, respectively. Precipitation and runoff are identified as the primary determinants of local drought and flood conditions. The occurrence of ENSO events correlates with sustained precipitation variations over the subsequent 2–3 months, resulting in corresponding changes in groundwater storage. Full article

Global navigation satellite system (GNSS) differential code bias (DCB) and topside ionosphere vertical electron content (VEC) can be estimated using onboard data from low-earth-orbit (LEO) satellites. These satellites provide the potential to make up for the lack of ground-based stations in the oceanic and polar regions and establish a high-precision global ionosphere model. In order to study the influences of different LEO-topside VEC processing methods on estimates, we creatively analyzed and compared the results and accuracy of the DCBs and LEO-topside VEC estimates using two topside VEC solutions—the SH-topside VEC (spherical harmonic-topside vertical electron content) and EP-topside VEC (epoch parameter-topside vertical electron content) methods. Some conclusions are drawn as follows. (1) Using GRACE-A data (400 km in 2016), the monthly stabilities (STDs) of GPS satellite DCBs and LEO receiver DCBs using the EP-topside VEC method are better than those using the SH-topside VEC method. For JASON-2 data (1350 km), the STD results of GPS DCBs using the SH-topside VEC method are slightly superior to those using the EP-topside VEC method, and LEO DCBs using the two methods have similar STD results. However, the root mean square (RMS) results for GPS DCBs using the SH-topside VEC model relative to the Center for Orbit Determination in Europe (CODE) products are slightly superior to those using the EP-topside VEC method. (2) The peak ranges of the actual GRACE-A-topside VEC results using the SH-topside VEC and EP-topside VEC methods are within 42 and 35 TECU, respectively, while the peak ranges of the JASON-2-topside VEC results are both within 6 TECU. Additionally, only the SH-topside VEC model results are displayed due to the EP-topside VEC method not modeling VEC. Due to the difference in orbital altitude, the results and distributions of the GRACE-topside VECs differ from those of the JASON-topside VECs, with the former being more consistent with the ground-based results, indicating that there may be different height structures in the LEO-topside VECs. In addition, we applied the IRI-GIM (International Reference Ionosphere model–Global Ionosphere Map) method to compare the LEO-based topside VEC results, which indicate that the accuracy of GRACE-A-topside VEC using the EP-topside VEC method is better than that using the SH-topside VEC method, whereas for JASON-2, the two methods have similar accuracy. Meanwhile, we note that the temporal and spatial resolutions of the SH-topside VEC method are higher than those of the EP-topside VEC method, and the former has a wide range of usability and predictive characteristics. The latter seems to correspond to the single-epoch VEC mean of the former to some extent. Full article

Space-based optical imaging detection serves as a crucial means for acquiring characteristic information of space objects, with the quality and resolution of images directly influencing the accuracy of subsequent missions. Addressing the scarcity of datasets in space-based optical imaging, this study introduces a method that combines an improved Phong model and higher-order spherical harmonics (HOSH) for the optical imaging computation of space objects. Utilizing HOSH to fit the light field distribution, this approach comprehensively considers direct sunlight, earthshine, reflected light from other extremely distant celestial bodies, and multiple scattering from object surfaces. Through spectral reflectance experiments, an improved Phong model is developed to calculate the optical scattering characteristics of space objects and to retrieve common material properties such as metallicity, roughness, index of refraction (IOR), and Alpha for four types of satellite surfaces. Additionally, this study designs two sampling methods: a random sampling based on the spherical Fibonacci function (RSSF) and a sequential frame sampling based on predefined trajectories (SSPT). Through numerical analysis of the geometric and radiative rendering pipeline, this method simulates multiple scenarios under both high-resolution and wide-field-of-view operational modes across a range of relative distances. Simulation results validate the effectiveness of the proposed approach, with average rendering speeds of 2.86 s per frame and 1.67 s per frame for the two methods, respectively, demonstrating the capability for real-time rapid imaging while maintaining low computational resource consumption. The data simulation process spans six distinct relative distance intervals, ensuring that multi-scale images retain substantial textural features and are accompanied by attitude labels, thereby providing robust support for algorithms aimed at space object attitude estimation, and 3D reconstruction. Full article

Conventional thermospheric density models are limited in their ability to capture solar-geomagnetic coupling dynamics and lack probabilistic uncertainty estimates. We present MSIS-UN (NRLMSISE-00 with Uncertainty Quantification), an innovative framework integrating sparse principal component analysis (sPCA) with heteroscedastic neural networks. Our methodology leverages multi-satellite density measurements from the CHAMP, GRACE, and SWARM missions, coupled with MSIS-00-derived exospheric temperature (tinf) data. The technical approach features three key innovations: (1) spherical harmonic decomposition of T∞ using spatiotemporally orthogonal basis functions, (2) sPCA-based extraction of dominant modes from sparse orbital sampling data, and (3) neural network prediction of temporal coefficients with built-in uncertainty quantification. This integrated framework significantly enhances the temperature calculation module in MSIS-00 while providing probabilistic density estimates. Validation against SWARM-C measurements demonstrates superior performance, reducing mean absolute error (MAE) during quiet periods from MSIS-00’s 44.1% to 23.7%, with uncertainty bounds (1σ) achieving an MAE of 8.4%. The model’s dynamic confidence intervals enable rigorous probabilistic risk assessment for LEO satellite collision avoidance systems, representing a paradigm shift from deterministic to probabilistic modeling of thermospheric density. Full article

The large lakes and reservoirs of the northeastern Tibetan Plateau play a key role in regional water resources, yet their influence on terrestrial water storage (TWS) changes at different spatial scales remains unclear. This study employed the constrained forward modeling (CFM) method to correct leakage errors in level-2 spherical harmonic (SH) coefficients from the Gravity Recovery and Climate Experiment and its follow-on missions (GRACE/GRACE-FO) at three spatial scales: two circular regions covering 90,000 km² and 200,000 km², respectively, and a 220,000 km² region based on the shape of mass concentration (Mascon). TWS changes derived from SH solutions after leakage correction through CFM were compared with level-3 Mascon solutions. Individual water storage components, including lake and reservoir water storage (LRWS), groundwater storage (GWS), and soil moisture storage (SMS), were quantified, and their relationships with precipitation were assessed. From 2003 to 2022, the CFM method effectively mitigated signal leakage, revealing an overall upward trend in TWS at all spatial scales. Signals from Qinghai Lake and Longyangxia Reservoir dominated the long-term trend and amplitude variations of LRWS, respectively. LRWS explained more than 47% of the TWS changes, and together with GWS, accounted for over 85% of the changes. Both CFM-based and Mascon-based TWS changes indicated a consistent upward trend from January 2003 to September 2012, followed by declines from November 2012 to May 2017 and October 2018 to December 2022. During the decline phases, GWS contributions increased, while LRWS contributions and component exchange intensity decreased. LRWS, SMS, and TWS changes were significantly correlated with precipitation, with varying time lags. These findings underscore the value of GRACE/GRACE-FO data for monitoring multiscale TWS dynamics and their climatic drivers in lake- and reservoir-dominated regions. Full article

This paper proposes a reconstruction framework for estimating the far-field (FF) radiation patterns of large, heavy, or non-rotatable wireless-enabled systems. The method combines a tilted orbital sampling (ToS) strategy with sparse spherical harmonic (SH) expansion, compressed sensing (CS), and convex optimization (CO), thereby linking a mechanically constrained acquisition scheme with a mathematically efficient recovery process. The purpose of this integration is not only to reduce the number of measurements but also to retrieve the radiation information most relevant to Internet of Things (IoT) devices and bulky equipment that cannot be easily rotated within anechoic chambers. The framework is validated on two representative cases: a canonical half-wave dipole and a commercial Wi-Fi-enabled device. In the latter and more challenging case, accurate reconstruction is achieved with fewer than 30 SH coefficients and using less than 20% of the measurements required by a conventional full-sphere scan, with the normalized root-mean-square error remaining below 5%. Although inaccessible angular regions may be partially uncharacterized, such directions are of minor relevance for the intended operational coverage. The resulting SH-based representation can be seamlessly integrated into ray-tracing propagation simulators and electromagnetic optimization workflows, enabling efficient and application-oriented OTA characterization under realistic chamber constraints. Full article

Electroencephalography (EEG) is a non-invasive biosensing platform with a spatial-frequency content that is of significant relevance for a multitude of aspects in the neurosciences, ranging from optimal spatial sampling of the EEG to the design of spatial filters and source reconstruction. In the past, simplified spherical head models had to be used for this analysis. We propose a method for spatial frequency analysis in EEG for realistically shaped volume conductors, and we exemplify our method with a five-compartment Boundary Element Method (BEM) model of the head. We employ the recently developed technique for spatial harmonic analysis (Sphara), which allows for spatial Fourier analysis on arbitrarily shaped surfaces in space. We first validate and compare Sphara with the established method for spatial Fourier analysis on spherical surfaces, discrete spherical harmonics, using a spherical volume conductor. We provide uncertainty limits for Sphara. We derive relationships between the signal-to-noise ratio (SNR) and the required spatial sampling of the EEG. Our results demonstrate that conventional 10–20 sampling might misestimate EEG power by up to 50%, and even 64 electrodes might misestimate EEG power by up to 15%. Our results also provide insights into the targeting problem of transcranial electric stimulation. Full article