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A family of spherical caps of the 2-dimensional unit sphere S2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S2. We solve this problem up to eight spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of unit balls in Euclidean 3-space is 8. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n>1 great circles in S2. Next, we prove dual bounds for TS-coverings of S2 by congruent spherical caps. Here a covering of S2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension >2.

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.

Integral formulas represent a methodological basis for the determination of gravitational fields generated by planetary bodies. In particular, spherical integral transformations are preferred for their symmetrical properties with the integration domain being the entire surface of the sphere. However, global coverage of boundary values is rarely guaranteed. In practical calculations, we therefore split the spherical surface into a near zone and a far zone, for convenience, by a spherical cap. While the gravitational effect in the near zone can be evaluated by numerical integration over available boundary values, the contribution of the far zone has to be precisely quantified by other means. Far-zone effects for the isotropic integral transformations and those depending on the direct azimuth have adequately been discussed. On the other hand, this subject has only marginally been addressed for the spherical integral formulas that are, except for other variables, also functions of the backward azimuth. In this article, we significantly advance the existing geodetic methodology by deriving the far-zone effects for the two classes of spherical integral transformations: (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations with arbitrary-order vertical derivative of respective boundary conditions and (2) spatial (vertical, horizontal, or mixed) derivatives of these generalised analytical solutions up to the third order. The integral and spectral forms of the far-zone effects are implemented in MATLAB software package, and their consistency is tested in closed-loop simulations. The presented methodology can be employed in upward/downward continuation of potential field observables or for a quantification of error propagation through spherical integral transformations.

There is the need for robust alternatives to the widely used spherical harmonic analysis when measurements are restricted to a region, or when high spatial frequency fields with much less parameters are required. Spherical cap harmonic analysis (SCHA) is one of the preferred alternative regional modelling techniques over the last decades. This paper presents a comprehensive and systematic review of the SCHA literature, underlining the respective merits and weaknesses of the ways in which the technique has been used since it was proposed in the context of geomagnetic field modelling. It reflects the multidisciplinary use of this technique and examines the evidences presented mainly in Earth and planetary science journals. Some bibliometric parameters are provided to understand how the technique and the knowledge of its limitations have progressed and improved, and some avenues for future research are highlighted.

The Earth’s magnetic field originated in the fluid core, the so-called core field, is the dominant contribution to the geomagnetic field. Since ancient times, the core geomagnetic field has been used primarily for geographical orientation and navigation by means of compasses. Nowadays, thanks to the large amount of geomagnetic data available, core field models can be developed on a global or regional scale. Global models resolve large-scale geomagnetic field features, while regional models can resolve greater detail over a particular region. The spherical harmonic cap analysis is a widely used technique for regional-scale modelling of the geomagnetic field. In this work we have developed a regional model of the core field and its secular variation between 2014.5 and 2020.5 over the Iberian Peninsula, based on data from Swarm satellites, geomagnetic observatories and repeat stations. Its performance has been validated by comparing the fit to the available geomagnetic data using the regional model and the global models IGRF and CHAOS over the whole spatio-temporal range studied. In order to optimise the model, a detailed study of its input parameters has been carried out, showing that not all parameters have an equal influence on the modelling. This new model reproduces the input data with a root mean square error of 2.9 nT, improving the outcome of global models on this region. The results of this work will allow the Spanish Instituto Geográfico Nacional to produce the magnetic cartography of Iberia and the Balearic Islands in 2020.0, which for the first time will be based on a regional core field model, replacing the polynomial variation method used in the past.

Concrete-filled steel tubes (CFST) are widely used due to their high strength, ductility, and energy dissipation capacity. However, gaps in between core concrete and steel tube adversely affect the mechanical performance of structures, thereby compromising the safety of the building. In this paper, four concrete-filled steel tube specimens with spherical-cap gaps were designed, and quasi-static tests were conducted to investigate the impact of gap depth on the seismic performance of concrete-filled steel tube columns. The test results indicate that the gap reduced the cumulative energy dissipation and initial stiffness of concrete-filled steel tubes. The gap weakened the compressing effect on the steel tube exerted by the expansion of core concrete, leading to premature yielding of the steel tube. As the gap’s depth increased from 0 mm to 30 mm, the load-bearing capacity and ductility of the concrete-filled steel tube columns decreased by 24.86% and 21.7%, respectively. This research quantified the extent to which gaps weaken the seismic performance of CFST columns, and the reduction coefficients of bearing capacity under different gap ratios were provided. This contributes to enhancing structural safety and lays a foundation for further research.

Spatio-temporal characteristics are the crucial conditions for Moon-based Earth observations. In this study, we established a Moon-based Earth observation geometric model by considering the intervisibility condition between a Moon-based platform and observed points on the Earth, which can analyze the spatio-temporal characteristics of the observations of Earth’s hemisphere. Furthermore, a formula for the spherical cap of the Earth visibility region on the Moon is analytically derived. The results show that: (1) the observed Earth spherical cap has a diurnal period and varies with the nadir point. (2) All the annual global observation durations in different years show two lines that almost coincide with the Arctic circle and the Antarctic circle. Regions between the two lines remain stable, but the observation duration of the South pole and North pole changes every 18.6 years. (3) With the increase of the line-of-sight minimum observation elevation angle, the area of an intervisible spherical cap on the lunar surface is obviously decreased, and this cap also varies with the distance between the barycenter of the Earth and the barycenter of the Moon. In general, this study reveals the effects of the elevation angle on the spatio-temporal characteristics and additionally determines the change of area where the Earth’s hemisphere can be observed on the lunar surface; this information can provide support for the accurate calculation of Moon-based Earth hemisphere observation times.

The global ionosphere map (GIM) is not capable of serving precise positioning and navigation for single frequency receivers in Australia due to sparse International GNSS Service (IGS) stations located in the vast land. This study proposes an approach to represent Australian total electron content (TEC) using the spherical cap harmonic analysis (SCHA) and artificial neural network (ANN). The new Australian TEC maps are released with an interval of 15 min for longitude and latitude in 0.5° × 0.5°. The validation results show that the Australian Ionospheric Maps (AIMs) well represent the hourly and seasonally ionospheric electrodynamic features over the Australian continent; the accuracy of the AIMs improves remarkably compared to the GIM and the model built only by the SCHA. The residual of the AIM is inversely proportional to the level of solar radiation. During the equinoxes and solstices in a solar minimum year, the residuals are 2.16, 1.57, 1.68, and 1.98 total electron content units (TECUs, 1 TECU = 1016 electron/m2), respectively. Furthermore, the AIM has a strong capability in capturing the adequate electrodynamic evolutions of the traveling ionospheric disturbances under severe geomagnetic storms. The results demonstrate that the ANN-aided SCHA method is an effective approach for mapping and investigating the TEC maps over Australia.

The particles found in diverse processes such as in pneumatic conveying, food processing, drilling operations, etc., may or may not be spherical in shape. Different types of non-spherical shapes are known to play an important role in fluid–particle interactions in terms of hydrodynamics and thermal behavior. The shape effect is studied in this work for a spherical cap and circular disc having the same projected area, in cylindrical confinement of λ (≡base diameter of particle to diameter of the tube) = 0.5 for the Poiseuille flow of air ( Pr  = 0.72) over a Reynold number range 1 ≤  Re  ≤ 100 in steady state regime. The momentum and energy equations are solved for this problem using finite element-based techniques using COMSOL Multiphysics. The obtained results for both spherical cap and circular disc are compared with a spherical shape under otherwise identical conditions. The results show that drag experienced by spherical cap is lowest in comparison to other considered shapes at low Reynolds numbers. However, this trend gets reversed at high inertial flow ( Re  = 100). Although, the heat transfer rate in the case of spherical cap is observed to be higher than that of the circular disc and sphere. Especially, at Re  = 1 rate of heat transfer from spherical cap is ~3 times higher than the sphere. Furthermore, correlations have been proposed for drag coefficient and average Nusselt number over the range of Reynold number 1 ≤  Re  ≤ 100 incorporating both the non-spherical shapes along with a sphere thereby enabling interpolation for the intermediate values in the various applications.

The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.