A continuous minimax problem for calculating minimum norm polynomial interpolation points on the sphere

This paper considers the calculation of the minimum norm points for polynomial interpolation over the sphere S 2 ? R 3 . The norm of the interpolation operator ? n , considered as a map from C(S 2 ) to C(S 2 ), is given by ? ? n ? = max x ? S 2 ?B -1 b (x)? 1 , where the nonsingular matrix B and vector b are determined by the fundamental system of points x j ? S 2 , j = 1,?, d n . The problem is to choose the fundamental system to minimise ? ? n ?. Algorithms for solving this continuous minimax problem must be able to handle many local maxima close to the global maximum, and local maxima which lie close to each other along ridges. A first order dual algorithm is used to find a spherical parametrisation of a normalised fundamental system. The results suggest that for these points the growth in ? ? n ?, for n ? 30, is less than c 0 + c 1 n, where c 0 ? 1.8 and c 1 ? 0.7

On the solution of a Riesz equilibrium problem and integral identities for special functions

The aim of this note is to provide a full space quadratic external field extension of a classical result of Marcel Riesz for the equilibrium measure on a ball with respect to Riesz s-kernels. We address the case s=d-3 for arbitrary dimension d, in particular the logarithmic kernel in dimension 3. The equilibrium measure for this full space external field problem turns out to be a radial arcsine distribution supported on a ball with a special radius. As a corollary, we obtain new integral identities involving special functions such as elliptic integrals and more generally hypergeometric functions. It seems that these identities are not found in the existing tables for series and integrals, and are not recognized by advanced mathematical software. Among other ingredients, our proofs involve the Euler-Lagrange variational characterization, the Funk-Hecke formula, the Weyl regularity lemma, the maximum principle, and special properties of hypergeometric functions.Comment: Minor modifications. To appear in Journal of Mathematical Analysis and Applications (JMAA

Removing the mask -- reconstructing a scalar field on the sphere from a masked field

The paper analyses a spectral approach to reconstructing %the image of a scalar field on the sphere, given only information about a masked version of the field together with precise information about the (smooth) mask. The theory is developed for a general mask, and later specialized to the case of an axially symmetric mask. Numerical experiments are given for the case of an axial mask motivated by the cosmic microwave background, assuming that the underlying field is a realization of a Gaussian random field with an artificial angular power spectrum of moderate degree ($\ell \le 100$). The recovery is highly satisfactory in the absence of noise and even in the presence of moderate noise

Extremal systems of points and numerical integration on the sphere

This paper considers extremal systems of points on the unit sphere S r ⊆ R r+1 , related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d n = dim P n points, where P n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S 2 of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments

Effect of Mesh Phasing on the Transmission Efficiency and Dynamic Performance of Wheel Hub Planetary Gear Sets

Transmission efficiency and refinement of planetary wheel hub gearing system are key design attributes for heavy and off-highway vehicles. Reduction of power loss, directly leading to the development of new generation ECO-axles requires analysis of gear contacting conditions for lubricated conjunctions to determine frictional performance. This is also affected by gear dynamics, which is a prerequisite for assessment of noise, vibration and harshness performance. Therefore, a combined tribo-dynamic analysis is essential. There is a dearth of such holistic analysis, particularly for the case of wheel hub planetary systems. The paper presents such an analysis, which has not hitherto been reported in literature. The inexorable interplay of transmission efficiency and noise, vibration and harshness refinement is demonstrated. The key attributes of noise, vibration and harshness refinement and transmission efficiency can pose contrary requirements and near-optimal conditions can be highlighted by mesh phasing of gearing contacts, thus alleviating the need for more complex gear teeth modifications entailing prohibitive manufacturing costs

Status of Muon Collider Research and Development and Future Plans

The status of the research on muon colliders is discussed and plans are outlined for future theoretical and experimental studies. Besides continued work on the parameters of a 3-4 and 0.5 TeV center-of-mass (CoM) energy collider, many studies are now concentrating on a machine near 0.1 TeV (CoM) that could be a factory for the s-channel production of Higgs particles. We discuss the research on the various components in such muon colliders, starting from the proton accelerator needed to generate pions from a heavy-Z target and proceeding through the phase rotation and decay ($\pi \to \mu \nu_{\mu}$) channel, muon cooling, acceleration, storage in a collider ring and the collider detector. We also present theoretical and experimental R & D plans for the next several years that should lead to a better understanding of the design and feasibility issues for all of the components. This report is an update of the progress on the R & D since the Feasibility Study of Muon Colliders presented at the Snowmass'96 Workshop [R. B. Palmer, A. Sessler and A. Tollestrup, Proceedings of the 1996 DPF/DPB Summer Study on High-Energy Physics (Stanford Linear Accelerator Center, Menlo Park, CA, 1997)].Comment: 95 pages, 75 figures. Submitted to Physical Review Special Topics, Accelerators and Beam

Global collision-risk hotspots of marine traffic and the world’s largest fish, the whale shark

© The Author(s), 2022. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Womersley, F. C., Humphries, N. E., Queiroz, N., Vedor, M., da Costa, I., Furtado, M., Tyminski, J. P., Abrantes, K., Araujo, G., Bach, S. S., Barnett, A., Berumen, M. L., Bessudo Lion, S., Braun, C. D., Clingham, E., Cochran, J. E. M., de la Parra, R., Diamant, S., Dove, A. D. M., Dudgeon, C. L., Erdmann, M. V., Espinoza, E., Fitzpatrick, R., González Cano, J., Green, J. R., Guzman, H. M., Hardenstine, R., Hasan, A., Hazin, F. H. V., Hearn, A. R., Hueter, R. E., Jaidah, M. Y., Labaja, J., Ladinol, F., Macena, B. C. L., Morris Jr., J. J., Norman, B. M., Peñaherrera-Palmav, C., Pierce, S. J., Quintero, L. M., Ramırez-Macías, D., Reynolds, S. D., Richardson, A. J., Robinson, D. P., Rohner, C. A., Rowat, D. R. L., Sheaves, M., Shivji, M. S., Sianipar, A. B., Skomal, G. B., Soler, G., Syakurachman, I., Thorrold, S. R., Webb, D. H., Wetherbee, B. M., White, T. D., Clavelle, T., Kroodsma, D. A., Thums, M., Ferreira, L. C., Meekan, M. G., Arrowsmith, L. M., Lester, E. K., Meyers, M. M., Peel, L. R., Sequeira, A. M. M., Eguıluz, V. M., Duarte, C. M., & Sims, D. W. Global collision-risk hotspots of marine traffic and the world’s largest fish, the whale shark. Proceedings of the National Academy of Sciences of the United States of America, 119(20), (2022): e2117440119, https://doi.org/10.1073/pnas.2117440119.Marine traffic is increasing globally yet collisions with endangered megafauna such as whales, sea turtles, and planktivorous sharks go largely undetected or unreported. Collisions leading to mortality can have population-level consequences for endangered species. Hence, identifying simultaneous space use of megafauna and shipping throughout ranges may reveal as-yet-unknown spatial targets requiring conservation. However, global studies tracking megafauna and shipping occurrences are lacking. Here we combine satellite-tracked movements of the whale shark, Rhincodon typus, and vessel activity to show that 92% of sharks’ horizontal space use and nearly 50% of vertical space use overlap with persistent large vessel (>300 gross tons) traffic. Collision-risk estimates correlated with reported whale shark mortality from ship strikes, indicating higher mortality in areas with greatest overlap. Hotspots of potential collision risk were evident in all major oceans, predominantly from overlap with cargo and tanker vessels, and were concentrated in gulf regions, where dense traffic co-occurred with seasonal shark movements. Nearly a third of whale shark hotspots overlapped with the highest collision-risk areas, with the last known locations of tracked sharks coinciding with busier shipping routes more often than expected. Depth-recording tags provided evidence for sinking, likely dead, whale sharks, suggesting substantial “cryptic” lethal ship strikes are possible, which could explain why whale shark population declines continue despite international protection and low fishing-induced mortality. Mitigation measures to reduce ship-strike risk should be considered to conserve this species and other ocean giants that are likely experiencing similar impacts from growing global vessel traffic.Funding for data analysis was provided by the UK Natural Environment Research Council (NERC) through a University of Southampton INSPIRE DTP PhD Studentship to F.C.W. Additional funding for data analysis was provided by NERC Discovery Science (NE/R00997/X/1) and the European Research Council (ERC-AdG-2019 883583 OCEAN DEOXYFISH) to D.W.S., Fundação para a Ciência e a Tecnologia (FCT) under PTDC/BIA/28855/2017 and COMPETE POCI-01–0145-FEDER-028855, and MARINFO–NORTE-01–0145-FEDER-000031 (funded by Norte Portugal Regional Operational Program [NORTE2020] under the PORTUGAL 2020 Partnership Agreement, through the European Regional Development Fund–ERDF) to N.Q. FCT also supported N.Q. (CEECIND/02857/2018) and M.V. (PTDC/BIA-COM/28855/2017). D.W.S. was supported by a Marine Biological Association Senior Research Fellowship. All tagging procedures were approved by institutional ethical review bodies and complied with all relevant ethical regulations in the jurisdictions in which they were performed. Details for individual research teams are given in SI Appendix, section 8. Full acknowledgments for tagging and field research are given in SI Appendix, section 7. This research is part of the Global Shark Movement Project (https://www.globalsharkmovement.org)