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Chinese Physics B 2011年第9期20卷 152-158页

作者：张继涛吴学健李岩 State Key Laboratory of Precision Measurement Technology & Instruments Department of Precision Instruments and MechanologyTsinghua University

To determine the Avogadro constant with a target relative uncertainty of 2 x 10-s, the uncertainty component of the silicon sphere＇s volume introduced by the spherical harmonics method, which is usually used in determining the sphere＇s volume, is reevaluated. By means of representing the shape of the silicon sphere by an ellipsoid with Gaussian white noise in its diameters, the uncertainty of the current mapping methods based on the spherical harmonics theory can be estimated theoretically. It is evidenced that the uncertainty component attributed to the current mapping method is underestimated. To eliminate this effect as much as possible, the number of mapping points should be increased to more than before. Moreover, a new mapping method is proposed to accomplish the equal-area mapping with large number points on the silicon sphere.

关键词： metrology Avogadro constant silicon sphere spherical harmonics

An efficient spherical mapping algorithm and its application on spherical harmonics

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Science China(Information Sciences) 2013年第9期56卷 28-37页

作者： WAN ShengHua YE TengFei LI MaoQing ZHANG HongChao LI Xin School of Electrical Engineering and Computer Science Louisiana State University (LSU) Department of Automation Xiamen University Department of Mathematics LSU

The sphere is a natural and seamless parametric domain for closed genus-0 surfaces. We introduce an efficient hierarchical optimization approach for the computation of spherical parametrization for closed genus-0 surfaces by minimizing a nonlinear energy balancing angle and area distortions. The mapping results are bijective and lowly distorted. Our algorithm converges efficiently and is suitable to manipulate large-scale geometric models. We demonstrate and analyze the effectiveness of our mapping in spherical harmonics decomposition.

关键词： spherical parametrization hierarchical optimization spherical harmonics

Molecular Simulations using spherical harmonics

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Chinese Journal of Chemistry 2003年第10期21卷 1252-1255页

作者：蔡文生徐佳维邵学广 MAIGRET Bernard Department of Chemistry UMR 7565 CNRS/UHP

Computer aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the interacting part of molecules and they should reflex the complementarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The complementarity of lobe hole is very important in protein ligand interactions. spherical harmonic models based on expansions of spherical harmonic functions were used as a fingerprint to approximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined separately. This method can be used as a filter to eliminate candidates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.

关键词： molecular surface shape complementarity molecular docking molecular recognition spherical harmonics

On Approximation by Reciprocals of spherical harmonics in L^p Norm

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Acta Mathematicae Applicatae Sinica 2005年第4期21卷 529-536页

作者： Bao-huai Sheng Department of Mathematics Shaoxing College of Arts and Sciences Shaoxing Zhejiang 312000 China

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.

关键词： spherical harmonics approximation by reciprocals of polynomials translation operator

Ionospheric Grid Modeling of Regional Satellite Navigation System with spherical harmonics

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Ionospheric Grid Modeling of Regional Satellite Navigation S...

CLEBSCH-GORDAN COEFFICIENTS,spherical harmonics,AND d FUNCTIONS

第四届中国卫星导航学术年会

作者： Jiachen Fan Xiaoli Wu Enqiang Dong He Zhao Haibo Kan Jinshi Xie Beijing Satellite Navigation Center

As an important component of differential navigation service, ionospheric grid can greatly improve the accuracy of single frequency users. Wide-Area Augmentation System （WAAS） broadcasts a set of ionospheric grids data for authorized single frequency GPS users every 300s. WAAS uses inverse distance weighting （IDW） algorithm to calculate the vertical total electron content （VTEC） at those ionosphere grid points （IGP）. IDW algorithm has good correction accuracy, but has low availability in the area with few ionosphere pierce points （IPP）. The availability and accuracy of ionospheric grid by IDW algorithm depend on the distribution of monitoring stations and the constellation of navigation system. As the constellation of regional satellite navigation system consists mainly of geostationary earth orbits （GEO） satellites and inclined geosynchronous orbits （IGSO） satellites, the IPPs have a concentrating distribution. Considering that the distribution of monitoring stations are all in China, using IDW algorithm to calculate ionospheric grid will not have good availability. This paper proposes a practical algorithm to calculate the regional ionospheric grid based on spherical harmonics （SH） model and International Reference Ionosphere （IRI-2012） model. Compared with IDW algorithm, the simulation data analysis indicates that the SH algorithm can greatly improve the availability from 82.62% to 99.83% for China’s mainland and from 34.66% to 94.25% for all grids, but the average correction RMS of ionospheric grid increases from 2.60TECU to 3.33TECU.

关键词： regional satellite navigation system ionospheric grid spherical harmonics inverse distance weighting

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Chinese Physics C 2014年第9期38卷 505-505页

作者： K.A.Olive K.Agashe C.Amsler M.Antonelli J.-F.Arguin D.M.Asner H.Baer H.R.Band R.M.Barnett T.Basaglia C.W.Bauer J.J.Beatty V.I.Belousov J.Beringer G.Bernardi S.Bethke H.Bichsel O.Biebe E.Blucher S.Blusk G.Brooijmans O.Buchmueller V.Burkert M.A.Bychkov R.N.Cahn M.Carena A.Ceccucci A.Cerr D.Chakraborty M.-C.Chen R.S.Chivukula K.Copic G.Cowan O.Dahl G.D'Ambrosio T.Damour D.de Florian A.de Gouvea T.DeGrand P.de Jong G.Dissertor B.A.Dobrescu M.Doser M.Drees H.K.Dreiner D.A.Edwards S.Eidelman J.Erler V.V.Ezhela W.Fetscher B.D.Fields B.Foster A.Freitas T.K.Gaisser H.Gallagher L.Garren H.-J.Gerber G.Gerbier T.Gershon T.Gherghetta S.Golwala M.Goodman C.Grab A.V.Gritsan C.Grojean D.E.Groom M.Grnewald A.Gurtu T.Gutsche H.E.Haber K.Hagiwara C.Hanhart S.Hashimoto Y.Hayato K.G.Hayes M.Heffner B.Heltsley J.J.Hernandez-Rey K.Hikasa A.Hocker J.Holder A.Holtkamp J.Huston J.D.Jackson K.F.Johnson T.Junk M.Kado D.Karlen U.F.Katz S.R.Klein E.Klempt R.V.Kowalewski F.Krauss M.Kreps B.Krusche Yu.V.Kuyanov Y.Kwon O.Lahav J.Laiho P.Langacker A.Liddle Z.Ligeti C.-J.Lin T.M.Liss L.Littenberg K.S.Lugovsky S.B.Lugovsky F.Maltoni T.Mannel A.V.Manohar W.J.Marciano A.D.Martin A.Masoni J.Matthews D.Milstead P.Molaro K.Monig F.Moortgat M.J.Mortonson H.Murayama K.Nakamura M.Narain P.Nason S.Navas M.Neubert P.Nevski Y.Nir L.Pape J.Parsons C.Patrignani J.A.Peacock M.Pennington S.T.Petcov Kavli IPMU A.Piepke A.Pomarol A.Quadt S.Raby J.Rademacker G.Raffel B.N.Ratcliff P.Richardson A.Ringwald S.Roesler S.Rolli A.Romaniouk L.J.Rosenberg J L.Rosner G.Rybka C.T.Sachrajda Y.Sakai G.P.Salam S.Sarkar F.Sauli O.Schneider K.Scholberg D.Scott V.Sharma S.R.Sharpe M.Silari T.Sjostrand P.Skands J.G.Smith G.F.Smoot S.Spanier H.Spieler C.Spiering A.Stahl T.Stanev S.L.Stone T.Sumiyoshi M.J.Syphers F.Takahashi M.Tanabashi J.Terning L.Tiator M.Titov N.P.Tkachenko N.A.Tornqvist D.Tovey G.Valencia G.Venanzoni M.G.Vincter P.Vogel A.Vogt S.P.Wakely W.Walkowiak C.W.Walter D.R.Ward G.Weiglein D.H.Weinberg E.J.Weinberg M.White L.R.Wiencke C.G.Wohl L.Wolfenstein J.Womersley C.L. University of Minnesota School of Physics and Astronomy116 Church St.S.E.MinneapolisMN 55455USA University of Maryland Department of PhysicsCollege ParkMD 20742-4111USA Albert Einstein Center for Fundamental Physics Universitat BernCH-3012 BernSwitzerland Lab.Nazionali di Frascati dell'INFN CP 13via E.Fermi40I-00044 Frascati(Roma)Italy Departement de physique Universite de MontrealC.P.6128succ.centre-villeMontreal(Quebec) H3C 3J7Canada Pacific Northwest National Laboratory 902 Battelle BoulevardRichlandWA 99352USA Department of Physics and Astronomy University of OklahomaNormanOK 73019USA Department of Physics University of WisconsinMadisonWI 53706USA Physics Division Lawrence Berkeley National Laboratory1 Cyclotron RoadBerkeleyCA 94720USA CERN European Organization for Nuclear ResearchCH-1211 Geneve 23Switzerland Department of Physics The Ohio State University191 W.Woodruff Ave.ColumbusOH 43210 COMPAS Group Institute for High Energy PhysicsRU- 142284ProtvinoRussia LPNHE IN2P3-CNRS et Universites de Paris 6 et 7F-75252 ParisFrance Max-Planck-Institute of Physics 80805 MunichGermany Department of Physics University of WashingtonSeattleWA 98195USA Ludwig-Maximilians-Universitat Fakultdt für PhysikSchellingstr.4D-80799 MünchenGermany Enrico Fermi Institute and Department of Physics University of ChicagoChicagoIL 60637-1433USA Department of Physics Syracuse UniversitySyracuseNY13244USA Department of Physics Columbia University538 W.120th StreetNew YorkNY10027 USA High Energy Physics Group Blackett LaboratoryImperial CollegePrince Consort RoadLondon SW7 2AZUK Jefferson Lab 12000 Jefferson Ave.Newport NewsVA 23606USA Department of Physics University of VirginiaPO Box 400714CharlottesvilleVA 22904USA Fermi National Accelerator Laboratory P.O.Box 500BataviaIL 60510USA Kavli Institute for Cosmological Physics University of ChicagoChicagoIL 60637-1433USA Department of Physics and Astronomy University of SussexFalmerBrighton BN1 9QHUK Department o

Note： A square-root sign is to be understood over every coefficient, e.g., for -8/15 read - √8√15.

关键词： CLEBSCH-GORDAN COEFFICIENTS spherical harmonics AND d FUNCTIONS

Simulation for MSS-2 low-perigee elliptical orbit satellites:an example of lithospheric magnetic field modelling

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Earth and Planetary Physics 2023年第1期7卷 151-160页

作者： Yi Jiang Nils Olsen JiaMing Ou Qing Yan State Key Laboratory of Lunar and Planetary Sciences Macao University of Science and TechnologyTaipa 999078MacaoChina Division of Geomagnetism DTU SpaceNational Space InstituteTechnical University of DenmarkDiplomvej 3712800 Kgs.LyngbyDenmark

A future constellation of at least four geomagnetic satellites(designated Macao Scientific Satellite-1(MSS-1)and Macao Scientific Satellite-2(MSS-2))was recently proposed,to continue high-quality geomagnetic observations in the post-Swarm period,focusing especially on collecting data that will provide a global,three-dimensional survey of the geomagnetic *** this paper,we present a simulation of two years of orbits(2020.01.01-2022.01.01)of two satellites(tentatively denoted as MSS-2)that are constellated in elliptical(200×5,300 km)low-perigee *** comparing error variances of Gauss coefficients,we investigate the sensitivity of lithospheric magnetic field modelling to data collected from various satellite orbits,including a near circular reference orbit of 300×350km,and elliptical orbit of 180×5,300 km,220×5,300 km,200×3,000 km and 200×1,500 *** find that in two years the two MSS-2 satellites can collect 35,000 observations at altitude below 250 km,data that will be useful in advancing the quality of lithospheric magnetic field modelling;this number of observations reflects the fact that only 4.5%of the flight time of these satellites will be below250 km(just 6.4%of their flight time below 300 km).By combining observations from the MSS-2 satellites’elliptical orbits of 200×5,300km with observations from a circular reference orbit,the variance of the geomagnetic model can be reduced by a factor of 285 at spherical harmonic degree n=200 and by a factor of 1,300 at n=*** planned lower perigee of their orbits allows the new satellites to collect data at unprecedentedly lower altitudes,thus dramatically improving the spatial resolution of satellite-derived lithospheric field models,(up to 80%at n=150).In addition,lowering the apogee increases the time interval during which the satellites fly at near-Earth altitudes,thus improving the model predictions at all spherical harmonic degrees(around 52%-62%at n=150).The upper limit of the expected improvement to t

关键词： elliptical orbit satellite orbit simulation lithosphere field modelling spherical harmonics

Near-Field Source Localization Using spherical Microphone Arrays

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Chinese Journal of Electronics 2016年第1期25卷 159-166页

作者： HUANG Qinghua ZHANG Guangfei LIU Kai Key Laboratory of Specialty Fiber Optics and Optical Access Networks Shanghai University

A new method is proposed for joint range and bearing(azimuth and elevation) estimation of multiple near-field acoustic sources using observations collected by a spherical microphone array. First, spherical Fourier transform(SFT) is used to construct the array signal model in the spherical harmonics domain to decouple range and bearing information. Then the relation among the spherical harmonics of three adjacent degrees is exploited to build the recursive relationship of the signal subspace. Using Eigenvalue decomposition(EVD), bearings are estimated based on the eigenvalues and simultaneously the steering matrix can be represented by the signal ***, range is estimated using the energy ratios of the elements of the steering matrix in the spherical harmonics domain. The algorithm can avoid parameter pairing and multi-dimensional searching. It has lower computational complexity than that of the Multiple signal classification(MUSIC) method. The performance is evaluated by Monte-Carlo simulations and the estimation root meansquare errors are compared to the corresponding CramerRao bounds(CRBs) and those of MUSIC range estimates,which demonstrate the validity of the proposed algorithm.

关键词： spherical microphone array Source localization Near-field spherical Fourier transform(SFT) spherical harmonics

Summability of Fourier-Laplace Series with the Method of Lacunary Arithmetical Means at Lebesgue Points

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Acta Mathematica Sinica,English Series 2001年第3期17卷 489-496页

作者： Feng DAI Kun Yang WANG Department of Mathematics.Beijing Normal University.Beijing 100875,P.R.China E-mail:wangky@bnu,*** ***

Let ∑n-1 be the unit sphere in the n-dimensional Euclidean space Rn.For a function f ∈L(∑n-1) denote by σNδ（f） the Cesàro means of order δ of the Fourier-Laplace series of *** special value λ∶=（n-2）/2 of δ is known as the critical *** the case when n is even,this paper proves the existence of the‘rare’sequence {nk} such that the summability 1/N sum from k=1 to N σnkλ（f）（x）→f（x）,N→∞ takes place at each Lebesgue point satisfying some antipole conditions.

关键词： Convergence Fourier-Laplace series spherical harmonics Lebesgue point

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H^P(S^(d-1))(0

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Acta Mathematica Sinica,English Series 2005年第2期21卷 439-448页

作者： FengDAI KunYangWANG DepartmentofMathematicalandStatisticalSciences CAB632UniversityofAlbertaEdmontonAlbertaT6G2G1Canada DepartmentofMathematics BeijingNormalUniversityBeijing100875P.R.China

Let S^(d-1) = {x : |x| = 1} be a unit sphere of the d-dimensional Euclideanspace R^d and let H^p = H^p(S^(d-1)) (0 -1 is denoted by σ_k~δ(f). For 0 < p ≤1, it is known that δ(p) := (d-1)/p - d/2 is the critical i... 详细信息

Let S^(d-1) = {x : |x| = 1} be a unit sphere of the d-dimensional Euclideanspace R^d and let H^p = H^p(S^(d-1)) (0 spherical polynomials of degree less than or equal to j, in the space H^p(S^(d-1)). Given adistribution f on S^(d-1), its Cesaro mean of order δ > -1 is denoted by σ_k~δ(f). For 0 < p ≤1, it is known that δ(p) := (d-1)/p - d/2 is the critical index for the uniform summability ofσ_k~δ(f) in the metric H^p.

关键词： H^p spaces spherical harmonics cesaro means K-functionals strongapproximation