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40
Yet another fast multipole method without multipoles  Pseudoparticle multipole method
, 1999
"... In this paper we describe a new approach to implement the O(N) fast multipole method and O(N log N) tree method, which uses pseudoparticles to express the potential field. The new method is similar to Anderson's method, which uses the values of potential at discrete points to represent the pote ..."
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Cited by 27 (1 self)
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In this paper we describe a new approach to implement the O(N) fast multipole method and O(N log N) tree method, which uses pseudoparticles to express the potential field. The new method is similar to Anderson's method, which uses the values of potential at discrete points to represent the potential field. However, for the same expansion order the new method is more accurate. 1 Introduction The tree algorithms [2, 3] are now widely used in astrophysical community. For astrophysical simulations, the tree algorithms are particularly suitable because of the adaptive nature of the algorithm. However, the use of tree algorithms in astrophysics has been limited to problems with relatively short timescales, such as collisions of two galaxies or large scale structure formation of the universe. This is mainly because of the high calculation cost associated with highaccuracy calculation. Existing implementations of BarnesHut treecode use only up to quadrupole moment. Therefore the calculati...
Extremal systems of points and numerical integration on the sphere
 Adv. Comput. Math
"... 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 ..."
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Cited by 27 (5 self)
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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.
Analysis and synthesis of soundradiation with spherical arrays
, 2009
"... ii This work demonstrates a comprehensive methodology for capture, analysis, manipulation, and reproduction of spatial soundradiation. As the challenge herein, acoustic events need to be captured and reproduced not only in one but in a preferably complete multiplicity of directions, instead. The s ..."
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Cited by 15 (6 self)
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ii This work demonstrates a comprehensive methodology for capture, analysis, manipulation, and reproduction of spatial soundradiation. As the challenge herein, acoustic events need to be captured and reproduced not only in one but in a preferably complete multiplicity of directions, instead. The solutions presented in this work are using the soapbubble model, a working hypothesis about soundradiation, and are based on fundamental mathematical descriptions of spherical acoustic holography and holophony. These descriptions enable a clear methodic approach of soundradiation capture and reproduction. In particular, this work illustrates the implementation of surrounding spherical microphone arrays for the capture of soundradiation, as well as the analysis of soundradiation with a functional model. Most essential, the thesis shows how to obtain holophonic reproduction of soundradiation. For this purpose, a physical model of compact spherical loudspeaker arrays is established alongside with its electronic control. iii iv
Optimal asymptotic bounds for spherical designs. arXiv:1009.4407v3 [math.MG
, 2011
"... Abstract In this paper we prove the conjecture of Korevaar and Meyers: for each ..."
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Cited by 13 (1 self)
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Abstract In this paper we prove the conjecture of Korevaar and Meyers: for each
Packing planes in four dimensions and other mysteries
 in Proceedings of the Conference on Algebraic Combinatorics and Related Topics
, 1997
"... How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, ..."
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How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains and P. W. Shor. We have found many nice examples of specific packings (70 4spaces in 8space, for instance), several general constructions, and an embedding theorem which shows that a packing in Grassmannian space G(m,n) is a subset of a sphere in R D, D = (m + 2)(m − 1)/2, and leads to a proof that many of our packings are optimal. There are a number of interesting unsolved problems. 1.
Construction of spherical cubature formulas using lattices
, 2005
"... We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked ..."
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Cited by 9 (0 self)
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We construct cubature formulas on spheres supported by homothetic images of shells in some Euclidian lattices. Our analysis of these cubature formulas uses results from the theory of modular forms. Examples are worked
Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
"... Abstract. This paper is concerned with proving the existence of solutions to an underdetermined system of equations, and the application to existence of spherical tdesigns with (t+1)2 points on the unit sphere S2 in R3. We show that the construction of spherical designs is equivalent to solution o ..."
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Cited by 8 (1 self)
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Abstract. This paper is concerned with proving the existence of solutions to an underdetermined system of equations, and the application to existence of spherical tdesigns with (t+1)2 points on the unit sphere S2 in R3. We show that the construction of spherical designs is equivalent to solution of underdetermined equations. A new verification method for underdetermined equations is derived using the Brouwer fixed point theorem. Application of the method provides spherical tdesigns which are close to extremal (maximum determinant) points and have the optimal order O(t2) for the number of points. An error bound for the computed spherical designs is provided. Key words. Verification, underdetermined system, spherical designs, extremal points, interpolation, numerical integration.
Nonexistence of certain spherical designs of odd strengths and cardinalities, Discrete Comput
 Geom
, 1999
"... A spherical τdesign on S n−1 is a finite set such that, for all polynomials f of degree at most τ, the average of f over the set is equal to the average of f over the sphere S n−1. In this paper we obtain some necessary conditions for the existence of designs of odd strengths and cardinalities. Thi ..."
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Cited by 7 (3 self)
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A spherical τdesign on S n−1 is a finite set such that, for all polynomials f of degree at most τ, the average of f over the set is equal to the average of f over the sphere S n−1. In this paper we obtain some necessary conditions for the existence of designs of odd strengths and cardinalities. This gives nonexistence results in many cases. Asymptotically, we derive a bound which is better than the corresponding estimation ensured by the DelsarteGoethalsSeidel bound. We consider in detail the strengths τ =3andτ = 5 and obtain further nonexistence results in these cases. When the nonexistence argument does not work, we obtain bounds on the minimum distance of such designs. 1
Flexible and optimal design of spherical microphone . . .
, 2007
"... This paper describes a methodology for designing a flexible and optimal spherical microphone array for beamforming. Using the approach presented, a spherical microphone array can have very flexible layouts of microphones on the spherical surface, yet optimally approximate a desired beampattern of h ..."
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Cited by 7 (0 self)
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This paper describes a methodology for designing a flexible and optimal spherical microphone array for beamforming. Using the approach presented, a spherical microphone array can have very flexible layouts of microphones on the spherical surface, yet optimally approximate a desired beampattern of higher order within a specified robustness constraint. Depending on the specified beampattern order, our approach automatically achieves optimal performances in two cases: when the specified beampattern order is reachable within the robustness constraint we achieve a beamformer with optimal approximation of the desired beampattern; otherwise we achieve a beamformer with maximum directivity, both robustly. For efficient implementation, we also developed an adaptive algorithm for computing the beamformer weights. It converges to the optimal performance quickly while exactly satisfying the specified frequency response and robustness constraint in each step. One application of the method is to allow the building of a realworld system, where microphones may not be placeable on regions, such as near cable outlets and/or a mounting base, while having a minimal effect on the performance. Simulation results are presented.
Angular and radial directivity control for spherical loudspeaker arrays. Master’s project
 University of Music and Performing Arts, Institute of Electronic Music and Acoustics
, 2008
"... Spherical loudspeaker arrays, as concerned within this work, are a finite set of transducers distributed on the surface of a sphere or platonic solid. The purpose of these arrays is to synthesize artificial acoustic radiation or to reproduce natural sound sources. This relatively recent research t ..."
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Spherical loudspeaker arrays, as concerned within this work, are a finite set of transducers distributed on the surface of a sphere or platonic solid. The purpose of these arrays is to synthesize artificial acoustic radiation or to reproduce natural sound sources. This relatively recent research topic is applicable in many fields, such as musical performances or acoustic measurements. The present thesis develops and discusses a control system for directivity pattern synthesis using an icosahedral loudspeaker array. In order to obtain sensible control parameters a surrounding spherical microphone array is used to measure the individual directivities of the array transducers. Basically, weighted combinations can be computed to create a variable directivity directly at the measurement radius. It is, however, advantageous to decompose these directivities into orthogonal spherical harmonic components. At this radius the spherical harmonics provide directivity synthesis with well defined angular resolution and simple relations for rotation of synthesis patterns. Arbitrarily, synthesis patterns will appear blurred at other radii due to sound propagation. Inherently, the spherical harmonics are affected by well