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Spherical and Spheroidal Harmonics: Examples and Computations Spherical and Spheroidal Harmonics: Examples and Computations Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Lin Zhao, B.S., M.S. Graduate Program in Mathematics The Ohio State University 2017 Thesis Committee: Ghaith A. Hiary, Advisor Roman Holowinsky © Copyright by Lin Zhao 2017 Abstract This thesis introduces the theory of spherical and spheroidal harmonics, with em- phasis on algorithms, computations, and explicit examples. In particular, we collect in one place several viewpoints on spherical and spheroidal harmonics, and clarify how the various viewpoints are related and how to transition between them. The ultimate goal is to be able to apply the spheroidal harmonics in calculating the spacing dis- tributions of eigenvalues of the Gaussian unitary ensemble (GUE) random matrices. These eigenvalue statistics are conjectured to model corresponding statistics for zeros of the Riemann zeta function. New results are presented for comparing statistics of zeros and eigenvalues. ii Acknowledgments First and foremost, I am deeply indebted to my advisor, Prof. Ghaith A. Hiary for his guidance over my master’s study in Department of Mathematics. This thesis would not have been possible without his consistent support, great patience, and continual encouragement. I feel very fortunate to be introduced to this interesting topic and it has been a wonderful journey delving into it. I had the opportunity to see broad connections to various aspects of mathematics, as well as interesting applications in many other disciplines. The many discussions with Prof. Hiary have been tremendously helpful. I benefited a lot from his extremely organized and rigorous way of thinking, as well as his meticulousness. I will also be amused by his jokes about mathematicians and physicists from time to time. I am also very thankful to Prof. Roman Holowinsky for serving on my committees and for trying to accommodate my schedule at his best. I owe many thanks to Mr. Roman Nitze, who has been remarkably patient and kind to help me on every step of applying the dual degree. I am also very grateful for all the professors who taught me from the math department. Their passion and dedication to the classes greatly inspired me to think deeply, thoroughly, and creatively. I am privileged to be able to learn from them. Finally, I would like to thank my family for their love, care, encouragement, and support over the years. This thesis is dedicated to them. iii Vita 2010 . B.S., Control Science and Engineering, Harbin Institute of Technology, China. 2012 . M.S., Control Science and Engineering, Harbin Institute of Technology, China. 2012–present . Graduate Research Associate, Electrical and Computer Engineering, The Ohio State University 2016-present . M.S. Student, Mathematics, The Ohio State University Publications Research Publications L. Zhao, W. Zhang, J. Hu, A. Abate, and C.J. Tomlin. “On the optimal solu- tions of the infinite-horizon linear sensor scheduling problem”, IEEE Transactions on Automatic Control, 59(10):2825-2830, March 2014. L. Zhao and W. Zhang. A unified stochastic hybrid system approach to aggregate modeling of responsive loads, IEEE Transaction on Automatic Control, 2018, to be published. L. Zhao, W. Zhang, H. Hao, and K. Kalsi. A geometric approach to aggregate flexibility modeling of thermostatically controlled loads, IEEE Transactions on Power Systems, 32(6):4721-4731, November 2017. L. Zhao, H. Hao, and W. Zhang. Extracting flexibility of heterogeneous deferrable loads via polytopic projection approximation, The 55th IEEE Conference on Decision and Control, Las Vegas, USA, 2016. iv L. Zhao, and W. Zhang. A geometric approach to virtual battery modeling of thermostatically controlled loads, The American Control Conference, Boston, MA, 2016. L. Zhao and W. Zhang. A unified stochastic hybrid system approach to aggregated load modeling for demand response, The 54th IEEE Conf. on Decision and Control, Osaka, Japan, 2015. S. Li, W. Zhang, L. Zhao, J. Lian and K. Kalsi. On social optima of non-cooperative mean field games, The 55th IEEE Conference on Decision and Control, Las Vegas, USA, 2016. Fields of Study Major Field: Mathematics v Table of Contents Page Abstract ....................................... ii Acknowledgments .................................. iii Vita ......................................... iv List of Tables .................................... viii List of Figures ................................... ix 1. Introduction .................................. 1 2. Harmonic Analysis on the Sphere Sn−1 in Euclidean Coordinates .... 5 2.1 Laplacian and Invariant Integrals on Spheres ............ 7 2.2 Eigenfunctions of the Spherical Laplacian .............. 10 2.3 Decomposition of Homogeneous Polynomials ............ 12 2.4 L2 Spectral Decomposition on Spheres ................ 15 3. Spherical Harmonics Basis in Euclidean Coordinates ........... 18 3.1 Kelvin Transform ............................ 18 3.2 Projection of Homogeneous Polynomials to Hd spaces ....... 23 3.3 Examples ................................ 25 4. Harmonic Analysis on the sphere Sn−1 in Spherical Coordinates ..... 29 4.1 Spherical Harmonics (n = 3) ..................... 29 4.1.1 Legendre Equation and Polynomials ............. 31 4.1.2 Rodrigues’ Formula ...................... 34 vi 4.1.3 General Solution of Spherical Laplace Equation ....... 35 4.2 Hyperspherical Harmonics (n > 3) .................. 41 4.2.1 Gegenbauer polynomials .................... 41 4.2.2 Hyperspherical Harmonics .................. 43 5. Harmonics Analysis on the Spheroid .................... 45 5.1 Prolate Spheroidal Coordinates .................... 45 5.2 Spheroidal Harmonics ......................... 48 5.3 Series Expansion Representation of Spheroidal Wave Functions .. 50 6. Application Example ............................. 58 6.1 Linear Prolate Function ........................ 59 6.2 GUE Eigenvalue Spacing Distribution ................ 61 6.3 Simulation Results ........................... 63 Bibliography .................................... 69 Appendices 75 A. Harmonic Analysis on the Flat Space Rm .................. 75 B. Harmonic Analysis on the Torus Rm=Zm .................. 80 C. Recurrence Formulas ............................. 82 D. Mathematica Code .............................. 85 vii List of Tables Table Page 3.1 Basis of Hd for different dimensions ................... 26 viii List of Figures Figure Page 2.1 Spherical coordinate systems (left) and Mathematica illustration with 2 8 ϕ [0; 5 π] (right) ............................. 6 4.1 Legendre Polynomials .......................... 34 5 5 3 4.2 Mathematica density plot for ReY10(θ; ϕ) (left) and ReY10(θ; ϕ)+ReY5 (θ; ϕ) (right) ................................... 40 4.3 Gegenbauer polynomials ......................... 43 5.1 Prolate spheroidal coordinate systems (left) and Mathematica illustra- 2 8 tion with ϕ [0; 5 π] (right) ....................... 47 2 8 5.2 Constant coordinate curves of equation (5.3) for ϕ [0; 5 π] with dif- ferent values of ξ = cosh µ, where µ 2 f0:4; 0:6; 0:9; 1:2; 1:5g. ..... 47 5.3 Examples of prolate spheroidal wave functions: angular functions (left) and radial functions (right) ....................... 55 ¯ 5.4 Renormalized angular functions Sml(5; η) ............... 56 5.5 R10;10(5; ξ) obtained from Mathematica built-in function (left) and Graham’s modified function (right) ................... 56 5.6 Mathematica density plot for R3;4(3; ξ)S3;4(3; η) cos 3ϕ on the left and R3;4(3; ξ)S3;4(3; η) cos 3ϕ + R2;4(3; ξ)S2;4(3; η) cos 2ϕ on the right ... 57 6.1 Probability density of the nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). 64 ix 6.2 Probability density of the next-nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). .................................. 65 6.3 Probability density of the 3rd nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). .................................. 67 6.4 Probability density of the 4th nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). .................................. 67 6.5 Difference of the probability densities between the empirical dataand GUE prediction. Left: k = 0, N0 = 4 with different number of Rie- mann zeta zeros; Right: k = 1 with N0 = 4 of the right figure in Fig. 6.2. ................................. 68 6.6 Difference of the probability densities between the empirical dataand GUE prediction. Left: k = 2 with N0 = 5; 6; Right: k = 3 with N0 = 5; 6; 7. ............................... 68 x Chapter 1: Introduction Harmonic analysis on the sphere is perhaps less familiar than the well-studied examples of harmonic analysis on the flat space Rn and the torus Rm=Zm. This is partly because Sn−1 does not have a natural group structure, in contrast to Rn or Rm=Zm which are abelian additive groups. Still, the sphere Sn−1, like the torus, is a compact space with a natural group action on it. Specifically, it is acted on transitively by the rotation
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