Survey of Spectra of Laplacians on Finite Symmetric Spaces

REFERENCES

• Adolphson , A. 1989 . “On the distribution of angles of Kloosterman sums” . J. reine angew. Math. , 395 : 214 – 220 . [Adolphson 1989]
   Web of Science ®Google Scholar
• Angel , J. 1993 . “Finite upper half planes over finite rings and their associated graphs” , Ph.D. Thesis San Diego : Univ. of California. . [Angel 1993]
   Google Scholar
• Angel , J. , Celniker , N. , Poulos , S. , Terras , A. , Trimble , C. and Velasquez , E. 1992 . “Special functions on finite upper half-planes” . Contemporary Math. , 138 : 1 – 26 . [Angel et al. 1992]
   Google Scholar
• Angel , J. , Poulos , S. , Terras , A. , Trimble , C. and Velasquez , E. 1994 . “Spherical functions and transforms on finite upper half-planes: eigenvalues of the combinatorial Laplacian, uncertainty, traces” . Contemporary Math. , 173 : 15 – 70 . [Angel et al. 1994]
   Google Scholar
• Angel , J. , Shook , B. , Terras , A. and Trimble , C. 1995 . “Graph spectra for finite upper half-planes over rings” . Linear Algebra and its Applications , 226–228 : 423 – 457 . [Angel et al. 1995]
   Web of Science ®Google Scholar
• Arthur , James . 1989 . “The trace formula and Hecke operators”. ” . In Number Theory, Trace Formulas, and Discrete Groups Edited by: Aubert , K. E. 11 – 27 . Boston : Academic Press. . [Arthur 1989]
   Google Scholar
• Bannai , E. and Ito , T. 1984 . Algebraic Combinatorics I: Association Schemes Menlo Park , CA : Benjamin/Cummings. . [Bannai and Ito 1984]
   Google Scholar
• Biggs , N. 1974 . Algebraic Graph Theory London : Cambridge U. Press. . [Biggs 1974]
   Google Scholar
• Buser , P. 1992 . Geometry and Spectra of Compact Riemann Surfaces Boston : Birkhauser. . [Buser 1992]
   Google Scholar
• Carlitz , L. 1953 . “Weighted quadratic partitions over a finite field” . Canadian J. Math. , 5 : 317 – 323 . [Carlitz 1953]
   Web of Science ®Google Scholar
• Celniker , N. 1994 . “Eigenvalue bounds and girths of graphs of finite, upper half-planes” . Pacific J. of Math. , 166 : 1 – 21 . [Celniker 1994]
   Web of Science ®Google Scholar
• Celniker , N. , Poulos , S. , Terras , A. , Trimble , C. and Velasquez , E. 1993 . “Is there life on finite upper half-planes?” . Contemporary Math. , 143 : 65 – 88 . [Celniker et al. 1993]
   Google Scholar
• Chung , F. 1991 . “ Constructing random-like graphs. ” . In Probabilistic Combinatorics and Its Applications Edited by: Bollobás , B. 21 – 55 . Providence : Amer. Math. Soc. . [Chung 1991], San Francisco, 1991, Proc. Symp. Appl. Math. 44
   Google Scholar
• Courant , R. and Hilbert , D. 1961 . Methods of Mathematical Physics I New York : Wiley. . [Courant and Hilbert 1961]
   Google Scholar
• Diaconis , P. 1988 . Group Representations in Probability and Statistics Hayward , CA : Inst. Math. Statistics. . [Diaconis 1988], Lecture Notes–Monograph Series 11
   Google Scholar
• Evans , R. 1995 . “Spherical functions for finite upper half-planes with characteristic two” . Finite Fields and Their Applications , 1 : 376 – 394 . [Evans 1995]
   Google Scholar
• Evans , R. 1994 . “Character sums as orthogonal eigenfunctions of adjacency operators for Cayley graphs” . Contemporary Math. , 168 : 33 – 50 . [Evans 1994]
   Google Scholar
• Friedman , J. 1993 . Expanding Graphs Providence : Amer. Math. Soc. . [Friedman 1993], Princeton, 1992, DIMACS Series in Discrete Math, and Theor. Comp. Sci. 10
   Google Scholar
• Goetgheluck , P. 1993 . “Fresnel zones on the screen” . Experimental Math. , 2 : 301 – 309 . [Goetgheluck 1993]
   Google Scholar
• Gutzwiller , M. 1990 . Chaos in Classical and Quantum Mechanics New York : Springer. . [Gutzwiller 1990]
   Google Scholar
• Hashimoto , K. 1989 . “Zeta functions of finite graphs and representations of p-adic groups”. ” . In Automorphic Forms and Geometry of Arithmetic Varieties Edited by: Hashimoto , K. and Namikawa , Y. 211 – 280 . Boston : Academic Press. . [Hashimoto 1989], Advanced Studies in Pure Math. 15
   Google Scholar
• Hejhal , D. and Rackner , B. 1992 . “On the topography of Maass waveforms for PSL(2,Z)” . Experimental Math. , 1 : 275 – 305 . [Hejhal and Rackner 1992]
   Google Scholar
• Katz , N. 1980 . Sommes Exponentielles 1 – 209 . Paris : Soc. Math. France. . [Katz 1980], Astérisque 79
   Google Scholar
• Katz , N. 1988 . Gauss Sums, Kloosterman Sums, and Monodromy Groups Princeton : Princeton U. Press. . [Katz 1988]
   Google Scholar
• Katz , N. 1993 . “Estimates for Soto-Andrade sums” . J. reine angew. Math. , 438 : 143 – 161 . [Katz 1993]
   Web of Science ®Google Scholar
• Katz , N. 1995 . “A note on exponential sums” . Finite Fields Appl. , 1 : 395 – 398 . [Katz 1995]
   Google Scholar
• Krieg , A. 1990 . Hecke Algebras . Memoirs Amer. Math. Soc. , 87 ( #435 ) : 158 [Krieg 1990]
   Web of Science ®Google Scholar
• Kuang , J. “Eigenfunctions on the finite Poincaré plane” [Kuang 1995], preprint available at http://www.msri.org/MSRI-preprints/online/1995-015.html
   Google Scholar
• Lafferty , J. D. and Rockmore , D. 1992 . “Fast Fourier analysis for SL2over a finite field and related numerical experiments” . Experimental Math. , 1 : 115 – 139 . [Lafferty and Rockmore 1992]
   Google Scholar
• Lafferty , J. D. and Rockmore , D. “Numerical investigation of the spectrum for certain families of Cayley graphs” 63 – 73 . [Lafferty and Rockmore 1993], in [Friedman 1993]
   Google Scholar
• Lang , S. 1965 . Algebra Reading , MA : Addison-Wesley. . [Lang 1965]
   Google Scholar
• Li , W.-C. W. 1992 . “Character sums and Abelian Ramanujan graphs” . J. Number Th. , 41 : 199 – 217 . [Li 1992]
   Web of Science ®Google Scholar
• Li , W.-C. W. 1996 . “A survey of Ramanujan graphs”. ” . In Arithmetic, Geometry, and Coding Theory Edited by: Pellikaan , R. Berlin : de Gruyter. . [Li 1996], Luminy, France, 1993
   Google Scholar
• Li , W.-C. W. “Some character sum estimates” [Li a], preprint
   Google Scholar
• Lubotzky , A. 1994 . Discrete Groups, Expanding Graphs, and Invariant Measures Basel : Birkhäuser. . [Lubotzky 1994]
   Google Scholar
• Lubotzky , A. , Phillips , R. and Sarnak , P. 1988 . “Ramanujan graphs” . Combinatorica , 8 : 261 – 277 . [Lubotzky et al. 1988]
   Web of Science ®Google Scholar
• McKay , B. D. 1981 . “The expected eigenvalue distribution of a large regular graph” . Linear Algebra Appl , 40 : 203 – 216 . [McKay 1981]
   Web of Science ®Google Scholar
• Lubotzky , A. 1995 . MATLAB Version 4 User's Guide Englewood Cliffs , NJ : Prentice Hall. . [MathWorks 1995]
   Google Scholar
• Medrano , A. , Myers , P. , Stark , H. and Terras , A. 1996 . “Finite analogues of Euclidean space” . J. Comp. Appl. Math. , 68 [Medrano et al. a]
   Web of Science ®Google Scholar
• Medrano , A. , Myers , P. , Stark , H. and Terras , A. “Finite Euclidean graphs over rings” [Medrano et al. b], preprint
   Google Scholar
• Mehta , M. L. 1967 . Random Matrices and the Statistical Theory of Energy Levels New York : Academic Press. . [Mehta 1967]
   Google Scholar
• Myers , P. 1995 . “Euclidean and Heisenberg graphs: spectral properties and applications” , Ph.D. Thesis San Diego : Univ. California. . [Myers 1995]
   Google Scholar
• Poulos , S. 1991 . “Graph theoretic and spectral properties of finite upper half-planes” , Ph.D. Thesis San Diego : Univ. California. . [Poulos 1991]
   Google Scholar
• Powers , D. 1987 . Boundary Value Problems, , 3rd ed. San Diego : Harcourt, Brace, Jovanovich. . [Powers 1987]
   Google Scholar
• Salié , H. 1932 . Über die Kloostermanschen Summen S(u, v; q) . Math. Z. , 34 : 91 – 109 . [Salié 1932]
   Google Scholar
• Sarnak , P. 1990 . Some Applications of Modular Forms Cambridge : Cambridge U. Press. . [Sarnak 1990]
   Google Scholar
• Sarnak , P. 1995 . “Arithmetic quantum chaos”. ” . In The Schur Lectures 183 – 236 . Ramat-Gan : Bar-Ilan University. . [Sarnak 1995], (Tel Aviv, 1992), Israel Math. Soc. Conf. Proc. 8
   Google Scholar
• Sarnak , P. 1995 . “Spectra of hyperbolic surfaces” Princeton University. . [Sarnak a], preprint
   Google Scholar
• Schmidt , W. 1976 . Equations over Finite Fields: An Elementary Approach New York : Springer. . [Schmidt 1976], Lecture Notes in Math. 536
   Google Scholar
• Small , C. 1991 . Arithmetic of Finite Fields New York : Dekker. . [Small 1991]
   Google Scholar
• Soto-Andrade , J. “Geometrical Gel'fand models, tensor quotients, and Weil representations” . Arcata Conference on Representations of Finite Groups . Arcata , CA . pp. 305 – 316 . Providence : Amer. Math. Soc. . [Soto-Andrade 1987], Proc. Symp. Pure Math. 47, part 2
   Google Scholar
• Soto-Andrade , J. and Vargas , J. “Harmonic analysis on the twisted finite Poincaré upper half plane” [Soto-Andrade and Vargas 1995], preprint available at http://www.msri.org/MSRI-preprints/online/1995-073.html
   Google Scholar
• Stanton , D. 1990 . “An introduction to group representations and orthogonal polynomials”. ” . In Orthogonal Polynomials: Theory and Practice Edited by: Nevai , P. and Ismail , M. E. H. 419 – 433 . Dordrecht : Kluwer. . [Stanton 1990]
   Google Scholar
• Stark , H. 1987 . “Modular forms and related objects” . Canadian Math. Soc. Conf. Proc. , 7 : 421 – 455 . [Stark 1987]
   Google Scholar
• Stark , H. and Terras , A. “Zeta functions of finite graphs and coverings” [Stark and Terras], to appear in Advances in Math.
   Google Scholar
• Sternberg , S. 1994 . Group Theory and Physics Cambridge : Cambridge U. Press. . [Sternberg 1994]
   Google Scholar
• Sunada , T. 1988 . “Fundamental groups and Laplacians”. ” . In Geometry and Analysis on Manifolds Edited by: Sunada , T. 248 – 277 . New York : Springer. . [Sunada 1988], Katata/Kyoto, 1987, Lecture Notes in Math. 1339
   Google Scholar
• Terras , A. 1985 . Harmonic Analysis on Symmetric Spaces and Applications I New York : Springer. . [Terras 1985]
   Google Scholar
• Terras , A. 1988 . Harmonic Analysis on Symmetric Spaces and Applications II New York : Springer. . [Terras 1988]
   Google Scholar
• Terras , A. 1991–1992 . “Fourier analysis on finite groups and applications” San Diego : Univ. California. . [Terras 1991], Lecture Notes
   Google Scholar
• Velasquez , E. 1991 . “The Radon transform on finite groups” , Ph.D. Thesis San Diego : Univ. California. . [Velasquez 1991]
   Google Scholar
• Venkov , A. B. and Nikitin , A. M. 1994 . “The Selberg trace formula, Ramanujan graphs, and some problems of mathematical physics” . Petersburg Math. J. , 5 : 419 – 484 . [Venkov and Nitikin 1994]
   Google Scholar
• Weil , A. 1948 . “On some exponential sums” . Proc. Nat. Acad. Sci. , 34 : 204 – 207 . [Weil 1948], Reprinted as pp. 386–389 of his Collected Works I, Springer, New York, 1980
   PubMed Web of Science ®Google Scholar
• Wolfram , S. 1991 . Mathematica: A System for Doing Mathematics by Computer, , 2nd ed. Reading , MA : Addison–Wesley. . [Wolfram 1992]
   Google Scholar