Sains Malaysiana 42(5)(2013): 655–660

Computation and Visualization of Cuspidal Waveforms for Modular Group Using GridMathematica

(Pengiraan dan Visualisasi Gelombang Berjuring untuk Kumpulan Modular

dengan Menggunakan GridMathematica)

Chan Kar Tim* & Hishamuddin Zainuddin

Department of Physics, Faculty of Science, Universiti Putra Malaysia

43400 Serdang, Selangor, Malaysia

Saeid Molladavoudi & Hishamuddin Zainuddin

Laboratory of Computational Sciences and Mathematical Physics

Institute for Mathematical Research, Universiti Putra Malaysia

43400 Serdang, Selangor, Malaysia

Received: 9 March 2012/Accepted: 20 October 2012

ABSTRACT

Spectral studies on the eigenfunctions of Laplace-Beltrami operator on a cusp manifold are known to contain both discrete and continuous eigenvalues. The discrete eigenfunctions are usually called Maass cusp forms where their eigenvalues are not known analytically. The aims of this report were to compute the eigenvalues λ = r2 + 1/4 for the modular group, PSL(2,Z) numerically and visualize the waveforms using GridMathematica. At the same time, we compared the performance of parallel programming (GridMathematica) and normal programming (Mathematica). This serves to show the feasibility and advantages of using the parallel version of commercially available software for complex computations of Maass cusp forms. In our computer search for 33 eigenvalues in the r-interval [9, 30.4], we found that the performance of the parallel programme is about six times faster than the normal programme.

Keywords: GridMathematica; Maass cusp forms; modular group

ABSTRAK

Kajian spektrum pada fungsi eigen operator Laplace-Beltrami di atas permukaan berjuring diketahui mempunyai nilai eigen yang diskrit dan selanjar. Fungsi eigen diskrit biasanya dikenali sebagai fungsi bentuk juring Maass dengan nilai eigennya tidak diketahui secara analitik. Tujuan kertas ini adalah untuk mengira nilai eigen λ = r2 + 1/4 bagi kumpulan modular, PSL(2,Z) secara berangka dan menggambarkan gelombangnya dengan menggunakan GridMathematica. Pada masa yang sama, kami juga membandingkan prestasi pengaturcaraan selari (GridMatematica) dengan pengaturcaraan biasa (Mathematica). Ini bertujuan untuk menunjukkan kebolehlaksanaan dan kelebihan menggunakan perisian komersial versi selari untuk pengiraan kompleks fungsi bentuk juring Maass. Dalam carian komputer untuk 33 nilai eigen dalam selang-r [9, 30.4], didapati bahawa prestasi pengaturcaraan selari adalah lebih kurang enam kali ganda lebih laju daripada pengaturcaraan biasa.

Kata kunci: Fungsi bentuk juring Maass; GridMathematica; kumpulan modular

REFERENCES

Booker, A.R., Strombergsson, A. & Venkatesh, A. 2006. Effective computation of Maass cusp forms. International Mathematics Research Notices IMRN pp. 1-34. doi: 10.1155/1MRN/2006/71281.

Gutzwiller, M.C. 1990. Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag.

Hejhal, D.A. & Rackner, B.N. 1992. On the topography of Maass waveforms for PSL(2,Z). Experimental Mathematics 1(4): 275-305.

Miyake, T. 1989. Modular Forms. Berlin: Springer-Verlag.

Risager, M.S. 2004. Asymptotic densities of Maass newforms. Journey of Number Theory 109: 96-119.

Siddig, A.A.M. & Zainuddin, H. 2009. Computation of Maass cusp forms on modular group in Mathematica. International Journal of Pure and Applied Mathematics 54(2): 279-295.

Stromberg, F. 2005. Computational aspects of Maass waveforms, Ph.D. Thesis. University of Uppsala, Sweden (unpublished).

Then, H. 2005. Maass cusp forms for large eigenvalues. Mathematics of Computation 74(249): 368-381.

*Corresponding author; email: ckartim3371@gmail.com