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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
Diserahkan: 9 Mac 2012/Diterima: 20 Oktober 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
RUJUKAN
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.
*Pengarang untuk surat-menyurat; email: ckartim3371@gmail.com
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