Rotation invariant patterns for a nonlinear Laplace-Beltrami equation: A Taylor-Chebyshev series approach
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Bibliographic record
Abstract
<p style='text-indent:20px;'>In this paper, we introduce a rigorous computational approach to prove existence of rotation invariant patterns for a nonlinear Laplace-Beltrami equation posed on the 2-sphere. After changing to spherical coordinates, the problem becomes a singular second order boundary value problem (BVP) on the interval <inline-formula><tex-math id="M1">\begin{document}$ (0,\frac{\pi}{2}] $\end{document}</tex-math></inline-formula> with a <i>removable</i> singularity at zero. The singularity is removed by solving the equation with Taylor series on <inline-formula><tex-math id="M2">\begin{document}$ (0,\delta] $\end{document}</tex-math></inline-formula> (with <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> small) while a Chebyshev series expansion is used to solve the problem on <inline-formula><tex-math id="M4">\begin{document}$ [\delta,\frac{\pi}{2}] $\end{document}</tex-math></inline-formula>. The two setups are incorporated in a larger zero-finding problem of the form <inline-formula><tex-math id="M5">\begin{document}$ F(a) = 0 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M6">\begin{document}$ a $\end{document}</tex-math></inline-formula> containing the coefficients of the Taylor and Chebyshev series. The problem <inline-formula><tex-math id="M7">\begin{document}$ F = 0 $\end{document}</tex-math></inline-formula> is solved rigorously using a Newton-Kantorovich argument.</p>
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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