School of Science Department of Mathematics 51 Quantum Groups Supervisor: IP Ivan Chi Ho / MATH Student: WANGSA Devandhira Wijaya / MATH-PMA Course: UROP 2100, Fall UROP 3200, Spring The Integral Ramanujan Identity for the Quantum Dilogarithm has been stated in many papers authored by notable figures in the field such as Faddeev, Kashaev, Teschner, Volkov; yet the rigorous proof of the identity is nowhere to be found in current literature. In this paper, we present the rigorous proof of the Integral Ramanujan Identity and its proper conditions where the identity holds. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: KUSUMO William Max / MAEC Course: UROP 1000, Summer This report examines the reverse use of the Method of Characteristics, in which systems of Ordinary Differential Equations (ODEs) are converted into Partial Differential Equations (PDEs), focusing on linear scalar and matrix-valued settings. Although the method is usually applied to simplify PDEs by reducing them to ODEs, we show that characteristic curves can also be used to express solutions of ODEs in PDE form. The scalar case leads to explicit and very direct solutions. The matrix case, illustrated through the computation of Lyapunov Characteristic Exponents, highlights how multidimensional transformations evolve along trajectories. While the discussion is restricted to linear systems, the approach captures the essential ideas behind the Method of Characteristics and suggests how similar ideas may be extended to nonlinear and more complex problems. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: LI Jiaran / DSCT Course: UROP 1100, Fall UROP 2100, Summer We present a neural network-based approach for approximating the flow map of dynamical systems and computing the Finite Time Lyapunov Exponent through gradients of the learned mapping. Our original design included a physics-informed loss term, but the training process failed to converge when this term was added. In contrast, a conventional loss function that relies only on observed data leads to stable convergence and produces FTLE fields of reasonable quality. In addition, we investigate the effect of sampling density and distribution on the accuracy of the resulting FTLE computation.
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