Incorporating Pólya’s Problem Solving Method in Remedial Math

György Pólya’s problem solving method has influenced generations of mathematicians and non-mathematicians alike. Though almost all math teachers have come across Pólya’s problem solving method, his ideas are not regularly implemented in the classroom. Few studies have examined the effectiveness of his approach in teaching remedial math. In this article we revisit this once well-known teaching method and show how it can be used in basic skills math classes to ease student fears of math, and potentially change their common misconceptions of the subject

From Ancient Greece to Beloch\u27s Crease: The Delian Problem and Origami

The problem of how to double the volume of a cube, also known as the Delian problem, has intrigued mathematicians for millennia. The great variety of solutions discovered over the centuries have used diverse tools, mathematical and otherwise. Recently, in 1936, Margherita Piazollo Beloch discovered a simple and elegant solution to this question. It uses a single piece of paper and a handful of folds. The solution has renewed interest in geometrical constructability problems, in particular those that incorporate origami. And ultimately, it\u27s given rise to the field of origami mathematics.Faculty Sponsor: Shenglan Yua

Large deviations for stochastic systems of slow-fast diffusions with non-Gaussian L\'evy noises

We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and L\'evy noises. The fast variables evolve at much faster time scale than the slow variables, but they are fully inter-dependent. We study the asymptotics of the logarithmic functionals of the slow variables in the three regimes based on viscosity solutions to the Cauchy problem for a sequence of partial integro-differential equations. We also verify the comparison principle for the related Cauchy problem to show the existence and uniqueness of the limit for viscosity solutions

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numerical experiments demonstrate that our control strategy is effective and sufficiently accurate