Jared Weed

Jared Weed, Bowei Wu, Jingfang Huang et al.

In this paper, we present an accurate numerical method for the time-harmonic Maxwell's equations for bi-periodic multilayered media with quasi-periodic incident waves using the Method of Fundamental Solutions in conjunction with a periodization scheme. Following an approach used in acoustic scattering problems, the electric and magnetic fields in each layer are expressed as a sum of near and distant interactions. The near interaction comprises interactions between the unit cell and its nearest neighboring copies, while the distant interaction is approximated by proxy source points placed on spheres surrounding the unit cell. Imposing continuity of tangential components at the layer interface, quasi-periodicity conditions on the walls of the unit cell, and Rayleigh-Bloch expansion for the radiation condition yields a system of equations for the unknown coefficients, which can be solved by Schur complement and a backward-stable solver. The scheme is verified with known solutions and exhibits exponential convergence close to $10^{-14}$ for both single and multiple interfaces. An example with 39 interfaces is presented to demonstrate the solver's performance. The paper provides promising results for extending this method to a fast and accurate boundary integral equation solver for many cutting-edge applications involving a large number of layers in electromagnetics and optics.

Jared Weed

Rubik's Revenge, a 4x4x4 variant of the Rubik's puzzles, remains to date as an unsolved puzzle. That is to say, we do not have a method or successful categorization to optimally solve every one of its approximately $7.401 \times 10^{45}$ possible configurations. Rubik's Cube, Rubik's Revenge's predecessor (3x3x3), with its approximately $4.33 \times 10^{19}$ possible configurations, has only recently been completely solved by Rokicki et. al, further finding that any configuration requires no more than 20 moves. With the sheer dimension of Rubik's Revenge and its total configuration space, a brute-force method of finding all optimal solutions would be in vain. Similar to the methods used by Rokicki et. al on Rubik's Cube, in this paper we develop a method for solving arbitrary configurations of Rubik's Revenge in phases, using a combination of a powerful algorithm known as IDA* and a useful definition of distance in the cube space. While time-series results were not successfully gathered, it will be shown that this method far outweighs current human-solving methods and can be used to determine loose upper bounds for the cube space. Discussion will suggest that this method can also be applied to other puzzles with the proper transformations.