Mathematics Seminar - 7 December 2012
This week the Mathematics Seminar will feature two short presentations by Marist students.
Hancock Center Room 2023, 3:30 PM
Samantha Sprague, Class of 2015
What do Math and Legos have in common?
Legend has it that Queen Dido used strips of an ox's hide to solve the classical isoperimetric problem, which involves finding the maximum area that can be enclosed given a fixed perimeter. Using LEGO bricks we describe our solution(s) to a simpler analogous question: What is the smallest number of non-overlapping 1x1 bricks needed to enclose a given area? Our problem assumes that all areas are integral, and a 1x1 square corresponds to one unit of area. Standard optimization strategies combined with discrete mathematics and number theory allow us to show that the smallest perimeter needed to enclose A units of area is twice the smallest integer greater than or equal to the square root of A, plus one. This question lends itself to interesting generalizations. Polyominoes can be used to similarly discretize the isoperimetric problem. However, when using LEGO blocks both area and perimeter have two dimensions: length and width. This aspect is lost when using polyominoes. Other generalizations include using different brick-shapes, imposing various weighting constraints, and the formulation of a LEGO double bubble problem comparable to the one solved by Dr. Frank Morgan.
Laura Tobak, Class of 2013
Modeling Elastic Interface Waves with the Parabolic Equation Method
Underwater acoustic waves can be modeled using the elastic parabolic equation in range dependent environments with elastic sediment layers. Here we use the rotated variable parabolic equation method, which is more accurate than either the mapping method or the coordinate transformation method. We generate solutions that allow investigation of underwater acoustics due to both compressional and shear seismic sources. In particular we will study the generation of Scholte and potential Stoneley interface waves. We examine the effects of varying frequency, sound speed, and sediment layer thickness on the Scholte and Stoneley wave amplitudes.