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Mathematical Sciences

Student Summer Research Reports

Date: 3:30pm - 4:30pm PDT September 20, 2018 Location: J.R. Howard Hall 259

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J.R. Howard Hall 259

Power Analysis of Two Biased Coin Designs in Clinical Trials

By Xiaohan Qiu - Mathematics Major And
Minho Choi – Computer Science & Mathematics Major

Power, as one of the most important statistical aspects in clinical trials, represents the ability of detecting treatment effects. Given limited sample size, clinicians hope to select more effective treatment, which often requires more powerful designs for allocating treatments. We present two adaptive biased coin designs and compare them with the complete randomization. We give a mathematical proof that the two adaptive biased coin designs are more powerful than the complete randomization.

Hearing the Local Orientability of Orbifolds

By Sean Richardson – Computer Science and Mathematics Major

 Given some object, such as a drum, there exists a spectrum of fundamental frequencies determined by physics. However, if the drum was in a neighboring room and you could only listen to these frequencies, is it possible to deduce the drum’s shape? In other words, “Can you hear the shape of a drum?”. In this research project, we ask a similar question but for abstract mathematical objects. The vibrational frequencies for abstract objects correspond to the eigenvalue spectrum of the Laplace operator associated to the object.

The abstract shapes we study are called orbifolds. An orbifold is a multidimensional object that is allowed to have some “trouble spots,” which are tied to the symmetries allowed in n-dimensional space. We ask: Given the Laplace spectrum of an unknown orbifold, what properties of the orbifold are determined? We show that one can hear the local orientability of an orbifold. That is, we can use the Laplace spectrum to detect trouble spots associated to orientation reversing symmetries of n-dimensional space.

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