May

26

2022

We consider the set M_n(Z; H) of n x n matrices with integer elements of size at most H and obtain upper and lower bounds on the number of s-tuples of matrices from M_n(Z; H), satisfying various multiplicative relations, including multiplicative dependence, commutativity and bounded generation of a subgroup of GL_n(Q). These problems generalise those studied in the scalar case n=1 by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices.

May

27

2022

Several Mathematics graduate students gave oral and poster presentations at the May 20-22 SIAM PNW Meeting in Vancouver, WA. They will share their research in a blitz with AMC Seminar audience.

May

31

2022

The exponential of the topological entropy of any dynamical system that admits a Markov partition is a special kind of number -- a weak Perron number. These are positive, real, algebraic integers that are at least as big as the modulus of all of their Galois conjugates. A question that remains mysterious is: which weak Perron numbers are realized as exp(entropy) by which families of dynamical systems?

May

31

2022

Stochastic differential equations are used to model various phenomena including biological models. In this paper we attempt to simulate a system of SDEs in order to use inverse problem techniques to estimate the parameters. We used a model built by researchers at Boston University as well as techniques from parameter estimation and sensitivity analysis. Ultimately, we found that there was no continuous dependence on the parameters for the solutions so we were not able to perform parameter estimation on the model.

Jun

02

2022

Machine learning models are powerful tools which may aid in the prediction of survival outcomes of cancer patients. This study evaluated sixteen supervised machine learning models, eight classification and eight regression, on their ability to predict survival outcomes on breast cancer and prostate cancer data sets from the SEER database. The most accurate models, based on the fraction of correct predictions, were found to be the Gradient Boosting models for both classification and regression. To maximize accuracy, the hyperparameters of these two models were optimized.