### Experimental Math

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### General Information

Student: |
Aniket Shah |

Office: |
Hill 323 |

School: |
Rutgers University |

E-mail: |
aniket dot shah6 at gmail dot com |

Project: |
Hybrid Dynamical Systems and Computational Topology |

### Project Description

A newer method of prostate cancer treatment is intermittent androgen suppresion. From a paper titled "Quantitative mathematical modeling of PSA dynamics of prostate cancer patients treated with intermittent androgen suppression," published in 2012 by K. Aihara et al., we have obtained two systems of first order homogeneous differential equations which model the on and off-treatment dynamics of cancer cells. From these equations we are trying to obtain a map on a manageable phase space corresponding to the levels of cancer cells, which we are trying to analyze computationally.

### Weekly Log

- Pre-REU:
- Met the mentor Professor Konstantin Mischaikow and post-doctoral researcher Shaun Harker, obtained the biology papers to read for background information and relevant mathematical models. Read background material on isolated invariant sets from "Isolated Invariant Sets and the Morse Index" by Charles Conley.
- Week 1:
- Moved in, obtained the differential equations from the paper. Learned from Shaun Harker about the conley-morse database and how the dynamics would be calculated in terms of combinatorial morse sets. Read about stable and unstable equilibria on smooth manifolds.
- Week 2:
- Solved for general form of solutions to the given systems of differential equations. Away.
- Week 3:
- Discussed with Shaun the need to code a zero-finding method in c++ that takes a particular solution particular to the earlier system of differential equations as an intermediate step in the computation of the dynamical map. Learned about c++.
- Week 4:
- Wrote a method in c++ that, given a solution and an initial value of the differential equations we are dealing with, finds the smallest positive root or indicates that one does not exist.

### Readings

Conley, C., 1978: *Isolated Invariant Sets and the Morse Index.* American Mathematical Society, 89 pp.
Coddington, E., Levinson, N., 1955: *Theory of Ordinary Differential Equations.* McGraw-Hill Book Company, Inc., 422 pp.

### Additional Information