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Research with Undergraduates at Polytechnic Institute of NYU
Research with Undergraduates at Polytechnic Institute of NYU
Note to students
As an undergraduate student in science or engineering, research experience in a frontier area can be fun and highly rewarding. If you have solid training in one or a few of the key subjects including ordinary differential equations, partial differential equations, multivariable calculus, numerical methods, probability, statistics, combinatorics, linear algebra, abstract algebra, and you are keenly interested in applying your mathematical knowledge to real-world problems arising in biology, chemisty, physics, engineering, and social science, you are encouraged to contact me to start a possible joint research project. The format of this study may consist of several phases: (a) having an initial consultation of interest and strength of student; (b) receiving reading assignment; (c) making presentations based on materials read; (d) proposing or being proposed of possible research subjects; (e) engaging in research and discussion; (f) writing up of research results; (g) submitting work for publication if it merits.
Experience and interest in scientific computation may be desired but not required.
Lecture given to Yeshiva University Student Mathematics Club (powerpoint slides and Youtube movies)
Research Projects Finished
Luciano Medina (a senior at the Department of Electrical Engineering) and I finished a joint research project entitled
Universal Curve, Biological Time, and Dynamically Varying Scaling Exponent in Growth Law
which had been accepted for publication in Journal of Nonlinear Analysis (Series B): Real-World Applications (http://ees.elsevier.com/nonrwa/).
In this project, we present an effective method for constructing universal curves arising in quantitative biology modeling tumor growth. As byproducts, we show that the universal curves are independent of the detailed properties of a dynamical parameter called the scaling exponent, hence, are truly universal, and we derive a monotonicity condition for the scaling exponent assuming that a dimensionless time variable called the biological time flows forwardly, which confirms experimental observations based on biomedical data. We also observe a superimposed period-doubling and transition-to-chaos picture when a finite time-delay is introduced and the scaling exponent oscillates periodically.
This work was partially supported by the Othmer Institute for Interdisciplinary Studies at Polytechnic University
Nafis Rahman (a sophomore at Department of Mathematics) carried out a study on Growth of Nanodroplets.
Jesse Wang (a sophomore at Department of Chemical and Biological Sciences) carried out a study on Species Viability and Microscopic Scales.
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