School of Mathematical Sciences
College of Science
School of Mathematical Sciences
College of Science
BA, Columbia College; Ph.D., New York University
This course will continue to expose students to the logical methodology of mathematical modeling. It will also provide them with numerous examples of mathematical models from various fields.
This course is an introduction to the basic concepts of linear algebra, and techniques of matrix manipulation. Topics include linear transformations, Gaussian elimination, matrix arithmetic, determinants, vector spaces, linear independence, basis, null space, row space, and column space of a matrix, eigenvalues, eigenvectors, change of basis, similarity and diagonalization. Various applications are studied throughout the course.
0 - 9 Credits
Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.
This course is an investigation and extension of the theoretical aspects of elementary calculus. Topics include mathematical induction, real numbers, sequences, functions, limits, and continuity. The workshop will focus on helping students develop skill in writing proofs.
This course introduces sample spaces and events, axioms of probability, counting techniques, conditional probability and independence, distributions of discrete and continuous random variables, joint distributions (discrete and continuous), the central limit theorem, descriptive statistics, interval estimation, and applications of probability and statistics to real-world problems. A statistical package such as Minitab or R is used for data analysis and statistical applications.
This course will introduce graduate students to the logical methodology of mathematical modeling. They will learn how to use an application field problem as a standard for defining equations that can be used to solve that problem, how to establish a nested hierarchy of models for an application field problem in order to clarify the problem’s context and facilitate its solution. Students will also learn how mathematical theory, closed-form solutions for special cases, and computational methods should be integrated into the modeling process in order to provide insight into application fields and solutions to particular problems. Students will study principles of model verification and validation, parameter identification and parameter sensitivity and their roles in mathematical modeling. In addition, students will be introduced to particular mathematical models of various types: stochastic models, PDE models, dynamical system models, graph-theoretic models, algebraic models, and perhaps other types of models. They will use these models to exemplify the broad principles and methods that they will learn in this course, and they will use these models to build up a stock of models that they can call upon as examples of good modeling practice.
Ross, David S., Khamir Mehta, and Antonio Cabal. "Mathematical Model of Bone Remodeling Captures the Antiresorptive and Anabolic Actions of Various Therapies." Bulletin of Mathematical Biology 79. 1 (2017): 117-142. Print.
Wahle, C.W., et al. "Model for Screened, Charge-Regulated Electrostatics of an Eye Lens Protein: Bovine GammaB-Crystallin." Physical Review E 96. 3 (2017): 1-25. Print.
Bell, Michael M., et al. "Statistical-Thermodynamic Model for Light Scattering from Eye Lens Protein Mixtures." Journal of Chemical Physics 146. (2017): 1-32. Print.
Ross, David S., Kara L. Maki, and Emily K. Holtz. "Existence Theory for the Radically Symmetric Contact Lens Equation." SIAM Journal on Applied Mathematics 76. 3 (2016): 827-844. Print.
Caniga, Michael, et al. "Preclinical Experimental and Mathematical Approaches for Assessing Effective Doses of Inhaled Drugs, Using Mometasone to Support Human Dose Predictions." Journal of Aerosol Medicine and Pulmonary Drug Delivery 29. 4 (2016): 362—377. Print.
Wahle, Chris, David S. Ross, and George Thurston. "Methods for Light Scattering Free Energy Determination for Restricted Composition Domains in Ternary Liquid Mixtures." Journal of Chemical Physics 139. (2013): 124114. Print.
Huang, Jinxin, et al. "Maximum-likelihood Estimation in Optical Coherence Tomography in the Context of the Tear Film Dynamics." Biomedical Optics Express 4. 10 (2013): 1806-1816. Print.
Brooks, B. P., N. DiFonzo, and D. S. Ross. "The GBN-Dialogue Model of Outgroup-Negative Rumor Transmission: Group Membership, Belief, and Novelty." Nonlinear Dynamics, Psychology, and Life Sciences 17. 2 (2013): 269-293. Print.
Huang, J., et al. "Quantitative Measurement of Tear Film Dynamics with Optical Coherence Tomography and a Maximum-Likelihood Estimator." Optics Letters 38. 10 (2013): 1721-1723. Print.
Cabal, A., et al. "A Semi-Mechanistic Model of the Time-Course of Release of PTH into Plasma Following Administration of the Calcilytic JTT-305/MK-5442 in Humans." Journal of Bone and Mineral Research 28. 8 (2013): 1830-1836. Print.
DiFonzo, N., et al. "Rumor Clustering, Consensus, and Polarization: Dynamic Social Impact and Self-Organization of Hearsay." Journal of Experimental Social Psychology 49. 3 (2013): 378-399. Print.
Golen, E., et al. "An Underwater Sensor Allocation Scheme for Non-Circular Sensing Coverage Regions." ISRN Sensor Networks 2013. (2013): 963029. Web.
Ross, David S., et al. "Dynamics of Cell Signaling and PTH Treatments for Ostoporosis." Discrete and Continuous Dynamical Systems, Series B 17. 6 (2012): 2185-2200. Print.
Wahle, Chris, David S. Ross, and George Thurston. "On the Design of Experiments for Determining Ternary Mixture Free Energies from Static Light Scattering Data using a Nonlinear Partial Differential Equation." Journal of Chemical Physics 137. (2012): 34201. Print.
Wahle, Chris, David S. Ross, and George Thurston. "On Inferring Liquid-Liquid Phase Boundaries and Tie Lines from Ternary Mixture Light Scattering." Journal of Chemical Physics 137. (2012): 34203. Print.
Wahle, Chris, David S. Ross, and George Thurston. "Mathematical and Computational Aspects of Quaternary Liquid Mixing Free Energy Measurement Using Light Scattering." Journal of Chemical Physics 137. (2012): 34202. Print.
Agyingi, E., D. S. Ross, and K. Bathena. "Transmission Dynamics of Leismaniasis." Journal of Biological Systems 19. 2 (2011): 237-251. Print.
Published Conference Proceedings
Agyingi, E., D. S. Ross, and S. Maggelakis. "Modeling the Effect of Topical Oxygen Therapy on Wound Healing." Proceedings of the Advances in Mathematical and Computational Methods. Ed. I. Kotsireas, R. Melnik, and B. West. Melville, NY: American Institute of Physics, 2011. Print.
Agyingi, Ephraim, S. Maggelakis, and D.S. Ross. “The Effect of Bacteria on Epidermal Wound Healing.” Mathematical Modeling of NaturalPhenomena, 5.3 (2010): 28-39. Print. *
Hollenbeck, Dawn., K. Martini, A. Langner, A. Harkin, D. Ross,G. Thurston. “Model for Evaluating the Patterned Charge Regulation Contributionto Electrostatic Interactions between LowDielectric Spheres”. Physical Review E, 82(2010): 0314021-03140213. Print. â‰ *
Ross, David. “The Inverse Trochoid Problem.” Journal of theFranklin Institute, 347 (2010): 1281-1308. Print. Â«
Lutzer, Carl., and D. Ross. “The Dynamics of Embedded-Charge Microenergy Harvesting.” Journal of Computational and Nonlinear Dynamics,5.2 (2010): 0210041-0210049. Print. â‰ Â«