Courses

The course will focus on the foundations of functional analysis and create a solid framework into which students will be able to build a strong research platform in applied mathematics. The course will begin with the basics of normed vector spaces, the Lebesgue integral, Hilbert spaces and operators. After providing the theoretical and computational basis, the course will provide students with applications into differential equations, Sobolev spaces, Distributions, and Fourier analysis. (3 units; Fall odd years)

Students will be introduced to concepts of the qualitative theory of differential equations such as existence and uniqueness, linear systems, autonomous systems, stability, bifurcations, chaos. Students will be introduced to dynamical systems and study the Poincaré-Bendixson theorem, limit cycles and the Hartman-Grobman theorem. Periodic and bounded solutions will be discussed briefly. Applications in modeling biological and social phenomena will also be discussed. Pre- or C- Requisite: MAT 503. (3 units; Fall odd years)

This course is designed to teach students how to analyze, solve and apply partial differential equations. Students will learn about well-posed and ill-posed problems; existence and uniqueness of solutions to PDEs (Partial Differential equation). Students will learn about linear, quasi-linear and non-linear PDEs of the first order in on spatial dimension. Students will learn about wave propagation, hyperbolic and parabolic PDEs in one spatial dimension. Students will learn and use different methods for solving PDEs such as separation of variables, method of characteristics, green functions, etc. Students will learn about Fourier series and integrals, higher order equations and vibrational methods. Applications will be discussed for most equations. Prerequisite: MAT 503. (3 units; Spring even years)

The study of difference equations to model different biological phenomena. An introduction to concepts of the qualitative theory of differential equations such as steady-state solutions, stability and linearization, Phase-Plane Methods and nullclines, classifying stability characteristics, global and local behavior, Limit cycles, oscillations, and the Poincaré-Bendixson theorem. An introduction to linear difference equations and how to solve these analytically while understanding the behavior of solutions. Students will learn how analyze the qualitative behavior of solutions for Nonlinear difference equations via linearization. The study of steady states, stability criteria and Cobwebbing and other numerical resources to solve difference equations. Prerequisites: MAT 403 and 413. (3 units; Spring even years)

In this course students will learn to apply techniques in numerical analysis to solutions of equations of one variable, interpolations and polynomial approximations, numerical integration and differentiation, numerical solutions of initial value problems, iterative methods solving linear systems, approximation theory, approximation Eigenvalues, systems of nonlinear equations, and boundary-value problems for ordinary differential equations. (3 units; Fall even years)

This course is a nonmeasure theoretic introduction to stochastic processes. A strong probabilistic foundation will be used to analyze appropriate probability models in order to predict the effects of randomness on systems studied. Applications of probability theory to the study of phenomena in various fields including engineering, computer science, management science, the physical and social sciences, and operations research. (3 units; Fall even years)

This course will enable the student to formulate, analyze, and simulate problems arising in the fields on natural and mathematical sciences. Computational skills will be coupled with development of stochastic models with emphasis upon project implementation, efficiency, and accuracy of algorithms and the interpreting results obtained. Specific topics include linear and nonlinear systems of equations, polynomial interpolation, numerical integration, and introduction to numerical solution of differential equations incorporated into real world applications of Markov Chains, Random Walks, Poisson Processes, Birth and Death Processes, Renewal Phenomena, Phase-type Distributions, Queuing systems, Brownian Motion, and SIR Modeling. (4 units; Spring, odd years)

This course provides an introduction to probability, discrete and continuous random variables, probability distributions, expected values, sampling distributions, point estimation, confidence intervals, hypothesis testing and general linear modeling, Specific topics include tools for describing central tendency and variability in data; methods for performing inference on population means and proportions via sample data; statistical hypothesis testing and its application to group comparisons; issues of power and sample size in study designs; and random sample and other study types. While there are some formulae and computational elements to the course, the emphasis is on qualitative nonlinear thinking, interpretation and concepts. (3 units; Spring, even years)