Courses

Methods of integration, applications of the integral, Taylor's theorem, infinite series.

Vectors in dimensions 2 and 3, complex numbers and the complex exponential function with applications to differential equations, Cramer's rule, vector-valued functions of one variable, scalar-valued functions of several variables, partial derivatives, gradients, surfaces, optimization, the method of Lagrange multipliers. 

Multiple integrals, Taylor's formula in several variables, line and surface integrals, calculus of vector fields, Fourier series.

Matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, canonical forms, applications. 

Special differential equations of order one. Linear differential equations with constant and variable coefficients. Systems of such equations. Transform and series solution techniques. Emphasis on applications.

Topics of linear and non-linear partial differential equations of second order, with particular emphasis to Elliptic and Parabolic equations and modern approaches.

Mathematical methods for economics. Quadratic forms, Hessian, implicit functions. Convex sets, convex functions. Optimization, constrained optimization, Kuhn-Tucker conditions. Elements of the calculus of variations and optimal control. 

The second term of this course may not be taken without the first. Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.

Introduction to Probability and Statistics

Probability is the foundation on which statistics is built. The purposes of this course are 1) to introduce you to probability and 2) prepare you to take a sequel course on statistical inference (Statistics 4204,5204).

We shall begin by covering the basic axioms of probability and using these in some simple settings. Then we will take up the idea of independence and conditional probability. Following we shall consider random variables, and the properties first of univariate discrete and continuous distributions. When we look at two or more variables, additional considerations arise, such as the relationship between the variables|conditional distributions and marginal distributions. Following these basics, we will then take up some ways of summarizing distributions, e.g., expectations and variances and, in summarizing relationships among variables, covariance and correlation. We then take up some of the more important distributions in statistics. In particular, for the discrete case, we will study the Bernoulli and binomial distributions and the generalization to the multinomial distribution, also the Poisson distribution. For continuous distributions, we take up the univariate and bivariate normal, the Gamma and the Beta distribution. In statistical applications, sums of independent random variables (for example, a sample average is such a sum, divided by sample size) are extremely important and characterizing the properties of these in large samples justifies many of the ways in which we make inferences in statistics. Thus, we take up the properties of these sums in large samples, focusing on stating laws of large numbers and also a simple central limit theorem.

The aim of the course is to describe the two aspects of statistics {estimation and inference {in some details. The topics will include maximum likelihood estimation, Bayesian inference, confidence intervals, bootstrap methods, some nonparametric tests, statistical hypothesis testing, linear regression models, ANOVA, etc.

This course is a foundation course for learning software programming using the Java language. The course will introduce the student to programming concepts, programming techniques, and other software development fundamentals. Students will learn the concepts of Object Oriented programming using Java. The course will present an extensive coverage of the Java programming language including how to write, compile and run Java applications.

The purpose of this course is to learn programming concept and Object Oriented fundamentals using Java. Students will receive a solid understanding of the Java language syntax and semantics including Java program structure, data types, program control flow, defining classes and instantiating objects, information hiding and encapsulations, inheritance, exception handling, input/output data streams, memory management, Applets and Swing window components.