Mathematics

This is the second semester in a mainstream calculus sequence. It covers the calculus of inverse trigonometric and hyperbolic functions; applications of the definite integral for finding areas and volumes of revolutions; techniques of integration; improper integrals; sequences and series: Convergence tests, power series, Taylor series with applications; vectors and the three-dimensional space: Dot and cross products, lines and planes.

This course familiarizes students with aspects of mathematics which are of importance for Physics and Research Skills. Students learn how certain mathematical techniques can be applied. After completion of the course, the student is able to: solve simple differential equations; use several basis mathematical techniques, particularly: exponential- and square root functions, algebra, solving equations, functions, goniometry, linear algebra, differentiating and integrating; use numerical integration techniques to solve differential equations; use the basics of system analysis as a tool to solve environmental problems; formulate mathematical models for simple real-world applications; operationalize and analyze mathematical models by doing computer simulations; and qualitatively analyze and construct a model independently.

This course covers the following topics: graph algorithms such as max flow; data structures such as van Emde Boas Trees; NP-completeness; exponential and parameterized algorithms for NP-hard problems; approximation algorithms; randomized algorithms; computational geometry; linear programming and optimization.

This course introduces regression analysis, one of the most widely used statistical techniques. Topics include simple and multiple linear regression, nonlinear regression, analysis of residuals and model selection, one-way and two-way factorial experiments, random and fixed effects models.

Topics in this Linear Algebra course include: linear systems and matrices; linear programming; vector spaces; linear transformations; ranks and determinants; linear systems; numerical solution of linear systems; diagonalization; Euclidean space; descriptive statistics.

This course covers advanced and basic mathematics, with guest speakers covering topics chosen from algebra, geometry, and analysis. The course covers:

• Algebra: Matrix Groups
• Geometry: Topology of surfaces
• Analysis: Fixed point theorems and applications

This course examines how optimization principles are of undisputed importance in modern design and system operation and illustrates how algorithms can be designed from mathematical theories for solving optimization problems. Topics include fundamentals, unconstrained optimization: one-dimensional search, Newton-Raphson method, gradient method, constrained optimization: Lagrangian multipliers method, Karush-Kuhn-Tucker optimality conditions, Lagrangian duality and saddle point optimality conditions, and convex programming: Frank-Wolfe method. The course requires students to take prerequisites.

This course familiarizes students with the fundamental techniques of linear algebra so that they can eventually master the "diagonalisation" of square matrices, which is one of the most important subjects in linear algebra. Upon completion, students understand the basic notions of linear algebra, such as matrices, determinants, eigenvalues, eigenvectors and diagonalisation, projections, and others, and be able to apply these techniques in various cases.

The course is an introduction to three important tools of applied mathematics, namely ordinary differential equations, Fourier-series, and partial differential equations. Some basic theoretical properties are proved and solution methods presented. Ordinary differential equations: linear differential equations of order n, the Cauchy problem, Picard's existence theorem, solution by power series and equations with singular points.  Fourier series: convergence point-wise, uniformly and in the mean-square, Parseval's equation. Partial differential equations: the heat equation and the wave equation solved on a finite interval by separation of variables and Fourier series and their solutions compared, the Dirichlet problem for the Laplace equation on the rectangle and the disc, the Poisson integral formula.

This course examines the methods frequently used to find numerical solutions to problems that arise in applied mathematics. The topics covered include methods for solving linear and nonlinear algebraic equations, interpolation, differentiation, integration and the numerical solution of ordinary differential equations.