Mathematics

This course covers analytic functions, special functions (gamma function, Bessel functions, Legendre polynomials and spherical harmonics), Fourier series and Fourier transforms, Laplace transforms, Ordinary differential equations, partial differential equations, and green functions.

This course examines groups, which are best understood as symmetries of mathematical objects. Students explore geometric group theory and the connection between the algebraic properties of a group and the geometric properties of the spaces it acts on.

This course examines the fundamental concepts of probability and statistics required for data analysis. Topics include sampling; introduction to experimental design; review of simple probability; estimation; confidence intervals; hypothesis testing including types of errors and power; inferences about means and proportions based on single and independent samples; matched pairs designs; introduction to nonparametric methods; contingency tables; regression; and analysis of variance.

This course covers the derivation and analysis of fundamental partial differential equations, including Laplace’s equation, the wave equation, and the diffusion equation. Emphasis is placed on analytical solution techniques such as separation of variables, Fourier series and integrals, and the method of characteristics. Additional topics include maximum principles and the use of Green’s functions for solving boundary value problems.

The overarching goal of the course is for the students to acquire basic knowledge of linear algebra which is necessary for further studies in mathematics and natural sciences. Special emphasis is placed on developing the mathematical theory for vector spaces in a systematic way that contributes to strengthening the students' ability to absorb mathematical text, to conduct mathematical reasoning, to solve problems of both theoretical and applied nature and to communicate mathematics. The course covers Matrices; Determinants; Linear spaces; Euclidean spaces; Linear mappings; Spectral theory; Systems of linear ordinary differential equations; and Quadratic forms.

This course explores the development of mathematics in relation to trends and philosophies that have changed over time and that have affected the conceptualization of mathematics. Mathematics and culture frequently meet at the crossroads of evolution of human intelligence. Mathematics had a huge impact on the development of civilization; conversely, mathematics has been influenced by the development of civilization. Topics discussed in this course include axioms for geometry by Euclid, calculus by Newton and Leibniz, concepts of computation by Turing and von Neumann, art and mathematics, society and mathematics, science/technology and mathematics, and Oriental/European culture and mathematics. 

This course examines information theory, including error-correcting codes, data compression and cryptography. Each of these subtopics are enhanced by the application of entropy functions, and more sophisticated error-correcting codes are provided by way of brief introduction to number theory and finite fields. 

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.