Mathematics

This course is designed to improve students' scientific literacy and recognition ability as well as enhance their ability to discriminate from a professional perspective by analyzing and critiquing many specious ideas. The course will be offered in collaboration with many experts in various disciplines, including mathematics, physics, chemistry, biology, management, computer science, medicine, literature, history, philosophy, political science, history, journalism, etc.

The course provides a theoretical understanding and practical skills related to object-oriented programming. Practical skills will be learnt using the C++ programming language. The course enables students to tackle complex programming problems, making good use of the object-oriented programming paradigm to simplify the design and implementation process.

This is an advanced course in linear and logistic regression, which expounds on the knowledge gained in introductory mathematical statistics courses. It covers matrix formulation of multivariate regression, methods for model validation, residuals, outliers, influential observations, construction and use of F- and t- tests, likelihood-ratio-test, confidence intervals and prediction, and applied implementation of various techniques in R software. Students also consider correlated errors, Poisson regression, multinominal and ordinal logistic regression. The first part of the course expands on previous study of linear regression to consider how to check if the model fits the data, what to do if it does not fit, how uncertain it is, and how to use it to draw conclusions about reality. The second part of the course explores logistic regression, which is used in surveys where the answers follow a categorical alternative pattern such as “yes/no,” “little/just fine/much,” or “car/bicycle/bus.” Students describe differences between continuous and discrete data, and the resulting consequences for the choice of statistical model. Students learn to give an account of the principles behind different estimation principles, and describe the statistical properties of such estimates as they appear in regression analysis. The interpretation of regression relations in terms of conditional distributions is studied. Odds and odds ration are presented, and students describe their relation to probabilities and to logistic regression. Students formulate both linear and logistic regression models for concrete problems, estimate and interpret the parameters, examine the validity of the model and make suitable modifications, use the model for prediction, utilize a statistical computer program for analysis, and present the analysis and conclusions of a practical problem in a written report and oral presentation. The course makes use of lectures, exercises, computer exercises, and project work.

This course builds up a toolbox of numerical optimization methods for building solutions in future studies, thereby making it an ideal supplement for students from many different fields in science. The course is taught both at a theoretical level that goes into deriving the math, and also on an implementation level with focus on computer science and good programming practice. Students participate in weekly programming exercises where they implement the algorithms and methods introduced from theory, and apply their own implementations to case-study problems like computing the motion of a robot hand or fitting a model to highly non-linear data. Topics include: first order optimality conditions, Karush-Kuhn-Tucker conditions, Taylors theorem, mean value theorem, nonlinear equation solving, linear search methods, trust region methods, linear least-squares fitting, regression problems, and normal equations.