Mathematics

Topics cover include: Bivariate probability, continuous densities, generating functions. The exponential densities, including normal, t-, χ2 and F. Simple parametric and nonparametric tests. Further topics include the consistency, efficiency and sufficiency of estimates, maximum likelihood estimation; the central limit theorem, Chebyshev's inequality, the Neyman-Pearson lemma and the likelihood ratio test; regression, and analysis of variance.

This course provides the analytic background behind quantum information theory in the framework of operators on Hilbert spaces and functional analysis. Topics include completely positive and completely bounded maps; operator systems and spaces; Choi representation and Kraus operators; Stinespring's representation theorem; tensor products; quantum measurements and related sets of correlations; entanglement; Schmidt decompositions; and factorizable channels and applications in quantum information theory.

Many phenomena in natural sciences and social sciences can be summarized as quantitative relationships expressed by differential equations. In particular, many laws in economics and finance can be described by differential equations. This course enables non-mathematics students to master the basic theories and methods of differential equations, and to cultivate their logical thinking ability and the ability to apply differential equations to solve practical problems. The course consists of two parts: ordinary differential equations and partial differential equations. The main topics include: first-order ordinary differential equations, high-order ordinary differential equations, linear ordinary differential equations, difference equations, basic partial differential equations (harmonic equation, heat conduction equation, wave equation), mathematical model of differential equations.