Mathematics

The course covers many of the following topics: Random events, sigma-algebras, monotone classes. Measurable spaces, random variables - measurable functions. Measures, probability measures, signed measures. Borel sets in R^d, Lebesgue measure. Caratheodory extension theorem. Sequences of events and random variables, Borel-Cantelli lemma. Distributions of random variables. Independence of random variables. Integral of measurable functions - mathematical expectation,.
Moments of random variables, L_p spaces. Convergence concepts of measurable functions. Limit theorems for integrals. Weak and strong laws of large numbers. Completeness of L_p spaces. Conditional expectation and conditional distribution of random variables. Fubini's theorem.

The Individual Research Training Senior (IRT Senior) Course is an advanced course of the Individual Research Training A (IRT A) course in the Tohoku University Junior Year Program in English (JYPE) in the fall semester. Though short-term international exchange students are not degree candidates at Tohoku University, a similar experience is offered by special arrangement. Students are required to submit: an abstract concerning the results of their IRT Senior project, a paper (A4, 20-30 pages) on their research at the end of the exchange term, and an oral presentation on the results of their IRT Senior project near the end of the term.

The course provides an introduction to selected topics in discrete mathematics; including graph theory, combinatorics, final bodies, and code theory.

Ordinary Differential Equations (ODEs) allow us to describe many natural phenomena and are therefore essentially important in many scientific areas. This course introduces the standard and most common tools to solve differential equations, particularly Laplace transform and linear algebra method. 

An introduction to linear algebra, mainly in R^n but concluding with an introduction to abstract vector spaces. The principal topics are vectors, systems of linear equations, matrices, eigenvalues and eigenvectors, and orthogonality. The important notions of linear independence, span and bases are introduced. This course is both a preparation for the practical use of vectors, matrices, and systems of equations and also lays the groundwork for a more abstract, pure-mathematical treatment of vector spaces. Students learn how to use a computer to calculate the results of some simple matrix operations and to visualize vectors.

Asset price in discrete time, random walks, conditional expectation, elements of discrete-time martingale theory, the binomial asset pricing model, option pricing in discrete time, and -time permitting- discrete time term structure models and/or discrete time portfolio theory.

This course focuses on the basic concepts and theorems of Ring Theory and Field Theory, which are about generalized and abstracted properties of the set of integers and the set of rational numbers with respect to the four elementary arithmetic operations. Recommended Prerequisite: MATH321.

The course covers the following topics:

• Homomorphisms and Factor Rings
• Prime and Maximal Ideals
• Introduction to Extension Fields, Vector Spaces
• Algebraic Extensions
• Geometric Constructions
• Finite Fields
• Unique Factorization Domains
• Euclidean Domains
• Isomorphism Theorems, Series of Groups
• Sylow Theorems and it applications
• Automorphism of Fields
• The Isomorphism Extension Theorem
• Splitting Fields, Separable Extensions
• Galois Theory
• Cyclotomic Extensions
• Insolvability of the Quintic 

This course covers basic topology, meaning of convergence, and continuity.

This course focuses on the fundamentals of modern numerical techniques for a wide range of linear and nonlinear elliptic, parabolic and hyperbolic partial differential equations and integral equations central to a wide variety of applications in science, engineering, and other fields. Topics include Mathematical Formulations; Finite Difference Method, Finite Volume Method, Collocation Method, Finite Element Discretization.

This course introduces and investigates fundamental concepts of linear algebra in the context of the Euclidean spaces R^n. Topics include systems of linear equations, matrices, determinants, Euclidean spaces, linear combinations and linear span, subspaces, linear independence, bases and dimension, rank of a matrix, inner products, eigenvalues and eigenvectors, diagonalization, linear transformations between Euclidean spaces, and applications.