Mathematics

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrolment is by permission of the instructor. At the end of the course, students know basic numerical methods for evolutive ordinary and partial differential problems, together with their main theoretical and computational properties. In particular, students are able to analyze the properties of numerical methods; constructively examine corresponding computational results; advance their scientific computing education in higher level courses; and employ the acquired numerical skills in a variety of application areas. The two main topics covered are: 1) numerical solution of Ordinary Differential Equations (ODEs): Initial Value Problems; and 2) numerical solution of ODEs: Boundary Value Problems.

Time series analysis concerns the mathematical modeling of time-varying phenomena, e.g., ocean waves, water levels in lakes and rivers, demand for electrical power, radar signals, muscular reactions, ECG signals, or option prices at the stock market. The structure of the model is chosen both concerning the physical knowledge of the process, as well as using observed data. Central problems are the properties of different models and their prediction ability, estimation of the model parameters, and the model's ability to accurately describe the data. Consideration must be given to both the need for fast calculations and the presence of measurement errors. The course gives a comprehensive presentation of stochastic models and methods in time series analysis. Time series problems appear in many subjects and knowledge from the course is used in, i.e., automatic control, signal processing, and econometrics.

This course introduces basic and advanced mathematics used in engineering to develop an awareness and an appreciation of the role of mathematics in engineering. This course deals with mathematical principles, methods, and modeling.  

This course covers a number of fundamental topics concerning groups of graph automorphisms, with an emphasis on group-theoretic notions and results. Topics include fundamentals of graph theory and of group theory; graph automorphisms, transitive graphs; group actions on graphs; Cayley graphs, Schreier graphs; fundamental group of a graph, coverings; free group: definition, elementary properties; subgroups of free groups; and Hanna Neumann conjecture.

The course gives an introduction to group and ring theory with emphasis on finite groups, polynomial rings, and field extensions.

This course offers a study of ordinary differential equations (ODEs). Topics include: origins of ODEs in applications; first order equations; linear order equations, higher order, and linear differential systems; existence, uniqueness, and continuation of solutions; resolution of ODE with power series; nonlinear equations-- autonomous systems, phase plane, classification of critical points, and stability theorems.

This course covers multiple linear regression and least squares methods; generalized linear models; survival regression models; nonlinear effects and basis expansions; parametric, semiparametric, and nonparametric likelihood methods; and aspects of practical regression analysis in R.

The course covers: Countability, measure spaces, σ-algebras, π-systems and uniqueness of extension. Construction of Lebesgue measure on R (proof non-examinable), Independence. The Borel-Cantelli lemmas, measurable functions and random variables, independence of random variables. Notions of probabilistic convergence. Construction of integral and expectation. Integration and limits. Density functions. Product measure and Fubini’s theorem. Laws of large numbers. Characteristic functions and weak convergence, Gaussian random variables. The central limit theorem. Conditional probability and expectation.

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.