Mathematics

The course covers some of the most prominent tools in modelling and simulation. Both deterministic and stochastic models are covered. These include mathematical optimization, the application of sophisticated mathematical methods to make optimal decisions, and simulation, the playing-out of real-life scenarios in a (computer-based) modelling environment. Topics may include formulation of management problems using linear/nonlinear and network models (these could include binary, integer, convex, and stochastic programming models) as well as solving these problems and analyzing the solutions; generating random variables using Monte Carlo simulation; discrete event simulation; variance reduction techniques; Markov Chain Monte Carlo methods. The course teaches students to use modelling and simulation computer packages.

This course introduces the mathematical formalism of quantum information theory. Topics include a review of probability theory and classical information theory (random variables, Shannon entropy, coding); formalism of quantum information theory (quantum states, density matrices, quantum channels, measurement); quantum versus classical correlations (entanglement, Bell inequalities, Tsirelson's bound); basic tools (distance measures, fidelity, quantum entropy); basic results (quantum teleportation, quantum error correction, Schumacher data compression); and quantum resource theory (quantum coding theory, entanglement theory, application: quantum cryptography). 

In this course, students are taught the foundational concepts of major stochastic fields and associated topics, including Statistics, probability, and combinatorics. The course is presented in “flipped-classroom” format, such that students are expected to learn concepts on their own, and then practice application in the classroom.

Introduce the concept, main types and basic methods of mathematical models through cases, so that students can master the thought and method of mathematical modeling initially.This course is also a basic course for students to participate in the mathematical contest in modeling.

Using cases as clues, this course introduces the basic principles and methods of building mathematical models, covering the fields of energy, communication, economy, ecology, aviation and aerospace.The main mathematical methods include elementary mathematics, elementary geometry, calculus, linear algebra, operations research, elementary number theory, graph theory, differential equations, etc.

The Individual Research Training Senior (IRT Senior) Course is an advanced course of the Individual Research Training B (IRT B) course in the Tohoku University Junior Year Program in English (JYPE) in the spring 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.

Information is a fundamental concept in the world around us that can be investigated from several perspectives. The mathematical theory of information provides a framework for a formal description and interpretation of information. In many ways, this mathematical framework (its applications and the interpretations it provides) is based on concepts from probability theory and statistics. This course provides students with an introduction to the field of information theory. Students will learn to apply and interpret a wide range of concepts from statistics and probability theory to develop, model, and understand the concept of information, as well as related ideas, in a structured and organized way. Many of the tools of statistics and probability theory students encounter in the course should be familiar to them from introductory or intermediate statistics courses, while other concepts might be new.

This course provides a detailed introduction to complex function theory which interrelates the geometric and analytic aspects. A principal goal is Cauchy’s famous integral theorem and its many consequences.

This is a second course in real analysis at the University of Edinburgh and builds on ideas in the analysis portion of Fundamentals of Pure Mathematics. The course begins with sequences and series of real numbers, introducing the concept of Cauchy sequences and results for bounded sequences. Subsequently, sequences and series of functions are introduced and concepts of uniform convergence and power series are discussed. The concept of Lebesgue integral on real line is then developed. Finally, the rudiments of Fourier series are introduced.
 

This course introduces students to the statistical computing and programming, with the main focus on R, Python, and SAS. Students learn basic computing and programming concepts including scripting, variables, expressions, assignments, control structures, and data structures. On the statistical side, they will learn to load raw data, make numerical and graphical summaries of data, and conduct various estimation and testing procedures. Topics include descriptive statistics, statistical estimation, robust estimation, categorical data analysis, testing hypotheses, ANOVA, regression analysis, performing resampling methods and simulations. Some basic knowledge of R is assumed.