Mathematics

This course will study the design and analysis of exact and approximation algorithms for various optimization problems using advanced techniques such as combinatorial methods, probabilistic methods, linear programming, semidefinite programming, and spectral methods. The course also covers spectral algorithms and related convex programs, SDP duality, multiplicative weight update, graph spectrum, eigenvalue interlacing, Cheeger–Alon–Milman inequality, random walks, local graph partitioning, expanders, Laplacian solver, effective resistance, sparsification, matrix scaling, abstract simplicial complex, and random spanning trees.

This course develops key mathematical and computational skills relevant to the wider mechanical engineering program. Topics include vector algebra, real analysis, limits, curve sketching, series, applications of integration, complex analysis, functions of more than one variable, matrix algebra, second order ordinary differential equations, and vector calculus. Practical implementation through programming is studied to solve problems selected from the topic areas.

This is a course in abstract algebra, although connections with other fields will be stressed as often as possible. It is a systematic study of the basic structure of groups, finite and infinite. Topics include homomorphisms, isomorphisms, and factor groups; group presentations and universal properties; Sylow theorems and applications; simple groups and composition series; classification of finite abelian groups and applications; and solvable groups and the derived series.

Full course description

To describe natural phenomena and processes, mathematical models are widely used. The focus in this course shall be on dynamical models (i.e., where time plays a role) in particular those that have interaction with the environment through inputs and outputs. Mathematical systems theory provides the framework to deal with such models in a systematic and useful way. First we consider some general aspects of mathematical modeling. Then we briefly address dynamical systems without inputs and outputs - but which may show nonlinear behavior. We study basic properties such as equilibrium points, linearization, and stability. We then switch to linear dynamical models with inputs and outputs. They are used in many different areas of the natural sciences and in engineering disciplines. We discuss the following topics and concepts. Linear difference and differential equations, Laplace transforms, transfer functions of linear systems; controllability, observability, minimality; system representations with an emphasis on state-space representations and canonical forms; stability; the interconnection of linear systems including feedback; frequency domain analysis and the relationship with filter theory, Fourier analysis, and time series analysis. To demonstrate the applicability of the techniques and concepts, many examples from science and engineering are mentioned and briefly discussed.

Course objectives

• To have the ability to interpret dynamical phenomena as mathematical systems and to cast them into such form. 
• To understand the basic concepts of linear systems theory. 
• To be familiar with analysis techniques for linear systems, to understand their behavior and interaction. 
• To become familiar with some application areas of mathematical systems and models. 

Prerequisites

SCI2019 Linear Algebra and SCI2018 Calculus