Courses tagged with "Information control" (59)

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

This is an introductory course in algebraic combinatorics. No prior knowledge of combinatorics is expected, but assumes a familiarity with linear algebra and finite groups. Topics were chosen to show the beauty and power of techniques in algebraic combinatorics. Rigorous mathematical proofs are expected.

Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.

This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis. The first half of the course covers multivariable calculus. The rest of the course covers the theory of differential forms in n-dimensional vector spaces and manifolds.

This is a variation on 18.02 Multivariable Calculus. It covers the same topics as in 18.02, but with more focus on mathematical concepts.

Acknowledgement

Prof. McKernan would like to acknowledge the contributions of Lars Hesselholt to the development of this course.

This is an undergraduate course on differential calculus in one and several dimensions. It is intended as a one and a half term course in calculus for students who have studied calculus in high school. The format allows it to be entirely self contained, so that it is possible to follow it without any background in calculus.

18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.

This course analyzes combinatorial problems and methods for their solution. Topics include: enumeration, generating functions, recurrence relations, construction of bijections, introduction to graph theory, network algorithms, and extremal combinatorics.

Combinatorial Optimization provides a thorough treatment of linear programming and combinatorial optimization. Topics include network flow, matching theory, matroid optimization, and approximation algorithms for NP-hard problems.

This course explored topics such as complex algebra and functions, analyticity, contour integration, Cauchy's theorem, singularities, Taylor and Laurent series, residues, evaluation of integrals, multivalued functions, potential theory in two dimensions, Fourier analysis and Laplace transforms.

The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.

Course Format

This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:

• Lecture Videos by Professor Arthur Mattuck.
• Course Notes on every topic.
• Practice Problems with Solutions.
• Problem Solving Videos taught by experienced MIT Recitation Instructors.
• Problem Sets to do on your own with Solutions to check your answers against when you're done.
• A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.
• A full set of Exams with Solutions, including practice exams to help you prepare.

Content Development

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time.

This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

Game Theory, also known as Multiperson Decision Theory, is the analysis of situations in which the payoff of a decision maker depends not only on his own actions but also on those of others. Game Theory has applications in several fields, such as economics, politics, law, biology, and computer science. In this course, I will introduce the basic tools of game theoretic analysis. In the process, I will outline some of the many applications of Game Theory, primarily in economics.

This course introduces students to iterative decoding algorithms and the codes to which they are applied, including Turbo Codes, Low-Density Parity-Check Codes, and Serially-Concatenated Codes. The course will begin with an introduction to the fundamental problems of Coding Theory and their mathematical formulations. This will be followed by a study of Belief Propagation--the probabilistic heuristic which underlies iterative decoding algorithms. Belief Propagation will then be applied to the decoding of Turbo, LDPC, and Serially-Concatenated codes. The technical portion of the course will conclude with a study of tools for explaining and predicting the behavior of iterative decoding algorithms, including EXIT charts and Density Evolution.

This course continues the content covered in 18.100 Analysis I. Roughly half of the subject is devoted to the theory of the Lebesgue integral with applications to probability, and the other half to Fourier series and Fourier integrals.

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.

This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.

Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.

This course covers the same material as Differential Equations (18.03) with more emphasis on theory. In addition, it treats mathematical aspects of ordinary differential equations such as existence theorems.