Courses tagged with "Information Theory" (118)
6.844 is a graduate introduction to programming theory, logic of programming, and computability, with the programming language Scheme used to crystallize computability constructions and as an object of study itself. Topics covered include: programming and computability theory based on a term-rewriting, "substitution" model of computation by Scheme programs with side-effects; computation as algebraic manipulation: Scheme evaluation as algebraic manipulation and term rewriting theory; paradoxes from self-application and introduction to formal programming semantics; undecidability of the Halting Problem for Scheme; properties of recursively enumerable sets, leading to Incompleteness Theorems for Scheme equivalences; logic for program specification and verification; and Hilbert's Tenth Problem.
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 linear optimization and its extensions emphasizing the underlying mathematical structures, geometrical ideas, algorithms and solutions of practical problems. The topics covered include: formulations, the geometry of linear optimization, duality theory, the simplex method, sensitivity analysis, robust optimization, large scale optimization network flows, solving problems with an exponential number of constraints and the ellipsoid method, interior point methods, semidefinite optimization, solving real world problems problems with computer software, discrete optimization formulations and algorithms.
Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.
Numerical methods for solving problems arising in heat and mass transfer, fluid mechanics, chemical reaction engineering, and molecular simulation. Topics: numerical linear algebra, solution of nonlinear algebraic equations and ordinary differential equations, solution of partial differential equations (e.g. Navier-Stokes), numerical methods in molecular simulation (dynamics, geometry optimization). All methods are presented within the context of chemical engineering problems. Familiarity with structured programming is assumed. The examples will use MATLAB®.
Acknowledgements
The instructor would like to thank Robert Ashcraft, Sandeep Sharma, David Weingeist, and Nikolay Zaborenko for their work in preparing materials for this course site.
This course is divided into two sections, Part I and Part II. Part I, found here, provides an introduction to statistical theory. A brief review of probability will be given mainly as background material, however, it is assumed to be known. Topics include normal distribution, limit theorems, Bayesian concepts, and testing, among others.
Part II prepares students for the remainder of the econometrics sequence and and can be found by visiting 14.381 Fall 2006.
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 teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations. (No epsilons or deltas are harmed by taking this course.) This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month.
This course covers differential, integral and vector calculus for functions of more than one variable. These mathematical tools and methods are used extensively in the physical sciences, engineering, economics and computer graphics.
Course Formats
The materials have been organized to support independent study. The website includes all of the materials you will need to understand the concepts covered in this subject. The materials in this course include:
- Lecture Videos recorded on the MIT campus
- Recitation Videos with problem-solving tips
- Examples of solutions to sample problems
- Problem for you to solve, with solutions
- Exams with solutions
- Interactive Java Applets ("Mathlets") to reinforce key concepts
Content Development
Denis Auroux
Arthur Mattuck
Jeremy Orloff
John Lewis
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.
This course covers matrix theory and linear algebra, emphasizing topics useful in other disciplines such as physics, economics and social sciences, natural sciences, and engineering. It parallels the combination of theory and applications in Professor Strang’s textbook Introduction to Linear Algebra.
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:
- A complete set of Lecture Videos by Professor Gilbert Strang.
- Summary Notes for all videos along with suggested readings in Prof. Strang's textbook Linear Algebra.
- Problem Solving Videos on every topic taught by an experienced MIT Recitation Instructor.
- Problem Sets to do on your own with Solutions to check your answers against when you're done.
- A selection of Java® Demonstrations to illustrate key concepts.
- A full set of Exams with Solutions, including review material to help you prepare.
Other Versions
Other OCW Versions
OCW has published multiple versions of this subject.
Related Content
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 subject offers an interactive introduction to discrete mathematics oriented toward computer science and engineering. The subject coverage divides roughly into thirds:
- Fundamental concepts of mathematics: Definitions, proofs, sets, functions, relations.
- Discrete structures: graphs, state machines, modular arithmetic, counting.
- Discrete probability theory.
On completion of 6.042J, students will be able to explain and apply the basic methods of discrete (noncontinuous) mathematics in computer science. They will be able to use these methods in subsequent courses in the design and analysis of algorithms, computability theory, software engineering, and computer systems.
Interactive site components can be found on the Unit pages in the left-hand navigational bar, starting with Unit 1: Proofs.
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