Courses tagged with "Information control" (59)

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.

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 an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.

This course begins with an introduction to the theory of computability, then proceeds to a detailed study of its most illustrious result: Kurt Gödel's theorem that, for any system of true arithmetical statements we might propose as an axiomatic basis for proving truths of arithmetic, there will be some arithmetical statements that we can recognize as true even though they don't follow from the system of axioms. In my opinion, which is widely shared, this is the most important single result in the entire history of logic, important not only on its own right but for the many applications of the technique by which it's proved. We'll discuss some of these applications, among them: Church's theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus; Tarski's theorem that the set of true sentence of a language isn't definable within that language; and Gödel's second incompleteness theorem, which says that no consistent system of axioms can prove its own consistency.

This subject offers an interactive introduction to discrete mathematics oriented toward computer science and engineering. The subject coverage divides roughly into thirds:

1. Fundamental concepts of mathematics: Definitions, proofs, sets, functions, relations.
2. Discrete structures: graphs, state machines, modular arithmetic, counting.
3. 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.

This course covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.

Principles of Applied Mathematics is a study of illustrative topics in discrete applied mathematics including sorting algorithms, information theory, coding theory, secret codes, generating functions, linear programming, game theory. There is an emphasis on topics that have direct application in the real world.

This course was recently revised to meet the MIT Undergraduate Communication Requirement (CR). It covers the same content as 18.310, but assignments are structured with an additional focus on writing.

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.

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

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.

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.

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 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.

Analysis I (18.100) in its various versions covers fundamentals of mathematical analysis: continuity, differentiability, some form of the Riemann integral, sequences and series of numbers and functions, uniform convergence with applications to interchange of limit operations, some point-set topology, including some work in Euclidean n-space.

MIT students may choose to take one of three versions of 18.100: Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible. Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology. Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication.

This course covers the fundamentals of mathematical analysis: convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, and the interchange of limit operations. It shows the utility of abstract concepts and teaches an understanding and construction of proofs. MIT students may choose to take one of three versions of Real Analysis; this version offers three additional units of credit for instruction and practice in written and oral presentation.

The three options for 18.100:

• Option A (18.100A) chooses less abstract definitions and proofs, and gives applications where possible.
• Option B (18.100B) is more demanding and for students with more mathematical maturity; it places more emphasis from the beginning on point-set topology and n-space, whereas Option A is concerned primarily with analysis on the real line, saving for the last weeks work in 2-space (the plane) and its point-set topology.
• Option C (18.100C) is a 15-unit variant of Option B, with further instruction and practice in written and oral communication. This fulfills the MIT CI requirement.