Computability theory

Maximal set

In recursion theory, the mathematical theory of computability, a maximal set is a coinfinite recursively enumerable subset A of the natural numbers such that for every further recursively enumerable subset B of the natural numbers, either B is cofinite or B is a finite variant of A or B is not a superset of A. This gives an easy definition within the lattice of the recursively enumerable sets. Maximal sets have many interesting properties: they are simple, hypersimple, and r-maximal; the latter property says that every recursive set R contains either only finitely many elements of the complement of A or almost all elements of the complement of A. There are r-maximal sets that are not maximal; some of them do even not have maximal supersets. Myhill (1956) asked whether maximal sets exist and Friedberg (1958) constructed one. Soare (1974) showed that the maximal sets form an orbit with respect to automorphism of the recursively enumerable sets under inclusion (modulo finite sets). On the one hand, every automorphism maps a maximal set A to another maximal set B; on the other hand, for every two maximal sets A, B there is an automorphism of the recursively enumerable sets such that A is mapped to B. (Wikipedia).

Video thumbnail

Find a Set with Greatest Cardinality that is a Subset of Two Given Sets (Lists)

This video explains how to determine a set with greatest cardinality that is a subset of two given sets.

From playlist Sets (Discrete Math)

Video thumbnail

How to Find a Minimal Generating Set

How to Find a Minimal Generating Set

From playlist Linear Algebra

Video thumbnail

Independent Vertex Sets | Graph Theory, Maximal and Maximum Independent Sets

What are independent vertex sets in graph theory? We'll go over independent sets, their definition and examples, and some related concepts in today's video graph theory lesson! A subset of the vertex set of a graph is an independent vertex set if and only if it contains no pair of adjace

From playlist Set Theory

Video thumbnail

Maximum and Minimum

Maximum and Minimum of a set In this video, I define the maximum and minimum of a set, and show that they don't always exist. Enjoy! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh

From playlist Real Numbers

Video thumbnail

Introduction to Sets and Set Notation

This video defines a set, special sets, and set notation.

From playlist Sets (Discrete Math)

Video thumbnail

Introduction to sets || Set theory Overview - Part 1

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other #sets. The #set with no element is the empty

From playlist Set Theory

Video thumbnail

Find a Set with Greatest Cardinality that is a Subset of Two Given Sets (Set Notation)

This video explains how to determine a set with greatest cardinality that is a subset of two given sets.

From playlist Sets (Discrete Math)

Video thumbnail

Rings 6 Prime and maximal ideals

This lecture is part of an online course on rings and modules. We discuss prime and maximal ideals of a (commutative) ring, use them to construct the spectrum of a ring, and give a few examples. For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj5

From playlist Rings and modules

Video thumbnail

Zorn’s Lemma and Basis

Why every vector space (not necessarily finite dimensional) has a basis, feat. Zorn's Lemma and the actual definition of a basis Check out my set theory playlist: Set theory https://www.youtube.com/playlist?list=PLJb1qAQIrmmDr_RYAtqY1MNgTgVNMJNIf Check out my vector space playlist: https

From playlist Vector Spaces

Video thumbnail

A Framework for Quadratic Form Maximization over Convex Sets -Vijay Bhattiprolu

Computer Science/Discrete Mathematics Seminar II Topic: A Framework for Quadratic Form Maximization over Convex Sets Speaker: Vijay Bhattiprolu Affiliation: Member, School of Mathematics Date: April 28, 2020 For more video please visit http://video.ias.edu

From playlist Mathematics

Video thumbnail

Lecture 2: A structure theorem for rooted binary phylogenetic networks and its various applications

This video is one of the two introductory lectures (Introduction to Discrete Mathematical Biology) given by Momoko Hayamizu as part of an omnibus lecture series "Advanced Modern Mathematical Sciences 2" for undergraduate mathematics majors at Waseda University. In this lecture, she gives a

From playlist 2020 Advanced Topic in Modern Mathematical Sciences 2

Video thumbnail

Additive Number Theory: Extremel Problems and the Combinatorics....(Lecture 3) by M. Nathanson

Program Workshop on Additive Combinatorics ORGANIZERS: S. D. Adhikari and D. S. Ramana DATE: 24 February 2020 to 06 March 2020 VENUE: Madhava Lecture Hall, ICTS Bangalore Additive combinatorics is an active branch of mathematics that interfaces with combinatorics, number theory, ergod

From playlist Workshop on Additive Combinatorics 2020

Video thumbnail

Schemes 5: Definition of a scheme

This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. We give some historical background, then give the definition of a scheme and some simple examples, and finish by explaining the origin of the word "spectrum".

From playlist Algebraic geometry II: Schemes

Video thumbnail

What are Maximal Circuits? | Graph Theory

What are maximal circuits of graphs? We'll go over the definition of maximal circuits and some examples and nonexamples in today's video graph theory lesson. A circuit is a closed trail: that is, a sequence of adjacent vertices, starting and ending at the same vertex, and traversing no ed

From playlist Graph Theory Exercises

Video thumbnail

Commutative algebra 11 (Spectrum of a ring)

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. In this lecture we define the spectrum of a ring as the space of prime ideals, and give a few examples. Reading: Lectures 9

From playlist Commutative algebra

Video thumbnail

Maximum and Maximal Cliques | Graph Theory, Clique Number

What are maximum cliques and maximal cliques in graph theory? We'll be defining both terms in today's video graph theory lesson, as well as going over an example of finding maximal and maximum cliques in a graph. These two terms can be a little confusing, so let's dig in and clarify our un

From playlist Graph Theory

Video thumbnail

Lecture 4: Production and Profit Maximization

MIT 14.04 Intermediate Microeconomic Theory, Fall 2020 Instructor: Prof. Robert Townsend View the complete course: https://ocw.mit.edu/courses/14-04-intermediate-microeconomic-theory-fall-2020/ YouTube Playlist: https://www.youtube.com/watch?v=XSTSfCs74bg&list=PLUl4u3cNGP63wnrKge9vllow3Y2

From playlist MIT 14.04 Intermediate Microeconomic Theory, Fall 2020

Video thumbnail

Anne de Roton: k-sum free sets in [0,1]

Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b

From playlist Combinatorics

Video thumbnail

Determine Sets Given Using Set Notation (Ex 2)

This video provides examples to describing a set given the set notation of a set.

From playlist Sets (Discrete Math)

Related pages

Natural number | Mathematics | Simple set | Automorphism | Computability | Lattice (order)