In mathematics, especially in order theory, a maximal element of a subset S of some preordered set is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some preordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of and the minimum of is again defined dually. In the particular case of a partially ordered set, while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements. Specializing further to totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide. As an example, in the collection ordered by containment, the element {d, o} is minimal as it contains no sets in the collection, the element {g, o, a, d} is maximal as there are no sets in the collection which contain it, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem, the Kirszbraun theorem, Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field. (Wikipedia).
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
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
How to Find a Minimal Generating Set
How to Find a Minimal Generating Set
From playlist Linear Algebra
Proof of the existence of the minimal polynomial. Every polynomial that annihilates an operator is a polynomial multiple of the minimal polynomial of the operator. The eigenvalues of an operator are precisely the zeros of the minimal polynomial of the operator.
From playlist Linear Algebra Done Right
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
What are the Maximum and Maximal Cliques of this Graph? | Graph Theory
How do we find the maximum and maximal cliques of a graph? We'll go over an example in today's graph theory lesson of doing just that! To use these elementary methods, we just need to remember our definitions. A clique of a graph G is a complete subgraph of G. We also call the vertex set
From playlist Graph Theory
Graph Theory: 50. Maximum vs Maximal
Here we describe the difference between two similar sounding words in mathematics: maximum and maximal. We use concepts in graph theory to highlight the difference. In particular, we define an independent set in a graph and a component in a graph and look at some examples. -- Bits of Gra
From playlist Graph Theory part-9
Maximum and Minimum Values (Closed interval method)
A review of techniques for finding local and absolute extremes, including an application of the closed interval method
From playlist 241Fall13Ex3
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
Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
Eulerianity of Fourier coefficients of automorphic forms - Henrik Gustafsson
Joint IAS/Princeton University Number Theory Seminar Topic: Eulerianity of Fourier coefficients of automorphic forms Speaker: Henrik Gustafsson Affiliation: Member, School of Mathematics Date: April 30, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
MATH1081 Discrete Maths: Chapter 2 Question 26
Here we look at a question about partially ordered set. We are given a set A consists positive integers, and for two numbers a and b in A, a is related to b if and only if a is a factor of b . Presented by Peter Brown of the School of Mathematics and Statistics, Faculty of Science, UNSW.
From playlist MATH1081 Discrete Mathematics
Existence Of Maximal Ideals - Feb 05, 2021- Rings and Modules
In this video we show using the axiom of choice that rings have maximal ideals.
From playlist Course on Rings and Modules (Abstract Algebra 4) [Graduate Course]
An analogue of o-minimality for valued fields - I. Halupczok - Workshop 2 - CEB T1 2018
Immanuel Halupczok (Universität Dässeldorf) / 08.03.2018 An analogue of o-minimality for valued fields. For first order structures on real closed fields, a very simple condition, namely o-minimality, implies strong tameness results about definable sets. In this talk, I will present an ana
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
Deep Learning Lecture 6.2 - PCA part 1
Principal Component Analysis: - Linear Projection - Maximal Projection Variance - Minimal Reconstruction Error - Rayleight Quotient of Covariance Matrix - Eigenvalue Decomposition.
From playlist Deep Learning Lecture
Additive Number Theory: Extremal Problems and the Combinatorics....(Lecture 2) 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
Calculus: Maximum-Minimum Problems With Two Variables
This video discusses how to find maximum and minimum values of a function of two variables using the second derivative test ("D-test").
From playlist Calculus
Elliptic Curves - Lecture 18a - Elliptic curves over local fields (the fundamental exact sequence)
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
From playlist An Introduction to the Arithmetic of Elliptic Curves