Functions and mappings | Equivalence (mathematics) | Singularity theory

In mathematics, -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let and be two manifolds, and let be two smooth map germs. We say that and are -equivalent if there exist diffeomorphism germs and such that In other words, two map germs are -equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. ) and the target (i.e. ). Let denote the space of smooth map germs Let be the group of diffeomorphism germs and be the group of diffeomorphism germs The group acts on in the natural way: Under this action we see that the map germs are -equivalent if, and only if, lies in the orbit of , i.e. (or vice versa). A map germ is called stable if its orbit under the action of is open relative to the Whitney topology. Since is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking -jets for every and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions ofthese base sets. Consider the orbit of some map germ The map germ is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs for are the infinite sequence, the infinite sequence, and (Wikipedia).

Equivalence Relations Definition and Examples

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Equivalence Relations Definition and Examples. This video starts by defining a relation, reflexive relation, symmetric relation, transitive relation, and then an equivalence relation. Several examples are given.

From playlist Abstract Algebra

Two Equivalence Classes [a] and [b] Are Equal If and Only If a is Related to b

In this video I prove a statement surrounding relations. We have an equivalence relation on a set A and we have to show that the equivalence class of a is equal to the equivalence class of b if and only if a is related to b. If you enjoyed this video please consider liking, sharing, and

From playlist Relations

This video is a full introduction to equivalence relations. Timestamps: 0:00 What is a relation? 3:02 Terminology - A Relation defined on a Set 4:02 Equivalence Relation Definition 7:18 Reflexive 9:18 Symmetric 11:48 Transitive Thanks for watching! Comment below with questions, and make

From playlist Proofs

The picture in the lecture was taken from Wikipedia: https://en.wikipedia.org/wiki/Demographics_of_the_United_States#/media/File:USA2020dec1.png

From playlist Abstract Algebra 1

Abstract Algebra | Equivalence Relations

We give the definition of an equivalence relation and some examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/

From playlist Abstract Algebra

Put all three properties of binary relations together and you have an equivalence relation.

From playlist Abstract algebra

Important Math Proof: The Set of Equivalence Classes Partition a Set

In this video I prove a very important result in mathematics. Given an equivalence relation R on a nonempty set A, the set S of equivalence classes of A is a partition of A. Stated another way, this result says we can write A as a disjoint union of equivalence classes. The pencils I used

From playlist Relations

Cosets and equivalence class proof

Now that we have shown that the relation on G is an equivalence relation ( https://www.youtube.com/watch?v=F7OgJi6o9po ), we can go on to prove that the equivalence class containing an element is the same as the corresponding set on H (a subset of G).

From playlist Abstract algebra

Equivalence Relation on a Group Two Proofs

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Equivalence Relation on a Group Two Proofs. Given a group G and a subgroup H of G, we prove that the relation x=y if xy^{-1} is in H is an equivalence relation on G. Then cosets are defined and we prove that s_1 = s_2 iff [s_1] = [s

From playlist Abstract Algebra

Equivalence Relations -- Proof Writing 17

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From playlist Proof Writing

Some important facts about ≡ (mod n)

We prove some important facts about a very important equivalence relation on the integers -- congruence modulo n. Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/micha

From playlist Proof Writing

Set Theory (Part 6): Equivalence Relations and Classes

Please feel free to leave comments/questions on the video and practice problems below! In this video, I set up equivalence relations and the canonical mapping. The idea of equivalence relation will return when we construct higher-level number systems, e.g.integers, from the natural number

From playlist Set Theory by Mathoma

Equivalence Relations and Partitions

We look at the connection between equivalence relations on a set and partitions of a set. Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal W

From playlist Proof Writing

What is a Manifold? Lesson 14: Quotient Spaces

I AM GOING TO REDO THIS VIDEO. I have made some annotations here and annotations are not visible on mobile devices. STAY TUNED. This is a long lesson about an important topological concept: quotient spaces.

From playlist What is a Manifold?

ITHT: Part 11- Quillen Adjunctions

Credits: nLab: https://ncatlab.org/nlab/show/Introduction+to+Homotopy+Theory#QuillenAdjunctions Animation library: https://github.com/3b1b/manim My own code/modified library: https://github.com/treemcgee42/youtub... Music: ► Artist Attribution • Music By: "KaizanBlu" • Track Na

From playlist Introduction to Homotopy Theory

Measure Equivalence, Negative Curvature, Rigidity (Lecture 1) by Camille Horbez

PROGRAM: PROBABILISTIC METHODS IN NEGATIVE CURVATURE ORGANIZERS: Riddhipratim Basu (ICTS - TIFR, India), Anish Ghosh (TIFR, Mumbai, India), Subhajit Goswami (TIFR, Mumbai, India) and Mahan M J (TIFR, Mumbai, India) DATE & TIME: 27 February 2023 to 10 March 2023 VENUE: Madhava Lecture Hall

From playlist PROBABILISTIC METHODS IN NEGATIVE CURVATURE - 2023

We look at the notion of an equivalence relation on a set, define an equivalence class, and consider several examples. Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/

From playlist Proof Writing

Equivalences and Partitions, Axiomatic Set Theory 2 2

Defining equivalences and partitions of sets, and proving some theorems about their relations to each other. My Twitter: https://twitter.com/KristapsBalodi3 Equivalence Relations:(0:00) Partitions:(9:22) Connecting Equivalence and Partitions:(14:09) Representatives:(27:04)

From playlist Axiomatic Set Theory

Univalent foundations and the equivalence principle - Benedikt Ahrens

Short Talks by Postdoctoral Members Benedikt Ahrens - September 21, 2015 http://www.math.ias.edu/calendar/event/88134/1442858400/1442859300 More videos on http://video.ias.edu

From playlist Short Talks by Postdoctoral Members