Transformation (function) | Projective geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations". For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative) field. Equivalently Pappus's hexagon theorem and Desargues's theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold. (Wikipedia).
Homomorphisms in abstract algebra
In this video we add some more definition to our toolbox before we go any further in our study into group theory and abstract algebra. The definition at hand is the homomorphism. A homomorphism is a function that maps the elements for one group to another whilst maintaining their structu
From playlist Abstract algebra
Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t
From playlist Introduction to Homotopy Theory
Group Homomorphisms - Abstract Algebra
A group homomorphism is a function between two groups that identifies similarities between them. This essential tool in abstract algebra lets you find two groups which are identical (but may not appear to be), only similar, or completely different from one another. Homomorphisms will be
From playlist Abstract Algebra
Homomorphisms in abstract algebra examples
Yesterday we took a look at the definition of a homomorphism. In today's lecture I want to show you a couple of example of homomorphisms. One example gives us a group, but I take the time to prove that it is a group just to remind ourselves of the properties of a group. In this video th
From playlist Abstract algebra
What is a Group Homomorphism? Definition and Example (Abstract Algebra)
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys What is a Group Homomorphism? Definition and Example (Abstract Algebra)
From playlist Abstract Algebra
Homomorphisms (Abstract Algebra)
A homomorphism is a function between two groups. It's a way to compare two groups for structural similarities. Homomorphisms are a powerful tool for studying and cataloging groups. Be sure to subscribe so you don't miss new lessons from Socratica: http://bit.ly/1ixuu9W ♦♦♦♦♦♦♦♦♦♦ W
From playlist Abstract Algebra
Stable Homotopy without Homotopy - Toni Mikael Annala
IAS/Princeton Arithmetic Geometry Seminar Topic: Stable Homotopy without Homotopy Speaker: Toni Mikael Annala Affiliation: Member, School of Mathematics Date: January 30, 2023 Many cohomology theories in algebraic geometry, such as crystalline and syntomic cohomology, are not homotopy in
From playlist Mathematics
Definition of a Group Homomorphism and Sample Proof
We define what it means for a function to be a group homomorphism. The intuition behind the definition is explained. We then do a simple proof to show that a specific function is a group homomorphism. Useful Math Supplies https://amzn.to/3Y5TGcv My Recording Gear https://amzn.to/3BFvcxp (
From playlist Group Theory
Viktoriya Ozornova: Equivalences in higher categories
CONFERENCE Recording during the thematic meeting : « Chromatic Homotopy, K-Theory and Functors» the January 24, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldwide mathematicians on CIR
From playlist Topology
Homotopy type theory: working invariantly in homotopy theory -Guillaume Brunerie
Short talks by postdoctoral members Topic: Homotopy type theory: working invariantly in homotopy theory Speaker: Guillaume Brunerie Affiliation: Member, School of Mathematics Date: September 26, 2017 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Feature Matching (Homography) Brute Force - OpenCV with Python for Image and Video Analysis 14
Welcome to a feature matching tutorial with OpenCV and Python. Feature matching is going to be a slightly more impressive version of template matching, where a perfect, or very close to perfect, match is required. We start with the image that we're hoping to find, and then we can search f
From playlist OpenCV with Python for Image and Video Analysis
Lucia Di Vizio : Méthodes galoisiennes appliquées aux équations fonctionnelles issues de la...
CONFERENCE Recording during the thematic meeting : « ALEA Days» the March 16, 2023 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker : Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathemat
From playlist Combinatorics
What do number theorists know about homotopy groups? - Piotr Pstragowski
Short Talks by Postdoctoral Members Topic: What do number theorists know about homotopy groups? Speaker: Piotr Pstragowski Affiliation: Member, School of Mathematics Date: September 29, 2022
From playlist Mathematics
Stefan Schwede: Equivariant stable homotopy - Lecture 2
I will use the orthogonal spectrum model to introduce the tensor triangulated category of genuine G-spectra, for compact Lie groups G. I will explain structural properties such as the smash product of G-spectra, and functors relating the categories for varying G (fixed points, geometric fi
From playlist Summer School: Spectral methods in algebra, geometry, and topology
Three-dimensional Anosov flows and non-Weinstein Liouville domains - Thomas Massoni
Joint IAS/Princeton University Symplectic Geometry Seminar Topic: Three-dimensional Anosov flows and non-Weinstein Liouville domains. Speaker: Thomas Massoni Affiliation: Princeton University Date: October 17, 2022
From playlist Mathematics
Computing homology groups | Algebraic Topology | NJ Wildberger
The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2-dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each
From playlist Algebraic Topology
Valérie Berthé - Fractions continues multidimensionnelles et dynamique (Part 3)
Le but de cet exposé est de présenter des généralisations multidimensionnelles des fractions continues et de l’algorithme d’Euclide d’un point de vue systèmes dynamiques, en nous concentrant sur les liens avec la numération et les substitutions. Nous allons considérer principalement deux t
From playlist École d’été 2013 - Théorie des nombres et dynamique
Valérie Berthé - Fractions continues multidimensionnelles et dynamique (Part 2)
Le but de cet exposé est de présenter des généralisations multidimensionnelles des fractions continues et de l’algorithme d’Euclide d’un point de vue systèmes dynamiques, en nous concentrant sur les liens avec la numération et les substitutions. Nous allons considérer principalement deux t
From playlist École d’été 2013 - Théorie des nombres et dynamique
This is lecture 3 of an online mathematics course on group theory. It gives a review of homomorphisms and isomorphisms and gives some examples of these.
From playlist Group theory
Valérie Berthé - Fractions continues multidimensionnelles et dynamique (Part 1)
Le but de cet exposé est de présenter des généralisations multidimensionnelles des fractions continues et de l’algorithme d’Euclide d’un point de vue systèmes dynamiques, en nous concentrant sur les liens avec la numération et les substitutions. Nous allons considérer principalement deux t
From playlist École d’été 2013 - Théorie des nombres et dynamique