Fixed points (mathematics) | Metric geometry
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number such that for all x and y in M, The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied fork ≤ 1, then the mapping is said to be a non-expansive map. More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M, d) and (N, d') are two metric spaces, then is a contractive mapping if there is a constant such that for all x and y in M. Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1). A contraction mapping has at most one fixed point. Moreover, the Banach fixed-point theorem states that every contraction mapping on a non-empty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed-point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem. Contraction mappings play an important role in dynamic programming problems. (Wikipedia).
What is length contraction? Length contraction gives the second piece (along with time dilation) of the puzzle that allows us to reconcile the fact that the speed of light is constant in all reference frames.
From playlist Relativity
Determine a Vertical Stretch or Vertical Compression
This video provides two examples of how to express a vertical stretch or compression using function notation. Site: http://mathispower4u.com
From playlist Determining Transformations of Functions
Ex: Function Notation for Horizontal and Vertical Stretches and Compressions
This video explains how to recognize a horizontal and vertical compression or stretch using function notation. Site: http://mathispower4u.com
From playlist Determining Transformations of Functions
Determine a Horizontal Stretch or Horizontal Compression
This video provides two examples of how to express a horizontal stretch or compression using function notation. Site: http://mathispower4u.com
From playlist Determining Transformations of Functions
How to find the horizontal and vertical compressions, stretches of multiple functions
👉 Learn how to identify transformations of functions. Transformation of a function involves alterations to the graph of the parent function. The transformations can be dilations, translations (shifts), reflection, stretches, shrinks, etc. To sketch the graph of a transformed function, we s
From playlist I don't know where to put it
Basic Methods: We define mappings (or functions) between sets and consider various examples. These include binary operations, projections, and quotient maps. We show how to construct the rational numbers from the integers and explain why division by zero is a forbidden operation.
From playlist Math Major Basics
Ex: Identify Horizontal and Vertical Stretches and Compressions -- Function Notation
This video explains how to recognize a horizontal and vertical compression or stretch using function notation. Site: http://mathispower4u.com
From playlist Determining Transformations of Functions
Overview of functions stretching and shrinking - Online Tutor - Free Math Videos
👉 Learn how to determine the transformation of a function. Transformations can be horizontal or vertical, cause stretching or shrinking or be a reflection about an axis. You will see how to look at an equation or graph and determine the transformation. You will also learn how to graph a t
From playlist Characteristics of Functions
Describing a transformation with vertical and horizontal stretch then graphing
👉 Learn how to identify transformations of functions. Transformation of a function involves alterations to the graph of the parent function. The transformations can be dilations, translations (shifts), reflection, stretches, shrinks, etc. To sketch the graph of a transformed function, we s
From playlist Characteristics of Functions
What is a Tensor? Lesson 12 (redux): Contraction and index gymnastics
What is a Tensor? Lesson 12 (redux): Contraction and index gymnastics I have redone the index gymnastics lecture to try and fill in the details regarding contractions. I will keep them both in the playlist for now.
From playlist What is a Tensor?
How To Bridge Tokens From Ethereum To Polygon With MetaMask | Session 04 | #blockchain
Don’t forget to subscribe! This project series is about how to bridge tokens from Ethereum to Polygon with MetaMask. This series will guide you through it all, creating an asset, deploying the token contract, and bridging the tokens to the Polygon network. Introduction: https://www
From playlist Bridge Tokens From Ethereum To Polygon With MetaMask
How To Bridge Assets From Ethereum To Binance Smart Chain | Session 04 | #blockchain
Don’t forget to subscribe! In this project series, you will learn how to bridge assets from Ethereum to Binance smart chain. You will learn all the important steps that you need to bridge the assets. Introduction: https://www.youtube.com/watch?v=Tkvn7D4t64A&list=PLQbzkJk10-f7hvXfE4U
From playlist Bridge Assets From Ethereum To Binance Smart Chain
How To Create Drug Authenticity dApp On OpenZeppelin | Session 03 | #ethereum | #blockchain
Don’t forget to subscribe! This project series is about how to create drug authenticity dApp on Openzeppelin. Through this series, we shall be creating 3 separate dApps that would leverage on openZeppelin smart contract to enable users to verify the authenticity of drugs they buy from t
From playlist Create Drug Authenticity dApp On OpenZeppelin
Paul Arne Østvær: A1 contractible varieties
The lecture was held within the framework of the Hausdorff Trimester Program : Workshop "K-theory in algebraic geometry and number theory"
From playlist HIM Lectures: Trimester Program "K-Theory and Related Fields"
What is General Relativity? Lesson 14: The covariant derivative of a covector
We start by demonstrating that contraction commutes with directional covariant derivative and then derive the CFREE and COMP expressions for the covariant derivative of a covector.
From playlist What is General Relativity?
Dylan Thurston: Characterizing rational maps positively using graphs
HYBRID EVENT Recorded during the meeting "Advancing Bridges in Complex Dynamics" the September 21, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematicians on CIRM'
From playlist Virtual Conference
Egbert Rijke: Daily applications of the univalence axiom - lecture 2
HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 22, 2022 by 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 M
From playlist Combinatorics
Category theory for JavaScript programmers #21: terminal and initial objects
http://jscategory.wordpress.com/source-code/
From playlist Category theory for JavaScript programmers
Egbert Rijke: Daily applications of the univalence axiom - lecture 1
HYBRID EVENT Recorded during the meeting "Logic and Interactions" the February 21, 2022 by 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 M
From playlist Combinatorics
This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2toQ.
From playlist 3D printing