Structures on manifolds | Lipschitz maps
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (or modulus of uniform continuity). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line: Continuously differentiable ⊂ Lipschitz continuous ⊂ -Hölder continuous, where . We also have Lipschitz continuous ⊂ absolutely continuous ⊂ uniformly continuous. (Wikipedia).
What is a Lipschitz condition?
This is a basic introduction to Lipschitz conditions within the context of differential equations. Lipschitz conditions are connected with `"contractive mappings'", which have important applications to the existence, uniqueness and approximation of solutions to equations -- including ordi
From playlist Mathematical analysis and applications
Every Lipschitz Function is Uniformly Continuous Proof
In this video I go through the proof that every Lipschitz function is uniformly continuous. I hope this video helps someone who is studying mathematical analysis/advanced calculus. If you enjoyed this video please consider liking, sharing, and subscribing. You can also help support my ch
From playlist Advanced Calculus
Uniform Continuity and Cauchy In this video, I answer a really interesting question about continuous functions: If sn is a Cauchy sequence and f is a continuous function, then is f(sn) Cauchy as well? Surprisingly this has to do with uniform continuity. Watch this video to find out why!
From playlist Limits and Continuity
Existence and Uniqueness of Solutions (Differential Equations 11)
https://www.patreon.com/ProfessorLeonard THIS VIDEO CAN SEEM VERY DECEIVING REGARDING CONTINUITY. As I watched this back, after I edited it of course, I noticed that I mentioned continuity is not possible at Endpoints. This is NOT true, as we can consider one-sided limits. What I MEANT
From playlist Differential Equations
In this video, I cover the notion of continuity, as used in topology. The beautiful thing is that this doesn't use epsilon-delta at all, and instead just something purely geometric. Enjoy this topology-adventure! Topology Playlist: https://youtube.com/playlist?list=PLJb1qAQIrmmA13vj9xkHGG
From playlist Topology
a neat fact about uniform continuity
Uniform Continuity and Derivatives In this video, I present a really neat test for uniform continuity, which has to do with derivatives. Check out this video to find out what it is! Uniform Continuity: https://youtu.be/PA0EJHYymLE Mean Value Theorem: https://youtu.be/PloNnv_DWas Continu
From playlist Limits and Continuity
Quantitative decompositions of Lipschitz mappings - Guy C. David
Analysis Seminar Topic: Quantitative decompositions of Lipschitz mappings Speaker: Guy C. David Affiliation: Ball State University Date: May 12, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Mod-04 Lec-16 Gronwall's Lemma
Ordinary Differential Equations and Applications by A. K. Nandakumaran,P. S. Datti & Raju K. George,Department of Mathematics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in.
From playlist IISc Bangalore: Ordinary Differential Equations and Applications | CosmoLearning.org Mathematics
Uniform Continuity and Compactness
Uniform Continuity and Compactness In this video, I show a super nice property of uniform continuity: Namely any continuous function on [a,b] is *automatically* uniformly continuous. This test definitely saves us lots of time, enjoy! Here is a proof that's valid for general metric spaces
From playlist Limits and Continuity
Continuity and Monotonicity In this video, I show a very interesting fact about functions: Namely, if a function f is continuous and one-to-one, then it is either strictly increasing, or strictly decreasing. Intuitively it makes sense, but can you prove it? Continuity Playlist: https://w
From playlist Limits and Continuity
[Quiz] Regularization in Deep Learning, Lipschitz continuity, Gradient regularization
Regularization, Lipschitz continuity, Gradient regularization, Adversarial Defense, Gradient Penalty, were all topics of our daily Quiz questions! Tim Elsner @daidailoh (Twitter) jumped in again to do Ms. Coffee Bean’s Job in explaining these concepts that were part of the AI Coffee Break
From playlist AI Coffee Break Quiz question answers
Mod-04 Lec-22 Continuation of Solutions
Ordinary Differential Equations and Applications by A. K. Nandakumaran,P. S. Datti & Raju K. George,Department of Mathematics,IISc Bangalore.For more details on NPTEL visit http://nptel.ac.in.
From playlist IISc Bangalore: Ordinary Differential Equations and Applications | CosmoLearning.org Mathematics
Gunther Leobacher: Stochastic Differential Equations
In the second part we show how the classical result can be used also for SDEs with drift that may be discontinuous and diffusion that may be degenerate. In that context I will present a concept of (multidimensional) piecewise Lipschitz drift where the set of discontinuities is a sufficient
From playlist Virtual Conference
Mini Course 1: Dynamics of Anosov representations (Lecture - 01 ) by Francois Labourie
Geometry, Groups and Dynamics (GGD) - 2017 DATE: 06 November 2017 to 24 November 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru The program focuses on geometry, dynamical systems and group actions. Topics are chosen to cover the modern aspects of these areas in which research has b
From playlist Geometry, Groups and Dynamics (GGD) - 2017
D. Vittone - Rectifiability issues in sub-Riemannian geometry
In this talk we discuss two problems concerning “rectifiability” in sub-Riemannian geometry and particularly in the model setting of Carnot groups. The first problem regards the rectifiability of boundaries of sets with finite perimeter in Carnot groups, while the second one concerns Radem
From playlist Journées Sous-Riemanniennes 2018
Stefan Wenger - 21 September 2016
Wenger, Stefan "“Plateau’s problem in metric spaces and applications”"
From playlist A Mathematical Tribute to Ennio De Giorgi
Math 101 Introduction to Analysis 110415: Continuity (two versions)
Continuity: definition of (actually sequential continuity); examples; standard definition involving neighborhoods; examples.
From playlist Course 6: Introduction to Analysis
Flows of vector fields: classical and modern - Camillo DeLellis
Analysis Seminar Topic: Flows of vector fields: classical and modern Speaker: Camillo DeLellis Affiliation: Faculty, School of Mathematics; IBM von Neumann Professor, School of Mathematics Date: April 13, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics