Fractals | Dimension theory | Metric geometry
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let A be a Borel subset of Rn, and let s > 0. Then the following are equivalent: * Hs(A) > 0, where Hs denotes the s-dimensional Hausdorff measure. * There is an (unsigned) Borel measure μ satisfying μ(A) > 0, and such thatholds for all x ∈ Rn and r>0. Otto Frostman proved this lemma for closed sets A as part of his PhD dissertation at Lund University in 1935. The generalization to Borel sets is more involved, and requires the theory of Suslin sets. A useful corollary of Frostman's lemma requires the notions of the s-capacity of a Borel set A ⊂ Rn, which is defined by (Here, we take inf ∅ = ∞ and 1⁄∞ = 0. As before, the measure is unsigned.) It follows from Frostman's lemma that for Borel A ⊂ Rn (Wikipedia).
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Problems, Paradoxes, and Sophisms
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Problems, Paradoxes, and Sophisms
Burnside's Lemma (Part 2) - combining math, science and music
Part 1 (previous video): https://youtu.be/6kfbotHL0fs Orbit-stabilizer theorem: https://youtu.be/BfgMdi0OkPU Burnside's lemma is an interesting result in group theory that helps us count things with symmetries considered, e.g. in some situations, we don't want to count things that can be
From playlist Traditional topics, explained in a new way
Toeplitz methods in completeness and spectral problems – Alexei Poltoratski – ICM2018
Analysis and Operator Algebras Invited Lecture 8.18 Toeplitz methods in completeness and spectral problems Alexei Poltoratski Abstract: We survey recent progress in the gap and type problems of Fourier analysis obtained via the use of Toeplitz operators in spaces of holomorphic functions
From playlist Analysis & Operator Algebras
Math 060 101317C Linear Transformations: Isomorphisms
Lemma: Linear transformations that agree on a basis are identical. Definition: one-to-one (injective). Examples and non-examples. Lemma: T is one-to-one iff its kernel is {0}. Definition: onto (surjective). Examples and non-examples. Definition: isomorphism; isomorphic. Theorem: T
From playlist Course 4: Linear Algebra (Fall 2017)
Stabilizer in abstract algebra
In the previous video we looked at the orbit of a set. To work towards the orbit stabilizer theorem, we take a look at what a stabilizer is in this video.
From playlist Abstract algebra
Linear Algebra: Continuing with function properties of linear transformations, we recall the definition of an onto function and give a rule for onto linear transformations.
From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics
Linear Algebra 19r: Translations, or How to Represent Nonlinear Transformations by Matrix Products
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Part 3 Linear Algebra: Linear Transformations
Drinfeld's lemma for schemes - Kiran Kedlaya
Joint IAS/Princeton University Algebraic Geometry Seminar Topic: Drinfeld's lemma for schemes Speaker: Kiran Kedlaya Affiliation: University of California, San Diego; Visiting Professor, School of Mathematics Date: February 4, 2019 For more video please visit http://video.ias.edu
From playlist Joint IAS/PU Algebraic Geometry
Linear Algebra Vignette 2a: RREF - What It's For
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes
Graph regularity and counting lemmas - Jacob Fox
Conference on Graphs and Analysis Jacob Fox June 5, 2012 More videos on http://video.ias.edu
From playlist Mathematics
Regularity methods in combinatorics, number theory, and computer science - Jacob Fox
Marston Morse Lectures Topic: Regularity methods in combinatorics, number theory, and computer science Speaker: Jacob Fox Affiliation: Stanford University Date: October 24, 2016 For more videos, visit http://video.ias.edu
From playlist Mathematics
9. Szemerédi's graph regularity lemma IV: induced removal lemma
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Prof. Zhao explains a strengthening of the graph regulari
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
6. Szemerédi's graph regularity lemma I: statement and proof
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Szemerédi's graph regularity lemma is a powerful tool in
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
A stable arithmetic regularity lemma in finite (...) - C. Terry - Workshop 1 - CEB T1 2018
Caroline Terry (Maryland) / 01.02.2018 A stable arithmetic regularity lemma in finite-dimensional vector spaces over fields of prime order In this talk we present a stable version of the arithmetic regularity lemma for vector spaces over fields of prime order. The arithmetic regularity l
From playlist 2018 - T1 - Model Theory, Combinatorics and Valued fields
7. Szemerédi's graph regularity lemma II: triangle removal lemma
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX Continuing the discussion of Szemerédi's graph regularity
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
László Lovász: The many facets of the Regularity Lemma
Abstract: The Regularity Lemma of Szemerédi, first obtained in the context of his theorem on arithmetic progressions in dense sequences, has become one of the most important and most powerful tools in graph theory. It is basic in extremal graph theory and in the theory of property testing.
From playlist Abel Lectures
10. Szemerédi's graph regularity lemma V: hypergraph removal and spectral proof
MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019 Instructor: Yufei Zhao View the complete course: https://ocw.mit.edu/18-217F19 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP62qauV_CpT1zKaGG_Vj5igX In this first half of this lecture, Prof. Zhao shows how
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Sylvia Serfaty (NYU) -- Microscopic description of Coulomb gases
We are interested in the statistical mechanics of systems of N points with Coulomb interactions in general dimension for a broad temperature range. We discuss local laws characterizing the rigidity of the system at the microscopic level, as well as free energy expansion and Central Limit T
From playlist Columbia Probability Seminar
Linear Algebra Vignette 3d: Easy Eigenvalues - Linearly Dependent Columns
This course is on Lemma: http://lem.ma Lemma looking for developers: http://lem.ma/jobs Other than http://lem.ma, I recommend Strang http://bit.ly/StrangYT, Gelfand http://bit.ly/GelfandYT, and my short book of essays http://bit.ly/HALAYT Questions and comments below will be prompt
From playlist Linear Algebra Vignettes