Planar graphs | Theorems in graph theory
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most vertices. A weaker form of the separator theorem with vertices in the separator instead of was originally proven by , and the form with the tight asymptotic bound on the separator size was first proven by . Since their work, the separator theorem has been reproven in several different ways, the constant in the term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs. Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch-decomposition of the graph. Separator hierarchies may be used to devise efficient divide and conquer algorithms for planar graphs, and dynamic programming on these hierarchies can be used to devise exponential time and fixed-parameter tractable algorithms for solving NP-hard optimization problems on these graphs. Separator hierarchies may also be used in nested dissection, an efficient variant of Gaussian elimination for solving sparse systems of linear equations arising from finite element methods. Beyond planar graphs, separator theorems have been applied to other classes of graphs including graphs excluding a fixed minor, nearest neighbor graphs, and finite element meshes. The existence of a separator theorem for a class of graphs can be formalized and quantified by the concepts of treewidth and polynomial expansion. (Wikipedia).
Convolution Theorem: Fourier Transforms
Free ebook https://bookboon.com/en/partial-differential-equations-ebook Statement and proof of the convolution theorem for Fourier transforms. Such ideas are very important in the solution of partial differential equations.
From playlist Partial differential equations
Differential Equations | Convolution: Definition and Examples
We give a definition as well as a few examples of the convolution of two functions. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Differential Equations
(3.2.3) The Determinant of Square Matrices and Properties
This video defines the determinant of a matrix and explains what a determinant means in terms of mapping and area. https://mathispower4u.com
From playlist Differential Equations: Complete Set of Course Videos
Multilinearity of the determinant In this video, I define the notion of a multilinear function and I show that the determinant is multilinear. Come and get a taste of the beauty of multilinear algebra :) Check out my Determinants Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIr
From playlist Determinants
In this video, I calculate the determinant of a block matrix and show that the answer is what you expect, namely the product of the determinants of the blocks. This is useful for instance in the proof of the Cayley Hamilton theorem, but also in the theory of Jordan Forms. Cayley-Hamilton
From playlist Determinants
Concavity and Parametric Equations Example
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Concavity and Parametric Equations Example. We find the open t-intervals on which the graph of the parametric equations is concave upward and concave downward.
From playlist Calculus
Characterization of the determinant
In this video, I show why the determinant is so special in math: Namely, it is the only function which is multilinear, alternating, and has the value 1 at the identity matrix. This is a generalization of a previous matrix puzzle for the 2 x 2 case. 2 x 2 case: https://youtu.be/lIMeIC1ZJO8
From playlist Determinants
AlgTop10: More on graphs and Euler's formula
We discuss applications of Euler's formula to various planar situations, in particular to planar graphs, including complete and complete bipartite graphs, the Five neighbours theorem, the Six colouring theorem, and to Pick's formula, which lets us compute the area of an integral polygonal
From playlist Algebraic Topology: a beginner's course - N J Wildberger
J. Aramayona - MCG and infinite MCG (Part 3)
The first part of the course will be devoted to some of the classical results about mapping class groups of finite-type surfaces. Topics may include: generation by twists, Nielsen-Thurston classification, abelianization, isomorphic rigidity, geometry of combinatorial models. In the secon
From playlist Ecole d'été 2018 - Teichmüller dynamics, mapping class groups and applications
The Four-Color Theorem and an Instanton Invariant for Spatial Graphs I - Peter Kronheimer
Peter Kronheimer Harvard University October 13, 2015 http://www.math.ias.edu/seminars/abstract?event=83214 Given a trivalent graph embedded in 3-space, we associate to it an instanton homology group, which is a finite-dimensional Z/2 vector space. The main result about the instanton hom
From playlist Geometric Structures on 3-manifolds
Michal Pilipczuk: Introduction to parameterized algorithms and applications, lecture III
The mini-course will provide a gentle introduction to the area of parameterized complexity, with a particular focus on methods connected to (integer) linear programming. We will start with basic techniques for the design of parameterized algorithms, such as branching, color coding, kerneli
From playlist Summer School on modern directions in discrete optimization
Louis Esperet: Coloring graphs on surfaces
Recording during the thematic meeting: "Graphs and surfaces: algorithms, combinatorics and topology" the May 11, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematici
From playlist Mathematical Aspects of Computer Science
Linear Algebra: Ch 2 - Determinants (2 of 48) What is a Determinant? (Part 2)
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the notation of and how to calculated a determinant. (Part 2) Next video in this series can be seen at: https://youtu.be/k3ZmxI267Zo
From playlist LINEAR ALGEBRA 2: DETERMINANTS
This is Lecture 22 of the CSE547 (Discrete Mathematics) taught by Professor Steven Skiena [http://www.cs.sunysb.edu/~skiena/] at Stony Brook University in 1999. The lecture slides are available at: http://www.cs.sunysb.edu/~algorith/math-video/slides/Lecture%2022.pdf More information may
From playlist CSE547 - Discrete Mathematics - 1999 SBU
This lecture is part of an online graduate course on Galois theory. We define the discriminant of a finite field extension, ans show that it is essentially the same as the discriminant of a minimal polynomial of a generator. We then give some applications to algebraic number fields. Corr
From playlist Galois theory
[Discrete Mathematics] Euler's Theorem
We introduce Euler's Theorem in graph theory and prove it. Visit our website: http://bit.ly/1zBPlvm Subscribe on YouTube: http://bit.ly/1vWiRxW *--Playlists--* Discrete Mathematics 1: https://www.youtube.com/playlist?list=PLDDGPdw7e6Ag1EIznZ-m-qXu4XX3A0cIz Discrete Mathematics 2: https:/
From playlist Discrete Math 2
26. Sum-product problem and incidence geometry
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 A famous open problem says that no set of integers can si
From playlist MIT 18.217 Graph Theory and Additive Combinatorics, Fall 2019
Jeff Erickson - Lecture 4 - Two-dimensional computational topology - 21/06/18
School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects (http://geomschool2018.univ-mlv.fr/) Jeff Erickson (University of Illinois at Urbana-Champaign, USA) Two-dimensional computational topology - Lecture 4 Abstract: This series of lectures will describe recent
From playlist Jeff Erickson - School on Low-Dimensional Geometry and Topology: Discrete and Algorithmic Aspects
Introduction to Parametric Equations
This video defines a parametric equations and shows how to graph a parametric equation by hand. http://mathispower4u.yolasite.com/
From playlist Parametric Equations
Laura Starkston: Unexpected symplectic fillings of links of rational surface singularities
HYBRID EVENT Recorded during the meeting "Milnor Fibrations, Degenerations and Deformations from Modern Perspectives" the September 09, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given
From playlist Virtual Conference