Matrices | Numerical differential equations | Algebraic graph theory
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph. Together with Kirchhoff's theorem, it can be used to calculate the number of spanning trees for a given graph. The sparsest cut of a graph can be approximated through the Fiedler vector — the eigenvector corresponding to the second smallest eigenvalue of the graph Laplacian — as established by Cheeger's inequality. The spectral decomposition of the Laplacian matrix allows constructing low dimensional embeddings that appear in many machine learning applications and determines a spectral layout in graph drawing. Graph-based signal processing is based on the graph Fourier transform that extends the traditional discrete Fourier transform by substituting the standard basis of complex sinusoids for eigenvectors of the Laplacian matrix of a graph corresponding to the signal. The Laplacian matrix is the easiest to define for a simple graph, but more common in applications for an edge-weighted graph, i.e., with weights on its edges — the entries of the graph adjacency matrix. Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries — resulting in normalized adjacency and Laplacian matrices. (Wikipedia).
Directed Laplacian Matrices - John Peebles
Short Talks by Postdoctoral Members Topic: Directed Laplacian Matrices Speaker: John Peebles Affiliation: Member, School of Mathematics Date: September 28, 2021
From playlist Mathematics
Physics - Advanced E&M: Ch 1 Math Concepts (13 of 55) What is the Laplacian of a Scalar (Field)?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain, develop the equation, and give examples of the Laplacian of a scalar (field). Next video in this series can be seen at: https://youtu.be/2VXFzhcGT3U
From playlist PHYSICS 67 ADVANCED ELECTRICITY & MAGNETISM
Physics - Advanced E&M: Ch 1 Math Concepts (33 of 55) Curl of a Cylindrical Vector Field
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is the Laplacian in a cylindrical vector field. Next video in this series can be seen at: https://youtu.be/OuSpKEg8KzU
From playlist PHYSICS 67 ADVANCED ELECTRICITY & MAGNETISM
Laplace Transform Explained and Visualized Intuitively
Laplace Transform explained and visualized with 3D animations, giving an intuitive understanding of the equations. My Patreon page is at https://www.patreon.com/EugeneK
From playlist Physics
C79 Linear properties of the Laplace transform
The linear properties of the Laplace transform.
From playlist Differential Equations
A Laplacian for Nonmanifold Triangle Meshes - SGP 2020
Authors: Nicholas Sharp and Keenan Crane presented at SGP 2020 https://sgp2020.sites.uu.nl https://github.com/nmwsharp/nonmanifold-laplacian Abstract: We describe a discrete Laplacian suitable for any triangle mesh, including those that are nonmanifold or nonorientable (with or without b
From playlist Research
Matrix models, Laplacian growth and Hurwitz numbers - Anton Zabrodin
Anton Zabrodin ITEP November 5, 2013 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Details: https://twitter.com/_tim_hutton_/status/1244736881989992449 Run it for yourself in Ready: https://github.com/GollyGang/ready
From playlist Ready
Laplacian of a scalar or vector field | Lecture 20 | Vector Calculus for Engineers
Definition of the Laplacian of a scalar or vector field. Join me on Coursera: https://www.coursera.org/learn/vector-calculus-engineers Lecture notes at http://www.math.ust.hk/~machas/vector-calculus-for-engineers.pdf Subscribe to my channel: http://www.youtube.com/user/jchasnov?sub_con
From playlist Vector Calculus for Engineers
Introduction to Laplacian Linear Systems for Undirected Graphs - John Peebles
Computer Science/Discrete Mathematics Seminar II Topic: Introduction to Laplacian Linear Systems for Undirected Graphs Speaker: John Peebles Affiliation: Member, School of Mathematics Date: February 23, 2021 For more video please visit http://video.ias.edu
From playlist Mathematics
Network Science. Lecture12 .Diffusion and random walks on graphs.
Diffusion and random walks on graphs. Lecture slides: http://www.leonidzhukov.net/hse/2020/networks/lectures/lecture12.pdf
From playlist Network Science. Module 2, 2020
Lecture 18: The Laplace Operator (Discrete Differential Geometry)
Full playlist: https://www.youtube.com/playlist?list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS For more information see http://geometry.cs.cmu.edu/ddg
From playlist Discrete Differential Geometry - CMU 15-458/858
Ginestra Bianconi (8/28/21): The topological Dirac operator and the dynamics of topological signals
Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. Typically, topological signals of a given dimension are investig
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Sparsification of graphs and matrices - Daniel Spielman
Daniel Spielman Yale University November 3, 2014 Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph
From playlist Mathematics
Francesca Da Lio: Analysis of nonlocal conformal invariant variational problems, Lecture III
There has been a lot of interest in recent years for the analysis of free-boundary minimal surfaces. In the first part of the course we will recall some facts of conformal invariant problems in 2D and some aspects of the integrability by compensation theory. In the second part we will show
From playlist Hausdorff School: Trending Tools
Hermitian and Non-Hermitian Laplacians and Wave Equaions by Andrey shafarevich
Non-Hermitian Physics - PHHQP XVIII DATE: 04 June 2018 to 13 June 2018 VENUE:Ramanujan Lecture Hall, ICTS Bangalore Non-Hermitian Physics-"Pseudo-Hermitian Hamiltonians in Quantum Physics (PHHQP) XVIII" is the 18th meeting in the series that is being held over the years in Quantum Phys
From playlist Non-Hermitian Physics - PHHQP XVIII
Raffaella Mulas - Spectral theory of hypergraphs
Hypergraphs are a generalization of graphs in which vertices are joined by edges of any size. In this talk, we generalize the graph normalized Laplace operators to the case of hypergraphs, and we discuss some properties of their spectra. We discuss the geometrical meaning of the largest an
From playlist Research Spotlight
Network Analysis. Lecture 11. Diffusion and random walks on graphs
Random walks on graph. Stationary distribution. Physical diffusion. Diffusion equation. Diffusion in networks. Discrete Laplace operator, Laplace matrix. Solution of the diffusion equation. Normalized Laplacian. Lecture slides: http://www.leonidzhukov.net/hse/2015/networks/lectures/lectu
From playlist Structural Analysis and Visualization of Networks.
Differential Equations | The Laplace Transform of a Derivative
We establish a formula involving the Laplace transform of the derivative of a function. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist The Laplace Transform