Ring Theory: As an application of all previous ideas on rings, we determine the primes in the Euclidean domain of Gaussian integers Z[i]. Not only is the answer somewhat elegant, but it contains a beautiful theorem on prime integers due to Fermat. We finish with examples of factorization
From playlist Abstract Algebra
(PP 6.3) Gaussian coordinates does not imply (multivariate) Gaussian
An example illustrating the fact that a vector of Gaussian random variables is not necessarily (multivariate) Gaussian.
From playlist Probability Theory
(PP 6.1) Multivariate Gaussian - definition
Introduction to the multivariate Gaussian (or multivariate Normal) distribution.
From playlist Probability Theory
Gaussian Integers and Infinitely many Primes of the form 4k+1
In this video we give a motivation for the Gaussian integers define them and use them to prove that there are infinitely many primes of the form 4k+1. 0:00 Introduction and primes that are the sum of two squares. 4:12 Definition and properties of the Gaussian Integers. 9:45 Infinitely many
From playlist Summer of Math Exposition 2 videos
(PP 6.5) Affine property, Constructing Gaussians, and Sphering
Any affine transformation of a (multivariate) Gaussian random variable is (multivariate) Gaussian. How to construct any (multivariate) Gaussian using an affine transformation of standard normals. How to "sphere" a Gaussian, i.e. transform it into a vector of independent standard normals.
From playlist Probability Theory
PUSHING A GAUSSIAN TO THE LIMIT
Integrating a gaussian is everyones favorite party trick. But it can be used to describe something else. Link to gaussian integral: https://www.youtube.com/watch?v=mcar5MDMd_A Link to my Skype Tutoring site: dotsontutoring.simplybook.me or email dotsontutoring@gmail.com if you have ques
From playlist Math/Derivation Videos
(ML 19.1) Gaussian processes - definition and first examples
Definition of a Gaussian process. Elementary examples of Gaussian processes.
From playlist Machine Learning
(ML 19.2) Existence of Gaussian processes
Statement of the theorem on existence of Gaussian processes, and an explanation of what it is saying.
From playlist Machine Learning
Gaussian Integral 6 Gamma Function
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I calculate the Gaussian integral by using properties of the gamma function, which
From playlist Gaussian Integral
Rings and modules 5 Examples of unique factorizations
This lecture is part of an online course on rings and modules. We give some examles to illustrate unique factorization. We use the fact that the Gaussian integers have unique factorization to prove Fermat's theorem about primes that are sums o 2 squares. Then we discuss a few other quadra
From playlist Rings and modules
Pi hiding in prime regularities
A story of pi, primes, complex numbers, and how number theory braids them together. Mathologer on why 4k + 1primes break down as sums of squares: https://youtu.be/DjI1NICfjOk Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply sh
From playlist Neat proofs/perspectives
Introduction to number theory lecture 43 Gaussian integers
This lecture is part of my Berkeley math 115 course "Introduction to number theory" For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53L8sMbzIhhXSAOpuZ1Fov8 We give some applications of Gaussian integers to the binary quadratic form x^2+y^2. The t
From playlist Introduction to number theory (Berkeley Math 115)
Zeev Rudnick: Angles of Gaussian primes
Abstract: Fermat showed that every prime p=1 mod 4 is a sum of two squares: p=a2+b2, and hence such a prime gives rise to an angle whose tangent is the ratio b/a. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given i
From playlist Number Theory
Orli Herscovici - Kohler-Jobin Meets Ehrhard - IPAM at UCLA
Recorded 07 February 2022. Orli Herscovici of the Georgia Institute of Technology presents "Kohler-Jobin Meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements" at IPAM's Calculus of Variations in Probab
From playlist Workshop: Calculus of Variations in Probability and Geometry
(ML 19.9) GP regression - introduction
Introduction to the application of Gaussian processes to regression. Bayesian linear regression as a special case of GP regression.
From playlist Machine Learning
Vincent Vargas - 3/4 Liouville conformal field theory and the DOZZ formula
Materials: http://marsweb.ihes.fr/Cours_Vargas.pdf Liouville conformal field theory (LCFT hereafter), introduced by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings", can be seen as a random version of the theory of Riemann surfaces. LCFT appears in Polyakov's work a
From playlist Vincent Vargas - Liouville conformal field theory and the DOZZ formula
Gaussian Integral 10 Fourier Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I show how the Gaussian integral appears in the Fourier transform: Namely if you t
From playlist Gaussian Integral
Gaussian Integral 8 Original Way
Welcome to the awesome 12-part series on the Gaussian integral. In this series of videos, I calculate the Gaussian integral in 12 different ways. Which method is the best? Watch and find out! In this video, I present the classical way using polar coordinates, the one that Laplace original
From playlist Gaussian Integral