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
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
Ex 2: Determine Factors of a Number
This is the second of three videos that provides examples of how to determine the factors of a number using a numbers prime factors. Search Video Library at http://www.mathispower4u.wordpress.com
From playlist Factors and Prime Factorization
Theory of numbers: Gauss's lemma
This lecture is part of an online undergraduate course on the theory of numbers. We describe Gauss's lemma which gives a useful criterion for whether a number n is a quadratic residue of a prime p. We work it out explicitly for n = -1, 2 and 3, and as an application prove some cases of Di
From playlist Theory of numbers
(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
[ANT09b] The Diophantus chord method
Cremona's tables: https://johncremona.github.io/ecdata/ Table for example 2: https://www.lmfdb.org/EllipticCurve/Q/576/c/3 Tables for example 3: https://www.lmfdb.org/EllipticCurve/Q/27/a/3 and https://www.lmfdb.org/EllipticCurve/Q/432/e/4 Table for the exercise: https://www.lmfdb.org/Elli
From playlist [ANT] An unorthodox introduction to algebraic number theory
(PP 6.6) Geometric intuition for the multivariate Gaussian (part 1)
How to visualize the effect of the eigenvalues (scaling), eigenvectors (rotation), and mean vector (shift) on the density of a multivariate Gaussian.
From playlist Probability Theory
(PP 6.2) Multivariate Gaussian - examples and independence
Degenerate multivariate Gaussians. Some sketches of examples and non-examples of Gaussians. The components of a Gaussian are independent if and only if they are uncorrelated.
From playlist Probability Theory
(PP 6.7) Geometric intuition for the multivariate Gaussian (part 2)
How to visualize the effect of the eigenvalues (scaling), eigenvectors (rotation), and mean vector (shift) on the density of a multivariate Gaussian.
From playlist Probability Theory
Mathematics of Post-Quantum Cryptograhy - Kristin Lauter
Woman and Mathematics - 2018 More videos on http"//video.ias.edu
From playlist My Collaborators
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
Pär Kurlberg: Class number statistics for imaginary quadratic fields Pär Kurlberg
Abstract: The number F(h) of imaginary quadratic fields with class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h x is less or equal to y 100 was completed by M. Watkins, a
From playlist Number Theory
12. Perturbative Renormalization Group Part 4
MIT 8.334 Statistical Mechanics II: Statistical Physics of Fields, Spring 2014 View the complete course: http://ocw.mit.edu/8-334S14 Instructor: Mehran Kardar In this lecture, Prof. Kardar continues his discussion on the Perturbative Renormalization Group, including the Irrelevance of Oth
From playlist MIT 8.334 Statistical Mechanics II, Spring 2014
Introduction to number theory lecture 2: Survey.
This lecture is part of my Berkeley math 115 course "Introduction to number theory" We continue the survey of some problems in number theory, and discuss congruences, quadratic reciprocity, additive number theory, recreational number theory, and partitions. For the other lectures in the
From playlist Introduction to number theory (Berkeley Math 115)
Lec 7 | MIT 6.451 Principles of Digital Communication II
Introduction to Finite Fields View the complete course: http://ocw.mit.edu/6-451S05 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.451 Principles of Digital Communication II
Andrea Sportiello: The challenge of linear-time Boltzmann sampling
Let Xn be an ensemble of combinatorial structures of size N, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with
From playlist Services numériques pour les mathématiques
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
This video explains how to determine the GCF of integers and expressions. http://mathispower4u.wordpress.com/
From playlist Integers