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
Proof of Lemma and Lagrange's Theorem
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Proof of Lemma and Lagrange's Theorem. This video starts by proving that any two right cosets have the same cardinality. Then we prove Lagrange's Theorem which says that if H is a subgroup of a finite group G then the order of H div
From playlist Abstract Algebra
In this video, I prove the famous Riemann-Lebesgue lemma, which states that the Fourier transform of an integrable function must go to 0 as |z| goes to infinity. This is one of the results where the proof is more important than the theorem, because it's a very classical Lebesgue integral
From playlist Real Analysis
Measure Theory 2.1 : Lebesgue Outer Measure
In this video, I introduce the Lebesgue outer measure, and prove that it is, in fact, an outer measure. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
The Frobenius Problem - Method for Finding the Frobenius Number of Two Numbers
Goes over how to find the Frobenius Number of two Numbers.
From playlist ℕumber Theory
In this video, I show how to calculate the integral of x^3 from 0 to 1 but using the Lebesgue integral instead of the Riemann integral. My hope is to show you that they indeed produce the same answer, and that in fact Riemann integrable functions are also Lebesgue integrable. Enjoy!
From playlist Real Analysis
Measure Theory 3.1 : Lebesgue Integral
In this video, I define the Lebesgue Integral, and give an intuition for such a definition. I also introduce indicator functions, simple functions, and measurable functions.
From playlist Measure Theory
In this video, I present an overview (without proofs) of the Lebesgue integral, which is a more general way of integrating a function. If you'd like to see proods of the statements, I recommend you look at fematika's channel, where he gives a more detailed look of the Lebesgue integral. In
From playlist Real Analysis
Measure Theory 2.4 : Sets of Measure Zero
In this video, I introduce the Cantor Set, and prove that it and countable sets (including the rationals) have measure zero. Email : fematikaqna@gmail.com Subreddit : reddit.com/r/fematika Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
2023 Number Challenge: Find sum of four squares that is equal to 2023
#mathonshorts #shorts check out wiki page: https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares.
From playlist Math Problems with Number 2023
Measure Theory 2.2 : Lebesgue Measure of the Intervals
In this video, I prove that the Lebesgue measure of [a, b] is equal to the Lebesgue measure of (a, b) is equal to b - a. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
Karma Dajani - An introduction to Ergodic Theory of Numbers (Part 2)
In this course we give an introduction to the ergodic theory behind common number expansions, like expansions to integer and non-integer bases, Luroth series and continued fraction expansion. Starting with basic ideas in ergodic theory such as ergodicity, the ergodic theorem and natural ex
From playlist École d’été 2013 - Théorie des nombres et dynamique
Dynamical systems, fractals and diophantine approximations – Carlos Gustavo Moreira – ICM2018
Plenary Lecture 6 Dynamical systems, fractal geometry and diophantine approximations Carlos Gustavo Moreira Abstract: We describe in this survey several results relating Fractal Geometry, Dynamical Systems and Diophantine Approximations, including a description of recent results related
From playlist Plenary Lectures
Vaughn Climenhaga: Closed geodesics and the measure of maximal entropy on surfaces without...
For negatively curved Riemannian manifolds, Margulis gave an asymptotic formula for the number of closed geodesics with length below a given threshold. I will describe joint work with Gerhard Knieper and Khadim War in which we obtain the corresponding result for surfaces without conjugate
From playlist Jean-Morlet Chair - Pollicott/Vaienti
Analysis III - Integration: Oxford Mathematics 1st Year Student Lecture:
The third in our popular series of filmed student lectures takes us to Integration. This is the opening lecture in the 1st Year course. Ben Green both links the course to the mathematics our students have already learnt at school and develops that knowledge, taking the students to the next
From playlist Oxford Mathematics 1st Year Student Lectures
Measure Theory 2.3 : Open and Closed Inervals are Lebesgue Measurable
In this video, I prove that the open and closed intervals (a, b) and [a, b] (as well as [a, b) and (a, b]) are in fact Lebesgue measurable, and thus validating the previous video in this series. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
Measure Theory 3.3 : Riemann Integral Equals Lebesgue Integral
In this video, I describe a new way of defining the Riemann Integral, and use that to prove that the Riemann and Lebesgue Integrals are the same for Riemann Integrable functions. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory
Measure Theory 3.4: Monotone Convergence Theorem
In this video, I will be proving the Monotone Convergence Theorem for Lebesgue Integrals. Email : fematikaqna@gmail.com Subreddit : https://www.reddit.com/r/fematika Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Measure Theory