Arithmetic functions | Integer partitions | Integer sequences
In number theory, the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5. (Wikipedia).
Theory of numbers: Multiplicative functions
This lecture is part of an online undergraduate course on the theory of numbers. Multiplicative functions are functions such that f(mn)=f(m)f(n) whenever m and n are coprime. We discuss some examples, such as the number of divisors, the sum of the divisors, and Euler's totient function.
From playlist Theory of numbers
Number theory Full Course [A to Z]
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure #mathematics devoted primarily to the study of the integers and integer-valued functions. Number theorists study prime numbers as well as the properties of objects made out of integers (for example, ratio
From playlist Number Theory
Partitions of a Set | Set Theory
What is a partition of a set? Partitions are very useful in many different areas of mathematics, so it's an important concept to understand. We'll define partitions of sets and give examples in today's lesson! A partition of a set is basically a way of splitting a set completely into disj
From playlist Set Theory
Abstract Algebra | Partitions and Equivalence Relations
We prove that there is a one-to-one correspondence between partitions of a set and equivalence relations on a set. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Introduction to number theory lecture 6. Multiplicative functions.
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 examples of multiplicative functions and show how to calculate them. As an app
From playlist Introduction to number theory (Berkeley Math 115)
We have seen an example of partitioning in the previous video. These partitioned sets are called equivalence sets or equivalence classes. In this video we look at some notation.
From playlist Abstract algebra
Determine Generating Functions of Sequences from Known Generating Functions (Part 2)
This video explains how to determine generating functions of sequences from known generating functions. mathispower4u.com
From playlist Additional Topics: Generating Functions and Intro to Number Theory (Discrete Math)
Partitions, Dyson, and Ramanujan - George Andrews
George Andrews The Pennsylvania State University September 27, 2013 More videos on http://video.ias.edu
From playlist Dreams of Earth and Sky
Rogers-Ramanujan Identities | Part 3: More on partition generating functions.
We look more at partition generating functions, focusing on what each part of the generating function means in terms of the allowed partitions given some restriction. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Rogers-Ramanujan Identities
Ole Warnaar: Cylindric partitions and character identities
Abstract: As was shown in the 1980s by Kac, Peterson and Wakimoto, the characters of infinite dimensional Lie algebras provide a rich source of modular forms. Finding manifestly positive expressions for such characters remains, however, a difficult open problem. In this talk I will describ
From playlist Number Theory Down Under 9
PiTP 2015 - "Fermion Path Integrals and Topological Phases" - Edward Witten
https://pitp2015.ias.edu/
From playlist 2015 Prospects in Theoretical Physics Program
Karl Mahlburg: Automorphic forms and classical partition identities
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Number Theory
Entropy Equipartition along almost Geodesics in Negatively Curved Groups by Amos Nevo
PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis
From playlist Ergodic Theory and Dynamical Systems 2022
Algebraic combinatorics: applications to statistical mechanics and complexity theory - Greta Panova
Short proofs are hard to find (joint work w/ Toni Pitassi and Hao Wei) - Ian Mertz Computer Science/Discrete Mathematics Seminar II Topic: Short proofs are hard to find (joint work w/ Toni Pitassi and Hao Wei) Speaker: Ian Mertz Affiliation: University of Toronto Date: December 5, 2017 F
From playlist Mathematics
Tutorial for Shiraz Minwalla lectures by Suman Kundu
PROGRAM KAVLI ASIAN WINTER SCHOOL (KAWS) ON STRINGS, PARTICLES AND COSMOLOGY (ONLINE) ORGANIZERS Francesco Benini (SISSA, Italy), Bartek Czech (Tsinghua University, China), Dongmin Gang (Seoul National University, South Korea), Sungjay Lee (Korea Institute for Advanced Study, South Korea
From playlist Kavli Asian Winter School (KAWS) on Strings, Particles and Cosmology (ONLINE) - 2022
Off-shell Partition Functions in 3d Gravity - Lorenz Eberhardt
IAS Physics Group Meeting Topic: Off-shell Partition Functions in 3d Gravity Speaker: Lorenz Eberhardt Affiliation: Member, School of Natural Sciences, IAS Date: May 25, 2022 I will discuss partition functions in three-dimensional quantum gravity with negative cosmological constant in ca
From playlist Physics Group Meeting
PiTP 2015 - "Hall Viscosity, Central Charge and Entanglement" - Nicholas Read
https://pitp2015.ias.edu/
From playlist 2015 Prospects in Theoretical Physics Program
Seok Kim - 6 dimensional superconformal field theories (2)
PROGRAM: THE 8TH ASIAN WINTER SCHOOL ON STRINGS, PARTICLES AND COSMOLOGY DATES: Thursday 09 Jan, 2014 - Saturday 18 Jan, 2014 VENUE: Blue Lily Hotel, Puri PROGRAM LINK: http://www.icts.res.in/program/asian8 The 8th Asian Winter School on Strings, Particles and Cosmology is part of a seri
From playlist The 8th Asian Winter School on Strings, Particles and Cosmology
In this video we cover some rational function fundamentals, including asymptotes and interecepts.
From playlist Polynomial Functions