Graph invariants | Generating functions | Topological graph theory | Algebraic topology
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if then , if then , if then , if then , etc. Note that only the ranks of infinite groups are considered, so for example if , where is the finite cyclic group of order 2, then . These finite components of the homology groups are their torsion subgroups, and they are denoted by torsion coefficients. The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. The modern formulation is due to Emmy Noether. Betti numbers are used today in fields such as simplicial homology, computer science, digital images, etc. (Wikipedia).
#MegaFavNumbers - 7,588,043,387,109,376 by Egi
87,109,376^2=7,588,043,387,109,376. The last 8 digits is the square rootđ, it's called an automorphic number which n^2 ends with n
From playlist MegaFavNumbers
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From playlist Number Theory
#MegaFavNumbers: 10,904,493,600 & Fibonacci Numbers
This is my #MegaFavNumber. Link to all the #MegaFavNumbers Videos: https://www.youtube.com/watch?v=R2eQVqdUQLI&list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo Channel Links: Website: https://sites.google.com/view/pentamath Channel: https://www.youtube.com/channel/UCervsuIC9pv4eQq98hAgOZA Subscri
From playlist MegaFavNumbers
Conversion Arcs and 2,916,485,648,612,232,232,816 (MegaFavNumbers)
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From playlist MegaFavNumbers
Algebra - Ch. 24: Complex Numbers (17 of 28) Simplify
Visit http://ilectureonline.com for more math and science lectures! We will simplify 3 imaginary numbers. To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 . Next video in this series can be seen at: https://youtu.be/HKyOUtYAja8
From playlist ALGEBRA CH 24 COMPLEX NUMBERS
Algebra - Ch. 6: Factoring (6 of 55) How to Determine a Prime Number? NOTE: 49 IS NOT A PRIME NUMBER
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain how to determine if a number is a prime number. To donate: http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 Next video in this series can be seen at: https://youtu
From playlist THE "HOW TO" PLAYLIST
The Magical Fraction 1/999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999
The number 1/999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 has the Fibonacci numbers in order for every group of 24 decimals. This video explains why the pattern emerges. (sources, proofs, and links below) Via Futility Closet: http://www.futilitycloset.com/2015/06/28/mad
From playlist Everyday Math
Complex Numbers - Basics | Don't Memorise
Now that we know what imaginary numbers are, we can move on to understanding Complex Numbers. â To access all videos related to Complex Numbers, enroll in our full course now: https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=bmsapLZM
From playlist Complex Numbers
The golden ratio | Lecture 3 | Fibonacci Numbers and the Golden Ratio
The classical definition of the golden ratio. Two positive numbers are said to be in the golden ratio if the ratio between the larger number and the smaller number is the same as the ratio between their sum and the larger number. Phi=(1+sqrt(5))/2 approx 1.618. Join me on Coursera: http
From playlist Fibonacci Numbers and the Golden Ratio
Letâs talk about random Cech and Vietoris-Rips complexes [Andrew M. Thomas]
In this tutorial, I introduce you to the behavior of random Cech and Vietoris-Rips complexes and show how a probabilistic approach in TDA can mirror real world phenomena. Even with a relatively simple approach like looking at Betti numbers, probability can give important insights into the
From playlist Tutorial-a-thon 2021 Spring
Joel Friedman - Sheaves on Graphs, L^2 Betti Numbers, and Applications.
Joel Friedman (University of British Columbia, Canada) Sheaf theory and (co)homology, in the generality developed by Grothendieck et al., seems to hold great promise for applications in discrete mathematics. We shall describe sheaves on graphs and their applications to (1) solving the
From playlist T1-2014 : Random walks and asymptopic geometry of groups.
Lewis Bowen - L2 invariants and Benjamini-Schramm convergence
October 30, 2015 - Princeton University Does there exist a sequence of free subgroups Fk of the isometry group of hyperbolic n-space such that the Cheeger constant of the quotient space Hn/Fk tends to zero as k tends to infinity? I will explain how to answer this (and related questions) w
From playlist Minerva Mini Course - Lewis Bowen
Growth of topological invariants of locally symmetric spaces - Mikolaj Fraczyk
Short talks by postdoctoral members Topic:Growth of topological invariants of locally symmetric spaces. Speaker: Mikolaj Fraczyk Affiliation: Member, School of Mathematics Date: September 24 For more video please visit http://video.ias.edu
From playlist Mathematics
Discrete Morse Theory -- math major seminar.
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From playlist MathMajor Seminar
An Amazing Connection Between the Riemann Hypothesis and Topology
https://gregoriousmaths.com/2021/08/19/a-couple-of-other-connections-between-number-theory-and-topology/ 0:00 Introduction and plan 2:32 The Riemann hypothesis 7:22 Introducing the complex we will study 19:41 Studying the asymptotic behaviour of \beta_k(\Delta_n) 22:54 Some number theoret
From playlist Summer of Math Exposition Youtube Videos
Courtney Gibbons, Mysterious Mathematical Object Syzygy, PCMI Ignite!
What makes mathematicians and mathematics educators passionate? What IGNITES us? Join Courtney Gibbons from Hamilton College for "Mysterious Mathematical Object Syzygy". Ignite presentations at the 28th annual PCMI Summer Session taking place July 1â21, 2018, at the Prospector Conference
From playlist Math
Colloquium MathAlp 2015 - Jean-Yves Welschinger
PolynÎmes aléatoires et topologie "Le lieu des zéros d'un polynÎme à coefficients réels de n variables est (en général) une hypersurface de l'espace affine réel de dimension n dont la topologie dépend du choix du polynÎme. à quelle topologie s'attendre lorsque le polynÎme est choisi au ha
From playlist Colloquiums MathAlp
Benjamini-Schramm Limits of Finite Volume Manifolds (Lecture-4) by Ian Biringer
PROGRAM: PROBABILISTIC METHODS IN NEGATIVE CURVATURE (ONLINE) ORGANIZERS: Riddhipratim Basu (ICTS - TIFR, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Mahan M J (TIFR, Mumbai) DATE & TIME: 01 March 2021 to 12 March 2021 VENUE: Online Due to the ongoing COVID pandemic, the meeting will
From playlist Probabilistic Methods in Negative Curvature (Online)
We discuss what imaginary numbers are and how they are part of the larger set of complex numbers in this free math video tutorial by Mario's Math Tutoring. This is a nice introduction to working with i. We also go through some examples. 0:26 A Hierarchy of Different Types of Numbers 1:03
From playlist Imaginary & Complex Numbers