Insights into Mathematics | Particle collisions, hypergroups and quadratic residues | Diffusion Symmetry 3 | N J Wildberger @njwildberger | Uploaded 2 years ago | Updated 23 hours ago
We introduce a simple particle collision model for the algebraic structure called a hypergroup, which is a normalization of a fusion rule algebra, and closely connected also to Bose Messner algebras and association schemes from algebraic combinatorics.
In this model, the results of particle collisions are probabilistic, there is a distinguished particle called a photon which is absorbed in any collision, and each particle has a unique antiparticle (which might in fact coincide with the given particle), characterized by the condition on yielding a photon when collided with the original particle.
We will be introducing the "Golden hypergroup", along with hypergroups coming from complete graphs, and also from complete bi-partite graphs.
A further interesting family of such algebraic objects goes back in some sense to C. F. Gauss and his study of quadratic residues and non-residues. For that we will need to review some basic facts about prime fields, including the idea of primitive roots.
Video Contents (thanks to Lucas Lofaro)
0:00 Intro
0:08 Overview
1:53 Describing Particle Collisions
5:39 The Golden Hypergroup
9:05 Complete Graphs
11:30 Complete Bipartite Graphs
20:30 Quadratic Residues and Non-Residues
22:52 Structure of R and N via Primitive Roots
26:03 Additive Arithmetic of R and N
29:59 Moving to a Multiplicative Arithmetic
37:00 A Different Example of Multiplicative Arithmetic
The Playlist for this series is at youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
************************
This is closely related to the Exceptional Structures in Mathematics and Physics lectures which are part of the Explore Research Level Mathematics video lectures. These are designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations --- naturally without any pretense of being "able to do an infinite number of things"!
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
We introduce a simple particle collision model for the algebraic structure called a hypergroup, which is a normalization of a fusion rule algebra, and closely connected also to Bose Messner algebras and association schemes from algebraic combinatorics.
In this model, the results of particle collisions are probabilistic, there is a distinguished particle called a photon which is absorbed in any collision, and each particle has a unique antiparticle (which might in fact coincide with the given particle), characterized by the condition on yielding a photon when collided with the original particle.
We will be introducing the "Golden hypergroup", along with hypergroups coming from complete graphs, and also from complete bi-partite graphs.
A further interesting family of such algebraic objects goes back in some sense to C. F. Gauss and his study of quadratic residues and non-residues. For that we will need to review some basic facts about prime fields, including the idea of primitive roots.
Video Contents (thanks to Lucas Lofaro)
0:00 Intro
0:08 Overview
1:53 Describing Particle Collisions
5:39 The Golden Hypergroup
9:05 Complete Graphs
11:30 Complete Bipartite Graphs
20:30 Quadratic Residues and Non-Residues
22:52 Structure of R and N via Primitive Roots
26:03 Additive Arithmetic of R and N
29:59 Moving to a Multiplicative Arithmetic
37:00 A Different Example of Multiplicative Arithmetic
The Playlist for this series is at youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
************************
This is closely related to the Exceptional Structures in Mathematics and Physics lectures which are part of the Explore Research Level Mathematics video lectures. These are designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations --- naturally without any pretense of being "able to do an infinite number of things"!
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
