Particle collisions, hypergroups and quadratic residues
Insights into Mathematics
100K subscribers
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.
The Playlist for this series is at https://www.youtube.com/playlist?list...
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 https://www.youtube.com/channel/UCriF..., or become a Patron at https://www.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.
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
Algebraic Calculus One: https://www.youtube.com/playlist?list...
Algebraic Calculus Two: https://www.youtube.com/playlist?list...