Particle collisions, hypergroups and quadratic residues | Diffusion Symmetry 3 | N J Wildberger

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    • Playlist   ************** Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Insights into Mathematics YT channel:    / njwildberger   Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos) Wild Egg Maths Playlists:    • Box Arithmetic Book: A finite multiset fou...   (NEW!!)    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)