The One Dimensional Random Walk Hypergroup | Diffusion Symmetry
Insights into Mathematics
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We introduce affine transformations, in the form of the ax+b group, to describe the classical (group theoretic) symmetries of the simplest lattice - the one dimensional integral lattice. We meet the fundamental idea of representation theory: assigning concrete matrices to abstract algebraic objects in order to facilitate computation and understanding.
In particular we assign 2 x 2 matrices to the affine transformations of the line, and then restrict these to study the space group of the one dimensional integer lattice. The notion of semi-direct product is involved here, along with the idea of one subgroup of a group normalizing another.
Then we introduce the one dimensional random walk associated to this lattice when viewed as a graph, and the associated hypergroup K. This is an unbounded analog of some polygon hypergroups we have considered already, such as the Golden Hypergroup.
We finish with some algebraic computations to tackle the interesting question of where we can expect to be after n steps of such a nearest neighbour random walk.
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...