It’s better with ‘theory’

It seems to me that adding the word theory tends to increase my interest in a subject.  Consider puddles, they’re fun…but ‘Puddle Theory’ what’s that?  Sounds intriguing.

Definition: An n-puddle is a n-dimensional manifold with boundary such that the boundary is the union of a sky and ground.  The ground is a n-1 manifold with boundary equal to the n-2 boundary of the sky.  The sky boundary resides within a hyperplane of dimension n-1.

We can use standard geometric structures on \mathbb{R}^n because in general an n-puddle will not be smooth.

Definition: Define a splash to be a family of functions F_{\alpha}:P\to P indexed by [0,1].  F_{\alpha} need not be continuous, in fact will seldom be so.

It’s possible for P to become disconnected during the splash.  Splashes can be divided into volume-preserving and non-volume-preserving splashes.  These are called dry and wet respectably.

Definition: A splash is dry if \lim_{\alpha\to 1} F_{\alpha}\to id_P.

It’s possible to compose a series of splashes, F^1_{\alpha_1}, F^2_{\alpha_2},...F^k_{\alpha_k},... k\in\mathbb{N}   such that F_1^k = F_0^{k+1}.  It was discovered by A. Child that for any such sequence there exists an integer N such that for all n>N F^n_{\alpha} is the empty splash. This is known as Child’s Theorem.  We leave this as an exercise.

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About because0fbeauty

Fascinated by the way mathematics and physics interact, captivated by visual and tactile mathematics and hoping to become a better expositor of these things is why I blog...occasionally...when I remember.
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2 Responses to It’s better with ‘theory’

  1. sheabrowne says:

    I love this! I’d like to see A. Child’s theory of Quantum Gravity…might be a Nobel Prize in there somewhere! BTW, I’ve pick up a book on basic Algebra (rings, groups etc.) for undergraduates … I’ve missed out on a lot in my Physics education, but it’s never too late to catch up…

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