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contraction & Inflation

CC-By 4.0 James B. Wilson, 2024

Contractions & Entropy

  • Data table \(T\)

  • User recipe of rows \(v\)

  • Contraction \(\langle u\mid T\rangle\) is newly generated data.

Contractions need recipes

  • Data table \(T\)

  • User recipe for columns \(v\)

  • Contraction \(\langle T\mid v\rangle\) is newly generated data.

Multiple Contractions allowed

Entropy of contractions

  • The order of contractions is immaterial to the result.
  • Entropic ("disorder") means operators that applied in any order are the same.
  • Commutative? Associative? No, these are different operators, not one operator with rewriting rules.

Entropy

\begin{pmatrix} \begin{bmatrix} u_{11} \\ \vdots \\ u_{m1} \end{bmatrix} \cdots \begin{bmatrix} u_{1n} \\ \vdots \\ u_{mn} \end{bmatrix} \end{pmatrix} =\begin{bmatrix} \begin{pmatrix} u_{11} & \cdots & u_{1n} \end{pmatrix}\\ \vdots \\ \begin{pmatrix} u_{m1} & \cdots & u_{mn} \end{pmatrix} \end{bmatrix}

Given products \((u_1,\ldots,u_n)\) and \(\begin{bmatrix} v_1\\ \vdots\\ v_m\end{bmatrix}\)

Entropy

=

Main Example

Inflation & Distribution

  • Recepies \(u,v,w\)

  • Build a product \(u\otimes v\otimes w\) whose partial evaluations are entropic.

Inflation produces tables

Product Notations

\begin{gather*} u*v,\quad uv, \quad \langle u|v\rangle,\quad [u,v]\\ \langle u_1,\ldots,u_{\ell}\rangle, \quad [u_1,\ldots,u_{\ell}] \end{gather*}

Functional notation \[u:\text{axes}\to \text{space}\] so partition axes to partition input.

\langle u_a+\acute{u}_a, u_{\bar{a}}\rangle = \langle u_1,\ldots, u_a+\acute{u}_a,\ldots, u_{\ell}\rangle

tensor spaces

  • Data table \(T\)

  • First recipe of rows \(u\)

  • Second recipe \(\acute{u}\)

  • Contract \[\langle u+\acute{u}\mid T\rangle=\langle u\mid T\rangle+\langle \acute{u}\mid T\rangle.\]

Multiple recipes

measurement distributes

Even in limits

\int_a^b (f+g) \text{d}x=\int_a^b f \text{d}x +\int_a^b g\text{d}x
\int_{I\sqcup J} f \text{d}x=\int_I f \text{d}x +\int_J f\text{d}x

measurement Is contextual

  • River's length is in miles/km
  • River's depth is in feet/meters.
  • Volume in gallons/liter 
  • \[\text{vol}(t\mid \text{length}, \text{depth}, \text{width})\]
  • USA \(t=39,500 \text{gal}/\text{mile}\times \text{ft}^2\) 
  • EU \(t=1 \text{l}/1000 \text{km}^2\times \text{m}\)

Convention is not the point.  In binary both conversions are bizzare choices.

Definition.

A tensor space is a distinguished term of a distributive product.

 

A tensor is an term/element of a tensor space.

Definition.

A cotensor space is the type of outputs of a distributive product.

 

A cotensor is an term/element of a cotensor space.

Interpretation Tensors Valence
Area/Volume/... Measure 2,3,...
Markov process Distribution 2
Cost functions costs 1
Distributed computing Thread pool any

Potential tensor spaces

Interpretation Tensors Valence
Logic/circuits gates (and/or...) 2,3,...
Foundations Distribution 2
hom, tensor product,... Categories 2,3...

Potential tensor spaces

The implication

Data Table

Multiplication Table

The point:

\begin{aligned} x^2 & = -2-3x\\ x^2+3x+2 & = 0\\ (x+1)(x+2) & = 0 \end{aligned}

Data Table

Multiplication Table

The point

Algebra has 1200 year head start on organizing data.

The requirement

(u+\acute{u})*(v+\acute{v})
u*(v+\acute{v})+\acute{u}*(v+\acute{v})
(u*v+u*\acute{v})+(\acute{u}*v+\acute{u}*\acute{v})
(a+b)+(c+d)=(a+b)+(c+d)
(u+\acute{u})*v + (u+\acute{u})*\acute{v}
(u*v+\acute{u}*v)+(u*\acute{v}+\acute{u}*\acute{v})

Medial Law: a special case of entropy

  • Current distributes proportional to resistance

  • \(I=R_1 I_1+R_2 I_2\)

  • New addition \[I_1\boxplus I_2=R_1 I_1 +R_2 I_2\] is medial.

Ohms law

Eckmann-Hilton Argument

A medial addition with a 0 is a both commutative and associative.

 

Commonoid = "Commutative Monoid"

To play nice with products add axiom:

\[\forall a \qquad \langle 0_a,u_{\bar{a}}\rangle=0\]

Grothendieck.

Every commoind has associated to it an abelian group to which it is maximally embedded.

Fact. If negatives exist then they already play nice \[\forall a \qquad \langle -u_a,u_{\bar{a}}\rangle=-\langle u_a,u_{\bar{a}}\rangle\]

Murdoch-Toyoda.

Every medial quasigroup is an affine twist of an abelian group. 

\[x\boxplus y=Mx+Ny+b\]

Quasi-group: \(a+x=b\) and \(x+a=b\) has unique solutions.

Tensor take products Cotensors make products