Tensors as area & volume

Volume basics

\[Vol(\ell, w,h)= \ell\times w\times h\]

Volume reality

Area...easy

Area...getting worse

New uses? Make up new measurements

New uses? Sales Volume?

Slightly more honest

Histogram Visualization

New uses? interactions probabilities and quantum information

Tensors algebra

Facts?

Facts?

Consequences?

Consequences?

Real Consequences?

Building blocks

• Some parts expand up:  \[a*b\qquad \partial_x X\partial_y Y\]
• Some parts accumulate: \[a*d-b*c\qquad \partial_x X\partial_y Y-\partial_x Y\partial_y X\]
• One version of decomposition focusses on this separation: "sums of products"

Products can grow bigger (inflation)

or get smaller (contraction)

Distributive dicta

\[a*e*\cdots *u\] \[a\otimes e\otimes \cdots \otimes u\] \[\langle a,e,\ldots, u\rangle\]

\[\langle v_1,\ldots, v_{\ell}\rangle= \langle v_a,v_{\bar{a}}\rangle\qquad \{1,\ldots,\ell\}=\{a\}\sqcup\bar{a}\]

Choose a heterogenous product notation

Choose a sums

\(\displaystyle \int_I v_a(i) d\mu\)     short for code    "sum(vs[a],method54)"

For \(\{v_a(i)\mid i\in I\}\)

\[\int_I \langle v_a(i),v_{\bar{a}}\rangle\,d\mu = \left\langle \int_I v_a(i)\,d\mu,v_{\bar{a}}\right\rangle\]

Entropic Edict / Fubini Formula

\[=\int_J\int_I \langle v_a(i), v_b(j),v_{\overline{ab}}\rangle d\nu d\mu\]

\[\int_I \int_J \langle v_a(i), v_b(j),v_{\overline{ab}}\rangle d\mu d\nu = \left\langle \int_I v_a(i) \,d\mu, \int_J v_b(j)\, d\nu,v_{\overline{ab}}\right\rangle\]

\[\int_I \int_J f\, d\mu d\nu=\int_J\int_I f \, d\nu d\mu\]

Theory

Murdoch-Toyoda Bruck

All entropic sums that can solve equations \(a+x=b\) are affine.

Eckmann-Hilton

Entropic sums with 0 are associative & commutative.

Grothendieck

You can add negatives.

Mal'cev, Miyasnikov

You can enrich distributive products with universal scalars.

Davey-Davies

Tensor algebra is ideal determined precisely for affine.

First-Maglione-Wilson

Full classification of algebraic enrichment: 

• If 3 or more positions must be Lie
• 2 can be associative or Jordan.

Back to decompositions

What are the markets?

Data table --> Multiplication table

Use algebra to factor

Forcing Good Algebra in Real Life

In real life the "multiplication" you get from a tensor is bonkers!

\[*:\mathbb{R}^2\times \mathbb{R}^3\to \mathbb{R}^3\]

Get to 100's 1000's of dimensions and you have no chance to know this algebra.

Study a function by its changes, i.e. derivatives.

Study multiplication by derivatives:

\[\partial (f·g ) = (\partial f )·g + f·(\partial g ).\]

In our context discretized. I.e. ∂ is a matrix D.

\[D(f ∗g ) = D(f ) ∗g + f ∗D(g )\]

For general tensors \[\langle t| : U_{1}\times \cdots \times U_{\ell}\to U_0\] there are many generalizations.

 E.g.

Or \[ D_0\langle t |u_1,u_2, u_3\rangle = \langle t| D_1 u_1, u_2,u_3\rangle + \langle t| u_1, D_2 u_2, u_3\rangle + \langle t|u_1, u_2, D_3 u_3\rangle.\]

For general tensors \[\langle t| : U_{1}\times \cdots \times U_{\ell}\to U_0\]

Choose a "chisel" (dleto) an augmented matrix C.

Write \[\langle \Gamma | C(D)|u\rangle =0\] to mean:

means chisel

And it's solutions are a Lie algebra

The main fact

Summary

Bonus

First-Maglione-W.

In other words, the Whitney "Associative" tensor product is not the right one.

Use the Lie derivation-tensor "densor"