Integrable

When is the whole the sum of its parts?

2025 James B. Wilson

https://slides.com/jameswilson-3/integrable/

When it the whole the sum of its parts?

  • Goal: count all the words in all the files on the University's website.

  • Breaks up as codata

    • Individual pages, with links to others

    • Words, followed by other words.

  • If we can count words as 1, then to count all the words is to add up "integrate" all the parts.

Integral

\(dx\)

 

\(f\)

 

Limit of sums

Domain

Codata is Integrable when...

  • There is a numerical measurement \(m\) that applies to the codata.

  • For each co-operator \[\begin{aligned} A & \rightarrow \bigsqcup_{\omega} A^{|\omega|}\\ a&\mapsto \text{case }h \quad (a_i)_{I\in I}, a_{|\omega|}]\end{aligned}\] we find \[m(a)=m(h)+\int_I a_i dm\]

  •  

Choose an integrable domain:

  1. Pick some form of codata

  2. Identify its co-operators

  3. Devise something to measure for about the parts of your codata.

  4. Make sure you measure something integrable: the whole values should be the sum of its parts.

Measuring codata

\(\mathbb{A}\to \mathbb{A}\) "meters to meter+meters", "seconds to second+seconds"

1

2

3

Measuring codata

\(\mathbb{A}\to \mathbb{A}\) "meters to meter+meters", "seconds to second+seconds"

Unfold \(\mathbb{A}\to \mathbb{A}\)

\[as->(a_x,a_y)+as\]

If the data for keeps unfolding

\[(a_x,a_y)+(b_x,b_y)+(c_x,c_y)+...\]

which we might plot on a grid.

(1,0)

(0,1)

(1,1)

(2,2)

(2,0)

(2,1)

(0,2)

(1,2)

If the data unwraps as indexed in an area that the total data measure is growing proportional to what?  

 

A function of the boundary.  

Estimate growth type

  1. Identify one co-operator

  2. Define a "center and radius" for how your data can expand from this co-operator.
  3. Estimate the total type of growth (length, area, volume, etc.)

 

 

Rates of Change

Measure: 2R, Perimeter: 2

Area: \((2R)^2\), Perimeter: \(4(2R)\)

Volume: \((2R)^3\), Surface Area: \(6(2R)^2\)

Rates of Change

Measure: 2R, Perimeter: 2

Area: \(\pi R^2\), Perimeter: \(2\pi R\)

Volume: \(\frac{4}{3}\pi R^3\), Surface Area: \(4\pi R^2\)

Predict Rate of change:

Given your integrable measurement, predict the rate of change of increasing the sample.

 

 

Your Codata's growth?

Measure: f(R), Perimeter: f(R)

\[\frac{d}{dR}\int_0^R perimeter(r) dr=measure(R)\]