SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

2021 James B. Wilson, Colorado State University

https://youtu.be/V4N7q2C-SxY

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Cause: Ordering what doesn't want to be ordered.

 

Γi1⋯iaj1⋯jb\Gamma_{i_1\cdots i_{a}}^{j_1\cdots j_b}

 

Kd1×⋯×dℓK^{d_1\times \cdots \times d_{\ell}}

 

xi1⊗…⊗xiℓx_{i_1}\otimes \ldots \otimes x_{i_{\ell}}

 

Kd1⊗⋯⊗KdℓK^{d_1}\otimes \cdots \otimes K^{d_{\ell}}

 

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

In Code:  we want a command like

def MakeSpace( K, d_1, ...., d_ell )  
	...

Above "ell" is a variable also,

but the program doesn't know this,

this cannot be programmed without very special features (dependent types, variadic inputs, etc.).

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Solution 1: assign a new variable.

 

Kd1×⋯×dℓK^{d_1\times \cdots \times d_{\ell}}

becomes

Kd,whered=d1×⋯×dℓK^{d}, \quad where\quad d=d_1\times \cdots \times d_{\ell}

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Solution 2: use a set

Kd1⊗⋯⊗KdℓK^{d_1}\otimes \cdots \otimes K^{d_{\ell}}

becomes

⨂d∈DKd,D={d1,…,dℓ}\bigotimes_{d\in D} K^d,\qquad D=\{d_1,\ldots,d_{\ell}\}

 

One step further let the set index itself:

Let V\mathcal{V} be a set of spaces and consider ⨂V∈VV=⊗V\bigotimes_{V\in \mathcal{V}} V=\otimes\mathcal{V}

 

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

// Original
def MakeSpace( K, d_1, ...., d_ell )  
	... 
   
// Create index set
ds = [1,2, 19,...]  // "..." means user filled in constants

def MakeSpace( K, ds)
	...

Solution 2 in code

x1+⋯+xℓx_1+\cdots+x_{\ell}

ΣX=∑x∈Xx,X={x1,…,xℓ}\Sigma X = \sum_{x\in X} x, \qquad X=\{x_1,\ldots, x_{\ell}\}

xs = [ 3, -9, 541 ]
sum(xs)

// Alternatives
xs.fold(+)     // head+tail.fold(+), recursive
+&xs           // "&"="forall" + = sum

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Solution 3−ε3-\varepsilon:  if sub-sub-scripted, e.g.

Γi1…ia\Gamma_{i_1\ldots i_a}

replace i∗=(i1,…,ia)i_*=(i_1,\ldots,i_a)

Γi∗\Gamma_{i_*}

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Asside: what is (i1,…,ia)(i_1,\ldots, i_a)?

(i1,…,ia)∈[d1]×⋯×[dℓ](i_1,\ldots,i_a)\in [d_1]\times \cdots \times [d_{\ell}]

where [n]={1,…,n}[n]=\{1,\ldots,n\}

 

Generally

(x1,…,xa)∈X1×⋯×Xa(x_1,\ldots,x_a)\in X_1\times \cdots \times X_{a}

That is...

x∈ΠX=∏X∈XXx\in \Pi \mathcal{X}=\prod_{X\in \mathcal{X}} X

SubscriptsonsubscriptsSubscripts_{on_{subscripts}}

Solution 3: use functions

Γi1…ia\Gamma_{i_1\ldots i_a}

Step I, replace iki_k with tuple ι∈ΠD\iota\in \Pi D

 

Step II, Γ\Gamma is now a function on tuples

Γ:ΠD→K\Gamma:\Pi D\to K

and we access it like any function, 

Γι∈K\Gamma_{\iota}\in K

Γi1…iaj1…jb\Gamma_{i_1\ldots i_a}^{j_1\ldots j_b}

Γιτ\Gamma_{\iota}^{\tau} where ι∈ΠLτ∈ΠU\iota\in \Pi L \quad\tau\in \Pi U

is = x -> ...  // A lambda, a little function 
js = y -> ...  // that given index, calcs the coordinate.
gamma(is, js)

Examine what you mean

 

A subscript in the end is just turning symbol into a function, aia_i means a function varying with ii, so a:I→Ta:I\to T.

 

All functions are defined on sets and so order of the domain never matters.

 

Let that reality sink in and soon your subscripts will become tidy and programable.

But above all be flexible with others, and yourself!