12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Wednesday May 01, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Cactus Language (view source)
Revision as of 01:04, 28 May 2009
, 01:04, 28 May 2009→Note 2: markup
We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely:
The first couple of operators that we need to consider are logical analogues
of those that occur in the classical "finite difference calculus", namely:
{| align="center" cellpadding="6" width="90%"
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
|-
| The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math>
|}
These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse, passing from the space <math>U = X \times Y</math> to its ''differential extension'', <math>\operatorname{E}U,</math> that has the following description:
{| align="center" cellpadding="6" width="90%"
| <math>\operatorname{E}U ~=~ U \times \operatorname{d}U ~=~ X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math>
|-
| with
|-
| <math>\operatorname{d}X = \{ \texttt{(} \operatorname{d}x \texttt{)}, \operatorname{d}x \}</math> and <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math>
|}
<pre>
The interpretations of these new symbols can be diverse, but the easiest
The interpretations of these new symbols can be diverse, but the easiest
for now is just to say that dx means "change x" and dy means "change y".
for now is just to say that dx means "change x" and dy means "change y".
