I understand the following sentence:
The rule was defined as ~~~~ by John.
It can be shortened by omitting "as ~~~~" or "by John", like
The rule was defined as ~~~~.
The rule was defined by John.
Now, my question is what is difference between "defined as" and "defined by" in the context of Mathematical papers or textbooks.
I saw many times they are used interchangeably, e.g.,
The penalty function is defined by
~~~~~~~~~~~~~~~ Equation ~~~~~~~~~~~~~~~~~~~, (1)
where ~ is ~~~~~~~~~~~~, ~ is ~~~~~~~~~~~, and ~ is ~~~~~~~~~~~~~~~. (variable description)
and
The penalty function is defined as
~~~~~~~~~~~~~~~ Equation ~~~~~~~~~~~~~~~~~~~, (1)
where ~ is ~~~~~~~~~~~~, ~ is ~~~~~~~~~~~, and ~ is ~~~~~~~~~~~~~~~. (variable description)
I guess that both are correct but the meaning is slightly different. What is exactly different???
Also, some people write
where ~ is ~~~~~, ~ is ~~~~~, and ~ is ~~~~~~~.
but the others write
where ~ is ~~~~~, ~ ~~~~~~, and ~ ~~~~~~~. (Only write the be-verb in the first one.)