Meta-singular numbers

Not quite clear exposition and definition. Need to be more exact.

In \href book I defined (generalized) limit of any (e.g. non-continuous) function.

But applying it, to say, a differential equationMathworldPlanetmath (replacing the limit in the definition of derivative with my generalized limit) does not work because the components of the equation may be of different types (for example generalized limit of a function is not a real number).

To overcome this shortcoming I propose what I call meta-singular numbers.

[TODO: introduce the term singularity level above for a given number system.]

Let a is a generalized limit. I will denote r(a) such number (or generalized limit of a lower rank) that a=xlim({r(a)}×𝖥𝖢𝖣x) where x is a filter (that is a is a limit of a constant function), if such r(a) exists. I will call reduced limit repeated applying r(r(a)) to a generalized limit a.

This definition is for now all I know about meta-singular numbers. It’s yet needed to formulate and prove some theorems.

See also \href rough draft on my site (in PDF format).

Title Meta-singular numbers
Canonical name MetasingularNumbers
Date of creation 2013-06-10 20:51:40
Last modified on 2013-06-10 20:51:40
Owner porton (9363)
Last modified by porton (9363)
Numerical id 10
Author porton (9363)
Entry type Definition