Super Numbers and Super calculus

[kurt.physics]

Hello

Does anyone know about super numbers and super calculus, could anyone recommend some good books on the subject.

I herd of super numbers and S. calculus when i was reading michio kakus book Hyperspace, its apparently used for super symmetry theory.

Thanks

[Gib Z]

This topic sure sounds super. Unfortunately, I've never heard of such a thing, and neither has amazon.com, or even Google. In fact, the results for it are so bad, the top rated result is this page. Most other links are things like "Super hard calculus" or "calculus made super easy".

The only thing i found was this, from some site i've never heard of;
http://www.chapters.indigo.ca/books/Learning-Use-Super-Calculus-3-Gary-B-Shelly-T-Cashman/9780538910545-item.html?pticket=yvmsdm45splsfwm1xntfot55PQSC5V9s NGRJTOU6n33SctbsJEA%3d

Learning to use Super Calculus 3 by Shelly and Cashman. Not sure if it's good, or if they're just badly trying to
describe Calculus.

EDIT: Oh, and nothing on Super numbers either, closest thing is a Super-Poulet Number, which I remember coming across before, its a number theoretic property, so probably not what your looking for.

[CRGreathouse]

Well, there are the super-d numbers, a silly property in recreational number theory:
http://mathworld.wolfram.com/Super-dNumber.html

Another odd thing on MathWorld are the super unitary amicable pairs and perfect numbers:
http://mathworld.wolfram.com/SuperUnitaryAmicablePair.html
http://mathworld.wolfram.com/SuperUnitaryPerfectNumber.html

There are the super-ballot numbers, which appears to be an interesting combinatorial property related to the Catalan numbers:
http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf
http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Gessel/xin.html
http://arxiv.org/abs/math.CO/0408117

1, 2, 4, 6, 12, 24, 36, 48, ... (http://www.research.att.com/~njas/sequences/A004394) are the superabundant numbers; 1, 3, 11, 45, 197, 903, ... (http://www.research.att.com/~njas/sequences/A001003) are the super-Catalan numbers; 341, 1387, 2047, 2701, 3277, ... (http://www.research.att.com/~njas/sequences/A050217) are the super-Poulet numbers; A074379 (http://www.research.att.com/~njas/sequences/A074379) is a subset of the super-Carmichael numbers (321197185 is the first super-Carmichael number not in A074379).

[Math Is Hard]

Not sure if this link will work, but here is Kaku's reference to "super numbers" from a Google book search:
http://books.google.com/books?id=_HBtAHuG6dwC&pg=PA145&lpg=PA145&dq=kaku+super+numbers&source=web&ots=yqtL9cagiR&sig=TwNm-zCBT1OUzX9bNN5-iIH_9lg

[CRGreathouse]

Not sure if this link will work, but here is Kaku's reference to "super numbers" from a Google book search:
http://books.google.com/books?id=_HBtAHuG6dwC&pg=PA145&lpg=PA145&dq=kaku+super+numbers&source=web&ots=yqtL9cagiR&sig=TwNm-zCBT1OUzX9bNN5-iIH_9lg

It looks like the super numbers are some kind of ring? that is not an integral domain. Ican't find a definition anywhere inthe book -- there are only two pages mentioning "super" and "numbers" (145 and 150), and neither page nor the span of pages between them has a definition.

[Math Is Hard]

Is he talking about this: Lie superalgebra (http://en.wikipedia.org/wiki/Lie_superalgebra)?

[CRGreathouse]

That looks like a fair possibility, although that would mean that (on p. 145) he confuses * and [ , ].

[kurt.physics]

Yeh, that google book, Hyperspace is where i read of super numbers, where a * a = -a * a e.t.c

funny no one has herd of it

[Gib Z]

Whatever super numbers are, they are not the only things to possess the property a* b = -b *a. MIH seems to have found what you want though, it looks like exactly what you were looking for.

[Hurkyl]

Wikipedia has a page on superalgebra (and a category on super linear algebra).

[eastside00_99]

I don't think it is lie superalgebras. At my school, there is a sp math course called supersymmetry which is all about super-numbers or whatever. I will find out what book they are going to use if they will use one. The prof that teaches this course was my multivariable analysis teacher and one day he did mention what the definition of super-numbers are and what the class is about (it is basically trying to do analogues of classical analysis with these super-numbers--the trick is that there are some pathologies). This has been stored in my memory about a year ago and I don't if it is exactly correct: super-numbers are a sequence of real numbers (x_k) together with some "objects" (n_k) that have aritmetic properties of sqrt(-1) such that sum(|x_kn_k|) converge in some sense. That is basically all I can remember from the top of my head and it maybe wrong. I believe there are analogues for big theorems like the Cauchy-Reimann equations. A lot of this stuff hasn't been completely worked out because the top people moved quickly to the scheaf-theoritic point of view ignoring possibly some hidden more classical flavored results (as the Ph.D. student of the person who teaches the course told me).

[eastside00_99]

well, they are not using a book...its probably all notes. Do you by chance live in the piedmont of NC..

[Hyperparadox]

Search Grassman Variables in google, and have a look at this:
http://books.google.com/books?id=E4qpedgudPEC&lpg=PA93&dq=supersymmetry%20in%20chaos&pg=PA8#v=onepage&q=&f=false
(http://books.google.com/books?id=E4qpedgudPEC&lpg=PA93&dq=supersymmetry%20in%20chaos&pg=PA8#v=onepage&q=&f=false)
(This is a link to page 8 of book "Supersymmetry in Disorder and Chaos" by K Efetov, so if the link doesn't work you may search it in Google Books.)