Re: Is e^x zero in some extended number system?
- From: israel@xxxxxxxxxxx (Robert Israel)
- Date: 30 Jan 2006 22:38:32 GMT
In article <1138634478.228518.54720@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Dave L. Renfro <renfr1dl@xxxxxxxxx> wrote:
>I came across the following a couple of days ago
>and I'm curious if any related work has been done
>along these lines.
>
>Paul Dienes, "The exponential function in linear algebras",
>Quarterly Journal of Mathematics (Oxford) 1 (1930), 300-309.
>http://www.emis.de/cgi-bin/JFM-item?56.0151.02
>
>The text of the first two paragraphs follows.
>
> "The introduction of complex numbers was chiefly
> suggested by the problem of determining the zeros
> of polynomials. Some integral functions, such as
> e^x, have no zero in the field of complex numbers.
> This fact suggests the following question. Can we
> generalize the idea of number to such an extent
> that the exponential function may have a zero in
> the extended field?"
>
> "We shall prove in this Note that the exponential
> function has no zero in the linear associative
> algebra to a finite base, and that it has no
> zero in finite non-associative linear algebras.
> [Dienes assumes, of course, that the algebra
> product has a multiplicative identity.] This
> result extends to a large class of algebras to
> an infinite base. In particular, the exponential
> function has no zero in the tensor algebra of
> relativity theory and it misses only singular
> tensor values which do not divide some tensor.
> Moreover, in Hilbert's [*] algebra of infinite
> bounded matrices, so important in atom mechanics,
> the exponential function has no absolutely
> bounded matrix zero."
What is he using as definition of the exponential
function? Maybe I'm missing something, but all
definitions of "exponential function" that I've seen
(e.g. on Lie algebras or Banach algebras) are either
explicitly mapping something into a group or make it
easy to prove that exp(x) has an inverse, namely
exp(-x).
Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.
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