Subject: Re: HELP! Who knows Oliver HEAVYSIDE?
From: hardy@umnstat.stat.umn.edu (Michael Hardy)
Date: 3 Jan 1997 01:59:34 GMT
In article <59qaki$c6d@boole.maths.tcd.ie>,
Timothy Murphy wrote:
> (Heaviside) was a man of strong views,
> with a particular dislike for Cambridge (UK) mathematicians.
> He felt (probably correctly) that his work had been spurned
> by these superior fellows, on the grounds of lack of rigour.
In article <5a7002$690@totara.its.vuw.ac.nz>,
John Harper answered:
> Jeffreys + Jeffreys "Methods of Mathematical Physics" is a well-known
> book by a pair of Cambridge mathematicians; they give proper credit to
^^^^^^^^^^^^^^
> Heaviside for his many contributions to their subject, and point out
> that his non-rigorous methods often gave correct answers, but they also
> point out where his results were wrong. (see J+J' 3rd ed 1956 p229)
One of the authors was Sir Harold Jeffreys, a professor of
_astronomy_. I am uncertain whether to call him a "mathematician" or
to say rather that he was one of Heaviside's fellow physical
scientists, perhaps as much entitled to complain about mathematicians'
insistence on logical rigor as Heaviside was. Jeffreys got around
enough that at one point I mistakenly thought he was a geologist.
In foundations of statistical inference he is one of the two Big Names
in "objective Bayesianism". (Edwin Jaynes, another physicist, is the
other.)
I wonder what those who are most insistent on logical rigor in
mathematics would think of his use of "improper priors"? These work
like this: one will observe some data X_1, .... , X_n whose conditional
distribution given a scale paramenter s is known and depends on s. But
s itself is not observable. Prior to observing s, s has a known
probability distribution, the "prior" distribution. After observing the
data one finds the conditional distribution of s given the data, the
"posterior" distribution. Jeffreys would often use something like
ds/s as the prior distribution. He was aware that this doesn't integrate
to 1 over the interval [0, oo]. But the posterior distribution does.
Similarly he would use dp/[p(1-p)] as the prior distribution of a
proportion known to lie in the interval (0,1). Etc. . . . .
Mike Hardy
PS: Note that I quoted Murphy and Harper _separately_ rather than doing
_nested_ quotes, a frequent and unpleasant practice on USENET.
Michael Hardy
hardy@stat.umn.edu
Subject: Re: fundamental group isomorphisms and continuous maps
From: Joerg.Winkelmann@rz.ruhr-uni-bochum.de (Joerg Winkelmann)
Date: 3 Jan 1997 10:03:14 GMT
Maurice Herlihy (mph@cs.brown.edu) wrote:
:
: Let K and L be compact, connected cell complexes, and \phi a fixed isomorphism
: between their fundamental groups.
:
: Is anything known about the circumstances under which there exists a continuous
: map f: K -> L that induces the isomorphism \phi on the fundamental groups?
special case:
if L is an Eilenberg-MacLane space of type K( ,1),
i.e. if all higher homotopy groups of L vanish (e.g. this is true for
Riemann surfaces) then the homotopy types of continuous maps
between K and L are classified by the homomorphisms between the
fundamental groups. In particular, for every homomorphisms between the
fundamental groups there exists a continuous map inducing this
homomorphism.
Joerg
