Newsgroup sci.physics 204929

In article <32747A3C.41C67EA6@mp-sun1.informatik.uni-mannheim.de>, Georg Gutermuth  writes:
>Hello out there!
>
>Perhaps someone can help me with my problem.
>
>I am looking for literature on x-ray. To be more precise, a formular
>describing the form of an artificially produced x-ray spectrum as well
>as a formular for the interaction cross section of x-ray with matter
>(photo-effect, compton-effect and e+/e- production)
>
>Thank you very much in advance
>
For some general info you may look up 
X-ray Diffraction, by B.E. Warren.
A bit dated but quite relevant and dirt cheap.  You can get it from 
Dover Publications, for $10.
For all you ever wanted to know and more, I would suggest
Handbook on Synchrotron Radiation, Published by North Holland.  Huge 
multi volume work and very expensive, but I'm sure that you can find a 
university library carrying it.
Mati Meron			| "When you argue with a fool,
meron@cars.uchicago.edu		|  chances are he is doing just the same"

In article <552lk2$nv7@sjx-ixn10.ix.netcom.com>, Alan \"Uncle Al\" Schwartz  writes:
>
>meron@cars3.uchicago.edu wrote:
>>In article <550d1q$suv@herald.concentric.net>, Hitech@cris.com (Hitech) writes:
>>>In article <54u957$bf1@geraldo.cc.utexas.edu>,
>>>Jeff Candy  wrote:
>>>>
>>>>exp(i pi) = -1 has no intrinsic physical content.  e=mc^2 does.
>>>>This is not to say that exp(ix) is an unimportant function in the 
>>>>physical sciences.  
>>>
>>>Great explanation!  I take it from your response there is no reasonably
>>>fundamental physical understanding of e^(i Pi) = -1 or else you would have
>>>provided it.
>>>
>>Why do you think there should be a physical understanding.  This is a 
>>mathematical statement, not a physical one.
>
>
>Euler's Equation unites algebra and analytic geometry. I'd be >real< 
>curcumspect about minimizing that.
>
I would not even dream of minimizing that.  It is one of the most 
beautiful (and surprising) things that ever came from math.  But it is 
not a statement about physical quantities, thus asking for 
"fundamental physical understanding" is missing the mark.  What is the 
fundamental physical understanding of x^2 - y^2 = (x - y)*(x + y) ?
Mati Meron			| "When you argue with a fool,
meron@cars.uchicago.edu		|  chances are he is doing just the same"

In article <3274E91F.5247@sdd.hp.com>, Steven Hines  writes:
>Russell Turpin wrote:
>> 
>> There is no comparable passage of faith in the sciences.  If a
>> student learns how to engage the relevant questions, perform the
>> relevant research, intelligently analyze the results, and write
>> up the results (all issues of performance, not belief), it does
>> not matter whether he is some kind of Platonist who believes that
>> he is investigating the ultimate Reality or a Buddhist who thinks
>> it is all illusion or a science skeptic who thinks it is just a
>> meaningless game.  Whatever one's metaphysics and epistemology,
>> as long as they do not reduce a person to dysfunction (hence, the
>> requirement about finding their way to the classroom), they are
>> enough to learn and do science.  There is no faith to which one
>> must leap, no metaphysical tenets that one must accept.
>
>Don't you have to believe that every event has a cause? Doesn't
>acceptance of this proposition constitute a "passage of faith"?
>
Assume, yes, believe, no.  There is a difference.  Science is self 
checking and self correcting.  It makes assumptions, draws conclusions 
and checks the conclusions versus experimantal data.  But the 
assumptions aren't sacred and if the data fails to support your 
conclusions you may be forced to modify your assumptions.  It is the 
continuous cross checking that distinguishes assumptions from beliefs.
Mati Meron			| "When you argue with a fool,
meron@cars.uchicago.edu		|  chances are he is doing just the same"

Ken Seto (kenseto@erinet.com) wrote:
: Christopher R Volpe  wrote:
: 
: >magnus.lidgren wrote:
: >> 
: >> A short summary of some of the questions and answers:
: 
: >> 
: >> Q3. As I understand it, Quantum mechanics states that when a photon makes
: >> its way through a glass body, it is absorbed and (a new?) (re?)-emitted a number of
: >> times before it passes through. Every time "the photon" actually is a "real" photon it
: >> travels at c speed and during "the absorption period" "it" stands still ??
: >> 
: >> Majority opinion was that the photon goes through repeated absorbtions
: >> and re-emissions (in the forward direction) thus delaying "its" passage
: >> through the glass in correlation with the index of refraction for glass.
: 
: >Anthony Potts disagrees with this. I thought the same thing, until he
: >offerred me the following explanation in email (which I hope he doesn't
: >mind my sharing)
: 
: >++++++++++++++++++++Begin included message++++++++++++++++++++++++
: >On Wed, 9 Oct 1996, Christopher R Volpe wrote:
: 
: >> Anthony Potts wrote:
: >> >
: >> >
: >> > The net effect is to make the photons travel slower, even though they are
: >> > not being absorbed and reemitted by the medium.
: >>
: >> Interesting. I was told that this absorbtion and reemission is precisely
: >> why it is slower.
: >>
: >Well, I don't know who told you that, but it seems to be a very dodgy
: >way
: >of looking at it. For a start, without the light being absorbed by a
: >single atom, maxwell's equations tell us that it will be slowed, because
: >of the change in the permittivity and permeability.
: 
: >Secondly, if the light were absorbed, and reemitted, why on earth would
: >it
: >keep going in the same direction? The atoms have no reason to emit the
: >light in the same way that it first hit them, so you would just get the
The atoms *do* have good reason to emit preferentially in the forward
directon: conservation of momentum. Remember that the abosorbtion/reemission
process is not done by a stationary atom.
But nonetheless, his point is well-taken; the elastic scattering of
waves is the main effect.
To quote J. D. Jacson [page 683]: [scattering by quasi-free charges]
" But in the forward direction the scattering per electron increases 
rapidly to quite large values compared to the Thomson (free charge)
cross section."
Also, see the remarks following eqn. 14.110 .
-- 
Lawrence R. Mead (lrmead@whale.st.usm.edu) 
ESCHEW OBFUSCATION ! ESPOUSE ELUCIDATION !
http://www-dept.usm.edu/~scitech/phy/mead.html 

tadchem (tadchem@arn.net) wrote:
: On Sun, 11 Aug 1996 05:00:17 GMT, Alan Douglas said...
: 
: >From what I've heard from those who know of such things, there has
: >been work done on this, but the results suggest Earth's gravity is too
: >great and the atmosphere too thick to make it likely that an impact
: >could propel a rock to escape velocities without vaporising it. 
: There is a theory running around that, in the early years of planet Earth, a 
: Mars-sized object struck it, seriously rearranging the crust, mantle ,and 
: even the core, and splashing a lot of mantle rock up into high orbit,  Thos 
: rock eventually cooled and solidified into the Moon.  I've even seen video 
: of a computer simulation of this.  Didn't get out of the gravity well, 
: however.
: 
: >Theoretical arguments aside, the question has been floated on usenet
: >as to whether there has ever been a meteorite found which was known to
: >have come from Earth in the first place.  I've not seen an answer to
: >that yet, but I strongly suspect it would be "no".
: Would you accept tektites (terrestrial impact ejecta that achieves 
: sub-orbital trajectories like Alan Shepard's)?
At the AAS Division for Planetary Sciences meeting last week here
in Tucson, there was a fair bit of this kind of discussion.  There is no
meteorite in our collections (that we know of) that originated
on the earth, got blown into interplanetary space for a few Ma,
then came back down to earth.  Tektites aren't classic "meteorites"
in that sense.  Most people attribute this fact to the Earth's thick
atmosphere and it's comparatively large escape velocity.  There
should be even fewer Venusian meteorites.
There's also the queston of orbital dynamics, where it is much easier
for material to go towards the sun than away from it, so Mars ejecta
has an easier time getting to the Earth, Venus, Mercury, etc. than
Earth ejecta (if any) has of getting to Mars.  Perhaps we could sample
the entire inner solar system by roving around Mercury for a while? :)
                Barbara Cohen
     Cosmochemical Cocktail Mixer, PhD to be
     ---------------------------------------
Madness takes its toll. Please have exact change.

In article <54vsgo$e7d@sjx-ixn7.ix.netcom.com>, uncleal0@ix.netcom.com 
says...
>
>nhb003@lightspeed.net wrote:
>>making sure I can post
>>
>
>Clueless newbie twits should be multiply posting to  news:alt.test  lest 
>those of us with some concept of netiquette and vigilante justice be 
>making sure we can flame.
>
>-- 
>Alan "Uncle Al" Schwartz
>UncleAl0@ix.netcom.com ("zero" before @)
>http://www.ultra.net.au/~wisby/uncleal.htm  (lots of + new)
> (Toxic URLs! Unsafe for children, Democrats, and most mammals)
>"Quis custodiet ipsos custodes?"  The Net!
>
Welcome- UncleAn-style. :-)
We're not all so bilious.

pausch@electra.saaf.se (Paul Schlyter) writes:
>  
> I agree.  Only FORTRAN and Ada have better support and/or efficiency
> for numerical programming than C/C++ does.  Thus C/C++ is a better
> choice than almost any other programming language for
> number-crunching programs.
Last time I checked FFT benchmark in CMU Common Lisp on my Linux
machine was 30%-20% faster than in C (both double precision with heavy
optimization)

David A. Cary (d.cary@ieee.org) wrote:
: In article <51kc4l$aga@thorn.cc.usm.edu>,
: lrmead@ocean.st.usm.edu (Lawrence R. Mead) wrote:
: +[snip]
: +: The uncertainty principle relates to measuring the values of variables
: +                                       ^^^^^^^^^^^^^^^^^^^^
: +: rather than the values that they can take.
: +                   ^^^^^^^^^^^^^^^^^^^^^^^^^
: +
: +What - operationally - is the difference ??
: 
: Operationally, one may measure that a particle has a position (x) 20 mm
: above the table. The uncertainty principle relates to how accurately (dx)
: (+-1 mm ? +-1 nm ?) that can be measured.
: 
: The Heisenberg uncertainty principle(s)
:   dE*dT =< hbar
:   dx*dp_x =< hbar
: 
: say nothing about what particular Energy, Time, Position, and Momentum will
: be measured. It only states to what accuracy (dE, dT, dx, dp_x) they can be
: measured.
I repeat my question: what is the *operational* difference? [note the
word operational - look up its meaning if you do not know it]
: Some people believe that particles and photons actually do have some exact
Some people "believe" in fairies too. In science, we believe in what we
can *measure* .
-- 
Lawrence R. Mead (lrmead@whale.st.usm.edu) 
ESCHEW OBFUSCATION ! ESPOUSE ELUCIDATION !
http://www-dept.usm.edu/~scitech/phy/mead.html 

Kelly Miller wrote:
> 
> davidelm  writes:
> 
> >Glass is one of those things which is always a liquid at any
> >temperature.
> 
> just another example that gross ignorance does not prevent the use of
> fingers on a keyboard...
> 
> Kelly Miller
> miller@aries.scs.uiuc.edu
Insults don't prove your point. The do make a point about you.
Take your insults elsewhere, flame-boy.
There are plenty of references to support both sides of this
argument. This is a matter of definition.

Darren Swersky (swersky@compusmart.ab.ca) wrote:
: A beam of light spreads by its distance squared. Therefore, its intensity
: must decrease by its distance squared. At some distance, the beam will be
: so spread out that the individual photons will no longer be touching.
What makes you think the photons were "touching" near the source? And
what does it *mean* to have photons touching??
: Relativists believe photons are indivisible, and while I don't agree, I
: don't see any reason for photons to split up in a beam anyhow. Therefore,
"Relativists" do not "believe" photons are indivisible: they note how
real photons behave and we never experience *half* of a photon or
*1/3* of a photon - always one or two or ... .
[long polemic deleted] According to GR at least, do travel in waves
at lightspeed. The waves of which you spoke [item deleted] is a
*spherical* wave from a *point* source. Indeed those can be broken up
into a superposition of "photons" (right and left cirularized waves) - but
the superposition is rather complicated: the description of what happens
to a spherical wave from the quantum (photon) point of view is not
easily stated in the simplified viewpoint you wish to espouse. The situation
is similar for gravity waves, and a bit more complicated still because 
gravitons are expected to carry spin two, rather than one (with its two base 
states) as for the photon.
: 	Let me know what you think.
: 
: Darren Swersky
: 
: 
-- 
Lawrence R. Mead (lrmead@whale.st.usm.edu) 
ESCHEW OBFUSCATION ! ESPOUSE ELUCIDATION !
http://www-dept.usm.edu/~scitech/phy/mead.html 

Mike,
If you're interested in particle physics take a look at the OnScreen
Science site at . It's mainly devoted to
Chamber Works, our particle-chamber simulation that requires no advanced
math; but it also has a page of annotated links to other sites of
interest.
Bob Estes
OnScreen Science, Inc.
800-617-6416
onscrn@aol.com

> 
> >By the way, in _French_ meter is spelled metre _not_ in English.
> 
> In English metre is spelled metre (at least in most places where English
> is spoken/written).  In America they spell the same word meter.
> In Canada, both French and English write "metre". In the late 70's, when
Canada began its metrification, the decision on how to spell metre was
made with the idea that by spelling it "metre", you could distinguish it
from a meter ( parking or volt variety ) by the spelling rather than the
context of the sentence.
Having both languages spell it the same way has its advantages also.
By the way, Canada is a "half and half" country. Part imperial and part
metric. Since our economy is so closely intertwined with the US, the country
could not go 100% metric. We are waiting for our southern neighbour to join
the rest of the world before we switch over 100%. As an engineer, I use both
systems of measurement each day. I much prefer to do calculations in metric.
-- 
           Robert Burgman, P. Eng.
           Sr. Project Engineer - Systems
           Avenor Pulp Mill - Gold River, BC

Hello readers,
Though I am not a physicist, I am very interested in the behaviour of 
physicists, especially the stereotypical, eccentric, absent minded Professor. 
I am doing a study on academics and the odd or unusual things they have got 
up to, and overall general nutty behaviour and attitudes.
Has anyone out there got any amusing/humorous anecdotes of their 
lecturers/professors/researchers which they would share with me? 
Alternatively, if you are one of the above, have you ever done anything which 
has raised a laugh (if only to yourself) and perpetuated the great Nutty 
Professor tradition?
All responses, however short or long, gratefully received -- I'll post some 
of the best back to the group.
Please send all replies to: F.Ali@Lboro.ac.uk
Many thanks,
Farooq Ali

In article <54r4h4$him@udevdiv.Unibase.COM>, john@petcom.com wrote:
-> Why do I never see Simon's posts first-hand? And I never seem to receive a Cc
-> either. Hmmph. Maybe my program needs debriefing.
-> Yes, thank you Simon. I'm sorry for the finger-bite. :-) Keep it out there,
-> please.
-> My statement that there are no black holes is not to say that there aren't
-> some extremely massive objects out there. The atomic model that I have
-> developed, and from which my concept of gravity just seemed to drop out,
-> postulates that energy assumes similar patterns at certain points as one
-> moves up the scale. Just like one finds the same musical note as one 
-> goes from one octave to the next. So, in comparing galaxies to atoms,
-> I developed my Galaxy Model, which treats the arms of a galaxy as if
-> they were the electrons of an atom. This means that the centers of
-> galaxies compare to us as aggregations of protons compare to electrons-
-> 1000 times denser and of opposite charge. Certainly these will be
-> extremely massive bodies, but they are not such because gravity is
-> crushing something which would otherwise be less dense- rather they
-> are dense because that is their nature. Call them black holes if you
-> want, but gravity will reach a limit before it breaks down atoms.
Don't you high-school kids have anything better to talk about?  Galaxies as
atoms! Hahahahah!!! Nice.
-- 

What would be the best way to obtain numerical solutions of
Maxwell's equations in an anisotropic medium?
Allan Adler
adler@pulsar.cs.wku.edu

Mikenew2@aol.com (Mike Birtel) writes:
>jti@santafe.edu (Jeff Inman) wrote:
[re: people who fear "ontological reduction" really just fear determinism]
>>  This needn't be a fear that "spirit" is
>> ultimately deterministic but might quite reasonably be a concern that
>> such a reduction to the inadequate tools and concepts that physicists
>> currently employ constitutes a loss of important aspects of the
>> original.
>> 
>> So, the complaint might quite well be that the inadequacy of reducing
>> experience to the kind of "material" that scientists typically mean,
>> seems to suggest that "material" is poorly understood.  In other
>> words, one can be a materialist without being reductive.  This
>> represents one kind of critique of physics.
>
>I think we essentially agree. I certainly didn't want to imply that any
>physics we have is capable of modeling the "ontological facts," simply that
>the assumption that it someday could is essential to the belief that there
>is nothing special about "experience" except that we aren't capable of
>modeling it with our limited tools and processing power.  
But this blurs your distinction between "ontological" and
"epistemological" reduction.  Changes in what can be known and
observed by science are accompanied by changes in the quality of what
is being observed, that is, in what "matter" is, etc.  Science is not
merely extended its power to explain somthing that remains unchanging,
it is describing a changing thing.
>The fact that we
>can't understant reductive models well says nothing about the world, but
>teels us a great deal about ourselves - this was the point of
>distinguishing epistemological from ontological reduction: I fully believe
>in ontological reduction but not epistemological reduction - we simply
>can't work with models when their complexity approaches that of even
>today's physics.
But vice versa, too.  As far as I know, for all the supposed
sophistication of the models, the theories of the *ontos* currently
entertained by scientists don't "expand" (i.e. the opposite of
reduction) to theories of experience, except through the premise you
make, that it might be (or must be) possible.  Hence "ontological
reduction" of experience remains for now untenable.  One werely offers
a promise, by suggesting that this is the result of epistemology.
Presumably, a transformation of the ontology will accompany any
advances against the epistemological limitations.
-- 
"But among those whom this story reached were were also the woman's
 in-laws, and they decided, without telling her a word, to find this
 angel and to see if knew how to fly ..."

I have seen references in books and articles that there are more
dimensions than our "normal" four. Maybe as many as 13 or 16; and that
perhaps the others "shrunk," or curled up on themselves, and are tiny
now. I would like to know how the dimensions and the number were
identified; how and why they shrunk; and what the implications are; can
something exist in these other dimensions as well as our own; will our
4 dimensions shrink? Where else might I find information on the
subject? Thanks.
BEM

In article <55232g$cus@newton.cc.rl.ac.uk>, tony.richards@rl.ac.uk says...
>
>laradex3@sj.znet.com (Larry Adams) wrote:
>>
>>A photon is the combination of two spin
>>waves of opposite helicity within matter
>>undergoing magnetic resonance.
>>
>>L.A.
>>
>BS
>-- 
>Tony Richards
So you haven't read Ferromagnetic Resonance by Vonsovskii,
so what?  L.A.

Silke-Maria  Weineck (weinecks@mail2.sas.upenn.edu) wrote:
]Michael Kagalenko (mkagalen@lynx.dac.neu.edu) wrote:
]: Silke-Maria  Weineck (weinecks@mail1.sas.upenn.edu) wrote:
]: ]Michael Kagalenko (mkagalen@lynx.dac.neu.edu) wrote:
]: ]: Silke-Maria  Weineck (weinecks@mail2.sas.upenn.edu) wrote:
]: ]: ]guilt. But here we have the "science" community, intoning in near-unison
]: ]: ]and near-ignorance, "where there's smoke, there must be fire."  
]: ]
]: ]:  You are lying. 
]: ]
]: ]Am I? You have so far demonstrated neither knowledge nor originality i 
]: ]your detractions of Derrida.
]
]:  Actually this is true sentence. since I have not posted any
]:  "detractions of Derrida", I have not "demonstrated either 
]:  knowledge or originality" by them.
]
]: ] Perhaps I'm "lying" about others -- I'm 
]: ]certainly right about you.
]
]
]:   That is false. I challenge you to produce a single post, claiming
]:   that Derrida must be wrong, because
]:  "where there's smoke, there must be fire." Failing that, I 
]:  expect  you to publically apologize for your lying.
]
]
]If you think you're worth an hour spent at altavista, your self-image has 
]re-inflated to an amazing degree. You have never posted a derogatory 
]remark about Derrida? If you say so. I apologize for having 
]misjudged you.
 That's welcome development, but it's not nearly good enough. Here's your 
 claim again:
 ]guilt. But here we have the "science" community, intoning in near-unison
]: ]: ]and near-ignorance, "where there's smoke, there must be fire."
 Produce the post(s) advancing such an argument, or apologize.
-- 
LAWFUL,adj. Compatible with the will of a judge having jurisdiction
                -- Ambrose Bierce, "The Devil's Dictionary"

On Mon, 28 Oct 1996 15:40:54 +0200, Richard Weideman
 wrote:
>Hi,
>
>I'm trying to find a lightweight material with a high radar cross
>section, or any other information about increasing the radar visibility
>of a small object.
>
>Can anyone point me in the right direction ?
>
>All the stuff I find on the net is about REDUCING the cross section for
>the purpose of hiding things away ...
>(various mass destruction devices etc.)
>
>I'm trying to make something MORE visible, not less
>(for recreational purposes, not mass destruction)
>
>Cheerz
>Richard
 Look up cornercube or retroreflector.  It reflects the signal back to
ita source regardless of relative orientations of the two.
Boris Mohar
Boris

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Joshua Burton and others have caught some errors:
In article <55153l$r58@charity.ucr.edu>, John Baez  wrote:
>    ELECTROMAGNETIC FORCE          WEAK FORCE         STRONG FORCE
>        
>    photon                          W+                  6 gluons 
>                                    W-
>                                    Z  
There are 8 gluons.  
>Right now, we believe that neutrinos only exist in a
>left-handed form, rotating one way but not the other around the axis
>they move along.  This is intimately related to their apparent
>masslessness.  In fact, for reasons that would take a while to explain,
>the lack of parity symmetry in the Standard Model forces us to assume
>all fermions acquire their mass only through interaction with the Higgs
>particle!  
Joshua notes that were there a right-handed neutrino, it would be
an SU(3) x SU(2) x U(1) singlet (or in math lingo: it'd transform
in the trivial representation), so it could acquire a Majorana mass.
One proposed method is the "seesaw mechanism".  For more on this sort
of thing see the recent thread on sci.physics.research entitled 
"Proof of neutrino rest mass?" and also 
Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/~www/neutrino/jhu/jhu.html 
>The Standard Model [...]
>would be ruined by "anomalies" --- certain nightmarish problems that
>can beset a quantum field theory --- if one idly tried to mess with the
>generations by leaving out a quark or the like.  Indeed, this is why the
>charm quark was first predicted, before the generation pattern was fully
>apparent.  
This is wrong.  Joshua notes that while the famous 1969 Adler-Bell-Jackiw
paper on anomalies predates the discovery of the charm quark, and perhaps 
*could* have been used to predict the charm quark, the actual prediction 
of the charm quark by Glashow-Iliopoulos-Maiani in 1970 was made for 
other reasons.  
They proposed this quark, and the so-called "GIM mechanism", to solve some
problems concerning kaon decay.  In the Standard Model as it then stood,
a neutral K-bar meson could decay into a muon-antimuon pair like this:
             W+
dbar--<----~~~~~~~----<---mu+ 
           |     |
          u^     V nu
           |     |
   s-->----~~~~~~~---->---mu-
            W-
(The neutral K-bar is a strange-antidown pair.)   But this decay was
not observed.  Glashow Iliopoulos and Maiani posited the new c quark
so that the process would be cancelled, or almost cancelled, by 
             W+
dbar--<----~~~~~~~----<---mu+
           |     |
          c^     V nu
           |     |
   s-->----~~~~~~~---->---mu-
            W-
Actually Joshua spoke of "neutral kaon mixing" rather than kaon decay,
and I'm getting the above stuff from Kerson Huang's book "Quarks Leptons
and Gauge Fields".  Perhaps the same general problem shows up in something
like the following? 
            W+
dbar--<----~~~~~~~----<---sbar
           |     |
          u^     V u 
           |     |
   s-->----~~~~~~~---->---d
            W-
In any event, Joshua says that his advisor, Mary Gaillard, noted that
the cancellation wouldn't be complete, since the up and charm have
different mass, and used the observed left-over kaon mixing to successfully
predict the charm mass six months before the charm quark --- or more
precisely, the J/psi, which is a c-cbar bound state --- was discovered
by Richter and Ting in 1974.  Ah, the good old days!  
>The Standard
>Model makes no attempt to explain this, although it does suggest that
>there had better be more than 17 quarks --- more, and the strong force
>would not be "asymptotically free" (weak at high energies), which would
>cause lots of problems for the theory. 
There had better NOT be more than 17 of something.  Joshua says
there had better not be more than 17 generations, but Huang's book
says there had better not be more than 17 flavors of quark.  In any
event, we are pretty sure there are no more than 3 generations, unless
the next generation has a darn heavy neutrino or something else really
weird like that happens.
>Gauge
>bosons have n - 2 physical degrees of freedom in n dimensions: for
>example, in dimension 4 the photon has 2 degrees of freedom, the spin-up
>and the spin-down states.  
It would be better to say, "the left and right polarized states".
A corrected version is now at
http://math.ucr.edu/home/baez/week93.html

Even though I've yet to get email, I suspect that someone is reading
this, at least I hope so.  Here is the correct proof without that
strange muddled ending that was in the first one.  I didn't have to
reach for this one either.  Like the rest of the proof it was just
staring at me in the face.
James S. Harris
Georgia, USA
--------------------------------------------
Introduction.
Fermat's Last Theorem has long been a magnet to the amateur and
professional mathematician alike because of its seeming simplicity; yet,
extraordinary difficulty.  Although there is a proof by Andrew Wiles, I
think it is understandable that the problem still would incite
curiosity.  I would also assume that a simpler solution would also be of
interest.
Note:  The following proof makes extensive use of Fermat's Little
Theorem which isn't usually stated.  I also make use of accepted results
which have came up in my previous posts on sci.math without going over
them in detail again.  
1.  Statement of the Problem:  Fermat's Last Theorem
Given x,y,z, relatively prime, n odd prime 
no solution exists for the equation x^n + y^n = z^n 
2.  Proof for Cases where x,y or z are divisible by n.
Let x=af, y=bg, z=ch  means that
 x+y=h^n or n^{n-1}h^n, z-x=g^n or
 n^{n-1}g^n, and z-y=f^n or n^{n-1}f^n.  
   For example, 
  x^n + y^n = (x+y)(x^{n-1}-x^{n-2}y+...+y^{n-1}) = z^n
Since x,y and z are relatively prime, (x+y) can only be divided out once
and the term it is      multiplied times can have no factors of (x+y)
except for maybe one n factor. 
And,
 (x+y-z)^n = n(z-x)(z-y)(x+y)Q 
where Q 
represents 
all those other terms that are hard to write out for the general case. 
For
 n=3 it is one.  And for n=5
  Q = z^2 - (x+y)z + x^2 + xy + y^2
.  
Using the above the following is always true.
  (x+y-z) = nfghq    (Using Q=q^n)
I can then use my earlier relations to write my equation in terms of
f,g,h,q, n only.  An intermediates step with z divisible by n is
   x + y-z = af - f^n = bg - g^n = n^{n-1}h^n - nch  = nfghq
From which I get
 f^n + 2nfghq + g^n = n^{n-1}h^n   and subtracting   f^n + nfgp + g^n =
(f+g^)n  
(p used for ease of writing the general case, for example, with n=3,
p=f+g)
  gives
   nfg(p-2hq) = (f+g)^n - n^{n-1}h^n
p is always divisible by (f+g) because nfgp=(f+g)^n - (f^n + g^n) 
Since (f+g) must be divisible by n, because in this case z is divisible
by n, requires that f,g,h or q be divisible by n which creates an
infinite regression similar to the one in Fermat's proof for n=3.
The same comes up with x or y divisible by n since you get
  g^n + 2nfghq - h^n = n^{n-1}f^n    subtracting 
  g^n + nghp - h^n = (g-h)^n
gives 
     ngh(p-2fq) = (g-h)^n - n^{n-1}f^n
which requires that f,g,h or q be divisible by n because (g-h) is
divisible by n which is again a contradiction for the reason given
before.
3.  Proof for Case x,y,z not divisible by n 
Extension of Fermat's Little Theorem:
Given a-b divisible by n, a^n - b^n must be divisible by n^2
So Fermat's Last Theorem can be written as
  Given x,y,z relative prime, none divisible by n; n odd prime
 for a solution to exist  (x+y)^n - (x^n + y^n) must be divisible by
n^2.
Now notice that x and y can be written in terms of n like 
x=jn+r and y=kn+s with r,s < n 
 If (x+y)^n - (x^n + y^n)
 were divisible by n^2 from a substitution it can be seen that this
would require that 
     (r+s)^n - (r^n + s^n)   be divisible by n^2  but we can also write
  u+r = c1n, v+s = c2n   which still requires that    (u+v)^n - (u^n +
v^n)   be divisible by n^2 .
But by varying c1 and c2 one can create u's and v's with any desired
modulus with respect to n.  Therefore, the above requires that for any
integers a,b
   (a+b)^n - (a^n + b^n)  must be divisible by n^2  
But  then I can use a=f, b=g which requires that (f+g)^n - (f^n + g^n)
must be divisible by n^2
(in working out the text version I've noticed that all of this isn't
required because f has the same modulus as x and g the same as y anyway)
Which from before equals nfgp and requires that p be divisible by n
But like before I can write
f^n + 2nfghq + g^n = h^n  and again subtract    f^n + nfgp + g^n =
(f+g)^n  
which gives
 nfg(p-2hq) = (f+g)^n - h^n
which conflicts with the requirement that x,y and z be relatively prime,
as before.
It is then seen that  (x+y)^n - (x^n + y^n)
   is not divisible by n^2  which completes the proof of Fermat's Last
Theorem.
Here is one example with an alternate proof of the above for one n.
Notice that for n=5,   (x+y)^n - (x^n + y^n) equals 5xy(x+y)(x^2 + xy +
y^2) 
which would mean that x^2 + xy + y^2 must be divisible by 5.  It's easy
enough
 to see that it can't be just by trying different r's and s'  with 
(r+s)^n - (r^n + s^n).
James S. Harris

if you write pi in base pi, i'm not going to have much trouble guessing
it now am i?
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