Subject: Re: Vietmath War: need help on physics history
From: Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium)
Date: 20 Dec 1996 04:28:39 GMT
In article <32B9BC0A.63A3@mindspring.com>
Richard Mentock writes:
> What about the old trisection of angle using compass and straightedge?
> That was long considered impossible, and was proven impossible in the
> nineteenth century. However, if you change the rules just a little bit
> (allow the solver to mark on the straightedge, or even to have had a
> mark on the straightedge), then the problem is solvable.
The purpose of analogies is to devise them to help you think. Not to
turn the switch off. For every 1 helpful analogy, a person can dream up
an infinity of useless analogies. But I am talking to deaf and dumb
ears here, can you read lips?
>
> That's what happens with FLT and the p-adics. Now, the bigger question,
> whether the p-adics are or should be the set of numbers that we use to
> view the world, seems to be a question in the domain of physics or
> metaphysics.
You use in wasting anymore of my time on you. You sound like a blue
collar worker who is faking to be a learned intellectual. Dumbo, your
above implies you believe that mathematics is utterly distinct from
physics. And you probably even think there are three distinct subjects
of math, physics and metaphysics. But you are not alone, for the
majority of people think that these subjects are distinct. I myself
realize that all subjects are somewhere inside of physics, everything,
from poetry to biology to physics itself. And there ain't no thing as
metaphysics. And your precious mathematics is just a subdepartment of
physics.
I don't write this reply for your edification Mentock, I write it for
everyone except you. I want you to stay in the weeds.
Subject: Re: holiday group question
From: Fred Galvin
Date: Thu, 19 Dec 1996 22:37:02 -0600
On Thu, 19 Dec 1996, Fred Galvin wrote:
> On 18 Dec 1996, Arturo Magidin wrote:
>
> > Aren't these HNN (Higman-Neumann-Neumann) Extensions? I seem to
> > remember one can prove even more: Given a group H, and two positive
> > integers n,m; with the proviso that n>1 and m>2, you can embed H into
> > a 2-generated group G, whose generators have exponent n and m
> > respectively.
>
> Yes, see F. Levin, Factor groups of the modular group, J. London Math.
> Soc. 43 (1968), 195-203.
Of course, the group H has to be *countable*.
Subject: Re: Eigenvalue placement problem
From: israel@math.ubc.ca (Robert Israel)
Date: 20 Dec 1996 00:01:18 GMT
In article <32B9328B.6349@usa.net>, Erwin Morales writes:
|> Does anybody know how to find the matrix G such that the eigenvalues
|> of [(I + BG)^(-1) A] are in specific locations on the complex plane?
|> The matrices A and B are given and I is the identity matrix.
|> In control theory, finding G such that the eigenvalues of [A + BG] are
|> specified is a standard problem. But how about if the matrix in
|> question is [(I + BG)^(-1) A]?
Well, if A is invertible the inverse of (I+BG)^(-1) A is A^(-1)(I + BG),
and the eigenvalues of M^(-1) are the reciprocals of the eigenvalues of
M. So that reduces your problem to the "standard" one. I don't know
if there's an easy solution in the case where A is not invertible
(but maybe you could take the limit as s -> 0 of the solution with
A replaced by A+sI).
Robert Israel israel@math.ubc.ca
Department of Mathematics (604) 822-3629
University of British Columbia fax 822-6074
Vancouver, BC, Canada V6T 1Y4
Subject: Re: Average
From: "Robert E Sawyer"
Date: 20 Dec 1996 07:32:22 GMT
"Average", at least as it's used in the US, is a term for any of a wide
variety of measures of location, including mode, median, and any of the
various means (arithmetic, harmonic, geometric, etc).
(This is both a "dictionary definition" and accepted statistical usage.)
--
Robert E Sawyer
soen@pacbell.net
Steven Johnson wrote in article <01bbeded$439b3da0$894277a1@pc66-137.state.ut.us>...
| Is an average always a mean in mathamatical terms. Or can a average be a
| median or a mode.
Subject: Re: comp.speech Frequently Asked Questions - part 2/3
From: James Salsman
Date: 20 Dec 1996 07:58:25 GMT
Jonathan Thompson wrote:
>
> ...You mean to say that a scaling factor has been added to
> the definition of the cepstrum for the sake of military secrets.
No, I mean to say that the cepstrum was redefined for the
sake of military secrets. What makes you say that the
difference between DFT(log|DFT(.)|) and IDFT(log|DFT(.)|)
is a scaling factor?
> I think you'll probably find that the reason for all subsequent
> references to to the cepstrum using an IDFT instead of a DFT is
> because the original paper was wrong.
How can the original use of a novel term be wrong? I've been
studying this topic for over a year, and I have no doubt that the
Schafer-Oppenheim cepstrum is very nearly the identity; it's only
a slight convolution. If you would read Appendix 2 of the
Bogert, Healy, and Tukey paper, it will be quite clear that their
definition is rock-solid.
> ... far more plausible than some conspiracy theory.
What's implausable about a government at war (c. 1970) and with
much earlier levels of encryption technology wanting to protect
cockpit and other high-noise-environment voice radio encryption
schemes from automated attacks which require a way to determine
when the plaintext speech has been recovered?
Could it be that H-P, one of the founders of which was
Secretary of Defense in that era, might have been involved?
See 37 USCFR 5.2, http://www.kuesterlaw.com/lawrule/rules9.htm#52
Sincerely,
:James Salsman
Subject: Re: Shortest distance question
From: Pertti Lounesto
Date: 19 Dec 1996 12:32:55 +0200
The functions y=exp(x) and y=log(x) are inverses of each other, so their
graphs are symmetric with respect to the line y=x, so the shortest line
segment connecting the two graphs is perpendicular to the line y=x, so
the slope/derivateive of the functions at the end-points of the line
segment equals 1, so the end points are (0,1) and (1,0).
--
Pertti Lounesto Pertti.Lounesto@hut.fi
Subject: Re: "What causes inertia?
From: Malkki Heikki
Date: Fri, 20 Dec 1996 11:11:13 -0800
Louis Savain wrote:
>
> In article <595qri$j05$1@learnet.freenet.hut.fi>,
> haporopu@mail.freenet.hut.fi (Hannu Poropudas,Oulu Suomi) wrote:
>
> >
> >I would like to ask Louis Savain one question.
> >I refer here to his "Re: What causes inertia", which was dated
> >Sat Dec 14 03:52:34 1996.
> >
> >
> >How does the definition of electron' s mass as follows:
> >
> >Electron's mass is only due expansion resistance of the Universe
> >
> >fit to your descriptions in your article.?
>
> Sorry. I've heard this definition of an electron's mass before but
> I'm sorry to say that it makes no sense to me. Enlighten me.
>
> Best regards,
>
> Louis Savain
Geometry of the Universe could be coordinated with aid of almost
instantaneous color electricity signals and mass changes that
color electricity to black color electricity (= no color electricity).
See README.see, README.mid, README.all and drawings of H-M in
http://www.funet.fi/pub/doc/misc/HannuPoropudas
Best Regards,
Hannu Poropudas.
Subject: Re: closed form soln of infinite product
From: ptwahl@aol.com (PTWahl)
Date: 20 Dec 1996 10:25:44 GMT
It is interesting to note that your question was asked
around 400 years ago by Johann Kepler ( the polymath
and astronomer. ) In the 1596 tome called Mysterium
Cosmographicum, Kepler considered a geometric form
of your question. Take an outer circle, then inscribe a
triangle, then inscribe a circle in the triangle, then a
square in that circle, another circle, a pentagon, and
repeat forever. Is there an innermost limiting circle?
The answer is yes, and the ratio of the radii of the two is
your number 0.11494 ... The calculation was beyond
Kepler, but he was first to note that the limit existed and
was greater than zero.
Kepler was seeking a geometric reason why his perfect
deity had chosen the ratios of the "crystalline spheres"
that held the orbits of known planets: Saturn, Jupiter,
Mars, Earth, Venus and Mercury. He later chose to nest,
not polygons, but polyhedrons, and obtained remarkably
good results. ( You may have seen his drawing, which
shows up in many Humanities texts. ) But I digress.
Because this is the fourth centennial of Kepler's great
work, I had hoped to write an article about all this. The
capstone would have been the closed-form solution you
seek. I failed. There is a rapidly-convergent expression
( in terms of Riemann's zeta function at even integers )
which is handy for computation. You may have noted
how slowly the original product converges!
Sinai Robins ( now at U.C.San Diego ) has turned my
result into some Fourier convolution integrals, and he
believes the integrals are "do-able" but we poured down
many pots of coffee together and finally gave up.
If you're not bored yet, here are the results. Let L be
the desired limit. Let its inverse be R = 8.7000366 ...
and M = ln ( R ). Use Z to denote the zeta function.
Now, M = ln( 11264 / ( 525 Pi )) + T
and, T = SUM{ n=1 ... infinity} of these terms:
( 2^(2n) - 1 ) [ Z(2n) - 1 - 2^(-2n) - 3^(-2n) ]^2 / n .
My notes say that the original product converged to
give 15 correct digits in about 300,000,000 terms.
The infinite sum I call T provides equal precision in just
eight terms. By the way, the exponential of T is a
pretty-looking doubly-infinite product: where m and n
each vary from 4 to infinity, multiply these terms:
[ ( mn - 1 ) ( mn + 1 ) ] / [ ( mn - 2 ) ( mn + 2 ) ] .
Further note: Richard K. Guy keeps a list of open
problems called "Western Number Theory Problems"
and this is number 91:24 on the list. ( # 24 from the
1991 conference. ) I presented the above at the 1993
conference, but haven't been back to see if anyone
beat my best effort.
Patrick T. Wahl
( no institutional affiliation )
"... the triangle is the first figure in geometry. Immediately
I tried to inscribe into the next interval between Jupiter and
Mars a square, between Mars and Earth a pentagon,
between Earth and Venus a hexagon ..." -- Johann Kepler
Subject: Re: Q: Solving recurrences with Mathematica
From: nahay@pluto.njcc.com (John Nahay)
Date: 20 Dec 1996 11:16:05 GMT
ahesham@batelco.com.bh wrote:
: Does anyone know of a way to solve a recurrence with two
: variables using Mathematica or other programs.
Rewrite your recurrence as a matrix equation. For two variables,
this should be an infinite matrix, indexed by a 2-tuple for columns
and a 2-tuple for rows. One must invert the upper block of this
matrix of size, n, for each n. I am doing just such a thing
right now (actually, I am trying to FIND a finite recursion, given
the INFINITE number of equations it satisfies).
Subject: Re: Solving cubic and quartic equations
From: ptwahl@aol.com (PTWahl)
Date: 20 Dec 1996 11:25:05 GMT
I don't know what you'd find on the Internet. The methods are
arcane and not very pretty. I studied this in the pre-calculator
days ( late sixties ) when numerical approximations were
expensive and algebra was cheap ( kind of the reverse of the
world today. )
For the cubic, pick up an old "CRC Handbook of Chemistry and
Physics." Mine is the 38th edition ... look for one of the old
editions that really fit in one hand. Check the table of contents
in the Math Formulas section. They don't get lost in theory and
just say how to do it.
For both the cubic and the biquadratic ( old word for quartic ),
most libraries ought to have the 1948 classic by J.V. Uspensky
called "Theory of Equations." This was the standard reference
for about thirty years, and is still useful despite the emphasis on
arithmetic and on looking up results in tables. Uspensky writes
at a high-school algebra level ( a little trigonometry, but you can
skip that part ) and devotes Chapter 5 to your subject. Worked
examples are included.
Don't expect to find a "formula" like we have for the quadratic
equation. I suppose you could write it that way, but in the real
world it's done step-by-step to avoid having cube roots of sums
of square roots and messy stuff like that.
Patrick T. Wahl
( no institutional affiliation )
Subject: Re: Vietmath War: war victims; blinded victims
From: Andre Engels
Date: 20 Dec 1996 11:38:21 GMT
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) wrote:
>In article <598ml5$am5@svin12.win.tue.nl>
>Andre Engels writes:
>
>> No, it's not. You may say ....5555 is an integer (I won't stop you), but
>> then your concept of integer is not the one that FLT is talking about.
>
> Define "finite" for finite integers. Math is the science of precision.
>If you cannot define finite without a componentry of infinity. Then
>finite integer does not exist.
>
Why not? As long as I can define them in ANY way, they exist, IMO.
My first attempt of defining them would be Peano's axioms, but, as Godel
showed, this defines an infinite number of different systems, so we can't
really use it. So I go for this one:
A number is an integer if and only if it is a member of some finite set,
all of whose members are either 0, or such that they are n+1 for some
number n in the set. I here use Dedekind's of finite: A set S is finite if
and only if it cannot be put in a 1-to-1 correspondence to a strict subset
of S.
> Who cares about the properties of ether when ether does not exist.
>
> Who cares about the behavior of the Higgs boson when the Higgs does
>not exist.
>
> Who cares about FLT and whether finite integers have a solution, when
>Finite Integers do not exist.
>
> You guys are poor at mathematical reasoning, but poorer still at
>understanding what I write.
I'm sorry, I think you made a typo here, you must have meant:
"I am poor at mathematical reasoning, but poorer still are you at
understanding what I write."
>I think this is because you do not know
>math well enough to see the full issues here.
Can't it be that instead of all mathematicians of the world, it is perhaps
you that doesn't understand math enough?
>But, it is to your credit
>that you are stupid enough to attack anyone who says something that is
>not printed in one of your textbooks. Congratulations.
No, we don't attack anyone who says something that is not printed in one
of our textbooks. But we do attack someone who repeatedly asserts such
things without proving them in anything like the correct way and calls us
stupid. We attack those, yes.
Andre Engels
Subject: Re: Generalised Lucas-Lehmer
From: numtheor@tiac.net (Bob Silverman)
Date: Fri, 20 Dec 1996 15:07:17 GMT
mert0236@sable.ox.ac.uk (Thomas Womack) wrote:
>Does anyone know if there's an algorithm like the Lucas-Lehmer one
>for finding prime repunits in other bases? I don't understand how
>Lucas-Lehmer works (I've got Hardy and Wright, I'll look up the proofs,
>but they don't look easy to extend), so I'm not sure if this is
>even a reasonable question.
Noone knows, but I doubt whether they exist. What makes Lucas-Lehmer work
for base 2 is that for N = 2^p-1, we have N+1 fully factored. N+1
happens to be the order of the twisted (sub) group of a finite field.
For base 10, we have N = (10^n-1)/9, and 9N+1 is fully factored. The problem
is to find a group whose order is (a priori) 9N+1. I don't know of any and
noone else does either. (Which is not to say that they don't exist). On the
other hand groups are known whose order is given by any cyclotomic
polynomial in N. 9N+1 is not one of them....
Subject: Re: Spivak's "Calculus on Manifolds"
From: weemba@sagi.wistar.upenn.edu (Matthew P Wiener)
Date: 20 Dec 1996 13:56:10 GMT
In article <59dcog$p47@nnrp1.farm.idt.net>, select information exchange You may be a future fields medalist, but I can't reccomend the book.
He didn't ask for your recommendation regarding Spivak COM.
>It's a great book if you know the stuff or if someone will guide you and
>explain the proofs.
Of if you are motivated and work it through. You'll learn a heck of a lot.
> However if you want some basic analysis check out
>Rudin's Principles of Mathematical Analysis.
And if he wants basic algebra there's Herstein TOPICS IN ALGEBRA. You
perhaps have some point, but it escapes me.
> This book is a classic and
>readable.
Indeed. But it has little to do with the request at hand.
> A good supplement might be Buck's Advanced Calculus.
He did not request advanced calculus either. A would-be reader of Spivak
COM should already be familiar with the material in Buck's book. Indeed,
Spivak is a supplement to Buck, not vice versa.
> Spivak's
>book seems very slick and unmotivated.
But if you just bear down and work your way through it, you will master it.
> You may need someone to fill in
>the gaps.
Or he may fill them in himself. His instructor recommended it to him,
so I presume he is not both alone and has already been judged ready to
make his first go at it by someone more acquainted with him than you
or I.
> If you already know Rudin, then you will follow Spivak.
Unless, of course, you have some great mental block regarding the modern
slick approach to calculus on manifolds. That's how it goes sometimes.
> A next
>book might then be Folland's Real Analysis or Royden's Real Analysis.
>Both have diferent approaches but are great in their own way.
They do not cover the same topic, although there is some overlap. You as
might as well say "a next book might be Spanier ALGEBRAIC TOPOLOGY".
> If you need
>any other advice, e-mail me.
You seem to be good with the "other", yes.
------------------------------------------------------------------------
Regarding the actually posted request, a warm up to Spivak COM, try
Harold Edwards ADVANCED CALCULUS: A Differential Forms Approach (3rd
edition), Birkhauser.
--
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
Subject: Re: Vietmath War: need help on physics history
From: Richard Mentock
Date: Fri, 20 Dec 1996 09:30:56 -0500
Archimedes Plutonium wrote:
>
> In article <32B9BC0A.63A3@mindspring.com>
> Richard Mentock writes:
>
> > What about the old trisection of angle using compass and straightedge?
> > That was long considered impossible, and was proven impossible in the
> > nineteenth century. However, if you change the rules just a little bit
> > (allow the solver to mark on the straightedge, or even to have had a
> > mark on the straightedge), then the problem is solvable.
>
> The purpose of analogies is to devise them to help you think. Not to
> turn the switch off. For every 1 helpful analogy, a person can dream up
> an infinity of useless analogies. But I am talking to deaf and dumb
> ears here, can you read lips?
Yes but we're not that close. Can you answer this: Do you consider
the trisection of the angle possible or impossible? Your answer will
tell me whether you understand mathematics or not.
--
D.
mentock@mindspring.com
http://www.mindspring.com/~mentock/index.htm
