Subject: Re: ABC conjecture
From: Bill Dubuque
Date: 03 Jan 1997 01:35:55 -0500
client@mediom.qc.ca (Nom du client) writes:
>
> Can someone could explain me clearly why ABC conjecture should be true?
See Lang's excellent survey (there is also a 5 page exposition in the
3rd (1993) edition of his Algebra).
Lang, Serge. Old and new conjectured Diophantine inequalities.
Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37--75.
MR 90k:11032 (Reviewer: D. J. Lewis) 11D75 (11-02 11D72 11J25)
Other useful surveys are
Oesterle, Joseph. Nouvelles approches du "theoreme" de Fermat. (French)
[New approaches to Fermat's ``theorem'']
Seminaire Bourbaki, Vol. 1987/88. Asterisque No. 161-162 (1988),
Exp. No. 694, 4, 165--186 (1989).
MR 90g:11038 (Reviewer: Toshihiro Hadano) 11D41 (11G05 11N37 14G25)
Frey, Gerhard. Links between solutions of $A-B=C$ and elliptic curves.
Number theory (Ulm, 1987), 31--62,
Lecture Notes in Math., 1380, Springer, New York-Berlin, 1989.
MR 90g:11069 (Reviewer: Kenneth A. Ribet) 11G05 (11D41 14G25 14K07)
Frey, G. Links between elliptic curves and solutions of $A-B=C$.
J. Indian Math. Soc. (N.S.) 51 (1987), 117--145 (1988).
MR 90i:11059 (Reviewer: Kenneth A. Ribet) 11G05 (11D41 14K07 14K15)
-Bill Dubuque
Subject: Re: SNOWFLAKE symmetry - who can explain it?
From: arteaga@cs.umd.edu (Santiago Arteaga)
Date: 3 Jan 1997 01:07:14 -0500
a bean wrote:
Maybe you should start being a little bit critical about your
own question. Let us try to put in a more general context.
Have you seen a crystal growing? They do grow symmetrically;
all faces and edges grow at the same rate. So maybe symmetry is not a
weird thing that happens to snowflakes, but rather it is a general
property of crystals. Including water crystals.
I know what you are thinking, that I am not addressing your
question, but think of this: what is it that, in a "typical" crystal,
forces each face and edge and vertex to grow in opposite direction but
same speed as the face/edge/vertex at the other side of the crystal?
And yet, not all crystals are symmetric. When they are large
enough the conditions change a little bit from one side to the other
and you get a not-so-perfect crystal, with an edge which is a little
bit longer than another. Hey, this sounds familiar, right? People have
said that something similar happens to the snowflakes if they grow too
large. Sorry, your snowflake.gif is not the end of the story. Somebody
put those pictures together to illustrate a point, and because they
were pretty. Maybe that person should have added another picture, this
one larger, of a snowflake which is visibly unsymmetric in your sense,
of not having all the branches with the same structure, to illustrate
another point.
The thing with snowflakes is that they aren't convex as
"typical" crystals, but "spiky". You must have seen how the ice that
sometimes forms on your car windshield tends to grow along lines
which sometimes branch. The point is, this branching is not
continuous but discrete; at some point, for some reason, a spike
branches. If they branched continuously, you would get a film of ice
covering your windshield, instead of the spiky growth.
Well, I don't know what is it that makes one of these spikes
to branch. I guess it is a combination of vibration/temperature/
humidity/pressure/somethingelse. But, whatever it is, it is likely to
happen simultaneously in the six sides of a _small_ snowflake. You
look at your windshield and you see that those spiky crystals do not
branch symmetrically at the scale of centimeters.
Now, since the six sides in a small snowflake were
symmetrical before they branched, the new spikes grow in symmetrical
directions. Let me remind you here of the question I made at the
beginning: "what is it that forces each edge to grow in opposite
direction but same speed as the edge at the other side of the
crystal?". The answer, that I guess you know but I'll say anyway,
is that since the whole snowflake is one and the same water crystal,
its molecules are oriented in the same directions all over the
snowflake. (At least if it is small enough.) Look at snowflake.gif
and check that all the branching happens at 60 degree angles.
Add the hypothesis that typically a snowflake falls to the
ground before it grows too large to stop being symmetric, and my
point is made. What we need now is somebody to pick up a few large,
unsymmetric snowflakes and put them in a bigsnowflake.gif that you
can see and ignore because it is not so pretty.
So there you have a mechanism which explains why small
snowflakes are symmetric. And you are right, I don't know the fine
details; I don't know quantum mechanics so I can't explain how
water molecules join in a crystal, nor why/how a spike branches.
I don't know how much time does it take for a snowflake to fall
to the fround, or how much has it grown by then, or what is the
typical size of a snowflake before it branches unsymmetrically.
I don't have all the answers, you are absolutely so right.
But I don't see a bigger question here; why do you think there is
one? Have you computed that radius of typical unsymmetry, and it
happens to be a lot larger than millimeters?
I am sorry if this message has a rather jerkish tone, but
somebody has to tell you that you can not say "your model can not
work just because". It would really help if you said something
more concrete than "there has to be something deeper". Could you
please say at which point in the previous explanation do you
disagree with us?
Santiago
Subject: Re: Math content course--2 hour lab
From: "J. Clarke"
Date: 3 Jan 1997 06:41:10 GMT
Del Stanton wrote in article
<5ah6kg$mla@news2.cais.com>...
> Posted to the news group and sent to
> "J. Clarke"
>
> Dear John Clark,
>
> Thank you for your response . . . .
>
> Hey - I am talking elementary school math.
>
> And - given truncated decimal values for e and pi - one can
> calculate the product using repeated addition. I am talking
> about numerical caculations here, and any numerical calculation
> must use rational numbers.
Given truncated values you can arrive at truncated answers. But a
truncated value of pi is no more pi than 3 is 4. Just a useful engineering
approximation.
>
> When I was taught multiplication its relationship to counting by
> "n" (counting by twos - counting by threes) was not taught, nor
> was the concept of multiplication as repeated addition. It
> was another process - separate from addition and subtraction -
> that was taught as a rote process. I didn't "understand"
> multiplication until years later.
Depends on your objectives. If one wants to calculate efficiently, then
memorizing the tables is better than knowing how to derive them--9 times 9
is 81 is a lot quicker than 9 plus 9 is 18 plus 9 is 27 plus 9 is 36 plus 9
is 45 plus 9 is 54 plus 9 is 63 plus nine is 72 plus (now let's see, how
many nines have I added here--I've lost track--oh, only 8--ok) 9 is 81.
Even with computers, table look up is often more efficient than repeated
addition. On the other hand, when dealing with computers, repeated
addition is a valid way to derive the tables. I was taught this
relationship and then tried to apply it instead of memorizing the tables,
with the result that I have never completely committed the multiplication
tables to memory. Which occasionally annoys me, but not enough to do
something about it.
--John
Subject: Re: 4 lines / Wlod-1996
From: Ryan Cormney
Date: Thu, 02 Jan 1997 23:25:01 -0800
Wlodzimierz Holsztynski wrote:
>
> ;
>
>
>
> Polynomials over finite Galois field
> spread so evenly across their finite affine space
> I wish for a network of friends
> to count on
>
> H.New Mexico
> 1996-03-05/06
One thing I hate about certain poets is that they hide their meaning in
unfamilar terms(or algebra in this case). Personally, I like the so
called generic style, sure more simplistic, yet just as powerful.
Consider it postmodernism. I wrote that poem(Inside) as a direct result
of reading Emily Dickenson(Miss Original). I appreciate your response to
my poem, but I would like to know exactly what you find generic(the
content, the style, rhyme). I admit I have a lot to learn, but I gain
nothing from your previous comment.
I was wondering what exactly are you trying to say in this above blurb.
?you want friends that are reliable as a function you plug numbers into?
never met anyone quiet like that. Nothing is consant in this world; all
exceptions are only human concoctions(numbers). Pretty generic, right!
RBC
Subject: Mathematica, Maple, Mathcad and similar
From: David Harding
Date: Fri, 03 Jan 1997 18:24:51 -0800
Dear mathematicians,
Is this newsgroup the forum to seek comments on the relative merits of
Mathematica, Maple, Mathcad and the like? If not, can people suggest a
newsgroup ?
I'd like to know if the products are orientated at different classes of
problems,
if they are easy to learn,
if they have loyal bands of users,
if they are available in the public domain,
and any other insights users would like to share with me,
thanks in advance,
David,
Melbourne Australia
Subject: Re: 0.999999999999999999999...= 1
From: sjmccaug@prairienet.org (Scott J. McCaughrin)
Date: 3 Jan 1997 07:44:54 GMT
In a previous article, abian@iastate.edu (Alexander Abian) says:
>In article <32C06F00.4FFB@slonet.org>, Casey Jones wrote:
>>Alexander Abian wrote:
>
>>> Now, if 0.99999999.... were less than 1.00000000... then
>>>
>>> (1) p + 0.999999999.... = 1.00000000....
>>>
>>> FOR SOME POSITIVE REAL NUMBER p.
I suspect at least some of the motivation for the debate on this posting is
perceived ambiguity in the notation "0.99999...". If 0.999999... < 1, then
1 - 0.999999... = p > 0, but how "big" should p be if we are permitted to
continue the 9's indefinitely?
Clearly no matter what positive p you choose
>>>
>>> p + 0.99999999.... > 1.00000000.... for any p > 0
>>>
>>> contradicting (1) and THEREFORE 0.99999999.... = 1.00000000....
>>>
No. You have implicitly defined p = 1 - 0.999999... (for the sake of the
argument) above, so p + 0.999999... = 1, not > 1. I'm not necessarily
questioning your conclusion, just how you got there.
Scott McC.
Subject: Re: Irrationality and Transcendence of e
From: thall@cs.unc.edu (Andrew Thall)
Date: 2 Jan 1997 18:01:30 -0500
In article <32C3F6A0.5811@bellsouth.net>,
harden wrote:
>I know of a very simple proof of the irrationality of e based on the
>factorial series( assume it is rational, set the rational number equal
>to the series for e, multiply by the factorial of the denominator( you
>can prove from the series that e is positive so you use a positive
>integer for both numerator and denominator ), separate the multiplied
>series after the term which equaled 1 after the multiplication, and show
>that the remainder of the series is between 0 and 1 and therefore not an
>integer, whereas everything else was an integer ), but are there any
>similar proofs of the transcendence of e based on the Taylor series for
>e^n: 1 + n + n^2/2! + n^3/3!... ? Comments would be greatly appreciated.
I don't know of any such approach, and doubt it would be possible, since
it's quite a different matter to prove a number transcendental.
The clearest proof of the transcendence of e that I've encountered was
a two-pager in Herstein's _Topics in Algebra_ (2nd ed., pg. 217), that
used Hermite polynomials, but required only some familiarity with calculus
to understand.
--Andrew.
--
"I shall drink beer and eat bread in the House of Life."
Subject: Re: Graph theory -- minimum diameter
From: ptwahl@aol.com (PTWahl)
Date: 3 Jan 1997 09:09:12 GMT
Eric Postpischil asked "What is the minimum diameter of an undirected
graph of n nodes of degree at most d?" (What interesting properties, and
other related questions.)
Some of your questions are covered in various graph theory texts. I seem
to recall one called _Algebraic Graph Theory_ by Biggs, which paid a lot
of attention to symmetry. He uses the notion of "girth" instead of
diameter, but you can make the adjustment in terminology. One class in
matrix theory and some elementary group theory suffice to get you through
most of it.
I recall the interesting result that there are at most four "regular"
graphs of diameter two. Here, regular means each vertex is of the same
degree. ( I am speaking of undirected graphs with at most one edge
between any two vertices. The degree of a vertex is the number of edges
at that vertex. )
Three of these are known, and have interesting symmetry properties. The
first is the pentagon, v = 5, d = 2. ( That is, 5 vertices, each of
degree 2. ) The next has v = 10 and d = 3. Called Petersen's graph, it
is often pictured as a 5-pointed star inside a larger pentagon, with 5
more edges connecting the inner figure to the outer figure in the most
direct routes. If you draw this figure, you will see that it indeed has
diameter 2, that is, a path of length 2 or less connects each pair of
points.
It's not hard to see that a regular graph of degree "d" that has diameter
2 will have v = 1 + d^2 vertices. Curiously, v = 5, 10, 50 and possibly
3250 are the only possible graphs of this type, corresponding to d = 2, 3,
7 and posibly 57. Folks (at IBM around 1960) named Hoffman and Singleton
discovered the Hoffman - Singleton graph with v = 50 and d = 7. I won't
pretend I can describe it, but it includes several copies of the Petersen
graph and intricate linkages, and has quite nice symmetry. ( Some texts
offer reasonable pictures, check the index. ) Biggs includes a proof of
the H-S result, that v = 3250 and d = 57 is the only other possibility.
This graph would have 92,625 edges if it exists. It remains undiscovered,
as far as I know, but must lack an important symmetry property called
"distance transitivity."
This is just a short introduction to some of the simplest results. Quite
a lot is known about regular graphs of small diameter, and the known ones
do have pleasing symmetry. I'm sure there's been work done on the minimum
diameter of any graph with v vertices and maximum degree d. (Such graphs
tend to be efficient and profitable to Ma Bell and family, so the research
is well-funded.)
You also asked about "multiple shortest paths." A regular graph of
minimal diameter can lack multiple shortest paths. In a sense, it's a
waste of edges to have two shortest paths between two nodes. However,
since only a few graphs can be perfectly efficient, the rest will have
multiple paths between some diametrically opposite nodes.
Hope this helps you get started!
Patrick T. Wahl
( no institutional affiliation )
Subject: Help : Covariant to contravariant and vice-versa
From: e9209h60@gamma.ntu.ac.sg
Date: 3 Jan 97 17:51:29 +0800
I am interested in knowing about algorithms for converting a
covariant tensor to a contravariant one and vice versa. I know that
for tensors of rank n and indices i_1, i_2, ..., i_n = 1..2,
this can be achieved by inner product of the tensor with the
permutation tensor, e, defined by,
_
| 1 if i=1, j=2
e_ij = { -1 if i=2, j=1
| 0 if i=j
-
In a general case when i_1, i_2, ..., i_n = 1..r,
how can we achieve this transformation from covariant to contravariant
tensor and vice versa ?
Thanks
Satish
Subject: Re: Probability (was re: math jokes)
From: christian.bau@isltd.insignia.com (Christian Bau)
Date: Fri, 3 Jan 1997 10:31:15 GMT
3.95.970103085014.470B-100000@rubin.but.auc.dk>
Organization: Insignia Solutions
In article ,
Nikolaj Toft Hansen <31nik@but.auc.dk> wrote:
> On Sat, 28 Dec 1996, Robert Lewis wrote:
>
> > Lenny Schafer wrote:
> > >
> > > Your chance of winning the typical big state lottery: 5 million to one.
> > > --
> >
> >
> > Aren't most lotteries (in big and small states) a matter of
> > picking six correct numbers out of a field of fifty?
> >
> > Assuming that you bought only one ticket, your chances are
> > closer to 16 million to one.
> >
> Which, incidentally, is exactly as probable as if a UFO would crash into
> the Eiffel Tower with Elvis as pilot (some brit' said)
The version that I heard was a british bookmaker saying that the only bet
that pays 14 million to one was that an UFO with Elvis on board crashes on
the Loch Ness monster.
My favorite comparison is: In Britain, if you buy twentyeight tickets each
week, your chances of winning in the national lottery and your chances of
being killed in a traffic accident are about the same.
Subject: Re: Ugly Mathematics?
From: nahay@pluto.njcc.com (John Nahay)
Date: 3 Jan 1997 10:39:50 GMT
U Lange (lange@gpu4.srv.ualberta.ca) wrote:
: : * How much agreement is there among mathematicians about beauty
: : and elegance of theorems and proofs? Where does this agreement
: : come from?
Zero. The "beauty" or "elegance" of a proof is strictly in the mind
of the person viewing the proof. All proofs can, subject to the conditions
of a sufficient quantity of time and paper, be reduced to formal, logical,
axiomatic statements. Hence, different proofs of the same theorem reduce
to the same set of symbols for a given formal system. Therefore, it is
only in a person's mind about "elegance". There is certainly no outside
objective measure of elegance and beauty.
Subject: ** structure of reality **
From: gary.forbat@hlos.com.au (Gary Forbat)
Date: 3 Jan 97 21:03:35
31-12-96
revised 1-1-97
Notes on the structure of reality - article 3
(first draft)
by Gary Forbat
Copyright (c) G. Forbat 1996
It may now be convenient to extend and qualify some of the main
concepts derived from the theory. In the previous essays I described
a process of material formation which provides the basis for the
observed material reality. The process operates through a building
procedure which involves a relationship between the physical
magnitudes of structures, that is, the volume they occupy, and the
rapidity of their internal cycles. Moreover, the process is universal,
ranging over an infinity of scale tranformations from the most
miniscule sizes to the most gigantic imaginable, in fact infinite in
both directions.
But it is not a single dimensional process involving only scale. What
is peculiar about the sequence is that the smaller structures of
the micro world are highly dynamic due to an extremely rapid internal
cycle operating to hold it together, and the smaller the structure,
the more dynamic it is. Dynamics refers to the rapidity of the
cyclical pulse. As particles break down to the cyclical funtion of a
number of smaller components, those components will have a
significantly more rapid internal cyclical rate than those of the
larger structure they contribute to forming. The atomic structure,
for instance, comes into being due to the cyclical function of the
electron in relation to the nucleus. The composition of the electron
has not yet been penetrated, but the possibilities are few. Either it
is composed of a very large number of tiny parts, or maybe fewer but
of a much higher dynamicity. The nucleus, on the other hand, is known
to break down to combinations of smaller, but much more dynamic parts
known as 'quarks'. Quarks themselves must reduce to even smaller
components, with cyclical rates of increasingly more rapidity. The
many qualities of quarks testify to a variance of configurations.
The quantum proportions testify to this very nature. With the
process of reduction infinite, so with it is the increase in
dynamicity.
We are fortunate enough to be able to observe two vastly different
aspect of the material process. The micro scales of phenomena present
an integrated view of average behaviour over many billions of cycles.
Imagine how the solar system would look if billions of planetary
cycles were pressed into a single second. Theoretically at least, it
would be possible to simulate the effect by taking a long term video
of the solar system in motion over many billions of years, and then
replaying the tape over a matter of seconds. Undoubtedly we could
make computer image simulations of it much more easily.
Then there is the almost static view of the process presented
by the structures of the large scale in their 'real time' cyclical
movements. Our viewpoint of stellar formations is fashioned from the
workings of the atomic structure, and compared to the speed and
capacity of the functioning of our instruments and sensing apparatus,
the stellar structures are both extremely large and so slowly evolving
as to be almost static. But now, let's venture to reconstruct in its
broadest principles the consequences of this infinite sequence of
structuring, not only to determine the status of our own viewpoint
within it, but to attempt to discover general principles that may be
directly affecting us and we are not yet aware of. Firstly, going up
or down in scale, the specific attributes of structure types that
occur depend on the interactive possibilities afforded on each
particular scale. Solar systems of one type or another, whether
binary or planetary are the almost exclusive forms that may be found
at the scale of the direct interaction between the most massive
atomic conglomerations. At this scale of consideration the universe
can be seen to be interspersed with stellar and planetary matter in
mutual interaction as solar systems. But we know that solar systems,
in turn, almost exclusively congregate in the larger massive
formations of galaxies, occuring in a small number of types. Galaxies
themseves form clusters with unique characteristics types of their own.
On the galactic scale of consideration the universe can be seen as
interspersed almost exclusively by galactic formations. Certainly they
are the only long term stable forms to be found at this scale.
In fact we can apply this principle at any level of magnitude. Thus
the universe is interspersed by atoms at the atomic scale of
consideration but with planetary/stellar matter on a larger scale.
So then, as the process builds to infinity, with each structure type
occuring in forms and attributes appropriate to interaction and
formation possibilities at that scale. Each transformation produces
unique structure types, and there is certainly no likelyhood of the
same structure type occuring at different levels either in the micro
and macro scales.
Both the reduction and its reverse process of expansion runs to infinity,
with the roots of each or any structure traceable in infinite steps
toward smaller scales. But this does not work in the reverse toward the
macro. The reason is that not all structures continue to build outward
forever. Large sections of it terminate at a certain level, as in the
case of the structures that intersperse in our seemingly empty spatial
regions. My findings are that these regions are far from empty.
The entire spatiality in fact contains a fine invisible mist of matter,
structured at its highest level to an interactive fabric to form
a micro infrastructure which sets the framework for the workings of
our atomic based matterial environment. But only those elements
which participate in further building processes to form the atomic
base can get through to build outward to form structures on larger
scales. The rest, indeed a very large portion of micro material,
is lost to further structuring. In this infinite chain of
expansions it should be expected that terminal stages are reached
from time to time. Nevertheless, what remains after each of these
mass terminations is still adequete to reconstruct other equally
thickly populated levels of structures on much larger scales.
So what is the status of our material system amid this infinity of
transformation levels ? On the micro end we observe the process through
a very high integration, but on the macro end it tends toward static.
With the two directions reflecting merely different aspects of a
single process, our observational access results from the circumstances
of our evolution as sensing beings and our relation to the material
interaction that brought it about. We are a direct product of our
micro infrastructure and the atomic base. The question remains
whether ours is the only material environment possible or whether
there may be others ? Perhaps other configurational circumstances can
exist among an infinity of types which produces alternative material
bases.
We need firstly to examine the general circumstances which must be
present for a material environment. Obviously the most evident
is the versatility of our atomic structure. It is extremely stable
and durabile with, stability, regularity, as well as variability in
chemical combination. It is truly like a wonder particle which goes on
to create a tremendously varied and interactive world of material
activity. Surely it would be fairly rare to find a scale level of
structuring where such a useful type of particle is found.
Nevertheless it stands to reason that in a infinite chain of
transformations other similarly efficient structure types are bound
to occur. some may indeed be even more flexible than the atom, or
perhaps somewhat less so, but still able to generate a causal
evolution in its conglomerate forms to create an alternative material
environment rivalling ours. Of course on the micro scales a funtional
world would evolve extremely rapidly compared to ours, and on the macro
scales the events would take on gigantic proportions, evolving very
slowly by our way of looking at it.
G. Forbat
to be continued in the next article
Subject: Re: EXTRAORDINARY PI
From: nahay@pluto.njcc.com (John Nahay)
Date: 3 Jan 1997 10:52:33 GMT
: My view is that math is largely independant from reality,
Number one: Math IS reality. It really DOES exist. Of course, I
am allowing ABTRACTIONS to exist. It exists regardless of whether
anyone is there to think it. I assume that this is what you mean by
"independent from reality".
Number two: ANY other subject (that might qualify as a major at a
university, for instance) ALSO using abstractions to describe it.
It's called LANGUAGE.
Number three: Math is, by far, infinitely better suited at describing
ALL of reality, not just part of it, than all the subjects referred to in
"Number two".
: it is best applied to the abstractions and mental models we
: made of real things.
That's because that's the only thing math CAN be applied to: the
abstractions and mental models we make of things. Again, EVERY
human subject, with its corresponding language and terminology,
can only be applied to an abstraction in our heads.
This is good because even the simplest things
: from reality are so complex that we wouldn't be able to cope with
: them.
Yes. Just as doctors and lawyers and judges make simplifying models
in their mind to achieve an end such as reaching a decision.
Reality is by nature three dimensional, so I don't it is
: possible to build a 2 dimensional figure in reality, the lines you
: draw will be made of atoms which are three dimensional.
: I don't think it is possible to build even ideal polyhedra.
Of course not. NO one can create EXACTLY what is in their mind.
The greatest sculptors and artists cannot do it. What they create
is their very best approximation to what's in their minds. "Best"
means "optimal subject to the condition that they wish to reach a
decision in a finite amount of time" (and effort).
Subject: Re: Ugly Mathematics?
From: hrubin@b.stat.purdue.edu (Herman Rubin)
Date: 3 Jan 1997 07:14:59 -0500
In article <5ainlm$j3m@earth.njcc.com>,
John Nahay wrote:
>U Lange (lange@gpu4.srv.ualberta.ca) wrote:
>: : * How much agreement is there among mathematicians about beauty
>: : and elegance of theorems and proofs? Where does this agreement
>: : come from?
>Zero. The "beauty" or "elegance" of a proof is strictly in the mind
>of the person viewing the proof. All proofs can, subject to the conditions
>of a sufficient quantity of time and paper, be reduced to formal, logical,
>axiomatic statements. Hence, different proofs of the same theorem reduce
>to the same set of symbols for a given formal system. Therefore, it is
>only in a person's mind about "elegance". There is certainly no outside
>objective measure of elegance and beauty.
There are different ideas of beauty and elegance. An elegant proof may
give little understanding of the theorem; witness the proof of the
Fundamental Theorem of Algebra using Liouville's Theorem. It is
extremely short, but a powerful deus from a powerful machina has
been invoked.
A good proof should provide insight. The proof using induction on
the highest power of 2 dividing the degree of the equation requires
a lot of machinery as well, and may very well be the most insightful
one I know. But if we taught the structure of the real numbers, and
the ideas of continuity needed even to understand infinite decimals,
we could just show that the minimum absolute value is attained and
it being non-zero leads to a contradiction.
In probability, I consider elegant proofs using transforms of any kind
as relatively unenlightening. This does not mean that I do not use them;
I consider these to be very important for obtaining numerical answers,
and it is often the case that proofs which give more insight are not
readily available.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558
Subject: Re: Ugly Mathematics?
From: Vincent Johns
Date: Fri, 03 Jan 1997 06:01:39 -0600
John Nahay wrote:
>
> U Lange (lange@gpu4.srv.ualberta.ca) wrote:
> : : * How much agreement is there among mathematicians about beauty
> : : and elegance of theorems and proofs? Where does this agreement
> : : come from?
>
> Zero. The "beauty" or "elegance" of a proof is strictly in the mind
> of the person viewing the proof. All proofs can, subject to the conditions
> of a sufficient quantity of time and paper, be reduced to formal, logical,
> axiomatic statements. Hence, different proofs of the same theorem reduce
> to the same set of symbols for a given formal system. Therefore, it is
> only in a person's mind about "elegance". There is certainly no outside
> objective measure of elegance and beauty.
I do not completely agree. There are criteria that are generally
accepted
for judging a proof "elegant". I can't formulate all of them, but you
are
already aware of some of them, such as conciseness and clarity. (I
am excluding correctness here, since otherwise you have a "poof", not
a proof.) Conciseness can be measured mechanically, by a computer --
all you need to do to compare two proofs on that dimension is to count
the symbols employed in expressing it. The lower number wins.
Although the relative importance of such criteria is a matter of
individual taste, their existence is less so (i.e., many of
these are shared values, and for good reason).
I would be interested in seeing the canonical "same set of symbols
for a given formal system" for, for example, the Pythagorean Theorem.
Hundreds of proofs have been published for this theorem (as I
understand it; many of them I have not read); they are all
equivalent, I suppose, in the sense that they are valid arguments,
but is it really true that each of them can be expressed in exactly
the same set of symbols? What do the diagrams look like, as expressed
in this common set of symbols?
Saying that there is "certainly" no outside objective measure of
elegance and beauty piques my curiosity; I would be interested in a
proof of this assertion. I understand that there are subjective parts
of a determination of beauty, but there are also aspects that are
widely recognized and shared. Editing a fashion magazine, IMHO, is
not a totally random process (beauty of the photographs to be published
must be considered), and neither is editing a mathematics journal.
I think I would say to U Lange that there is definitely some
agreement (not necessarily total) about which proofs are most
pleasing, and that this agreement originates in personal feelings
about how easy the proof is to understand, how quickly it can be
read, etc. It then migrates (via personal discussions or published
papers) to a consensus within the interested community, as people
who have seen the proof and are not shy about revealing their
emotional reactions to it share their feelings with each other.
For example, someone might say, "Did you see John Doe's new proof
of Exwyezee's Conjecture? It looks correct, but I think he
could have used the ___ Theorem to shorten his proof of Lemma 2."
BTW, the same sorts of criteria that I mentioned as applying to
people's emotional responses to proofs apply as well to computer
algorithms. Of two correct ones, one might be far easier to
understand, shorter, etc., than the other one, and (assuming that
it has been demonstrated that both are correct) it should not take
an expert to determine which is better written.
--
-- Vincent Johns
Please feel free to quote anything I say here.
Subject: Re: Fourier transform of non-linearities...
From: Gary Hampson
Date: Fri, 3 Jan 1997 13:08:00 +0000
In article <32cc24f6.7396859@news>, Greg Ratzel
writes
>On Thu, 2 Jan 1997 16:07:04 +0000, Gary Hampson
> wrote:
>>1) The function you describe as a time/space domain operation is simply
>>some weighting function of f(t). Similar to say Hamming or Hanning
>>window functions. On that basis its frequency domain equivalent is a
>>convolution with its fourier transform.
>
>I disagree. g(sigma) is more complicated than a window function.
>Window functions are multiplied by the time-domain function, so the
>corresponding frequency domain operation is convolution.
On more than 3 seconds reflection I agree, as they say "engage brain
before opening mouth", gone off half cocked.... and all that stuff.
--
Gary Hampson
Subject: Re: Probability (was re: math jokes)
From: Gary Hampson
Date: Fri, 3 Jan 1997 13:32:30 +0000
In article ,
Nikolaj Toft Hansen <31nik@but.auc.dk> writes
>On Sat, 28 Dec 1996, Robert Lewis wrote:
>
>> Lenny Schafer wrote:
>> >
>> > Your chance of winning the typical big state lottery: 5 million to one.
>> > --
>>
>>
>> Aren't most lotteries (in big and small states) a matter of
>> picking six correct numbers out of a field of fifty?
>>
>> Assuming that you bought only one ticket, your chances are
>> closer to 16 million to one.
>>
6 from 50 possible numbers is actually 1 in 13,983,816 which is as
stated earlier, slightly less than 1 in 14 million. Where does your 16
million figure come from ? Or could it be that you are more of a
pessimist than I .... ;)
--
Gary Hampson
Subject: Re: Need to handle Big Matrix (800x800) to use optimization algorithms
From: Gary Hampson
Date: Fri, 3 Jan 1997 13:18:35 +0000
In article <32C90B5C.64B1@public.ibercaja.es>, benigno
writes
>Hi,
> I need to handle Big matrix of around 800 x 800 to implement
> some optimization algorithms, I would use C++ libraries if
> possible to use on BorlandC++ 4.5, but if there is any shareware
Unless the matrix has some structure which allows many short cuts and
reduced storage (eg Toeplitz), then just get on with coding it. 800*800
is not that big (unless of course in the optimisation you need to
evaluate A.x many times, or you have some time critical conditions.
--
Gary Hampson
Subject: Re: Qu: pdf of a/b ??
From: Gary Hampson
Date: Fri, 3 Jan 1997 13:46:32 +0000
In article , Christopher Gordon
writes
>Hi,
>
>Is it possible to analytically express the probability distribution for
>
>c = a / b
>
>where a and b are univariate independent normally distributed random variables.
>
>I worked out the case when they both have 0 means, then c is Cauchy
>distributed.
>However when b has a zero mean and a has a nonzero mean, I end up with
>having to evaluate integrals of the form
>
>\int_0^\infinity exp(-A x^2 + B x) dx
>
>with A > 0.
>
Dear Chris,
I had the same problem. I found 2 ways around it.
1) Get help with the stinking integrals (the first time "help" enjoys
the novelty, if i>1 then "help" mysteriously disappears)
2) I generated a few thousand samples of a(i)/b(i) with a and b
appropriately distributed and then plotted the histogram of c and
assumed it was the populations pdf.
--
Gary Hampson
Subject: Numbered Word Forms To Algebra Equations.
From: davk@netcom.com (David Kaufman)
Date: Fri, 3 Jan 1997 08:51:47 GMT
Note: ^L indicates a line break on the line above.
The line breaks divide this post into 4 pages below
The first 2 pages have been changed somewhat from
before. Pages 3 and 4 show the numbered words and
their algebra for 4 variables related to car travel.
------------------------------------------------------------
The Numbered Word Solution Form And Its Algebra
Solves Constant Rate Problems Effectively.
Introduction:
------------
Numbered words--in fractions--is a powerful way to
examine the numerous physical properties of the world that
form constant rates and to solve related problems clearly,
easily and concisely.
Unfortunately, many k-12 math texts have numerous
unrelated forms to solve various constant rate problems.
By solving one simple type of problem in many
disconnected ways not only creates unnecessary hardships for
students and teachers, but makes it unlikely to connect
these key ideas about constant rates to geometry, algebra
and physical science.
A Brief Description Of Numbered Words:
-------------------------------------
A numbered word is usually a number next to the left of
a word (or abbreviation). The number tells how many, while
the word reveals what items are under consideration. For
example, "5 pounds" (5 lb) or "4 fruit" are numbered words.
A pound is a standard unit of measurement that's
counted. Each unit is the same amount of weight. A fruit is
a unit that's counted also, but it is not a standard and
each piece could be different or identical. In terms of
fractions, the pound is uniformly subdivided while a fruit
could be cut into any sized pieces. These pieces are counted
also to make fractions.
Show 2 times 3 with numbered words and related pictures:
-------------------------------------------------------
Given: 2 ones 1
------- 1
1 group
3 groups 1 1 1
1 1 1
Find: Multiply 2 X 3 using numbered words in Solution Form.
Show the Found Rate.
2 ones (3 groups)
------- = 6 ones
1 group
Found rate: 6 ones / 3 groups
The 3 Model Word Forms Below Explain Numbers In Division:
---------------------------------------------------------
Name | Arithmetic | Numbered Word Forms
-----------------|-------------|----------------------------
Solution Form | 3 (8) | 3 ones 8 groups
| --- = ? | -------- = 12 ones
(For calculation)| 2 | 2 groups
_________________|_____________|____________________________
Proportion Form | 3 ? | 3 ones 12 ones
| --- = --- | -------- = --------
| 2 8 | 2 groups 8 groups
(For meaning by | |----------------------------
naming rates) | | Given Rate = Found Rate
_________________|_____________|____________________________
Division Form | 12 ? | 12 ones 1.5 ones
| --- = --- | -------- = ----------
(For unit rate) | 8 1 | 8 groups 1 group
-----------------|-------------|----------------------------
Note: | 1.5 | 1.5 ones
| ---- = 1.5 | -------- = 1.5 ones
| 1 | group
-----------------|-------------|----------------------------
Note: The Found Rate shown below usually need not be written
but should be clearly understood.
Given: 3 dollars ($) per 2 gallons (gal)
Find: 1. How many dollars for 7 gallons?
2. How many gallons for $5?
3. Find the 2 possible unit rates.
Answer Found Rate
|---------|
Solution: 1. $ 3 (7 gal) |--------| | $ 10.50 |
----- = | $10.50 | | ------- |
2 gal |--------| | 7 gal |
|---------|
|---------|
2. 2 gal ($ 5) |---------| | 3.33 gal|
----- = | 3.33 gal| | ------- |
$ 3 |---------| | $5 |
|---------|
|-------------|
3a. $ 3 / 2 gal = | $ 1.50 / gal|
|-------------|
The unit rate above is called,
"the Unit Price".
|---------------|
3b. 2 gal / $ 3 = |.667 gal / $ 1 |
|---------------|
The unit rate above reads as follows:
".667 pounds per dollar".
Note that the numbered word solution form above not
only easily solves constant rate problems clearly and
concisely, but it also reveals the problem solved.
The numbered word solution form is a valuable tool for
thinking efficiently about relationships between physical
properties, presenting solutions to others effectively, and
understanding and creating equations as shown below.
Given: Speed: S = d / t = [S] miles / 1 hour = [S] mi/hr
(Note: Speed is a Unit Rate.)
Variables: Distance (d) Time (t)
Values: 205 mi 3.25 hr
Find: Use the numbered word solution form first, then write
below each numbered word part, the proper variable.
1. Find the distance traveled in 4.5 hours.
2. Find the time to go 365 miles.
3. Then find the average speed in miles per hour.
Solution: 1. 205 mi (4.5 hours) [d] mi
------- = = d
3.25 hr
S t = d
2. 3.25 hr (365 mi)
------- = [t] hr = t
205 mi
(1/S) d = t or d/S = t
3. 205 mi
------ = [S] mi/hr = S
3.25 hr
d/t = S
----------------------------------------------------------
Given: Gasoline Unit Price: U = C / V = [U] $/gal
Variables: | Cost (C) | Volume (V)
Values: | $15.93 | 7.85 gal
Find: 1. Find cost to fill a 12 gal tank. Use solution form.
2. Find the unit price in dollars per gallon.
Solution: Solve Below.
1.
2.
------------------------------------------------------------
Given: Car efficiency: E = d / V
Variables: | Volume (V) | Distance (d)
Values: | 14.95 gal | 205 mi
Find: Find the efficiency in miles per gallon.
Solution:
------------------------------------------------------------
Constant rates can be multiplied together to create new
constant rates. Note below how the same word in top of the
fractions (multiplied together) cancels the same word in
bottom of the fractions. This is true for the same variable
(or same letter) too.
Given: The answers to the 3 above unit rates follow:
S = d / t U = C / V E = d / V
S = 63.07 mi/hr U = $ 2.029 /gal E = 13.71 mi/gal
Find: 1. Volume per time = V/t = F = [F] gal/hr = Flow
2. Cost per distance = C/d = O = [O] $/mi
3. Cost per time = C/t = X = [X] $/hr
1a. 63.07 mi gal [F] gal
-------- --------- = -------
hr 13.71 mi hr
1b. d V V
--- --- = --- = F
t d t
S / E = F
Challenge: Solve 2 and 3 like 1 above.
Given:
S = d / t U = C / V E = d / V
S = 63.07 mi/hr U = $ 2.029 /gal E = 13.71 mi/gal
----------------------------------------------------------
Given: 2 boys (B) per 3 girls (G)
2 B and 3 G per group (R),
5 people (P) per group
group structure G-B-G-B-G or B-B-G-G-G
2 aluminum atoms (Al) per 3 oxygen atoms (O)
2 Al and 3 O atoms per molecule (Al203)
Note: The numbers 2 and 3 in Al2O3 should be subscripts.
5 atoms per molecule
Find: Using the numbered word solution form, find:
1. How many girls if 6 boys present?
2. How many people if 6 boys present?
3. How many groups if 6 boys present?
4. Check answers 1 to 3 by drawing the number
of groups present in the form B-B-G-G-G .
Then Consider 5 next:
5. If 6 boys represent 6 Al atoms then how many O
atoms, total atoms and molecules are present?
6. Find all 12 different unit rates among B, G, R, & P.
------------------------------------------------------------
If you are interested in promoting constant rate
solutions in the K-12 school system (or elsewhere) using
numbered words as outlined above (and more to come), please
e-mail me your interests on this matter. Thanks.
-----------------------------------------------------------
C by David Kaufman, Jan. 3, 1997
Remember: Appreciate Each Moment's Opportunities To
BE Good, Do Good, Be One, And Go Jolly.
--
davk@netcom.com
Subject: Re: 4 lines / Wlod-1996
From: everest@netcom.com (Wlodzimierz Holsztynski)
Date: Fri, 3 Jan 1997 10:43:01 GMT
In article <32CCB44D.12D7@scf.usc.edu>,
Ryan Cormney wrote:
>Wlodzimierz Holsztynski wrote:
>>
>> ;
>>
>>
>>
>> Polynomials over finite Galois field
>> spread so evenly across their finite affine space
>> I wish for a network of friends
>> to count on
>>
>> H.New Mexico
>> 1996-03-05/06
>One thing I hate about certain poets
Hate? Take it easy.
>is that they hide their meaning in
>unfamilar terms(or algebra in this case).
You show lack of understanding (of poetry I mean, not mathematics).
> Personally, I like the so
>called generic style, sure more simplistic, yet just as powerful.
>Consider it postmodernism. I wrote that poem(Inside) as a direct result
>of reading Emily Dickenson(Miss Original). I appreciate your response to
>my poem, but I would like to know exactly what you find generic(the
>content, the style, rhyme).
Where do you have your poem?
OK, I'll provide a copy for rap:
Ryan Cormney wrote:
>
> Inside
>
> Lying next to heart of heat
> Waiting for the moment we meet
> Side by Side the wall divides
> A pulse of blood...we collide
> Next to you my muscle stiffens
> Inside you my pulse quickens
> I look to you for hope and pride
> Inside I shall reside...
> For Inside you I can confide. RBC 1995
>
> This is one of the many poems I wrote for my girlfriend.
For all we know from your poem,
your girlfriend can be a sheep.
> I admit I have a lot to learn,
I agree.
> but I gain nothing from your previous comment.
Poetry is not for everybody.
> I was wondering what exactly are you trying to say in this above blurb.
>?you want friends that are reliable as a function you plug numbers into?
>never met anyone quiet like that. Nothing is consant in this world; all
>exceptions are only human concoctions(numbers). Pretty generic, right!
>RBC
Wrong.
Don't worry about what *I* am trying to say.
Who cares? Just read.
And don't worry about math. My blurb is not
any algebra texbook. My blurb is a poem.
I might say more if there is any interest...
but it's no fun. Enough to say that
there is more in my poem
than meets the eye even of a reader much
more trained and sensitive to poetry than you.
Wlod
