Subject: Continued Fractions
From: hack@swamp-thing.watson.ibm.com (Michel Hack)
Date: 14 Jan 1997 19:41:46 GMT
Patrick T. Wahl (ptwahl@aol.com) reminded me that there's more in Knuth
Vol 2 than I thought: the exercises.
The Hurwitz doubling formula was intriguing, but I couldn't squeeze out
the info I needed. At first blush, doubling the value halves the early
partial quotients and doubles others -- but trying to follow through on
the recursion succumbs to combinatorial explosion based on odd/even of
the quotients. When I checked my vast collection of CF expansions for
powers of 2 and 5, I came across the following sequence of doublings:
pwr of 5 pwr of 2 #terms max q
16026 37209 21807 178546
16026 37210 21700 1074639800
16026 37211 21821 43355
So doubling 2^37209/5^16026 multiplies some partial quotient by more
than 6000; doubling again causes the largest partial quotient to shrink
by a factor of nearly 25000.
I then checked the Yao&Knuth; reference on the sum of partial quotients,
"A.C. Yao and D.E.Knuth, Proc. Nat. Acad. Sci. 72 (1975), 4720-4722".
Their formula gives an unusably-large bound for CF(m/n) where m < n:
sum < 1.4 * n (log n)^2 + O(n log n log log n)
When n has 10000 digits, the bound on the sum exceeds n by a factor
of over 20, yet I'm looking for a bound that is *much* smaller. The
Yao&Knuth; bound is for all m < n, which includes 1/n, so the bound
is certainly going to exceed n.
For the particular case of 2^a/5^b where the ratio is nearly 1, the
largest partial quotients I came across had 10 digits, even though
the original numerator and denominator had up to 23000 digits.
The behaviour of CF(2^a/5^b) reminds me of the generalised Catalan
conjecture and Tijdeman's bounds -- I wonder if this might be related.
(I've heard of Ribenboim's book on the subject, but haven't located a
copy yet. I have Alf vanderPoorten's Notes on FLT.)
Michel.
Subject: Re: Problomatic Teacher
From: hetherwi@math.wisc.edu (Brent Hetherwick)
Date: 15 Jan 1997 03:07:59 GMT
Keith Pitcher (kpitcher@weirdness.com) wrote:
:
: My sister realized that this was a trick question, as she knew a piece
: of paper can
: not be folded that many times in half, and so far every question had
: been based in reality. She came up with this description of her answer :
Okay, so I made a few calculations on the feasibility of folding a sheet
25 times, and if my rough estimate is correct, the resulting stack would
be no less than 3 miles high, under the presumption that we can get paper
of thickness 2000 sheets/foot. I admit, such a folding is a mite
"impractical".
Anyone care to check my figures?
$$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666
hetherwi@math.wisc.edu
$$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666 $$$ 666
Subject: Re: Why can't 1/0 be defined???
From: Darrell Ryan
Date: Tue, 14 Jan 1997 20:43:48 -0600
Wilbert Dijkhof wrote:
>
> Martijn Dekker wrote:
> >
> > pausch@electra.saaf.se (Paul Schlyter) wrote:
> >
> > :In article ,
> > :David Kastrup wrote:
> > :
> > :> Which is why oo is *not* a real number. All expressions involving
> > :> real numbers are either undefined, or equal exactly one real number.
> > :
> > :You mean expressions like:
> > :
> > :sqrt(4) = +2 or -2
The notation sqrt(4) means only the principal (positive) square root.
When we mean the negative square root, we write -sqrt (4). So sqrt(4)
is one number, 2.
> > :
> > :sqrt(sqrt(16)) = +2, -2, +2i or -2i
Since the principal square root of 16 is 4 and the principal square root
of 4 is 2, this expression represents one number, 2.
> > :
> > :arctan(1) = pi/4 + n*pi/2 where n is any integer
> > :
> > :?????
The range of the arctangent function is (-pi/2,pi/2) because that is the
largest interval (centered about the origin) where the tangent function
is one-to-one, therefore the tangent function can only have an inverse
on this restricted domain. True, there are an infinite number of angles
whose tangent is 1, but the notation arctan(1) means "The angle between
-pi/2 and pi/2 whose tangent is 1." In general, (IMHO) the notation y =
arctan x implies y is between -pi/2 and pi/2.
____________________________________________________________
Darrell Ryan
e-mail dryan@edge.net
personal website http://edge.edge.net/~dryan
company website http://www.edge.net/stmc
Subject: Re: Gaussian Elimination
From: dogmat@aol.com (Dogmat)
Date: 15 Jan 1997 03:56:30 GMT
Gaussian Elimination in this case means solving 3 equations in 3 unknowns.
With these linear equations, you may not have to worry about Gauss -- you
could just write the functions out and go at it analytically. However,
Gaussian elimination is basically an approach whereby you write the 3x3
matrix of the equations, i.e., with the equations in the rows and the
variables in the columns:
Ax=b
where A is the 3x3 matrix of coefficients of the equation
x are the 3 variables to be solved for
b is a [3,1] vector of the values of the functions when solved.
[Qualitative concept]
You basically take the first row of A, normalize it so that the top left
value in A is 1, then subtract it out of all the lower rows, so that all
values in the first column are 0 (except for the top row). Then, you do
this for the second row and thrid row. What you then have is a transformed
A' where all the values in the lower left are 0. The lowest row is now
c*x3=d, so you can solve for x3. Knowing x3, you can also solve the second
row for x2 from e*x2 + f*x3=g. Then you can solve for x1 in the first row.
[Modifications]
Often,the diagonals in A may be 0, which ruins the whole thing. Also, you
need to worry about roundoff. So an added feature is "maximum row-column
pivoting", where for each forward iteration, instead of normalizing
(pivoting) on the diagonal, you find the next largest value in the matrix
and pivot on that.
Any practical matrix book should explain this (much better than I did).
Also, there is lots of code around. For a Fortran code, look at Carnahan,
Luther, Wilkes. If you need more help, let me know. I would advise not
writing your own code, and there are some subtle things to worry about in
coding. However, there is a lot of off-the-shelf code on this.
Subject: Re: Problomatic Teacher
From: Vincent Johns
Date: Tue, 14 Jan 1997 16:18:24 -0600
(posted & emailed)
Brent Hetherwick wrote:
>
> Keith Pitcher (kpitcher@weirdness.com) wrote:
> : Q) Take a square piece of paper. Fold it in half. Do it again. Repeat
25
> : times. How many sheets thick is the final folded piece of paper.
> :
> : My sister realized that this was a trick question, as she knew a piece
> : of paper can
> : not be folded that many times in half, and so far every question had
> : been based in reality. [...]
>
> What if the sheet of paper had been 25' x 25'? Or large enough to carry
> this out?
I assume you mean conceptually, as real paper could not be folded that
many times. (To see this, consider the thickness of the thinnest paper
commercially available, and consider the number of layers even assuming
that the paper were cut in half and piled up, instead of being folded,
etc.) How is such an answer related to reality?
> Why didn't she ASK the teacher if it was a "trick question"?
It sounded as if this was one of a series of questions on a test,
and students don't always have the option of asking during a test.
(Standardized achievement tests come to mind.) I don't understand
how one wrong answer would produce an "F", though, unless most of
the rest of the answers were wrong as well.
> Why didn't she ASK for clarification of the problem? Doesn't it seem
> implausible that an honest teacher would try to trick students with a
> problem anyway?
My daughter has suffered through a few ignorant or careless
teachers, so dishonesty is not the only explanation for a
poorly worded question. (Fortunately, we've encountered many
excellent teachers as well.)
> In any case, trying to must professional opinion on your
> side is a very bad idea; at best, it can only serve to falsely enhance
> your ego at the expense of good pupil-teacher relations.
*What* good pupil-teacher relations? It sounds as if the teacher has
already been consulted and is inflexible. Some teachers are like that.
If I were the student, I would already have lost most of my respect for
this teacher.
> The matter truly
> is a judgement call, and you're only asking for trouble by trying to "go
> over" the teachers head, so to speak, in appealing to a greater
> authority.
There is risk here, but this statement is asserted without proof. My
wife and I complained to the school principal about a fifth-grade
teacher's erroneous disqualification of our daughter's answer on an
arithmetic problem. The eventual result was that the teacher, the
next summer, took an algebra class as part of her
professional-enhancement
course work and, from what we heard, became a much more effective
arithmetic teacher. We derived the satisfaction that other children
would not learn false ideas from her that they would later have to
unlearn.
Unfortunately, there are poor teachers everywhere, but one can learn
even from a poor teacher. Part of being a successful student is
knowing how to guess what kind of answer the teacher wants. Often,
this involves having learned the subject matter, but sometimes
other considerations come into play. Some of the game playing that
a student must do to be successful is even useful in later life, as
when one becomes involved in office politics.
Probably the best strategy for answering the "folding" question would
have been to say something like, "Assuming that the total number of
folds is 26, the number of sheets is..." If the guess at the total
number of folds is wrong, at least the student demonstrates an
understanding of how to do the calculation, and should get partial
credit. To me, the question is ambiguous, and its meaning depends
on what is repeated and when the repetition starts. I get total numbers
of folds of 1, 25, 26, 27, 50, and 52 (this last assumes that the
original 2 folds are followed by 25 more pairs of folds).
--
-- Vincent Johns
Please feel free to quote anything I say here.
Subject: Re: QUESTION: BOTANY & DIFFERENTIAL EQUATIONS
From: harper@kauri.vuw.ac.nz (John Harper)
Date: 15 Jan 1997 03:58:37 GMT
In article ,
Jessica T. Fried wrote:
>
> I'm looking for a topic to write a short paper on for my 300 level
>Diff.Eqn. class(undergrad). I'm really interested in plants, and would
>love to know if there are any SIMPLE applications of Diff.Eqn. in botany.
>If anyone out there has any ideas please drop me a line.
If the interest in plants extends to fungi hunt out Buller's "Researches
on Fungi" (an old but famous book, several volumes) and see his
discussion of the "sporabola" = path followed by a fungus spore when
shot off at a low Reynolds number. (It's what a parabola becomes when
there's a lot of air resistance.)
C T Ingold wrote a couple of later books taking such points up.
Then there's
W.K. Silk and R.O. Erickson "Kinematics of plant growth"
J.Theoret.Biol. 76 481-501, 1979 and
P.W.Gandar "The analysis of growth and cell production in root apices"
Bot.Gaz. 141(2) 131-138, 1980, and whatever the Science Citation Index
leads you to: this isn't my field so I don't know what's been done
since. It does involve some simple DEs though.
John Harper School of Math+Comp Sci Victoria Univ Wellington New Zealand
Subject: Re: I challenge anyone to solve this one
From: jzs@europa.com (Justin)
Date: 15 Jan 1997 03:41:53 GMT
In article <5bgu7h$1m2a@inst.augie.edu>, (Augie) wrote:
* "Nathan Crowder" wrote:
*
* >How many real solutions does the equation sin(x)=(x/100) have?
X‰15.5, 6.34, 3.11, 0, -3.11, -6.34, -15.5
--
JZS 3=)
Subject: Re: I challenge anyone to solve this one
From: jzs@europa.com (Justin)
Date: 15 Jan 1997 03:42:37 GMT
In article <5bgu7h$1m2a@inst.augie.edu>, (Augie) wrote:
* "Nathan Crowder" wrote:
*
* >How many real solutions does the equation sin(x)=(x/100) have?
X‰15.5, 6.34, 3.11, 0, -3.11, -6.34, -15.5
--
JZS 3=)
Subject: Re: Probability and Wheels: Connections and Closing the Gap
From: rhoads@sceloporus.rutgers.edu (Glenn Rhoads)
Date: 14 Jan 1997 23:24:01 -0500
bm373592@muenchen.org (Uenal Mutlu) writes:
>LOTSIM - Simulation-Program for all pick-X type Lottery Games
>[text deleted]
> Draw numbers are generated by the standard RNG, ie. the rand()
> function. Seed (srand(time)) is done once at pgmstart.
Is this in C?
You should note that the pseudo-random number generator in most
implementations of C is flawed. C returns an unsigned integer but
the rightmost bits of the number returned are NOT RANDOM! Suppose
C returns an integer in the range from 1 through 4,000,000,000 and
you want to convert this to a number in the range from 1 through 49.
Typically, programmers use the formula x = (n mod 49) + 1 to convert
n, the number returned by pseudo-random number generator, to the
number x of the desired range. This is a bad thing to do in C.
This formula emphasizes the bits on the right end, namely those bits
that are not random. To use the C's random number generator properly,
you have to first get rid of the rightmost bits. (e.g. x >>= 8 will
get rid of the 8 rightmost bits) If you want to do some serious
simulations with lots of samples, you really shouldn't use any
language's built-in generator and instead use a stronger generator.
-- Glenn Rhoads
Subject: Re: Proof of infinite prime #'s
From: T.Moore@massey.ac.nz (Terry Moore)
Date: 15 Jan 1997 02:36:50 GMT
In article <01bc026d$054f38a0$267e6bcf@rauhala.tyenet.com>, "Daryl Rauhala"
wrote:
>
>
> I am in need of a proof by contradiction that the set of prime numbers is
> infinite.
>
> Starts by assuming that the set of primes is finite and the largest prime
> is P.
> Let x = P! and let y = x + 1. From here is where I can't get things
> straight.
> I think we want to find the lowest number that will divide x and also
> show that because of our assumprion it also divides x which can't be
> possible since x and y are consecuctive integers. This is our
> contradiction that show our original assumption is wrong and the set in
> infinite.
Strangely enough there has been a long, tedious debate on this forum
on just this. One notorious author came up with a new improvement
to the usual approach. But he wrongly insists that the usual argument
is incorrect while others wrongly insist that his argument is incorrect.
Your approach of finding y is valid. (An alternative is to use
z = (product of primes <= P) + 1). The next step is to note that y cannot
be
divided by any number less than or equal to P. As you assumed the only
prime numbers are less than or equal to P, this is divisible by no primes
so it must be prime. But it's a prime larger than P - contradiciton.
If your teacher sides with the cranks who didn't like this argument
you may be able to keep the US law courts tied up for a few
millenia if you choose to sue.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
Subject: Re: Why do Black Holes Form at all?
From: odessey2@ix.netcom.com (Allen Meisner)
Date: 15 Jan 1997 05:18:40 GMT
In <5bfneu$3v1@nntp1.u.washington.edu> hillman@math.washington.edu
(Christopher Hillman) writes:
>
>In article <32DB0B90.3A6F@quadrant.net>,
>"Bruce C. Fielder" writes:
>
>|> If the gravitation of a black hole is such that anything falling
into a
>|> black hole will have its "time" slowed the closer it comes to the
event
>|> horizon, how does the thing form in the first place? Surely as the
>|> original mass contracts, it should slow (from our point of view)
until
>|> the original mass remains "waiting" (sorry about all the quotation
>|> marks) at the event horizon?
>|>
>|> As far as I can see, the same should hold true with the mass inside
the
>|> (soon to be) event horizon; the acceleration and gravity increases
and
>|> slows the time to infinity. So how does the thing ever form in our
>|> universe?
>
>The "picture" of a black hole you probably have in mind (really a sort
of
>map of a particular closed space-time, in the same sense that a
Mercator
>projection is a particular map of a certain curved surface) are the
>Scharwzchild coordinates, in which the "event" horizon appears as
>a cylindrical coordinate singularity. Geometrically, this cylinder
>(in the map) is really a circle (i.e. a two-sphere). There are other
>coordinate systems in which this coordinate singularity is removed.
>The best is a conformal map (preserving small shapes, like the
Mercator
>projection does for the surface of the earth) called the
Kruskal-Szekeres
>coordinates.
>
>It is true that an exterior observer (usually assumed to be stationary
>wrt to the black hole) observes nothing of the history of a particle
>after it passes through the event horizon. Moreover, as a particle
>approaches the horizon, signals from it back to more distant observers
>are extremely redshifted and also fade in intensity (exponentially in
the
>time of a distant observer, in fact, contrary to the impression left
>by the Schwarzchild coordinates that a distant observer will observe
>particles "hanging" suspended near the event horizon.
>
>Nonetheless, a particle falling into the BH (or the matter of the star
>itself as the hole is being formed) experiences nothing strange as it
passes
>through the event horizon. The event horizon is an artificial mental
construction
>(like the international date line) which has a GLOBAL significance
(this is
>the point of no return) but no LOCAL (physical) meaning. Indeed, by
>a remarkable coincidence, it turns out that you can obtain the correct
>experience according to gtr by a simple Newtonian analysis.
Specifically:
>
>Consider two particles falling straight into a gravitational source of
mass M.
>Suppose one is at radius R and the other at radius R+L (L small wrt
R).
>Then they accelerate apart relative to one another as
>
> -GM/R^2 + GM/(R+L)^2 ~ 2GML/R^3
>
>(where we expand in a power series in L, neglecting all but the first
order term).
>If we have two particles both at radius R and seperated tangentially
by L,
>they accelerate toward one another as
>
> -GM/R^2 (L/R) = -GML/R^3
>
>(by similar triangles). That is, the curvature coefficients are
2GM/R^3 radially
>and -GM/R^2 tangentially. Someone falling into a black hole is
therefore
>compressed tangentially and expanded radially by the force of gravity,
this effect
>increasing smoothly as R^(-3) right through the event horizon and down
to
>the true singularity at R=0.
>
>It is not obvious but true that these Newtonian values are in fact
correct
>according to standard gtr for a non-rotating non-charged black hole.
>I have modeled this discussion on the first few pages of the beautiful
>book Gravitation, by Misner, Thorne, and Wheeler, Freeman 1970, which
>also contains a thorough discussion of many coordinate systems for
>black holes including the Kruskal-Szekeres coordinates, and various
>techniques for calculating the curvatures and verifying that the
values
>given here are correct.
>
>Another way to visualize the situation is to consider a sphere of
particles
>"at infinity". They begin to fall slowly into the hole, carving a
three
>dimensional surface out in the four dimensional space-time as they do
so.
>You can readily determine the intrinsic geometry of this section using
>methods dicussed in MTW and then it turns out you can embedd this
"world-surface"
>as a sort of half-football in R^4. Again, the event horizon is simply
one of many
>spherical "latitude surfaces" on this football, and is not
distinguished in any
>way from its brethern. Incidently, such "world surfaces" form an
entire family
>of surfaces carving up the space time. There is a family of
"orthogonal" surfaces
>defined in the same way that potential curves determine streamlines in
the
>conformal mapping method of solving hydrodynamical flow problems.
These
>orthogonal surfaces are flat R^3 planes, flat right down to the
singularity!
>That is, the Scharzchild universe is a sort of four dimensonal "ruled
surface".
>A more familiar example of a (two dimensional) ruled surface is
obtained by
>taking a twisting curve in R^3 and considering the surface carved out
by its
>tangents. Typically this surface has a sharp cusp along the curve
itself;
>the true singularity as the center of a black hole arises
geometrically
>in an analogous fashion.
>
>Hope this helps!
>
>Chris Hillman
Mr. Hillman, you explained in another post that mass and kinetic
energy both contribute to the mass-energy of a particle. A body is
therefore its total mass-energy. You stated in another post that the
velocity space of a body is a Lobachevsky geometry. Mr. Archimedes
Plutonium has stated that the Lobachevsky geometry does not have a zero
reference point. Since a body with constant velocity has a non-zero
slope in the Loba geometry, it therefore has a potential energy? The
start metric of the potential energy can then be calculated because the
Loba geometry does not have a zero reference point? Could the relation
between the potential energy and inertial energy be the same as the
relation between the electric and magnetic fields? The potential field
induces an inertial field and the inertial field induces a potential
field: potential flux thereby inducing inertial flux? Since a body is
nothing but the mass-energy given by the sum of mass and kinetic
energy, then the motion of a macroscopic body is therefore the
potential-inertial propagation of the mass?
Regards,
Edward Meisner
Subject: Re: Vietmath War: If US had been parliamentary, no Vietnam war?
From: coriolan@ix.netcom.com(Caius Marcius)
Date: 15 Jan 1997 05:45:28 GMT
In <5bc184$q32$1@dartvax.dartmouth.edu>
Archimedes.Plutonium@dartmouth.edu (Archimedes Plutonium) writes:
>
> If the US had been a parliamentary form of government where all
>politicians are elected and not these cabinets that linger from one
>administration to another and really run the government. Then,
>hypothetically, is it highly likely that the Vietnam War would have
>never occurred? Or if it had, would not a parliamentary form of
>government gotten the US out quicker? One can argue that the US
Vietnam
>War was chiefly the result of foolish advisors to the president.
>
> Perhaps this is a great research inquiry as to see which form of
>democracy is superior-- the US or the UK parliamentary.
>
> In a parliamentary system, the likelihood of foolish advisors doing
>so much damage is minimized, I suspect.
>
> Same thing in mathematics, where math is run by the old geezers who
>control the math journals. They print and publish the pipsqueak little
>progress. And they do their utmost best to keep out anything that is
>big, new and exciting and important. In fact, they mostly publish
that
>which furthers their own self interests or
>you-rub-my-hand-I-rub-your-hand.
>
> The clowns that got the US into Vietnam are the same sort of
>intellectual clowns that control the mathematics publishing journals
>and who hate an idea such as Naturals = P-adics = Infinite
Integers.
Although the Democrats bear the ultimate responsibility for bogging the
US down in Vietnam, they did so in no small part to avoid looking
"soft" in the face of Republican criticism; Republicans were inclined
to hold a harder line.
I;m not sure this dynamic would have been any different under a
parliamentary system.
- CMC
Subject: Re: How many unlabled trees with d(v)<=4?
From: ptwahl@aol.com (PTWahl)
Date: 15 Jan 1997 06:38:50 GMT
Dear Richard,
On December 26, 1996, you had asked, "how many unlabeled trees are there
on n vertices where the valency of each vertex is ... the case v=4. ... in
more chemical terms, how many isomers of an n-alkane are there?"
I had replied earlier to comment that real-life carbon chemistry doesn't
match the mathematics of unlabeled trees too well.
Tonight in the library I encountered _Combinatorial Enumeration of Groups,
Graphs and Chemical Compounds_ by Polya and Read, Springer-Verlag (1987).
I had no time to look for your answer, but suspect that this source covers
much of what is known.
Patrick T. Wahl
( no institutional affiliation )
Subject: Re: Year 10,000?
From: ptwahl@aol.com (PTWahl)
Date: 15 Jan 1997 08:05:43 GMT
On January 8, Monte J. Zerger asked, "With all this focus on the coming
millenium, I am curious what we will call the year 10,000. ... What about
... 100,000 years?"
This adds to my January 13 posting on the topic of words for the
"deka-millenium" or whatever. I spent a little time in the library
tonight with the big dictionaries in Latin, Greek and English (O.E.D.)
I'm not going to try to write in Greek. My transliterations here map the
"x" or "chi" letter to "ch" and both "epsilon" and "eta" to "e." Please
forgive my ignorance of classic grammar; perhaps someone more educated can
get the declensions right.
Classical Greek seems to be the best source. My earlier guess at the word
is wrong. The Greek word "myrioi" for 10,000 is the source of "myrietes"
and "myrieteris," which mean "a period of 10,000 years." Similarly, there
is "chilieteris," a period of 1,000 years, which uses the "chili-" prefix
that became our "kilo-." By the way, there's a very long word for a
myriad of myriads = 10^8 in Greek. Nothing like these spellings seems to
have entered English.
Classical Latin seems to have had a wealth of "-ennium" words, including
some that I didn't suspect ( like triennium, tricennium, tricentennium for
periods of 3, 30 and 300 years respectively.) The word "millenium" is the
biggest I found. It appears that a modifier got stuck on the front if
there was more than a thousand of anything. Something like "decei
millenii" for ten millenia seems to be what they used.
Consulting the Oxford English Dictionary, I found no word for "10,000
years" that survived into English. ( Particularly NOT "myriennium" or
"myriayore": those are not in the O.E.D. ) I was surprised to find the
Greek "chili-" word above as the English word "chiliad." It means "a
group of 1000," but also "1000 years." Might "myriad" have the alternate
meaning, too? Only a scholar can say, and the O.E.D. gave no citation for
such a use.
In short, one had best look to Hindi/Sanskrit/Indo-European Ursprache. If
English ever had the word we seek, it seems to have been forgotten. But
if all else fails, my vote goes to "myriad."
Pedantically yours,
Patrick T. Wahl
( no institutional affiliation )
Subject: Re: A Puzzle regarding number arrangements
From: rjc@maths.ex.ac.uk
Date: Wed, 15 Jan 1997 02:11:09 -0600
In article <32DB9FDA.71E7@mindspring.com>,
R M Mentock wrote:
>
> rjc@maths.ex.ac.uk wrote:
>
> > I can show non-existence for n = 3, 4, 7, 8, 11, 12 etc. as follows.
>
> Very nice! but shouldn't that be "non-existence for 1,2,5,6,9,10 etc.",
> instead?
>
Errrr yeah, of course :-)
Robin
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Subject: Re: Why do Black Holes Form at all?
From: hillman@math.washington.edu (Christopher Hillman)
Date: 15 Jan 1997 08:49:13 GMT
In article <5bh4ud$oe9@geraldo.cc.utexas.edu>, mlerma@math.utexas.edu (Miguel Lerma) writes:
|> Christopher Hillman (hillman@math.washington.edu) wrote:
|> [...]
|> > Nonetheless, a particle falling into the BH (or the matter of the star
|> > itself as the hole is being formed) experiences nothing strange as it passes
|> > through the event horizon. The event horizon is an artificial mental construction
|>
|> Let me ask a follow up question. In Schwarchild's coordinates the
|> particle passes through the event horizon at time t = infinity.
|> However, Hawking has shown that a black hole cannot last until
|> t = infinity, it will evaporate first. If by the time the particle
|> enters the black hole it does not exist any more, how can it do it?
|> Someone told me that the "paradox" comes from an improper mixture of
|> classical and quantum physics,
I would agree with that, as far as it goes. Another way of putting this would be
to guess that Schwarchild coordinates are inappropriate for analyzing events near
the horizon, because of the coordinate singularity there.
|> but I would appreciate any more detailed
|> explanation about how matter can fall inside an evaporating (non rotating
|> and non charged) black hole.
Sorry, can't help you there. I have a good geometric understanding of the purely
classical theory (attained through self study of Misner-Thorne-Wheeler), but I
cannot claim to know much about quantum mechanics in general or Hawking's work
in particular. Hopefully someone who is familar with Hawking's computations will
be tempted to attempt an answer to your question.
Chris Hillman
Subject: Re: Evidence for God's Existence - TRY Math
From: "John P. Boatwright"
Date: Wed, 15 Jan 1997 00:50:58 +0000
Michael A. Stueben wrote:
>
> Not only can I give you a short easy-to-understand proof of the
> existence of God. I can do better I can prove to you that I am
> God.
>
> PROOF: Everytime I find myself talking to God, I realize that I
> am only talking to myself. Q.E.D.
Really? And every time you have sex with a woman you find that
it was only with yourself? Q.E.D.
>
> COMMENT: The existence of God is independent of the question is
> God the Christian God?
There is only one God, he's made it quite plain (the God of Abraham,
Isaac, Jacob, ect...).
>
> CONVERSATION: I believe in God. Why do you believe your belief
> is correct? I believe my belief is correct. Well then, why do
> you believe your belief of your belief is correct? I believe my
> belief of my belief is correct. well why do you believe your
> belief of your belief of your belief is correct? i believe my
> belief . . .
>
> The above conversation is nuts.
I agree, quit talking with yourself. The bible gives quite a few
predictions that have and others are about to come true. Why waste
time talking with yourself? Go to the source, read the bible. Start
with the chapter of Matthew, ask Jesus to forgive your sins and
quit worring about the trivial garbage above. Jesus's forgiveness
is FREE, you only have to ask for it, or don't. Best of luck.
Subject: Re: Numbers
From: davis_d@spcunb.spc.edu (David K. Davis)
Date: Wed, 15 Jan 1997 05:19:44 GMT
Leonard Timmons (ltimmons@mindspring.com) wrote:
: Alan "Uncle Al" Schwartz wrote:
: >
: > Leonard Timmons wrote:
: > >Is the duality between mind and matter equivalent
: > >to the duality between numbers and numerals?
: >
: > The duality between mind and matter is isomorphous to the duality between
: > fish and bicycles.
:
: Hey, I think you are making fun of me. Someday, when I start taking myself
: seriously, I'm going to be upset. ;-)
:
: In the mean time, though ...
:
: Does anyone out there believe that numbers (not numerals) actually
: exist (what ever that means) and on what basis are you making that
: claim?
:
: My second question: Does anyone out there believe that numerals
: actually exist and on what basis are you making that claim?
:
: Go ahead, make fun of me. I can take it.
I would say he's spoking fun at you. But you have to expect this if you
ask metafishical questions. At least you didn't fall for it hook line and
sinker.
If you mean to be serious, I would say first that numerals are just
notation for numbers and are therefore social constructs. Numbers,
however, have a significance beyond social convention. Pi, I believe
will be discovered by intelligent life where ever and whenever it arises.
Matter and reality inhabit the space of abstract logical possibility. The
most elementary and accessible parts of this space are known to us as
mathematics.
-Dave D.
Subject: Re: Euler-Maclaurin-Series
From: trobb12044@aol.com (TRobb12044)
Date: 15 Jan 1997 09:29:53 GMT
The folowing three references should give you all you need.
DIVERGENT SERIES, HARDY,CHELSEA PUB. CO.
INFINITE SERIES, BROMWICH,CHELSEA
THEORY AND APPLICATION OF INFINITE SERIES,KNOPP,BLACKIE&SON; LTD. (ALSO
AVAILABLE FROM DOVER IN PAPER BACK)
Tom Robbins (TRobb10244@aol.com)
Subject: Re: Why do Black Holes Form at all?
From: hillman@math.washington.edu (Christopher Hillman)
Date: 15 Jan 1997 09:32:09 GMT
In article <5bhpbg$hfq@dfw-ixnews9.ix.netcom.com>,
odessey2@ix.netcom.com (Allen Meisner) writes:
|> Mr. Hillman, you explained in another post that mass and kinetic
|> energy both contribute to the mass-energy of a particle.
Yes, the total energy (expanded in a power series in v) is
m/Sqrt{1-v^2} ~ m + (1/2) m v^2 + (3/8) m v^4 + ...
where the first term is the mass, the second term is the Newtonian
value for the kinetic energy, and the remaining terms may be considered
relativistic corrections to the kinetic energy (which are important only
for values of v close to 1).
|> You stated in another post that the
|> velocity space of a body is a Lobachevsky geometry.
Yes, The velocity space is space of forward pointing unit vectors, which
can act as tangent vectors to world lines; the spacelike components of such
vectors are interpreted as the components of the velocity and the timelike
component gives the time dilation rate at that event (for a clock carried
with the particle, relative to the rest frame).
|> Mr. Archimedes
|> Plutonium has stated that the Lobachevsky geometry does not have a zero
|> reference point.
In the same sense that ordinary euclidean space does not have any distinguished
points, he is correct. The euclidean plane, the ordinary sphere, the Lobachevsky
"hyperbolic" space (topologically a plane and thus often called "the hyperbolic
plane") and the velocity space of tachyons (topologically a cylinder) are all
surfaces of constant curvature and thus have no geometrically distinguished points.
Thus, the choice of an origin for any coordinate system is arbitrary. The
euclidean plane has constant curvature zero, and can be given the familiar
Cartesian coordinates. The remaining surfaces have constant nonzero curvature
and cannot be given a Cartesian coordinate system; in fact, the sphere cannot
be given ANY global coordinate system (i.e. one which avoids coordinate
singularities at all points) whereas the others can be given global, nicely
behaved conformal coordinate systems. One popular conformal system for
the Lobachevsky space was introduced by Poincare and maps this space onto
a disk of unit radius (with the geodesics represented as circular arcs
whose ends are orthogonal to the bounding circle). A good conformal
system for the tachyon velocity space is the exact analog of the Mercator
projection for the sphere (it represents lightlike geodesics as straight
line segments).
|> Since a body with constant velocity has a non-zero slope in the Loba geometry,
Unfortunately present technology does not support the drawing of a freehand
picture or two which would have greatly clarified my posting discussing
velocity spaces. In fact, a body with constant velocity (i.e. whose world
line has a constant unit tangent vector all along the world line) is
represented in the velocity space by a POINT. On the other hand, a body
with a curved world line experiences accelerations and such a world line
corresponds in the velocity space to a curve; in the case of constant
acceleration this curve is a geodesic (topologically a line) on the
Lobachevsky space. "Velocity space" is called that because its POINTS
correspond to possible values for the velocity associated with a
particular event on a given body's world line.
What you wrote after the quoted remark seemed pretty far off the mark to
me--- possibly because of the misunderstanding just noted.
Chris Hillman
Subject: Re: 1 / 2^.5 or 2^.5 / 2?
From: jac@ibms46.scri.fsu.edu (Jim Carr)
Date: 9 Jan 1997 18:57:48 GMT
jac@ibms46.scri.fsu.edu (Jim Carr) wrote:
| davk@netcom.com (David Kaufman) writes:
| >
| >The square root of 2 can be written in Basic computer language
| >as follows: 2^.5 or 2^(1/2) or SQR(2).
| =======
|
| If this is valid Basic, no wonder it is commonly said that those
| who learn Basic first are often crippled for life as programmers.
| That expression is equal to 1 in other high-level languages.
Pursuant to Eric's clever observation about the literal use of this
pseudo-Tex expression in C, it was always my intent that one first
express it in the usual way of doing exponentiation. And, of course,
the main reason BASIC does not even make the top 8 of "first languages"
used for CS teaching is more due to its discouragement of structured
programming methods than any differences in how it interprets
arithmetic statements. Actually, I don't know what it does, but I
can spot BASIC written in C (only global variables!) in my student's
code from 10 paces away.
Simon Read writes:
>
>rubbish nonsense rubbish nonsense drivel nonsense garbage
>spew bilge tosh tripe rhubarb moonshine nonsense hogwash
>
>2^(1/2) is the square root of two unless you deliberately do
>something strange, like using integer variables.
But that is exactly what the original author did, using a real
constant (0.5) in the first formula and an integer expression
(1/2) in the second. I will add that it is not unusual for semi-
experienced programmers (CS majors, after the C++ course) to make
the mistake of directly transcribing the printed equation that
results from that TeX expression into code that uses 1/2 = 0 as
the exponent in the pow function. More than half, usually.
>There are some finer points of FORTRAN where you can ask for integers,
>or possibly get things evaluated as integers
>_by not using any decimal points_
Of course, decimal points were not used in the above statements, but
my reason for this response is the reference to Fortran, an urban
legend that might explain why students who write in C might think
they are immune to it. They are not. Ditto for Ada, altough the
strong-typing might make you notice what you were doing.
>No way does BASIC cause people to be crippled for life as programmers.
>BASIC bears a strong resemblance to FORTRAN and Pascal, as a matter
>of fact,
There are modern versions of it that have logical structures, independent
subprogram modules with private data, etc, but the original language is
an abomination.
>It was invented as a teaching tool.
Pascal was invented as a teaching tool. BASIC was invented so you
could write very crude programs on exceedingly primitive computers,
AFAIK. If it was invented as a teaching tool, who is guilty?
--
James A. Carr | "The half of knowledge is knowing
http://www.scri.fsu.edu/~jac/ | where to find knowledge" - Anon.
Supercomputer Computations Res. Inst. | Motto over the entrance to Dodd
Florida State, Tallahassee FL 32306 | Hall, former library at FSCW.
Subject: Re: Empty Set Questions
From: mfinney@lynchburg.net
Date: 15 Jan 97 10:03:18 GMT
In <5belj4$2ng$1@nuke.csu.net>, ikastan@sol.uucp (ilias kastanas 08-14-90) writes:
> I just see no reason for empty domains
Consider the classic example of an empty domains -- quantification
over the set of all Unicorns. Sounds good, except that Unicorns *do*
exist (modified goat stock, I believe), and with genetic engineering
*might* exist in the future (in other forms).
The problem with assuming non-empty domains, is that frequently
one is unaware of the fact that the domain is empty. Or perhaps
cannot even know if the domain is empty -- how about sets whose
defining predicates require the solution of something like the 4-color
map problem? Or the domain may be empty today, but not
tomorrow. Or perhaps the reverse.
Reasoning should be correct *regardless* of the cardinality of the
set being reasoned about. If the reasoning is correct, it is correct
regardless if the set is later found to contain one element or none.
But classical logic only allows discussion of non-empty domains.
So, much of actual mathematical practice is therefore unjustified
from the standpoint of the underlying logic. This is clearly
unacceptable.
Subject: Re: Why do stars collapse?
From: hillman@math.washington.edu (Christopher Hillman)
Date: 15 Jan 1997 09:58:30 GMT
In article <32DBBDEB.42415D54@alcyone.com>,
Erik Max Francis writes:
|> Peter Diehr wrote:
|>
|> > > Black Holes in the GR sense remain hypothetical.
|> >
|> > You haven't been following the news very closely, have you?
|>
|> There is still no positive, undeniable evidence that a black hole exists.
|> I think it's safe to say that most physicists are pretty sure there exist,
|> an we have some convincing candidates, but there isn't quite the degree of
|> certainty yet that would warrant your objection.
By coincidence Ramesh Narayan (Harvard-Smithsonian Center for Astrophysics)
has just announced the apparent direct observation of hot gas disappearing
into the event horizon of a black hole (one member of the double star
V404Cyg in the constellation Cygnus, only 10,000 light years from Earth.)
Apparently their observations confirm a recent theoretical prediction
that gas being sucked into a hole can become superheated. Narayan
said that object "seems to be swallowing nearly a hundred times as
much energy as it radiates". In comparison, in the case of several
other double stars his team studied, hot gas was observed flashing as
it impacted the surface of a dense object interpreted as a neutron star.
Furthermore, Douglas Richstone (University of Michigan) just announced
the discovery of three new supercompact dark objects (presumably
black holes) with masses in the range 50-500 x 10^6 solar masses.
These objects were detected by their violent gravitational effects
on nearby stars, and they are all within 50 million light years of
the Earth.
So evidence continues to accumulate that black holes not only exist but
are quite common.
Chris Hillman
Subject: Re: Happy Birthday, HAL!
From: Jon Haugsand
Date: 15 Jan 1997 11:18:35 +0100
kfoster@rainbow.rmii.com (Kurt Foster) writes:
> : And if you shift right the letters HAL you will get his dady's name.
> :
> "Hal (for *H*euristically programmed *AL*gorithmic computer, no less)
> was a masterwork of the third computer breakthrough. ..." -- "2001 a space
> odyssey" -- a novel by Arthur C. Clarke
> The fact that shifting ther letters by 1 gives "IBM" is simply a
> coincidence. Arthur C. Clarke says so.
And do *you* seriously believe Clarke in this question?
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway, mailto:jonhaug@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Pho/fax: +47-22852441/+47-22852401
Subject: Re: Connected sets
From: jmcgowan@metric.inch.com (John McGowan)
Date: 15 Jan 1997 03:11:46 GMT
andrew harford (cs4-03@cslan.ud.ie) wrote:
> Does anyone know if the following is true?
> If A is a connected subset of a real vector
> space then for any elements x,y in A, there
> exists a continuous function f:[0,1]->A so
> that f(x)=0 and f(1)=y (using the usual
> topology).
No. In the plane the standard counter example is the topologist's sine
curve (y=sin(1/x) for x>0) along with the point (0,0).
It is not path connected
The point (0,0) cannot be connected to a point on the sine curve by a
continuous function (for if x=g(t),y=h(t) is a continuous function
connecting (0,0) to some point (a,sin(1/a) then consider just x=g(t) with
g(0)=0, g(1)=a... there exists a sequence t_j->t_0 (decreasing sequence)
with g(t_0)=0 (zero may not be the only value of t for which g(t)=0!) and
g(t_j)>0 and h(t_j)=1 (let x_j=1/(pi/2 + 2*j*pi) which decreases to
zero... for j>some J, x_j are smaller than a and by the intermediate value
theorem, there is a t_J<1 with g(t_J)=x_J. Again, by the intermediate value
theorem, there is a t_(J+1)0},
h(t_j) must lie on the curve, so h(t_j)=sin(x_j)=1. But then lim(h(t_j))=1
while h(t_0)=0, so h cannot be continuous... viz. there cannot be a
continuous function in this set mapping (0,0) to another point)
It is connected
If E,F are disjoint relatively open sets (relative to the topologist's
sine curve plus the origin) of A whose union is A (=top. sine curve plus
the origin) which separate A (vis. they are disjoint and their union is A)
then... the origin is in one of them, say, E. E must contain an open
square containing the origin intersected with A. Thus it must contain some
point on the topologist's sine curve as well. Just restricting to the
topologist's sine curve (F does not contain the origin and so is contained
just in the sine curve) E intersect the sine curve and F intersect the
sine curve would disconnect it (unless F is the null set), but that curve
is a path (connected!) so F is the null set (since it cannot contain the
origin since it is disjoint from E). So.. writing A as the union of
disjoint relatively open sets is only possible if one of them is the null
set... or A is connected.
Thus.. A is connected but not path connected.
On the other hand, if A is *open* and connected in a locally path connected
space, then it IS path connected.
not true for CLOSED sets...
(replace A by {(0,y::-1<=y<=1}UNION{(x,y):y=sin(1/x) for x>0}, viz. the
topologist's sine curve along with the vertical line segment of length 2
centered at the origin... now the set is closed and connected in this
case we would show two sequences, t_j and v_j, interleaved with
t_(j+1)
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