An efficient way to compute the variance uses the raw-score formula of
statistics. If you take the defn. of variance formula and expand it out and
replace mu with its estimate (sample mean) you get the raw-score formula.In
terms of computation, you will only require the sum of the sample and the
sum of the squares of the sample along with the number of samples.
Clay
> Dear all,
 In one of my program, I need to compute the variance of a 8x8 data matrix
as
> fast as possible...
 Any fast algorithm with lowest complexity?
 By the way, since what I actually need is an indication of signal local
> activity, is there any other measure which can also be a indicator of
local
> activity and is less complex than computing variance?

> -Walala

====
> In one of my program, I need to compute
> the variance of a 8x8 data matrix as
> fast as possible...
What does it means ?
You just have 16 samples of a signal and you want the variance ?
For me it is not clear why deviation from
the mean is a good indicator of activity.
Obviously, there is no fast algorithm for computing a sum
of 16 terms !
BUT if you compute the variance on a moving windows, there
is a very fast algorithm, that you can figure out pretty easily.
Your just have to perform :
1. compute mean m and replace your image I by M=(I-m)^2
2. compute moving average along X direction. This can be done
by a recursive algorithm, because if S(k) is the moving average
and F(k) is the initial function (e.g. a row of M), you have
S(k) = S(k-1)+F(k)-F(k-9)   (here I assume window of size 8)
3. compute moving average along Y direction.
All this stuff needs about O(n) operations where n is the number of pixels
in the original image and it's independant of the size of the window.
Gabriel
====
> You just have 16 samples of a signal and you want the variance ?
Sorry, I meant 64 samples.
> Obviously, there is no fast algorithm for computing a sum
> of 16 terms !
Sorrrry once again 8x8=64 not 16 ...
G.
====
myfile.> DON02. Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
subject:Mersenne numbers, Mp163 
> D.Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
We look at the usual long multiplication of, for 
example, 123*48.
 123 
x 48 
------- 
 984 
+492
------- 
 5904. 
 
However, I wish to concentrate just on the tens and 
units figures of the resulting product. This is called 
'multiplying modulo 100.' 
That is, any whole number of 100s can be thrown away in 
the process. 
 
Then, (100+23)*48 == 48*100+23*48 (is congruent to)
== (20+3)*(40+8) == 2*4*100+3*40+8*20+3*8 
== 120+160+24 
== 104 == 04 modulo 100.
 
In calculating the remainder of product (integers w*z) 
after dividing by positive integer, t, it can often be 
made easier by taking away convenient multiples of the 
modulus, t, from w or z respectively before executing 
the multiplication. 
 
Let w = at+b, say, and z = ct+d, say. 
 
Then, wz = (at+b)(ct+d) 
= at(ct+d) +b(ct+d) 
= t(act +ad +bc) +bd 
== bd modulo t. 
 
Bd should usually be less than w*z, but it may require 
to be reduced further. 
 
I photographed car number plate 'MP163' at Hanson St, 
Newtown, Wellington on today, 26-5-2002. This number 
plate could be shorthand for Mersenne numbers, 
2^prime-1. 
(Figure 2 raised to a prime index, subtract 1.) 
 
The world's largest known prime number in 2001 often is
2^13.4million-1. (Greater than 13 million 2s 
(Dominion Post, NZ Herald, GIMPS great
internet mersenne prime search.)
By the way, the 40th known Mersenne prime exponent is
a twin prime, 10x(23x63)^2 +/-1. And a previous record
holder was virtually a palindrome, Table Mountain, increasing
decreasing.
Exponent (1*2*34*44432+1)= 3021377. [sic.]
 _____
 / 
 
A theorem of mathematics number theory says if
(a is a positive integer )..
a^prime-1 has factor/s they must be of the form 
2*k*p+1. 
 
However, the Penguin Dictionary of Curious and 
Interesting Numbers (1997), entry 28, table of perfect 
numbers, does not give M_163 as a Mersenne prime. 
 
Therefore, I have written myprogram 'powabc1' in BBC 
interpreted Basic64, which tests for just such factors.
(On Acorn A5000 computer, UK 1990, RISC OS-reduced instruction
set computer, 4 Megabytes RAM randomaccess memory.)
In this case, several programs found 150287 and 704161, 
etc. divide M_163. 
Or 2^163 == 1 modulo 150287. 
 
Have I found a factor of M_163? 
The following, I believe, shows that I have.
Check. 2^163= 2^3*(2^20)^8 (is congruent to)
== 8*1048576^8
2^20==1048576-6*150287 
== 146854 mod 150287 
== -3433. 
Squaring 2^40= 11785489
== 63103 
Squaring 2^80==3,981,988,609 
==134544== -15743 
Squaring 2^160==247,842,049 
==18786 
2^163==8*18786 
==150288==1 mod 150287. 
 
Q.e.d.
By the way, my methods found (the least prime) factor
1580,187,223 of (10^9)^(10^9)+3. (sci.math sensation 2001.)
This is, digit 1 followed by 9 billion zeroes. Surpassed
only by Graham's number. I suspect gigaplex =
10^billion. (There is a typo error in Penguin Dictionary
of Curious and Interesting Numbers, 1997.)
 
yours sincerely, 
 
/ Donald S. McDonald (Wellington, New Zealand)
====
There was a time when people spoke their minds, and were not afraid to 
offend -
and that since then, too many truths have been buried.
Mark Kurlansky, 1968: The Year That Rocked The World
(Ballantine, 2004) thanks to Frank Lauria
In 1968 I was in La Jolla CA at UCSD during time Greg Benford describes in
Timescape living on Bonair Street Wind an Sea Beach near Unicorn
Theater with Ken Kesey's Merry Prankster Bus parked nearby frequently.
I was also teaching at San Diego State with Fred Alan Wolf.
...
PZ: I can't imagine anything more wrong-headed. And you say Rovelli is a 
big
shot?
That is why I say Rovelli's position is incoherent.
JS: Is coherence in the mind of the beholder?
PZ: If you want to treat g_uv as a physical field, based on its 
dynamical character
(i.e. matter-dependence), then the natural thing to do is separate the 
background
generalized Minkowski kinematical metric (which is NOT matter dependent) as
chronogeometric, and treat the gravitational g_uv alone as physical -- 
by
Rovelli's own argument.
JS: That will not work. It will not tip the light cones. It will not 
give the correct bending of light.
Also you are too vague on what you mean by matter on the RHS of
Guv(Einstein) = -(alpha)(alpha')Tuv(Matter)
alpha = e^2/hc ~ 1/137
alpha' = 8pi(Witten's reciprocal string tension)
alpha' ~ (10^-32 cm)^2  in a common convention.
I also include on RHS
tuv(Vacuum) ~ [(alpha)(alpha')]^-1/zpfguv
/zpf = Lp^-1[Lp^3|Vacuum Coherence|^2 - 1]
Repulsive dark energy is /zpf > 0
Attractive dark matter is /zpf < 0
FRW Omega(Dark Energy + Dark Matter) ~ 0.96
With Omega(Total) = 1
  i.e. FLAT SPACE
Preferred foliation is where CMB is maximally isotropic to ~ 10^-5.
This allows accurate navigation with weightless warp drive and 
traversable wormholes both supported by configurations of dark energy 
and dark matter exotic w = -1 vacua.
PZ: Yet he doesn't even mention this.
JS: There was a time when people spoke their minds, and were not afraid 
to offend -
and that since then, too many truths have been buried.
Mark Kurlansky, 1968: The Year That Rocked The World
...
JS: Also at quantum level Rovelli
mention's Dirac's insight that the Heisenberg Picture is better than
Schrodinger Picture and that time as in the flow of time's 
arrow
is not dynamical time but is a statistical thermodynamic construct.
PZ: Sure, if you want to quantize everything and abandon time.
Unruh's for example, and some of the others.
Unruh has some important ideas however.
PZ: But then I fail to see any material distinction between relabeling
the bare unindividuated
spacetime points (passive diffeomorphism), on the one hand, and
shifting all physical
fields (including the GR metric field) with respect to such a raw
manifold (active
diffeomorphism), on the other. Kretschman's point appears to be fully
valid in both cases:
how can physics in general -- any sensible physics -- possibly depend
on a mere re-labeling
of raw unindividuated spacetime points; or, for that matter, on a
common shift of all physical
systems, including the *physical* metric field g_uv, with respect to
such a set of unindividuated
points?
JS: Admittedly a sticky wicket that I also need to understand more
deeply.
PZ: It's an artificial model that has nothing to do with physical 
relativity IMO.
JS: I found this remark by Goldstein helpful:
In the ADM formulation 4-diffeomorphism invariance amounts to the 
requirement that one ends up with the same space-time, up to coordinate 
transformations, regardless of which path in multi-fingered space-time 
is followed, i.e. which lapse function N is uses. p.278
OK so this idea goes back to the archetypal notions of classical 
thermodynamics with a state function, to holonomic integrability of 

equality of mixed partial derivatives with not multiply-connected 
manifolds, to a closed Cartan exterior differential form on a cycle (no 
boundary), no topological defects and all that.
The passive coordinate transformations are like EM gauge transformation 
(e/c)Au -> (e/c)Au + hChi,u  in a fixed gauge constraint.
PZ: Rotating a system in isotropic space is connected with a true 
physical symmetry
of the system including the vacuum in which it is embedded. Yes, the 
symmetry
of the system Hamiltonian under such a transformation is an active 
transformational
symmetry that is characteristic of the particular system -- unlike the 
invariance of the
proper formal expression of any physical law under a *mere* coordinate 
transformation.
JS: Note also 12.2.2 p.279 alluding to your digging up Kretchmann from 
Dr. Frankenstein's favorite graveyard. ;-)
The fundamental symmetry at the heart of general relativity is its 
invariance under general coordinate transformations of spacetime. It is 
important to stress that almost any theory can be formulated in such a 
4-diffeomorphism invariant manner by adding further structure to the 
theory (e.g. a preferred foliation of spacetime as a dynamical object). 
General relativity has what is sometimes called serious diffeomorphism 
invariance, meaning that it involves no spacetime structure beyond the 
4-metric and, in particular, singles out no special foliation of 
spacetime.
Goldstein and Teufel then knock standard QGrav including even Ashtekar 
-> Loop Spin Foams that are perhaps near to being falsified by NASA's 
EINSTEIN in
Jack better check this story out:
Gary S. Bekkum
...
How to get
localization in space and the flow of time as we experience it in our
immediate inner consciousness
has nothing to do with the particular local coordinate representation
like r and t in, for example,
K = e^2GM/c^2r
dr/dt = c/K
for null geodesic
and in his Tables generally in the context of potentially practical
metric engineering of the guv field using the EM Au field in spite of
the enormous gravity string tension ~ c^4/G ~ 10^19 Gev per 10^-33 cm.
PZ: You are blocking his actual definition of r. Classic operationalism 
does not
apply to a theory of this type. That is a critical point. Miss that and 
of course
nothing makes sense.
JS: I am not ready to renounce PW Bridgman's Operationalism. Indeed, 
nothing Hal Puthoff says about the foundations of his PV makes any sense 
to my mind. If it ain't broke, don't fix it. Of course I am not a 
doctrinaire positivist like Stephen Hawking proudly proclaims he is.
...
JS: Correct, with the proviso that matter includes both real, i.e.
on mass shell, sources as well as virtual, i.e. off mass 
shell
sources. The virtual sources divide into two classes:
I. Non-exotic near EM field Fuv giant coherent quantum states of
virtual photons that contribute to Omega(Matter) of the FRW metric and
to Tuv in Einstein's local field equation.
II. Exotic vacuum w = -1 zero point stress-energy density local tensor
~ (String Tension)/zpfguv for both repulsive dark energy /zpf > 0 of
negative pressure and attractive dark matter /zpf < 0 of positive
pressure.
These exotic vacuum virtual sources contribute Omega(Exotic Vacua) ~
0.96 to Omega(Total) = 1 in our large-scale spatially flat
post-inflationary local Level I Hubble sphere brane world as in
Lenny Susskind's megalopolis Landscape subject to the natural
selection of the Weak Anthropic Principle (WAP)
OK, I think I made an error above including brane worlds in sense of 
parallel worlds?
Here is why I think I made an error (If I did so did Hawking and 
Scientific American in their popular science reports):
D-Branes are extended surfaces without edges. In order that the black 
hole be a localized object, it is assumed that our ordinary four 
dimensions (three space and one time) are all orthogonal to these 
D-Brane surfaces ... Thus to us these D-Branes would look as though they 
were located at a point (or at least a very small region) of our 
observable three dimensions of space. W. G. Unruh p. 168 Black holes, 
dumb holes, and entropy, i.e. the D-Branes are in the compactified 
Calabi-Yau space. I have to look again at Hawking's The Universe in a 
Nutshell that seems to give the wrong idea here? Perhaps I mis-remember?
Note also Ed Witten's formula generalizing Heisenberg's quantum 
uncertainty principle, i.e. eq. (5.9) p. 136
Delta X > h/DeltaP + alpha'(DeltaP)/h
The second gravity-string source of uncertainty should give the 
irreversible statistical arrow of time not found when alpha' = 0 i.e. 
infinite string tension, or infinite space-time stiffness of action 
without reaction as is also found in the signal locality of orthodox 
quantum theory in sense of Antony Valentini's papers.
Mass without mass, but with a strong micro-gravity
G* ~ 10^40G on 1 fermi scale.  The wormhole has an attractive dark 
matter exotic vacuum core where
Vacuum Coherence --> 0
just like inside a quantized vortex of circulation in superfluid HeII.
Therefore, in the core
/zpf ~ - (1 fermi)^-2
w = -1
Therefore the quantum pressure is positive and the exotic vacuum core 
gravitates as
Grad^2V(Exotic Vacuum) ~ -c^2(1 fermi)^-2
this prevents the spread out electric charge from exploding and also 
compensates any quantized rotation centrifugal forces. For now keep 
charge and rotation zero for simplicity. That is I here only model a 
spin 0 neutral micro-geon.  That means I would need a high-power 
graviton laser as the Heisenberg uncertainty scattering probe 
microscope. That's OK since this is only a gedankenexperiment.
The SSS metric then has the factor
1 - 2GM/c2r
Where G(mass density) is replaced by c^2/zpf
GM is replaced by ~ c^2/zpfR^3
for a micro-geon of size R
r is DeltaX as in Witten's formula above with DeltaP as the scattering 
momentum transfer between gravitons and the micro-geon.
Therefore, the Schwarzschild factor is (neglecting factors of 2, pi etc 
all lumped into dimensionless parameter b
1 - bc^2/zpfR^3/DeltaX
The critical value of Delta X is then the event horizon where the 
Schwarzschild factor vanishes - can it be reached?
This micro-geon is a solution of the exotic vacuum local field equation
Guv(Einstein) + /zpfguv = 0
At critical Delta X, the geon looks like a POINT PARTICLE from the huge 
space-warp induced by the probe's momentum transfer Delta P.
C is the circumference
dC/dR = 2pi (1 - bc^2/zpfR^3/DeltaX)^1/2
Where one considers the radial size fixed at R the scale of the throat 
of the wormhole.
So when can we have
DeltaX = bc^2/zpfR^3  ?
Note that alpha' may be large of order (10^-11 cm)^2 not (10^-32 cm)^2 
as in Witten's idea.
====
Interesting!
Jack,
Superstring theory does not view the Planck scale of spacetime as a quantum
foam but rather as the Planck scale is approached the dimensionality of
spacetime goes to 10-d.  In string theory the spacetime does fluctuate but
this fluctuation is harmonic rather than chaotic. In fact the assumption
that the harmonics of the strings absorbs all the quantum fluctuation can 
be
used to derive the dimensionality of spacetime. Also Lorentz invariance is
assumed in this calculation. This was first done by L. Brink and H.B.
Nielsen in 1973 (A Simple Physical Interpretation of the Critical 
Dimension
of Space-time in Dual Models, *Physics Letters* 45B:4 (1973) 332-336. 
This
paper is also included in the anthology edited by John Schwartz,
*Superstrings: the First 15 Years of Superstring Theory, Vol. 1* (World
Scientific, 1985). [Note that Dual Models is the old name for string
theory, when it was still evolving away from the terminology of s-matrix
theory. However, the Mandelstam labels of S, T, and U duality have
resurfaced in membrane theory!]
  I mentioned the Brink-Nielsen view of string theory in my paper Notes 
on
pdf.]
     Also on page 6 of F.W. Stecker's pdf paper [referred to in the NASA
report], he says We note that there are variants of quantum gravity and
large extra dimension models which do not violate Lorentz invariance and 
for
which the constraints considered here do not apply.
     BTW: In the Acknowledgments on page 8, Serge Rudaz is one of four
people thanked for helpful discussions.  I remember meeting Serge Rudaz 
in
seminars. He was then a young physics student (at Cornell?) who was an old
acquaintance of yours. He told us about instantons -- which was a new 
idea
then. I have not heard of him since 1976 -- until seeing this
acknowledgement!
     Nuff said!
     Saul-Paul
JS: Yes, Serge was with me at UCSC in Summer 1973 when I went to see 
Jean Cocteau's Orphee on campus with Helen Quinn (who was close to 
having he baby at that time) and I think Serge and a few others. This 
was days before I went to SRI to meet with Puthoff and Targ on the tape 
you have. The story is in the book Destiny Matrix.
----------
Therefore quantum foam is suspect. Also Bohm's quantum potential
view of vacuum fluctuations is relevant in context of the recent
paper from Teheran.
I need to follow the experiment below more carefully.
Jack better check this story out:
Gary S. Bekkum
====
i have realised a few mathematical functions?, properties?, lately and
was wondering about them, such as whether they are already known, or
are even important.
if anyone is interested, hit me up,AIM:heartxenocide
====
I really hope no one takes this as spam, I'm a student member of the
MAA, and I've posted here a few, brief times, however since I'm still
a lower division undergraduate most of the subject matter is far above
my head, so I'm not a regular (yet!)
With that said, I have created some math t-shirt designs that just
make me smile and I'd like to share them with the math community, with
the hopes that they delight someone else as much as they delight me.
the url is http://www.cafeshops.com/subjective and the designs are
under Math and Science (click on the Einstein picture)
All comments, critiques, commendations, cat-calls and creative input
====
I am stuck in high school maths mode, and can't seem to get into 
university level maths.  This might be because I am entirely self 
taught, but I don't know.  Does anyone else have this problem?
I am at a level where I understand most high school maths, and I've 
studied Calculus Made Easy.  It would be helpful to have some kind of 
way to check my knowledge.
Does anyone know of any good textbooks that cover high school maths 
with worked exercises, and any texts that help the transition from 
high school maths to the more exciting stuff at university level?  A 
book that takes time to explain things, point out applications and say 
why rather than just how.  I am trying to understand maths, not just 
learn some techniques or shortcuts.
By the way, I would prefer internet resources and small paperback 
books.  There's no way I can afford college textbooks unfortunately.
Johnathan
====
 I am stuck in high school maths mode, and can't seem to get into
> university level maths.  This might be because I am entirely self
> taught, but I don't know.  Does anyone else have this problem?
 I am at a level where I understand most high school maths, and I've
> studied Calculus Made Easy.  It would be helpful to have some kind of
> way to check my knowledge.
 Does anyone know of any good textbooks that cover high school maths
> with worked exercises, and any texts that help the transition from
> high school maths to the more exciting stuff at university level?  A
> book that takes time to explain things, point out applications and say
> why rather than just how.  I am trying to understand maths, not just
> learn some techniques or shortcuts.
 By the way, I would prefer internet resources and small paperback
> books.  There's no way I can afford college textbooks unfortunately.
 Johnathan
You could check out:
http://store.doverpublications.com/by-subject-science-and-mathematics-mathem
atics-calculus.html
they are cheap and useful.  You could also try Schaum's outlines; they are
also very good.
For web content, try:
http://www.math.temple.edu/~cow/
If you run a google search, you will find a plethora of math related
websites.  Many, many, many more than I care to list.  Make google your 
best
friend.  You can find anything you could possibly want to know using google
in about 30 seconds.
Lurch
====
>  > I am stuck in high school maths mode, and can't seem to get into
> university level maths.  This might be because I am entirely self
> taught, but I don't know.  Does anyone else have this problem?
>  > I am at a level where I understand most high school maths, and I've
> studied Calculus Made Easy.  It would be helpful to have some kind of
> way to check my knowledge.
>  > Does anyone know of any good textbooks that cover high school maths
> with worked exercises, and any texts that help the transition from
> high school maths to the more exciting stuff at university level?  A
> book that takes time to explain things, point out applications and say
> why rather than just how.  I am trying to understand maths, not just
> learn some techniques or shortcuts.
>  > By the way, I would prefer internet resources and small paperback
> books.  There's no way I can afford college textbooks unfortunately.
>  > Johnathan
You could check out:
> 
http://store.doverpublications.com/by-subject-science-and-mathematics-mathema
tics-calculus.html
they are cheap and useful.  You could also try Schaum's outlines; they 
are
> also very good.
For web content, try:
http://www.math.temple.edu/~cow/
If you run a google search, you will find a plethora of math related
> websites.  Many, many, many more than I care to list.  Make google your 
best
> friend.  You can find anything you could possibly want to know using 
google
> in about 30 seconds.
Lurch
In particular, include ext:pdf site:.edu in your searches.  You'll
just get postscript notes, usually written up by professors.
'cid
====
There was a time when people spoke their minds, and were not afraid to 
offend -
and that since then, too many truths have been buried.
Mark Kurlansky, 1968: The Year That Rocked The World
(Ballantine, 2004) thanks to Frank Lauria
In 1968  was in La Jolla CA at UCSD during time Greg Benford describes in
Timescape living on Bonair Street Wind an Sea Beach near Unicorn
Theater with Ken Kesey's Merry Prankster Bus parked nearby frequently.
I was also teaching at San Diego State with Fred Alan Wolf.
OK, now I have looked at the rest of Rovelli's argument, I'll work through
this one more time.
PZ: Rovelli's position is very odd. First he says:
...Of course, nothing [in GR] prevents us... from singling out the
gravitational
field as 'the more equal among equals', and declaring that location is
absolute in
GR, because it can be defined with respect to it.  (p 108)
JS: But he rejects that Paul. He says doing that misses the great
Einsteinian insight.
The great Einsteinian insight being the conditional and physical nature 
of
the gravitational metric field -- which conflicts with the great 
Einsteinian insight
strict equivalence.
The opposite of a profound truth is another profound truth. -- Niels 
Bohr
What Rovelli doesn't seem to understand is that this all makes perfect 
sense once
you give up strict equivalence and distinguish the background and 
physical metrics.
JS: I do not understand this distinction. Please give more details what 
you mean.
Have you read pp. 112 - 114 that completely demolishes Hal Puthoff' s
use of
dr/dt = c' = c/K   radial null geodesic
in his Tables.
PZ: It does no such thing. I would not even characterize pp 112-114 as 
an argument.
It is simply a sketch of a model in which *everything* is quantized 
except the raw
manifold.
JS: It shows no intrinsic meaning to Puthoff's r and t as he means it in 
his Tables.
PZ: He wants to throw away time in order to keep a unified g_uv.
Why do you think this is an argument against c' = c/K? As far as I can 
see it is simply
a different theory.
Rovelli briefly considers an alternative approach in which we retain the 
Minkowski
background of standard QFT. But there he has
g_uv = n_uv + fluctuations
which makes no sense to me. He does not seem to understand the 
distinction between
kinematical g_uv and dynamic gravitational g_uv. He cannot get his head 
over the unified
metric.
What does he mean by fluctuations?
JS: What do you mean by kinematical g_uv and dynamic gravitational 
g_uv apart from Ruvwl = 0 in  the former and not in the latter.
JZ: Sounds like sheer nonsense to me. Nonsense because it is divorced 
from the physical
fundamentals.
JS: No argument from me on that one.
PZ: This of course is exactly what the classic Einstein
chronogeometric model does,
going all the way back to special relativity.
JS: I think you are misreading Rovelli.
PZ: The Einsteinian model is a chronogeometric model, in which the 
metric g_uv reflects the
fundamental structure of spacetime.
JS: If you mean, for example,
dT = goo^1/2dt
dR = grr^1/2dr
Then I agree that dT and dR are physical and dt and dr are not.
However, to get a gravity shift of light frequency do
dT'/goo'^1/2 = dT/goo^1'2
treating dt as a kind of nonlocal invariant.
If you mean more than this, then explain with detailed examples.
PZ: That is the great Einsteinian insight -- which is.
unfortunately, based on strict Einstein equivalence, which is fictitious.
JS: Again I really do not understand what you mean by this sentence.
PZ: Now Rovelli wants to pretend that the great Einsteinian insight is 
something else entirely
-- that the gravitational field is a physical field.
Loony tunes. He is tying himself up in knots.
JS: Yes, in global special
relativity, NO in local general relativity.
PZ: In Einstein general relativity, the unified metric g_uv represents 
the fundamental structure
of spacetime, and in inseparable from it. That's Einstein.
It doesn't even make sense to me to say that this chronogeometric model 
holds only locally.
PZ: That is precisely what distinguishes the
Einsteinian from the Lorentzian model.  The transformational and
metric structure
is not, in the classic Einsteinian model, separable from spacetime
itself.
But then he says:
There is no absolute referent of motion in GR; the dynamical fields
move with
respect to each other. (p 108)
JS: Again you are misreading. There is no contradiction here in
Rovelli's argument.
PZ: This is like arguing that everything is relative because everything 
is relative to the
absolute, including the absolute.
Sounds like Lewis Carroll.
JS: ;-)
PZ: None of this has anything to do with Einstein general relativity, 
which treats the
inertial field as real. The unified metric derives its physical meaning 
from this
equivalence.
Rovelli ignores all this and simply takes the unified Einstein g_uv as 
given.
PZ: This second assertion seems to approach the metric field of GR as
just another field, which
is very close to and even indistinguishable from the physical rubber
rod and clock model
of PV and Yilmaz -- just another physical field which happens to have
metric properties.
And of course such a field is fully relational with respect to
unindividuated spacetime points
on a raw manifold stripped of all coordinate systems, transformation
properties, and metrics.
JS: Yes on just another field. But NO that it's like PV and Yilmaz. Not
true at all because,
at least in PV, Hal uses an absolute non-dynamical background global
Minkowski space
PZ: That is what Rovelli *should* be doing, but he doesn't even consider 
this
possibility. He seems to think you can treat unified g_uv as a physical 
field.
JS: Why do you think you cannot?
PZ: I can't imagine anything more wrong-headed. And you say Rovelli is a 
big
shot?
That is why I say Rovelli's position is incoherent.
JS: Is coherence in the mind of the beholder?
PZ: If you want to treat g_uv as a physical field, based on its 
dynamical character
(i.e. matter-dependence), then the natural thing to do is separate the 
background
generalized Minkowski kinematical metric (which is NOT matter dependent) as
chronogeometric, and treat the gravitational g_uv alone as physical -- 
by
Rovelli's own argument.
JS: That will not work. It will not tip the light cones. It will not 
give the correct bending of light.
Also you are too vague on what you mean by matter on the RHS of
Guv(Einstein) = -(alpha)(alpha')Tuv(Matter)
alpha = e^2/hc ~ 1/137
alpha' = 8piWitten's reciprocal string tension)
alpha' ~ (10^-32 cm)^2  in a common convention.
I also include on RHS
tuv(Vacuum) ~ [(alpha)(alpha')]^-1/zpfguv
/zpf = Lp^-1[Lp^3|Vacuum Coherence|^2 - 1]
Repulsive dark energy is /zpf > 0
Attractive dark matter is /zpf < 0
FRW Omega(Dark Energy + Dark Matter) ~ 0.96
With Omega(Total) = 1 i.e. FLAT SPACE
Preferred foliation is where CMB is maximally isotropic to ~ 10^-5.
This allows accurate navigation with weightless warp drive and 
traversable wormholes both supported by configurations of dark energy 
and dark matter exotic w = -1 vacua.
PZ: Yet he doesn't even mention this.
JS: There was a time when people spoke their minds, and were not afraid 
to offend -
and that since then, too many truths have been buried.
Mark Kurlansky, 1968: The Year That Rocked The World
PZ: So he has not made any argument at all against a bimetric approach. 
He has simply
ignored it, even though his own argument points to it.
JS: with a literal meaning for coordinates r & t, otherwise his formula
dr/dt = c/K
is physically meaningless, which it is in GR as explained by Rovelli on
pp. 112 - 114.
PZ: Only if you insist exclusively on operational meaning -- which would 
also knock out
F = ma, which by a similar argument is empirically meaningless.
JS: Yes, until you also add things like
F = - GMmr/r^3
F = eE + e(v/c)xB
etc.
That is, Hal's use of engineering is precisely missing the third
step on Rovelli's p. 108.
PZ: Rovelli's third step is merely a sketch of a proposal to quantize 
everything, while
retaining a unified g_uv treated in its entirety as a physical field.
This is not an argument.
Rovelli's argument for his proposal, such as it is, is to be found on 
p 109:
In my *opinion*, this is the right way to go.
Well, in my *opinion*, it's fundamentally wrong-headed. In addition, 
Rovelli has
grossly mischaracterized the Feynman-type alternative -- which shows 
that he has
not understood it; although he is right that under this alternative, the 
GR conceptual
revolution is indeed dis-valued, but to a much greater extent than he 

seems to realize.
JS: Also p. 112
Recall that Einstein described his great intellectual struggle to find
GR as 'understanding the meaning of coordinates'.
This is my key objection to Puthoff's and Ibison's use of r ant t
in their PV modeling.
They do not IMHO understand the meaning of coordinates: in GR. Neither
does Yilmaz apparently.
PZ: I have no idea of what you are getting at here. Einstein's point 
about coordinates
depends entirely on his strict equivalence thesis, which is untenable.
JS: I guess I have never understood your idea here. Can you try to 
explain it again with
complete clarity?
PZ: Once you abandon strict equivalence, the unification of 
gravitational and inertial
g_uv is formal and arbitrary. It has no deep physical meaning. Then we 
recover
chronogeometric background coordinates together with a physical rubber-rod-
and-clock non-inertial g_uv.
Rovelli doesn't seem to understand that.
JS: Neither do I.
pp 112 - 114 make this clear IMHO.
PZ: This is not an argument -- it's merely a proposal: In my 
opinion....
JS: Also at quantum level Rovelli
mention's Dirac's insight that the Heisenberg Picture is better than
Schrodinger Picture and that time as in the flow of time's 
arrow
is not dynamical time but is a statistical thermodynamic construct.
PZ: Sure, if you want to quantize everything and abandon time.
Unruh's for example, and some of the others.
Unruh has some important ideas however.
PZ: But then I fail to see any material distinction between relabeling
the bare unindividuated
spacetime points (passive diffeomorphism), on the one hand, and
shifting all physical
fields (including the GR metric field) with respect to such a raw
manifold (active
diffeomorphism), on the other. Kretschman's point appears to be fully
valid in both cases:
how can physics in general -- any sensible physics -- possibly depend
on a mere re-labeling
of raw unindividuated spacetime points; or, for that matter, on a
common shift of all physical
systems, including the *physical* metric field g_uv, with respect to
such a set of unindividuated
points?
JS: Admittedly a sticky wicket that I also need to understand more
deeply.
PZ: It's an artificial model that has nothing to do with physical 
relativity IMO.
JS: I found this remark by Goldstein helpful:
In the ADM formulation 4-diffeomorphism invariance amounts to the 
requirement that one ends up with the same space-time, up to coordinate 
transformations, regardless of which path in multi-fingered space-time 
is followed, i.e. which lapse function N is uses. p.278
OK so this idea goes back to the archetypal notions of classical 
thermodynamics with a state function, to holonomic integrability of 

equality of mixed partial derivatives with not multiply-connected 
manifolds, to a closed Cartan exterior differential form on a cycle (no 
boundary), no topological defects and all that.
The passive coordinate transformations are like EM gauge transformation 
(e/c)Au -> (e/c)Au + hChi,u  in a fixed gauge constraint.
PZ: Rotating a system in isotropic space is connected with a true 
physical symmetry
of the system including the vacuum in which it is embedded. Yes, the 
symmetry
of the system Hamiltonian under such a transformation is an active 
transformational
symmetry that is characteristic of the particular system -- unlike the 
invariance of the
proper formal expression of any physical law under a *mere* coordinate 
transformation.
JS: Note also 12.2.2 p.279 alluding to your digging up Kretchmann from 
Dr. Frankenstein's favorite graveyard. ;-)
The fundamental symmetry at the heart of general relativity is its 
invariance under general coordinate transformations of spacetime. It is 
important to stress that almost any theory can be formulated in such a 
4-diffeomorphism invariant manner by adding further structure to the 
theory (e.g. a preferred foliation of spacetime as a dynamical object). 
General relativity has what is sometimes called serious diffeomorphism 
invariance, meaning that it involves no spacetime structure beyond the 
4-metric and, in particular, singles out no special foliation of 
spacetime.
Goldstein and Teufel then knock standard QGrav including even Ashtekar 
-> Loop Spin Foams that are perhaps near to being falsified by NASA's 
EINSTEIN in
Jack better check this story out:
Gary S. Bekkum
PZ: However, moving *everything physical* -- including the unified g_uv 
-- along the raw
spacetime manifold is quite another kettle of fish. It is devoid of 
physical content IMHO.
But again, a space-time frame of reference is not *merely* a coordinate 
transformation,
since it represents the *motion* of a possible observer. Thus there is 
no *a priori* reason
why physical laws should take the same form in different frames of 
reference, contrary
to Einsteinian doctrine.  Of course they *may* as a matter of fact, but 
that is very different
from insisting a priori that they always *should*.
That's yet another Einsteinian red herring.
PZ: What is physically significant here is not this abstract and
artificial definition of diffeomorphism
with respect to a raw manifold conceived as an unindividuated point
set, but a shift of physical
fields with respect to the metric-transformational structure of
spacetime, which of course in GR
depends on the distribution of matter.
JS: Precisely, yes that is the idea I think. It's a return to I think
Leibniz's relationism?
PZ: Descartes. He is almost literally putting Descartes before the horse of
physical relativity.
JS: ;-)
How to get
localization in space and the flow of time as we experience it in our
immediate inner consciousness
has nothing to do with the particular local coordinate representation
like r and t in, for example,
K = e^2GM/c^2r
dr/dt = c/K
for null geodesic
and in his Tables generally in the context of potentially practical
metric engineering of the guv field using the EM Au field in spite of
the enormous gravity string tension ~ c^4/G ~ 10^19 Gev per 10^-33 cm.
PZ: You are blocking his actual definition of r. Classic operationalism 
does not
apply to a theory of this type. That is a critical point. Miss that and 
of course
nothing makes sense.
JS: I am not ready to renounce PW Bridgman's Operationalism. Indeed, 
nothing Hal Puthoff says about the foundations of his PV makes any sense 
to my mind. If it ain't broke, don't fix it. Of course I am not a 
doctrinaire positivist like Stephen Hawking proudly proclaims he is.
PZ: So it appears that Rovelli ends up in a position where he is
essentially arguing that the matter-
dependence of the metric field implies that the GR metric is really a
physical metric, and the GR
metric field is to be regarded as a physical field like any other
physical field.
That is, the *unified* metric field. That is Rovelli's tacit, yet 
arbitrary, assumption
IMO.
Then, the physics
depends only on the relative disposition of the various physical
fields with respect to each other.
JS: Correct, with the proviso that matter includes both real, i.e.
on mass shell, sources as well as virtual, i.e. off mass 
shell
sources. The virtual sources divide into two classes:
I. Non-exotic near EM field Fuv giant coherent quantum states of
virtual photons that contribute to Omega(Matter) of the FRW metric and
to Tuv in Einstein's local field equation.
II. Exotic vacuum w = -1 zero point stress-energy density local tensor
~ (String Tension)/zpfguv for both repulsive dark energy /zpf > 0 of
negative pressure and attractive dark matter /zpf < 0 of positive
pressure.
These exotic vacuum virtual sources contribute Omega(Exotic Vacua) ~
0.96 to Omega(Total) = 1 in our large-scale spatially flat
post-inflationary local Level I Hubble sphere brane world as in
Lenny Susskind's megalopolis Landscape subject to the natural
selection of the Weak Anthropic Principle (WAP)
OK, I think I made an error above including brane worlds in sense of 
parallel worlds?
Here is why I think I made an error (If I did so did Hawking and 
Scientific American in their popular science reports):
D-Branes are extended surfaces without edges. In order that the black 
hole be a localized object, it is assumed that our ordinary four 
dimensions (three space and one time) are all orthogonal to these 
D-Brane surfaces ... Thus to us these D-Branes would look as though they 
were located at a point (or at least a very small region) of our 
observable three dimensions of space. W. G. Unruh p. 168 Black holes, 
dumb holes, and entropy, i.e. the D-Branes are in the compactified 
Calabi-Yau space. I have to look again at Hawking's The Universe in a 
Nutshell that seems to give the wrong idea here? Perhaps I mis-remember?
Note also Ed Witten's formula generalizing Heisenberg's quantum 
uncertainty principle, i.e. eq. (5.9) p. 136
Delta X > h/DeltaP + alpha'(DeltaP)/h
The second gravity-string source of uncertainty should give the 
irreversible statistical arrow of time not found when alpha' = 0 i.e. 
infinite string tension, or infinite space-time stiffness of action 
without reaction as is also found in the signal locality of orthodox 
quantum theory in sense of Antony Valentini's papers.
PZ: But while this may allow or even imply a relational view of the
raw spacetime manifold, in the
Cartesian sense, it is clearly not physical general relativity, in
the classic Einsteinian sense,
which requires fundamental identification of the inertial and
gravitational metrics.
JS: I do not understand what you just said.  EEP in every significant
sense still holds IMHO.
PZ: You are in a loop.
JS: Ground Hog Day. Help let me out of my bottle Oh Thief of Baghdad. 
The Magic Flying Carpet is obviously the weightless Alcubierre timelike 
geodesic faster than light warp drive powered by dark energy and dark 
matter exotic vacua configurations. See my animated picture of this in 
http://qedcorp.com/APS/StarGate1.mov
PZ: EEP is NOT Einstein equivalence. EEP is merely a correspondence 

principle.
Einstein strict equivalence is NOT valid. And even EEP is not strictly 
valid as
advertised (see e.g. Ohanian and Ruffini Ch. 1).
given guv field still works for example.
PZ: This simply reflects weak equivalence -- equality of gravitational 
and inertial mass.
That does not imply Einstein equivalence. It works whether we use a 
unified g_uv
or go to a bimetric formalism, and regardless of whether we consider 
inertial forces
as real or fictitious.
JS: The so-called fictitious or inertial forces,  e.g. G-Force when 

a jet takes off (0 to 120 mph in 2 sec) from an aircraft carrier are 
physically real. I can tell you that from first-hand experience. The use 
of fictitious like the use of hidden variable are unfortunate 
choices of words. Extra variable is better for the latter.
PZ: So this is not a valid argument IMO. It is an example of the 
classical logical fallacy
affirming the antecedent.
JS: The tidal stretch-squeeze
works etc.
PZ: Of course. So?
The point is that you can always tell the difference between a real 
gravity field and
an inertial field. They are NOT completely physically equivalent.
JS: If you mean Ruvwl =/= 0 locally for a real gravity field, and is 
zero for an inertial field, I agree.
However, I do not see how you can write
guv =  guv(inertial) + guv(real)
That is, apart from
guv = Globally Flat Metric + du,v + dv,u
I do not claim we can physically distinguish the two terms on RHS nor is 
the
second term small compared to first.
The Bohm pilot constraint for the extra variable, i.e. actual 
distortion of Hagen Kleinert's  elastic-plastic 4D world crystal 
lattice
i.e. density of vortex line topological defects of curvature 
disclination and possibly torsion dislocations is
du = Lp^2(Goldstone Phase of Vacuum Coherence),u
,u is ordinary partial derivative.
Compare my equation here to Goldstein's eq. 12.4 on p. 282.
PZ: Consequently,
unified g_uv has no deep physical meaning or necessity -- as Feynman 
argued.
PZ: Which all seems to contradict his statement,
...Of course, nothing [in GR] to prevents us... from singling out the
gravitational field as 'the
more equal among equals', and declaring that location is absolute in
GR, because it can be
defined with respect to it.
JS: No, you have simply misread the context of Rovelli's remark. Read
it again more carefully. This is not a valid objection at all to
Rovelli's argument IMHO.  The way I read him, his argument is
splendidly globally self-consistent.
PZ: The way I read him, he doesn't actually *have* an argument. He 
simply *prefers*
to quantize space and time intervals rather than unpack Einstein's 
unified g_uv.
PZ: So IMO Rovelli's position is incoherent. If anything, he is
arguing for an *alternative*
non-Einsteinian interpretation of GR in which matter-dependent g_uv
is to be treated as a
*physical field*, which means it can in principle be distinguished
from the kinematical inertial
field with its trivially valid transformable metric tensor.
JS: We are back to Square One. The Rock has rolled back down on
Sisyphus. I cannot pinpoint the precise misconception that is driving
your position here.
PZ: It is no wonder you are having this difficulty, since it is 
*Rovelli's* misconception
that is causing the problem.
Rovelli doesn't seem to understand that recognition of the dynamical 
character of
non-inertial g_uv fundamentally distinguishes it from the kinematical 
g_uv, which
latter doesn't depend on the distribution of matter. So the natural 
thing to do is
to split unified g_uv and thus clearly separate the dynamical and 
kinematical metric
fields.
JS: Show me an algorithm to do that in any problem.
PZ: Yet he doesn't even consider this.
In other words, Rovelli is really a *Yilmazian*.
JS: Not in my book.
PZ: Actually, you are right -- because he does not follow the natural 
development
of his own argument to its logical conclusions. If he did, his first 
alternative (p 109)
would fall squarely in the Feynman-Yilmaz category, with rubber rods and 
clocks
and a flat background inertial metric.
JS: It seems to me the objective is to eliminate any flat background 
non-dynamical metric.
Nondynamical means ACTION WITHOUT REACTION.
Note, I have it formally in my equations but it is not physically 
measurable separately.
Indeed, the pre-inflationary vacuum is the perfectly flat world  crystal 
with no vortex string topological
defects of effective multiple-connectivity in the space of the vacuum 
coherence order parameter which
is zero everywhere prior to the inflationary vacuum phase transition 
like a normal metal going super-conducting.
PZ: Obviously it is a field that depends on the distribution of
matter, if that's what he
means. But it is the only field that is defined in terms of a
spacetime metric, so it
is obviously unique in that sense.
JS: Read Rovelli's paper in the book. Then tell me what you think.
PZ: I read it. Several times. I've been scratching my head for several
days now.
See above.
====
> I can't find a proof of the (multivariable) chain rule that makes sense.
Assume g is differentiable at x and f is differentiable at g(x), where 
differentiable means in the total derivative sense. Let  dg(x) and 
df(g(x)) denote the associated linear differential approximants. Then
   f(g(x+h)) - f(g(x)) = df(g(x))[g(x+h)) - g(x)] + o(g(x+h)) - g(x))
                     = df(g(x))[dg(x)(h) + o(h)] + o(g(x+h)) - g(x))
                      = df(g(x))[dg(x)(h)] + df(g(x))[o(h)]
                              + o(g(x+h)) - g(x)).
The first term in the last sum is what we want to see, namely the 
composition of  df(g(x)) and dg(x) evaluated at h. So we're done if we can 
show both df(g(x))[o(h)] and o(g(x+h)) - g(x)) are o(h), which is not too 
hard.
====
> I can't find a proof of the (multivariable) chain rule that makes sense.
Assume g is differentiable at x and f is differentiable at g(x), where 
>differentiable means in the total derivative sense. Let  dg(x) and 

>df(g(x)) denote the associated linear differential approximants. Then
   f(g(x+h)) - f(g(x)) = df(g(x))[g(x+h)) - g(x)] + o(g(x+h)) - g(x))
                     = df(g(x))[dg(x)(h) + o(h)] + o(g(x+h)) - g(x))
                      = df(g(x))[dg(x)(h)] + df(g(x))[o(h)]
                              + o(g(x+h)) - g(x)).
The first term in the last sum is what we want to see, namely the 
>composition of  df(g(x)) and dg(x) evaluated at h. So we're done if we can 

>show both df(g(x))[o(h)] and o(g(x+h)) - g(x)) are o(h), which is not too 
>hard.
Fine, but she said I want to see a real proof that uses 
definitions and little greek letters...
I suppose we can leave it to the reader to change a few letters
to greek to make things easier to follow. But if she's never seen
this proof then possibly she's never seen the definition:
(Def: o(h) means some function e(h) with the property
that e(h)/||h|| -> 0 as h -> 0.)
Def: f: R^n -> R^m is _differentiable_ at x if there exists a
linear map T: R^n -> R^m such that
  f(x + h) = f(x) + Th + o(h);
if so then T = df(x).
(Then we should also point out how the notation she may
be looking at follows from this: If df(x) exists then the matrix
corresponding to that linear map has the partial derivatives
of the components of f for its entries, and the formula you
see for the chain rule in some calculus books is just
giving the product of two matrices.)
************************
David C. Ullrich
====
> 
> I can't find a proof of the (multivariable) chain rule that makes 
sense.
>>Assume g is differentiable at x and f is differentiable at g(x), where 
>>differentiable means in the total derivative sense. Let  dg(x) and 

>>df(g(x)) denote the associated linear differential approximants. Then
>>   f(g(x+h)) - f(g(x)) = df(g(x))[g(x+h)) - g(x)] + o(g(x+h)) - g(x))
>>                     = df(g(x))[dg(x)(h) + o(h)] + o(g(x+h)) - g(x))
>>                      = df(g(x))[dg(x)(h)] + df(g(x))[o(h)]
>>                              + o(g(x+h)) - g(x)).
>>The first term in the last sum is what we want to see, namely the 
>>composition of  df(g(x)) and dg(x) evaluated at h. So we're done if we can 

>>show both df(g(x))[o(h)] and o(g(x+h)) - g(x)) are o(h), which is not too 

>>hard.
Fine, but she said I want to see a real proof that uses 
> definitions and little greek letters...
> 
How many students have you found who've said that? Not many?
It reads to me as though this person is taking a second year course at an
American University. The text books taught from can be pretty dire when it
comes to formal proof. Indeed one teaches three 'different' (at least) 
chain
rules for special cases, rather than just _the_ chain rule.
I'm sure that the course instructor would be happy
to provide some way for the OP to satisfy her curiosity, even if it's
pointing her towards the library and a relevant book. But just in case, as
you're the Analyst round here, which ones would be useful in demonstrating
that actually you don't want too many greek letters floating around? Is
Rudin any good? Lot's of grad programs seem to prefer it.
> I suppose we can leave it to the reader to change a few letters
> to greek to make things easier to follow. But if she's never seen
> this proof then possibly she's never seen the definition:
(Def: o(h) means some function e(h) with the property
> that e(h)/||h|| -> 0 as h -> 0.)
Def: f: R^n -> R^m is _differentiable_ at x if there exists a
> linear map T: R^n -> R^m such that
  f(x + h) = f(x) + Th + o(h);
if so then T = df(x).
(Then we should also point out how the notation she may
> be looking at follows from this: If df(x) exists then the matrix
> corresponding to that linear map has the partial derivatives
> of the components of f for its entries, and the formula you
> see for the chain rule in some calculus books is just
> giving the product of two matrices.)

> ************************
David C. Ullrich
====
> 
>>> I can't find a proof of the (multivariable) chain rule that makes 
sense.
>>Assume g is differentiable at x and f is differentiable at g(x), where 
>differentiable means in the total derivative sense. Let  dg(x) 
and 
>df(g(x)) denote the associated linear differential approximants. Then
>>   f(g(x+h)) - f(g(x)) = df(g(x))[g(x+h)) - g(x)] + o(g(x+h)) - g(x))
>>                     = df(g(x))[dg(x)(h) + o(h)] + o(g(x+h)) - g(x))
>>                      = df(g(x))[dg(x)(h)] + df(g(x))[o(h)]
>>                              + o(g(x+h)) - g(x)).
>>The first term in the last sum is what we want to see, namely the 
>composition of  df(g(x)) and dg(x) evaluated at h. So we're done if we 
can 
>show both df(g(x))[o(h)] and o(g(x+h)) - g(x)) are o(h), which is not too 

>hard.
>> 
>> Fine, but she said I want to see a real proof that uses 
>> definitions and little greek letters...
>> 

>How many students have you found who've said that? Not many?
It reads to me as though this person is taking a second year course at an
>American University. The text books taught from can be pretty dire when it
>comes to formal proof. Indeed one teaches three 'different' (at least) 
chain
>rules for special cases, rather than just _the_ chain rule.
I'm sure that the course instructor would be happy
>to provide some way for the OP to satisfy her curiosity, even if it's
>pointing her towards the library and a relevant book. But just in case, as
>you're the Analyst round here, which ones would be useful in demonstrating
>that actually you don't want too many greek letters floating around? 
Um, all the comments about greek letters have been tongue in cheek.
I don't know what text would be useful for demonstrating that you
don't want too many greek letters floating around, because the
idea that you don't want too many greek letters floating around
is news to me.
I suspect that you were being facetious, actually asking about
demonstrating something else. But I can't figure out what the
something else might be.
>Is
>Rudin any good? Lot's of grad programs seem to prefer it.

>> I suppose we can leave it to the reader to change a few letters
>> to greek to make things easier to follow. But if she's never seen
>> this proof then possibly she's never seen the definition:
>> 
>> (Def: o(h) means some function e(h) with the property
>> that e(h)/||h|| -> 0 as h -> 0.)
>> 
>> Def: f: R^n -> R^m is _differentiable_ at x if there exists a
>> linear map T: R^n -> R^m such that
>> 
>>   f(x + h) = f(x) + Th + o(h);
>> 
>> if so then T = df(x).
>> 
>> (Then we should also point out how the notation she may
>> be looking at follows from this: If df(x) exists then the matrix
>> corresponding to that linear map has the partial derivatives
>> of the components of f for its entries, and the formula you
>> see for the chain rule in some calculus books is just
>> giving the product of two matrices.)
>> 
>> 
>> ************************
>> 
>> David C. Ullrich
************************
David C. Ullrich
====
Need help on math, visit www.helptosolve.com Just try.
Want to help others with math, visit www.helptosolve.com and join the team.
====
> 
>>Dean said that Northeasterners don't talk about religion. 
>>That's not true, because I know of Northeasterners who spend almost
>>all day talking about The Virgin Mary, Hearing Confession, who's Rabbi
>>is where, what Temple this person goes to, I have another Bar Mitzvah,
>>Episcopal Parishoners this, American Baptists convention that.
>>What Northeast is Dean from ??

> What planet are you from?
We have people from every planet on Earth in California. --- Former
governor Gray Davis
-- 
http://hertzlinger.blogspot.com
====
> Many of the problems with Bush's agenda is the dishonest
> reinterpretation of reality that is done so often by the
> Democrats.
Citations, please.
====
> Dean said that Northeasterners don't talk about religion. 
That's not true, because I know of Northeasterners who spend almost
> all day talking about The Virgin Mary, Hearing Confession, who's Rabbi
> is where, what Temple this person goes to, I have another Bar Mitzvah,
> Episcopal Parishoners this, American Baptists convention that.
What Northeast is Dean from ??
What does this have to do with mathematics?
-- 
http://hertzlinger.blogspot.com
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
>Dean said that Northeasterners don't talk about religion. 
K3wl? How is that on topic for sci.math, sci.skeptic, alt.atheism or
alt.christnet? Of the 5 groups that you posted to, only
alt.politics.democrats is relevant; clearly your intent was to troll
for a crss-posted flame war.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
> Dean said that Northeasterners don't talk about religion. 
Maybe you should furnish an actual quote before you build a castle in 
the 
> air upon what you think he might have said.  
''My father used to tell us how much strength he got from religion, but 
> we   didn't have Bible readings. There are traditions where people do 
> that. We   didn't,'' he said. ''People in the Northeast don't talk about 
> their religion. It's a   very personal private matter, and that's the 
> tradition I was brought up in.''   Howard Dean, to Boston Globe last 
> week.  Quoted in countless other papers.
You gotta interpret that in context.  It means northeastern politicians
don't talk to the papers about their religion (they leave that to 
preachers).  It was definitely more polite than what I would have said, 
which would be, I didn't come here today to talk about religion, so 
what's your next question?
====
> 
>> Dean said that Northeasterners don't talk about religion. 
>> 
>> Maybe you should furnish an actual quote before you build a castle in 
the 
>> air upon what you think he might have said.  
>> 
>> ''My father used to tell us how much strength he got from religion, but 
>> we   didn't have Bible readings. There are traditions where people do 
>> that. We   didn't,'' he said. ''People in the Northeast don't talk about 

>> their religion. It's a   very personal private matter, and that's the 
>> tradition I was brought up in.''   Howard Dean, to Boston Globe last 
>> week.  Quoted in countless other papers.
You gotta interpret that in context.  It means northeastern politicians
>don't talk to the papers about their religion (they leave that to 
>preachers).  It was definitely more polite than what I would have said, 
>which would be, I didn't come here today to talk about religion, so 
>what's your next question?
By the same logic the previous poster used, Bush has also told lies.
President Bush's compassionate agenda resonates with the people of
both New York and California, 
        -Tracey Schmitt,spokeswoman for the Bush-Cheney '04
campaign
That's not true, because I know of Californians who are not impressed
with Bush's compassionate agenda.
 
 
-----
Yang 
a.a. #28
a.a. pastor #-273.15, the most frigid church of Celcius nee Kelvin 
EAC Econometric Forecast and Socerey Division
Proudly plonked by Lani Girl and Crazyalec
The Bush 'balanced' budget:             -525 billion and worsening
The Bush 'economic' policy:             -3 million jobs and counting
The Bush Iraq lie:                      -472 GIs, one friend's co-worker's 
son and mounting
Having Bush fuck up my country:           Worthless
====
> Dean said that Northeasterners don't talk about religion.
We don't, certainly not when compared to other parts of
the country.
Most people I know consider it an impolite subject, at
least in public & amongst strangers.
====
 > Dean said that Northeasterners don't talk about religion.
>  > Maybe you should furnish an actual quote before you build a castle in
the
> air upon what you think he might have said.
>  ''My father used to tell us how much strength he got from religion, but
> we   didn't have Bible readings. There are traditions where people do
> that. We   didn't,'' he said. ''People in the Northeast don't talk about
> their religion. It's a   very personal private matter, and that's the
> tradition I was brought up in.''   Howard Dean, to Boston Globe last
> week.  Quoted in countless other papers.
I heard my father once say that people in the South are
very conservative religiously. Well I live in Atlanta now
and am friends with some folk at the Unitarian Church.
My father. What a dirty, rotten liar.
====
|Did you see my reply to the OP?
Yes, although the contents were not still clear in mind when I last
Amanda's acquaintance objected to her proof based on its being a
proof by contradiction. I guess I think the distinction between a
proof by proving the contrapositive and a proof by contradiction is
fuzzy enough not to make this strictly incorrect. For the question
of whether the proof is valid, I think you'd agree it's also not a
distinction that could be relevant. People consider the proof okay
not because it's crossed this fuzzy line; people consider the proof
okay because we're considering a context where proof by contradiction
and the whole complex of related forms of reasoning are all accepted.
You had two main reasons why proof by contradiction was considered
worse:
|(i) it's easy to give a wrong proof by contradiction, where 
|the contradiction arises from some error 
|
|(ii) if you give a direct proof that A implies B, by assuming
|A and then deducing B, the steps in the proof can give
|some insight into what A really entails - you show A implies
|B by showing A implies C and C implies B, and along the
|way you've shown two facts that might be interesting and
|useful elsewhere, that A implies C and that C implies B.
|You don't get this sort of bonus from a proof by contradiction,
|since in the course of the proof you're assuming things
|which it turns out never actually hold.
I think the practice of constructive proof may shed some light on
this. The conclusion of a constructive proof tells you more about
the nature of the proof than the conclusion of a classical proof,
generally. In particular, when the conclusion is negative, the proof
was a proof by contradiction (essentially) and when the conclusion
is positive it wasn't.
I can't tell whether the circumstances reducing the risk of (i) and
enhancing the prospects of (ii) are different from the circumstances
where the theorem (as it would be stated in constructive terms) is
more solidly stated.
In constructive mathematics one has the general advice of Bishop
to cut down on negations. If we seek to tighten up our theorems by
getting rid of flabby negations, then we also as a side-effect
refrain from proofs by contradiction.
Lots of negations and implications can be upgraded by supplying them
with more solid content. Bishop has a paper which uses Goedel's
Dialectica paper for this. Goedel has a method of rewriting
statements which puts them into a form
   (there exists w) (for all t) P(w,t)
where w ranges over a set of possible witnesses to the truth of
the statement, and t ranges over a set of possible cases. If this
statement is false, one would like to do more than just negate it;
one would like to say something about the function f:w->t for which
P(w, f(w)) fails for all w. A proof of the negative statement
implicitly provides such an f, as witness. Even when a statement
is negative, then, there can be implicit solid content in the
construction demonstrating the fact, in terms of information on
and around this f.
I get the impression that in cases where a theorem is negative,
but has relatively tangible constructive content, one is not so
especially prone to err in the way described in (i), because one
is regarding the construction in the proof (of f, say), which
actually *does* exist, and trying to see whether *it* is as
advertised. In the proof that there are infinitely many primes,
to pick a simple example, the engine is the method for getting
a new prime out of a finite set of old primes. An argument which
focusses on this kind of concrete construction doesn't seem to
suffer from the kind of problem that some attempts at proof by
contradiction do.
I'm going on impressions, though, and it seems like it would
take somewhat tricky casework to get a more solid sense for what
kinds of proof do and do not tend to suffer from these problems.
Keith Ramsay
====
|Did you see my reply to the OP?
Yes, although the contents were not still clear in mind when I last
Amanda's acquaintance objected to her proof based on its being a
>proof by contradiction. 
Although I don't think that even he was saying the proof was
therefore _wrong_, just that it was not the best possible proof.
(Cf for example the title of the thread; it's about finding a
beautiful proof, not a _valid_ proof...)
>I guess I think the distinction between a
>proof by proving the contrapositive and a proof by contradiction is
>fuzzy enough not to make this strictly incorrect. 
I'm honestly not sure what you mean by that sentence, but
regarding the minor premise, I don't see what's fuzzy about
the distinction. To prove A implies B:
(a) Assume ~B. Prove ~A
or
(b) Assume A and ~B. Prove P and ~P for some P.
Some proofs of A implies B have the form (a); some
have the form (b). What's fuzzy here?
Oh. You're claiming that the distinction is fuzzy enough
that those of us who are saying he was wrong in
stating the proof was a proof by contradiction are not
on solid ground? I disagree. Um, lemme put that a
little more strongly: that's not so. The proof Amanda
gave is of the form (a), not of the form (b).
I mean that's just a mathematical _fact_. You can say
you don't see why it matters, fine. But you say above
you think the distinction between (a) and (b) is
fuzzy - I don't see any fuzziness, and it's simply a
fact that (a) is an outline of the proof she gave, for
appropriate A and B, while (b) simply isn't.
>For the question
>of whether the proof is valid, I think you'd agree it's also not a
>distinction that could be relevant. People consider the proof okay
>not because it's crossed this fuzzy line; people consider the proof
>okay because we're considering a context where proof by contradiction
>and the whole complex of related forms of reasoning are all accepted.
Yes of _course_. Of the people in this thread who do see the 
difference between (a) and (b) above, some of whom have been
saying that (a) is preferable, _none_ of them have stated or
even hinted that (b) is invalid.
>You had two main reasons why proof by contradiction was considered
>worse:
|(i) it's easy to give a wrong proof by contradiction, where 
>|the contradiction arises from some error 
>|
>|(ii) if you give a direct proof that A implies B, by assuming
>|A and then deducing B, the steps in the proof can give
>|some insight into what A really entails - you show A implies
>|B by showing A implies C and C implies B, and along the
>|way you've shown two facts that might be interesting and
>|useful elsewhere, that A implies C and that C implies B.
>|You don't get this sort of bonus from a proof by contradiction,
>|since in the course of the proof you're assuming things
>|which it turns out never actually hold.
I think the practice of constructive proof may shed some light on
>this. The conclusion of a constructive proof tells you more about
>the nature of the proof than the conclusion of a classical proof,
>generally. In particular, when the conclusion is negative, the proof
>was a proof by contradiction (essentially) and when the conclusion
>is positive it wasn't.
I can't tell whether the circumstances reducing the risk of (i) and
>enhancing the prospects of (ii) are different from the circumstances
>where the theorem (as it would be stated in constructive terms) is
>more solidly stated.
In constructive mathematics one has the general advice of Bishop
>to cut down on negations. If we seek to tighten up our theorems by
>getting rid of flabby negations, then we also as a side-effect
>refrain from proofs by contradiction.
Lots of negations and implications can be upgraded by supplying them
>with more solid content. Bishop has a paper which uses Goedel's
>Dialectica paper for this. Goedel has a method of rewriting
>statements which puts them into a form
   (there exists w) (for all t) P(w,t)
where w ranges over a set of possible witnesses to the truth of
>the statement, and t ranges over a set of possible cases. If this
>statement is false, one would like to do more than just negate it;
>one would like to say something about the function f:w->t for which
>P(w, f(w)) fails for all w. A proof of the negative statement
>implicitly provides such an f, as witness. Even when a statement
>is negative, then, there can be implicit solid content in the
>construction demonstrating the fact, in terms of information on
>and around this f.
I get the impression that in cases where a theorem is negative,
>but has relatively tangible constructive content, one is not so
>especially prone to err in the way described in (i), because one
>is regarding the construction in the proof (of f, say), which
>actually *does* exist, and trying to see whether *it* is as
>advertised. In the proof that there are infinitely many primes,
>to pick a simple example, the engine is the method for getting
>a new prime out of a finite set of old primes. An argument which
>focusses on this kind of concrete construction doesn't seem to
>suffer from the kind of problem that some attempts at proof by
>contradiction do.
I'm going on impressions, though, and it seems like it would
>take somewhat tricky casework to get a more solid sense for what
>kinds of proof do and do not tend to suffer from these problems.
The context here, or at least the context I had in mind, was
proofs by students in classes where half the point to the class
is learning to read and write proofs (beginning algebra,
analysis, topology classes are often in this category).
In that context I can tell you from experience that (i)
is a real danger. Of course it's much less of a problem
when we're talking about proofs in the real world written
by grownups - one has an idea what _sort_ of contradiction
to expect, so when one gets a totally irrelevant contradiction
one looks for the error instead of saying qed.
>Keith Ramsay
************************
David C. Ullrich
====
>  > For the greater cogency and obviousness in your paper THERE
> SHOULD BE a TABLE with componentries of system, which one are
> sorted out on their influence to precision of system GPS as a whole.
Such a table was posted by Sam Wormley on the 22nd, see
Then any layman can see that we can neglect  neglible small
> relativistic corrections  as contrasted to by other factors
> defining and restricting limiting accuracies GPS as a whole.

>   VI. Summary
>  >   Excluding the deliberate degradation of SA, the dominant error source
>   for satellite ranging with single frequency receivers is usually the
>   ionosphere. It is on the order of four meters, depending on the
>   quality of the single-frequency model. For dual-frequency (P/Y-code)
>   receivers (which eliminate SA) the Standard Error Model of Table I
>   has one principal change (in addition to the elimination of the SA
>   error). The ionospheric error is reduced from four meters to about
>   one meter.

> The GR correction is about 44 microseconds or 13km per day
> and would be cumulative. You can hardly call that negligible.
1. What periodicity of corrections of parameters in GPS? 
2. What parameters are adjusted in GPS? 
3. What medial - statistical values have parameters adjusted 
   in GPS in each session of corrections?
--
Aleksandr
====
>  > For the greater cogency and obviousness in your paper THERE
> SHOULD BE a TABLE with componentries of system, which one are
> sorted out on their influence to precision of system GPS as a whole.
>  > Such a table was posted by Sam Wormley on the 22nd, see
>  > Then any layman can see that we can neglect  neglible small
> relativistic corrections  as contrasted to by other factors
> defining and restricting limiting accuracies GPS as a whole.
>   >   VI. Summary
>  >   Excluding the deliberate degradation of SA, the dominant error
source
>   for satellite ranging with single frequency receivers is usually 
the
>   ionosphere. It is on the order of four meters, depending on the
>   quality of the single-frequency model. For dual-frequency 
(P/Y-code)
>   receivers (which eliminate SA) the Standard Error Model of Table I
>   has one principal change (in addition to the elimination of the SA
>   error). The ionospheric error is reduced from four meters to about
>   one meter.
>   > The GR correction is about 44 microseconds or 13km per day
> and would be cumulative. You can hardly call that negligible.
 1. What periodicity of corrections of parameters in GPS?
The corrections are of the order of nanoseconds per day so
confirm GR to roughly one part in 10,000 by that simplistic
comparison. There is much more information available in web
pages if you want to look for more details.
> 2. What parameters are adjusted in GPS?
> 3. What medial - statistical values have parameters adjusted
>    in GPS in each session of corrections?
Happy New Year
George
====
To base 10:
STEP 1: For the characteristic: carry out repeated division of the number 
by
10 until the result falls below 10 and count the number of times.
STEP 2: For the mantissa: raise the result to the power of 10 using
multiplication only, then divide by 10 again as in Step 1. Continue to Step
2 for further digits.
This process generates the logarithm digit by digit, to the same accuracy 
as
the other arithmetical functions. Also works efficiently in binary.
This has almost certainly been tried before. I would be interested in any
====
equation.
The equation is
Ax=0
A is a very large sparse matrix which have 1473-by-1473.
Sum(x_i) is 1.0
All x are not zero.
Please, would you like to let me know how to solve the simultaneous
equation by fortran.
Additionally, My major is not mathmatics. Thus, it is difficult to
solve the equation in a method of pseudoinverse, SVD, etc.. Please,
let me know that, easily and in detail.
====
> equation.
 The equation is
 Ax=0
 A is a very large sparse matrix which have 1473-by-1473.
> Sum(x_i) is 1.0
> All x are not zero.
 Please, would you like to let me know how to solve the simultaneous
> equation by fortran.
 Additionally, My major is not mathmatics. Thus, it is difficult to
> solve the equation in a method of pseudoinverse, SVD, etc.. Please,
> let me know that, easily and in detail.
>
Well, I haven't dealt with matrices that large, but maybe you could try
Cramer's rule.  As for the code, you are on your own, I don't know Fortran.
You could try:
http://www.library.cornell.edu/nr/bookfpdf.html
Lurch
====
> The equation is
>> Ax=0
>> A is a very large sparse matrix which have 1473-by-1473.
>Well, I haven't dealt with matrices that large, but maybe you could try
>Cramer's rule. 
You know, I think all of us have a tendency to think we can answer
more questions well than we really can. I don't know a whole heckuva lot
about numerical analysis, but this strikes me as a really, really bad
answer. For one thing, Cramer's rule gives an explicit solution to
Ax=b  when  A  is invertible; the OP's matrix  A  is surely not
invertible, lest the only solution be  x = 0. For another thing,
Cramer's rule expresses the answer in terms of (here) 1474 determinants
of 1473x1473 matrices. How do you propose that the determinants be
evaluated? A naive algorithm sums 1473! products and introduces the
possibility of tremendous quantities of round-off error and massive
cancellations. Even using row operations, while much faster, is still
time consuming and prone to error if the entries are not, say, 
small integers. As useful as Cramer's rule can be for theoretical work,
I imagine it's essentially _never_ used for computing numerical solutions
to linear equations with more than a handful of variables.
So what response might be better for the OP?
It's probably true that any good matrix package can handle your
matrix (a couple thousand rows is not considered huge these days)
but since you say your matrix is sparse, you would probably benefit
from using routines specifically written for this common special case.
Here is a link which was posted the other day in sci.math.num-analysis
(which is a much more appropriate group for technical questions
of this type):
    http://vlsicad.cs.ucla.edu/sparse.html
Of course the equation  Ax=0 has no solution with sum(x_i)=1 if  A  is
nonsingular, so it would be good to think about how you know that
your problem has a solution. More generally, it is likely to help
those who are helping you if you can indicate where your matrix comes
from, since that may signal the use of special routines e.g. for
solving differential equations numerically.
dave
====
> 
>> The equation is
>> Ax=0
>> A is a very large sparse matrix which have 1473-by-1473.
 >Well, I haven't dealt with matrices that large, but maybe you could try
>Cramer's rule.
 You know, I think all of us have a tendency to think we can answer
> more questions well than we really can. I don't know a whole heckuva lot
> about numerical analysis, but this strikes me as a really, really bad
> answer. For one thing, Cramer's rule gives an explicit solution to
> Ax=b  when  A  is invertible; the OP's matrix  A  is surely not
> invertible, lest the only solution be  x = 0. For another thing,
> Cramer's rule expresses the answer in terms of (here) 1474 determinants
> of 1473x1473 matrices. How do you propose that the determinants be
> evaluated? A naive algorithm sums 1473! products and introduces the
> possibility of tremendous quantities of round-off error and massive
> cancellations. Even using row operations, while much faster, is still
> time consuming and prone to error if the entries are not, say,
> small integers. As useful as Cramer's rule can be for theoretical work,
> I imagine it's essentially _never_ used for computing numerical solutions
> to linear equations with more than a handful of variables.
 So what response might be better for the OP?
 It's probably true that any good matrix package can handle your
> matrix (a couple thousand rows is not considered huge these days)
> but since you say your matrix is sparse, you would probably benefit
> from using routines specifically written for this common special case.
> Here is a link which was posted the other day in sci.math.num-analysis
> (which is a much more appropriate group for technical questions
> of this type):
>     http://vlsicad.cs.ucla.edu/sparse.html
 Of course the equation  Ax=0 has no solution with sum(x_i)=1 if  A  is
> nonsingular, so it would be good to think about how you know that
> your problem has a solution. More generally, it is likely to help
> those who are helping you if you can indicate where your matrix comes
> from, since that may signal the use of special routines e.g. for
> solving differential equations numerically.
 dave
Hey, it's an open forum!  I consider this NG like an online math club.  I
offer suggestions, and they aren't always useful, or correct.  None of this
stuff is being published, so relax.  He isn't paying me for my help; so, he
is free to take my advice, or not.  It is up to him.  I also provided a 
link
to an online book of Numerical analysis in Fortran, which should answer his
question.
Lurch
====
>> The equation is
>> Ax=0
>> A is a very large sparse matrix which have 1473-by-1473.
>Well, I haven't dealt with matrices that large, but maybe you could try
>>Cramer's rule. 
You know, I think all of us have a tendency to think we can answer
>more questions well than we really can. [...]
I have an explanation for that... hmm, no I don't.
************************
David C. Ullrich
====
I am looking for a formula for the calculation of the midpoint for the
smallest, an area A circumscribing circle. Is there like for triangles
a connection between the center of gravity and the center of that
circle? Are any formulas for such a circle known?
Carolin Hau§ner
====
hi
wikipedia needs our help to carry on:
http://www.wikimedia.org/letter.html
thanks to all
bye
max
====
What's happened to you ?
I get up early every morning to read your diatribes and I haven't seen 
anything for a few days now.
Does this mean I have to learn maths if I want to be a part of this 
newsgroup ?
Does this mean I have to rely on comics for my cackles each morning ?
I hope the FBI didn't turn on you and that the Evil Mathematical 
Establishment (tm) haven't imprisioned you in an infinite series !
Please hurry back, my humour is suffering and my education lacking - at 
least I was being *forced* to learn maths while you posted for no other 
reason than trying to understand the people who replied to you!
Ivan.
So this poster who's opinion, by his own admissions in terms of
mathematics is worthless, still feels that his opinion is of value,
clearly because he's at least learned the true nature of the math
community. JSH writing about me in the Focus on point of dispute, more 
math thread ... I've been villified by James, does this make me famous?
====
the application of nonlinear systems (particularly catastrophe
theory) to human behavior. Through the years I have struggled with
balancing the criticisms of catastrophe methodology and the heuristic
merits of the theory. I've read most of the catastrophe literature and
theory and singularities.
Although the proponents of catastrophe theory are also somewhat
convincing, I question its applicability and implementation in
psychological research. I'm at a point where I need to make a
decision:
Do I drop this line of research or do I continue working toward the
development of catastrophe applications (e.g., write grants to develop
software for analyzing catastrophes/bifurcations in psychological
data)?
Kate
====
the application of nonlinear systems (particularly catastrophe
> theory) to human behavior. Through the years I have struggled with
> balancing the criticisms of catastrophe methodology and the heuristic
> merits of the theory. I've read most of the catastrophe literature and
> theory and singularities.
Although the proponents of catastrophe theory are also somewhat
> convincing, I question its applicability and implementation in
> psychological research. I'm at a point where I need to make a
> decision:
Do I drop this line of research or do I continue working toward the
> development of catastrophe applications (e.g., write grants to develop
> software for analyzing catastrophes/bifurcations in psychological
> data)?
Kate
It would seem that this is more a question for your advisor/committee/
faculty to help you answer.  IMO, as a grad student it is generally
advisable to do something that gives you a high probability of
producing a respectable dissertation ASAP.  A speculative fishing
expedition (i.e. one where you're not sure there are any fish in the
lake as opposed to the standard risk of just not catching any of fish
there are) is not what you should undertake early in your career.
However, if you still want to pursue it, ask around about funding
prospects, that is is anyone supporing this type of work?  Maybe
there is all kinds of homeland securtiy money being thrown at
this area in an attempt to figure out terrorists or something (don't
laugh, they're spending billions on a missle defense system that
hasn't worked yet).  This is where faculty should be able, through
their connections, to help you with information on research and
funding trends.  But unless your advisor knows about this area and
is supportive of the idea, you're asking for problems, IMO.
Good luck,
Russell
====
> the application of nonlinear systems (particularly catastrophe
> theory) to human behavior. Through the years I have struggled with
> balancing the criticisms of catastrophe methodology and the heuristic
> merits of the theory. I've read most of the catastrophe literature and
> theory and singularities.
 Although the proponents of catastrophe theory are also somewhat
> convincing, I question its applicability and implementation in
> psychological research. I'm at a point where I need to make a
> decision:
 Do I drop this line of research or do I continue working toward the
> development of catastrophe applications (e.g., write grants to develop
> software for analyzing catastrophes/bifurcations in psychological
> data)?
 Kate
Why study pseudo-science anyway?
====
} Why study pseudo-science anyway?
All the better to refute asinine one-liners, my dear.
====

> } Why study pseudo-science anyway?
 All the better to refute asinine one-liners, my dear.
>
Are you claiming it isn't a pseudo-science?
Lurch
====
the application of nonlinear systems (particularly catastrophe
> theory) to human behavior. Through the years I have struggled with
> balancing the criticisms of catastrophe methodology and the heuristic
> merits of the theory. I've read most of the catastrophe literature and
> theory and singularities.
>  > Although the proponents of catastrophe theory are also somewhat
> convincing, I question its applicability and implementation in
> psychological research. I'm at a point where I need to make a
> decision:
>  > Do I drop this line of research or do I continue working toward the
> development of catastrophe applications (e.g., write grants to develop
> software for analyzing catastrophes/bifurcations in psychological
> data)?
>  > Kate
Why study pseudo-science anyway?
Which one? ;-)
Russell
====
> Does the natural density of all positive integers that are
the sum of two odd
> primes = 1/2? How about the sum of two squares? The sum of
two squarefree
> integers?
**************************************************
If the Goldbach conjecture is true, then the answer to your
first question is yes.
If a number is the sum of two squares then it can't have a
prime factor that is 3 modulo 4 and whose highest power is
odd.  Since the series  1 + 1/3 + 1/7 + 1/11 + 1/19 + . . .
diverges, this would imply that the natural density of the
numbers that are the sum of two squares is 0.
    ( [ (1+1/3)(1+1/7)(1+1/11) . . . ] ^ (-1)  =  0.)
_________________________________________________________
Eric J. Wingler  (wingler@math.ysu.edu)
Dept. of Mathematics and Statistics
Youngstown State University
One University Plaza
Youngstown, OH  44555-0001
330-941-1817
====
>> Does the natural density of all positive integers that are
>the sum of two odd
>> primes = 1/2? How about the sum of two squares? The sum of
>two squarefree
>> integers?
**************************************************
>If the Goldbach conjecture is true, then the answer to your
>first question is yes.
If a number is the sum of two squares then it can't have a
>prime factor that is 3 modulo 4 and whose highest power is
>odd.  Since the series  1 + 1/3 + 1/7 + 1/11 + 1/19 + . . .
>diverges, this would imply that the natural density of the
>numbers that are the sum of two squares is 0.
>    ( [ (1+1/3)(1+1/7)(1+1/11) . . . ] ^ (-1)  =  0.)
I give up. Supposing that everything you say is true, which
I imagine it is, how does it follow that the sums of two squares
have density 0?
(My guess is that you deduce somehow that the sum of the
reciprocals of the sums of two squares is finite. If so that's
interesting because the sum of 1/(j^2 + k^2) is infinite...)
_________________________________________________________
Eric J. Wingler  (wingler@math.ysu.edu)
>Dept. of Mathematics and Statistics
>Youngstown State University
>One University Plaza
>Youngstown, OH  44555-0001
>330-941-1817
>
************************
David C. Ullrich
====
>>If a number is the sum of two squares then it can't have a
>>prime factor that is 3 modulo 4 and whose highest power is
>>odd.  Since the series  1 + 1/3 + 1/7 + 1/11 + 1/19 + . . .
>>diverges, this would imply that the natural density of the
>>numbers that are the sum of two squares is 0.
>>    ( [ (1+1/3)(1+1/7)(1+1/11) . . . ] ^ (-1)  =  0.)
>I give up. Supposing that everything you say is true, which
>I imagine it is, how does it follow that the sums of two squares
>have density 0?
Let S(p) be the set of positive integers that are either not divisible by p
or are divisible by p^2.  The natural density of S is 1 - 1/p + 1/p^2.
Any positive integer that is the sum of two squares must be in S(p) for
all primes p = 3 mod 4.  If these primes are p_1, p_2, p_3, ..., the
natural density of intersection_{j=1}^n S(p_j) is 
product_{j=1}^n (1 - 1/p_j + 1/p_j^2) 
  = product_{j=1}^n exp(-1/p_j) (1+O(1/p_j^2))
  <= C exp(-sum_{j=1}^n 1/p_j))
which goes to 0 as n -> infinity because the series sum_{j=1}^infty 1/p_j 
diverges.
Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2
====
>>If a number is the sum of two squares then it can't have a
>prime factor that is 3 modulo 4 and whose highest power is
>odd.  Since the series  1 + 1/3 + 1/7 + 1/11 + 1/19 + . . .
>diverges, this would imply that the natural density of the
>numbers that are the sum of two squares is 0.
>    ( [ (1+1/3)(1+1/7)(1+1/11) . . . ] ^ (-1)  =  0.)
>I give up. Supposing that everything you say is true, which
>>I imagine it is, how does it follow that the sums of two squares
>>have density 0?
Let S(p) be the set of positive integers that are either not divisible by 
p
>or are divisible by p^2.  The natural density of S is 1 - 1/p + 1/p^2.
Ah. That's the bit that hadn't clicked - couldn't see where the minus
signs in the expansion of 1/(1+1/p) were going to come in, but there
it is. Duh.
>Any positive integer that is the sum of two squares must be in S(p) for
>all primes p = 3 mod 4.  If these primes are p_1, p_2, p_3, ..., the
>natural density of intersection_{j=1}^n S(p_j) is 
>product_{j=1}^n (1 - 1/p_j + 1/p_j^2) 
>  = product_{j=1}^n exp(-1/p_j) (1+O(1/p_j^2))
>  <= C exp(-sum_{j=1}^n 1/p_j))
>which goes to 0 as n -> infinity because the series sum_{j=1}^infty 1/p_j 
>diverges.
Robert Israel                                israel@math.ubc.ca
>Department of Mathematics        http://www.math.ubc.ca/~israel 
>University of British Columbia            
>Vancouver, BC, Canada V6T 1Z2
************************
David C. Ullrich
====
THe cost of a long-distance telphone call is determined by aflat fee for 
teh first 5 minutes and a fixed amount for each additional minute. If a 15-
minute telephone call costs $3.25 and a 23-minute call costs $5.17, find 
the cost of a 30-minute call.
        I was thinking of setting it up as a system of equations but im not 

really sure, can someone explain how i would set this problem up, cause 
alwyas get stumped on these type of questions. TIA
---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption 
=---
====
> THe cost of a long-distance telphone call is determined by aflat fee for
> teh first 5 minutes and a fixed amount for each additional minute. If a
15-
> minute telephone call costs $3.25 and a 23-minute call costs $5.17, find
> the cost of a 30-minute call.

>     I was thinking of setting it up as a system of equations but im not
> really sure, can someone explain how i would set this problem up, cause
> alwyas get stumped on these type of questions. TIA

News==----
> http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000
> ---= 19 East/West-Coast Specialized Servers - Total Privacy via 
Encryption
=---
Let x = the flat fee
and y = the fixed amount after the first five
,then
5x + 18y = 5.17
5x + 10y = 3.25
You get x = .17 and y = .24.
Lurch
====
> THe cost of a long-distance telphone call is determined by aflat fee for
>> teh first 5 minutes and a fixed amount for each additional minute. If a
>15-
>> minute telephone call costs $3.25 and a 23-minute call costs $5.17, find
>> the cost of a 30-minute call.
Let x = the flat fee
>and y = the fixed amount after the first five
>,then
5x + 18y = 5.17
>5x + 10y = 3.25
You get x = .17 and y = .24.
Lurch
>
No, a flat fee is fixed and pays for anything up to including 5
minutes, not a per minute fee. You should have x in the equations
instead of 5x. The first five minutes cost .85 (the flat fee) and the
additional minutes are .24 each. A 2 minute call would also cost .85
because you don't apply the flat fee on a per minute basis.
--Lynn
--Lynn
====
  >> THe cost of a long-distance telphone call is determined by aflat fee
for
>> teh first 5 minutes and a fixed amount for each additional minute. If 
a
>15-
>> minute telephone call costs $3.25 and a 23-minute call costs $5.17,
find
>> the cost of a 30-minute call.
  >Let x = the flat fee
>and y = the fixed amount after the first five
>,then
>  >5x + 18y = 5.17
>5x + 10y = 3.25
>  >You get x = .17 and y = .24.
>  >Lurch
No, a flat fee is fixed and pays for anything up to including 5
> minutes, not a per minute fee. You should have x in the equations
> instead of 5x. The first five minutes cost .85 (the flat fee) and the
> additional minutes are .24 each. A 2 minute call would also cost .85
> because you don't apply the flat fee on a per minute basis.
 --Lynn
 --Lynn
>
Let c=cost m=cost per minute over 5 and x=call length
c=.85+m(x-5)
c=.85+m(x-5)
c=.85+mx-5m, x>5
As Lynn said, anything below 5 remains .85
A 15 minute call costs $3.25 so x=15 and c=3.25
3.25=.85+15m-5m
2.4=10m
m=$.24
So Now we know m and our formula becomes c=.24x-.35
So when x=30, c=6.85. Therefore a 30 minute call costs $6.85
David Moran
====
Can the equation for c be transformed into an elliptic curve?
> The Barcelona conjecture:
>  > Let c=(x+y+z)^p/(pxyz2^p)
>  > for integer c,x,y,z and p prime greater than or equal to 5, the
> Barcelona conjecture is that no solutions exist with gcd(c,xyz)=1 (no
> c exist that shares no factor with x or y or z).
> I haven't seen this conjecture before, but compare the Beal conjecture:
> www.math.unt.edu/~mauldin/beal.html
Yes, I was aware of the Beal conjecture and briefly attempted to prove
> it also :)
The reason you hadn't seen the Barcelona conjecture is that it is
> virtually unknown outside of this newsgroup since I've only posted it
> here and in the research math group. I came up with it while
> attempting to prove FLT using elementary techniques ( ok, once we are
> done laughing the question remains - who hasn't? I mean even JSH keeps
> on trying.)
In some respects it should be easier to prove FLT using the Barcelona
> conjecture as the latter places less restrictions on the value of c -
> maybe even Fermat was working on this approach as it only requires
> elementary methods, though I really doubt it as it isn't documented in
> Ribenboim's book on FLT.
If anyone can lend me a hand I'm trying to get a grip on how FLT was
> tied to elliptic curves, my goal being to see how feasible it is to
> apply the same methods to the Barcelona conjecture - for the moment it
> is way too difficult for me.
> 
====
BTW; did anyone here bother to check out David Sereda, and of his
http://www.ufonasa.com/ EVIDENCE the case for NASA UFOs
In some recent additions to my MAZDA like Internal Rocket Rotary
Combustion Engine (IRRCE sfc = 15+KW/kg), there seems we also have
ourselves a wee bit of lunar He3 to burn off.
http://guthvenus.tripod.com/gv-h2o2-irrce.htm
The taking of lunar He3 is first come first served, that could be
the likes of China or Russia because, we're obviously not smart
enough.
Venus still offers life; via moon He3 could help turn the trick
I've got a few more words of wisdom to offer on behalf of the ARTEMIS
PROJECT (lunar He3) http://guthvenus.tripod.com/gv-lse-he3.htm
The rest of this report isn't entirely related to energy so much as it
relates to truth or consequences. Such as for this next topic of there
being other life NOT as we know it on Venus, that's obviously opposing
the sorts of anti-humanity folks that couldn't care less if our entire
world was destroyed by their resident warlord. Perhaps these folks can
get their next level of future funding from the same source as Bush,
Salem Laden.
As for making policy look like happenstance, and/or vice versa, is key
to snookering folks. http://guthvenus.tripod.com/moon-04.htm
Though as for we humans need not, and perhaps we should not venture
ourselves much beyond Venus L2 (VL2). Wouldn't want to contaminate a
perfectly good planet with our inferior DNA nor lack of morals,
especially of this group that's bashing honest research just out of
spite. Besides, their stealth donkey-carts could be far more lethal
than what our WMD donkey-carts can manage.
As far as our human physiology being adaptable to pressure. Under such
pressure things are not nearly as hot as we've been told, and you wont
need but a fraction of a percent of O2. Of course, that degree of
adaptation might have to be accommodated at a modus rate of a few bars
per day.
http://guthvenus.tripod.com/venus-air.htm
I have a few other recent/ongoing comments on H2O2/C12H26 and of He3:
http://guthvenus.tripod.com/gv-irrce.htm
http://guthvenus.tripod.com/gv-hybrid-irc.htm
http://guthvenus.tripod.com/gv-cm-ccm-01.htm
http://guthvenus.tripod.com/gv-lm-1.htm
http://guthvenus.tripod.com/radio-maybe.htm
====
Portfolio of PAF as of 28DEC03
BCE  400  21.90  $8,760.00
BLS 50  27.92  $1,396.00
BMY  100   27.80  $2,780.00 
Q   50,000  3.92 $196,000.00
SBC   11,600  25.68  $297,888.00
realestate land 3APR03 of 3 lots $19,000
science-art of pictures,porcelain etc starting JAN03 for $12,160.
realestate land 30JUL03 another lot $11,500.
and Wyeth at a small profit and with the proceeds bought 200 more
shares of BCE.
Sold DT because it is hard to play DT as a Crossover switching
campaign. Sold Verizon because it is overpriced compared to SBC. And
sold Wyeth because I was never really aware of the history of phen
fen (excuse the spelling) until I saw a PBS program on that drug and
the history of the company of Wyeth in relation to that drug. After
seeing the history of phen fen, I could not help but think that Wyeth
is a company that fosters a culture of money grubbing ever more than
it fosters a culture that they would seek health medicines to make
the world better. For a company to know that phenfen was dangerous in
Europe and yet bring that drug over into the USA to sell as a diet
drug, even knowing that it had a past history of dangerousness, tells
me that Wyeth entire corporate upper management needs to be fired. And
I do not want to own a drug company of that sort of history of placing
money-grubbing over that of good science.
Wyeth should emulate Johnson & Johnson or Merck as companies that get
the science before they ever think about money.
I am wanting and trying to get the PAF portfolio positioned for the
year 2004 such that it follows the VonNeumann Game Theory of the
Optimal Strategy for Playing the StockMarket. The key and central
theme of the OS of Stockmarket is switching campaigns of the
Crossover.
Through these many years of playing the stockmarket since 1978, I
beleive I have found the OS of StockMarket and am trying to arrange
the portfolio so that I can faithfully abide by this OS for 2004 and
many years beyond.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
====
Here I thought in this GOOGLE site, of whatever God was restricted
and/or moderated to NASA/NSA/DoD, now DHS.
Seems there's some folks that certainly talk and act the part like
God, as they certainly are those willing and able to terminate
whatever life as we know it, especially mine. Though I guess I'm just
happy that I'm not one of those nice Cathars having to dodge another
round of exterminations by the Pope, though our resident warlord has
certainly been doing his fair share of mastering carnage based upon
lies similar to but not even nearly as good as what the Catholic
church used to justify their actions. Although, we've got the so
what's the difference as our ultimate qualifier that trumps anything
Pope.
BTW; did anyone here bother to check out David Sereda, and of his
http://www.ufonasa.com/ EVIDENCE the case for NASA UFOs
In some of my recent additions to the MAZDA like Internal Rocket
Rotary Combustion Engine (IRRCE sfc = 15+KW/kg), there seems we also
have ourselves a wee bit of lunar He3 to burn off.
http://guthvenus.tripod.com/gv-h2o2-irrce.htm
The taking of lunar He3 is first come first served, that could be
the likes of China or Russia because, we're obviously not smart
enough.
Venus, not Mars, still offers life; via moon He3 could help turn the
trick
I've got a few more words of wisdom to offer on behalf of the ARTEMIS
PROJECT (lunar He3) http://guthvenus.tripod.com/gv-lse-he3.htm
The rest of this report isn't entirely related to energy so much as it
relates to truth or consequences. Such as for this next topic of there
being other life NOT as we know it on Venus, that's obviously opposing
the sorts of anti-humanity folks that couldn't care less if our entire
world was destroyed by their resident warlord. Perhaps these folks can
get their next level of future funding from the same source as Bush,
Salem Laden.
As for making policy look like happenstance, and/or vice versa, is key
to snookering folks. http://guthvenus.tripod.com/moon-04.htm
Though as for we humans need not, and perhaps we should not venture
ourselves much beyond Venus L2 (VL2). Wouldn't want to contaminate a
perfectly good planet with our inferior DNA nor lack of morals,
especially of this group that's bashing honest research just out of
spite. Besides, their stealth donkey-carts could be far more lethal
than what our WMD donkey-carts can manage.
As far as our human physiology being adaptable to pressure. Under such
pressure things are not nearly as hot as we've been told, and you wont
need but a fraction of a percent of O2. Of course, that degree of
adaptation might have to be accommodated at a modus rate of a few bars
per day.
http://guthvenus.tripod.com/venus-air.htm
I have a few other recent/ongoing comments on H2O2/C12H26 and of He3:
http://guthvenus.tripod.com/gv-irrce.htm
http://guthvenus.tripod.com/gv-hybrid-irc.htm
http://guthvenus.tripod.com/gv-cm-ccm-01.htm
http://guthvenus.tripod.com/gv-lm-1.htm
http://guthvenus.tripod.com/radio-maybe.htm
====
Dear all,
I want to ask what is the inverse operation of Kroneck product?
More specifically, we know that matrix operation
A*X*B=kron(A, B')*vec(X)
where kron is the Kronecker product of matrices as defined in matlab; 
vec(X)
is the stacked vector version of matrix X. All matrices are square...
Now I want to reverse the operation, suppose I have a big matrix C,
how to find A and B to get C=kron(A, B')?
Under what condition these A and B cannot be found? Then how to find them
approximately, i.e., optimal in the mean-square sense or under other
criteria? That's to say, find A and B, such that kron(A, B')=C1 where C1 is
a reasonably good approximation to C?
-Walala
====
walala:
Maybe if you gave the homework questions verbatim from the Prof, we could 
be
of more help. It seems your interpretation of your tasks is confusing some
people. Then again, if you really knew the questions, you wouldn't be
posting for answers.
Jim
> Dear all,
 I want to ask what is the inverse operation of Kroneck product?
 More specifically, we know that matrix operation
 A*X*B=kron(A, B')*vec(X)
 where kron is the Kronecker product of matrices as defined in matlab;
vec(X)
> is the stacked vector version of matrix X. All matrices are square...
 Now I want to reverse the operation, suppose I have a big matrix C,
 how to find A and B to get C=kron(A, B')?
 Under what condition these A and B cannot be found? Then how to find them
> approximately, i.e., optimal in the mean-square sense or under other
> criteria? That's to say, find A and B, such that kron(A, B')=C1 where C1
is
> a reasonably good approximation to C?

> -Walala

====
> walala:
 Maybe if you gave the homework questions verbatim from the Prof, we could
be
> of more help. It seems your interpretation of your tasks is confusing 
some
> people. Then again, if you really knew the questions, you wouldn't be
> posting for answers.
 Jim
 > Dear all,
>  > I want to ask what is the inverse operation of Kroneck product?
>  > More specifically, we know that matrix operation
>  > A*X*B=kron(A, B')*vec(X)
>  > where kron is the Kronecker product of matrices as defined in matlab;
> vec(X)
> is the stacked vector version of matrix X. All matrices are square...
>  > Now I want to reverse the operation, suppose I have a big matrix C,
>  > how to find A and B to get C=kron(A, B')?
>  > Under what condition these A and B cannot be found? Then how to find
them
> approximately, i.e., optimal in the mean-square sense or under other
> criteria? That's to say, find A and B, such that kron(A, B')=C1 where 
C1
> is
> a reasonably good approximation to C?
>   > -Walala
>
Dear Jim,
This really isn't my professor's homework problem... our school is now in
break so there are no damn homeworks right now...
This is a problem I am currently interested in... please tell me which part
of my description confuses you? Then I can make myself clearer...
Rgs,
-Walala
====
How would I compute the probability of a draw in tac tac toe (noughts 
and crosses) assuming each player was playing randomly (ie no thought 
involved). What about the probability that player 1 wins?
--
Interbang
change nospam to optimusprime to reply
====
>How would I compute the probability of a draw in tac tac toe (noughts 
>and crosses) assuming each player was playing randomly (ie no thought 
>involved). What about the probability that player 1 wins?
Assuming X goes first and that the players alternate until the grid is
full, the number of final complete grids can be found by computing 9C4
- this is the number of ways of choosing the 4 cells that get Os
Of course the 9C4 includes many games that are merely rotations or
reflections of each other. It also ignores the fact that many of the
games would really be over before the grid is full. But you may find
that ignoring such issues provides the most straightforward approach.
In particular, all the positions among the 9C4 can be regarded as
equally likely, which will not be the case if you consider positions
at the real end of the game.
What you have to do next is find the number of drawn positions, i.e.
the number of filled-in grids which do not contain a winning line
either for X or for O. You could approach this by enumerating the
places in the grid where a winning line can arise and then considering
the possible arrangements of the other symbols. You will need to be
very careful to avoid counting twice the positions with more than one
winning line.
Hope this helps
====
> 
>> 
>> 
> 
> 
>> So you know that tan' pi/4 = whatever.
>> Now what does that *mean*?
>> (Can you recall the *definition* of derivative?)
> 
> It means that you can apply l'Hopital's rule without knowing in
> advance what lim_{x -> pi/4}  (tan(x) - 1)/(x - pi/4) is.
> 
> Try again! That's no definition of derivative.
> 
> How do you use L'H's rule to calculate this limit without
> already knowing what it is?
>> 
>> The definition of derivative of tan(x) may be deduced from the
>> derivatives of sin(x) and cos(x) and the quotient rule for
>> derivatives, the derivative of the numerator and denominator of
>> (tan(x) - 1)/(x - pi/4) can be found without being aware that
>> it is a difference quotient.
>> 
>> If f(x) = tan(x) - pi/4 = sin(x)/cos(x), then
>> f'(x) = (cos(x)*cos(x) - sin(x)(-sin(x))/cos(x)^2 + 0
>>       = 1/cos(x)^2
>> 
>> Excellent: you can compute derivatives. Alas you seem
>> to have forgotten what they are. :-(
>> 
>> You have (d/dx)(tan x - pi/4) = 1/cos^2 x [dunno what the pi/4 is doing,
>> but it's irrelevant anyway]. Now what does that mean?
>> 
>> (Heavy hint: apply the definition of derivative.)
>> 
>> and if g(x) = x -pi/4 then g'(x) = 1
>> 
>> Any point to this?
>>  
>> Since f(pi/4) = g(pi/4) = 0 but g'(pi/4)  and f'(pi/4) are
>> continuous  and nonzero at x = pi/4, L'Hopital applies.
>> 
>> But why waste time using L'H?
It is not a matter of whether it is optimal to do it the way I
> showed, but merely a question of whether it is possible,  and I have
> shown it to be possible.
No you didn't --- you used the limit itself in the process
of applying L'H :-(
-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
 Alan Partridge, _Bouncing Back_ (14 times)
====
> [snip]
>>In his Commutative Algebra, D. Eisenbud has another definition:
>>Definition 2. Let R be a commutative ring. An element p of R is prime 
if
>>  p is not a unit and the following is true: If a and b are elements of
>>R such that p divides ab, then p divides a or b.
>>Is Definition 2 common in the commutative algebra community?
>> It is common in algebraic number theory as well.
Then I must wonder what the common definition of divides is. 
According
> to the definition which I typically use, 0 divides 0. And then according
> to Definition 2, we would have 0 being a prime, which surely we don't
> want.
Not necessarily --- the zero ideal isn't prime in every ring.
Def 2 just says that p is prime iff p generates a prime ideal.
Of course, in algebra, it's prime ideals, not prime elements, that
are of major interest.
-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
 Alan Partridge, _Bouncing Back_ (14 times)
====
bcc
Dear Gary, Jack and Tony  :
worked for some time on the notion of Self Referential
Noise as models of reality. Wheelers's spacetime FOAM
is just another interpretation of Noise. You could
take that Noise as your  absolute   frame of
reference.
JS: The new EINSTEIN results seem to argue against this model?
I am not sure of that of course. Sirag says the foam is not really
randomly chaotic but coherently harmonic and that sets the
number of extra dimensions in Calabi-Yau space? On the other hand
Christian Beck seems to agree with you and says he can determine
the 25 epicycles of the standard model (some say only 17) from
chaotic strings. This would seem choose a definite location on 
Susskind's Landscape in opposition to the WAP ideas of chaotic 
papers and books by Lee Smolin.
CC: One day you may want to look at :
Carlos Castro `` The String Uncertainty Relations
Follow from the New Relativity Principle  .
Foundations of Physics. {bf 8} ( 2000 ) page 1301.
for a way to derive the stringy uncertainty relations
from first principles.
JS: Is the claim being made that the new term beyond Heisenberg 
uncertainty is the source of irreversibility as in the arrow of time? 
Remember Hawking talks about a new source of uncertainty although 
Susskind seems to think that is wrong?
Note also Ed Witten's formula generalizing Heisenberg's quantum 
uncertainty principle, i.e. eq. (5.9) p. 136
Delta X > h/DeltaP + alpha'(DeltaP)/h
The second gravity-string source of uncertainty should give the 
irreversible statistical arrow of time not found when alpha' = 0, i.e. 
infinite string tension, or infinite space-time stiffness of action 
without reaction as is also found in the signal locality of orthodox 
quantum theory in sense of Antony Valentini's papers.
Note the conformal look
w = 1/z + alpha'z
====
> proofs

> why do authors usually consider the proofs so important in mathematical
> texts ?
Because that is what mathematics is without.
> Isn't it usually better when only a few people read and check the proofs,
No. If you refuse study proofs then you are cravenly deferring to 
authority.
-- 
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Needless to say, I had the last laugh.
 Alan Partridge, _Bouncing Back_ (14 times)
====
> proofs
>> 
>> 
>> why do authors usually consider the proofs so important in mathematical
>> texts ?
Because that is what mathematics is without.
Is this sentence a word without or something?
>> Isn't it usually better when only a few people read and check the 
proofs,
No. If you refuse study proofs then you are cravenly deferring to 
authority.
************************
David C. Ullrich
====
 : >         ----------------------------- <^> 
<(åáåÀåá)>
 : <^> -----------------------------
 : >
 : >
 : > Any number having this form can be produced by an RM.
 : > Some examples of computable numbers:
 : >
 : > (0)
 : > (1)
 : > 11(0)
 : > 01(110)
 : >
 : > The length of the initial string plus the length of the repeating 
string
 : > must be less than or equal to the number of states of the RM that
 : > produced the number.
 : >
 : >
 : > I don't see this.  Here is a simplified conceptual diagram of a 
Russell
 : Machine.
 : >
 : >       s1      s2         s6
 : > 0         s5
 : > ---------------------
 : > 1           s3
 : >        s4            s7
 : >              s8
 : >
 : > There is one piece of information missing, which state s8 goes to.
 : > Add to this s8 -> s7
 : >
 : > This RM will output  0 0 1 1 0 0 1 1 1 1 1 1 1 1
 : > notation is  0 0 1 1 0 0 (1 1)
 : >
 : > If it can't halt then the states get reused.
 : 
 : I assume that s1->s2->s3->s4->s5->s6->s7->s8->s8.
 : The ouput is 001100(1).
Then it DOES halt, DUMBASS.  s8 *IS* a halt state.
Are you now going to retract your claim that RMs don't halt because
they don't have halt states?
====
 : > What kinds of numbers are we talking about here?
 : > Rational? Real? Complex? Supernatural?  Infinitesimal?
 : >
 : I have made the proof a little more rigorous.
But you haven't answered the question.
WHAT IS a number???
 : First, I define how to compute a computable number.
That is ENTIRELY premature.  FIRST, you have to define NUMBER!
 : I do this with a very non-standard type of Turing machine.
 : Let's call these Russell machines (RM).
 : A number is computable if it can be represented by
 : the infinite binary string produced by a RM.
This is just stupid.  A TM can do anything you  need to do.
 : A RM has a single, infinitely long tape.
 : Every position of the tape is initially set to 0.
 : A RM starts at the leftmost end of the tape.
 : RMs can perform two operations, labeled 0 and 1.
 : 
 : 0 = move write head one position to the right
 : 1 = write a 1 to current location and move one position right
 : 
 : RMs never read from the tape because they would always read 0.
 : RMs never halt because they have no halt states.
The non-halting is stupid.  It basically makes it impossible to
have a well-defined result.  If you ever DO have a well-defined
result, observers can always back-allege that the machine actually
DID halt, as soon as the result BECAME well-defined.  You can allege
that it is still running but WE can allege that what it is now doing
no longer deserves to be CALLED running.
 : The state of an n-state RM can be specified with
 : (1 + Ceil( log2( n ) ) ) bits where the first bit represents the
 : operation to perform and the next Ceil(log(2(n)) bits
 : represent which state to switch to.
This is ungrammatical.  If the bits represent which state to switch
to then they DO NOT represent the state you are CURRENTLY in!
But you SAID the state of an n-state RM can be specified with....
I guess specified is just ambiguous.
 : An n-state RM can be fully specified by listing all states
 : which requires n * (1 + Ceil(log2(n))) bits.
OK.
 : Some examples of RMs and their output:
 : 
 : States  RM                Output
 : 1       0                 (0)...
 : 1       1                 (1)...
 : 2       01 11             01(1)...
 : 2       11 00             (10)....
 : 3       001 010 100       (001)...
 : 
 : The output of a RM has the following form:
 : There is a finite, possibly empty initial string that is written once,
 : and a finite, never empty repeating string that is written over and 
over.
 : A computable number can be written as the initial string followed by
 : the repeating string in parenthesis.
In other words, this thing only produces rational numbers.
That is a stupid limitation.
 : Any number having this form can be produced by an RM.
 : Some examples of computable numbers:
 : 
 : (0)
 : (1)
 : 11(0)
 : 01(110)
 : 
 : The length of the initial string plus the length of the repeating string
 : must be less than or equal to the number of states of the RM that
 : produced the number.
 : 
 : Now, prove that no set can contain every RM computable number.
But it can; this is just the rational numbers.  They are a set.
 : The standard diagonal argument has a problem - it is usually
 : impossible to determine if the diagonal number produced is
 : computable.
No, it isn't; that is NOT the problem.
I just threw the rest of it away.
Bothering to create an alternative paradigm IS STUPID.
You should have STUCK to TMs.  You relieve yourself of the
burden of explanation to everybody  (of what non-standard
thing you mean) -- everybody ALREADY knows what a TM is.
And whenEVER you are asked a direct question,
JUST ANSWER IT, if you are going to respond.
WHAT KIND OF *NUMBERS* are you talking about (for
judging computability)?  Natural?  Real? Complex?
Rational?  Supernatural?  Infinitesimal?   *ANSWER* THE FUCKING 
*QUESTION*, DUMBASS!
====
>  : Let S be the set of all computable numbers
>  : and assume S is countable.
I have made the proof a little more rigorous.
First, I define how to compute a computable number.
> I do this with a very non-standard type of Turing machine.
> Let's call these Russell machines (RM). 
> A number is computable if it can be represented by
> the infinite binary string produced by a RM. [...]
Let C be the set of all RM-computable binary strings, according to your 
definition.  To show that C is countable, it suffices to demonstrate a 
one-to-one function f : C -> N, where N is the set of natural numbers { 
0, 1, 2, ... }.  This is not difficult.
First, let's agree that any RM M with n states can be uniquely encoded 
as a natural number.  How we choose to do so isn't important, as long as 
we agree upon the rule -- so let's say for argument's sake that we use 
your idea as a basis.  To encode M, write 1, followed by n zeroes, 
followed by another 1, and then a sequence of n strings with ceil(lg(n)) 
+ 1 bits each, encoding the transition rules in ascending numerical 
order as you described.  Treat the resulting string as a natural number 
written in binary, and let that be the encoding for M.  Whatever this 
number is, I'll denote it E(M).  It should be obvious that if M1 and M2 
are distinct RM's, then E(M1) <> E(M2).
Now for any RM-computable string c., let RM(c) be the set of all RM's 
that compute c.  There must be AT LEAST one member of this set, since c 
is RM-computable*.  Let M(c) be the unique element of RM(c) that fulfils 
these two properties:
 1. M(c) in RM(c)
 2. For all M in RM(c), E(M(c)) <= E(M).
In other words, M(c) is the RM for c that maps to the smallest natural 
number among all the RM's that compute c, under the RM-encoding rule.  
It's not important that we pick this one in particular, but doing so 
eliminates all ambiguity.
So:  We can now define a one-to-one function f : C -> N as follows:
  f(c) := E(M(c))
Quickly, let's argue that this is one-to-one:  Let c1 and c2 be 
RM-computable strings with c1 <> c2.  Then M(c1) <> M(c2)**, and by the 
definition of E, E(M(c1)) <> E(M(c2)), so we're all set.
This suffices to show that the false diagonalization proof you sketched 
out is inapplicable here.
-M
*  And, actually, there are infinitely many RM's for any RM-computable
   string.  The proof is left as an exercise for the reader. ;)
** I hold this truth to be self-evident.  But if you disagree, it's not
   so terrible to prove it from your RM definitions.
P.S.- I mean for <> to denote the  does not equal relation.
P.P.S.- This level of detail is overkill for just getting the intuition, 
but imprecision can be even MORE confusing, I find.
-- 
http://www.dartmouth.edu/~sting/  | Dartmouth College, Hanover, NH, USA
====
in message
 >  : Let S be the set of all computable numbers
>  : and assume S is countable.
>  > I have made the proof a little more rigorous.
>  > First, I define how to compute a computable number.
> I do this with a very non-standard type of Turing machine.
> Let's call these Russell machines (RM).
> A number is computable if it can be represented by
> the infinite binary string produced by a RM. [...]
 Let C be the set of all RM-computable binary strings, according to your
> definition.  To show that C is countable, it suffices to demonstrate a
> one-to-one function f : C -> N, where N is the set of natural numbers {
> 0, 1, 2, ... }.  This is not difficult.
I didn't think so until a few days ago.
It's harder than it looks.
> So:  We can now define a one-to-one function f : C -> N as follows:
   f(c) := E(M(c))
 Quickly, let's argue that this is one-to-one:  Let c1 and c2 be
> RM-computable strings with c1 <> c2.  Then M(c1) <> M(c2)**, and by the
> definition of E, E(M(c1)) <> E(M(c2)), so we're all set.
Let x be the string produced by the comparator.
By construction, x was not produced by f().
E() depends on f(). E(M(x)) is undefined.
x represents a RM computable natural number.
Russell
- 2 many 2 count
====
 > Let C be the set of all RM-computable binary strings, according to your
> definition.  To show that C is countable, it suffices to demonstrate a
> one-to-one function f : C -> N, where N is the set of natural numbers {
> 0, 1, 2, ... }.  This is not difficult.
I didn't think so until a few days ago.
> It's harder than it looks.
So:  We can now define a one-to-one function f : C -> N as follows:
>  >   f(c) := E(M(c))
>  > Quickly, let's argue that this is one-to-one:  Let c1 and c2 be
> RM-computable strings with c1 <> c2.  Then M(c1) <> M(c2)**, and by the
> definition of E, E(M(c1)) <> E(M(c2)), so we're all set.
Let x be the string produced by the comparator.
I'm sorry, perhaps I missed something.  What comparator?
> By construction, x was not produced by f().
> E() depends on f(). E(M(x)) is undefined.
> x represents a RM computable natural number.
Setting aside for the moment that I do not know what string you mean by 
x, I can safely assert that E does not depend upon f.  I chose E to be a 
simple encoding rule, so that we could be easily convinced that it is a 
one-to-one mapping from RM's to natural numbers.  Its definition does 
not depend upon f's definition.
What do you mean by an RM-computable natural number?  Your definition 
of RM-computability talks about binary strings of infinite length.  Due 
ot their regular structure, you can ENCODE such a string uniquely as a 
natural number.  That's essentially what I proved above.  But you did 
not show a one-to-one mapping from RM-computable strings to natural 
numbers in your original definition!  
The obvious mapping -- just dropping the parentheses and gluing the 
bits together) doesn't quite work, because (for example) all these 
distinct RM-computable strings map to the same number:
  010(10), 0010(10), 00010(10), ...
If you don't like referring to a specific RM, an alternate proof 
formulation would be to first prove that RM-computable strings have the 
structure you described:  wx+ where w in (0+1)* and x in (0+1)(0+1)*.  
Once you prove that, you can encode any RM-computable string in binary 
by mapping 0 => 0, 1 => 11, and . to 10 and writing .w.x (the dots 
guarantee a unique natural number interpretation even when w begins with 
one or more zeroes).  Again, the choice of code is unimportant except 
insofar as we have to agree what it is.
-M
-- 
http://www.dartmouth.edu/~sting/  | Dartmouth College, Hanover, NH, USA
====
 Let x initally be (0).
> If s has the form 111...111(0) and the length of the initial
> segment of s is longer or equal to the length of the initial
> segment of x then take the initial segment of s, append a 1, and
> make this new string the initial segment of x.
 Examples:
 s           x
> (0)       1(0)
> 1(0)     11(0)
> 11(0)   111(0)
 Prove that x differs from every member of S.
 x differs from every s that does not end with (0).
> x differs from every s with an inital segment not of the form
> 111.111. x differs from every s with form 111...111(0).
> Therefore, x differs from every member of S.
> Well ... let S be the set of al RM computable numbers of the form
>> 1*(0) (any number of sequential ones followed by a repeating of
>> zeros). In this way, you will not be able to compute a x which
>> differs from every possible member of S (at least not using your
>> technique).
> Why not?
> x will differ from every member of S and x will be a RM computable
> number.
> Another way to look at his :
>> if all the numbers in a set S or of the form 111...11(0), you can define
a
>> set S' (containing natural numbers) such that an element of S which has 
a
>> sequence of m 1's corresponds to an element m of S'. Your x' would then
be
>> m+1. Now if you consider the set S I used in my previous post, it would
>> correspond tot the entire set of natural numbers. As for each m,
contained
>> in this set, m+1 is also contained, there cannot exist an x' which does
not
>> belong to this set.
Earlier in this post, someone said the natural numbers are computable.
Let's use your set as an example. We can interpret the strings right to
left.
Now, each binary string represents a natural number in base 1.
Let's call the set S.
S(0) = (0)
S(1) = (0)1
S(2) = (0)11
...
I give a constructive method of creating a RM computable number, x,
that is not in S. How can my method fail to produce x?
You assume that every string that corresponds to a finite number is in S.
This is not true if my proof is correct.
I can only think of two reasons why I would not be able to compute x.
1) S contains a member with infinitely many 1's
2) S contains a member with so many 1's that adding
one more 1 results in an infinitely long string of 1's.
If you can think of another reason why x can not be computed,
please post it.
Can you show that x is impossible to compute?
Russell
- 2 many 2 count
====
> 
>> Let x initally be (0).
>> If s has the form 111...111(0) and the length of the initial
>> segment of s is longer or equal to the length of the initial
>> segment of x then take the initial segment of s, append a 1, and
>> make this new string the initial segment of x.
>> Examples:
>> s           x
>> (0)       1(0)
>> 1(0)     11(0)
>> 11(0)   111(0)
>> Prove that x differs from every member of S.
>> x differs from every s that does not end with (0).
>> x differs from every s with an inital segment not of the form
>> 111.111. x differs from every s with form 111...111(0).
>> Therefore, x differs from every member of S.
>> Well ... let S be the set of al RM computable numbers of the form
> 1*(0) (any number of sequential ones followed by a repeating of
> zeros). In this way, you will not be able to compute a x which
> differs from every possible member of S (at least not using your
> technique).
> Why not?
>> x will differ from every member of S and x will be a RM computable
>> number.
> Another way to look at his :
> if all the numbers in a set S or of the form 111...11(0), you can
> define 
> a
> set S' (containing natural numbers) such that an element of S which
> has a sequence of m 1's corresponds to an element m of S'. Your x'
> would then 
> be
> m+1. Now if you consider the set S I used in my previous post, it
> would correspond tot the entire set of natural numbers. As for each
> m, 
> contained
> in this set, m+1 is also contained, there cannot exist an x' which
> does 
> not
> belong to this set.
Earlier in this post, someone said the natural numbers are computable.
> Let's use your set as an example. We can interpret the strings right
> to left.
> Now, each binary string represents a natural number in base 1.
> Let's call the set S.
S(0) = (0)
> S(1) = (0)1
> S(2) = (0)11
> ...
I give a constructive method of creating a RM computable number, x,
> that is not in S. How can my method fail to produce x?
Because every x would be of the form (0)11..1, and because of the 
definition of S, any such RM number is contained in S.
> You assume that every string that corresponds to a finite number is in
> S. This is not true if my proof is correct.
That was how S was defined : it contains all the strings of the form 
(0)11..1 (this can be considered as all unary representations of natural 
numbers).
> I can only think of two reasons why I would not be able to compute x.
1) S contains a member with infinitely many 1's
> 2) S contains a member with so many 1's that adding
> one more 1 results in an infinitely long string of 1's.
If you can think of another reason why x can not be computed,
> please post it.
3) for any s in S, s-with-a-1-added-to-the-back is also contained in S
Can you show that x is impossible to compute?
As there can not exist an element which is both member of a set and not 
contained in the set, yes.

Russell
> - 2 many 2 count

> 
-- 
Pento
De wereld was soep, en het denken meestal een vork,
tot smakelijk eten leidde dat zelden. - H. Mulisch
====
>  > I give a constructive method of creating a RM computable number, x,
> that is not in S. How can my method fail to produce x?
 Because every x would be of the form (0)11..1, and because of the
> definition of S, any such RM number is contained in S.
The definition of S leads to contradiction.
S can not contain every natural number.
> You assume that every string that corresponds to a finite number is in
> S. This is not true if my proof is correct.
 That was how S was defined : it contains all the strings of the form
> (0)11..1 (this can be considered as all unary representations of natural
> numbers).
I show that there exists an RM computable number of the form (0)11...1
that is not in set S. I can even show how this number can be constructed
by a Turing machine.
RM's are a subset of TMs.
Any RM can be emulated by a universal Turing machine (UTM).
We are only concerned with the subset of RM's that output an initial,
finite and contiguous string of 1's followed by an infinite string of 0's.
The UTM can generate the output tape for each of these RM's.
These tapes are then read by a second TM I will call a Comparator (CTM).
The CTM compares each input tape with the CTM's output tape.
If the input tape has a longer initial string of 1's, the CTM rewinds and
1.
After all of the RM tapes have beed read by the CTM we examine the
output of the CTM. This tape must contain the representation of a
natural number.
Every tape read by the CTM represents a natural number.
The CTM output tape contains the representation of the successor
of some member of S. The successor of a natural number is a
natural number.
S can not contain a representation of every natural number.
> I can only think of two reasons why I would not be able to compute x.
>  > 1) S contains a member with infinitely many 1's
> 2) S contains a member with so many 1's that adding
> one more 1 results in an infinitely long string of 1's.
>  > If you can think of another reason why x can not be computed,
> please post it.
 3) for any s in S, s-with-a-1-added-to-the-back is also contained in S
Not true. I show how to construct such a number that is not in S.
> Can you show that x is impossible to compute?
 As there can not exist an element which is both member of a set and not
> contained in the set, yes.
The definition of S leads to contradiction.
S can not exist.
Russell
- 2 many 2 count
====
>  
>  > I give a constructive method of creating a RM computable number, x,
> that is not in S. How can my method fail to produce x?
You have yet to say What KIND of numbers RM numbers CAN OR CAN'T be.
Are they natural?  Rational?  Real? Complex?  Supernatural?  Infinitesimal?
IT MATTERS. ANSWER the question!
>  > Because every x would be of the form (0)11..1, and because of the
> definition of S, any such RM number is contained in S.
The definition of S leads to contradiction.
No, it doesn't.
> S can not contain every natural number.
It can and it does.
YOU defined it.  Do you have any idea how
STUPID this makes you look, defining a set and then
saying that it can't contain things it obviously contains?
> You assume that every string that corresponds to a finite number is 
in
> S. 
No, we don't assume it, we just NOTICE it.
You defined S as the class of all outputs of RMs that printed
some natural number of 1s and then stopped.
> This is not true if my proof is correct.
Ergo, your proof is not one.
> That was how S was defined : it contains all the strings of the form
> (0)11..1 (this can be considered as all unary representations of 
natural
> numbers).
YOU defined S that way.  So S contains all natural numbers.
> I show that there exists an RM computable number of the form (0)11...1
> that is not in set S. 
No, you don't.
> I can even show how this number can be constructed
> by a Turing machine.
No, you can't.
 
> RM's are a subset of TMs.
No, they're not.  For one thing, you alleged that they never halted.
Are you going to retract that?
> Any RM can be emulated by a universal Turing machine (UTM).
> We are only concerned with the subset of RM's that output an initial,
> finite and contiguous string of 1's followed by an infinite string of 
0's.
Since RM output tapes were supposed to START OUT as all 0's, can't the
the RM just HALT after it gets through printing all its 1's?
> The UTM can generate the output tape for each of these RM's.
And put it where, exactly?  TM's, by default, only have ONE output tape.
> These tapes are then read by a second TM I will call a Comparator (CTM).
No, they aren't.  There are an infinite number of these tapes and the CTM
never finishes reading them all.
 
> The CTM compares each input tape with the CTM's output tape.
The CTM does NOT HAVE an output tape, other than its input tape.
That's just how TMs ARE DEFINED.
====
           ----------------------------- <^> 
<(áÀá)> <^> -----------------------------
> RM's are a subset of TMs.
> Any RM can be emulated by a universal Turing machine (UTM).
> We are only concerned with the subset of RM's that output an initial,
> finite and contiguous string of 1's followed by an infinite string of 
0's.
>
So you're not proving there is no complete RM list, 1st you are
proving there is no complete restricted RM list?
Is this based on modelling the diagonal number from your previous post?
<quote > and then prove that it is not in my list.
>  > 0.000...
> 0.1000...
> 0.11000...
> 0.111000...
> 0.1111000...
> 0.11111000...
> 0.111111000...
> ...
>  > Good luck trying to prove the diagonal number is
> not in the list using a countable number of operations.
 Seems too easy, surely? If I understand you correctly the diagonal
> number is:
  0.11111....
 Every number in your list consists of a finite string of 1s followed
> by zeros (000...). The diagonal number doesn't. Ergo it isn't in the
> list.
I can find an entry in my list that matches the diagonal number to any
finite number, n, of positions. This is true for all n.
My list is infinite.
There is at least one entry in my list that has a 1 in every finite
position.
How can you prove the diagonal number is not in my list?
Russell
</quote>
<quote>
convenience, I've changed the layout very slightly...
   > 0.111000...
  > 0.1111000...
 > 0.11111000...
> 0.111111000...
           ^--------
</quote>
Herc
</bah>
====
            ----------------------------- <^> 
<(áÀá)>
<^> -----------------------------

> RM's are a subset of TMs.
> Any RM can be emulated by a universal Turing machine (UTM).
> We are only concerned with the subset of RM's that output an initial,
> finite and contiguous string of 1's followed by an infinite string of
0's.
So you're not proving there is no complete RM list, 1st you are
> proving there is no complete restricted RM list?
This is the only subset of RM's anyone has concerns about.
I think everyone agrees my method works for all the other RMs.
> Is this based on modelling the diagonal number from your previous post?
Partly.
> <quote
> and then prove that it is not in my list.
>  > 0.000...
> 0.1000...
> 0.11000...
> 0.111000...
> 0.1111000...
> 0.11111000...
> 0.111111000...
> ...
>  > Good luck trying to prove the diagonal number is
> not in the list using a countable number of operations.
>  > Seems too easy, surely? If I understand you correctly the diagonal
> number is:
>  >  0.11111....
The diagonal method can be converted into the computable number proof.
Of course, when I responded to your thread I was arguing against Cantor's
diagonal proof. I am arguing for it in this thread.
Forcing the diagonal number to be computable shows that invoking
infinity is like killing a fly with a machine gun.
All we really need to prove it that the diagonal has more 1's than
any of the real numbers in the list above. The diagonal can have
a finite number of 1's and still not be in the list.
I suspect that the diagonal proof can be modified to show that
the rational numbers are uncountable.
Let R be the set of all rational numbers, r, such that 0 <= r < 1.
If R can be ordered such that the diagonal is a rational number,
the diagonal proof shows the rationals to be uncountable.
Most people will argue that the diagonal of this set must be
an irrational number. I have never seen a proof of this.
It is easy to come up with a list of rationals that have a rational
diagonal. The set I give above is one such set.
Russell
- 2 many 2 count
====
> Let R be the set of all rational numbers, r, such that 0 <= r < 1.
> If R can be ordered such that the diagonal is a rational number,
> the diagonal proof shows the rationals to be uncountable.
And if my grandmother had wheels, she'd be a wagon.
There are several ways to put that subset of the rational numbers into
one-to-one correspondence with the positive integers.
> Most people will argue that the diagonal of this set must be
> an irrational number. I have never seen a proof of this.
> It is easy to come up with a list of rationals that have a rational
> diagonal. The set I give above is one such set.
Of course, that set was not all of the rationals in that range.
Here is an example of the beginning of a complete list:
0
1/2
1/3
2/3
1/4
3/4
1/5
2/5
3/5
4/5
1/6
5/6
1/7
...
Identifying a rational given its position (or vice-versa) is probably
going to require generating the list up to that position, but this list
is obviously complete.  Therefore, the set of rational numbers in [0,1)
is countable.
You want a proof that any such complete list has an irrational diagonal?
Okay:
Assume that a complete list with a rational diagonal can be found.  The
diagonalization process would generated a rational number not in the
list.  This would contradict the assumption.  Therefore, either the list
is incomplete or the diagonal is not rational.  (My list is complete, so
the diagonal must not be rational.  Your list had a rational diagonal,
but was obviously incomplete.)
When applied to an allegedly complete list of the reals in [0,1), the
diagonally-generated number does not have the option of being other than
real, so the only possible conclusion is that the list was incomplete. 
-- 
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
====
 > Let R be the set of all rational numbers, r, such that 0 <= r < 1.
> If R can be ordered such that the diagonal is a rational number,
> the diagonal proof shows the rationals to be uncountable.
 And if my grandmother had wheels, she'd be a wagon.
 There are several ways to put that subset of the rational numbers into
> one-to-one correspondence with the positive integers.
 > Most people will argue that the diagonal of this set must be
> an irrational number. I have never seen a proof of this.
> It is easy to come up with a list of rationals that have a rational
> diagonal. The set I give above is one such set.
 Of course, that set was not all of the rationals in that range.
Of course.
> Here is an example of the beginning of a complete list:
 0
> 1/2
> 1/3
> 2/3
> 1/4
> 3/4
> 1/5
> 2/5
> 3/5
> 4/5
> 1/6
> 5/6
> 1/7
> ...
 Identifying a rational given its position (or vice-versa) is probably
> going to require generating the list up to that position, but this list
> is obviously complete.  Therefore, the set of rational numbers in [0,1)
> is countable.
Maybe. That is the point of the proof.
> You want a proof that any such complete list has an irrational diagonal?
> Okay:
 Assume that a complete list with a rational diagonal can be found.  The
> diagonalization process would generated a rational number not in the
> list.  This would contradict the assumption.  Therefore, either the list
> is incomplete or the diagonal is not rational.  (My list is complete, so
> the diagonal must not be rational.  Your list had a rational diagonal,
> but was obviously incomplete.)
How are you proving your list is complete?
To prove that the set contains all rational numbers
you will have to show there is no way to order
the set such that the diagonal is rational.
There are a lot of ways to order a set of rationals.
Russell
- 2 many 2 count
====
> How are you proving your list is complete?
Any rational in [0,1) can be uniquely written as a ratio between two
coprime nonnegative integers m/n.  That is an entry among the first
n(n-1)/2 + 1 entries on the list, although specifying the exact location
involves a summation on the Euler phi-function.  Anyway, that entry on
the list corresponds to no other rational.  (This last proviso is
moderately important, because some people like to present complete
lists of reals in which a given list entry can have more than one real
number associated with it.)
> To prove that the set contains all rational numbers
> you will have to show there is no way to order
> the set such that the diagonal is rational.
I just proved that the set contains all rational numbers in [0,1).
To prove that a given integer is even, is it necessary to show both that
its units digit in base two is 0 and that its units digit in base ten is
in {0,2,4,6,8}?
> There are a lot of ways to order a set of rationals.
If you are talking about the complete set of rationals, the number of
ways is the same as the number of real numbers.  If you think a relevant
ordering exists, either point out a flaw in my proof or describe the
ordering.
Otherwise, you might as well quibble with a proof that the sum of any
finite set if even numbers is not odd by pointing out that there are a
lot of ways to choose a set of even numbers.
-- 
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
    <-sOdnVAq4PkNeXGiRVn-sA@comcast.com> 
<Xns945E7EF166C5Brobbygoetschalckxstu@134.58.127.12>
    <8aGdnb0k8ZEW73CiRVn-jA@comcast.com> 
<Xns945E983979A02robbygoetschalckxstu@134.58.127.12>
    <q8CdnfPGJ8E4knOiRVn-tw@comcast.com> 
<Xns945F37E9E9C54robbygoetschalckxstu@134.58.127.12>
    <ts6dnYCx2LHp4nKiRVn-sw@comcast.com> 
<1g6p2lp.1nq0zj71e3df92N%panoptes@iquest.net>
    <9tCdnYk507_tD3KiRVn-hQ@comcast.com>
====
  > Let R be the set of all rational numbers, r, such that 0 <= r < 1.
> If R can be ordered such that the diagonal is a rational number,
> the diagonal proof shows the rationals to be uncountable.
>  > And if my grandmother had wheels, she'd be a wagon.
>  > There are several ways to put that subset of the rational numbers into
> one-to-one correspondence with the positive integers.
> Here is an example of the beginning of a complete list:
>  > 0
> 1/2
> 1/3
> 2/3
> 1/4
> 3/4
> 1/5
> ...
>  > Identifying a rational given its position (or vice-versa) is probably
> going to require generating the list up to that position, but this list
> is obviously complete.  Therefore, the set of rational numbers in [0,1)
> is countable.
 Maybe. That is the point of the proof.
et cetera, et cetera.  It's equivalent to the abbreviation Q.E.D.
There's no reason to say Maybe.
  [I repeat my opinion that you should search Google before continuing
to post in this thread.  Pedagogy is fun, and I'll be the first to admit
I have benefitted greatly from it, but it tends to clutter up the groups
if it's let run amok, IMHO.]
> How are you proving your list is complete?
  Actually, he didn't -- he just said, It's obvious, and moved on.
A rigorous proof would use the traditional diagonal method -- NOT
Cantor's diagonalization, but a different kind of diagonal analogy
I think attributed to Godel.  It involves laying out the plane of
(positive) rational numbers as follows:
   1/1--1/2  1/3--1/4 ...
      .'   .'   .'
   2/1  2/2  2/3  2/4 ...
    | .'   .'
   3/1  3/2  3/3  3/4 ...
      .'
   4/1  4/2  4/3  4/4 ...
    |
   ...  ...
You'll need a fixed-width font to see the ASCII-art line that
I've drawn zigzagging across the diagonals of the grid.  This line
obviously passes through every number in the grid.  And every
positive rational number is *somewhere* on that grid, as you can
see from the way it's laid out.  So the line passes through every
positive rational number, in sequence.  Straighten out the
line, and remove duplicates, and add zero and the negative rationals,
and you're done -- you have a complete list of all the rationals.
It begins
  0, 1, -1, 1/2, -1/2, 2, -2, 3, -3, 1/3, -1/3, 1/4, -1/4, 2/3, ...
and continues to infinity.  Now match up the elements of that
list, one-to-one, with the positive integers
  1, 2,  3,  4,    5,  6,  7, 8,  9,  10,   11,  12,   13,  14, ...
Ta-da!
  [The next bit has been re-ordered for clarity.]
> To prove that the set contains all rational numbers
> you will have to show there is no way to order
> the set such that the diagonal is rational.
> There are a lot of ways to order a set of rationals.
 > You want a proof that any such complete list has an irrational 
diagonal?
> Okay:
>  > Assume that a complete list with a rational diagonal can be found.  The
> diagonalization process would generated a rational number not in the
> list.  This would contradict the assumption.  Therefore, either the 
list
> is incomplete or the diagonal is not rational.
[Q.E.D.]
-Arthur
====
         ----------------------------- <^> 
<(áÀá)> <^> -----------------------------
> RM's are a subset of TMs.
> Any RM can be emulated by a universal Turing machine (UTM).
> We are only concerned with the subset of RM's that output an initial,
> finite and contiguous string of 1's followed by an infinite string of
> 0's.
>   > So you're not proving there is no complete RM list, 1st you are
> proving there is no complete restricted RM list?
 This is the only subset of RM's anyone has concerns about.
> I think everyone agrees my method works for all the other RMs.
 > Is this based on modelling the diagonal number from your previous post?
 Partly.
 > <quote  > and then prove that it is not in my list.
>  > 0.000...
> 0.1000...
> 0.11000...
> 0.111000...
> 0.1111000...
> 0.11111000...
> 0.111111000...
> ...
>  > Good luck trying to prove the diagonal number is
> not in the list using a countable number of operations.
>  > Seems too easy, surely? If I understand you correctly the diagonal
> number is:
>  >  0.11111....
 The diagonal method can be converted into the computable number proof.
 Of course, when I responded to your thread I was arguing against Cantor's
> diagonal proof. I am arguing for it in this thread.
 Forcing the diagonal number to be computable shows that invoking
> infinity is like killing a fly with a machine gun.
 All we really need to prove it that the diagonal has more 1's than
> any of the real numbers in the list above. The diagonal can have
> a finite number of 1's and still not be in the list.
 I suspect that the diagonal proof can be modified to show that
> the rational numbers are uncountable.
 Let R be the set of all rational numbers, r, such that 0 <= r < 1.
> If R can be ordered such that the diagonal is a rational number,
> the diagonal proof shows the rationals to be uncountable.
 Most people will argue that the diagonal of this set must be
> an irrational number. I have never seen a proof of this.
This is my argument against the diagonal.  I can find a rational
on the list of computables that equals any specific representation (finite)
of the 'irrational' diagonal construct number.
If the rational number equals the irrational number then its the same 
number.
I'm pretty sure Cantor falls on its head, I can give an algorithm for all 
numbers
UTM(Z), and increasing portions of the list it makes.  That is all anyone 
can
ask, and from that noone can make an 'infinite' diagonal construct at all in 
practicality.
Every number is on the list because the most they can differ by is 
(0..9)/oo.
Herc
> It is easy to come up with a list of rationals that have a rational
> diagonal. The set I give above is one such set.

> Russell
> - 2 many 2 count


>
====
> 
>> I give a constructive method of creating a RM computable number, x,
>> that is not in S. How can my method fail to produce x?
>> Because every x would be of the form (0)11..1, and because of the
>> definition of S, any such RM number is contained in S.
The definition of S leads to contradiction.
> S can not contain every natural number.
> 
>> You assume that every string that corresponds to a finite number is
>> in S. This is not true if my proof is correct.
>> That was how S was defined : it contains all the strings of the form
>> (0)11..1 (this can be considered as all unary representations of
>> natural numbers).
I show that there exists an RM computable number of the form (0)11...1
> that is not in set S. I can even show how this number can be
> constructed by a Turing machine.
RM's are a subset of TMs.
> Any RM can be emulated by a universal Turing machine (UTM).
> We are only concerned with the subset of RM's that output an initial,
> finite and contiguous string of 1's followed by an infinite string of
> 0's. 
The UTM can generate the output tape for each of these RM's.
> These tapes are then read by a second TM I will call a Comparator
> (CTM). 
The CTM compares each input tape with the CTM's output tape.
> If the input tape has a longer initial string of 1's, the CTM rewinds
> additional 1.
After all of the RM tapes have beed read by the CTM we examine the
> output of the CTM. This tape must contain the representation of a
> natural number.
Every tape read by the CTM represents a natural number.
> The CTM output tape contains the representation of the successor
> of some member of S. The successor of a natural number is a
> natural number.
S can not contain a representation of every natural number.
> 
>> I can only think of two reasons why I would not be able to compute
>> x. 
>> 1) S contains a member with infinitely many 1's
>> 2) S contains a member with so many 1's that adding
>> one more 1 results in an infinitely long string of 1's.
>> If you can think of another reason why x can not be computed,
>> please post it.
>> 3) for any s in S, s-with-a-1-added-to-the-back is also contained in
>> S 
Not true. I show how to construct such a number that is not in S.
> 
>> Can you show that x is impossible to compute?
>> As there can not exist an element which is both member of a set and
>> not contained in the set, yes.
The definition of S leads to contradiction.
> S can not exist.
OK, consider the set of natural numbers, N.
Also consider the operation + 1.
Is it not true that for each n in N, n + 1 is also in N ?
Now, consider the function f : N -> RM-numbers :
f(0) = (0)
f(1) = (0)1
f(2) = (0)11
...
and so on, with f(n) corresponding to its own RM-number for each natural 
number n.
Clearly, not all the different RM-numbers are reached by this function f. 
Consider the set F = f(N) of numbers which are. For each element s of F, we 

can find a unique natural number n such that s = f(n). This means we can 
give the inverse function of f, call this g.
Now, for each s in F, there can not be an x which is s followed by an extra 

1, which is not already in F : this would mean that g(x)=g(s)+1, with g(s) 
a natural number, would not be a natural number.
If you still consider this line of thinking to be wrong, please give me the 

exact x for which value it goes wrong.
-- 
Pento
De wereld was soep, en het denken meestal een vork,
tot smakelijk eten leidde dat zelden. - H. Mulisch
====
 OK, consider the set of natural numbers, N.
> Also consider the operation + 1.
 Is it not true that for each n in N, n + 1 is also in N ?
Maybe.
I might not be the person you want to ask this question to.
My proof shows that no set can contain every computable
natural number. If N does not contain every computable
natural number, how can we say N contains all natural numbers?
> Now, consider the function f : N -> RM-numbers :
 f(0) = (0)
> f(1) = (0)1
> f(2) = (0)11
> ...
> and so on, with f(n) corresponding to its own RM-number for each natural
> number n.
 Clearly, not all the different RM-numbers are reached by this function f.
> Consider the set F = f(N) of numbers which are. For each element s of F,
we
> can find a unique natural number n such that s = f(n). This means we can
> give the inverse function of f, call this g.
> Now, for each s in F, there can not be an x which is s followed by an
extra
> 1, which is not already in F : this would mean that g(x)=g(s)+1, with 
g(s)
> a natural number, would not be a natural number.
Since g() is the inverse of f(), and x was not generated by f(),
g(x) may not be defined.
x is RM computable so there exists a RM that will output it.
This means f() can not emulate every possible RM.
Russell
- 2 many 2 count
====
> OK, consider the set of natural numbers, N.
>> Also consider the operation + 1.
>> Is it not true that for each n in N, n + 1 is also in N ?
Maybe.
>I might not be the person you want to ask this question to.
>My proof shows that no set can contain every computable
>natural number. If N does not contain every computable
>natural number, how can we say N contains all natural numbers?
(I haven't been following this thread, so please make allowances if what 
I say here is either redundant or irrelevant.)
The standard definitions of computable are equivalent to computable 
by some Turing machine.  Now clearly, each particular natural number K 
can be generated by some TM; indeed, it's even computable by a 
finite-state machine (with K states).
Furthermore, the infinite sequence of all natural numbers together 
(e.g., in base 1, 010110111011110111110...) can be generated by a 
(non-halting) TM -- the algorithm repeatedly copies the last string of 
1's (which can be done finitely by temporarily replacing the most recent 
1 copied with some other symbol, to keep track of how much has already 
So in what sense isn't every natural number computable?
[snip]
-- 
---------------------------
|  B  B   aa     rrr  b     |
|  BBB   a  a   r     bbb   |   
|  B  B  a  a   r     b  b  |   
|  BBB    aa a  r     bbb   |   
-----------------------------
====
         ----------------------------- <^> 
<(áÀá)> <^> -----------------------------
 >  >> OK, consider the set of natural numbers, N.
>> Also consider the operation + 1.
>> Is it not true that for each n in N, n + 1 is also in N ?
>  >Maybe.
>I might not be the person you want to ask this question to.
>My proof shows that no set can contain every computable
>natural number. If N does not contain every computable
>natural number, how can we say N contains all natural numbers?
 (I haven't been following this thread, so please make allowances if what
> I say here is either redundant or irrelevant.)
 The standard definitions of computable are equivalent to computable
> by some Turing machine.  Now clearly, each particular natural number K
> can be generated by some TM; indeed, it's even computable by a
> finite-state machine (with K states).
 Furthermore, the infinite sequence of all natural numbers together
> (e.g., in base 1, 010110111011110111110...) can be generated by a
> (non-halting) TM -- the algorithm repeatedly copies the last string of
> 1's (which can be done finitely by temporarily replacing the most recent
> 1 copied with some other symbol, to keep track of how much has already
 So in what sense isn't every natural number computable?
>
Check out this 3 state TM that counts from 0 upwards in binary.  Every
second position on the tape represents the number.
Herc
knock knock knock on the door,  feet still bleeding
====
>
message
>  >> OK, consider the set of natural numbers, N.
>> Also consider the operation + 1.
>> Is it not true that for each n in N, n + 1 is also in N ?
>  >Maybe.
>I might not be the person you want to ask this question to.
>My proof shows that no set can contain every computable
>natural number. If N does not contain every computable
>natural number, how can we say N contains all natural numbers?
 (I haven't been following this thread, so please make allowances if what
> I say here is either redundant or irrelevant.)
 The standard definitions of computable are equivalent to computable
> by some Turing machine.  Now clearly, each particular natural number K
> can be generated by some TM; indeed, it's even computable by a
> finite-state machine (with K states).
 Furthermore, the infinite sequence of all natural numbers together
> (e.g., in base 1, 010110111011110111110...) can be generated by a
> (non-halting) TM -- the algorithm repeatedly copies the last string of
> 1's (which can be done finitely by temporarily replacing the most recent
> 1 copied with some other symbol, to keep track of how much has already
 So in what sense isn't every natural number computable?
Consider all TM's that write an initial, finite and contiguous strings of
1's
and then halt. Let S be the set of all output tapes from these TM's.
Define another TM I call a comparator (CTM).
CTM compares the input tape to its output tape.
If the input tape has more 1's, the CTM rewinds
to the beginning of its output tape, copies the input tape,
We let CTM read all the tapes in S.
What is on the output tape of CTS?
The output of CTM must have a finite number of 1's.
Every tape read by CTM was finite in length.
The output of CTM has exactly one more 1 than
some input tape in S.
S can not contain every possible string that
represents a natural number.
Russell
- 2 many 2 count
====
comp.theory and sci.math need to come off the list of newsgroups
for this thread.  Those groups surely have more important things
to discuss.
 : 
 : Consider all TM's that write an initial, finite and contiguous strings 
of
 : 1's
 : and then halt.
OK.  There is one of these TMs for every natnum.
 : Let S be the set of all output tapes from these TM's.
That's basically the set of all natural numbers.
 : Define another TM I call a comparator (CTM).
 : CTM compares the input tape to its output tape.
CTM *DOES* *NOT* *HAVE* an output tape.
CTM is a TM.  In general, TMs DO NOT HAVE output tapes.
TMs have ONE tape.  They use it for input.
They also write on it, but that does not make it an output tape.
It is at best an input/output tape.
 : If the input tape has more 1's,
This is ridiculous.  The input tape IS the output tape, if CTM
is a TM.  It can NEVER have a DIFFERENT number of 1's from itself.
 : the CTM rewinds to the beginning of its output tape, copies the input 
tape,
That is not how TMs work.  What actually happens is that the TM reads
at the end, and then halts.
 : We let CTM read all the tapes in S.
No, we don't, because S has an infinite number of tapes, so CTM
(if it really is a TM, which, as you have defined it, IT ISN'T,
because it has an output tape) will never finish reading all
these tapes.
 : What is on the output tape of CTS?
That's a typo; CTS does not exist; do you mean CTM?
 : The output of CTM must have a finite number of 1's.
Right.
 : Every tape read by CTM was finite in length.
Right.
 : The output of CTM has exactly one more 1 than
 : some input tape in S.
Wrong.
S has infinitely many input tapes and there is no upper bound on
how long they are.
 : S can not contain every possible string that
 : represents a natural number.
As you defined it, it does indeed contain exactly that;
it represents the natnums in unary.
-- 
 ---
  It's difficult ... you need to be united to have any 
   strength, but internal issues have to  be addressed.
 --- E. Ray Lewis, on  liberalism in America
    <-sOdnVAq4PkNeXGiRVn-sA@comcast.com> 
<Xns945E7EF166C5Brobbygoetschalckxstu@134.58.127.12>
    <8aGdnb0k8ZEW73CiRVn-jA@comcast.com> 
<Xns945E983979A02robbygoetschalckxstu@134.58.127.12>
    <q8CdnfPGJ8E4knOiRVn-tw@comcast.com> 
<Xns945F37E9E9C54robbygoetschalckxstu@134.58.127.12>
    <Bu-dnYP-tIz1JHOiRVn-sA@comcast.com> 
<Xns945F820BCBD74robbygoetschalckxstu@134.58.127.12>
    <19idnd3Ywuot6HKiRVn-sA@comcast.com> <bsnvo0$q5k$1@lust.ihug.co.nz> 
<V7GdnTOguOLYF3KiRVn-vw@comcast.com>
====
>
  [ Obviously, every natural number N is computable by an N-state FSM. ]
> So in what sense isn't every natural number computable?
 Consider all TM's that write an initial, finite and contiguous strings of
> 1's and then halt. Let S be the set of all output tapes from these TM's.
  S is an infinite set which is also countable.
> Define another TM I call a comparator (CTM).
> CTM compares the input tape to its output tape.
> If the input tape has more 1's, the CTM rewinds
> to the beginning of its output tape, copies the input tape,
 We let CTM read all the tapes in S.
  What do you mean read all the tapes in S?  S has infinitely
many members.  If CTM tries to read all the members of S, it
will never finish.
> What is on the output tape of CTS?
  Nothing -- CTM never halts, as it never finishes reading its
input tapes.
> The output of CTM must have a finite number of 1's.
> Every tape read by CTM was finite in length.
> The output of CTM has exactly one more 1 than
> some input tape in S.
  Non sequitur.
> S can not contain every possible string that
> represents a natural number.
  Non sequitur.
  Surely you've had this explained to you in the past,
about how some infinities are bigger than others, and which
ones, and how?  Check Google Groups if you haven't, or don't
remember -- I'm sure there's plenty of tutorial information
on there.  Check sci.math's archives, too.
-Arthur
====
What are the parameteric coordinates of closed loops generated  by 
intersection point F between b and c and 
intersection point I between a and c extensions 
in a quadrilateral of sides a,b,c,d ( side a smallest,side d largest,
angle th variable/parameter between a and d )? These occur in a four
bar linkage mechanism . The book 4 Bar Atlas(by Hrones and Nelson)
does not include parameteric equations. TIA.
====
I need to know how to calculate x^y where y is not an integer.  Any
help in this matter would be appreciated.
====
> I need to know how to calculate x^y where y is not an integer.  Any
> help in this matter would be appreciated.
If y is rational, i.e. y = a/b, then b th root (x^a).  For example,
x^2/3 = cube root of (x^2).  Otherwise, check out:
http://mathworld.wolfram.com/Power.html
Lurch
====
> I need to know how to calculate x^y where y is not an integer.  Any
>> help in this matter would be appreciated.
If y is rational, i.e. y = a/b, then b th root (x^a).  For example,
x^2/3 = cube root of (x^2).  Otherwise, check out:
http://mathworld.wolfram.com/Power.html
??? I don't see a mathematical definition of x^y on that
page, nor anything else that might be what he's looking
for when he asks how to calculate it...
>Lurch
>
************************
David C. Ullrich
====
>>I have another more general question, if you don't mind : is there an
>>intuitive meaning for a nilpotent group ? 
Nilpotent groups are closely connected to nilpotent lie algebras,
>which is where the name actually comes from, as I understand it, and
>where nilpotent actually makes some sense.
>As to intuitive meaning, that will depend a lot on you, I guess.
[...]
>Others think of them as groups constructed through central extensions,
>which is not a bad way to think about them either.
Some of us just instinctively think, Product of p-groups. Realistically
I just think about p-groups, period, and then mutter something about
direct products at the end of the story. (And if you want an intuitive
meaning of p-groups, think about layers of small vector spaces over
the field of characteristic  p, forming the quotients of chains of
normal subgroups. I don't necessarily think only of central extensions,
by the way; in fact one of the prototypes I keep in mind is a semidirect
product of a  Z/p  acting on a  p-dimensional vector space over  GF(p).
I'll bet most people don't think about the dihedral group of order 8
this way...)
Of course, some of us carry along a related intuitive notion that
group means finite group. You'll have to modify the previous 
paragraph if you think otherwise.
dave
X-Received: (from approve@localhost)
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 
1.9  primary) id hBSDc1T02534;
====
   [.snip.]
J. Willekens
====
> Notice that the in Noble Verse 57:25 above, Allah Almighty says clearly 
We
> sent down iron.... and He didn't say We created iron from earth....
> Allah Almighty's claim was very accurate and precise.  We sent down
> iron..... clearly states that iron was created outside the earth and 
was
> brought down by the Will of Allah Almighty for a purpose, and that is
> (material for) Mighty war, as well as many benefits for mankind, that 
Allah
> Noble Quran, 57:25)
>
Hypothetically, if this is proven incorrect, and the iron did not come 
down,
would you be willing to give up on the Quran?
Bill
      <vuop8jtg11tge3@corp.supernews.com> 
      <bsia8e$pbm$1@neptun.beotel.net> 
      <vuraj3sm7q1e87@corp.supernews.com> 
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 02:12 PM, Mark K. Bilbo <noem@il.huh> said:
>There is no down in space. 
There is if you send a goose up. I don't know if that's enough down
for a jacket.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
speak thusly:
>    at 02:12 PM, Mark K. Bilbo <noem@il.huh> said:
> 
>>There is no down in space. 
There is if you send a goose up. I don't know if that's enough down
> for a jacket.
The poor goose...
-- 
Mark K. Bilbo
      <vuop8jtg11tge3@corp.supernews.com> 
      <bsia8e$pbm$1@neptun.beotel.net> 
      <vuraj3sm7q1e87@corp.supernews.com> 
      <bskjme$vpe$1@neptun.beotel.net>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 07:30 PM, Goran Jakupovic <jg01495p@kiklop.etf.bg.ac.yu> said:
>Like somebody else already said not just iron, but all atoms starting
>with helium and heavier now in solar system were products of
>supernova.
No. Helium is predominantly primordial, Beryllium and Lithium are
almost entirely primordial. The conditions for light element
nucleosynthesis are very different from the conditions for heavier
elements. Stars produce Helium, but they destroy Lithium and
Beryllium. There's more Helium in the Universe now than right after
the big bang, but not a lot more.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
>  >  > down is relative like meaning closer to core like when you get 
down
> from
> bed. but the quran has things being created on earth but for some odd
> reason
> it diverges with iron. it says we sent down iron. iron was not 
created
> on
> earth according to quran. how did the quran know that iron came down?
> down
> meaning from elsewhere. it was not made on this earth. god caused 
iron
> to
> descend. coincidence?
>  >   > It's untrue. Iron was here when the earth was formed. It did not come
> later.
> And if anything, it came up from the core from volcanoes and the 
like.
you are mistaken in a small way. iron not even from this solar system. it
> came from other stars and had to land on earth. maybe it was primitive
> molten earth but earth was here. if earth was not here what iron land on?
> simple logics no offense. the miracle is that quran say many things 
created
> on earth and if this book from man who forging gods word man just say 
iron
> like people and animals and plants and mountains was created on earth. 
but
> for some reason quran treats iron different and say that it is sent down 
to
> earth. why not just say it was created on earth? because that would be
> wrong.
 --
> saab siddiqui al mujahed
> but you have to change the (a) to @ for it to work


Iron is an element, copper is an element, carbon is an element, and so on.
What scientific basis do you have for your belief that iron came latter 
than
the others? What made up the earth that the iron landed on?
Bill
====
thusly:
> 
>> thusly:
>> right. i did not say it came to earth after it finished with animals
> plants
>> and all that. i only noted that the quran has many things created on
> earth
>> but for some reason does not have iron created on earth.
>> What about all the *other elements that weren't formed here?
what about them? are you trying to raise an argument ex silentio? if it
> turns out on usenet you never once say your mothers maiden name does that
> necessarily mean you did not know your mothers maiden name? as for the
> elements other than iron i will be agnostic on what the author of the 
quran
> knew about them for now.
So this is like one of those christian arguments. God failed to mention
anything *useful like penicillin but seems to have mumbled something vague
about iron.
-- 
Mark K. Bilbo
      <vuopj0f71q4c37@corp.supernews.com> 
      <245fb07.0312261913.713872dc@posting.google.com>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
>Atheism is simply a lack of belief in gods.
No, that's Agnosticism. Atheism is the belief in the lack of a god.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
>    at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
> 
>Atheism is simply a lack of belief in gods.
No, that's Agnosticism. Atheism is the belief in the lack of a god.
How about un-theist. Does that work for you? I think the god-concept
is a nonexistant mind created reality that it is used as a tool to
dupe the gullible.
I think religious mysticism is a form of mental illness and madness
that is unique to the human mind on this planet. This still work for
you?
I think the god-concept and religion have held back the development of
the human race by 2000 to 3000 years. Instead of looking for the
nonexistent afterlife the human race should be working to build a
better life here and now. If not for religion and the god-concept it
is a possible that we could be on other planets by now and have
lifespans of over 500 years. The loss and waste of potential in the
human race because of the god-concept is immense.
Still working for you?
If all of this does not fit within your definition parameters please
let me know about my ill-defined thought processess.
Jefferson Rourke
Laissez-Faire!
      <vuopj0f71q4c37@corp.supernews.com> 
      <245fb07.0312261913.713872dc@posting.google.com> 
      <3fee0f44$21$fuzhry+tra$mr2ice@news.patriot.net> 
      <245fb07.0312272002.3afa03c1@posting.google.com>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 08:02 PM, raynand@netzero.net (Jefferson Rourke) said:
>How about un-theist. Does that work for you?
It's not a question of whether it works for me; it's a question of
whether it works for the English language. The English terms are
agnostic and atheist. What would work for me is honesty, which you are
not exhibiting.
>I think religious mysticism is a form of mental illness and madness
>that is unique to the human mind on this planet. 
Then you were lying when you stated that Atheism is simply a lack of
belief in gods. You have gone beyond not believing in a god to
believing that there is no god. 
>I think the god-concept and religion have held back the development
>of the human race by 2000 to 3000 years.
Perhaps it did. And perhaps it had the oposite effect. Do you have
evidence, or is it just a matter of faith for you?
>Still working for you?
Nope. But if your faith works for you, . . .
>If all of this does not fit within your definition parameters please
>let me know about my ill-defined thought processess.
You might start with post hoc ergo propter hoc.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
speak thusly:
>    at 08:02 PM, raynand@netzero.net (Jefferson Rourke) said:
> 
>>How about un-theist. Does that work for you?
It's not a question of whether it works for me; it's a question of
> whether it works for the English language. The English terms are
> agnostic and atheist. What would work for me is honesty, which you are
> not exhibiting.
Nope. Wrong.
Some of the colloquial connotations that have accreted to the words are
similar to what you're claiming but colloquial meaning shifts around all
the time.
Atheism is lacking belief in any gods. The word was coined to mean that.
That is what atheists use the word to mean. That is the meaning.
-- 
Mark K. Bilbo
====
thusly:
>>    at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
>> 
>>Atheism is simply a lack of belief in gods.
>> 
>> No, that's Agnosticism. Atheism is the belief in the lack of a god.
How about un-theist. Does that work for you? I think the god-concept
> is a nonexistant mind created reality that it is used as a tool to
> dupe the gullible.
Actually, there's no point to trying to change terms. Theists would trash
any term used.
And atheism actually *is un- or maybe more accurately non- theism.
Since a- means without. As in amoral means *without morals as
contrasted to immoral which is *not moral. The in- prefix meaning
opposite of or not.
What they try to claim is atheism is, is more something you might call
intheism (to coin a word). That would be opposite of theism. 
Atheism works fine. Fits the way we do things in the language (hence 
the
word was brought into the language to mean without theism). Theist
misunderstanding or even deliberate obfuscation notwithstanding...
-- 
Mark K. Bilbo
====
> thusly:
> 
>>    at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
>> 
>>Atheism is simply a lack of belief in gods.
>> 
>> No, that's Agnosticism. Atheism is the belief in the lack of a god.
How about un-theist. Does that work for you? I think the god-concept
> is a nonexistant mind created reality that it is used as a tool to
> dupe the gullible.
Actually, there's no point to trying to change terms. Theists would trash
> any term used.
And atheism actually *is un- or maybe more accurately non- 
theism.
> Since a- means without. As in amoral means *without morals as
> contrasted to immoral which is *not moral. The in- prefix meaning
> opposite of or not.
What they try to claim is atheism is, is more something you might call
> intheism (to coin a word). That would be opposite of theism. 
Atheism works fine. Fits the way we do things in the language (hence 
the
> word was brought into the language to mean without theism). Theist
> misunderstanding or even deliberate obfuscation notwithstanding...
Hey Mark:
I was just trying to have a bit of fun with the guy and see what his
response would be. I thought I had the definition right the first time
and I was writing off of the top of my head without double checking.
Hope you had a happy Winter Solstice,
Jefferson Rourke
====
thusly:
>> thusly:
>> 
>    at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
> 
>Atheism is simply a lack of belief in gods.
> 
> No, that's Agnosticism. Atheism is the belief in the lack of a god.
>> 
>> How about un-theist. Does that work for you? I think the god-concept
>> is a nonexistant mind created reality that it is used as a tool to
>> dupe the gullible.
>> 
>> Actually, there's no point to trying to change terms. Theists would 
trash
>> any term used.
>> 
>> And atheism actually *is un- or maybe more accurately non- 
theism.
>> Since a- means without. As in amoral means *without morals 
as
>> contrasted to immoral which is *not moral. The in- prefix 
meaning
>> opposite of or not.
>> 
>> What they try to claim is atheism is, is more something you might call
>> intheism (to coin a word). That would be opposite of theism. 
>> 
>> Atheism works fine. Fits the way we do things in the language (hence 
the
>> word was brought into the language to mean without theism). Theist
>> misunderstanding or even deliberate obfuscation notwithstanding...
Hey Mark:
I was just trying to have a bit of fun with the guy and see what his
> response would be. I thought I had the definition right the first time
> and I was writing off of the top of my head without double checking.
No big thing. I was just rattling along on an interesting (to me at least)
subject.
I can't help it. I'm moving more and more into linguistics as (if things
hold together) will be my next field.
Words fascinate me to no end...
-- 
Mark K. Bilbo
====
speak thusly:
>    at 07:13 PM, raynand@netzero.net (Jefferson Rourke) said:
> 
>>Atheism is simply a lack of belief in gods.
No, that's Agnosticism. Atheism is the belief in the lack of a god.
No, atheism is lacking belief in gods.
We know, we're atheists.
-- 
Mark K. Bilbo
      <vuop8jtg11tge3@corp.supernews.com>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 11:42 AM, Saab Siddiqui <p@O.g> said:
>im snipping mr metz points that i have no response to at this
>time.
Good; that is proper quoting style for Usenet. You'll see people
is generally something to avoid and not to imitate.
>not that star collide with earth but that matter from star like iron
>collide with earth.
The Earth was formed by the accretion of matter. Much of that matter
was Iron. There was already a substantial amount of Iron when the
Earth was very small. The material that fell later included a lot of
elements besides Iron, and had no higher percentage of Iron than the
initial material. Given the outgassing of lighter elements, the
primordial proto-Earth probably had a higher concentration of Iron
than the new material falling on it did.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
      <vuhkggpve1n19b@corp.supernews.com> 
      <nI4Gb.44221$aP.11492@newssvr31.news.prodigy.com> 
      <vuhmdd8f638563@corp.supernews.com> 
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 04:56 PM, Mark K. Bilbo <noem@il.huh> said:
>The process is that iron forms in stars. The stars go nova and eject
>material which includes iron. 
No. Iron forms in significant quantities only in stars that go
supernova. A simple nova is not hot enough or dense enough.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
      <L5bGb.32630$6e6.11772@news02.roc.ny>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
>Not just large, but dense. A white dwarf star is about the size of
>the Earth,
Well, I prefer a yellow dwarf, but would rather that it remain a safe
93,000,000 miles away ;-)
>I do wonder what would happen if the impacting object was a 
neutron star,
Lethal. The details would depend on the size, but I imagine that the
radiation would kill us before we had a chance to observe the rest.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 01:51 AM, Steve Knight <wooooly@sonic.net> said:
>   Great. Some camel fucking, rag head, sand muncher, 
At least he is not a racist xenophobe like you. 
*PLONK*
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
>How do you prove what the (imaginary) zeroes of the Fibonacci polynomials
>are?
http://mathworld.wolfram.com/FibonacciPolynomial.html
F(1,x)=1
>F(2,x)=x
>F(k,x)=xF(k-1,x)+F(k-2,x)
Let  G(T,x) = sum F(k,x) T^k, so that (1 - x T - T^2) G(T,x) = T, so
that  G(T,x) = T/(1-xT-T^2)  which we can expand using partial frations:
G(T,x) = a(x)/(1-r(x)T) + b(x)/(1-s(x)T)  where  r(x) and s(x)  are
the roots of the quadratic  Z^2 - x Z - 1 .  Then  F(k,x), which
is the coefficient of  T^k  in  G,  is simply  a(x) r(x)^k + b(x) s(x)^k.
So the roots are the values of  x  which make
   ( r(x)/s(x) )^k = -b(x)/a(x).
I make it out to be that  a(x) = 1/sqrt(x^2+4)  and  b(x) = - a(x),
so -b(x)/a(x) = 1.  So the roots are the values of  x  which make
    r(x) = zeta s(x)
where  zeta  is any k-th root of unity.  I further find  r  and  s  to be
x/2 +- sqrt(x^2+4)/2, so this is equivalent to the equations
    x + sqrt(x^2+4) = zeta ( x - sqrt(x^2+4) )   [Note that zeta <> 1]
    (zeta + 1) sqrt(x^2+4) = (zeta - 1) x
    (zeta + 1)^2 (x^2+4) = (zeta - 1)^2 x^2
    (4 zeta) x^2 + 4 (zeta + 1)^2 = 0
    x^2 = - (zeta + 1)^2 / zeta
Writing  zeta = exp( 2 pi i p/k )  we can then rewrite the last line as
    x = ( +- i) (exp( 2 pi i p/(2k) ) + exp( - 2 pi i p/(2k) )
      = ( +- i) 2 cos( pi p/k ).
dave
====
>How do you prove what the (imaginary) zeroes of the Fibonacci 
polynomials
>are?
>  >http://mathworld.wolfram.com/FibonacciPolynomial.html
>  >F(1,x)=1
>F(2,x)=x
>F(k,x)=xF(k-1,x)+F(k-2,x)
 Let  G(T,x) = sum F(k,x) T^k, so that (1 - x T - T^2) G(T,x) = T, so
> that  G(T,x) = T/(1-xT-T^2)  which we can expand using partial frations:
> G(T,x) = a(x)/(1-r(x)T) + b(x)/(1-s(x)T)  where  r(x) and s(x)  are
> the roots of the quadratic  Z^2 - x Z - 1 .  Then  F(k,x), which
> is the coefficient of  T^k  in  G,  is simply  a(x) r(x)^k + b(x) s(x)^k.
> So the roots are the values of  x  which make
>    ( r(x)/s(x) )^k = -b(x)/a(x).
 I make it out to be that  a(x) = 1/sqrt(x^2+4)  and  b(x) = - a(x),
> so -b(x)/a(x) = 1.  So the roots are the values of  x  which make
>     r(x) = zeta s(x)
> where  zeta  is any k-th root of unity.  I further find  r  and  s  to be
> x/2 +- sqrt(x^2+4)/2, so this is equivalent to the equations
>     x + sqrt(x^2+4) = zeta ( x - sqrt(x^2+4) )   [Note that zeta <> 1]
>     (zeta + 1) sqrt(x^2+4) = (zeta - 1) x
>     (zeta + 1)^2 (x^2+4) = (zeta - 1)^2 x^2
>     (4 zeta) x^2 + 4 (zeta + 1)^2 = 0
>     x^2 = - (zeta + 1)^2 / zeta
> Writing  zeta = exp( 2 pi i p/k )  we can then rewrite the last line as
>     x = ( +- i) (exp( 2 pi i p/(2k) ) + exp( - 2 pi i p/(2k) )
>       = ( +- i) 2 cos( pi p/k ).
>
it yet.
====
> Don
Have you looked at
http://www.utm.edu/research/primes/mersenne/
163 does not appear here either.
Mike
Yes, thanks Mike.
Tom, Greetings,
Re: Mersenne numbers, Mp163
.MZD03-Oct.MPTH163
The search for Mersenne primes begins by looking for
small factors of (mersenne numbers) 2^prime -1.
That's what I did.
My letter proves a low factor for 2^163-1.
Showing (on a pocket calculator) that it ISN't a Mersenne PRIME.
I have photographed mersenne number plates MP163 and MP67
in my (one street away.) And Mercurial factorisation HG 7817 etc.
LN 2718, PI 315, PI 180. All in Newtown.
= 70^2 + 14^2 == 5200 -2*52.
= 14^2 x(5^2+1.)
I have also eliminated some exponents about recent world
records on sci.math , groups.google.com.
About 2^(13.4 million) -1.
e.g. 'twin primes about mersenne prime exponent.'
or 'factors of mersenne numbers.' ?? 80mill... divides M_13mill...
search don.lotto@paradise.net.nz
Gimps require one supercomputer week to prove
the world's largest known prime number. After it has been
selected by 200,ooo distributed personal computers.(??)
Penguin dictionary of curious +interesting numbers states (1987)
29th mersenne prime = M_132049.
This held me up for a while. The revised edn (David Wells 1997)
gives that Mp as the 30th and inserts a more recent
M_110503 as 29th. Unfortunately I missed that
interesting date 11.05.03 and 25th = 21.7.01.
29th = 1.3.2049 etc.
there seem to be lots.
They are supposed to say (probably-possibly the 40th Mersenne prime)
if not all indicies have been checked twice.
Readers Guide abstracts reported M_858433 is prime?.
I confirmed a FACTOR and possibly reported a typo.
In fact it should be M_859433 may be prime.
I advised author David Wells of a number of errors in [1987]
and he acknowledged me in Revised Edn [1997.]
(I claim 1/*61 and 7^*510 and 1215306625.eis)
I have about 45 sequences in On-line encyclopedia of integer sequences.
search google njas research eis lookup.
Don.McDonald
29.12.03 23:13
myfile.> DON02. Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
> subject:Mersenne numbers, Mp163 
> D.Calc.Factors.FermatMers.Mersenne.CARMP163.SPMP163
We look at the usual long multiplication of, for 
> example, 123*48.
...
In message <mnEYpEj034n@paradise.net.nz> you write:
> Dear
> nzlc
 teletext powerball#860 total prize pool $34million lies.
> teletext pball#860 total prize pool $34mill lies.
10 lotto always add twice..lies. (teleph keypad)
> very deceptive and misleading.
how many ticket sells?.
> $7.248 588 million powerball.

> don.mcdonald
 27.12.03 23:23
> 04/389-6820.
> 
>>D.LOTTO.lotoadvice.clients.NZLCLotter.pball+twic
check--

> (prob) -- #formula,-- value, -- FACTORS -- , (centiseconds).
> ---
> 2 860 # draw powerball..=860= 2^2*5*43*all 3cs
> 27.12.03
divis 1 roll down.
> winners x $ dividend.
3 11*1558133 div 2..=17139463= 11*19*82007*all 7cs
> 4 95*714 div 3..=67830= 2*3*5*7*17*19*all 4cs
> 5 725*151 divis 4..=109475= 5*5*29*151*all 4cs
> 6 3328*55 divis 5..=183040= 2^8*5*11*13*all 4cs
> 7 8899*28 divis 6..=249172= 2^2*7*11*809*all 4cs
9 p(3)+p(4)+p(5)+p(6)+..p(7)..
> total dividends should be $17.7 mill
=17748980= 2^2*5*887449*all 11cs
10 lotto always add twice..lies. (teleph keypad)
> =568625929723389440= Accy?(2^2*3*7*167* 47s.
11 34748980 tot prize pool
> too big by $ 17.ooo,ooo m..=34748980= 2^2*5*7*47*5281*all 7cs
14 210*323 primorial..=67830= 2*3*5*7*17*19*all 5cs
> 15 double digit bounce..(teleph keypad
> =36825334448268624= Accy?(2^2*3*19*2^2*53*761861437609prime 1mn
16 55331155338899..=55331155338899= 11*43*116979186763prime 39s.
e n d. prog c241Q 24.2.03 close *spool
>
====
The opposite of a profound truth is another profound truth. -- Niels
Bohr
What Rovelli doesn't seem to understand is that this all makes perfect
sense once
you give up strict equivalence and distinguish the background and
physical metrics.
JS: I do not understand this distinction. Please give more details what
you mean.
PZ: In that case you don't understand Newtonian physics either, which 
makes precisely
this distinction: you don't understand the Newtonian distinction between 
real and
fictitious forces.
But at least you are honest enough to admit it. :-)
JS: What I understand is that fictitious or inertial forces are 
artifacts of the non-geodesic timelike motion of the local frames of 
reference. I understand Coriolis, centrifugal, standing on a scale in an 
elevator as inertial forces. I also understand that LOCALLY there is, 
APPROXIMATELY, no way to distinguish the inertial force from a gravity 
force or G-force on a SINGLE TEST PARTICLE 1 NOT ON A TIMELIKE GEODESIC 
in sense of connection field for parallel transport (Experiment A), IF 
one MAKES NO ATTEMPT to measure the relative tidal acceleration between 
TWO OTHER TEST PARTICLES 2 & 3 BOTH ON TIMELIKE GEODESICS with ZERO 
G-FORCE (Experiment B).  My OPERATIONALISM is showing, which you ignore 
in your too abstract formal analysis. Therefore, you end up in a false 
comparison comparing apples to oranges so to speak by confounding the 
essentially different, indeed, complementary in Bohr's sense of total 
experimental arrangements - even of macro relevance, Experiments A & B.
Further, I do not see how you tie that to strict equivalence, which, 
if I understand you, you say is fundamentally wrong in some way? I do 
not understand how you mean background and physical above. Do you 
mean nondynamical and dynamical. The problem is that you introduce 
key terms without enough contextual background to understand what you 
mean. In many cases an equation would eliminate the ambiguity.
Now if you mean by strict equivalence that Einstein did not include 
Experiment B as a matter of principle in his early formulations, then 
if indeed, that is historically correct, then he may have made an error 
that was later corrected and is completely corrected in MTW (1973), 
which I suppose you say EEP is a correspondance, which is always the 
way I viewed it to begin with. If indeed your history of the evolution 
of Einstein's thought on his own theory is correct, I do not know if it 
is, then it is a minor footnote only. I am sure similar stories exist in 
the evolution of all the great theories of physics from Newton on.
Have you read pp. 112 - 114 that completely demolishes Hal Puthoff' s
use of
dr/dt = c' = c/K   radial null geodesic
in his Tables.
PZ: It does no such thing. I would not even characterize pp 112-114 as
an argument.
It is simply a sketch of a model in which *everything* is quantized
except the raw
manifold.
JS: It shows no intrinsic meaning to Puthoff's r and t as he means it
in his Tables.
PZ: In Rovelli's approach, almost everything is quantized and time 
itself has no fundamental
meaning.
So, OK,  things are VERY different in Rovelli's theory. No argument there.
He wants to dig down to the raw manifold so he can quantize the 
stripped-off
Einsteinian chronogeometric structure of spacetime, replete with its 
unified metric,
thinking this may be the real solution to the quantum gravity conundrum.
I say he has not properly understood the status and meaning of the 
unified metric.
He has simply skated over this. He is trying to run before he can walk.
...
What does he mean by fluctuations?
JS: What do you mean by kinematical g_uv and dynamic gravitational
g_uv apart from Ruvwl = 0 in  the former and not in the latter.
PZ: I mean what it means in Newtonian physics.
JS: Huh? Newton uses forces with action at a distance. He never invokes 
any geometrodynamical
replacement of forces the way Einstein does. Newton never talks of a 
metric so what do you mean?
Do you simply mean again the distinction between inertial and 
non-inertial frames of reference?
There are no fictitious or inertial forces in inertial frames. 
Newton only had implicitly
the idea of a global frame of reference not local frames of reference on 
a rigid Euclidean space with a
rigid absolute time.
Einstein in 1905 unified rigid space and rigid time into a rigid 
space-time in which space and time separately were no longer rigid.
Special Relativity uses a NONDYNAMICAL background RIGID 4D space-time 
that ACTS on MATTER WITHOUT BACK-ACTION of MATTER on space-time.
Einstein by 1915 corrects that approximation in General Relativity. 
Space-time GEOMETRY is now DYNAMICAL in TWO WAY RELATION (Bohm and 
(MASS-ENERGY).
Similarly, nonlocal linear unitary evolving orthodox micro-quantum 
theory with signal locality has a NON-DYNAMICAL BIT pilot wave 
relative to its IT extra-variable. The BIT is of course DYNAMICAL 
relative to its ENVIRONMENT via boundary conditions, stochastic pumps, 
semi-classical couplings etc. I am only here talking SELF-REFERENTIAL 
DYNAMICS of a kind not even recognized in other interpretations of 
micro-QM where
IT FROM BIT (Wheeler)
BIT is complete description of micro-quantum reality.
This includes all collapse models with the possible exception of 
Penrose's OR and all many-worlds models from Everett to 
Gell-Mann/Hartle to David Deutsch's multi-verse and also Cramer's 
transactional.
Shelly Godstein takes a wrong turn IMHO in his Bohmian Quantum Gravity 
paper in Physics Meets Philosophy at the Planck Scale in rejecting a 
source for the pilot wave where it is most important on the vast 
scale of the Universe in the FRW limit.
In contrast to micro-quantum theory, MACRO-QUANTUM THEORY is P.W. 
Anderson's More is different in action IMHO.
MACRO-QUANTUM THEORY is local, non-unitary nonlinear with presponse 
(Dick Bierman) signal nonlocality in the sense of Antony Valentini's 
violation of sub-quantal heat death.
The nonlocal linear micro-quantum Schrodinger equation in the 
configuration space of entangled parts of the whole is replaced by a 
local nonlinear MACRO-QUANTUM Landau-Ginzburg equation coupled to a 
residual micro-quantum Schrodinger equation in the sense of the old 
two-fluid model of Tiza but now generalized. This seems to go against 
some of Lenny Susskind's and t'Hooft's ideas and seems to support some 
of Hawking's older ideas on information loss in black holes. However, I 
am not sure of that. Lenny et-al seems to want to misapply micro-quantum 
theory in the MACRO-domain ignoring PW Anderson's More is different? I 
could be wrong. We shall see.
The phase-transition from an unstable completely random white zero point 
noise micro-quantum vacuum to a metastable MACRO-QUANTUM VACUUM with 
colored zero point noise controlled by Vacuum Coherence has a lower 
q-entropy defined as log of the phase space needed by the vacuum.
96% of the stuff of The World, we have been forced by the weight of 
FACTS to expand our notion of MATTER as MASS-ENERGY to include VIRTUAL 
ZERO POINT ENERGY or EXOTIC VACUA.  Zero Point energy has w = 
Pressure/Energy Density = -1.  Dark energy is exotic vacuum with 
negative micro-quantum pressure and dark matter is the same, but with 
positive pressure. All lepto-quarks have dark matter vortex string 
cores which prevent the distributed electric charge of the IT 
extra-variable from exploding. This is consistent with J.P. Vigier's 
notion of tight atomic states and it solves the old 
Abraham-Becker-Lorentz self-energy of the electron problem from ~ 100 
years ago. The smallness of the cosmological constant is not solved by 
string theory as Ed Witten admits, but it is, IMHO, solved by 
MACRO-QUANTUM VACUUM COHERENCE.
http://qedcorp.com/APS/EmergentGravity.pdf
Key prediction: No dark matter detector will click with the right 
dark stuff because all dark stuff is virtual not real. Dark stuff 

looks like w ~ 0 at a distance but up close it is w = -1 as one day 
interstellar space probes using dark energy weightless warp (Alcubierre) 
drives will confirm.
What is interesting about Lenny Susskind's theory however is the 
connection between black holes and elementary lepto-quarks and gauge 
force bosons as merely a matter of the complexity or bit length of the 
strings in which string has dual meaning as vibrating strings of 
energy and strings of computer theory in the sense of algorithmic 
complexity and all that. This is already seen in black hole 
thermodynamics where
Area/Lp^2 ~ number of bits
and the world hologram idea.
====
> Sure, math has little to do with actual numbers.  But sometimes you
> need to look at literals to look for patterns.  Even when you're
> result is radix neutral, the elegance of hexadecimal will set your
> brain in a mode of logic and intelligence.  Unless you're examining
> bowling scores, use hexadecimal.
Re: Hexadecimal leads to insight. google 4e6 in hex (= 3d 0900.)
don.mcdonald
3 dimension phone calls.
30.12.03
      <vzYGb.2175$HR.5464@news.indigo.ie>
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 03:36 PM, Timothy Murphy <tim@birdsnest.maths.tcd.ie> said:
>Isn't that all equipment?
Only if 18 is a multiple of 4.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 05:54 AM, mensanator@aol.compost (Mensanator) said:
>Which explains why DEC used octal for 16-bit registers and 20-bit
>addressing?
When did I ever say that everything DEC did with the PDP-11 was
reasonable? For that matter, when did I ever say that anything DEC did
with the PDP-11 was reasonable?
More to the point, A implies B is not equivalent to B implies A.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
>Message-id: <3fee0d35$13$fuzhry+tra$mr2ice@news.patriot.net
>   at 05:54 AM, mensanator@aol.compost (Mensanator) said:
>Which explains why DEC used octal for 16-bit registers and 20-bit
>>addressing?
When did I ever say that everything DEC did with the PDP-11 was
>reasonable? For that matter, when did I ever say that anything DEC did
>with the PDP-11 was reasonable?
When did I say it was unreasonable? My point apparently went completely 
over
your head. What you _did_ say was
<quote>
Headecimal is only reasonable for equipment that deals
with numbers in bytes that are multiples of 4 bits.
</quote>
It would be unreasonable for the PDP-11 to use hex because the index 
register
designations are three bits, thus making the machine language more easily
interpreted when the dump is in octal. If the PDP-11 had 16 registers 
instead
of 8, then it would be reasonable to use hex. Note that this has nothing to 
do
with word size or addressing.
Of course, a real programmer would understand that.
More to the point, A implies B is not equivalent to B implies A.
--
Mensanator
Ace of Clubs
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 01:17 AM, mensanator@aol.compost (Mensanator) said:
>My point apparently went completely over your head.
No, your point was simply irrelevant.
>What you _did_ say was
><quoteHeadecimal is only reasonable for equipment that deals
>with numbers in bytes that are multiples of 4 bits.
></quote>
K3wl. No read it very carefully and note what I did *NOT* write in it.
>It would be unreasonable for the PDP-11 to use hex
>thus making the machine language more easily
>interpreted when the dump is in octal. 
Only for people whose ability to do mental arithmetic is impaired.
>Note that this has nothing to do with word size or addressing.
I note your belief to that effect.
>Of course, a real programmer would understand that.
ROTF,LMAO! There is a difference between understanding why somebody
did something stupid and pretending that it wasn't stupid. The fact
that I don't agree with the decision doesn't mean that I didn't
understand it before you were born.
I repeat, 
>>More to the point, A implies B is not equivalent to B implies A.
You quoted it, but appear to have not read and understood it.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
> Making a mistake on sci.math is a good way to attract _corrections_.
> You could have learned what you've learned about analytic 
> continuation a lot faster if you'd simply tried to avoid being
> certain you were right and everyone else was wrong, _even
> though_ (as you _said_) you did not understand the construction
> that others were talking about.
  To all: please excuse my not being good at following what others
have done.
  So, there are two approaches to the construction of a Riemann
surface from a given relation f(w,z)=0: the original which goes from
relation to system of branches to Riemann surface, and the modern
which bypasses the introduction of branches. But one may think that
branches are worth studying for there own sake, and then in context
just think of Riemann surfaces as a byproduct. This is about branches.
  Branches cannot be studied from the given equation on its own.
Auxillary entities have to be introduced, first an ordinary point z0
in the z-plane to serve as an initial value for z. This done the
solutions of the equation f(w,z0)=0, say w1, w2, w3... will be initial
values for branches. Now, permutations on branches produced by
analytic continuation round closed circuits are already determined
without the branches as yet being fully specified. A circuit from z0
passing among but not through singularities and returning to z0 sends
each initial value into an initial value which may be the same or
different. The permutation depends only on the homotopy class of the
circuit. These classes depend in turn on the branch points which are
determined by the given equation. So one gets the idea that associated
with a specific branch point there may be a specific branch
permutation. It turns out that this is only so when another auxillary
entity, namely a set of branch boundaries, is introduced.
  So one arrives at a theory but one which depends on auxillary
entities which are arbitrary. At this point it is useful to take a
short digression into the philosophy of language. A statement may be
made in different languages but must be made in some language. So it
can be claimed that a statement is language-independent only in the
sense that it is translatable. This is the best that can be done. By
analogy what is needed in the present theory is to show that from a
result for one set of auxillary entities a result for any other can be
derived. This is easy and well known for an alternative initial point.
The two points are joined by an arc and the arc added to the circuit.
It is not quite as simple for an alternative set of boundaries but it
can be done. With this, branches and permutations on branches are
treated in a way which is as general as it can be.
====
[lots of stuff]
I think you've got it.  Good show.
Lee Rudolph
====
<snip>
 It's just pretty depressing to
> see a book tell you that you have learned next to nothing that you should 
know
> coming into graduate school.  Any comments or advice would be 
appreciated,
> including book suggestions (the author tends to say I've heard that
> such-and-such book is good though I have not seen it which is pretty 
odd
> considering the focus of the book).
You may have trouble getting into a top ranked graduate program if
your background is deficient.  However, what's far more important is
what you do in the graduate school you do get into.  Many will allow
you to take a few undergraduate courses to firm up your background. 
Another clue to what you will eventually need to know is the book of
problems from UCBerkely qualifying exams published by Springer.
Don't let snobs tell you that a PhD from a less than top-ranked school
is worthless.  It isn't.  What is important is to work with an active
and productive researcher with a national (or international)
reputation in his/her field of research.  Such people will make sure
you do a good thesis and will be able to write letters that say what
prospective employers want to hear.  There are such people even at
lower ranked schools due to the bad job market of recent years.  You
may even do better than you might at a top school because as a good
student you'll get a lot more attention than you would otherwise.
====
> I checked out a book called All the Mathematics You Missed [But Need to 
Know
> for Graduate School] from the library and was surprised by its contents. 
 The
> book is divided into 16 sections that I am supposed to know before I 
get
> into
> graduate school.  This is my last year and I can check off very little.
Here are the sixteen topics that I need to know along with whether or 
not I
> will have completed them by the end of the year:
> 
[...]
> I know that looks awful, even beyond awful, with 5/16.  I don't think 
it's
> realistic that I could learn that much material over the Summer.  Which 
areas
> do I absolutely need to know?  
> 
That kind of depends on what the grad school you go to expects.  For
example, some schools will expect you to have some familiarity with
undergraduate complex analysis in their graduate complex analysis
course, and some won't.  
What may be most helpful is to take a look at Lebesgue integration.  At
least get a rough outline of what it's about.  And in particular, you
can be more careful about checking details and stuff for the
preliminary stuff like measures and sigma algebras.  This is probably
one of the harder topics in a first year grad course, and it helps to
get a start on it.
Another thing is to learn some point set topology before you get to
grad school.  It shouldn't be too hard to learn, given that you've had
real analysis.  This will really help in learning graduate level
analysis.  Even though there is such a thing as graduate point set
topology, few places offer a course in it, and many assume you've
learned undergrad point set topology somehow.  There's nothing like
walking into an analysis or algebraic topology class and learning that
you're *supposed* to know what a locally compact, connected, Hausdorff
space is.
You might as well learn about Gaussian curvature of surfaces in R^3. 
It motivates a lot of very advanced material that you may very well run
into in some of your classes.  And it's a relatively simple topic that
is fun to learn about.  It should offer a nice respite from Lebesgue
integration or whatnot.  
There's more stuff you could learn (like the generalized Stokes
theorem), but probably it not essential, and you have more than enough
on your hands already.
> Is this book very accurate in what I SHOULD know for graduate school?  
Almost
> everything seems to roughly fall under analysis/applied math.  
[...]
I think so.  Note your ideas about applie math are rather misguided.
The strange thing is that, besides applied math classes, I'm taking or 
have
> taken what they offer in terms of pure math.  It's just pretty depressing 
to
> see a book tell you that you have learned next to nothing that you should 
know
> coming into graduate school.  
Cheer up; you have a jump on all your future classmates who will not
have a clue that they should know these things.  Sometimes it takes
people a year or two in grad school before they realize they need to
remedy their ignorance in some of these topics, and then it's very late
and hard to fix.
Besides, doesn't the author say he doesn't expect the reader to know
all these things?  After all, look at the title; clearly he expects you
to have missed some of the topics.  The emphasis should be on the
phrase *should know*.  I could go on all day about what you should'a,
but it won't help you much.
> Any comments or advice would be appreciated,
> including book suggestions (the author tends to say I've heard that
> such-and-such book is good though I have not seen it which is pretty 
odd
> considering the focus of the book).

I remember flipping through this book at Borders, and thinking the
references were rather scanty in some spots.
with the list of references for those topics in the book, so I can
comment on whether I think they are sufficient or lacking.  I remember
most seemed sufficient.
====
I'll try to clear a few things up.  The lowest grade I've made in a math 
course
is a B+.  Normally I just get A's.  For what it's worth, my school is a 
Tier
2 school according to US News and World Reports.  It also has a PhD 
program. 
Some of the professors that are well known do not want to teach 
undergraduates,
so that is partly (I think) why topology is not offered.
I'm interested in algebra and number theory.  That is why I was surprised 
when
the book had only 2 sections on algebra and none on number theory.  I've 
taken
the regular number theory course and an algebraic number theory one as well. 
 
Two of the Professors that are writing my recommendations were disappointed
with where I wanted to apply.  One of them has told me that I have 
research
potential but I guess that can be interpreted as just being nice.  
====
[...]
| Two of the Professors that are writing my recommendations were 
disappointed
| with where I wanted to apply.  One of them has told me that I have 
research
| potential but I guess that can be interpreted as just being nice.  
Take it seriously if they think you should try applying to some better
schools.
It's too bad you didn't get better opportunities as an undergraduate, but
consider that water under the bridge.
It's still some years from when you have to do anything, but one thing you
might want to keep in mind for when you get there: If you wind up in a PhD 
program which is only okay, but you have reason to believe you're doing
thesis work of a higher caliber (so it seems like you might be on your way
to being underrated), I'd recommend trying to get letters of recommendation
from bigger names in addition to your advisor. I was always crummy at
self-promotion, but even I know that it does your career good to be well-
connected. So as you get into a field, get acquainted with some of the 
people 
already established in it. When you get good results, see if you can get 
favorable letters.
It was heartening when I was looking for mathematical jobs when someone from 

one of the schools I'd had an interview with came up to me afterward, and 
said 
something like, Hey, Professor [name] was just telling us about you, and 

evidently it was something good he'd told them. Not that I got the job :-(
but it sure can help the odds.
I remember a guy who managed to get a letter of recommendation from Deligne
saying something like, this guy solved a problem we tried to solve. He did
get the job!
Keith Ramsay
====
> I'll try to clear a few things up.  The lowest grade I've made in a math 
course
> is a B+.  Normally I just get A's.  For what it's worth, my school is a
> Tier 2 school according to US News and World Reports.  It also has a
> PhD program. Some of the professors that are well known do not want to
> teach undergraduates, so that is partly (I think) why topology is not
> offered.
your school does not seem impressive at all.
even in cases where prefessors do not like teaching undergrads, an
undergrad  topology course offering is a must. of course that can be
resolved by offering independent studies. does your school offer such?
if it does, why didn't you study undergrad topology independently?
> I'm interested in algebra and number theory.  That is why I was surprised
> when the book had only 2 sections on algebra and none on number theory.
> I've taken the regular number theory course and an algebraic number
> theory one as well.  
in general, a gap in undergrad algebra, advanced calculus, and
topology is a very serious one for anyone pursuing a graduate math
degree. a gap in number theory is not.
> Two of the Professors that are writing my recommendations were 
disappointed
> with where I wanted to apply.
assuming your report is accurate, your professors have good reasons to
be dissapointed - graduate math degrees from unranked programmes are
generally worthless in the job market.
> One of them has told me that I have research potential but I guess 
that
> can be interpreted as just being nice.  
true. but if his/her assessemnt is accurate, then you should endeavour
to fill in the gaps in your undergrad math preparation and then shoot
high for a ranked math graduate school.
====
>I'll try to clear a few things up.  The lowest grade I've made in a math 
course
>is a B+.  Normally I just get A's.  For what it's worth, my school is a 
Tier
>2 school according to US News and World Reports.  It also has a PhD 
program. 
>Some of the professors that are well known do not want to teach 
undergraduates,
>so that is partly (I think) why topology is not offered.
I'm interested in algebra and number theory.  That is why I was surprised 
when
>the book had only 2 sections on algebra and none on number theory.  I've 
taken
>the regular number theory course and an algebraic number theory one as 
well.  
Two of the Professors that are writing my recommendations were 
disappointed
>with where I wanted to apply.  One of them has told me that I have 
research
>potential  
That's different. You didn't mention those letters in your original
post.
You should apply to a few good schools. Those professors have a
better idea than you do about your ability compared to the students
those places usually admit. The worst that could happen is they
turn you down...
>but I guess that can be interpreted as just being nice. 
No. Math professors are not nice. It's a condition of employment.
************************
David C. Ullrich
====
> I'll try to clear a few things up.  The lowest grade I've made in a math
> course
> is a B+.  Normally I just get A's.  For what it's worth, my school is a 
Tier
> 2 school according to US News and World Reports.  It also has a PhD 
program. 
> Some of the professors that are well known do not want to teach
> undergraduates,
> so that is partly (I think) why topology is not offered.
I'm interested in algebra and number theory.  That is why I was surprised 
when
> the book had only 2 sections on algebra and none on number theory.  I've 
taken
> the regular number theory course and an algebraic number theory one as 
well.  
Two of the Professors that are writing my recommendations were 
disappointed
> with where I wanted to apply.  One of them has told me that I have 
research
> potential but I guess that can be interpreted as just being nice.  
> 
Hmm...this is interesting.  It sounds like to me that they think you
could do better.  Perhaps you are being overly pessimistic.  Granted,
the discussion on sci.math thus far has not been encouraging, but it
has mostly centered around getting into really hard schools.
Your professors probably have a better idea than you of where you can
get in.  If they think you can get in somewhere better, then follow
their advice!  Your professors may also have a better reputation than
you think.  Even if they're not hotshots (that don't teach, as you
mention above), you shouldn't dismiss them; if they have decent
reputations, and/or have connections, you could do very well with their
recommendations.
Am I right in thinking that you've decided to apply to some masters
programs, instead of some doctoral programs?  I urge you to read my
other posts in this discussion.  Basically, you may very well be
shooting yourself in the foot if you decide to go the masters route.
====
> Am I right in thinking that you've decided to apply to some masters
> programs, instead of some doctoral programs?  I urge you to read my
> other posts in this discussion.  Basically, you may very well be
> shooting yourself in the foot if you decide to go the masters route.
I should say that I agree with Chan-Ho Suh here.  (Surprise!)
I think that the best reason to go to a masters program, for someone
who someday hopes to get a doctorate, is that they have failed to get
into the sort of doctoral programs they think they need in order to
succeed.  I think this is true for all disciplines, actually.  Some of
the comments here suggest that a masters may be a positive detriment
to some math PhD programs; but if you could have gotten a decent
doctorate anyway, then in any discipline, math or not, the masters is
really only a waste of a couple years.
Note that this is the hidden message of the Philosophical Gourmet
excerpt I quoted.  The only reason to get a masters is if without it,
you would not be able to get into a decent doctoral program.  Perhaps
it doesn't help in math as much as it might in some fields, but
regardless, if you can get into a decent doctoral program, then you
shouldn't get a separate masters first.
And the only way to know if you can get into a decent doctoral program
is to apply.  
Thomas
====
> Two of the Professors that are writing my recommendations were
> disappointed with where I wanted to apply.  One of them has told me
> that I have research potential but I guess that can be interpreted
> as just being nice.
You mean, in the sense that they thought you should apply to better
schools?  Take that advice!  I think one should aim high, while
being prudent, in this as everything.  I think it's always a good idea
to apply to several places that you think are beyond your reach.
====
>Message-id: <bcnHb.3073$d4.1048@newsread1.news.atl.earthlink.net>
[...]
>If you are aiming for a ph.d, then I would suggest getting into the best 
M.S
>program you can, and then apply to a great ph.d program when you get your
>M.S..  This is essentially what I am doing, for I am in the same 
situation.
>Good luck!

Lurch
This is basically what I plan on doing.  Is it bad to get a masters at a 
school
with a PhD program and then transfer to another school afterwards, even if 
you
get good recommendations?  I'm applying to a few pretty low-ranked PhD 
programs
and one masters program.  I just don't know of any really GREAT masters
programs but I'm sure they are out there.  Do you know of any?
====
I think there are several mathematics programs for people with weak
backgrounds at interesting schools ,e.g. , Mathematics Opportunity 
Committee
at Berkeley , and perhaps Princeton or Harvard have something similar .
Best of Luck.
> I checked out a book called All the Mathematics You Missed [But Need to
Know
> for Graduate School] from the library and was surprised by its 
contents.
The
> book is divided into 16 sections that I am supposed to know before I 
get
into
> graduate school.  This is my last year and I can check off very little.
 Here are the sixteen topics that I need to know along with whether or
not I
> will have completed them by the end of the year:
 1.  Linear Algebra - Yes
> 2.  Real Analysis - Yes
> 3. Differentiating Vector Valued Functions (jacobians, inverse function
> theorem) - No (nothing like this taught at my school)
> 4.  Point Set Topolgy - No (not offered here)
> 5. Classical Stokes Theorems - Yes
> 6. Differential Forms and Stokes Theorem - No (nothing like that here)
> 7. Curvature for Curves and Surfaces (differential geometry) - No (not
offered)
> 8. Geometry - No (only course offered is one for future high school
teachers
> and was advised not to take it)
> 9.  Complex Analysis - No (schedule conflicts last year and this year)
> 10.  Countability and the Axiom of Choice - No (not offered but I have
looked
> into it a bit)
> 11. Algebra - Yes
> 12. Lebesgue Integration - No (not undergrad here)
> 13. Fourier Analysis - No (I thought this was for engineers)
> 14. Differential Equations - Yes
> 15. Combinatorics and Probability - No (combinatorics not offered;
probability
> only after calc-based statistics is taken)
> 16. Algorithms - No (the closest thing to what is described here is a
mid-level
> computer science course).
 I know that looks awful, even beyond awful, with 5/16.  I don't think 
it's
> realistic that I could learn that much material over the Summer.  Which
areas
> do I absolutely need to know?
 Is this book very accurate in what I SHOULD know for graduate school?
Almost
> everything seems to roughly fall under analysis/applied math.  The math
> department has no one that does any research whatsoever in geometry (for
those
> areas listed here).  Only 2 sections are devoted to algebra and there is
> nothing about number theory.
 The strange thing is that, besides applied math classes, I'm taking or
have
> taken what they offer in terms of pure math.  It's just pretty depressing
to
> see a book tell you that you have learned next to nothing that you should
know
> coming into graduate school.  Any comments or advice would be 
appreciated,
> including book suggestions (the author tends to say I've heard that
> such-and-such book is good though I have not seen it which is pretty 
odd
> considering the focus of the book).

====
> I checked out a book called All the Mathematics You Missed [But Need to
> Know for Graduate School] from the library and was surprised by its
> contents.  The book is divided into 16 sections that I am supposed to
> know before I get into graduate school.  This is my last year and I
> can check off very little.
Here are the sixteen topics that I need to know along with whether or
> not I will have completed them by the end of the year:
1.  Linear Algebra - Yes
> 2.  Real Analysis - Yes
> 3. Differentiating Vector Valued Functions (jacobians, inverse
> function theorem) - No (nothing like this taught at my school)
> 4.  Point Set Topolgy - No (not offered here)
> 5. Classical Stokes Theorems - Yes
> 6. Differential Forms and Stokes Theorem - No (nothing like that here)
> 7. Curvature for Curves and Surfaces (differential geometry) - No
> (not offered)
> 8. Geometry - No (only course offered is one for future high school
> teachers and was advised not to take it)
> 9.  Complex Analysis - No (schedule conflicts last year and this year)
> 10.  Countability and the Axiom of Choice - No (not offered but I have
> looked into it a bit)
> 11. Algebra - Yes
> 12. Lebesgue Integration - No (not undergrad here)
> 13. Fourier Analysis - No (I thought this was for engineers)
> 14. Differential Equations - Yes
> 15. Combinatorics and Probability - No (combinatorics not offered;
> probability only after calc-based statistics is taken)
> 16. Algorithms - No (the closest thing to what is described here is a
> mid-level computer science course).
I know that looks awful, even beyond awful, with 5/16.  I don't think
> it's realistic that I could learn that much material over the Summer.
> Which areas do I absolutely need to know?
strong background in in areas 1-11. at least moderate working
knowledge in areas 11-15.
> Is this book very accurate in what I SHOULD know for graduate school?  
Almost
> everything seems to roughly fall under analysis/applied math.  The math
> department has no one that does any research whatsoever in geometry (for 
those
> areas listed here).  Only 2 sections are devoted to algebra and there is
> nothing about number theory. 
The strange thing is that, besides applied math classes, I'm taking or 
have
> taken what they offer in terms of pure math.  It's just pretty depressing 
to
> see a book tell you that you have learned next to nothing that you should 
know
> coming into graduate school.  Any comments or advice would be 
appreciated,
> including book suggestions (the author tends to say I've heard that
> such-and-such book is good though I have not seen it which is pretty 
odd
> considering the focus of the book).
it seems clear that you wasted your undergrad years in a bad math
school (typical.) that alone is good reason for you to quit while you
are not so far behind, assuming that a major reason for you to pursue
math is a job, including academic math jobs beyond k-12. now, if the
main reason for you to pursue math is for it's own sake (extremely
unlikely,) then you will need to patch the huge gaps your undergrad
institution left behind, and then engage in graduate math studies,
preferably in a highly ranked programme.
====
>>What you need to know before starting grad school varies considerably
>>from place to place - you probably don't have any chance at all of 
>>getting into one of the best graduate programs in the country, but
>>there are plenty of graduate math departments around where most
>>of the incoming students will have a background more or less like
>>yours (and plenty of places where the majority of incoming grad
>>students will be entering with what the department considers an
>>inadequate background, because they can't get the students they
>>want - it happens a lot that incoming students at medium-level
>>grad programs start by taking a lot of undergraduate classes
>>that didn't exist where the student came from.)
>>One bit of advice would be next time, if you intend to go to
>>grad school, do your undergrad work at a place with a slightly
>>stronger program. Maybe a little late for that now...
I realized that I had no shot at any top program about a year ago when I 
looked
>at the course offerings at the Ivy League schools and MIT.  This is my 
senior
>year so it's too late to go to a stronger program.  I know that my 
department
>is bad but now it's just could have and should have.  How realistic 
is it
>to get a masters degree and transfer to a better school?  
Hard to say. I don't know of any examples (in math) of someone with
a BS from a mediocre place who got a master's from some place
better and then a PhD from a top school. The fact that I don't
know of any examples doesn't prove there are none - Bushnell
has stated that he knows of plenty of such examples (otoh he's
failed to name any, after being asked twice...)
But it really doesn't follow that you're screwed. You should be able
to get into a PhD program _somewhere_ if that's what you want.
Then later when you're applying for jobs the fact that the place
you got your degree was not a top school will not be in your
favor, but if you've written a really fabulous thesis people will
overlook that - if you settle the Riemann hypothesis people 
will be interested regardless of where you got your degree.
>Are you just screwed
>if you did not go to the right school and take the right courses with the 
right
>professors?
When we're talking about undergraduate work probably right courses
is the main thing. And the right grades in those courses.
(You could arrange to flunk philosophy or something, so you get to
stick around another year and fill in a few gaps...)
>>As far as what you should really know, probably the most
>>important thing is that you have a good idea what a _proof_
>>is, and you have some facility writing correct proofs of relatively
>>easy facts (by this I mean most of the exercises in a beginning
>>abstract algebra course, say. The algebra you say you've
>>taken was about groups and rings and things, not like
>>a high-school algebra course, right?)
Of course. 
>By the way, many of the topics you're thinking of as
>>applied math are _also_ very important in pure math -
>>it may not look that way from the course offerings where
>>you went to school. (In particular, although there is
>>certainly such a thing as a _course_ in Fourier analysis
>>that's meant for engineers, Fourier analysis itself is
>>incredibly important in many fields of math. Same for
>>complex.)
The applied math courses here are almost exclusively for engineering or 
physics
>majors.  There is a class that covers Fourier Analysis
>and Partial Differential Equations but I (mistakenly?) thought that it was 
for
>the engineering and physics people.  
>************************
>>David C. Ullrich
>
************************
David C. Ullrich
====
> Hard to say. I don't know of any examples (in math) of someone with
> a BS from a mediocre place who got a master's from some place
> better and then a PhD from a top school. The fact that I don't
> know of any examples doesn't prove there are none - Bushnell
> has stated that he knows of plenty of such examples (otoh he's
> failed to name any, after being asked twice...)
We can also look at catalogues from schools that list all the degrees
of their faculty.  Of course, that is often somewhat dated, but it's
at least published information so I don't need to be reticent about
names.
UMass/Amherst's math department includes one Nathaniel Whitaker, who
is BA, Hampton Institute, 1974; MS, Cincinatti, 1981; PhD, California,
1987.  I assume that California means UC Berkeley.
I looked through the list of current grad students at Cornell.  I
found one Liang Chen, who has a BS from Peking University and a MS
from the University of Wisconsin at Madison, and is now at Cornell.
And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
Autonoma de Madrid, and an MA from Duke and is now at Cornell.
Jennifer Fawcett is BA from Rice, MA from UC Davis, and now at
Cornell.  Lee Gibson, BS from the University of Kentucky, MS from the
University of Louisville, now at Cornell.  Farkhod Eshmatov, BS from
Tashkent State University, MA from SUNY Binghamton, now at Cornell.  
That's six, in about ten minutes of web looking.
Thomas
====
> Hard to say. I don't know of any examples (in math) of someone with
>> a BS from a mediocre place who got a master's from some place
>> better and then a PhD from a top school. The fact that I don't
>> know of any examples doesn't prove there are none - Bushnell
>> has stated that he knows of plenty of such examples (otoh he's
>> failed to name any, after being asked twice...)
We can also look at catalogues from schools that list all the degrees
>of their faculty.  Of course, that is often somewhat dated, but it's
>at least published information so I don't need to be reticent about
>names.
UMass/Amherst's math department includes one Nathaniel Whitaker, who
>is BA, Hampton Institute, 1974; MS, Cincinatti, 1981; PhD, California,
>1987.  I assume that California means UC Berkeley.
That's _one_ example that seems valid, at least if California does
mean Berkeley. My reaction to most of the examples below is the
same as the people who've already replied - the majority of the
places that you seem to be regarding as mediocre seem like
pretty good schools to me. This is certainly true of most of the
places you say people got their master's degrees.
Take the person with the BA from Rice. You think that here
undergraduate transcript looks anything like the transcript of
the OP's?
>I looked through the list of current grad students at Cornell.  I
>found one Liang Chen, who has a BS from Peking University and a MS
>from the University of Wisconsin at Madison, and is now at Cornell.
>And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
>Autonoma de Madrid, and an MA from Duke and is now at Cornell.
>Jennifer Fawcett is BA from Rice, MA from UC Davis, and now at
>Cornell.  Lee Gibson, BS from the University of Kentucky, MS from the
>University of Louisville, now at Cornell.  Farkhod Eshmatov, BS from
>Tashkent State University, MA from SUNY Binghamton, now at Cornell.  
That's six, in about ten minutes of web looking.
Thomas
************************
David C. Ullrich
====
> Take the person with the BA from Rice. You think that here
> undergraduate transcript looks anything like the transcript of
> the OP's?
Who knows?  It's possible to have bad grades from a good school, and
good grades from a bad school.  I'm speaking only about the latter.
Thomas
====
> Take the person with the BA from Rice. You think that here
>> undergraduate transcript looks anything like the transcript of
>> the OP's?
Who knows?  
It's easy to find out with a little web searching. At
we read that an undergad degree at Rice requires 8 courses 
at 300 level or above. To get an idea what that means we look
at the course offerings in three recent semesters (details below).
And we conclude that the transcript of someone with a degree
from Rice looks _nothing_ like the OP's transcript - it's possible
that he hasn't taken even _one_ course that's anything like
a 300+ course at Rice, much less 8. (He says he's taken algebra - 
it's hard to know how the content of that course compares
with the content of a similarly-named course at Rice. But
it's easy to guess what the typical admissions committee
is going to guess about the comparison between the
similarly named courses at the two schools...)
Citing a person with a BA from Rice as an example of someone
with a mediocre BA who nonetheless got into a good PhD program
by getting a master's from a good place turns out to be as
ridiculous as it seemed to me yesterday, before I looked
up the details.
Details:
  
http://math.rice.edu/Courses/webpages.html
   Math 401: Differential Geometry 
   Math 423: Partial Differential Equations 
   Math 444: Geometric Topology 
   Math 468: Potpourri (Dynamical Systems) 
   Math 499/699:Math. Sciences VIGRE
   Seminar: Computational Algebraic
   Geometry 
   Math 590: Current Mathematics Seminar 
   Math 591: Graduate Teaching Seminar 
   
MATH 356   ABSTRACT ALGEBRA                         Credits 3.00
Spring 03
       * DISTRIBUTION COURSE: GROUP III
       Groups: normal subgroups, factor groups, Abelian groups.
Rings: ideals,
       Euclidean rings, and unique factorization.  Fields: algebraic
extensions,
       finite fields.  Students may not take this course and Math 463.
       001 HB 227 - MWF 02:00PM - 02:50PM      Hyeon, Donghoon David
Enr: 12 Max: 0
       MATH 366   GEOMETRY                                 Credits
3.00  Spring 03
       * DISTRIBUTION COURSE: GROUP III
       Topics chosen from Euclidean, spherical, hyperbolic, and
projective geometry,
       with emphasis on the similarities and differences found in
various geometries.
       Isometries and other transformations are studied and used
throughout.  The
       history of the development of geometric ideas is discussed.
This course is
       strongly recommended for prospective high school teachers.
       001 HB 227 - MWF 03:00PM - 03:50PM      Hassett, Brendan
Enr: 32 Max: 0
       MATH 382   COMPLEX ANALYSIS                         Credits
3.00  Spring 03
       * DISTRIBUTION COURSE: GROUP III
       Study of the Cauchy integral theorem, Taylor series, residues,
as well as the
       evaluation of integrals by means of residues, conformal
mapping, and
       application to two-dimensional fluid flow.  May not receive
credit for this
       course and Math 427.
       001 HB 227 - MWF 01:00PM - 01:50PM      Song, Joung Min Jaime
Enr: 26 Max: 0
       MATH 390   UNDERGRADUATE COLLOQUIUM                 Credits
1.00  Spring 03
       * DISTRIBUTION COURSE: GROUP III
       Lectures by undergraduate students on mathematical topics not
usually covered
       in other courses.  Each student is required to give one lecture
and to attend
       all sessions.
       001 HB 423 - MF 12:05PM - 12:50PM       Jones, Frank
Enr: 7 Max: NA
       MATH 402   DIFFERENTIAL GEOMETRY                    Credits
3.00  Spring 03
to year.
       Prereq- Math 401 and either Math 443 or permission of the
instructor.
       001 HB 423 - TTH 09:25AM - 10:40AM      Wolf, Michael
Enr: 3 Max: 0
       MATH 424   PARTIAL DIFFERENTIAL EQUATIONS           Credits
3.00  Spring 03
       See Math 423.
       001 HB 423 - MWF 02:00PM - 02:50PM      Jones, Frank
Enr: 8 Max: 0
       MATH 426   TOPICS IN REAL ANALYSIS                  Credits
3.00  Spring 03
harmonic
       analysis, probabilty theory, advanced topics in measure theory,
ergodic theory,
       and elliptic
       integrals.
       001 HB 423 - TTH 01:00PM - 02:20PM      Veech, William A.
Enr: 2 Max: 0
       MATH 427   COMPLEX ANALYSIS                         Credits
3.00  Spring 03
       Study of the Cauchy-Riemann equation, power series, Cauchy's
integral formula,
       residue calculus, and conformal
       mappings.
       001 HB 427 - MWF 10:00AM - 10:50AM      Boshernitzan, Michael
Enr: 9 Max: 0
       MATH 443   GENERAL TOPOLOGY                         Credits
3.00  Spring 03
       Study of basic point set topology.  Includes a treatment of
cardinality and
       well ordering, as well as metrization.
       Prereq: MATH 321 or permission of instructor recommended.
       001 HB 427 - MWF 02:00PM - 02:50PM      Clark, Gregory
Enr: 8 Max: 0
       MATH 445   ALGEBRAIC TOPOLOGY                       Credits
3.00  Spring 03
       Introduction to the theory of homology.  Includes simplicial
complexes, cell
       complexes and cellular homology and cohomology, as well as
manifolds, and
       Poincare Duality.
       Prereq- Math 444 and either Math 356 or Math 463 or permission
of instructor.
       001 HB 427 - MWF 01:00PM - 01:50PM      Cochran, Tim D.
Enr: 10 Max: 0
       MATH 464   ALGEBRA II                               Credits
3.00  Spring 03
       See Math 463.
       001 HB 423 - MWF 11:00AM - 11:50AM      Boshernitzan, Michael
Enr: 8 Max: 0
       MATH 465   TOPICS IN ALGEBRA                        Credits
3.00  Spring 03
       Pre-req- Math 356 or Math 463 or permission of the instructor.
       001 HB 427 - MWF 09:00AM - 09:50AM      Hassett, Brendan
Enr: 5 Max: 0
       MATH 468   POTPOURRI                                Credits
3.00  Spring 03
       This course will consider power series, real analytic
functions, and some
       related matters.
       Prereq- Math 321 and either Math 382 or Math 427 (which may be
taken
       concurrently), or permission of the instructor.
       001 TBA - TTH 01:00PM - 02:20PM         Semmes, Stephen
Enr: 3 Max: NA
       MATH 490   SUPERVISED READING                       Credits
Spring 03
       No description.
       001 TBA - TBA                           Staff
Enr: 6 Max: NA
       002 TBA - TBA                           Staff
Enr: 1 Max: NA
       003 TBA - TBA                           Staff
Enr: 0 Max: NA
       MATH 502   TOPICS IN DIFFERENTIAL GEOMETRY          Credits
3.00  Spring 03
       The Atiyah-Singer theorem, secondary invariants, and related
topics.
       001 HB 427 - TTH 10:50AM - 12:05PM      Forman, Robin
Enr: 20 Max: 0
       MATH 522   TOPICS IN  REAL ANALYSIS                 Credits
3.00  Spring 03
       Geometric Measure Theory treats measure-theoretic properties of
geometrically
       defined sets of various dimensions. Some of the critical
notions are  Hausdorff
       measure, rectifiable sets, and rectifiable currents.  The k
dimensional
       Hausdorff (outer) measure  H k(A) gives, for every nonnegative
number  k ,  a
       precise notion of the k  dimensional size of  A .  Rectifiable
sets and
       currents arise as limits of k dimensional manifolds. these
occur in the
       solution of the Plateau Problem of finding a  k  dimensional
surface of least
       k dimensional area having a given boundary.
       Graduate student standing or permission of instructor.
       001 TBA - MWF 12:00PM - 12:50PM         Staff
Enr: 7 Max: NA
       MATH 590   CURRENT MATHEMATICS SEMINAR              Credits
1.00  Spring 03
       Lectures on topics of recent research in mathematics
delivered by
       mathematics graduate students and faculty.
       Prereq: graduate student standing or permission of the
department.
       001 HB 227 - TTH 02:30PM - 03:45PM      Staff
Enr: 25 Max: NA
       MATH 591   GRADUATE TEACHING SEMINAR                Credits
1.00  Spring 03
       Discussion on teaching issues and practice lectures by
participants as
       preparation for classroom teaching of mathematics.
       Graduate student status or permission of department.
       001 HB 427 - T 02:30PM - 03:30PM        Staff
Enr: 15 Max: NA
       MATH 800   THESIS AND RESEARCH                      Credits
Spring 03
       
http://www.rice.edu/projects/courses/2002fall/MATH.html
MATH 321   INTRODUCTION TO ANALYSIS I               Credits 3.00  Fall
02
       * DISTRIBUTION COURSE: GROUP III
       A thorough treatment of basic methods of analysis such as
metric spaces,
       compactness, sequences and series of functions.  Also further
topics in
Liouville theory.
       Prereq- Math 222 or permission of department.
       001 HB 427 - MWF 03:00PM - 03:50PM      Semmes, Stephen
Enr: 21 Max: NA
       MATH 355   LINEAR ALGEBRA                           Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Linear transformations and matrices, solution of linear
equations, eigenvalues
       and eigenvectors, quadratic forms, rational canonical form,
Jordan canonical
       form.
       001 HZ 210 - MWF 02:00PM - 02:50PM      Clark, Gregory
Enr: 108 Max: NA
       MATH 368   TOPICS IN COMBINATORICS                  Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Study of combinatorics and discrete mathematics.  Topics that
may be covered
       include graph theory, Ramsey theory, finite geometries,
combinatorial
       enumeration, combinatorial games.
       Prereq- Math 211.
       001 HB 227 - MWF 03:00PM - 03:50PM      Stong, Richard A.
Enr: 26 Max: NA
       MATH 381   INTRODUCTION TO PARTIAL DIFFERENTIAL EQU Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Laplace transform:  inverse transform, applications to constant
coefficient
       differential equations.  Boundary value problems:  Fourier
series, Bessel
       functions, Legendre polynomials.
       001 HZ AMP - MWF 01:00PM - 01:50PM      Evans, Richard
Enr: 73 Max: NA
       MATH 390   UNDERGRADUATE COLLOQUIUM                 Credits
1.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Lectures by undergraduate students on mathematical topics not
usually covered
       in other courses.  Each student is required to give one lecture
and to attend
       all sessions.
       001 HB 227 - MF 12:05PM - 12:55PM       Jones, Frank
Enr: 6 Max: 0
       MATH 401   DIFFERENTIAL GEOMETRY                    Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Study of the differential geometry of curves and surfaces in
R3.  Includes an
       introduction to the concept of curvature and thorough treatment
of the
       Gauss-Bonnet theorem.
       001 HB 427 - TTH 09:25AM - 10:40AM      Wolf, Michael
Enr: 12 Max: NA
       MATH 423   PARTIAL DIFFERENTIAL EQUATIONS           Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Theory of distributions.  Wave equation, Laplace's equation,
heat equation.
       Fundamental solutions.  Other topics include first order
hyperbolic systems,
       Cauchy-Kowalewski theorem, potential theory, Dirichlet and
Neumann problems,
       integral equations, elliptic equations.
       001 HB 427 - MWF 02:00PM - 02:50PM      Jones, Frank
Enr: 9 Max: NA
       MATH 425   REAL ANALYSIS                            Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Lebesgue theory of measure and integration.
       001 HB 423 - TTH 01:00PM - 02:20PM      Wiandt, Tamas
Enr: 13 Max: NA
       MATH 428   TOPICS IN COMPLEX ANALYSIS               Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Special topics include Riemann mapping theorem, Runge's
Theorem, elliptic
       function theory, prime number theorem, Riemann surfaces.
       001 HB 453 - MWF 10:00AM - 10:50AM      Hassett, Brendan
Enr: 4 Max: NA
       MATH 444   GEOMETRIC TOPOLOGY                       Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Introduction to algebraic methods in topology and differential
topology.
       Elementary homotopy theory.  Covering spaces.
       001 HB 427 - MWF 09:00AM - 09:50AM      Hempel, John P.
Enr: 10 Max: NA
       MATH 463   ALGEBRA I                                Credits
3.00  Fall 02
       * DISTRIBUTION COURSE: GROUP III
       Groups, rings, fields, vector spaces.  Matrices, determinants,
eigenvalues,
       canonical forms, multilinear algebra.  Structure theorem for
finitely generated
       abelian groups. Galois theory.
       001 HB 453 - MWF 11:00AM - 11:50AM      Hempel, John P.
Enr: 12 Max: NA
       MATH 490   SUPERVISED READING                       Credits
Fall 02
       No description.
       001 TBA - TBA                           Staff
Enr: 3 Max: 0
       002 TBA - TBA                           Staff
Enr: 2 Max: NA
       003 TBA - TBA                           Staff
Enr: 0 Max: NA
       MATH 521   ADVANCED TOPICS IN REAL ANALYSIS         Credits
3.00  Fall 02
       Topic TBA.
       001 GRB 212W - MWF 03:00PM - 03:50PM    Hardt, Robert M.
Enr: 7 Max: NA
       MATH 527   ERGODIC THEORY AND TOPOLOGICAL DYNAMICS  Credits
3.00  Fall 02
       No description
       001 HB 427 - TTH 10:50AM - 12:05PM      Veech, William A.
Enr: 3 Max: NA
       MATH 541   TOPICS IN TOPOLOGY                       Credits
3.00  Fall 02
       No description.
       001 HB 423 - MWF 01:00PM - 01:50PM      Cochran, Tim D.
Enr: 9 Max: NA
       MATH 590   CURRENT MATHEMATICS SEMINAR              Credits
1.00  Fall 02
       Lectures on topics of recent research in mathematics
delivered by
       mathematics graduate students and faculty.
       Prereq: graduate student standing or permission of the
department.
       001 HB 427 - T 02:30PM - 03:50PM        Song, Joung Min Jaime
Enr: 25 Max: 0
       MATH 591   GRADUATE TEACHING SEMINAR                Credits
1.00  Fall 02
       Discussion on teaching issues and practice lectures by
participants as
       preparation for classroom teaching of mathematics.
       Graduate student status or permission of department.
       001 HB 453 - TH 02:30PM - 03:50PM       Masters, Joseph
Enr: 14 Max: 0
       MATH 800   THESIS AND RESEARCH                      Credits
Fall 02          
>It's possible to have bad grades from a good school, and
>good grades from a bad school.  I'm speaking only about the latter.
Thomas
************************
David C. Ullrich
====
>  >> Take the person with the BA from Rice. You think that here
>> undergraduate transcript looks anything like the transcript of
>> the OP's?
>  >Who knows?  
It's easy to find out with a little web searching.
When I said who knows I meant not which classes did she take but
did she do well in them.
Thomas
====
> Take the person with the BA from Rice. You think that here
>> undergraduate transcript looks anything like the transcript of
>> the OP's?
Who knows?  It's possible to have bad grades from a good school, and
>good grades from a bad school.  I'm speaking only about the latter.
I think it has been mentioned in this thread that grades (and also
GRE scores) are maybe not too important; the list of courses taken
might matter much more (assuming the grades average at least a B
or so) and the extra-curricular indicators may be even more telling
(someone else mentioned things like undergrad publications). In all
these ways the weak student at the strong school has more opportunities
than the strong student at the weak school, IMHO.
It would be very interesting to find data about pipeline issues
like this. Assume for a moment the (actually quite bogus) premise 
that every high-school math-lover dreams of a full professorship.
Assume as well that institutions can be broken into equivalence
classes which can be linearly ordered (maybe not too bad an assumption).
Fix a cohort of people (say, US citizens born in one year) and 
assume that they operate in a closed system of opportunities and
feeder stages. (Not really accurate, since so many students cross
borders, but we could assume the number of positions to be held by
members of the cohort will stay fixed for a while.)
Then one can track a cohort of people through various stages:
 admission to university
 graduation with bachelor's degree in math
 admission to grad school
 attainment of PhD
 postdoc
 tenure-track job
 tenured professorship
At each stage, one could take a census of how many from the cohort
are in Harvard-class schools; how many at a Rice-class school;
how many at Big State U; how many at Local Community College, etc.
This would provide a statistical answer to the OP's questions
(which may or may not be applicable to that one person, of course).
My hunch is that the genius comments made elsewhere are pretty
accurate, which is to say that the main job of schools is to stand
out of the way and let the bright students flower and identify 
themselves. That would be reflected in the data if we found that
each institution tends to be fed only be institutions of
comparable or superior rank: the passages from stage to stage
are simply filters which let the best progress.
Actual data are hard to come by but for example the AMS regularly
publishes data about where the Group-I PhDs get their first jobs,
the Group-IIs, etc. These data tend on the whole to support the
hunch I just made.
Also available are the column totals. A typical age cohort in
the US is nowadays about 4 million. About 2-3 million will enter
college (including community college), 1-2 million will complete
college, of whom about 1% will be math majors. In some years almost
every one of them who wants it can find a spot in a grad school,
but that's not true when the economy is bad, and it's harder to
do when the number of mathematically-strong foreign students is
high. At the output end there, we know there are only something like
1000-1200 new PhD's per year in the US, only half of whom are
originally from the US. Officially-named post-doc positions must
number in the low hundreds (counted as openings per year, not
total inhabitants). Regular academic positions advertised per year number
in the high hundreds, but the bulk of those are at the lower-ranked
schools. (You can peruse the annual compendium of job openings to
quantify this if you wish.) Finally, the number of tenure cases
decided positively per year has to be in the mid-hundreds, again
far more at the lower schools (where turnover is higher) than at
Ivy level. In short: only a tiny fraction of youngsters who are good
at math will make it all the way to the end.
One can always be optimistic and hope for some upward migration,
but at almost every level this seems unlikely. For example, there
may be hundred openings every year for postdocs at Ivy-class schools.
Where do you think the successful applicants will come from?
Some will come from, say, Kansas or Wyoming, but the great majority
are surely from Ivy-class grad schools. Those same schools produced
several times that number of PhDs the previous year, so where will
the rest of their students go? They will be grabbing the tenure-track
openings at Kansas and Wyoming, forcing _their_ students to look
for jobs at schools with no graduate program at all, etc.
Seems to me this would make for a dandy study by a budding economist!
dave
====
Hard to say. I don't know of any examples (in math) of someone with
> a BS from a mediocre place who got a master's from some place
> better and then a PhD from a top school. The fact that I don't
> know of any examples doesn't prove there are none - Bushnell
> has stated that he knows of plenty of such examples (otoh he's
> failed to name any, after being asked twice...)
We can also look at catalogues from schools that list all the degrees
> of their faculty.  Of course, that is often somewhat dated, but it's
> at least published information so I don't need to be reticent about
> names.
UMass/Amherst's math department includes one Nathaniel Whitaker, who
> is BA, Hampton Institute, 1974; MS, Cincinatti, 1981; PhD, California,
> 1987.  I assume that California means UC Berkeley.
I looked through the list of current grad students at Cornell.  I
> found one Liang Chen, who has a BS from Peking University and a MS
> from the University of Wisconsin at Madison, and is now at Cornell.
And Peking University is mediocre??  Not to mention, Wisconsin is
comparable to Cornell, so it doesn't really prove anything.  Cornell is
more selective, but reputation-wise it's very similar.
> And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
> Autonoma de Madrid, and an MA from Duke and is now at Cornell.
Sigh...as someone else has mentioned, do you regard any place from
outside the U.S. as mediocre?  
> Jennifer Fawcett is BA from Rice, MA from UC Davis, and now at
> Cornell.  
Rice is a pretty good place for an undergraduate math degree,
especially if you're planning to specialize in low-dimensional
topology.  Getting a recommendation from Hempel is a good thing; and
she did in fact do that.
As for Jennifer (nickname - Sunny), she is not a good example for the
point you are making because she transferred over with Thurston when he
moved from Davis.  Also, let me add that Sunny was good enough to get
into Berkeley when she applied to grad schools.  So this is really a
not very good example, is it?
> Lee Gibson, BS from the University of Kentucky, MS from the
> University of Louisville, now at Cornell.  Farkhod Eshmatov, BS from
> Tashkent State University, MA from SUNY Binghamton, now at Cornell.  
That's six, in about ten minutes of web looking.
> 
I think I've spent enough time on this also.
====
> And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
> Autonoma de Madrid, and an MA from Duke and is now at Cornell.
Sigh...as someone else has mentioned, do you regard any place from
> outside the U.S. as mediocre?  
Not at all.  I am simply giving people who earned an intermediate
masters from a third institution.
Thomas
====
> And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
> Autonoma de Madrid, and an MA from Duke and is now at Cornell.
Sigh...as someone else has mentioned, do you regard any place from
> outside the U.S. as mediocre?  
Not at all.  I am simply giving people who earned an intermediate
> masters from a third institution.
Thomas
Why?  David Ullrich specifically said he didn't know any example of
people getting a degree from a *mediocre* undergrad, then an M.S. at a
better school, and then a Ph.D. at a *top* school.  He then mentioned
you had not given any examples of such.  To which you responded by
listing examples, one of which is quoted above.
If you are indeed simply giving examples of those who earned an
intermediate masters from a third institution, that's irrelevant.
====
> Why?  David Ullrich specifically said he didn't know any example of
> people getting a degree from a *mediocre* undergrad, then an M.S. at a
> better school, and then a Ph.D. at a *top* school.  He then mentioned
> you had not given any examples of such.  To which you responded by
> listing examples, one of which is quoted above.
What I *said*, originally, was that for someone who went to a
nothing-special school, it was not crazy to get a good masters if you
can't get into topnotch PhD programs.  
I made a fair prima facie case for why this is so, including examples
from other fields.  I was asked do you know anyone who did that and
I said yes (more than one) at MIT.  
I certainly don't claim I have given conclusive examples.  On the
other hand, people claiming the contrary have given no examples of any
kind, nor have they given any reasons to the contrary.
At some point, you folks need to provide *some* kind of reason beyond
your common agreement.
Thomas
====
Why?  David Ullrich specifically said he didn't know any example of
> people getting a degree from a *mediocre* undergrad, then an M.S. at a
> better school, and then a Ph.D. at a *top* school.  He then mentioned
> you had not given any examples of such.  To which you responded by
> listing examples, one of which is quoted above.
What I *said*, originally, was that for someone who went to a
> nothing-special school, it was not crazy to get a good masters if you
> can't get into topnotch PhD programs.  
I made a fair prima facie case for why this is so, including examples
> from other fields.  I was asked do you know anyone who did that and
> I said yes (more than one) at MIT.  
I certainly don't claim I have given conclusive examples.  On the
> other hand, people claiming the contrary have given no examples of any
> kind, nor have they given any reasons to the contrary.
At some point, you folks need to provide *some* kind of reason beyond
> your common agreement.
Well, as someone who stepped into this thread rather late in the game,
I know my impressions of what is going on may be very different from
people actively engaged in the thread; however, it appears to me there
are several related, but different, discussions going on.  There is
one, started by Lee Rudolph, in which the topic of 'bootstrapping'
oneself up the prestige ladder, through a masters, is raised.  Your
response, with examples (*not* the one about MIT), was issued directly
to David Ullrich's response in this discussion.  That's why I, and I
think several others, interpreted your issuing of examples as being
related to this idea of 'bootstrapping'.
So my comment above was in that context.  
i believe what said Lee off in the first place was your comment that:
Then with a respected masters under your belt, you are an excellent
competitor with the people who are coming straight out of the best
undergrad programs.
As I've explained in another post (regarding genius), this doesn't
appear to be so.  But your point is taken, obviously, one has to do
what one can to improve one's chances of getting into a better Ph.D.
program than one would initially, in light of a mediocre (or worse)
undergrad.
In any case, I think this subthread is dying out, since it's rather
clear now that nobody is really arguing about anything.  I'll post a
response to your response to my genius post, since I think that still
has some life in it.
====
 > Hard to say. I don't know of any examples (in math) of someone with
> a BS from a mediocre place who got a master's from some place
> better and then a PhD from a top school. The fact that I don't
> know of any examples doesn't prove there are none - Bushnell
> has stated that he knows of plenty of such examples (otoh he's
> failed to name any, after being asked twice...)
 We can also look at catalogues from schools that list all the degrees
> of their faculty.  Of course, that is often somewhat dated, but it's
> at least published information so I don't need to be reticent about
> names.
 UMass/Amherst's math department includes one Nathaniel Whitaker, who
> is BA, Hampton Institute, 1974; MS, Cincinatti, 1981; PhD, California,
> 1987.  I assume that California means UC Berkeley.
 I looked through the list of current grad students at Cornell.  I
> found one Liang Chen, who has a BS from Peking University and a MS
> from the University of Wisconsin at Madison, and is now at Cornell.
> And Jose Trujillo Ferreras, who has a Licenciado from the Universidad
> Autonoma de Madrid, and an MA from Duke and is now at Cornell.
> Jennifer Fawcett is BA from Rice, MA from UC Davis, and now at
> Cornell.  Lee Gibson, BS from the University of Kentucky, MS from the
> University of Louisville, now at Cornell.  Farkhod Eshmatov, BS from
> Tashkent State University, MA from SUNY Binghamton, now at Cornell.
 That's six, in about ten minutes of web looking.
>
Do you want tp say  that any university outside US (Madrid, Beijing in your
examples) are mediocre places?
> Thomas
====
> I realized that I had no shot at any top program about a year ago
>> when I looked at the course offerings at the Ivy League schools and
>> MIT.  This is my senior year so it's too late to go to a stronger
>> program.  I know that my department is bad but now it's just could
>> have and should have.  How realistic is it to get a masters
>> degree and transfer to a better school?  Are you just screwed if you
>> did not go to the right school and take the right courses with the
>> right professors?
No.  The typical advice in your place is to apply to a good masters
>program.  This generally means a school that offers only the masters
>and not the doctorate in your field.  Find the best ones, and if you
>did well as an undergrad and have decent GREs, you should be able to
>get in without a problem.  Then with a respected masters under your
>belt, you are an excellent competitor with the people who are coming
>straight out of the best undergrad programs.
Hey, just call me a starry-eyed old cynic, but I find it really,
*really* hard to believe that from the point of view of the
Ivy League schools and MIT, or any top program, there even
*exist* any respected masters degrees.  Do you know of a single
example of someone who got a (respected or otherwise) masters
degree in mathematics at a school that offers only the masters
and not the doctorate in mathematics, and was then accepted
to any top program, specifically, one of the Ivy League
schools and MIT?  
Lee Rudolph
====
I asked tb+usenet@becket.net (Thomas Bushnell, BSG) 
>Do you know of a single
>example of someone who got a (respected or otherwise) masters
>degree in mathematics at a school that offers only the masters
>and not the doctorate in mathematics, and was then accepted
>to any top program, specifically, one of the Ivy League
>schools and MIT?  
He produced (not in direct response to that question) the
following. 
(1) A faculty member at UMass--Amherst with an MS from Cincinatti
(1981) and a Ph. D. from Berkeley (1987).  Since Cincinatti is not
a school that offers only the masters and not the doctorate in
mathematics, this is not an example.
(2) A grad student at Cornell with an MS from the University of
Wisconsin at Madison.  Since UWM is not a school that offers
only the masters and not the doctorate, this is not an example.
(3) A grad student at Cornell with an MA from Duke.  Since Duke
is not a school that offers only the masters and not the doctorate
in mathematics, this is not an example.
(4)  A grad student at Cornell with an MA from UC Davis.  Since UC 
Davis is not a school that offers only the masters and not the 
doctorate in mathematics, this is not an example.
(5) A grad student at Cornell with a master's from the University of 
Louisville.  Finally, an example! for, indeed, the University of
Louisville does not offer a doctorate in mathematics.  (However,
his BA was from the University of Kentucky, which *does*.)  
 
(6) A grad student at Cornell with a master's from SUNY Binghampton.
Since SUNY Binghampton is not a school that offers only the masters 
and not the doctorate in mathematics, this is not an example.
Well, I only *asked* for a single example, so I can't say
I didn't get what I asked for.  The tare/wheat ratio was a bit
high, though.
>That's six, in about ten minutes of web looking.
Only for very small values of six.
Lee Rudolph
====
> Well, I only *asked* for a single example, so I can't say
> I didn't get what I asked for.  The tare/wheat ratio was a bit
> high, though.
I gave examples of people with intermediate masters from third
institutions, which is what I thought you said you wanted.
Still, as I said, I have made a prima facie case, and so far, you
haven't given *any* reasons to think I'm wrong.  I really am
interested, but just blank assertion isn't worth much, is it?
*Some* kind of explanation would be appropriate, don't you think?
rightly or wrongly, is expected by admissions committees in math--and
not so much in other subjects.  This is an excellent point, but the
same correspondent mixed up people with mediocre records and people
with excellent records from second-class schools.  The latter do not
necessarily look like non-hotshots who are making up for their
earlier failures.
Thomas
====
>Message-id: <bsl479$1dm$1@panix2.panix.com
>> I realized that I had no shot at any top program about a year ago
> when I looked at the course offerings at the Ivy League schools and
> MIT.  This is my senior year so it's too late to go to a stronger
> program.  I know that my department is bad but now it's just could
> have and should have.  How realistic is it to get a masters
> degree and transfer to a better school?  Are you just screwed if you
> did not go to the right school and take the right courses with the
> right professors?
>>No.  The typical advice in your place is to apply to a good masters
>>program.  This generally means a school that offers only the masters
>>and not the doctorate in your field.  Find the best ones, and if you
>>did well as an undergrad and have decent GREs, you should be able to
>>get in without a problem.  Then with a respected masters under your
>>belt, you are an excellent competitor with the people who are coming
>>straight out of the best undergrad programs.
Hey, just call me a starry-eyed old cynic, but I find it really,
>*really* hard to believe that from the point of view of the
>Ivy League schools and MIT, or any top program, there even
>*exist* any respected masters degrees.  Do you know of a single
>example of someone who got a (respected or otherwise) masters
>degree in mathematics at a school that offers only the masters
>and not the doctorate in mathematics, and was then accepted
>to any top program, specifically, one of the Ivy League
>schools and MIT?  
Lee Rudolph
Just to clear this up, I meant that I knew that I had no shot at any top
program without going to one of the best schools.  It seems like my program
would roughly put me around the junior level at one of the best schools when 
I
graduate.  
Furthermore, I'm sure there are more math majors at the very best schools 
than
there are spots in PhD programs at those schools, so I assumed that they 
could
even filter down to some of the 'great' schools that aren't quite the best,
making it even harder to get into those.
Basically, I want to be able to get into a very good state school.  
====
> Hey, just call me a starry-eyed old cynic, but I find it really,
> *really* hard to believe that from the point of view of the
> Ivy League schools and MIT, or any top program, there even
> *exist* any respected masters degrees.  Do you know of a single
> example of someone who got a (respected or otherwise) masters
> degree in mathematics at a school that offers only the masters
> and not the doctorate in mathematics, and was then accepted
> to any top program, specifically, one of the Ivy League
> schools and MIT?  
Yes, in math, physics, and in philosophy.
A master's degree from a doctoral program generally means we let him
go because he couldn't hack it.  But a good master's degree (say in
philosophy from Tufts) means a lot.  It means this person has done
well in an advanced program, carries no implied failure, and
indicates to most people reviewing applications that they can do
graduate work well.
This is exactly what is not clear when an applicant comes from a
second-rate undergrad school.
All your annoying scare quotes serve only to obscure the point:
There are three different kinds of masters degrees out there in
strictly academic subjects (subjects like math, physics, or
philosophy):
* There are the ones which are kind exit for someone washing out of a
  doctoral program;
* There are the ones that smaller schools can offer to try and boost
  their profile;
* There are ones that are taught by top-notch faculty, well respected
  in their field, and which carry weight.
The latter two generally exist only in departments that do not offer a
doctorate.  Both the latter two have very similar promotional
materials, and it's important to figure out the difference.
Those who receive the last sort are generally better off in their
application than if they had not gone, most especially when their
undergraduate degree is from a less stellar school.
I quote, for example, from the Philosophical Gourmet Report:
  Who should consider an M.A. program in philosophy?  Three categories
  of students who ultimately want to get a Ph.D. and pursue an
  academic career might benefit from such programs: (i) students whose
  undergraduate major was not philosophy; (ii) students who majored in
  philosophy at universities with philosophy departments outside the
  mainstream of the profession; and (iii) students who majored in
  philosophy, have a solid grounding in the various areas of
  philosophy, but who studied philosophy at smaller colleges and
  universities, or at institutions with weak academic
  reputations....Students in each category may be both qualified and
  able to get into the Ph.D. programs of their choice; but students
  who fit into one of these categories may be more likely to have
  trouble getting into Ph.D. programs and may be good candidates to
  benefit from M.A. programs.
  A good M.A. program will provide many benefits: it will allow a
  student to get a basic grounding in philosophy or expand the breadth
  of her existing knowledge; to develop increased familiarity with
  current debates in philosophy; to prepare and polish written work in
  philosophy that will be useful in the applications process for
  Ph.D. programs; and to get to know some established philosophers who
  can then provide meaningful letters of recommendation for
  Ph.D. programs.
This advice applies pretty much to any strictly academic subject, for
which there isn't an industry demand for master's degrees.  In
computer science, for example, a master's degree is a real job boost
all by itself, and the schools have adjusted to suit, and so the
advice doesn't carry over so easily.
But for a strictly academic subject, where the job qualification is
really a doctorate, this is what a terminal master's program is good
for.
And to repeat, yes, I know people in a variety of fields who were
admitted to top-notch graduate programs upon receiving a master's at
such a school.
Thomas
====
> Hey, just call me a starry-eyed old cynic, but I find it really,
>> *really* hard to believe that from the point of view of the
>> Ivy League schools and MIT, or any top program, there even
>> *exist* any respected masters degrees.  Do you know of a single
>> example of someone who got a (respected or otherwise) masters
>> degree in mathematics at a school that offers only the masters
>> and not the doctorate in mathematics, and was then accepted
>> to any top program, specifically, one of the Ivy League
>> schools and MIT?  
Yes, in math, physics, and in philosophy.
A master's degree from a doctoral program generally means we let him
>go because he couldn't hack it.  But a good master's degree (say in
>philosophy from Tufts) means a lot.  It means this person has done
>well in an advanced program, carries no implied failure, and
>indicates to most people reviewing applications that they can do
>graduate work well.
Well, I'll have to say I'm amazed to hear about the math example.  
But I believe you.  (Would it be within the limits of your discretion
to say which doctoral program *in math*, in particular, took this
person?   Was it one of the Ivy League schools and MIT?)
What, by the way, is the nature of the evidence on which you 
would base a statement (which you didn't make, explicitly)
that a master's degree *in mathematics* indicates to most
people reviewing applications that the person with the degree
can do graduate work *towards a doctorate in mathematics* well?
I freely admit that I have never reviewed applications for a graduate
program in mathematics.  Have you?  I know a lot of people who have,
and have sometimes talked to them about the process.  Have you?
>This is exactly what is not clear when an applicant comes from a
>second-rate undergrad school.
All your annoying scare quotes serve only to obscure the point:
They aren't scare quotes, they are quotation marks that mark
direct quotations.  I regret that you find them annoying (but 
I'm not sorry I put them in; quite the contrary, I'm glad I 
put them in, and I hope you will learn to find them, not 
annoying, but a positive joy, once you have perceived their
function and appreciated that it is very useful).
>There are three different kinds of masters degrees out there in
>strictly academic subjects (subjects like math, physics, or
>philosophy):
* There are the ones which are kind exit for someone washing out of a
>  doctoral program;
Right.
>* There are the ones that smaller schools can offer to try and boost
>  their profile;
* There are ones that are taught by top-notch faculty, well respected
>  in their field, and which carry weight.
The latter two generally exist only in departments that do not offer a
>doctorate.  Both the latter two have very similar promotional
>materials, and it's important to figure out the difference.
So that I can compare my ideas of quality with yours, could you
give examples--not in physics or philsophy, but in math--of a
few master's programs of the latter two types (in departments
that do not offer a doctorate)?  Would, for example, Boston
College's master's program in mathematics count (in the second 
of the latter two types)?  They certainly have some top-notch 
faculty, well-respected in their field teaching master's students,
but I have not heard that the graduates of their master's program 
are getting in to top schools (because, indeed, I have not heard 
that they have any ambitions beyond the master's degree, with its 
concomitant payoff if you're a school teacher in Massachusetts)--
I don't hear everything, naturally.  If not BC, perhaps some 
other school in the Boston area?
 
>Those who receive the last sort are generally better off in their
>application than if they had not gone, most especially when their
>undergraduate degree is from a less stellar school.
I quote, for example, from the Philosophical Gourmet Report:
[deleted]
Philosophy is a different kettle of fish entirely from mathematics,
and the material I deleted seems to me quite irrelevant to a
discussion of master's degrees in mathematics.
>This advice applies pretty much to any strictly academic subject, 
So you assert.  I have my doubts.  You haven't yet given me
much reason (by my standards) to lose my doubts.
>for
>which there isn't an industry demand for master's degrees.  In
>computer science, for example, a master's degree is a real job boost
>all by itself, and the schools have adjusted to suit, and so the
>advice doesn't carry over so easily.
But for a strictly academic subject, where the job qualification is
>really a doctorate, this is what a terminal master's program is good
>for.
And to repeat, yes, I know people in a variety of fields 
Including, you say, mathematics--the only one I'm interested in
talking about.  Let's stick to that one in the sequel, if you
don't mind.  
>who were
>admitted to top-notch graduate programs upon receiving a master's at
>such a school.
How'd they do?
Lee Rudolph
====
> Well, I'll have to say I'm amazed to hear about the math example.  
> But I believe you.  (Would it be within the limits of your discretion
> to say which doctoral program *in math*, in particular, took this
> person?   Was it one of the Ivy League schools and MIT?)
MIT. 
> What, by the way, is the nature of the evidence on which you 
> would base a statement (which you didn't make, explicitly)
> that a master's degree *in mathematics* indicates to most
> people reviewing applications that the person with the degree
> can do graduate work *towards a doctorate in mathematics* well?
Can you explain why you think that mathematics should be different
than other disciplines?  
Your continual protestations that you think it amazing and unheard of
suggests to me simply that you can't think of anything you could say
other than your open mouthed amazement.  
So far, all you can say is I can't believe it!  In response to
which, unless you can say why, is simply repeat what I have said.  Or,
if you have a criticism of what I've said, then to make it, rather
than allude vaguely to it.
> Philosophy is a different kettle of fish entirely from mathematics,
> and the material I deleted seems to me quite irrelevant to a
> discussion of master's degrees in mathematics.
Can you say what the differences are?  
>This advice applies pretty much to any strictly academic subject, 
So you assert.  I have my doubts.  You haven't yet given me
> much reason (by my standards) to lose my doubts.
Well, so far you haven't even explained the doubt, other than said
it's there.  I'm afraid I don't give much credence to doubts of that
degree.  
In other words, if you want the conversation to continue, you gotta do
more than just say you still haven't proven it to me.  I've made a
fair prima facie case, and if you have an objection to register to it,
do so.
Perhaps you have misunderstood what I'm saying.
I'm saying that a bachelors from Podunk U, plus a masters from 
Really-Cool-U,
is worth as much as a bachelors from Really-Cool-U.
Thomas
====
> Well, I'll have to say I'm amazed to hear about the math example.  
>> But I believe you.  (Would it be within the limits of your discretion
>> to say which doctoral program *in math*, in particular, took this
>> person?   Was it one of the Ivy League schools and MIT?)
MIT. 
> What, by the way, is the nature of the evidence on which you 
>> would base a statement (which you didn't make, explicitly)
>> that a master's degree *in mathematics* indicates to most
>> people reviewing applications that the person with the degree
>> can do graduate work *towards a doctorate in mathematics* well?
Can you explain why you think that mathematics should be different
>than other disciplines?  
Your continual protestations that you think it amazing and unheard of
>suggests to me simply that you can't think of anything you could say
>other than your open mouthed amazement.  
So far, all you can say is I can't believe it!  In response to
>which, unless you can say why, is simply repeat what I have said.  Or,
>if you have a criticism of what I've said, then to make it, rather
>than allude vaguely to it.
> Philosophy is a different kettle of fish entirely from mathematics,
>> and the material I deleted seems to me quite irrelevant to a
>> discussion of master's degrees in mathematics.
Can you say what the differences are?  
>This advice applies pretty much to any strictly academic subject, 
>> 
>> So you assert.  I have my doubts.  You haven't yet given me
>> much reason (by my standards) to lose my doubts.
Well, so far you haven't even explained the doubt, other than said
>it's there.  I'm afraid I don't give much credence to doubts of that
>degree.  
In other words, if you want the conversation to continue, you gotta do
>more than just say you still haven't proven it to me.  I've made a
>fair prima facie case, and if you have an objection to register to it,
>do so.
Perhaps you have misunderstood what I'm saying.
I'm saying that a bachelors from Podunk U, plus a masters from 
Really-Cool-U,
>is worth as much as a bachelors from Really-Cool-U.
It doesn't look to you like he's misunderstood your claim, it looks
like he simply doesn't believe it's so. Neither do I.
You've _stated_ that you know of plenty of examples _in math_.
He's asked you to _give_ such examples. He's asked this
_twice_ by my count. So far you haven't given any examples.
(Speaking of if you want the conversation to continue...
you don't give much credence to his doubts? He hasn't
even asserted he's _right_, he's just expressed _doubts_
about what you're asserting. You're the one who's
making actual assertions and then refusing to back
them up with examples that you say you know of. Me,
I don't give much credence to people who say they
have examples of something but fail to produce them
when asked.)
>Thomas
>
************************
David C. Ullrich
====
> You've _stated_ that you know of plenty of examples _in math_.
> He's asked you to _give_ such examples. He's asked this
> _twice_ by my count. So far you haven't given any examples.
Huh?  No, I did mention I knew first hand examples at MIT.
You are free to doubt, the original poster asked am I screwed because
I didn't go to a first rate school.
Your doubts are nothing but doubts if you have nothing to do but say
I doubt it.  
I *am* interested in *why* you doubt it, something neither of you have
deigned to say.
Thomas
====
You've _stated_ that you know of plenty of examples _in math_.
> He's asked you to _give_ such examples. He's asked this
> _twice_ by my count. So far you haven't given any examples.
Huh?  No, I did mention I knew first hand examples at MIT.
You are free to doubt, the original poster asked am I screwed because
> I didn't go to a first rate school.
Your doubts are nothing but doubts if you have nothing to do but say
> I doubt it.  
I *am* interested in *why* you doubt it, something neither of you have
> deigned to say.
Thomas

Can you explain why you think that mathematics should be different
than other disciplines?  
Your continual protestations that you think it amazing and unheard of
suggests to me simply that you can't think of anything you could say
other than your open mouthed amazement.  
So far, all you can say is I can't believe it!  In response to
which, unless you can say why, is simply repeat what I have said.  Or,
if you have a criticism of what I've said, then to make it, rather
than allude vaguely to it.
--------------
Well, I really didn't think I was going to entangle myself in this
thread, but I've already put a foot in (my other post), and then I got
sucked into reading the whole thread.
However, I think I can offer some words of explanation of why I think
Lee and David both seem reluctant to accept your conclusions.  This is
only my own idea of what I think they are thinking/feeling, and is
based on my much more limited experience.  Clearly, Lee and David have
much more experience in these matters and have interacted significantly
more with the people in charge of making these kinds of admissions
decisions.  Yet I think the fact that in my handful of years in the
mathematical community I've picked up on the kinds of things (that I
believe) are influencing Lee and David, show how pervasive these
elements can be.  
Basically, in the mathematics world, there is a culture of genius
that is not in the subjects you've mentioned, like philosophy.  Now, I
need to explain what I mean by genius, because after all, we don't
normally regard geniuses as being limited to mathematics.
In mathematics, it is clear what genius is and who has it.  If
someone is a genius, there is no doubt.  This is different from
philosophy, where you might say, I think he's got some good ideas, but
I disagree...  In mathematics, you can't solve some problem, and
someone else can.  And if he was the only one who could, and could do
it near instantaneously, there is no more arguing or discussion if this
person has genius.  
This unambiguity (or near-unambiguity) of the quality of genius
pervades the whole institution of mathematics.  Especially when it
comes to recommendation letters.  Even if two people had a similar
performance (in terms of being able to do the work, etc.) in a class,
one may, in the professor's eyes, have exhibited a genius-like
quality in his/her performance.  Consequently, this person will get a
significantly better recommendation.  The other person may be seen as a
good worker, but obviously not a genius.  Even if they could solve
the same problems, etc., the prof might think one was just better
somehow.  More of a genius.
What does this have to do with grad school admissions?  Well, think
about it: if you're a mathematician, who are you going to admit? 
Applicant A who has done similar work (but at a much slower rate, a big
indication of non-genius ability), as applicant B, who graduated
college at age 16?  I don't think I'm wrong as saying that even getting
a masters (in the U.S.) is seen as a sign of weakness.  At the least,
it shows you didn't feel ready to plunge into a Ph.D. program
straightaway or you weren't strong enough.  Either identifies you as a
non-genius.
A note about the masters: I know at least one top ten school (in the
U.S.) that offers a masters that will *never*, as official policy,
admit someone who graduates from their own masters program into the
Ph.D. program.  The masters students there are also treated much worse
than the doctoral students in some very obvious ways.  I believe this
is reflective of the disdain many top graduate programs have toward the
masters.  A masters from some other country, I think, is given more
respect than one from the U.S.
The point you are missing is that admissions committees are NOT looking
at an applicant's record to see if he/she has done well.  Rather they
are looking for indications of genius.  A transcript, etc. can help,
but if they have one recommendation that is from a highly eminent
mathematician saying, this person is a genius, that's *all* they
need.  
I think you are operating under the assumption that the committees are
looking at prior performance as an indicator of future performance. 
That's not really so.  They want to see if the applicant has any genius
or near-genius qualities.  
Now of course, there aren't that many genius-like applicants.  So the
best schools will follow the above pattern of behavior closest.  Lesser
schools will be more willing to compromise and select people, partly
based on what they believe future performance will be.  Will they be
able to graduate with a decent thesis?  Will they do well enough to
reflect back favorably on the school?  And so on.  But even these
schools (especially the schools that are top 25 -- whatever that
means) will be influenced by the cult, er I mean culture, of genius. 
This is why you are facing such amazement from Lee and David.  They are
more than familiar with the culture of genius.  And they know that
the very best schools, which include MIT and some of the Ivy Leagues,
want geniuses more than anything else.  They prefer them to the
hard-workers.   Because the latter are unlikely to be Fields
Medalists, Wolf Prize winners, etc.
====
> This is why you are facing such amazement from Lee and David.  They are
> more than familiar with the culture of genius.  And they know that
> the very best schools, which include MIT and some of the Ivy Leagues,
> want geniuses more than anything else.  They prefer them to the
> hard-workers.   Because the latter are unlikely to be Fields
> Medalists, Wolf Prize winners, etc.
I am inclined to agree Chan-ho, but check this out:
 It is important to keep in mind that no technique has been or ever will 
be
discovered for teaching students to have ideas. All that the faculty can do
is to provide an ambience in which one's nascent abilities and insights can
blossom. Moreover, Ph.D. theses vary enormously in quality, from hard
exercises to highly original advances. Finally, many very good research
mathematicians begin very slowly, and their theses and first few papers
could be of minor interest. On the whole, we feel that the ideal attitude
is: (1) a love of the subject for its own sake, accompanied by
inquisitiveness about things which aren't known; and (2) a somewhat
fatalistic attitude concerning creative ability, and recognition that 
hard
work is, in the end, much more important. 
Taken directly from
http://www.math.harvard.edu/graduate/index.html#admission
So, it may also be that math people from Podunk U are looking to ride the
coat-tails of Geniuses.  What ever happened to studying math for math's
sake?  I say if someone is willing to go into debt for the majority of 
their
adult life, devote their life to the study and progress of a subject, then
let them do it.  This university b.s. is so ridiculous!
Lurch
====
Basically, in the mathematics world, there is a culture of genius
 that is not in the subjects you've mentioned, like philosophy.  
Maybe, maybe not...
  > You've _stated_ that you know of plenty of examples _in math_.
> He's asked you to _give_ such examples. He's asked this
> _twice_ by my count. So far you haven't given any examples.
>  > Huh?  No, I did mention I knew first hand examples at MIT.
>  > You are free to doubt, the original poster asked am I screwed because
> I didn't go to a first rate school.
>  > Your doubts are nothing but doubts if you have nothing to do but say
> I doubt it.
>  > I *am* interested in *why* you doubt it, something neither of you have
> deigned to say.
>  > Thomas
>  
 Can you explain why you think that mathematics should be different
> than other disciplines?
 Your continual protestations that you think it amazing and unheard of
> suggests to me simply that you can't think of anything you could say
> other than your open mouthed amazement.
 So far, all you can say is I can't believe it!  In response to
> which, unless you can say why, is simply repeat what I have said.  Or,
> if you have a criticism of what I've said, then to make it, rather
> than allude vaguely to it.
 --------------
 Well, I really didn't think I was going to entangle myself in this
> thread, but I've already put a foot in (my other post), and then I got
> sucked into reading the whole thread.
 However, I think I can offer some words of explanation of why I think
> Lee and David both seem reluctant to accept your conclusions.  This is
> only my own idea of what I think they are thinking/feeling, and is
> based on my much more limited experience.  Clearly, Lee and David have
> much more experience in these matters and have interacted significantly
> more with the people in charge of making these kinds of admissions
> decisions.  Yet I think the fact that in my handful of years in the
> mathematical community I've picked up on the kinds of things (that I
> believe) are influencing Lee and David, show how pervasive these
> elements can be.
 Basically, in the mathematics world, there is a culture of genius
> that is not in the subjects you've mentioned, like philosophy.  Now, I
> need to explain what I mean by genius, because after all, we don't
> normally regard geniuses as being limited to mathematics.
 In mathematics, it is clear what genius is and who has it.  If
> someone is a genius, there is no doubt.  This is different from
> philosophy, where you might say, I think he's got some good ideas, but
> I disagree...  In mathematics, you can't solve some problem, and
> someone else can.  And if he was the only one who could, and could do
> it near instantaneously, there is no more arguing or discussion if this
> person has genius.
 This unambiguity (or near-unambiguity) of the quality of genius
> pervades the whole institution of mathematics.  Especially when it
> comes to recommendation letters.  Even if two people had a similar
> performance (in terms of being able to do the work, etc.) in a class,
> one may, in the professor's eyes, have exhibited a genius-like
> quality in his/her performance.  Consequently, this person will get a
> significantly better recommendation.  The other person may be seen as a
> good worker, but obviously not a genius.  Even if they could solve
> the same problems, etc., the prof might think one was just better
> somehow.  More of a genius.
 What does this have to do with grad school admissions?  Well, think
> about it: if you're a mathematician, who are you going to admit?
> Applicant A who has done similar work (but at a much slower rate, a big
> indication of non-genius ability), as applicant B, who graduated
> college at age 16?  I don't think I'm wrong as saying that even getting
> a masters (in the U.S.) is seen as a sign of weakness.  At the least,
> it shows you didn't feel ready to plunge into a Ph.D. program
> straightaway or you weren't strong enough.  Either identifies you as a
> non-genius.
 A note about the masters: I know at least one top ten school (in the
> U.S.) that offers a masters that will *never*, as official policy,
> admit someone who graduates from their own masters program into the
> Ph.D. program.  The masters students there are also treated much worse
> than the doctoral students in some very obvious ways.  I believe this
> is reflective of the disdain many top graduate programs have toward the
> masters.  A masters from some other country, I think, is given more
> respect than one from the U.S.
 The point you are missing is that admissions committees are NOT looking
> at an applicant's record to see if he/she has done well.  Rather they
> are looking for indications of genius.  A transcript, etc. can help,
> but if they have one recommendation that is from a highly eminent
> mathematician saying, this person is a genius, that's *all* they
> need.
 I think you are operating under the assumption that the committees are
> looking at prior performance as an indicator of future performance.
> That's not really so.  They want to see if the applicant has any genius
> or near-genius qualities.
 Now of course, there aren't that many genius-like applicants.  So the
> best schools will follow the above pattern of behavior closest.  Lesser
> schools will be more willing to compromise and select people, partly
> based on what they believe future performance will be.  Will they be
> able to graduate with a decent thesis?  Will they do well enough to
> reflect back favorably on the school?  And so on.  But even these
> schools (especially the schools that are top 25 -- whatever that
> means) will be influenced by the cult, er I mean culture, of genius.

> This is why you are facing such amazement from Lee and David.  They are
> more than familiar with the culture of genius.  And they know that
> the very best schools, which include MIT and some of the Ivy Leagues,
> want geniuses more than anything else.  They prefer them to the
> hard-workers.   Because the latter are unlikely to be Fields
> Medalists, Wolf Prize winners, etc.
====
> Basically, in the mathematics world, there is a culture of genius
> that is not in the subjects you've mentioned, like philosophy.  Now, I
> need to explain what I mean by genius, because after all, we don't
> normally regard geniuses as being limited to mathematics.
I think I understand clearly what you mean here, and it's is nearly a
good answer.  
The question is: what about the genius who goes to the fair-to-middlin
undergrad school?  Does the culture recognize that such people really
exist?
Someone who does a mediocre job and then gets a stunning masters will
not trigger the genius reaction, and I agree completely that the
stunning masters has not helped at all.
So if you are faced with a student who says I'm not that hot, but if
I go get a masters, will I get a leg up? the answer is probably no.
And this is a difference between math and many other fields, precisely
because of the genious factor.
But if you are faced with a student who *is* that hot, but is going to
a fair-to-middlin undergrad school, my claim is that *they* can
certainly benefit by going to a good masters program.  For example, at
that good masters program, for the first time, they have the chance to
get a genius recommendation from a recognized name in the field.
> I think you are operating under the assumption that the committees are
> looking at prior performance as an indicator of future performance.
> That's not really so.  They want to see if the applicant has any genius
> or near-genius qualities.
I think this as a very important point, and I don't disagree.  It's a
big difference between the math culture and many other fields.  (I
don't mean to imply that the math culture is wrong; I assume that it
works fine for math.)
My point is that there is a set of people who *do* have genius or
near-genius qualities, who are at less-regarded undergrad schools, and
that there are people who know this.
Thomas
====
> The question is: what about the genius who goes to the fair-to-middlin
> undergrad school?  Does the culture recognize that such people really
> exist?
When I was applying to grad school around 30 years ago the answer was
definitely yes; I'd bet it hasn't changed.  My undergrad degree is
from a land-grant university far better known for football than for
academics, with a math department that sent certainly less than
a half-dozen graduates on to grad school in an average year (and granted
about the same number of Ph.D.s) -- but I made A's in the introductory
graduate courses in algebra, analysis, and (point-set) topology, had 
recommendations from the people who taught them, and got a decent though
not stellar result on the Putnam exam one year.  I got offers from
two top-25 schools but dropped out for a couple of years ... and
subsequently went to a different top-25 school.  
====
Basically, in the mathematics world, there is a culture of genius
> that is not in the subjects you've mentioned, like philosophy.  Now, I
> need to explain what I mean by genius, because after all, we don't
> normally regard geniuses as being limited to mathematics.
I think I understand clearly what you mean here, and it's is nearly a
> good answer.  
The question is: what about the genius who goes to the fair-to-middlin
> undergrad school?  Does the culture recognize that such people really
> exist?
> 
In my experience, yes.  I know of such people and know some personally. 
I'm pretty confident in saying every mathematician knows of at least
one such case.  
The caveat is that these people are recognized to exist because they
are so damn brilliant.  It's hard to hide that kind of brilliance.  So
to give advice based on the fact that there are such people is rather
misleading and useless.  The people to whom such advice applies don't
need it!
I suppose if one goes to an incredibly bad undergrad, then even if one
is a genius, the profs' recs won't be given much weight, if they are
not held in any esteem by the letter readers.  But I would say there
are enough independent channels, like the Putnam, or other contests, or
the possibility of publishing in undergrad journals, or collaborating
with some mathematician who will be held in some esteem, that it seems
unlikely a genius-type will slip through the cracks.  Of course, there
are very capable people who will not appear to be geniuses or
near-geniuses, and I think it's fair to say going to a bad undergrad is
a big handicap.
> Someone who does a mediocre job and then gets a stunning masters will
> not trigger the genius reaction, and I agree completely that the
> stunning masters has not helped at all.
> So if you are faced with a student who says I'm not that hot, but if
> I go get a masters, will I get a leg up? the answer is probably no.
> And this is a difference between math and many other fields, precisely
> because of the genious factor.
> 
 
I think this is the point that others are trying to make.
> But if you are faced with a student who *is* that hot, but is going to
> a fair-to-middlin undergrad school, my claim is that *they* can
> certainly benefit by going to a good masters program.  For example, at
> that good masters program, for the first time, they have the chance to
> get a genius recommendation from a recognized name in the field.
> 
Well, the thing is that even a fair, middle-level undergrad school will
often have some very good mathematicians.  Especially state schools. 
Just because the undergrad education sucks, doesn't mean the department
as a whole does!  
The whole trickle-down effect caused by there not being enough slots
for very good mathematicians means that if you're a genius, someone
will recognize you as one, and that someone or someone else that s/he
knows will have a good reputation.  
I fear your point, while correct, is more academic than practical.  If
there is a very brilliant, genius-like, student that somehow got
hoodwinked into going to a mediocre undergrad, and somehow s/he is not
recognized as such, even through all the independent channels I
mentioned above, then sure, go to a respectable state school (or
wherever they will have some well-known mathematicians) and make your
genius known.  In fact, if the student is *that* good, after just one
semester, s/he should be able to transfer to some fine academic
institution.  
But I think in practice, this is a rare situation, and in terms of
giving advice, etc., one can't expect this to be the case.
> I think you are operating under the assumption that the committees are
> looking at prior performance as an indicator of future performance.
> That's not really so.  They want to see if the applicant has any genius
> or near-genius qualities.
I think this as a very important point, and I don't disagree.  It's a
> big difference between the math culture and many other fields.  (I
> don't mean to imply that the math culture is wrong; I assume that it
> works fine for math.)
> 
I assume so too.  I have doubts occasionally, but that's for another
thread, I guess ;-)
> My point is that there is a set of people who *do* have genius or
> near-genius qualities, who are at less-regarded undergrad schools, and
> that there are people who know this.
>
====
> You've _stated_ that you know of plenty of examples _in math_.
>> He's asked you to _give_ such examples. He's asked this
>> _twice_ by my count. So far you haven't given any examples.
Huh?  No, I did mention I knew first hand examples at MIT.
I guess you did, sorry.
>You are free to doubt, the original poster asked am I screwed because
>I didn't go to a first rate school.
Your doubts are nothing but doubts if you have nothing to do but say
>I doubt it.  
I *am* interested in *why* you doubt it, something neither of you have
>deigned to say.
The reason I doubt he's going to get into a first-rate doctoral 
program was not specifically because of where he did his
undergraduate work, it was because of what he said about
what courses he'd taken and not taken.
>Thomas
>
************************
David C. Ullrich
X-Cise: tanbanso@iinet.net.au
X-CompuServe-Customer: Yes
X-Coriate: admin@interspeed.co.nz
X-Ecrate: tanandtanlawyers.com
X-Pose: george_cox@btinternet.com
X-Punge: Micro$oft
====
   at 08:17 AM, porker899@aol.com (Porker899) said:
>I checked out a book called All the Mathematics You Missed [But Need
>to Know for Graduate School] from the library and was surprised by
>its contents.  The book is divided into 16 sections that I am
>supposed to know before I get into graduate school.  This is my
>last year and I can check off very little.
Check the catalogs of the specific schools thqt you want to apply to.
That sort of general checklist is at best a guide.
>3. Differentiating Vector Valued Functions
There wasn't a course called something like Advanced Calculus?
>4.  Point Set Topolgy - No (not offered here)
>6. Differential Forms and Stokes Theorem - No (nothing like that
>here)
Ouch!
>8. Geometry - No (only course offered is one for future high school
>teachers and was advised not to take it)
The advice was sound. The lack of a real Geometry course was
unfortunate.
>11. Algebra - Yes
I hope that you mean Abstract Algebra; it's essential.
>Which areas do I absolutely need to know?  
That depends on the school. But I would expect at least basic
knowledge of
 Set theory
 Groups, rings and fields
 Topology, including the Topolgy of the Real Line
 Real and Complex Analysis
Along with classes that you've already taken. It might turn out that
some of your course work was too shallow.
-- 
     Shmuel (Seymour J.) Metz, SysProg and JOAT
not reply to spamtrap@library.lspace.org
====
>Message-id: <3fee0f34$20$fuzhry+tra$mr2ice@news.patriot.net>
[...]
>>3. Differentiating Vector Valued Functions
There wasn't a course called something like Advanced Calculus?
Yes, I'm currently taking it.  We are not using a book and I don't have a
syllabus for the next semester yet.  I guess I will have that covered.  My
mistake.
[...]
>>11. Algebra - Yes
I hope that you mean Abstract Algebra; it's essential.
Yes.
>>Which areas do I absolutely need to know?  
That depends on the school. But I would expect at least basic
>knowledge of
 Set theory
 Groups, rings and fields
 Topology, including the Topolgy of the Real Line
 Real and Complex Analysis
Along with classes that you've already taken. It might turn out that
>some of your course work was too shallow.
-- 
>     Shmuel (Seymour J.) Metz, SysProg and JOAT
====
> The problem is that the length functional, length, is not continuous
> on this function space and so does not commute with limits.
I often wondered if there is a 2D analogy to the 1D limit concept
whereby in the 1D case, a point approaches a point, and in the 2D case
a curve approaches a curve. Naturally this limit concept can be
expanded (no pun intended) to include higher dimensions, ie a volume
approaching a volume... etc.
====
|Isn't this the same problem as needing 2 pi rho ds as the integrand 
for
|surface area rather than 2 pi rho dx?
Not really, but I'd agree that the mistake which leads people sometimes
to think that they need dx rather than ds is of a similar flavor.
Archimedes solved some problems which today we would solve by doing
an integral. One rigorous method was known as the method of exhaustion.
There were other, less rigorous methods termed mechanical, which
sometimes looked a bit like considering the solid to be sliced into
infinitesimal slices. The ancients knew then that this kind of method
required care, and they're still right.
Keith Ramsay
====
> The problem is that the length functional, length,
> is not continuous on this function space and so
> does not commute with limits.
Hmmm... the length-function is such a nice function,
that we better look for a limit-definition such that
length *is* continuous.
This has been emphasized by others in this thread already
and it goes into the heart of the matter, as I think and
as the OP seems to feel by now, too.
Rainer Rosenthal
r.rosenthal@web.de
====
The problem is that the length functional, length,
> is not continuous on this function space and so
> does not commute with limits.
Hmmm... the length-function is such a nice function,
> that we better look for a limit-definition such that
> length *is* continuous.
This has been emphasized by others in this thread already
> and it goes into the heart of the matter, as I think and
> as the OP seems to feel by now, too.
But the length function being nice is the problem.  The 
intuition that length is continuous is wrong and your request 
to change the topology so that it becomes continuous can't work 
since if it did then length and lim would commute in which case
the two lengths are the same.  You are going to have to 
change your defintion of length but if you do that, the 
essential Euclidean nature of the problem is changed.
I think the best we can do is to notice that 
   lim length(f_n) = 2 > length(lim f_n) = sqrt(2)
suggests that even if length is not continuous it
may be lower semicontinuous and, in fact, that is 
the case. (A function g on a metric space is continuous 
iff g and -g are lower semicontinuous so being lower 
semicontinuous gets you part way to continuity.)
This can all be made precise in the metric space of 
bounded real functions on [0,1] with the metric:
   d(g,h) = sup|g(x)-h(x)| 
   
and sup is over x in [0,1].
In this setting, define, length(g), the length of a bounded 
real function g as:
   sup sum sqrt( [x_i - x_(i-1)]^2 + [g(x_i) - g(x_(i-1))]^2 )
where 
- sup is over all partitions 0 = x_0 < x_1 < ... < x_n = 1 
  of [0,1] for all n
- sum is over i=1,...,n
====
> The intuition that length is continuous is wrong and
> your request to change the topology so that it becomes
> continuous can't work ...
This is clearly wrong: Let's take the discrete topology
and then every function is continuous.
Well, this is just the extreme of what I had in mind:
Looking for a definition of convergence of function
sequences, which excludes such zappel candidates.
There *must* be some definition of convergence, which
is in accordance with intuition. I did enough topology
once so as to succeed in finding out how to do that ...
But it would be nice to get a hint by some learned
people around here (Dave? Robert? Rob? ...)
> since if it did then length and lim would commute
> in which case the two lengths are the same.
No. This is correct for the discrete topology (see above)
but in a richer topology this is a non sequitur.
BTW why did you write twe two lengths? You have to
compare length(line) to length(n-th convergent), so you
have two lengths in every step n, but infinitely many
lengths in all.
> You are going to have to change your defintion of length
No. I don't have to. On the contrary I want to hold on to
the original lovely and well known length. And it's my
Don Quixote's aim to kill these ugly wood-be-convergences.
 This can all be made precise in the metric space of
> bounded real functions on [0,1] with the metric:
    d(g,h) = sup|g(x)-h(x)|
 and sup is over x in [0,1].
>
Well, if we restrict our function space to piecewise differentiable
function, then the line and the stepfunctions fall into that space.
And if we include some epsilon-demands like
     d_better(g,h) := max( sup|g(x)-h(x)|, sup|g'(x)-h'(x)| )
then we should get a more decent definition of convergence.
Rainer Rosenthal
r.rosenthal@web.de
====
The intuition that length is continuous is wrong and
> your request to change the topology so that it becomes
> continuous can't work ...
This is clearly wrong: Let's take the discrete topology
> and then every function is continuous.
Firstly, you have deleted part of my response making it seem 
wrong. I didn't say no topology could make it continuous.  I said
that you are not going to be able to make it work without
changing the essential nature of the problem. 
Secondly, in the discrete topology, lim f_n does not equal 
f so the essential nature of the original problem is not
preserved.
Thirdly, I think it needs to be placed in a reasonably 
elementary common framework to stay within the spirit of
the original problem and bounded functions on [0,1] with 
the sup|f-g| metric seems to satisfy that pretty well.
====
>  > The intuition that length is continuous is wrong and
> your request to change the topology so that it becomes
> continuous can't work ...
> This is clearly wrong: Let's take the discrete topology
> and then every function is continuous.

> Firstly, you have deleted part of my response
... in order to get hold of the relevant part. I have the
idea of a natural topology for function sequences, where
the length is a continuous function.
> ... you are not going to be able to make it work without
> changing the essential nature of the problem.
I disagree again. The essential nature of the problem is IMO
the proper definition of convergence. The OP was perplexed,
since the lengths of seemingly convergent functions did not
converge to the length of the limit function.
> Secondly, in the discrete topology, lim f_n does not equal
> f so the essential nature of the original problem is not
> preserved.
The essential nature is well preserved, since this radical
change of topology sheds light on the role of topology itself.
And the essential nature is wonderfully exhibited insofar as
lim f_n does not equal f in any topology, where length shall
be a meaningful term. The discrete topology is just an extreme
example. The topology, which I pointed to by the metric generating
formula (skipped by you):
    d_better(g,h) := max( sup|g(x)-h(x)|, sup|g'(x)-h'(x)| )
is quite natural and not as pathological as the discrete topology.
And here we have again:  lim f_n <> f.
> Thirdly, I think it needs to be placed in a reasonably
> elementary common framework to stay within the spirit of
> the original problem and bounded functions on [0,1] with
> the sup|f-g| metric seems to satisfy that pretty well.
Not really. The sup|f-g| metric is too rich, too far into the
direction of the trivial topology, where each sequence converges
against each function.
How about the enhancement (d_better from above)? The spirit of the
original problem is preserved, as I think. The OP could be happy.
Rainer Rosenthal
r.rosenthal@web.de
====
Perhaps we are just arguing over what we regard as
the essential nature of the problem.
To me the problem can be stated as: If f_n
converges to f then why does length(f_n) not
converge to length(f)?   (*)
Although its slightly less obvious in the poster's
original formulation, when stated this way it
becomes clear that the answer is that our
intuitive notion of length being continuous is
wrong.
To me, the if part of (*) is the essential
nature of the problem -- maybe not to you (since
this if part is false in all your examples).    
I'll give you the benefit of the doubt and
interpret your response as saying that the source
of the problem is that the intuitive notion that
f_n converges to f is wrong and a topology 
so defined can make it possible for length to 
be continuous.
====
> Perhaps we are just arguing over what we regard as
> the essential nature of the problem.
Well, in a way. Yes, indeed. True, true.
> To me the problem can be stated as: If f_n
> converges to f then why does length(f_n) not
> converge to length(f)?   (*)
I agree. And I am completeley happy with Rob Johnson's
answer, which says:
1. Our staircase f_n functions are converging
   to f in the C^0 norm. And one has to live
   with the fact that length is not continuous.
2. Our staircase f_n functions do not converge
   to f in the C^1 norm (my d_better() *proud*)
   and so there is no need for the lengths
   |f_n| to converge to |f|.
3. If a sequence f_n converges to f in the C^1
   norm, then |f_n| are converging against |f|.
Best wishes for the New Year
Rainer
Rainer Rosenthal
r.rosenthal@web.de
====
>> The intuition that length is continuous is wrong and
>> your request to change the topology so that it becomes
>> continuous can't work ...
> This is clearly wrong: Let's take the discrete topology
>> and then every function is continuous.
>> Firstly, you have deleted part of my response
... in order to get hold of the relevant part. I have the
>idea of a natural topology for function sequences, where
>the length is a continuous function.
Define the C^0(I->R^n) and C^1(I->R^n) norms to be
    ||f||  = sup|f|
         0    I
    ||f||  = sup|f| + sup|f'|
         1    I        I
See <http://mathworld.wolfram.com/C-kFunction.html>.
Suppose a sequence of curves {f_n} converges to {f} under the C^1 norm:
    |L(f ) - L(f)|
        n
      |  |                        |
    = |  |   |f'(t)| - |f'(t)| dt  |
      | | I   n                   |
        | 
    <=  |   |f'(t) - f'(t)| dt
       | I   n
    <= ||f - f||
          n     1
Thus, if a sequence of curves converges under the C^1 norm, then the
lengths of the sequence will also converge to the length of the limit.
Thus, length is continuous under the C^1 norm.  There is no way to
control length using only the C^0 norm.
Even though the staircase curves converge under the C^0 norm to the
diagonal, they don't converge under the C^1 norm.
>> ... you are not going to be able to make it work without
>> changing the essential nature of the problem.
I disagree again. The essential nature of the problem is IMO
>the proper definition of convergence. The OP was perplexed,
>since the lengths of seemingly convergent functions did not
>converge to the length of the limit function.
> Secondly, in the discrete topology, lim f_n does not equal
>> f so the essential nature of the original problem is not
>> preserved.
The essential nature is well preserved, since this radical
>change of topology sheds light on the role of topology itself.
>And the essential nature is wonderfully exhibited insofar as
>lim f_n does not equal f in any topology, where length shall
>be a meaningful term. The discrete topology is just an extreme
>example. The topology, which I pointed to by the metric generating
>formula (skipped by you):
    d_better(g,h) := max( sup|g(x)-h(x)|, sup|g'(x)-h'(x)| )
is quite natural and not as pathological as the discrete topology.
>And here we have again:  lim f_n <> f.
> Thirdly, I think it needs to be placed in a reasonably
>> elementary common framework to stay within the spirit of
>> the original problem and bounded functions on [0,1] with
>> the sup|f-g| metric seems to satisfy that pretty well.
Not really. The sup|f-g| metric is too rich, too far into the
>direction of the trivial topology, where each sequence converges
>against each function.
>How about the enhancement (d_better from above)? The spirit of the
>original problem is preserved, as I think. The OP could be happy.
Your d_better is the C^1 norm.  This norm makes length a continuous
function on curves.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying
====
> Your d_better is the C^1 norm.  This norm makes length
> a continuous function on curves.
So finally my exclamation worked Dave? Robert? Rob? ...) :-)
Best wishes for 2004
Rainer
Rainer Rosenthal
r.rosenthal@web.de
====
> Your d_better is the C^1 norm.  This norm makes length
> a continuous function on curves.
So finally my exclamation worked Dave? Robert? Rob? ...) :-)
No one was arguing that one could not define a topology that makes
length continuous or that different notions of convergence exist.
The problem is that none of these different notions of convergence
can result in length being continuous without violating the assumptions
of the problem.  Thus to make length continuous you either have to give
up on f_n converging to f or change the definition of length, etc.
====
>> The problem is that the length functional, length,
>> is not continuous on this function space and so
>> does not commute with limits.
> Hmmm... the length-function is such a nice function,
> that we better look for a limit-definition such that
> length *is* continuous.
In theory, same as we might have in theory that the speed of light is not 
limiting.
====
>If we have the analytic (about x = 0) real -> real function 
>  >f(x) = sum{k=0 to oo}  a(k) x^k /k!,
>  >we might, for whatever reason,
>want to define g(x) as
>  >sum{k=0 to oo} b(k) x^k /k!,
>   >where {b(k)} is a permutation of {a(k)}.
>   >I am wondering if taking the permutations of terms of exponential
>generating functions, or of ordinary generating functions, has been
>studied (or has any applications).
Applications? Does it matter?
Not really. 
:)
I was wondering more specifically if there were any applications among
other branches of *pure* mathematics, or in combinatorics anyway.
Studied? The closest thing that comes to mind is Cameron's concept of 
> oligomorphic permutations (google that and journal of integer sequences)
It depends on what kinds of permutations you're thinking of: e.g. a simple 

> involution 
>   b(2k) = a(2k+1)
>   b(2k+1) = a(2k)
it is easy to compute gfs:
            f(x) + f(-x)   f(x) - f(-x)
>   g(x) =  x ------------ + ------------
>                  2              2x
(which can be easily generalized for more similar constant bounded 
> distance permutations)
But what about the gf for the Gray code permutation? EIS A003188 (take 
the
> basic binary reflected Gray code, convert back to integers:  
> <0,1,3,2,6,7,5,4,..>) you get a permutation of the integers where the
> distance is unbounded. and the gf for it is ..er... not nice (IMHO) (which 

> makes it interesting).
Mitch Harris
thanks,
 Leroy
Quet
====
> I think what I need to do now is to take a 10 cm long, 10 gram
> weighing elastic wooden stick. I will put this stick on rather
> frictionless tiles of my room. I would like to bend this stick using
> thumb and middle finger of my right hand. I would like to make sure
> that elastic potential energy stored in this curved stick is maximum.
> Now I will tie a thread to a small stone weighing about 15 gram and
> other end of loose thread to center point of curved stick. I will
> place this stone near center point of bended, curved stick making sure
> that when I release both ends of stick, center point of stick strikes
> to center of gravity of stone. Now I will release both ends of stick
> to allow center point of bended stick to strike to center of gravity
> of stone. Now I will see stone propelled on rather frictionless tiles
> of my room. If the propelled stone pulls the stick through tied thread
> in direction of motion of stone, then I have propelled the stick,
> thread and stone in only one direction without having to expel
> reaction mass or using propellant.
You hope to use this in space, don't you?
You have a problem: what happens if you use your device twice?
This problem will arise in any situation in which friction forces are
the same in all directions. In particular, this includes space.
Think about the following experiment: you are in a little boat, on a
lake. You have big, heavy stones on board. You stay in the back of the
boat: you throw a stone backwards. What happens? Your boat goes
forward! And quite fast. But there is a problem, you have lost your
stone.
You think a little and say: why not attach a rope to the stone? So
that I can take it back on board! Methinks I'm a genus! We are to
enter into an age of stones&ropes! And what happens if you try? When
you'll use the rope to take back the stone the boat will go
backward...
In the end you will have not moved.
You are trying to do exactly the same thing. You don't have a chance.
There is a case in which you can do something, but this is no miracle
(and no apocalypse, btw). This is when friction forces are big in one
way and small in the other one. Say you are in the following
configuration:
_________________________________________________
/  /  /  /  /  /  /  /  / ,/  /  /  /  /  /  /  /
                          |
                        +---------+
                        |         |
                        | device  |
                        |         |
                        +---------+
                          |
                 '              
-------------------------------------------------
The rails on both side are here to represent (or to implement) the
oriented friction forces. You can easily move the device to the left
(with some click-click), but you cannot go in the other way. In such
a case, you could have bow and arrow in your device, a thread to
attach the arrow to the device, some motor to bring the arrow back
into the device. And now, if the friction forces are small enough
compared to the weight of your arrow, and the strenght of your bow,
you'll have a chance. But be sure of one thing: it won't be
significantly more efficient that anything else. And probably less.
Try to understand this last experiment, this is probably where you are
in trouble. Your intuition tells you it _has_ to work. In some
particular cases, yes. But not in general, and not when it would be
great.
To come back to the experiment you proposed: holding the bow, you play
the role of the rail. The bow cannot go backward: you hold it. But it
can go forward: you allow it to go this way.
I hope this has helped you to understand your mistake, I wish you a
Merry Christmas, and a happy New Year.
             /er.
====
Let
q(r,m) =
---
     1
/    ---
---  k^r
k|m
k>= sqrt(m)
which is, in linear-mode,
sum{k|m, k>=sqrt(m)}  1/k^r,
where the sum is over the divisors of m which are >= the squareroot of
m.
  
If we again sum over the divisors, but now over every positive divisor
of n, so that we have:
Q(r,n) =
---

/    q(r,m) 
--
m|n
= 
 sum{m|n}  q(r,m),
then the average of all the Q(r,n)'s, taken over the n's, approaches a
limit.
ie.
limit{q-> oo} 
      q
 1   ---
---  
 q   /     Q(r,n)      = A(r)
     ---
     n=1
= limit{q -> oo} 
(1/q) sum{n=1 to q}   Q(r,n)
And A(r) is, if I am right, ...
 oo
---   (1+1/2+1/3+...+1/k)
     -------------------
/           k^(r+1)
---
k=1
= sum{k=1 to inf} (1+1/2+1/3+...+1/k)/ k^(1+r),
which is an Euler sum.
[ http://mathworld.wolfram.com/EulerSum.html ]
(For example,
A(1) = 2*zeta(3).)
(Right?)
So, since q(r,m) = 
sum{k|m,k<=sqrt(m)} k^r  /m^r
(note inequality's direction here, as opposed to in q()'s definition),
I wonder naively
if there is some kind of zeta-function-like reflection formula for the
Euler-sum analytical continuation which relates
E(r) and E(2-r),
where E(r) is 
sum{k=1 to inf} (1+1/2+1/3+...+1/k)/ k^r.
(I know of no study regarding analytical continuating Euler sums at
all, nor anything beyond the consideration of such at r = integers >=
2.)
thanks,
Leroy Quet
====
My son bought us a couple of Pepsi drinks today, which feature a
Caps for caps promotion. You buy the bottle and look inside the
cap for an imprint; save the caps and you can win a team hat (cap)
bearing the logo of an NFL (American football) team.
The imprint inside the bottle cap is usually the name of one of
these teams. I don't follow sports, but the rules inside specify the
names of 32 teams (which I believe is the complete set of teams in
the league). When you have collected two copies of one team name,
you may claim a free hat as your prize (and no, it doesn't have to
bear the imprint of the team whose name you found twice).
Alternatively, some of the caps carry a Buy-one-get-one-free
message (according to the rules sheet). The label of the bottle
states the odds of this happening to be one in six. (Shouldn't
that be _odds_ of five to one? But I digress...)
I believe every cap carries exactly one imprint, although it isn't
clear from the rules that there are no Sorry, try again imprints.
    Here is my question: The rules state, Once you have your first
    bottle cap [with a team logo], odds of matching such cap are 1 in 36.
    I would like to know how they derive this number.
It appears that 1/6 of the caps win a free drink, and I am guessing
that the other 5/6 of the caps are equally distributed with the
imprints of the 32 teams, i.e. 5/192 of the caps say Bears, 5/192
say Bengals, etc. So once you have a cap showing Bears, isn't
your probability of getting a match next time  5/192 ?  That's about
1/38, not 1/36. What am I missing?
I considered the possibility that they meant that the odds of matching
a cap you already own, _given that_ the next cap is not a free-drink cap,
were 1/36. But that's obviously wrong; it's 1/32.
I considered the possibility that the 1/6 was just rounded off.
Repeating the calculation with 1/(5.5) of the imprints offering a
free drink, the chances that the second cap is a match to a team you
already have would be  (1 - 1/5.5)/32, about 1/39; with only 1/(6.5)
of the caps winning a free drink, a cap matches your team's 
(1 - 1/6.5)/32 of the time -- about 1/37.8 . So a rounding error is
not ennough.
Then I considered the possibility that the team names were not printed
equally often. (That sounds pretty impolitic to me, but then, I'm not
in marketing!) This sort of thing is common; for example, McDonalds
restaurants used to have a promotion in which customers were given
a free game ticket bearing the name of an ingredient in the Big Mac
sandwich; collect them all (seven, IIRC) and you get a free sandwich.
The company printed far fewer tickets showing Special Sauce so
the odds of getting a free sandwich were lower than 1/7 even if 
you were given a bundle of the other six ingredients by someone else.
So assume that the fraction of the caps imprinted with the name of
team  i  is  p_i = 5/192 + x_i. Then we know  sum p_i = 5/6, which is
to say  sum x_i  = 0.  Also evidently the claim is that among those 
people with a non-free-drink cap first, 1/36 of those people will have two 
identical caps; I make this out to say that  sum (p_i)^2 = (5/6) (1/36),
which is equivalent to  sum x_i^2 = (5/6) (1/36) - 32 (5/192)^2 = 5/3456.
Well, that's only 2 equations in 32 unknowns. We can find some of the
solutions by looking for a 2-variable problem to solve which specializes
the general case. For example, suppose there are  k  teams with a
different probability from the other  32-k  teams  (k=0, 1, 2, ...)
Then there are just two different probabilities to compute. I find that
the equations allow for the greater and smaller probabilities to be
one of many different combinations:
 k      p1             p2   
 1, .06347892127, .02483401328
 2, .05208333333, .02430555555
 3, .04694721281, .02387902397
 4, .04383151175, .02350026023
 5, .04166666667, .02314814815
 6, .04003864192, .02281159545
[...]
12, .03472222222, .02083333333
[...]
16, .03276559609, .01931773725
[...]
27, .02893518519, .01041666663
28, .02858307311, .00825182154
29, .02820430937, .00513612055
30, .02777777778, 0
The last case corresponds to using 30 team names each 1/36 of the time
and not using the other two teams at all. The cases k=12 and k=16
show the most equitable distributions of this type, minimizing
respectively the ratio and the difference between the high and the
low probabilities. I don't know whether we can improve these measures
by allowing the  p_i  to assume more than just two values.
So I don't know whether I'm missing a possible alternative model, but
it rather looks to me that either they or I have made an arithmetic
error, or else there is a gross disparity in the distribution of 
the team names!
Mathematically, this situation is obviously the birthday paradox.
Suppose again that the 32 team names are printed equally often.
Once you have 33 team imprints, you must have a match; the probability  p
that you have at least one match once you have  k  imprints is shown
in the table below. Also shown is the probability  q  that you have
at least one match after  k  draws (about 1/6 of which should garner
a free drink);  q_k  is  sum (5/6)^i (1/6)^(k-i) binomial(k,i) p_i.
 k,  p           q
 1,  0           0  
 2,  .0312500000 .0217013889
 3,  .0917968750 .0639738860
 4,  .1769409180 .1243626630
 5,  .2798233032 .1993244322
 6,  .3923509121 .2845765171
 7,  .5062851161 .3755105100
 8,  .6142852469 .4676137835
 9,  .7107139352 .5568439157
...
27,  .9999999497 .9997708172
28,  .9999999921 .9998813336
29,  .9999999990 .9999400091
30,  .9999999999 .9999703784
31, 1.0000000000 .9999857090
32, 1.0000000000 .9999932605
33, 1            .9999968919
(So if you really, really want a free team hat, probably 8 or 9
bottles of Pepsi will get you one.)
But these probabilities are affected by an unequal distribution of
the team names. For example, if as above we really only use 30 of
the team names (equally often), then the chances of an early match
go up. I didn't work out the probabilities in other cases, but
it seems clear that any choice of the p_i  which makes the chance
of a win with two caps equal to 1/36 rather than  5/192  will
increase all the other numbers in the table too, that is, the
chance of having a match when you possess  3  caps should be
greater than  0.091796875  no matter what solution  { p_i }  is
chosen for the pair of equations. I didn't see how to prove this.
Such is the curse of being a mathematician: even the insignificant
event can lead to quite a bit of pondering!
dave
====
>My son bought us a couple of Pepsi drinks today,
<snipSuch is the curse of being a mathematician: even the insignificant
>event can lead to quite a bit of pondering!
dave
Pepsi has caffeine, doesn't it?
--
Mensanator
Ace of Clubs
====
Given Two lines whose quations are 3x+y-8=0 and -2x+by+9=0, determine the 
value of b such that the two lines are perpendiclar.
 Ok so i know that they have to be negative reciprocals and i get stuck at
y=-3x+8
y=2x-9/b
What exactly do i do here? TIA for helping out guys, really appreciate it!
---= 19 East/West-Coast Specialized Servers - Total Privacy via Encryption 
=---
====
> Given Two lines whose quations are 3x+y-8=0 and -2x+by+9=0, determine the
> value of b such that the two lines are perpendiclar.
  Ok so i know that they have to be negative reciprocals and i get stuck 
at
> y=-3x+8
> y=2x-9/b
 What exactly do i do here? TIA for helping out guys, really appreciate 
it!

News==----
> http://www.newsfeed.com The #1 Newsgroup Service in the World! >100,000
> ---= 19 East/West-Coast Specialized Servers - Total Privacy via 
Encryption
=---
So you know that the slope of the second line has to be 1/3
So we can say 2/b=1/3; which means b=6.
-- 
David Moran
Chief Meteorologist
Oklahoma Storm Team
====
> y=2x-9/b
You've made a mistake here....
This should be y= 2x/b - 9/b
What exactly do i do here? TIA for helping out guys, really appreciate 
it!

> News==---- http://www.newsfeed.com The #1 Newsgroup Service in the World!
> Privacy via Encryption =---
====
A Covering Design c(v,k,t,m,b) is a pair (V,B), where V is a set of v 
elements (called points)
and B is a collection of b k-subsets of V (called blocks) such that 
every m-subset of V
intersects at least one member of B in at least t points.
It is required that v >= k >= t and v >= m >= t.
Given some parameters of a design and some subsets,
how can you determine fast the number of Covered/Uncoverd blocks
within these parameters?
I'm looking for an algorithm or pseudocode to count the number
of blocks that are covered/uncovered, with a method other than brute force.
Some examples of partially covered designs below:
Example #1,
Parameters: v=12 k=5 t=4 m=6 b=9
  1  3  5  6 12
  1  4  7  9 12
  1  8  9 10 11
  2  3  4  7  8
  2  3  8  9 12
  2  4  5  6  9
  2  7 10 11 12
  3  4  5 10 11
  5  6  7  8 10
The above design covers 770 blocks of the possible 924.
It leaves uncovered 154 blocks.
My question again is as of methods/algorithms to count the number of
covered/uncoverd blocks as fast as possible.
Example #2,
Parameters: v=18 k=7 t=4 m=7 b=11
  1  2  3  4  9 14 18
  1  2  3  4 11 12 13
  1  2  5  6  7  8  9
  1  2  8 15 16 17 18
  1  7  8 10 11 15 16
  2 10 12 13 14 15 16
  3  4  5  6  7 14 17
  3  4  5  6  8 10 18
  3  4  9 10 15 16 17
  5  6  7 11 12 13 18
  8  9 11 12 13 14 17
Covered blocks=31,692 of the possible 31,824.
Uncovred blocks=132
Example #3,
Parameters: v=15 k=6 t=3 m=4 b=14
  1  2  3  4  5  6
  1  2  3  7 10 13
  1  3  6  8 12 13
  1  4  7  8 10 14
  1  5  9 12 13 14
  1  6  7  9 11 13
  2  4 10 11 12 15
  2  5  6  7 12 14
  2  8  9 11 14 15
  3  4  7  9 12 15
  3  5 10 11 14 15
  4  5  7  8 11 13
  4  6  9 13 14 15
  5  6  8  9 10 15
Covered blocks=1363 of the possible 1365.
Uncovred blocks=2
Your ideas or comments are welcome.
====
> 
>> Dear Group,
>>              I`m trying to calculate a 2D fourier transform
>>              (2DFFT) of 
>> a 2d exponential decay. Upon comparisions with a lorentzian form
>> ( i compare the normalized values of the real and the imaginary
>> parts seperately) i see that with my best efforts i can achieve
>> an accuracy of only 10^(-3). Even this is possible only if go to
>> a totally unacceptable time step (In order to get a reasonable
>> resolution in the frequency domain i would need to do a 8192x8192
>> FFT calculation). 
>>             I`m unable to locate where the error is coming from.
>>             I 
>> don`t think my problem is related to zero padding as i can choose
>> a decay const which ensure that the value goes to zero within the
>> interval. I would be really grateful to any comments and pointers
>> as i`m really stuck. (i need an accuracy of at least 10^(-5) to
>> proceed). 
>> 
>> Ravi

Have you remembered to halve the t=0 input datum?
-- 
====
> On Sat, 10 Jan 
Have you remembered to halve the t=0 input datum?
          I`m sorry i don`t understand. why is this necessary? 
Ravi
====
>> On Sat, 10 Jan > 
>> 
>> 
>> Have you remembered to halve the t=0 input datum?
          I`m sorry i don`t understand. why is this necessary? 
Ravi
> 
Think of the finite fft as an approximation to the integral fourier 
transform. If you make a trapezoidal approximation to the integral, 
the first and last points must have weight 0.5 relative to the 
internal points. For a signal that decays to 0, the last point doesn't 
matter, but the first one does.
-- 
====
Download beta 1 - www.master-graph.com/mgraph20b1.exe (only 477KB).
All who will help me to improve this program will get a free
registration key.
You can do the following:
-find bugs
-feedback about you impression by the program
-advice me to change or add some new features
-translate it to the other language (see
www.master-graph.com/instructions/interface.html)
Feel free to contact with me - roman@master-graph.com
Sincerely,
Roman.
====
> I have been trying to figure out why there is a /constrained/ quadratic
> progamming step in SQP methods. I don't understand why the QP isn't
> formulated without constraints, instead using Langrange multipliers
> since it would be much easier to solve.
Three reasons:
1. with lagrange multipliers one has to solve for a stationary
point, not a minimum,
2. (worse) one needs to know the multipliers beforehand,
while the QP solver provides the correct ones.
3. One usually has inequalities and the QP is there to improve the
guess of which ones are active.
Arnold Neumaier
====
Dear all,
Convex optimization seems hot today... anybody can tell me what is the
relationship and difference between linear/nonlinear, discrete, and convex
optimization?
-Walala
====
 I haven't looked into ASM for years, so I can only repeat what I have
> read elsewhere: that current-day compilers are quite good at generating
> decent machine code.
That mantra has enough wiggle room in it to get out of anything.  Define
decent.  Decent to run MS-Word?
> For whatever definition of quite good. (You might
> want to direct this kind of question to comp.compilers.)
I did actually.  No surprises there.
> Anyway, optimization isn't just generating good ASM code, it's also
> stuff like loop unrolling, common subexpression elimination, etc.
> C and C++ are notoriously hard to optimize this, since it's often very
> difficult to trace which pointer may refer to what array. Fortran
> compilers have a far easier job since Fortran programs don't use
> pointers, they use indexes (at least that used to be true for older
> Fortran code, and I'm pretty sure that all the numeric libraries are
> still written in that style).
Yes, some languages have more burdens than others.  The simpler the
programming model, the easier to optimize.
>  > The DEC Alpha compiler was the exception.  Anyways
> this is imperative stuff, for mainstream languages like C and C++, so
pretty
> much the natural fit to the task of optimizing the CPU instruction
stream.
 Not really.
I suppose I didn't state my point clearly.  CPU instruction streams are
fundamentally imperative.  That is my point.  I don't care about C or C++,
that's a sideshow.  The point is, functional programming is inherently a
layer of complication on top of the imperative machine architecture.
> Still, since functional
> languages tend to have a far simpler semantics than imperative ones,
> it's less work to get the compiler right, so there's more time for doing
> a good code generation backend *g*)
This is all theory.  Look at what happens in industrial practice.  Nothing
much.
-- 
Brandon Van Every           Seattle, WA
20% of the world is real.
80% is gobbledygook we make up inside our own heads.
X-Received: (from approve@localhost)
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 
1.9  primary) id i08K3Bc26460;
====
Dear reader,
Based on the arithmetic of computers,
could someone please tell me which of the 
following two iterative functions is more eficient? :
( Both yield the fifth root of any positive number P,
 that is: P^(1/5) )
First one:
(4xP^2 + 17Px^6 + 4x^11) / (2P^2 + 16Px^5 + 7x^10)
(Householder's iterative function, fourth order convergence)
Second one:
(10xP^2 + 15Px^6)/(6P^2 + 18Px^5 + x^10)
My vote is for the second one. This is not homework.
Many thanks for your comments.
Domingo Gomez
X-Received: (from approve@localhost)
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 
1.9  primary) id i0AD3W216557;
====
can anybody give me any guidelines to using the chaboche material
model in ansys.
X-Received: (from approve@localhost)
        by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 
1.9  primary) id i0AKr8H16701;
====
I have 2 groups of data that come in several continuous batches of
equal size. I wish to compute their covariance. Certainly I can view
all batches in each group as one piece of data. Therefore computing
their covariance is as usual. However I do not want to do it that way
because each batch may have different mean and I do not wish all
batches to carry one mean value (for some reasons.) Instead, I
calculate covariance for each batch using,
BATCH_COV(A,B) = 1/M * SUM_i {(A_i - mean(A)) * (B_i - mean(B))}
where M = number of elements in a batch and i = element index. Then
calculate the complete covariance as
COV(X,Y) = 1/N * SUM_j {BATCH_COV(X^j,Y^j)}
where N = number of batches and j = batch index.
Is this mathematically correct? Please advice.