mm-3919
===
Subject: Re: Formula to combine physical measurements to one?
I thought one could incooperate this by correct weighting with the errors 
of
each partial mean.
What if I have different measurement methods, so that I get error bounds 
for
each, but I can't just take the normal mean of these measurements without
proper weighting (smaller error measurements have higher weight).
===
Subject: Re: Formula to combine physical measurements to one?
I misunderstood your first post. If you are saying that from the 
first set of measurements you have an estimate x1 and an estimate s1 of
its standard deviation (rather than the sample standard deviation of the
measurements) and similarly for the second set, then a reasonable way
to combine the estimates is to use x and s (the estimated sd of x) where
x = a*x1 + (1-a)*x2
where a = s2*s2/(s1*s1 + s2*s2)
s = s1*s2/sqrt( s1*s1 + s2*s2).
You can derive this, for example, by assuming that x1 and x2 are
independent, and by looking for the estimate of the form a*x1 + (1-a)*x2
(and so has the same mean as x1 and x2) which has minimum variance.
===
Subject: Re: Formula to combine physical measurements to one?
Thats sounds reasonable to me, if even though I have experience in maths 
but
not statistics to justify the last argument.
Meanwhile I was thinking of something and now I wonder why the following
gave me a different formula:
Suppose I have a Poisson distribution. I perform a measurement N times and
get a success rate of C counts. Now my scaled result is
x=C/N   (rate of positive events)
s(x)=sqrt(C)/N   (that's what I'm using for the error. correct?)
If I had split one such experiment into two (say having a coffee break),
then I had two estimates
x1=C1/N1, x2=C2/N2
s1=sqrt(C1)/N1, s2=sqrt(C2)/N2
which should combine logically to
x=(C1+C2)/(N1+N2)
s=sqrt(C1+C2)/(N1+N2)
if I had done the experiment in one go.
This result is only obtained with the formula:
x=(x1^2/s1^2+x2^2/s2^2) / (x1/s1^2+x2/s2^2)
s=sqrt(x1^2/s1^2+x2^2/s2^2) / (x1/s1^2+x2/s2^2)
which is different than above. Where's the mistake?
===
Subject: Re: Formula to combine physical measurements to one?
Well if you observed k1 .. kN and were assuming a Poisson
distribution, Sum{ k[i]}/N is the maximum likelihood estimator for
the parameter of the distribution. The rule I mentioned before is how
to get a minimum variance combination of two estimators. There's no
particular reason why this should be the maximum likelihood estimator.
I'm afraid the moral of the story is that there's many meanings of best
in best estimator...
Duncan
===
Subject: Re: Formula to combine physical measurements to one?
Any book on Probability and Distribution Theory, I would have thought.
Maybe see http://www.sph.umich.edu/syllabi/BIOSTAT601.pdf
Nick 
===
Subject: identity v tautology
Does 'identity' and 'tautology' mean the same thing, or is there an
essential difference between the two?
===
Subject: Re: identity v tautology
Identity is the result of a tautology.
~v~~
===
Subject: Re: identity v tautology
No it does not mean the same thing, yes there's an essential
difference between this two notions.
F.
-- 
===
Subject: Re: identity v tautology
See http://www.wittgenstein.internet-today.co.uk/plato.html
The basic meaning of the term tautology in mathematics and philosophy 
is 
that a proposition is tautologous if it is true whatever the facts in the 
empirical world turn out to be. This is not the normal English meaning, but 

an example will show how the meanings relate: the tautology all stallions 
are horses. It is tautologous because of the meaning of its words, which 
preclude the possibility of an animal being a stallion but not a horse. 
Thus, an explorer who finds an adult male okapi will have to decide whether 

to refrain from calling it a stallion or to agree to calling an okapi a type 

of horse. So from the question of the meaning of words, we arrive at a 
situation where what might actually be found in the world must be 
considered.
Nick 
===
Subject: Re: identity v tautology
Unfortunately this isn't the definition for anything anywhere in math,
philosophy, or ordinary language because at best it just tells us when
something is something and not what the something is. Didn't your
fifth grade teacher ever tell you not to define things by the is when
. . . technique because it's not exhaustive?
A tautology is combination of alternatives exhaustive of the
possibilities of truth as in A not A provided A is not self
contradictory.
A truism is problematic syllogistic inference as in If A then B.
That's a proposition not a tautology.
Says who? It would depend on the meaning of the terms. There aren't
any pre emptive meanings to such words so the idea that ordinary
propositions are exhaustive of the possibilities for truth is absurd.
Ordinary propositions are either true, false, or problematic but they
certainly don't exhaust the possibilities for truth.
~v~~
===
Subject: Re: Simultaneous Equations
Cc:  deepkdeb@yahoo.com
opinion on the following
Consider (A-1) under the given conditions.
x^2 + y^2 = z           (A-1)
My assertion:
(A-1) can be satisfied only if:
(1) both x and y are integers, (2) x = sqrt(m) and y = sqrt(n) where
both m and n are not perfect squares (3) x = sqrt(m), y = sqrt(n)
where only one of m or n is a perfect squre, (4) x = a + sqrt(b) and y
= a - sqrt(b), b is not a perfect square.
Is there any other conditions possible to make z an integer?
Your reply will be appreciated.
===
Subject: Re: Simultaneous Equations
You seem to keep missing the same fundamental concept.
I'll phrase it this way.
Free variables hate to be tied down.
If you choose z, one of the variables x,y is still free.
Thus, let z be any positive integer, and choose y such that
(1)   0<y<z
(2)   y is not the root of a quadratic equation with rational
coefficients. For example, you could choose y=1/Pi or y=1/e or
y=1/(2^(1/3)).
Then there is a unique positive real number x such that x^2+y^2=z, but
x and y do not satisfy your conditions.
quasi
===
Subject: Re: Simultaneous Equations
***3
You are completely right. In an equation of three variables two can
freely be chosen. These two will then determine the third. But my last
question to you I believe was in relations to all the previous
discusions. Kindly refer to (11.3), (11.4) etc only if  you have some
interest and your time permits.
***3
===
Subject: Re: Simultaneous Equations
Didn't I already provide a counterexample to the (11.3), (11.4)
system?
In any case, if there is a question, please state a complete problem. 
But this, time please state _all_ requirements on your variables.
For each variable, indicate the type and whether or not there are any
size restrictions. For example, for a given variable, is it required
to be a real number? A rational number? An integer? Must it be
positive? Nonnegative? Nonzero? Are all variables initially free or
are some to be regarded as fixed parameters?
Precise statements are worth the effort.
quasi
===
Subject: Re: #10 published examples of mistaken proofs by various authors; 
monograph-book: Correcting the Logical Flaws of Euclid's Infinitude of 
Primes Proof
Dude, what is wrong with you?
Leave him to his monographical bubble,
his posterity for the young physicists up-and-coming;
maybe *they* will crack his AP CODE-pieces ...
at the high school Advanced Placement level!
thus quoth:
You win the new Academy Award for Supertriviality,
only because Hardy&Wright and Heckman et al
of the aforegoing ancient astronomers were *invariably* going
beyond their finite primes to show that, ultimately,
N+1 may or may not be prime -- but, only as an afterthought;
the proof stands without it.  With it,
all of the foregoing authors become targets for your ape-,
I mean, serial monographical output.
Crack the code, Dude.
That seems to be your entire edifice.  If
you could actually comprehend the adics,
you might have a whole, new perspective, but
this also seems to beyond your ken.  At least,
you don't want to bother to learn any numbertheorie, or
you would actually have some examples,
of how this result of yours bears upon the rest
of the subject.
How many more of these authors are you going
to attack with horse-feathers, who are all
doing more or less the same thing as Euclid, but
you couldn't see that -- couldn't *read* that
from the text?
thus:
if you can't prove that all Fermat numbers are pairwise coprime,
again!
Darfur 'Mini-Summit'
http://www.larouchepub.com/other/1998/rice_2546.html
thus:
uh yeah; Borat wants you in Sudan,
why, Baby?...  Harry Potter wants you in Iran --
yeah, Baby; shag'US with a spoon?
--DARFURIA CONSISTS OF ARABs & nonARABs; NEWS-ITEM:
  we are marching to Darfuria, Darfuria, Darfuria!
Harry Potter IIX, ?Ordeal @ Oxford//Sudan ^ Aircraft Carrier!
http://larouchepub.com/other/2007/3410caymans_hedges.html
ALgoreTHEmovieFORpresident.COM:
http://larouchepub.com/eirtoc/site_packages/2007/al_gore.html
===
Subject: Re: #9 published examples of mistaken proofs by various authors; 
monograph-book: Correcting the Logical Flaws of Euclid's Infinitude of 
Primes Proof
Yes, Archie Poo's record has been stuck in the same
groove for years.
What I find amusing is the importance he attaches
to w+1 and w-1 being necessarily prime and the supposed
bearing this has on twin primes. This is why
he obsessivley clings to necessary primeness.
A particularly delusional idea as the proof is one
by contradiction. The mathematical universe in which
has yet has  no prime divisors.
w+1 has contradictory properties
it can be prime,  composite and a unit.
What relevance he thinks this inconsistent
mathematical universe might have  to
 a  mathematical universe in which
and there is no  last prime is beyond me.
He can never admit that any integer
greater than the last prime pn is composite by assumption.
Even a loony of AP's magnitude would find it
difficult to maintain that w+1 is  a  composite number
that is necessarily prime and go on to claim
it was on his list of primes.
===
Subject: Re: On Triangular number modulo k.
On 23 Apr 2007 21:51:48 +0300, Phil Carmody
Not exactly.
What you have actually shown is that k can't have any prime factors
other than 2.
In fact, {T(n)} covers all values mod k iff k is a power of 2.
By your argument, if {T(n)} covers all values mod k, then k can't have
an odd prime factor, hence k must be a power of 2.
To show the converse, assume k is a power of 2,
Let Z_2k={0,1,...,2k-1} and Z_k={0,1,...,k}
   f(n)=(n^2+n)/2 mod k
Claim f is exactly 2 to 1.
To see this, suppose f(a)=f(b).
   f(a) = f(b)
If a-b is even then a+b+1 is odd, hence a=b mod 2k, so a=b.
If a-b is odd then a+b+1=0 mod 2k
Conversely, if a=b or a+b+1=0 mod 2k then
(a-b)(a+b+1)=0 mod 2k
Thus, f(a)=f(b) iff a=b or a+b+1=0 mod 2k.
For every a in Z_2k there is a unique b in Z_2k such that a+b+1=0 mod
2k, (namely b=2k-a-1), moreover b is never equal to a.
It follows that f is exactly 2 to 1, as claimed.
Since the domain of f has 2k elements, the image of f must have
exactly k elements.
Therefore, when k is a power of 2, {T(n)} covers all residues mod k.
quasi
===
Subject: Re: On Triangular number modulo k.
Not at all, in fact. Given that I said it works for k=2 5 lines earlier, 
there was obviously someone else in control of my typing at that final 
line!
8-I
Phil
-- 
Home taping is killing big business profits. We left this side blank 
so you can help. -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
===
Subject: Re: On Triangular number modulo k.
Quasi, I do agree with your argument as your result matches with my
observation that indeed if k is a power of 2 then it visits every
number between 0 and k-1.
But to prove that it is the only form of representation of K that
exists, how do we do that? I could agree with all the periodicities by
observing that,
T(2k+r) = T(2k-r) = T(r),  0 <= r < k
T(2k) = 0
Also,
If k is odd, T(k) = T(k-1) = 0 with, T[ (k-1)/2 ] = T(m) = m(m+1)/2
(mod k)
Calling 'm' to be a middle value in the sequence, it is also that,
T(m+r) = T(m-r). Which literally means that if k is odd, the sequence
is bisected and there would be either (k-1)/2 or (k+1)/2 unique values
(atmost!).
If k is even, T(k) = k/2.
So, how does the above property lead one to find the number of unique
elements in the cycle in case if the k is not a power of 2? To
simplify we could have T(0) = 0.
===
Subject: Re: On Triangular number modulo k.
A few hints:
(1) a(a+1)/2 - b(b+1)/2 == 0 (mod k)
(2) If every number from 0 to k-1 is visited, then given a, there
should be exactly two distinct b in [0, 2k-1] for which (1) is
satisfied - why?
(3) What happens if k is an odd prime?
(4) What happens if k is a multiple of an odd prime?
---
J K Haugland
http://home.no.net/zamunda
===
Subject: Re: On Triangular number modulo k.
Haugland,
I also do agree with the property (1) that there have to exist two
distinct b's. Should the below argument do?
Working from, (a-b)(a+b+1) = 0 (mod 2k), we find that if lets say for
some a, b we are visiting a number twice, the above property would
hold, and since we visit uniquely, we should not be able to find the
same when we consider the same in the range of 0..k, that is,
For the unique visit,
(a-b)(a+b+1) != 0 (mod k)
a+b+1 != q'.k, for some q, q'.
This further upon addition/subtraction, implies,
2*a + 1 != q''.k  and
2*b + 1 != q'''.k
For the above is literally true if and only if k has an even factor.
Moreover, if k = k'.P where k'=2^x and P product of all other odd
primes, again the above property would fail for some a, b since a and
b are choosen to be arbitary in the range [0, k-1], right?
So, can we conclude that if the visit has to be done on all numbers,
the k has to be a complete even number, meaning  a power of 2??
===
Subject: Help understanding simpleconversion
[URL=http://img406.imageshack.us/my.php?image=mathspicto1.jpg]
[IMG]http://img406.imageshack.us/img406/1885/mathspicto1.th.jpg[/IMG][/
URL]
Hi I am trying to understand how this region is converted from
cartesian to polar co-ordinates. Since it is a circle, I was under the
impression the range should be 0<=theta<=2*pi and the magnitude should
be 0<=r<=1
===
Subject: Re: Help understanding simpleconversion
It says it is the solution, what is the question (see the first webpage).
As stated in the solution it only represents the bottom right and top right 

quadrants on an x-y graph.
The RH has a radius of 0 when theta=-pi/2 and theta=pi/2, and 2 (2 
cos(theta)) when theta=0
As you imply the LH side is a circle of radius 1 with the centre at (1,0)
Nick
===
Subject: Re: Collatz Nodes Are Not Randomly Distributed!
TIP: Never underestimate your opponent!
SUBTIP: Never underestimate yourself! I am convinced that you are capable of 

the formal mathematical proof! Because of  the plausability and logic of 
your proof by demonstration. But do not forget there are over 2 billion 
zeroes found by numerical demonstration all on the critical line of the 
complex plane, no counter example yet, still they want the formal 
mathematical proof of the Riemann hypothesis. Because there could still be a 

suprise in petto.
Can there be a fading away of the hailstone effect as one goes to infinity? 

Because the numbers tend to get so large that the hailstone effect 
relatively does not matter anymore and that the distribution of the nodes 
absolutely looks like totaly ordered?
What do you think?
Johan van der Galien
===
Subject: Re: Collatz Nodes Are Not Randomly Distributed!
Ok, I won't. It's entirely possible someone could learn to run
before they learn to walk, but I wouldn't bet on that. My point was
that if you can't follow the simple demonstration in plain English,
I have doubts that you would understand it in formal mathematical
terms. I'm not sure *I* would understand it in mathematical formalism.
But I never make such formalities and would be uncertain on how to do
it properly. A bad attempt just gives ammunition to strawman
arguments.
The logic is absolutely correct, not just plausible.
That's different. Here you're trying to use empirical evidence as
proof.
I'm not trying to do that. If I had said I calculated the even/odd
ratio
for all levels up to 84 (beyond which it is intractable) and I tried
to
deduce what the ratio would be at infinity from that empirical
evidence,
THEN you would have a point.
BUT...my demonstartion was by induction, no empirical evidence
whatsoever.
So your reference to the Riemann hypothesis is inappropriate.
And that's why empirical evidence is not accepted as proof in
mathematics.
People don't accept that there are an infinite number of primes
because
no one has found them all, it's accepted because it can be shown that
the
premise of a largest prime causes a contradiction, so the premise
must
be false if math is consistent.
No, none whatsoever. The Sequence Vectors and the Hailstone Functions
derived
from them are structure-centric, not value-centric. They're all about
the
PATTERN of 3n+1 vs. n/2 operations, not the VALUES of the nodes.
For example, there is a closed form formula to find the ith, kth
Generation
Type [1,2] Mersenne Hailstone. This means find a seed (S) that results
in
a Hailstone (H) that is a Mersenne Number after exactly six iterations
of
the pattern (3n+1)(n/2)(3n+1)(n/2)(n/2).
H = a(k,i) = 2**(6*((i-1)*9**(k-1) + (9**(k-1) - 1)/2 + 1) - 1) - 1
For i=1, k=6, a(k,i) = 2**177149-1 or a number with 177149 bits (53328
decimal
digits). Your mind simply cannot conceive of a number that large. See
Yet, it is trival to show that this is indeed
 - a Mersenne Number
 - is preceeded by Sequence Vector [1,2]
 - the Sequence Vector repeats exactly 6 times
And, since there are only 177148 Mersenne Numbers less than that, it
can
be empirically shown that it is, in fact, the first such number of
that
type.
And the reason it works perfectly for any number your computer is
capable of
producing (the 11th generation is intractable because the language
considers
an EXPONENT of 32 bits to be outrageous) is because it was derived
algebraically. Therefore, it is valid for ALL numbers regardless of
size with
absolutely no diminishing effects. Empirical testing merely confirms
what I
already knew and verifies that I implemented the algorithm correctly.
That cannot happen. The Hailstone effect is due to RATIOS of numbers
which
does not change based on the size involved, so there can be NO
diminishing
effect.
I'm not sure what you mean by totally ordered, we were talking about
the
even/odd ratio.
It's quite simple. You can prove inductively that the successor of n
under
the operation n*2 is
infinity.
Likewise, all branches are started by odd numbers and every branch
connects
to an even 1(mod3) number.
Now for there to be an even/odd ratio asymtotically close to 1, there
must
be nearly as many odd numbers as even numbers on that level, which
means
EVERY number on the previous level must have been an even 1(mod3).
But the induction shows that such a premise leads to a contradiction,
therefore, the premise that every number on a level can be 1(mod3) is
false
and the even/odd ratio cannot asymptocically approach 1.
===
Subject: Re: Ham-Sandwich-like Problem
for the case n=2 in the right way, the statement is true for the
plane. Unfortunately the probem of the continuity of the 'operation'
in the proof (mentioned above) is not clear to me. Is that a real
problem or just foolish of me?
===
Subject: Re: What Is a Mathematician?
When asked to cosign for your deadbeat friend's car loan, you tried
hard for a substitution.
===
Subject: Re: What Is a Mathematician?
Yes, that's my point.  Peer review is self-correcting,
a negative feedback loop.  
Tom
===
Subject: Re: What Is a Mathematician?
Then you have made a mistake.  If you continue to
do so too often, you cannot be a mathematician.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: What Is a Mathematician?
On 23 Apr 2007 17:00:04 -0400, hrubin@odds.stat.purdue.edu (Herman
A mathematician is one who practices mathematics. If you can tell me
what mathematics is I can tell you what a mathematician is.
A mathematician is one who proves theorems based on assumptions of
explicit predicates by exhaustion of alternatives for truth.
~v~~
===
Subject: Re: What Is a Mathematician?
Luckily, I am a philosopher, so that doesn't matter so much.
-- 
Jesse F. Hughes
Part of the problem here, Peter, is that you are an idiot.
                       -- Daryl McCullough gives a diagnosis
===
Subject: Re: What Is a Mathematician?
8.  Knows how to spell and pronounce Ramanujan and knows what he did.
9.  Supports bringing back an NFL team to Houston and calling it the Houston 

Eulers.
10.  Believes that e is a number and not a letter.
===
Subject: Re: What Is a Mathematician?
11. Knows what mathemtics is.
~v~~
===
Subject: Re: Wanted: Block design (28,7,1) with 12 blocks
So balanced with lambda = 1 means every pair occurs exactly once, so a
design can be balanced only if my counting, as below except with correct
arithmetic, of pairs available and used comes to equality.  This one
doesn't, so it's not balanced.
OK.  Where might I look for more information on this?
-- 
Randy Hudson
===
Subject: Re: Wanted: Block design (28,7,1) with 12 blocks
I posted a fairly lengthy response with a lot of links,
but Google Groups seems to have lost it.  Sometimes
it takes a day or so for posts to show up, but recently
the posting process is kind of haphazard.
The bad news is that I don't think what you want can
exist.  Here's a discussion of terminology:
[What is a design? How should we classify them?]
By R. A. Bailey and Peter J. Cameron
http://www.maths.qmul.ac.uk/~pjc/preprints/whatis.pdf
If all the blocks have equal size k, and if every point
appears (replicates) r times, then vr = bk where v is
the number of points and b the number of blocks.
Try to impose k = 7 and r = 3 on v = 28 points (so
that necessarily b = 12), and you quickly run into an
obstacle.
If we adopt your compact labelling of the points as
A-Z plus 8 and 9, then relabel as necessary for the
three blocks containing A:
{A,B,C,D,E,F,G}
{A,H,I,J,K,L,M}
{A,N,O,P,Q,R,S}
We then need two more blocks containing B, and
these must be otherwise disjoint from the first block
above, say:
{B,H,N,T,U,V,W}
{B,I,O,X,Y,Z,8}
There we run out of steam, because we cannot put
together even one block of size 7 containing C and
otherwise disjoint from our first block, without
repeating some pair, say:
{C,J,P,T,X,9, ?? }
If we abandon entirely the restriction on
having no pair appear more than once, then we
can provide a block design with v = 28, k = 7,
and r = 3 consisting of b = 12 blocks.  However
this might consistent merely of repeating four
disjoint blocks three times:
{A,B,C,D,E,F,G} (three times)
{H,I,J,K,L,M,N} (three times)
{O,P,Q,R,S,T,U} (three times)
{V,W,X,Y,Z,8,9} (three times)
None of the online references I can find for you
on partially balanced incomplete block designs
treat as many as v = 28 points with block size
[Constructing some PBIBD(2)s by Tabu Search Algorithm]
http://uxmcc1.iimas.unam.mx/~lbm/ts_pbibd2.pdf
which describes something of a dual to that,
namely v = 12 points with b = 28 blocks of
size k = 3 (r = 7).  It would be interesting to
know more details about that design.
===
Subject: Re: Wanted: Block design (28,7,1) with 12 blocks
Yes, I had just come to the same conclusion by a different path:
The first block has seven elements.  Each of those elements will
have to appear twice more among the remaining blocks, and no more
than one can appear in each of those blocks without repeating a
pairing from that first block.  So we need 14 more blocks for all
their appearances.  Unfortunately there are only 11 remaining
blocks available.
similar design with 35 elements in 15 blocks of 7, each element
appearing in 3 blocks and no pair appearing in more than 1 block.
I haven't even tried it yet, so it may be simple.
My online searches tend to dead-end at abstract-only with a charge
-- 
Randy Hudson
===
Subject: A question about groups and colimits
Given that every module over some ring is the colimit of its finitely
generated submodules, it follows that every Abelian group is the
colimit of its finitely generated subgroups. In general, is an
arbitrary group equal to the colimit of its finitely generated
===
Subject: Re: A question about groups and colimits
            days. My association with the Department is that of an alumnus.
The colimit would be the quotient of the set of all pairs of the form
(h,H), where h is in H and H is a finitely generated subgroup of G,
under the equivalence relation (h,H) ~ (h',H') if and only if there
exists a finitely generated subgroup K such that K contains H and H',
and h=h' in K. That is, if and only if h=h' (since we can always let
There is trivially a map from the colimit to G, induced by the
G; this is explicitly given by mapping the class [(h,H)] to h. This
map is one-to-one, since [(h,H)]=[(k,K)] if and only if h=k, and the
finitely generated, contains g, and so the colimit contains the
one to one and onto, giving an isomorphism.
If I'm not mistaken, this is true for any algebra (in the sense of
universal algebra) which has only finitary operations. 
--
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Re: A question about groups and colimits
            days. My association with the Department is that of an alumnus.
A short addendum:
It may not be obvious where the assumption of finitary operations was
used: I never defined the group operations on the the set of
equivalence classes. The operation can be defined by letting
subalgebras of a given algebra form a directed set under inclusion,
and if all operations are finitary, then the direct limit has an
obvious algebra structure such as the one just given, making the
colimit equal to the direct limit.
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Re: Naturalism redux
:
Sure, especially if they are not afraid of contradicting themselves.
Why does virtually everybody nowadays insist on calling it natural
language, then -- unless it's because the natural part is more
interesting? Linguists are otherwise not known for being reluctant to
neologizing.
So I can speak because my environment (in childhood, presumably) spoke
to me. But where did my environment have that from? It only pushes the
question back in time and raises the vexed question of the
(non)existence of primitive language.
I must admit my own use of trunk in this context is quite peculiar.
Still, if I ask if Newtonian mechanics should still be considered the
trunk of physics, a physicist would tend to have an opinion about it
rather than discuss my choice of terms. Why shouldn't something
similar be possible in any science, including linguistics?
Think of it as corpus linguistics.
Something different than your in addition, apparently.
Just to split even finer hairs: If they are in phonology, I think
that would mean they are already being studied!
If the linguist puts his work forward, introspections can be
immediately tested by any reader, at least if the linguist uses his
native language both in the text and the examples. That's about as
efficient and cheap as any research could be done. If he has brain
damage or genetic defects, the world has a chance to discover it (if
the linguist's immediate surroundings haven't already noticed). If he
doesn't have a pet (or monster, for that matter) theory to test, why
should he be doing research in the first place?
Is language inessential, then?
Jens S. Larsen
===
Subject: Re: Naturalism redux
Natural is opposed to artificial or formal, not 
environmental.
Besides, it's best not to get hung up on compositional meanings of 
jargon.
Fine... historical linguistics is the trunk.  And?
My in addition means and or furthermore.  The only way in which 
the social aspects of phonology were added was temporally in the 
current discussion.  In synchronic phonology itself, they co-exist 
with the articulatory aspects.
By quite a few people, in fact.
Efficient?  He wastes time writing a paper and constructing an absurd 
theory of language based upon his own idiosyncratic use of the 
language, and if he's influential enough, his followers will blindly 
accept his data as fact, despite repeated evidence from others  
showing that the misguided linguist's introspective data contradicts 
the language use of the vast majority of actual speakers, bogging down 
the field in decades of bickering over empirical claims and half-baked 
theories built on dubious data, rather than allowing the theory evolve 
from soundly collected data following proper scientific procedures.
No, that would never happen!
It's one thing to test a theory; it's quite another to 
(unintentionally or otherwise) allow your theory to bias your data 
from the outset.
Just today, I compared MP3s encoded at 128Kbps and 256Kbps.  At first, 
I could easily tell the difference; the 256 encoding sounded clearer 
and fuller.  Then I had someone else randomize them and play them for 
me so that I didn't know which file was being played, and I couldn't 
hear any difference at all, even after having just heard the files.  
The obvious instinctive theory that the 256Kbps should sound better 
affected my ability collect accurate data when I knew which file was 
which.
A linguist who collects data from a non-naive speaker (which 
necessarily includes himself) is collecting tainted data and is doing 
a disservice to the field if he publishes results based on it.
Inessential to what?
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: Naturalism redux
:
...
Good suggestion. Saussure introduced the distinction between
diachronic and synchronic linguistics, and Jakobson proposed to
reunite them as panchronic linguistics. Is this project of
reunification making any progress today, in your opinion?
...
A 10-line period? That is, I believe, what's technically known as a
rant.
No, that's the same.
That means the absurd theory was right in this case, doesn't it?
Naive speakers are much more likely to accept one's intentions than
one's colleagues are.
Let me rephrase: Is language essential to anything?
Jens S. Larsen
===
Subject: Re: Naturalism redux
On 24 Apr 2007 01:42:20 -0700, Jens S. Larsen
in sci.math,sci.lang,alt.philosophy:
[...]
If you really believe that testing a hypothesis is the same
as allowing the theory to bias the data from the start,
you're simply unqualified to do science.  Obviously the
nature of the hypothesis determines the kind of data to be
collected, but that's altogether different from biassing the
data (as, for instance, Nathan did when he evaluated the
quality of the non-randomized MP3s).
[...]
Naive speakers are unlikely to know the relevant technical
concepts, let alone what hypothesis is being tested.  (If
any: one may simply be collecting initial data on which to
base a hypothesis.)
[...]
Brian
===
Subject: Re: Naturalism redux
My own work deals with the interface between synchronic and diachronic 
phonology, so for me, the answer is yes.
When someone publishes tainted data and causes decades of unrest in my 
field because of his pernicious combination of carelessness and 
self-importance, I think I have every right to rant.
Remind me *not* to cite data from any of your work!
My initial theory (that the double bit rate would easily sound better) 
turned out to be wrong with the better constructed experiment.  The 
only way I heard a difference between the files was when I knew in 
advance which file was which.  When my knowledge of the file type was 
eliminated, the data become less tainted, and truer results emerged.
To be absolutely untainted, I would need to run experiments on naive 
subjects who don't know that I'm testing the quality of bit rates 
specifically, by mixing the actual test items with dummy items to mask 
the purpose of the experiment.
Only if you tell them!  Don't.  (At least, not until after you've 
collected data.  Invariably, my informants always ask what the purpose 
of the experiment was after it's over.)
I don't really see how that question is related to what I said.  What 
I said can be paraphrased as syntax is not the essence of language; 
phonology is not the essence of language; morphology is not the 
essence of language; etc. (the implication being that all of them are 
equally essential components to language).  I said nothing about what 
language itself may or may not be the essence of.
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: Naturalism redux
Because the annoying people who think that computer programs are
language and mathematics is language and who knows what-all else
is language are forcing the use of a retronym on us.
It existed 150,000 or so years ago. Why isn't that good enough for
you?
Because the diagrams used to lay out the subdivisions of disciplines
don't generally have trunks, but only nodes and branches?
===
Subject: Re: Naturalism redux
Peter T. Daniels:
Is that the academic version of mass rape?
Because environment apparently hasn't played any role in the evolution
of language since then.
You mean like this <http://www.seedmagazine.com/news/2007/03/
JSL.
===
Subject: Re: Naturalism redux
Serves me right for clicking a link. That's the ToC of something
called Seed Magazine, with no tree diagrams, trunked or otherwise.
===
Subject: Re: Naturalism redux
===
Subject:    Naturalism redux
You probably saw the ToC because the URL got line-wrapped and your mail 
client/browser cut the end off, so you got the main site contents page. 
What you _should_ have seen was a page entitled Scientific Method: 
Relationships Among Scientific Paradigms.
Try clicking this one instead:
-- 
Richard Herring
===
Subject: Re: Being Doctor
And you might have thought that someone doing premed would know the right 
place or people to ask.
Nick 
===
Subject: Re: Being Doctor
Hopefully, the OP will know enough not to apply for admission to a
math PhD program.
quasi
===
Subject: Re: Being Doctor
Of course, if the OP insists on applying to math departments, maybe
choose one where they specialize in surgery on manifolds. After
becoming a doctor, the cost of malpractice insurance will be much
less.
quasi
===
Subject: Re: looking for some help.
But surely yours is different from the first - you have a Math degree (like 

me, I have a Maths degree) and therefore one would have thought that you 
have the skills that with application that you can pick up where you left 
off. After all, what was the point of applying to go back to grad school to 

take math if you didn't think it possible.
Naturally if you have a break it will be more difficult than for someone who 

goes straight on from college to grad school.
I did a postgraduate degree in statistics with a year's break after 
university. I had to apply myself as I had had that single year's break. 
Alternatively one might presumably take some refresher courses maybe to go 
over old stuff.
Nick 
===
Subject: Probability Theory
Suppose that a biased coin that lands on heads with probability p is flipped 

10 times. Given that the total number of 6 heads results, find the 
probability that the first three outcomes were
(a) HTT
(b) THT
Here is how I started part (a), since we are dealing with a coin, binomial 
probabilities must play a part somewhere. 
P(A and B) = P(HTT and 6 heads) 
= prob of getting 1 head, then 2 tails, then 5 heads out of 7 
= p(1-p)^2 * 7C5(p^5)(1-p)^2. Am I correct?
===
Subject: Re: Probability Theory
No. You correctly calculated P(A and B), but the problem is asking for
the conditional probability of A given B.
You can use the formula P(A|B) = P(A and B)/P(B).
Since you already have P(A and B), you only need to compute P(B).
After simplifying, you will find, perhaps surprisingly, that the
answer is independent of p. Once you see that, it suggests that there
is probably an even a simpler way to arrive at the answer. There is,
so start again, using the answer itself to guide you to an easier
method.
quasi
===
Subject: Re: turning into NOR
 NOR(x,y) = (x + y)'
write a logic table
            z
xy     0     1
00    0      0
01    0      1
10    1     0
11    1     1
the xy term in your orgional is redundant, not needed
===
Subject: Re: turning into NOR
What do you mean when you say turn this function into NOR? Tell us
clearly what you are looking for.
Perhaps you want to use the fact that:
xy = ((xy)')' = (x' + y')' = NOR(x', y'), which follows from
DeMorgan's law.
===
Subject: Number Theory Question on Abundant Numbers
This is probably some trivial, known result.    I thought I'd ask the
question here.
A perfect number is equal to the sum of its divisors (not including
itself).   Examples of perfect numbers are 6 and 28.   If the sum of
the divisors exceeds the number itself, the number is an abundant
number.   The number 24 is abundant.
My question is:  If the sum of the divisors of a number (excluding the
number itself)  is S(n),  is there a maximum value to the ratio S(n)/
n?
A quick and dirty Mathematica program shows that the maximum value for
n<1,000,000 occurs for 720720, and the ratio is 3.50909.  I can't
think of a good reason why this ratio should have a maximum, but it
certainly seems to be a slowly increasing function of n.
Kieran Mullen
University of Oklahoma
Homer L. Dodge Dept. of Physics and Astronomy
===
Subject: Re: Number Theory Question on Abundant Numbers
kieran@ou.edu a .8ecrit :
   If you include the number itself you get the 'sigma function' (or 
sigma_1 function) instead of S(n) : 
http://en.wikipedia.org/wiki/Sum-of-divisors_function
   We have (Gronwall's theorem) :
   (with gamma the Euler constant)
   For deep links with the Riemann hypothesis follow the links (Robin's 
criterion) or have a look at these papers :
    'On Robin's criterion for the Riemann Hypothesis' :
http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Amath%2F0604314
    and Lagarias 'An Elementary Problem Equivalent to the Riemann 
Hypothesis' : 
http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Amath%2F0008177
    Hoping it helped,
                Raymond
===
Subject: Re: Number Theory Question on Abundant Numbers
Let n = a b c ... where a, b, c, ... are distinct primes. 
Then S(n) = (a + 1) (b + 1) (c + 1) ....
so S(n) / n = (1 + (1 / a)) (1 + (1 / b)) (1 + (1 / c)) .... 
which exceeds 1 + (1 / a) + (1 / b) + (1 / c) + ....
Now it's well-known that the sum of 1 / p over all primes p diverges, 
so S(n) / n can be made arbitrarily large. 
More quantitatively, the sum of 1 / p over all primes up to n 
is roughly log log n.
-- 
===
Subject: Re: Can solve this function equation?
On 4À.9d23Ëì, 
ÀËË.9f2.bd[CapitalATi
lde]45.bc[CapitalEth], Robert Israel
===
Subject: Re: Can solve this function equation?
Oh! I must admit that something is not clear. That is, does it
establsih f(e)+f(e^(-1))=2*f(1)?
is x<1 and each case has different slope. It dose not seem to good.
Can above quality be proved?
===
Subject: Re: Can solve this function equation?
The equation says g(t c) = c g(t) for all real t and all c in [0,1].
g(t) = b t for t <= 0, but a and b are arbitrary.  There is no reason
for a and b to be related.  In terms of f, this corresponds to 
f(x) = f(1) + b ln(x) for 0 < x <= 1
In particular, f(e) + f(e^(-1)) = 2 f(1) + a-b.  
Not under the assumptions you've given us.
-- 
Robert Israel              israel@math.MyUniversitysInitials.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: Help finding green function
I am having a bit of trouble finding information on Greens functions,
i.e. what they are, what they are for and how to calculate them.
I have several questions here:
[URL=http://img257.imageshack.us/my.php?
image=greenfuncquestionsan7.jpg][IMG]http://img257.imageshack.us/
img257/2592/greenfuncquestionsan7.th.jpg[/IMG][/URL]
Along with solutions to
1)[URL=http://img257.imageshack.us/my.php?image=greensol1jg6.jpg]
[IMG]http://img257.imageshack.us/img257/3974/greensol1jg6.th.jpg[/IMG]
[/URL]
2)[URL=http://img411.imageshack.us/my.php?image=greensol2zb9.jpg]
[IMG]http://img411.imageshack.us/img411/4171/greensol2zb9.th.jpg[/IMG]
[/URL]
Any help will be appreciated.
===
Subject: local max,
the partial derivatives exist as a point c, and f has a local max at
c, then all the partial derivatives equal 0 at c.
Is verifying this just a simple matter of appealing to the 1D case?
0, then f is increasing locally in some neighborhood along that axis.
Thus f cannot have a local max at c.
===
Subject: Re: local max,
Yes of course.
No, because that's false.
===
Subject: Re: local max,
in the x-axis direction.  Then the function must have positive slope
in that direction at c.  In the 1D section, my book has a theorem that
c and f(x) < f(c) for x < c.
===
Subject: Re: local max,
Yes, but that does not imply f is increasing in B(c).
===
Subject: Re: local max,
and not:
and is enough to prove the original assertion.
===
Subject: Re: local max,
Yes. It does not say that, so it does not say that f is increasing.
is increasing in some neighborhgood of c is just that f' need not
be continuous.
Yes - that's why he said Yes of course above...
************************
David C. Ullrich
===
Subject: Re: local max,
[...]
|
f(c),
|and not:
|
and extended by continuity to x=0 (i.e. f(0)=0). It's derivative
at x=0 is 1, but it oscillates in such a way that it's
decreasing on some intervals that come arbitrarily close
to x=0. As you observed, it's still enough to prevent x=0
from being a local maximum of f.
Keith Ramsay
===
Subject: Bayes' theorem question
I'm having a difficult time w/ Bayes' theorem; mostly it's figuring
out what to assign to each variable. I know the formula: P(A|B)=(Pr(B|
A)*Pr(A))/Pr(B).
Here's my problem: 1% of people have hemochromatosis. Your brand new
medical test that you --yes, you!-- developed is  99% accurate --
that's why they called it EPT --the Error Proof Test--. What's the
percentage of the population w/ hemochromatosis who have a positive
test result?
Pr(D|+)=(Pr(+|D)*Pr(D))/Pr(+)
Pr(+|D)=99%
Pr(D)=1%
Pr(+|N)=1%
Pr(N)=99%
Pr(+)=((Pr(+|D)*Pr(D)+(Pr(+|N)*Pr(N)))
Pr(D|+)=(.99*.01)/.0198
Pr(D|+)=50%
Amazingly, the probability of having hemochromatosis AND having a
positive test result is only 50%!  Not such an Error Proof Test
after all. Better do some follow up tests!
I hope this is correct.
BTW, can you explain what Bayesian inference is? And give me a simple
example as well. What about a problem involving linear programming and
either Bayes' theorem or Bayesian inference?
===
Subject: directional derivative,
A problem asks me to show that no real-valued function f has a
positive directional derivative at a fixed point c for every possible
direction u.
But wouldn't a paraboloid work?  Fix c corresponding to the vertex.
Then for any direction u, the slope will always be positive
(increasing).
To be clear, the exact problem is worded: Prove that there is no real-
every nonzero vector u in R^n.
===
Subject: Re: directional derivative,
No, the slope is 0 in every direction.
************************
David C. Ullrich
===
Subject: Re: directional derivative,
If f is differentiable at c, then some directional derivative of f at c 
is <= 0. If f is not differentiable at c, then it's possible that all 
about n = 1 to start), for the second, think about f(x) = |x|.
===
Subject: Re: directional derivative,-
What about the well-known rectangular circular half-cone,
At first sight the problem seems underspecified? incompletely posed? badly 
posed? 
provocative, i.e. aimed at putting in motion the reader's creativity?
Or did the author mean to ask:
Prove that no real-valued function f ==that has a continuous derivative at 
a point C==
has a positive directional derivative at C for every possible direction 
u.
is zero in every direction.
This example shows that zero local slope is not in contradiction to positive 

average slope.
Ciao: Johan E. Mebius
===
Subject: Re: directional derivative,-
That has no directional derivative in _any_ (non-zero) direction 
(at the origin, which is presumably the point you're talking about.)
At least not according to what I've always thought was the
standard definition, as at
 http://en.wikipedia.org/wiki/Directional_derivative
************************
David C. Ullrich
===
Subject: Re: directional derivative,--
That is absolutely correct, if it is indeed standard to consider only entire 

lines through 
the point in question. Is it standard in university and college math 
curricula? - I don't 
know.
When defining mathematical concepts I want to stay as closely as possible to 

everyday 
physical reality. So I identified direction with half-line rather 
than with line.
In my opinion it is partly a matter of convention, partly a matter of to 
what degree one 
wants to change the common meaning of an everyday word into a precise 
mathematical 
meaning, and partly a matter of convenience whether one considers half-lines 

or entire 
lines through the point in question.
Johan E. Mebius
===
Subject: Re: directional derivative,--
I think it is, one reason being you want partial derivatives to be the 
same directional derivatives in the directions of the axes.
I did the same thing.
===
Subject: Re: directional derivative,
Revert back to 2D to shake off the delusion.
For f(x)=x^2 what are the 2 directional derivatives at x=0?
Pick some other value of x. Now what are the 2 directional
derivatives?
quasi
===
Subject: Re: directional derivative,
directions.  I also see that if you pick a direction and the slope is
positive, then if you go in the opposite direction the slope would be
negative.  So I am trying to use this in my proof:
Now I get stuck how to combine these to get a contradiction.  I can
add the inequalities, but do not get any cancellation.
===
Subject: Simple algorithm question
I'm having a brain fart over this simple algorithm, but I hope someone
can help me with this forehead-knocker.
Suppose you have resources r_1, ..., r_n, as well as criterion
s_1, ..., s_n (the minimums), t_1, ..., t_n (the maximums), and R (the
total) - all of which are nonnegative integers.  I am trying to get an
algorithm to see if there exists x_1, ..., x_n such that the sum of
the x_i is R, and the x_i are such that x_i <= r_i, s_i <= x_i <= t_i
(ie. whether the resources satisfy the criterion), and whether it
exists uniquely if it does.
I am convinced that this algorithm exists, but what I have so far is
t_i).  Beyond that I have no clues.  What should I do?
===
Subject: Re: Simple algorithm question
Even without the given minima and the maxima this problem is hard (np-
complete).
(I will not specify this any further...)
If n is small use a brutforce method.
If you whish the SUM to be close to R there are (acceptable)
approximation algorithms.
The problem is called SUBSET-SUM.
http://en.wikipedia.org/wiki/Subset_sum_problem
Benjamin A. B.
===
Subject: Re: Simple algorithm question
On Apr 23, 8:30 pm, Benjamin A. Bartsch
I don't exactly get why (or how) my problem reduces to subset-sum, but
just from thinking about how it could relate to subset-sum did make me
realize that it was NP-complete.
Having said that, I'd better mention my original intent: my original
intent was some form of resource management (eg. you have n different
types of resources, and the cost of doing something in terms of
resources are bounded by the constraints s i, t i, and R.  You have
resources r i, so you have to determine whether some part of r i
satisfies the conditions - ie. you have enough resources to pay up).
What I was looking for is whether there was a way r i can be used to
satisfy the resource constraints, and if so, whether it was unique.
As was mentioned before, this is NP-complete (unless I have my
semantics wrong - I merely restated the problem).
This problem is sometimes found in board games (eg. in Caylus <http://
castle part, which has the most complicated resource requirements in
the game, you have s i = {1,0,0,0,0}, t i = {1,1,1,1,1}, and R=3 -
literally one each of three different resources, for which one is
fixed, so r i = {2,0,1,0,1} would satisfy the conditions uniquely, r i
= {1,1,1,1,0} would satisfy the conditions three ways, and r i =
{0,0,1,1,1} would not satisfy the conditions at all) and trading card
games (eg. the Pok.8emon trading card game has energy cards for which
each card provides resources satisfying conditions of this form, and
you have to find out whether the cards combined meets conditions of
this form).
Because of this, I need to work around the problem.  Now for most
instances of my problem n <= 10, s i <= t i <= s i + 5, and s i <=
10.  Does a heurestic exist for this given the additional restrictions?
===
Subject: Re: Simple algorithm question
Ok forget my last post...
First I thought its a SUBSET-SUM problem,
But what about the following algorithm:
xi_i = MIN (r_i, t_i); // initialize each with maximum
for (i = 1 to n) do DECREASE x_i as much as possible with respect to
IF (SUM(x_i) = R) success ELSE not possible
This algorithm prefers x_i with smaller index beeing as close to their
minimum as possible.
The solution is only unique if every x_i is the mininimum or every x_i
is the maximum.
Benjamin A. B.
===
Subject: Re: Mathematica 1, 2, 3, 4, 5, 6 (Individual Price: $1500)
Good One - Very Funny :-) 
===
Subject: Isn't it bizarre?
Hi.
f(z) = z^z
(z is complex)
This seemingly innocent-looking little thing is actually quite
bizarre. Especially the busines around the negative real axis.
What is going on with this thing? What would a graph of it look like
in the complex numbers? Is there any deep mathematical significance
to what's going on with this function at all?
===
Subject: Re: Isn't it bizarre?
To see what's going on around the negative real axis, consider a ray of
complex numbers around it, parametrized as:
z = x + epsilon*i, -n < x < 0, n in N, epsilon small.
Then when you plot f(z) for x < 0 and -epsilon < 0 < epsilon things become
clear:
Here's some Maple code:
Here are the two graphs:
http://misc.virtualcomposer2000.com/z2z1.gif
http://misc.virtualcomposer2000.com/z2z2.gif
It is clear that the two graphs are conjugate. It is also clear that:
f(-n)=(-n)^(-n), n in N, is real valued, which immediately implies:
The two previous facts mean that the principal branch of f(z) has a branch 
cut
at the negative real axis, which tears apart at all real z in 
(-oo,0){-n},
nin N.
If one uses the the notion of counterclockwise continuity around the branch
values of f(z) are shown in:
http://misc.virtualcomposer2000.com/z2z.gif
-- 
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/
===
Subject: Re: Isn't it bizarre?
For complex z, z ^ z is defined to be Exp(z * Log(z)).  Log(z) is
multi-valued, so the value of z ^ z will depend on which branch of Log
you choose.
Can you explain specifically what behavior of this function is bizarre
around the negative real axis?
===
Subject: Re: Isn't it bizarre?
Sure, but that's not what I'm after. arcsin() and sqrt() are also
multi-
valued, but that's not what I'm talking about.
Like the fact that it is fuzzy there? Esp. _on_ the neg. real axis.
For example, f(-1/3) is real, f(-1/4) is complex, f(5.8) is real,
f(5.81)
is complex, etc. (at least on the principal value.) That would
definitely rank up there on my list of weirdo things, that's for
sure!
Although I'm sure there are weirder things, but this is still weird...
What would a full complex graph of this function look like (of course
a strictly correct graph cannot be drawn since we do not have
enough dimensions, but you could project a shadow of the 4D
graph into 3D space and then display that just like is done with a
hypercube) around that area? Or a 3D-only magnitude/phase plot
(with phase as color)? Or separate real/imaginary graphs? Choose
whatever complex graphing method you like. But I'd really love to
know.
===
Subject: Re: Isn't it bizarre?
It is still unclear to me which branch of z ^ z you are considering.
You say f(-1 / 3) is real.  This means that -1 / 3 * Log(-1 / 3) has
an imaginary part of pi * i + n *pi * i, or that we chose a branch of
Log so that Log(-1 / 3) has an imaginary part of 3 * pi * i, which is
not normally taken to be the principle branch.  Maybe I'm just being
slow, but it would help me to understand this function if you would
explicitly state which branch you are considering.
===
Subject: Re: Isn't it bizarre?
Here is a Mathematica CartesianMap of z^z - it represents the square
[-1,1] x [-1,1] in the complex plane mapped using the function exp(z
log(z)) and the principal branch of the logarithm:
http://math.tkk.fi/~tlassila/images/zpowerz.GIF
Here are each of the four quadrants
1) [0,1] x [0,1]
2) [-1,0] x [0,1]
3) [-1,0] x [-1,0]
4) [0,1] x [-1, 0]
mapped separately for clarity:
http://math.tkk.fi/~tlassila/images/quads.GIF
===
Subject: Re: Isn't it bizarre?
mike3 a .8ecrit :
Are you sure ?
But this is *not* the principal value ! To be consistent, you must use 
the *same* branch of Log, and you are not doing so...
) That would
Answer : if you use the correct branches for Log, you get a perfectly 
ordinary smooth (multivalued) function, with the same behaviour as Log 
around the 0 branch poiint
===
Subject: Re: upper bounds on Dedekind sums
Here are some papers that mention cotangent sums. 
(Reviewer: Paul E. Gunnells) 11F20 (11L03 33B10)
MR1980640 (2005g:11061) Beck, Matthias Dedekind cotangent sums. Acta 
(11L03 11P21)
no. 2, 236--253. (Reviewer: Bruce C. Berndt) 11F20 (11A15 11F67)
lemma and Vasyunin's formulas. Zap. Nauchn. Sem. S.-Peterburg. Otdel. 
Mat. Inst. Steklov. (POMI) 282 (2001), Issled. po Linein. Oper. i Teor. 
Funkts. 29, 20--25, 276; translation in J. Math. Sci. (N. Y.) 120 
MR1763117 (2001d:11082) Cvijovi'c, Djurdje; Klinowski, Jacek Finite 
cotangent sums and the Riemann zeta function. Math. Slovaca 50 (2000), 
no. 2, 149--157. (Reviewer: Matti Jutila) 11L03 (11M06)
MR1453228 (98g:11098) Sczech, Robert Gaussian sums, Dedekind sums and 
the Jacobi triple product identity. Kyushu J. Math. 49 (1995), no. 2, 
233--241. (Reviewer: Bruce C. Berndt) 11L05 (11F20 11F27)
MR1174919 (93i:11053) Barner, Klaus .86ber das 
Theta-Multiplikatoren-System. (German) [On the theta multiplier system] 
Results Math. 22 (1992), no. 1-2, 445--469. (Reviewer: Bruce C. Berndt) 
11F27 (11F20)
MR0746865 (85i:11042) Dieter, Ulrich Cotangent sums, a further 
generalization of Dedekind sums. J. Number Theory 18 (1984), no. 3, 
289--305. (Reviewer: Bruce C. Berndt) 11F20
MR0389729 (52 #10560) Berndt, Bruce C. Periodic Bernoulli numbers, 
summation formulas and applications. Theory and application of special 
functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, 
Madison, Wis., 1975), pp. 143--189. Math. Res. Center, Univ. Wisconsin, 
Publ. No. 35, Academic Press, New York, 1975. (Reviewer: L. Carlitz) 
10A10 (10G05 33A10)
MR0313202 (47 #1757) Berndt, Bruce C. On Gaussian sums and other 
exponential sums with periodic coefficients. Duke Math. J. 40 (1973), 
145--156. (Reviewer: T. M. Apostol) 10G05
-- 
===
Subject: Re: Why is Maple able to give me RREF without encountering divide 
by zero?
I've got that misunderstanding wrapped up, sorta speak.
I don't really care what cause it, it's behind me now.
I admit you can do RREF without division:
I've got a demostration here:
http://www.cap-lore.com/MathPhys/Field/RREF.html
I don't want to touch or think about gaussian elimination right now.
The way I had to deal with it was whenever there is division by
zero, return singular. That was probably wrong.
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
I simply mean they are less like numbers then arguments (tools) to do
(derived) things with. I was naive to believe them were numbers.
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
On 23 Apr 2007 16:32:41 -0700, dillogimp@gmail.com
Well, since they _are_ numbers it's still a little hard to see how
they can be less _like_ numbers.
************************
David C. Ullrich
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
Transcendental functions are numbers? What do you know!
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
Actually, the original question mentioned sin(x), cos(x), and exp(x),
which are numbers, not functions.
-- 
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
sin(2) is a number.  sin(x) is a formula that represents a function.
-- 
G. A. Edgar                              
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
Since the original poster called them functions it is natural to take him 
to
rather than the numbers sin(x), cos(x) and exp(x) for some unspecified x.
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: All the transcendental numbers and functions like sin(x) are 
basically arguments right?
Yes, and I took that to be his meaning in my original answer, in which I
mentioned that functions can also be arguments.
-- 
===
Subject: Re: very complex numbers
Oof, my bad. But the proposed very complex numbers still do not
do this, and I'd therefore hesitate to call them a generalization of
the complex numbers. A generalization must also include what it
generalizes.
===
Subject: Mathematical terminology origins
Hi.
I've been wondering about the origins of some mathematical
terminology, such as the names of the various abstract algebraic
structures. Like, for example, why is a group called a group, anyway?
The colloquial use of the term group would be more akin to just a
set. What about a loop? (like a group, but without the associative
law) How about a monoid? This latter one is the only one I have a real
guess at. I'd assume it comes from mono- meaning one and -oid
meaning like, so it means one-like or oneoid, I guess referring to
the identity element, which under the binary operation behaves like
one does with the numbers and multiplication. How about rings and
fields?
===
Subject: Re: Mathematical terminology origins
Perhaps this can help.
http://members.aol.com/jeff570/mathword.html
In some cases, I don't think the answer is something meaningful, but
in general, there is probably reason for such choices.
The important thing in papers is to define what is meant by the terms
so the reader has a clear understanding.
Also, it has always bothered me that there is no consistency in
mathematical terms or usage and this makes life very difficult.
There is also a book on the matter:
<http://www.amazon.com/Words-Mathematics-Etymological-Dictionary-
Lastly, the history sites provide some insight, for example (many
others exist), the wonderful sites:
1.  http://en.wikipedia.org/wiki/History_of_mathematics
2.  http://www-groups.dcs.st-and.ac.uk/~history/
3.  http://members.aol.com/jeff570/mathsym.html
HTH, A
===
Subject: Re: Mathematical terminology origins
Mathword:   http://members.aol.com/jeff570/mathword.html
-- 
G. A. Edgar                              
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: counting connected subgraphs
Try
How many ways can an mxn grid be tiled by a single unholey polyomino of 
any
size, where rotations and reflections are considered distinct?
-- 
Patrick Hamlyn   posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
===
Subject: Re: Fourier transform of a finite measure
Originator: grubb@lola
Most certainly. Look up Bochner's theorem that relates positive
definite functions to Fourier transforms of positive measures.
Convolutions of finite measures can be defined and the Fourier
transform is an algebra homomorphism from the Measure algebra
into the space of uniformly continuous bounded functions. 
Invertibility of measures under convolution can be studied, although
the results aren't as good as might be desired because there
are other multiplicative linear functionals on the Measure
algebra. 
Paul Cohen (of set theory fame) classified the measures whose
fourier transform is either 0 or 1 in the context of LCA groups.
A good reference is Hewitt&Ross: Abstract Harmonic Analysis I&II.
--Dan Grubb
===
Subject: Re: Cantor's Powers.
My question. Is this correct?
I would appreciate it if one answers this simple question?
Zuhair
===
Subject: Re: Cantor's Powers.
It seems that nobody in this forum has the curtesy of even making a
simple answer to this question, by YES or NO.
It is a disgrace for this forum really.
Zuhair
===
Subject: Re: Cantor's Powers.
zuhair says...
Yes.
--
Daryl McCullough
Ithaca, NY
===
Subject: Re: Cantor's Powers.
Why?
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: What will be the next paradigm shift in Math?
1. When you say infinitesimals, do you just mean writing eg
        dy = sin x dx
rather than
        dy/dx = sin x ?
If so I would have thought the use of dx, dy, etc has become more popular,
rather than less.
2. Conway's Surreal Numbers (as they are called by Knuth)
give an alternative way of introducing infinitesimal quantities
as real objects.
-- 
Timothy Murphy  
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
===
Subject: Re: What will be the next paradigm shift in Math?
Non-standard analysis has been used to teach beginning
calculus.  How well, I do not know.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Green functions
I am having a bit of trouble finding information on Greens functions,
i.e. what they are, what they are for and how to calculate them.
I have several questions here:
[URL=http://img257.imageshack.us/my.php?
image=greenfuncquestionsan7.jpg][IMG]http://img257.imageshack.us/
img257/2592/greenfuncquestionsan7.th.jpg[/IMG][/URL]
Along with solutions to
1)[URL=http://img257.imageshack.us/my.php?image=greensol1jg6.jpg]
[IMG]http://img257.imageshack.us/img257/3974/greensol1jg6.th.jpg[/IMG]
[/URL]
2)[URL=http://img411.imageshack.us/my.php?image=greensol2zb9.jpg]
[IMG]http://img411.imageshack.us/img411/4171/greensol2zb9.th.jpg[/IMG]
[/URL]
Any help will be appreciated.
===
Subject: Re: Green functions
I think you can get some books about that, it's more efficient than
posting here.
Simply, find 'Green function' on wikipedia.
===
Subject: Re: English is a mess. was Re: ?? what about the niggers ??
                        ................
No, the game would be replayed, if it had not already
been so.  Tie games are not resumed unless there is an
error in the umpires' call which caused it to be tied,
or even if an error caused it to be scored one way or
the other, and it has to be a MAJOR misinterpretation.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: English is a mess. was Re: ?? what about the niggers ??
So far as I can tell, Peter's confusion lies in that he is aware that
a game can suspended and resumed later if they are forced to stop play
while the score is tied.  He believes, for reasons which are unclear,
that this is always the case:  that a game cannot end with a tie
score, but rather can only be suspended.  The last go-around I quoted
the relevant rules, as well as citing actual examples of tie games,
but he seems immune to such things.
Richard R. Hershberger
===
Subject: Re: English is a mess.
It certainly has interrogatives, and exclamatives are used.
Ellipsis is widely used in proofs.  I am not quite sure of
some of the other words.  Tenses have not been included, but
as mathematics can deal with time, the procedure is there.
Mathematical notation was never intended as an all-purpose
language, but as a limited language which can be used with
almost any other language.  Yet it is a language in which
precise communication is possible and brief, in contrast
with the natural languages, in which this is rarely so.
 I could list dozens of 
This is questionable.  It has not been tried, and many
aspects of mathematical pedagogy have been found simple
after they have been tried.  Whether natural language is
useful in describing mathematical concepts is rather
debatable; the issue is unlikely to arise.
 Every mathematical 
I am not convinced that the papers would get longer,
but additions would have to be made to the mathematical
notation language to shorten them.  There is no notation
for in a similar manner, for example.
However, there are papers which could be written in
mathematical notation including a natural deduction
style of proof which would be straightforward.  Many
of the usual papers have interspersed lines of 
commentary which add nothing.  Euclid's geometry,
written well before mathematical language, is close
to this.
Mathematics has never claimed to cover everything.
Which meaning of that?
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: English is a mess.
How does one ask What time is it? in mathematics?  How does one say 
Ouch! in mathematics?  If these statements cannot be made in 
mathematics, then this is yet another property that differs between 
mathematics and actual languages.
I'm not talking about and so on/... ellipsis.  I'm talking about 
linguistic ellipsis, which is a specific kind of syntactic gapping, as 
in John will visit his mother, but Bill won't.
Does mathematics deal with time in the same way ordinary languages 
do?  Can you say x was equal to 9 on the previous page, but now x is 
equal to 7?  Does mathematics even have equivalents to then and 
now?  (Those are examples of deixis, by the way.)
Used with in a way that no other language works.  English isn't 
used with French.  Yet another property that differs between 
mathematics and actual languages.
Yet another property that differs between mathematics and actual 
languages.
So you can't say this in mathematics?  Yet another property that 
differs between mathematics and actual languages.
The deictic meaning, of course.  But mathematics doesn't seem to have 
deixis. Yet another property that differs between mathematics and 
actual languages.
In the face of so many properties that differ between mathematics and 
actual languages, why do you continue to insist that they form a 
logical grouping?
As I recall, you refuse to 'define' integers, except by their 
properties.  If properties are valid ways to define such fundamental 
concepts, why are you unable to accept a definition of language based 
on the properties languages share?
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
-- 
===
Subject: Re: English is a mess.
Am Mon, 23 Apr 2007 22:29:36 -0400 schrieb :
So if someone is a specialist on fish, he might tell the concept of
vertebrates worthless, because gnus have neither fins nor gills?
Joachim
===
Subject: Re: English is a mess.
Fish are vertebrates, but would he know what a gnu is?
Neither would an ancient Greek or Roman; were they not
using languages?
In how many of the thousands of languages of Australia
and the Pacific islands can one discuss microbes or
photons or the Riemann Hypothesis?  
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: English is a mess.
Why would an ichthyologist care about gnus?
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: English is a mess.
Am Tue, 24 Apr 2007 01:13:05 -0400 schrieb :
You did not just state that math formula language is out of scope of
linguistics, which is ok - you claimed that it wasn't worthwile
looking at common properties of natural languages, programming
languages, and math languages and having a common concept for that.
This is particularly difficult to accept for a trained mathematician,
because it's second nature to mathematicians to find common features
and find meaningful concepts/categories for them.
Joachim
Joachim
===
Subject: Re: English is a mess.
Would you say a little more about ` verb-controlled preposition 
choice'
Peter T. Daniels said I was talking `Utter nonsense' when I 
described a conversation with a native German speaker's 
difficulties with English prepositions. I mentioned
* How to place it so the referents are clear.
* Which preposition to choose among two or more equally good 
choices.
* Avoiding overuse of a few prepositions so as to keep prose 
fresh. 
Was I? 
-- 
===
Subject: Re: English is a mess.
I don't recall that you said anything about describing a conversation.
What I recall is that you said she complained about prepositions (not
that she had trouble using prepositions?), and while this is perfectly
sense that you insisted she meant prepositions -- but you never gave
an example.
If there's a preference between them, then obviously the two or more
aren't equally good.
===
Subject: Re: English is a mess.
For example:
      We wished/longed/hankered for a hamburger.
     *We wished/longed/hankered a hamburger.
     *We desired/wanted/craved for a hamburger.
      We desired/wanted/craved a hamburger.
There's nothing inherent to wishing, longing, hankering, desiring, 
wanting, or craving that should either prevent or require for, yet 
in English, this is the pattern.  Compare also decide on, opt for, 
and choose, or track down, search for, and locate.
Any of these patterns can be different in other languages (we say 
depend on, but Russian says depend from), because 
subcategorization is often arbitrary with no logical connection to 
meaning.  Since it can't be derived logically, it must be memorized, 
something that native speakers have done without even trying.  
However, non-natives have to learn a completely new system that is 
arbitrarily different from their own.
Some verb+preposition combinations are called phrasal verbs, with the 
prepositions called participles rather than prepositions.  I'm not a 
syntactician, and I confess to not having a very good eye for telling 
quite possible that some of the examples I gave above are actually 
phrasal verbs.
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: English is a mess.
:
If the particular biologist has a special interest in flying animals,
why not?
Then again, if his interest is in speaking animals, the only species
to study is Homo sapiens.
Considering that linguistics is more like a political party than a
branch of science, that's probably good advice.
Jens S. Larsen
===
Subject: Re: English is a mess.
Am 23 Apr 2007 13:50:59 -0700 schrieb Jens S. Larsen:
I bet there are biologists that do so!
Joachim
===
Subject: Re: English is a mess.
Because the biological adaptations that allow them to fly are 
completely different (hollow bones for sparrows, high frequency wing 
beats for bees, flexible finger bones for bats).
Assuming the biologist is at all interested in biology, of course.
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: English is a mess.
Am Mon, 23 Apr 2007 18:10:42 -0400 schrieb :
Maybe he is interested in biological adaptation that allow different
organisms to fly?
Assuming he is not a bigot.
Joachim
===
Subject: Re: English is a mess.
Am Mon, 23 Apr 2007 10:00:46 -0400 schrieb :
Well, if you are looking for low-level phonology in programming
language, then you'll probably won't find much. Small wonder.
Joachim
===
Subject: Re: Is language a mess? (was: English is a mess.)
My definition of language fits this.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Is language a mess?
Your definition of language, if it includes both natural languages and 
mathematics, ignores dozens, probably hundreds, if not thousands, of 
properties common to every single natural language, and gains us 
absolutely nothing in return.
It's like me deciding to define the integers to include not only 
classic integers, but pi as well.  They're all numbers, so why not?  
That's at least one shared property they have (at least I didn't 
decide to add something outrageous like & to the set!).
Who cares about the thousands of shared properties I have to throw out 
to make pi fit in?  Who cares about the logical, natural, and 
painfully obvious separation between integers and pi?
If I'm so inclined to call pi an integer just to suit my own demented 
agenda, then who's to say I'm wrong?  Surely not mathematicians, 
because they're too entrenched in their own narrow world to see the 
benefit of my interdisciplinary and revolutionary proposal!
Nathan
-- 
Linguistics Program
Williams College
http://wso.williams.edu/~nsanders/
===
Subject: Re: how to weed out the unique solution from the infinite 
solution matrix?
But I managed to learn a lot by being stupid and saying false things.
===
Subject: Re: how to weed out the unique solution from the infinite 
solution matrix?
Really?
;; The number of nonzero rows in the reduced row echelon form of a
matrix A is called the rank
augmented matrix.
;; 1. If rank(A)=rank(M)=n then unique solution
;; 2. If rank(A)=rank(M)<n then infinite number of solution
;; 3. If rank(A)<rank(M)   then no solution
;; Most linear algebra books will tell you how to find a basis for
;; the column space of a matrix. The matrix with just those columns
;; will have full rank & thus unique solutions.
I was able to find the answer to my problem with these two theorems.
(Plus I learn that the model I am using has infinite solutions
sometimes)
But beyong that, do I any know anything more about the problem I am
solving? None.
===
Subject: Re: how to weed out the unique solution from the infinite 
solution matrix?
Actually, that's not entirely true, from the infinite solution case,
and the interpretation of the model I realize some recurrence has
multiple forms.
for example:
f(n)=3f(n-2)+2f(n-1)
f(n)= 6*f(n-3) + 7*f(n-2)
are all valid solutions gained from the insights of linear algebra.
But beyong that there is nothing else that would help me further.
(Or rather I've forgot all about them, because I never had any
opportunity for applying them.) (That is not saying I think they are
of any use to begin with.)(I will admit they are useful, once I found
a use for them.)
===
Subject: Re: how to weed out the unique solution from the infinite 
solution matrix?
I got the trick back sorta speak, I've known how to do this a long
time ago, but forgot about it.
I learned trick again from this page:
http://www.adeptscience.co.uk/products/mathsim/maple/powertools/linearalgebr

a/html/OnLine2-2-Dimension.html
===
Subject: Re: Thousands of solutions manuals
hi i need solutions to the following books
1.vector mechanics for engineers
statics and dynamics
beer and johnston
2.engineering mechanics II
    dynamics
    merriam and kraige.
thank u.
===
Subject: Re: Recovering color hue of electric arc
  Get the 1931 CIE color matching functions X-bar, Y-bar, and Z-bar, and 
then:
1.  Start by multiplying the X-bar, Y-bar and Z-bar values of 410 nm by 
30.
2.  Multiply the X-bar, Y-bar, and Z-bar values of 434 nm by 60.
3.  Multiply the X-bar, Y-bar, and Z-bar values of 486 nm by 160.
4.  Multiply the X-bar, Y-bar and Z-bar values of 656 nm by 300.
5.  Add up all of your X-bar, Y-bar, and Z-bar results into an X-bar sum, 
a Y-bar sum, and Z-bar sum.
6.  Divide each sum by the sum of the X-bar, Y-bar and Z-bar sums to get 
1931 CIE chromaticity coordinates x, y, and z of the combination of 
wavelengths.
7.  Hope the intensity data is linear and from a spectrometer that has 
flat spectral response or is corrected to represent flat spectral 
response.  I would not get optimistic that both parts of this item are 
true!
8.  In the unlikely event 7 is true for the lamp or other source that the 
spectrum was obtained from, hope the intensities are the same for your 
lamp or source.  There is some fair chance that will not be true.  For 
example, in my experience a usual Plucker tube hydergen spectrum tube 
has the narrow capillary region glowing a pinkish magenta, while the wider 
end regions glow a lavendar (pale purple) color.  Who knows what a 
hydrogen Geissler tube did way back when?
 - Don Klipstein (don@misty.com)
===
Subject: Re: Recovering color hue of electric arc
Yes.  No need to see the light.  Just an instrumental output of relative 
spectral radiant data will do.  Number of lines can be anything.  Here 
we are assuming infinitesimal width spectral lines, but a continuous 
spectrum with some finite nm resolution would also work.
Here's the CIE 1931 2-degree observer coordinates for your data:
Calculated as follows (Mathematica):
In[47]:=
H2RelativeSpectralRadiance={{410.174,30},{434.047,60},{486.133,160},{656.2,
         300}};
H2=H2RelativeSpectralRadiance
Out[48]=
{{410.174,30},{434.047,60},{486.133,160},{656.2,300}}
In[49]:=
X = Sum[ H2[[i, 2]] * CMFx[ H2[[i, 1]] ], {i, 1, Length[H2]} ]
Y = Sum[ H2[[i, 2]] * CMFy[ H2[[i, 1]] ], {i, 1, Length[H2]} ]
Z = Sum[ H2[[i, 2]] * CMFz[ H2[[i, 1]] ], {i, 1, Length[H2]} ]
Out[49]=
90.27701259978946
Out[50]=
52.20959422171659
Out[51]=
194.0450216610586
Out[52]=
The CMFx, CMFy, and CMFz were interpolations of the CIE data for the 
color matching functions downloaded from http://cvrl.ioo.ucl.ac.uk/
The following MMA code imported the file (need to delete extra space at 
end of file):
Module[ {i, [Lambda]},
   CMFxyzCIE1931Deg2Table5nm=
     Import[ CIEdata/cie-xyz-1931-2deg-5nm.txt, CSV ];
   CMFxCIE1931Deg2Table5nm ={};
   CMFyCIE1931Deg2Table5nm ={};
   CMFzCIE1931Deg2Table5nm ={};
   For[ i=1, i<=Length[CMFxyzCIE1931Deg2Table5nm], ++i,
       [Lambda]=CMFxyzCIE1931Deg2Table5nm[[i,1]];
       CMFxCIE1931Deg2Table5nm =
         Append[CMFxCIE1931Deg2Table5nm,{[Lambda],
             CMFxyzCIE1931Deg2Table5nm[[i,2]]} ];
       CMFyCIE1931Deg2Table5nm =
         Append[CMFyCIE1931Deg2Table5nm,{[Lambda],
             CMFxyzCIE1931Deg2Table5nm[[i,3]]} ];
       CMFzCIE1931Deg2Table5nm =
         Append[CMFzCIE1931Deg2Table5nm,{[Lambda],
             CMFxyzCIE1931Deg2Table5nm[[i,4]]} ] ]
     Print[CMFxCIE1931Deg2Table5nm];
   Print[CMFyCIE1931Deg2Table5nm];
   Print[CMFzCIE1931Deg2Table5nm]
   ]
CMFxCIE1931Deg2 = Interpolation[ CMFxCIE1931Deg2Table5nm ];
CMFyCIE1931Deg2 = Interpolation[ CMFyCIE1931Deg2Table5nm ];
CMFzCIE1931Deg2 = Interpolation[ CMFzCIE1931Deg2Table5nm ];
CMFx=CMFxCIE1931Deg2
CMFy=CMFyCIE1931Deg2
CMFz=CMFzCIE1931Deg2
-- 
Good day!
________________________________________
Christopher R. Carlen
Principal Laser&Electronics Technologist
Sandia National Laboratories CA USA
crcarleRemoveThis@BOGUSsandia.gov
NOTE, delete texts: RemoveThis and
BOGUS from email address to reply.
===
Subject: Re: Recovering color hue of electric arc
one
lines
that
(or
[snip mathematica code for brevity...]
code, it looks very similar to the code of John Walker pointed out by Vic
Roberts:
http://www.fourmilab.ch/documents/specrend/specrend.c
from this page:
http://www.fourmilab.ch/documents/specrend/
Porting the above proc spectrum_to_xyz to Maple, the results differ
slightly.
Here's my translated code in Maple:
I broke the range 380nm..,780nm into 5nm bins (81 bins) and when I load
wavelengths and relative intensities, I round to nearest integer as 
follows:
cieColorMatch is the authors table as he gives it on the program example. I
hope I haven't introduced any bugs.
For hydrogen, I loaded only the 4 prominent lines:
I got the following:
x=0.2734587737, y=0.2062572982, z=0.9976330621
One notices that the x coordinate is very close to yours, but the y 
coordinate
is somewhat off.
Which brings me to another strange example. That of Thallium. Loading the
spectral data for Thallium from NIST:
I get:
x=0.1929621428, y=0.9684470275,z=0.9978394855
Now, we have a problem: The above spectrum is practically monochromatic, so
shouldn't its x and y coordinates land exactly on the edge of a CIE
chromaticity diagram?
Indeed, x=0.192962...crosses the edge of the CIE diagram almost exactly at
535nm. But the y coordinate is way off, much higher than the approximate
y~=0.78.. needed for the pure edge color. y=0.9684470275 falls off the CIE
diagram.
Is there any explanation for this? The author of the above webpage mentions
and I quote:
The CIE Y value is a measure of the perceived luminosity of the light
source--how bright it appears to an observer (to a first approximation;
nonlinearities in the human visual system create a host of higher-order
complexities). In rendering, we're usually interested only in relative
luminosities so we can ignore absolute values of Y and simply scale
luminosities between user-defined minimum and maximum brightnesses.
So what does such a high value of y for Thallium denote? Is it safe to 
assume
that if the y value falls off the CIE diagram, and we have a monochromatic
spectrum, we can project it to the edge of the diagram?
What about when either value falls off the diagram and the spectrum is not
monochromatic?
-- 
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/
===
Subject: Re: Recovering color hue of electric arc
And lo and behold, the above is buggy. The equations for the chromaticity
coordinates should be:
x=X/(X+Y+Z);
y=Y/(X+Y+Z);
z=Z/(X+Y+Z);
chormaticity coordinates x,y,z are NOT the same as the CIE X,Y,Z 
components.
Duh!
With the corrected code I get:
x=0.2734587737, y=0.1500152547, z=0.5765259716, which is very close to 
your:
for hydrogen.
For Thallium, I get:
x=0.1929621428, y=0.7815822821,z=0.254555751e-1,
which indeed, falls exactly on the edge of the CIE diagram.
---
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/
===
Subject: Re: ? complex dot product
It seems that many ppl suggested using distance as a measure for how close
the complex vectors are. But in determining the intersection of two 
subspaces
A (belongs to R^(mxp)) and B (belongs to R^(mxq)), one needs to compute
the angles or cosines to determine the orthonormal basis of the 
intersection
of A and B. I wonder how this is extended to handle complex spaces if one 
cannot
define a meaningful dot or cosine operation.
by Cheng Cosine
   Apr/23/2k7 NC
===
Subject: Re: Skolem Paradox Question
What does formal mean? Does your use of the qualifier `formal' 
imply that there are ways to derive theorems of real analysis that 
are not formal? Because this is what I conceive of. That real 
numbers can be defined well enough independently of set theory, 
and well enough defined to fulfill our notion or real number. This 
is in contra-distinction with things like inaccessible ordinals 
that have no conception outside set theory.
-- 
===
Subject: Re: Skolem Paradox Question
What it means as defined in various textbooks in mathematical logic,
which definitions usually specify the property of being recursively
axiomatized and other such specifications as to the inference rules of
the system, etc. Most basically, that there is an algorithm to
determine whether a sequence of formulas is or is not a proof in the
system.
Yes, of course, in the sense that one might speak of theorems in an
informal sense of 'theorem'. However, it is hoped (for lack of a
better word that comes to my tongue right now) that any informal proof
in analysis can be formalized in some formal theory such as a formal
set theory.
That depends on your definitions of 'defined' and 'well defined'. Yes,
of course, all kinds of things can be rather loosely defined outside
of formal theories.
They can be conceived in an informal set theory. Perhaps they might
not be of much relevence in an informal set theory, but they can be
conceived. You use sets informally to conceive of real numbers, and
one can use sets informally to conceive of inaccessible cardinals.
MoeBlee
===
Subject: Re: Skolem Paradox Question
It is important to remember where I entered the
discussion. Aatu mentioned the real natural numbers and
the real real numbers. I propose that the real natural
numbers and the real real numbers are the ones that
exist independently of set theory. The natural numbers
are the counting numbers: 0 and its successors. The
real numbers are a field with the lub property. So I am
not trying to formalize in the spirit of set theory,
for set theory does the job. I propose that the real
numbers and the natural numbers are independent of set
theory.
-- 
===
Subject: Re: Skolem Paradox Question
That's fair enough; and I understand that. My point though concerns a
juncture of the conversation that came more recently. I recognize that
we can do analysis informally and that we can informally conceive of
natural numbers and real numbers. However, I don't find that such mere
informal conception in itself endows mathematical existence, at least
not in my own (admittedly tentative) view of mathematical existence.
You propose to say what real numbers are in terms of sets (least upper
bound property, for example) but not formally. I don't find that
satisfactory for reasons of vagueness and non-specificity (I don't
think you can say what a mathematical object is merely by saying that
it is one of a species of which all others are isomorphic to it).
Thus, I do think formalized theory plays a part in specifying
mathematical objects.
MoeBlee
===
Subject: Re: Skolem Paradox Question
According to your knowledge above then, a model of ZF theory is
a model of ZFC theory, since both are of the same language L(ZF) = L(ZFC)!
Whatever that means!
===
Subject: Re: Skolem Paradox Question
No, it's a model for L(ZF) and for L(ZFC), since L(ZF) = L(ZFC), but
it might be a model of ZF but not a model of ZFC.
The model is a certain mathematical object and it does not change
type. That object is not a different object nor different type
depending on context. It is the same object. Meanwhile, that object
happens to satisfy the predicates 'is a model for L(ZF)' and 'is a
model for L(ZFC)', is a model of ZF, but it might not satisfy the
predicate 'is a model of ZFC'.
It's been explained to you already, and you can look it up in
Shoenfield for yourself, as has already been mentioned.
You have a nasty habit of answering in the tone of a real smartass on
matters of which you are yourself confused, ignorant, and continually
obtuse.
Both (1) and (2) describe relations. (1) describes the relation
defined by 'M is a model for L' and (2) describes the relation defined
by 'M is a model of T'.
The model M is a certain mathematical object. It does not change
identity nor type. What differs is whether it satisfies 'M is a
model for L' while satisfying or not satisying 'M is a model of T'.
That difference depends on T. To put it another way: The model M
remains itself in all contexts and remains the same type of object,
and M is a model for a certain language L, but depending on T, it will
differ whether or not M is a model of T.
MoeBlee
===
Subject: Re: Skolem Paradox Question
Originator: grubb@lola
Fair enough. Just bump up one level then.
--Dan Grubb
===
Subject: Re: Skolem Paradox Question
If we escalate to yet another formal theory, then that is on the way
to infinitely many escalations and thus perpetual deferment of the
question of what mathematical objects are. And if we escalate to an
informal theory, then the question of what mathematical objects are is
raised anew since for an informal theory we don't have recourse to
your explanaton of 'it's all in a formal theory' (to paraphrase
tersely). I mentioned my own approach (infinite escalation, but
viewing it as a kind of successor operation), but with that I
recognize that the matter is not so simple as just declaring that
mathematical objects don't exist except per some formal theory.
MoeBlee
===
Subject: Re: Skolem Paradox Question
That sounds just like the cumulative hierarchy of ordinals (infinite
escalation).
I think a problem with defining necessary sets as non-sets is that the
reason they're defined is because they're necessary, in for example
quantification over sets and set-building with that quantifier.
Quantify over sets:  all of them.
ZF's universe is the Russell set so it contains itself.  Consider the
quantum pot that (under observation) never boils, similarly the
specification of the universe of ZF, in a subset of the axioms of ZF,
that is exactly the universe of ZF, contains itself.
Then that gets into considerations of infinity in the natural
numbers of real numbers, in the projective extension where for example
there is the wide use in the meromorphic.  Similarly of questions of
potential vis-a-vis actual infinity, where the potential infinity
might be seen as an iterative counting process where the actual
infinity includes all the numbers, plus one, a variety of transforms
nimbly equate about the mean.
I appreciate Press' correction about the complete ordered field vis-a-
vis field with least upper bound property, except the domain of reals
has no GLB, greatest lower bound, nor LUB, least upper bound, (it is
not bounded above nor below).  In the standard reals there is no
element that is greater than or equal to each element of each subset
nor is there an element less than or equal to each element of each
subset.  I think R's completeness is more along the lines of that
infinitely iterated non-asymptotic basic operations on the integers
don't yield products outside the field, in any possible way that they
could be taken.  For example as integer coefficient doubly infinite
power series over non-zero integers the real numbers contain all their
evaluations, and each real number can be represented in that base/
radix representation, except for infinitely many coefficients in the
positive exponents.
Some do divide by zero, then multiply back by that zero and retrieve
the previous result.  L'Hospitale's rule is widely used.
It's a common conception that the reals are complete, the complete
ordered field.  Generally that doesn't include division by zero.  The
Dedekind completeness condition, if that is the correct appellation,
is that every set of reals bounded above has a least upper bound that
is an element of the reals.  Yet, there are certainly subsets of the
reals with no least upper bound or greatest lower bound.  Indeed, how
is bounded above defined?  Basically that is to say a set is bounded
above when there exist no elements in the subset above a given element
of the entire superset in the total ordering, ie greater-than in the
total ordering of the reals.
(Well order the reals.)
To be bounded above (each of the elements in the subset), a subset of
the reals is less than some bound.  Yet, how does the bound,
boundary, exist?  For it to exist it must as well be a subset of the
reals bounded above, and where there are not sequential reals, that is
arbitrarily larger, as is its boundary above, ad infinitum.  With no
maximal element in the reals, eg standardly, the boundary above isn't
defined.
The objects exist.  There's only one theory with no axioms.
Ross
--
Finlayson Consulting
Clock Accurate
===
Subject: Re: Skolem Paradox Question
The cumulative hierarchy is not just of ordinals. The ordinals index
the hierarchy, but the hierarchy itself is not a hierarchy of
ordinals.
What I mentioned in my post is analogous to greater and greater
ordinals and analogous to the greater and greater levels of the
hierarchy, but those are not what I'm not referring to.
MoeBlee
===
Subject: probability and die simulation
I have a question that I need help on. Given that I have a fair coin, I need 

to simulate a die, that is to produce 6 outcomes with probabilities 1/6 
each? 
So far I have the following:
If we flip the coin 6 times we can let 6 be the number of bits in a binary 
representation. If we let the sequence of coin flips be the 6 bit binary # 
then we can keep the number if it is between 1 and 6 (where heads will 
represent 1 and tails 0), otherwise we repeat. Does this provide me with my 
6 
outcomes of 1/6 each. Also, is there another way to approach a problem like 

this?
===
Subject: Re: probability and die simulation
For more fun, you can simulate that fair six-sided die with four flips of a
coin that may not be fair; that is, the coin comes up heads with 
probability
p on every flip, but p may not be 1/2.
Flip it four times, having assigned in advance the sequences HHTT, HTHT,
THHT, THTH, TTHH, and HTTH to your six sides of the die.  If your four 
flips
result in a different sequence (that is, one with unequal numbers of H and 
T
outcomes), you re-flip, until you have one of the acceptable sequences.
-- 
Randy Hudson
===
Subject: Re: probability and die simulation
otherwise we repeat.
There are 2^6 = 64 possible outcomes. In 6 of them, you will
stop and accept the result. In the other 58/64 you will try again.
If you stop, each of 1-6 is equally likely.
I believe so.
Well, any method by which you come up with 6 equally-probable
events would be acceptable.
I don't see why you need 6 flips with a 58/64 probability of having
to flip again. Why not just 3 flips? Keep if 1-6, discard if 0 or 7.
               - Randy
===
Subject: Re: probability and die simulation
Actually with 6 flips, you can group 60 of them in tens to have 6 
equally likely outcomes and only  have to redo 4 out of 64.
Or even more efficiently, flip thrice getting binary 0 through 7 and 
redo 0 or 7.
===
Subject: Re: probability and die simulation
The latter would have an expected number of flips equal to 4
(ie 3*1/p where p is the probability of success, ie 3/4)
You could flip once to decide whether your final result is eg
odd or even, and then repeat flipping two to choose which odd
or even; this would lead to to an expected number of flips 11/3.
I wonder what the minimum is.
===
Subject: Re: probability and die simulation
The minimum is log_2 6, which is about 2.58. You can get within 0.1% of that 

with an extension of your approach: flip the coin 485 times to produce an 
integer in [0, 2^485); if the result is less than 3^306, flip the coin an 
additional 306 times for parity to get 306 integers in the range [1,6]; 
otherwise discard the flips and repeat. This takes about 791.5 flips on 
average for 306 results. If that's not close enough, replace 306 by 190537 
and 485 by 301994.
-- Ben
===
Subject: Re: FLTMA: Footprints of odd prime powers in FLT
Additionally, with (m) meaning modulus m, the matrix:
[
x/x(x)   x/z(y)   x/y(z)
z/x(y)   y/y(y)   x/z(y)
y/x(z)   z/x(y)   z/z(z) ] ^p =
[
1   1   -1
1   1   1
-1   1   1 ]. This seems applicable to the Eckl conjecture.
Doug
===
Subject: Solving 35 nonlinear equations
I've got a system of about 35 nonlinear equations that I want to solve
numerically. To do this I am passing my (Matlab) program a vector x
containing some initial guesses for the variables' values, and it
tries to minimize the sum of squares of vector y =
[f1(x1,x2,...,xn)
f2(x1,x2,...,xn)
f3(x1,x2,...,xn)
... ]
with a line-search method. However, since the system of equations is
so big, I want to make it smaller so that my program can solve it
efficiently with the line-search method. For example, if I know that
x3=x1+x2, I can eliminate x3 from vector x, since it is explicitly
redundant with x1 and x2, and take this equation out of my vector y.
My question is: how can I make my vector y as small as possible,
without doing heavy-duty manipulation of my equations? I've got my
original system of 35 variables down to 11 now, but I'm wondering if I
could have easily done better. Are there any general tricks I can use
to pick which variables I should pass in to my optimizer, and which I
should calculate based on the others?
Any insights would be appreciated.
===
Subject: Phase Plane - when the linear part of the eigenvalue is Zero
In this picture : http://img411.imageshack.us/img411/9598/21576104mq1.png
you can see that eigenvalues of the matrix of (d) and (e) has 0 or
zero linear part. (d)'s graph is #1 and (e)'s graph is #2
What I don't understand is - what value/idea/notion determines #1 or
#2 graph to be slanted (in two different positions -  /    or      
#1 is like this /
and
#2 is like 
Note: Both #1 and #2 are stable clockwise.
===
Subject: Re: Phase Plane - when the linear part of the eigenvalue is Zero
Well, I'd just insert x=(1,1) and note that for (d) we have Ax = (2,1), i.e. 

pointing in the '/' direction and for (e) we have Ax = (8,-5), i.e. pointing 

in the '' direction and with a much larger magnitude. So indeed (d) 
corresponds to figure 1 and (3) to figure 2, as you have found out.
Not sure, whether that's what you were looking for though.
-Michael.
===
Subject: Phase Plane - when the linear part of the eigenvalue is Zero
In this picture : http://img411.imageshack.us/img411/9598/21576104mq1.png
you can see that eigenvalues of the matrix of (d) and (e) has 0 or
zero linear part. (d)'s graph is #1 and (e)'s graph is #2
What I don't understand is - what value/idea/notion determines #1 or
#2 graph to be slanted (in two different positions -  /    or      
#1 is like this /
and
#2 is like 
Note: Both #1 and #2 are stable clockwise.
===
Subject: Phase Plane - when the linear part of the eigenvalue is Zero
In this picture : http://img411.imageshack.us/img411/9598/21576104mq1.png
you can see that eigenvalues of the matrix of (d) and (e) has 0 or
zero linear part. (d)'s graph is #1 and (e)'s graph is #2
What I don't understand is - what value/idea/notion determines #1 or
#2 graph to be slanted (in two different positions -  /    or      
#1 is like this /
and
#2 is like 
Note: Both #1 and #2 are stable clockwise.
===
Subject: Re: Hundreds of solutions manuals(18/3)
hey can you please send me the solutions manual for Can you please
send me the solutions manual pdf file for Material science and
engineering an Introduction 6E callister  and Fundamentals of
Engineering Thermodynamics (M. J. Moran & H. N.
===
Subject: Re: Big O equation
|
which
|
|One can easily get through an undergraduate mathematics education
|without ever seeing big-oh notation. Computer Science students, on
|the other hand, might already see it in first year.
Indeed, I don't think I ever saw it in a course. I saw it
first either in Hardy and Wright's number theory book
or in a discussion of NP completeness. I only really
used it in graduate school. It gets used in certain kinds
of computer science, number theory, and real analysis.
Keith Ramsay
===
Subject: Re: Big O equation
The little-o notation is introduced in chapter 7 of Apostol's
Calculus, Volume 1 2nd Edition.
===
Subject: Re: Elliptic Curve
Char 3 is excluded here because you cannot always bring a general cubic 
into this form in char 3. Non-singularity calculation probably works out
all right for this subset of cubics, but the result is not as universal 
as in the case 6neq 0.
Jyrki
===
Subject: Re: How to solve this regression problem?
Right, i'll type it out, thank you for your suggestion!
===
Subject: Re: Coloring problem
I believe he meant they must differ by at least 1 in each coordinate.
Are you looking for an n x n Latin square?  You can put the n^2 points into
n sets of n elements each, with no two points in any set sharing the same
value in either coordinate as any other member of the set.  But it's not
random; it's a specific construction.
-- 
Randy Hudson
===
Subject: Correct propagation of MC errors
lately some colleagues and I had a lot of discussions about the following
problem and therefore, I'd be very interested to hear your opinions about
it. In principle the problem set is the following:
We have a set of values organized in a 3D matrix that represent the spatial
distribution of values that were calculated by Monte Carlo simulations. 
This
matrix is basically the representation of a 3D mesh binning and as we're
dealing with MC simulations, it naturally consists of average values and
their associated errors (sigma/sqrt(N)). Sometimes the mesh binning is
actually too fine for some purposes and thus, the task would be to re-mesh
the 3D matrix with a lower resolution by merging a specified number of
individual bins into one respectively larger bin. For example a 100x100x100
matrix could become a 25x25x25 matrix by merging 4 bins into 1 single bin.
For this one would simply calculate the average value, which is the easy
part of the job. Now we're getting to the trickier part which we've been
discussing and this is the correct propagation of the error.
For the new low-res matrix one needs to calculate the error for the
individual bins and the question is on how to do this correctly. Basically
one could ignore that the original values already had errors and calculate
the standard deviation of the mean for each now enlarged bin. However, IMHO
this would be outright wrong as one cannot simply neglect that the original
values had associated errors. To my mind one has to take the error (or
better put the deviation) into account that is due to the averaging during
the re-meshing process, as well as the original errors of the individual
small bins.
In the original data neighboring bins have a low correlation due to the
underlying physical processes of the simulation, however, this information
cannot be retrieved. Thus, one cannot determine a covariance matrix for the
error propagation and as the best way out I'd consider these values in a
first approximation as uncorrelated. Consequently, the final error would be
determined by the quadratic sum of the sigmas of the individual bins that
were re-meshed as well as the sigma due to the averaging during the
re-meshing.
I'd be very interested to hear your opinion on this or which way you would
propose to calculate the error propagation for this case.
Chris
===
Subject: EigenGap?
I have heard of the term EigenGap of a matrix, but I don't really know
what is or how to calculate it. I would be extremely grateful if
somebody could explain this to me or point me in the right direction,
Adam
===
Subject: Re: EigenGap?
Sorry I should also add, that I want to use EigenGap to calculate the
ideal number of segments as part of a spectural clustering procedure.
So what I would like to know if what id EigenGap, how to calculate it,
and how to use this to find the ideal number of clusters for use in a
spectural clustering algorithm.
===
Subject: LOVE POEMS & QUOTES
Love Poems, a great way to express your feeling whenever you feel so
splendid beautiful feeling at falling in love, heartbreaked,
disappointed, or at any time you just want to say I LOVE YOU to
someone you love in an amazing-different way.
http://lovepoemsfree.blogspot.com/
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Well, notice the typo of consistent instead of sound or w-
consistent
Read: Rosser
Don't some people say we don't know if PA is consistent (wrongly)?
(Not that I necessarily think that you might know anything about
that.)
And which systems are/are not w-consistent?
C-B
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Yes.
The observation applies in case of omega-consistency, arithmetical
soundness, soundness and so on: in case of some theories to which the first
incompleteness theorem applies we have no idea whether they're
omega-consistent, sound for arithmetical statements, sound, etc.
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Variable n is a skolem constant in Godel 1931, Section 2.6: This
yields: forall(17,r) is not k-PROVABLE because if that were the case,
there would (by (7)) exist an n such that proofFork(n,forall(17,r)).
By (16) we would hence have provablek(not(subst(r,17,number(n)))),
while on the other hand the k-PROVABILITY of forall(17,r) also implies
that of substr(r,17,number(n)).  Therefore k would be inconsistent and
in particular w-inconsistent.
See http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf
On Formally Undecidable Propositions of Principia Mathematica and
Related Systems I page 16.
C-B
In CBL, formal proofs in Recursion Theory are full of skolem
constants.  There is a program that outputs itself., There is a
program that is equivalent to a given recursive function applied to
it. etc. can all be formalized as skloem constants.  For example M
in:
    Thm. There is a program that outputs itself.
1. s11(I,J) : Substitution is recursive
2. s11(I,I) : Substitute I for J in 1.
3. M # s11(I,I) : Let M be a program that computes s11(I,I).
    (M is a skloem constant.)
f(N)
5. N # N : Substitute N for s11(M,M) in 4.
   qed
[Notice how I give not only the reference but also:
1. A link to an on-line copy.
2. Quote the passage.
3. The page and section number.
4. A formalization in CBL.
Now compare that to the Conservatives who give only the reference and
claim that something is in there somewhere.  And if you ask where they
say Everywhere or Do your own homework!  See e.g. Zenith.]
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Response to Jonathan's part here: I don't think flames though are
good at all, rather I think that it is best to discuss things in a
civil
manner. If someone does not want to discuss civilly, then don't
discuss.
Even if CB cannot decieve, there is first of all a judgment call
here that he intends to decieve (at least that's how I'm interpreting
this), and that he just wants attention, both of which are
assumptions that may be wrong. The arguments are what take
center stage in a real debate, and the person presenting it goes
off to the side. Argument vs argument, simple, no?
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
But wait: why is flaming is good a default position, despite
the fact that flaming can be seen to created hatred, bad
blood, hurt feelings, pure anger, and turn an otherwise
civilized discussion into a brawl? Are you saying those
type of things are good too?
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
On Sun, 22 Apr 2007 05:01:23 -0500, David C. Ullrich
Not so!
It is ubiquitous mathematical practice to identify a property with a
set, namely with set of all objects that possess the property [...].
(Paul R. Halmos, Naive Set Theory)
F.
-- 
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Um. Sometimes you have to read more than one sentence in order
to see what point is being made. Here all you need to do is
read the very _next_ sentence. What I said was this:
'And your suggestion that we define property to be 
synonymous with set is utterly stupid. You _said_
this was supposed to be about ZFC - in ZFC property
is _not_ synonymous with set.'
What it says in Halmos is irrelevant here since Halmos
is not a book about ZFC.
************************
David C. Ullrich
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
On Tue, 24 Apr 2007 04:48:04 -0500, David C. Ullrich
Nope. It _is_ a book about ZFC, _despite_ its (rather misleading)
title.
F.
-- 
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
        ZFC-Foundation
to be precise.
F.
-- 
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Or point out that Frege's quote only says that all properties are sets
and not viser-verser!
What does the following exchange mean?
I am referring to the purported proof that na.95ve set theory is
inconsistent.
Yeah, it's only about ubiquitous mathematical practice - so there,
Frege! :-p
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Nope. David C. Ullrich is in error here. Actually, it _is_ a book
about ZFC (-Foundation), _despite_ its (rather misleading) title.
F.
-- 
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Does x=x describe a property of x?  Is there a ZFC set of all objects
possessing this property?
-- 
===
Subject: Re: ZFC is Consistent - No Axiom Infers the Negation of Any Axiom
Typo corrected.
F.
-- 
===
Subject: Need Solutions ASAP
Do you have the solutions for Fundamentals of Financial Management,
concise 5th edition by Brigham & Houston?
===
Subject: Re: Comprehensive Solution Manual for Textbooks
Do you have the solutions for Fundamentals of Financial Management,
concise 5th edition, by Brigham & Houston? I need it ASAP please email
===
Subject: Re: Is language a mess?
I don't know why I didn't see this yesterday.
That's called language acquisition.
If you want to call it adaptation, then I suppose adaptation means
something different in mathematics than it does in English or
biology.
Who would suggest such idiocy?
Howcome you can NEVER provide a reference for the idiotic statements
you so disingenuouosly attribute to the passive voice?
===
Subject: Re: Is language a mess?
        
        
The general writing parameters are flexible: draw a picture, write 
something (this varies from scribbles to labeling to conventional writing, 
depending on each child's development) ...
from http://www.indiana.edu/~reading/ieo/digests/d63.html
Reading and Writing in a Kindergarten Classroom
Prepared by: Bobbi Fisher
ERIC Clearinghouse on Reading, English, and Communication Digest #63
To foster emergent reading and writing in particular, whole language 
teachers attempt to replicate the strategies parents use successfully to 
stimulate the acquisition of language and the 'natural' acquisition of 
literacy  (Weaver, 1990 p. 23).
Practice. Portsmouth, NH: Heinemann. ED 326 847
Adults cannot actually teach children how to read and write, though they 
can demonstrate or model reading and writing for them, collaborate with 
them,  and discuss reading and writing strategies with them, and guide them 

in reading and writing.  In all these ways, adults facilitate children's 
developing ability to read and write.  But they cannot effectively teach 
children to read and write, any more than they can effectively teach babies 

and toddlers the rules for putting togethers sounds to make words, and words 

to make sentences.
(Constance Weaver.  1994 )
Unfortunately, I can't find the exact book in which Weaver made that 
statement.
However, I did find a list of Weaver's publications:
http://www.wmich.edu/english/fac/Weaver.html 
===
Subject: Re: Is language a mess?
You couldn't have made Peter's point better if you'd tried.
-- 
Richard Herring
===
Subject: Re: Is language a mess?
Similarly may introduce a valid argument in mathematics, because there 
you can systematically identify the points of similarity. Elsewhere it's 
often an invitation to the lazy reader to abandon critical thinking. 
What, other than the superficial, is similar about these processes, 
assuming that they occur at all?
Similarly (!) it has been suggested is often a device for tentatively 
introducing a claim you aren't prepared to support when challenged. Who 
suggested it, and why (absent a reasoned argument from evidence) should 
we take any notice of them?
-- 
Richard Herring
===
Subject: Re: Is language a mess?
The early stages in writing may well have been on more
perishable materials.  Sumer used clay tablets for its
early writing, and many of these were baked in fires.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Is language a mess?
Have you never even seen a picture of Mesopotamia? What perishable
material, praytell, do you suppose they used? Why on earth would they
have hit on indentations in lumps of clay if something more convenient
and portable had readily been to hand?
===
Subject: Re: Is language a mess?
In the marshes of southern Mesopotamia, and the adjacent
part of Iran in those days, clay was widely available.
Also, clay was an easily erasable and reusable medium.
Recent studies show that little of it was deliberately
baked, but just got caught in fires.
What else did they have which was more convenient and
portable?  One could prepare the reeds, but this would
be less convenient.  They MAY have used them as well,
but there does not seem to be any evidence of that.
-- 
This address is for information only.  I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu         Phone: (765)494-6054   FAX: (765)494-0558
===
Subject: Re: Is language a mess?
So you KNOW that you were lying when you said they could have been
using something other than clay of which no trace has survived.
===
Subject: Re: Fine Structure Constant
hopefully the link below is valid:
http://unifiedphysics.com/Analytic%20Estimate%20of%20the%20Mass%20of%20the%2

0Photon.pdf
===
Subject: Help with formal definition
Cc:  helltone@gmail.com
Hi all!
I need to write a formal definition what I call a signal, which is
in fact a (finite) list or sequence of booleans. For example, s =
true, false, true, true, ... is a signal.
I've never done such kind of thing... so the skeleton of the
definition or even an example of something that resembles would be
surely welcome.
How can I do it? A friend told me this has something to do with
regular expressions.
Additionally I would also want to be able to write s(n), to get the
nth-element.
Helltone
===
Subject: Re: Help with formal definition
Since your context is not clear, does this help?
1.  http://en.wikipedia.org/wiki/Boolean_algebra
2.  http://en.wikipedia.org/wiki/Tautology_(logic)
3.  http://mathworld.wolfram.com/BooleanAlgebra.html
Maybe you can provide more context of what specifically you need an
answer to.
~A
===
Subject: Re: Help with formal definition
For what purpose? To use in what way?
I'm not sure exactly what your friend meant, but that suggests
to me something more like Backus-Naur Form (which is often used
to capture the syntax of computer languages) than a mathematical
definition.
http://en.wikipedia.org/wiki/Backus-Naur_form
Who or what is going to use these formal definitions?
                - Randy
===
Subject: Re: Help with formal definition
That's one way. EBNF would probably be my chosen method.
 
What's wrong with just saying 'define V_n to be the n-th element in 
signal V' or similar?
Phil
-- 
Home taping is killing big business profits. We left this side blank 
so you can help. -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
===
Subject: Re: Help with formal definition
You need to provide more context; it is not clear what you mean by a formal
definition. An ordinary mathematical definition would look something like
 A /signal/ is a finite sequence a_1, ..., a_n of members in {false, true}.
 Given a signal s = a_1, ..., a_n we denote a_i -- 0 < i <= n -- by s(i).
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: Help with formal definition
My try:
A (finite) sequence (list, tuple or totally ordered set) is a total
The sequence s = s(0); s(1); s(2); ...; s(n) has length |s| = n and
terms forall i, s(i) in S.
The empty sequence varepsilon (with n = 0) is an empty function
(whose domain is the empty set).
A signal is a sequence.
Would it be acceptable? Any suggestions?
Helltone
===
Subject: Re: Help with formal definition
Maybe I've been talking axioms and set theory too much, but my thoughts 
go something like this:
The length of the string is a natural number:
L e N
Every natural less than the length denotes a symbol in the string:
There are no other symbols:
Every symbol in the string is either 1 or 0:
Aatu, what do you think?
Tony Orlow
===
Subject: Re: Help with formal definition
It's a bit cumbersome but as a mathematical definition there's nothing 
wrong
with it, provided you specify the set S to be the set of Boolean truth
values. Whether it's appropriate for your purposes is another matter, and 
as
others have pointed out you need to explain what you need the definition 
for
if we are to say anything about that.
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: Fermat's Last theorem short proof
It is indeed FERMAT'S CURSE, upon mathematicians, that is STILL UNFELT, they 

can't see it, despite it is here in front of their eyes, and the eyes alone 

can't think 
MATHEMATICIANS- said-
But why the curse Fermat , we are the most innocent people, we only enjoy 
thinking that others can't, ACTUALLY WE ARE THE HEAD, and we are very happy 

with it, despite we are the most poor people on earth, beside we didn't 
change anything, every thing is the same since you left, and more over we 
generally stay in the tail,
 OTHERS are ONLY RESPONSIBLE for all the CRIMES you hear about, even the 
PROPHETS, PHILOSOPHERS, REFORMERS, POLITICIANS... didn't resolve the issue, 

DID THEY?
FERMAT-said-
  of course not, and really you are so innocent to the limit you stay in the 

TAIL and soon they will find you a more suitable place as an award, I hope 
do 
enjoy it,
  MY CURSE, yes, because you left your TASKS for the TAILS,
 You have prevented the QUEEN'S TRUE SONS,
 You have made the QUEEN FOR SALE,
 But the KING will come back with the STOLEN CROWN to retain the dignity of 

the QUEEN, and alas if it is too LATE, then you will be taken for another, 
but BIG LESSON, and the sun will RISE AGAIN
You are INVADERS of the queen's land
......
.......
************
B.Karzeddin
===
Subject: Re: Fermat's Last theorem short proof
AND HOW ARE THE provers. 
 and
===
Subject: Re: Fermat's Last theorem short proof
This very much looks like parts of my proof :), but I don't need Rad fn.
gcd (x+y, (x^n+y^n)/(x+y)) = gcd (x+y, n) if x and y are coprime and n 
x^n+y^n = gcd(x+y,n)^2 * (x+y)/gcd(x+y,n) * ((x^n+y^n)/(x+y))/gcd(x+y,n)
x^n+y^n is divisible by (x+y) and gcd(x+y,((x^n+y^n)/(x+y))/gcd(x+y,n))=1
For c^n to be a solution, c needs to be a multiple of (x+y), but (x+y)^n 
is already larger than x^n+y^n, so there cannot be a solution for c^n.
It is quite easy to prove that if x and y are not coprime, there cannot 
be a solution either, because the existance of a solution for the FLT 
equation with x and y not coprime implies the existance of a solution 
with x and y coprime and vice versa.
-- 
Vik
===
Subject: Re: Fermat's Last theorem short proof
The error is right here (the above line).
None of your above analysis concerning common factors justifies your
claim that c is a multiple of x+y. All you know is that c^n is a
multiple of x+y.
Let's take the easy case where x+y and (x^n+y^n)/(x+y) have no common
factor. Letting a=x+y and b=(x^n+y^n)/(x+y), you have
c^n=ab where a,b are coprime.
That does not imply that c is a multiple of a (unless a is prime).
So your proof fails.
quasi
===
Subject: Re: Fermat's Last theorem short proof
There's already an error above that. 
x = 1, y = 2, n = 27. 
gcd( x + y, (x^n + y^n) / ((x + y) gcd( x + y, n )) = 3.
-- 
===
Subject: Re: Fermat's Last theorem short proof
Threaten to cry if they refuse to accept the proof. If that doesn't work,
lock yourself up in your room, play HIM as loud as possible and draw
pictures of gothic fairies in your diary.
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: Fermat's Last theorem short proof
Note that you may need to hurry to be first because I am going to post 
here a simple seventeenth century math proof of FLT (only for odd 
powers of n), which will hopefully not contain an fallacy (but I'm 
confident it does not).
-- 
Vik
===
Subject: Pressure (force) from wind on a building. Help needed ASAP!
I am trying to work with an equation I have for force from wind on an
enclosed building. The equation is two parts, consisting of Qz which
relates the the gound structure to the building
qz = CONSTANT * CONSTANT * CONSTANT * CONSTANT * V^2 * CONSTANT
V being the velocity
You then have the force equation
P = Qz * CONSTANT*CONSTANT - Qh(CONSTANT)
Qh being the avg height of the building.
Could someone give me a little direction on how to for a differential
equation with this?
Any help would be great!
===
Subject: Re: Pressure (force) from wind on a building. Help needed ASAP!
In other words,
P = a * v^2 - b
where a and b are constants.
A differential equation for what in terms of what? It looks like
everything of interest is a constant.
                       - Randy
===
Subject: Re: Cantor Confusion
False is the implication that n nodes can separate more than n+1 path
bunches (where a path is a path bunch with one element).
===
Subject: Re: Cantor Confusion
 
False is the implication that such separtions are at all relevant to the 
issue of the cardinality of the set of paths i a CIBT.
The levels of a CIBT are indexed by N, and for every different subset of 
N there is a path different from all other paths, that path ranching 
left at just the levels indexed by that subset of N and branching right 
WM, being incapable of faulting this proof of the uncountability of the 
set of paths of any CIBT, always chooses always to ignore it.
===
Subject: Re: Cantor Confusion
I use bunch, Dik uses bundle. I see no difference.
Which sets do you have in mind?
That is nonsense. These uncountability proofs are wrong. Further, if
they were correct, I show a contradiction in set theory. This cannot
be rebutted by a different proof.
===
Subject: Re: Cantor Confusion
For example, the set of all baths that branch left at level 5 cannot be 
separated from the set of all paths which branch right at level 5 by any 
one node.
 
Saying that they are wrong does not make them wrong. You must show that 
some step in their proof is wrong in order to establish that they even 
can be wrong, much less that they are wrong
Here it is again: 
   (1) The set of levels in a CIBT is indexed by N.
   (2) For each subset of N there is a unique path which branches left 
      at all the levels indexed by that set and right elsewhere.
   (3) this establishes a bijection between the set of all paths in the
      CIBT and the set of all subsets of N
   (6) The set of all paths in a CIBT is uncountable, 
      meaning of cardinality greater than that of N.
Which step is wrong, WM? 
If None are, then WM is wrong.
  
Your alleged rebuttal does not occur in any set theory recognized by 
mathematics , nor in any set theory which does not presume the contrary 
a priori, nor in any set theory all of whose presumptions (axioms) have 
been clearly stated so that one can determine unambiguously what does or 
does not follow from those axioms.
Each such attempted proof has been validly rebutted by showing one or 
more flaws in that proof, frequently by different critics. 
Whereas WM has not validly rebutted any of the several proofs of 
uncountability, such as the one above.
Wm is in over his head.
===
Subject: Re: Cantor Confusion
Try again
Which of these statements is false?
    A path can be separated from all other paths by a set of nodes
    There are uncountably many sets of nodes
                  - William Hughes
===
Subject: Re: Cantor Confusion
 Try again
Which of these statements is false?
   A path can be separated from all other paths by a set of nodes
  There are uncountably many sets of nodes
                      - William Hughes
===
Subject: is circle a multi polygon
is circle a multi polygon any equation to prove it?
===
Subject: Re: is circle a multi polygon
it is a limit case, solved before Newton 
===
Subject: maths problem
three freinds go to shop with 10 bucks each and pay 30 bucks to the
shop attendant and when the owner comes to shop he says to attendant
that the price is only 25 bucks and gives the attendant 5 bucks to be
returned to the three people.the attendant keeps 2 bucks with him and
returns them one buck each.so each paid 9 bucks and shop attendant
kept 2 bucks where is the one buck
please explain
gopal
===
Subject: Re: maths problem
The following week, two other people went to the same shop and paid $15
each.  However, the price was only $25 and therefore the owner gave the
attendant $5 to refund to the customers.  The attendant gave each
customer $1 and kept the other $3 for himself.  So the customers paid $14
each and the attendant kept $3, making a total of $31.
Looks like the missing dollar has just reappeared.
-- 
===
Subject: Re: maths problem
three friends = 0
SA = $30
SO = 25
SA = 5
three friends = 0
SO = 25
SA = 2
three friends = $1 each
you forgot to devide the $2 by three, each friend paid $9 + $2/3
===
Subject: Re: maths problem
The three friends each paid 9 bucks, total 27.
The attendant has 2 bucks, the owner has 25 bucks, total 27.
Which one buck are you referring to?
===
Subject: Re: maths problem
Simple explanation - you're new to Usenet.
Phil
-- 
Home taping is killing big business profits. We left this side blank 
so you can help. -- Dead Kennedys, written upon the B-side of tapes of
/In God We Trust, Inc./.
===
Subject: Equivalent norms question
suppose || || is a norm in an infinite dimensional vector space V.
Suppose that 1 <= c < infinity is given. What requirements must || ||
fulfill for there to be an inner product norm ||| ||| over V with the
(equivalency) property:
1/c*||x|| <= |||x||| <= c*||x||, for all x in V  ?
-Kasimir Blomstedt
===
Subject: Re: Equivalent norms question
This is the problem of isomorphic characterization of Hilbert spaces.
The most famous result is due to Lindenstrauss and Tzafriri:
a Banach space is isomorphic to a Hilbert space
iff every closed subspace is complemented.
V. Anisiu
===
Subject: Using EigenGap To Calculate Number of Clusters in Spectral 
Clustering
I have heard of the term EigenGap of a matrix, but I don't really know
what is or how to calculate it. I also heard that it can be used to
determine the number of clusters as part of a spectral clustering
approach. I would be extremely grateful if
somebody could explain this to me or point me in the right direction,
Adam
===
Subject: ZF question
If I understand correctly the ZF universe consists of sets only.  Is
it, then, possible to prove using the axioms that any x is a set?
That is
Ax  (x=0)  v  Ey  (y e x)       (e means belongs to)
===
Subject: Re: ZF question
Yes.
-- 
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
 - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: ZF question
Yes.  Also:
===
Subject: Re: ZF question
equivalent to:
 (Ax)(~(Ey)(y e x) v (Ey)(y e x))
-- 
Puts his arm around you, fiddles with your hair.  You know, and he says, 
come
on, you know, just because you like a bit of a kiss and a cuddle with 
another
man doesn't make you gay.  Which, you know, I've thought a lot about.  But 
I
think it does.  I think it does.                --- The Office 
(interviews)
===
Subject: Calculating EigenGap For Use In A Spectral Clustering Algorithm
I have heard of the term EigenGap of a matrix, but I don't really know
what is or how to calculate it. I also heard that it can be used to
determine the number of clusters as part of a spectral clustering
approach. I would be extremely grateful if
somebody could explain this to me or point me in the right direction,
Adam
===
Subject: Mean Distance Between Failure (MDBF)
Just a minor problem if anyone can help. I need to convert MDBF given
in Miles to Kilometers. I understand that is not simply multiplying
through by 1.609. Does anyone know the relevant formula?
===
Subject: Re: Mean Distance Between Failure (MDBF)
Why would a distance change simply because of the
units used to measure it?
===
Subject: Re: Mean Distance Between Failure (MDBF)
yes, it is that simple. 
===
Subject: Re: No Order Preserving Bijection
I think Russell Easterly is claiming that set theory is inconsistent
and to that effect he argues that it can be shown that there is a
bijection between x and y and that there is no such bijection.
He is wrong, of course, and his mistake lies in his claim that M1 is
the only possible bijection between x and y.
===
Subject: Re: The most simple configuration for FLT proof  ???
 snip.
         I am afraid so well,
but I am trying to find some other parameters or any
other relation concerned with the first statement:
  let for n=3  X^3 + Y^3 = Z^3..................(1)  
for integers X;Y;Z
then we'll come to:
               x^3 + y^3 = z^3..................(2)
for x;y;z integers of gcd=1 where let x<y<z
and from these to:
   t^3 = 2 3^u abt + 3^(3u-1) a^3 + b^3.....eq.(3.1)
 3^(3u-1) t^3 = 2 3^u abt + a^3 + b^3.......eq.(3.2)
where 3;a;b;t are natural numbers of gcd=1 and u
also some natural number but at least u=2
  We used to take firstly eq.(1) as true and
had come to eq.(2) as a new true.
Next we can choose eq.(3.1) or eq.(3.2)as true or
both of them as true but must to remember,
that parameters from eq.(3.1) can not be
equivalent to parameters from eq.(3.2).
Now for eq.(3.1) we can seek separately
for new integer or natural values parameters:
  let t = r+p+q   
  3^u a = r+q  
      b = r+p   
     or  t = f+g+h
 3^(u-1) a = f+h
         b = f+q 
Then once we'd like to belive of some even single
set of a;b;t;u is true for eq(3.1) so also
we'd have some integer parameters r;p;q
and natural parameters f;g;h
  Why? Because from eq.(3.1) we can be only sure
  You can be astonish why so many other integer numbers
could be involved in eq.(3.1) or eq.(3.2) but this is
the case of some identity: whenever some equation
works for integers so it works for some identity also or 
at least for one set of integers. Such set of integers
could be express with infinite and properly constructed
parameters:   let our unique solution will be:
a=11;b=38;t=101;u=2
        t = 101 = r+p+q
    3^u a = 99 = r+q
        b = 38 = r+p
 then r = 99+38-101 = 36
      q = 101-38 = 63
      p = 101-99 = 2
     
        t = 101 = f+g+h
3^(u-1) a = 33 = f+h
        b = 38 = f+g
 then f = 38+33-101 = -30
      h = 101-38 = 63
      g = 101-33 = 68
 the pity is once inputting such a;b;t;u into eg(3.1):
Ls = t^3 = 101^3 = 1030301
Rs = 2*9*11*38*101 + 3^5 *11^3 + 38^3 =
     759924 + 323433 + 54872 = 1138229
so Rs - Ls = 107928
Of course such a=11; b=38; t=101; u=2 
but they are natural and complying to some
basic rules: t-b = 101 - 38 = 63  is divided by 3^2
once it should be only 3 ; 
3^(3u-1) a^3 + b^3 = 323433 + 54872 = 378305 
where 378305 = 5*29*2609 is at least composite but not
divided by t = 101
  Could we have more luck ? 
For sure not but I think some use of new 
integer parameters generated in any proper way
could help us ...
  Coming back to my topic there are yet some
questionable points but for example r;p;q
input and for q odd or only divided by 2 
pointed by me Eisenstein criterion is working.
  Anyhow thank you for Your
argumentation.
Ro-Bin
===
Subject: A game with arrows
You have 2n points and connect them all with arrows,
*except* all points i and 2n+1-i.
Rules:
- If i points to j then 2n+1-j must point to 2n+1-i.
- If i points to j and j to k, i must point to k.
If the missing arrows wouldn't be, you could
immediately say that you can renumber the points such
that i points to j iff i<j. (Or so I think!)
 
Is this the case also in this version?
-- 
Hauke Reddmann <:-EX8    fc3a501@uni-hamburg.de
I got a lack of passion since watching television and it hurts
I got a great invention and nobody pays attention and it hurts
===
Subject: derivative of integral - textbook proof
Folland, ADVANCED CALCULUS, Theorem 4.47
A proof seems to use (something more general than):
If f(x,y) continuous on (0,1) x [0,1] and for each fixed y,
f(x,y) is continuously differentiable as a function of x,
then partial f/partial x is continuous on (0,1) x [0,1].
Is that correct?
-- 
G. A. Edgar                              
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: derivative of integral - textbook proof
OK here is a counterexample,  f(x,y) = xy^3/(x^2+y^4)
this is continuous everywhere, in particular on (-1,1) x [0,1],
but the partial derivative f_x has f_x(0,y) = 1/y for y not 0,
so certainly f_x is not continuous at (0,0).
-- 
G. A. Edgar                              
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: derivative of integral - textbook proof
Another one is ysin(x/y).
===
Subject: Re: derivative of integral - textbook proof
??  It only had to be continuous on (0,1) x [0,1].  The origin isn't in 
that domain.  Also, note that your f(x,y) is not defined at the origin, 
so even the original function isn't continuous there.
Did you mean to say something else?
-- 
Chris Smith
===
Subject: Re: derivative of integral - textbook proof
It is OK, the problem was modified from 
V. Anisiu
===
Subject: Re: derivative of integral - textbook proof
In my example, I used (-1,1) x [0,1], and (0,0) is in that domain.
[You could change variables to arrange (0,1) x [0,1] if you wanted to,
with the problem at x=1/2, y=0.]
Add to the definition f(0,0) = 0, then the extended f is continuous
there.
I find on Folland's web site his errata for the book, and it includes a
stronger hypothesis for the theorem so that the partial derivative
f_x will be continuous.
-- 
G. A. Edgar                              
http://www.math.ohio-state.edu/~edgar/
===
Subject: Re: derivative of integral - textbook proof
The statement seems OK since you need continuity to have differentiability, 

but I could be overlooking something.
Dave 
===
Subject: guys wanna see BEAUTIFUL PICS OF BOLLYWOOD ACTERSS SEE IN THIS 
SITE
see pic of actress very very sexy
===
Subject: Positive Orientation
I was hoping someone could explain to me what is the postitive orientation 
of a tetrahedra and why it's important in triangulating a knot complement?
===
Subject: Re: Positive Orientation-
To fix thoughts, think of a regular tetrahedron lying on the floor with 
vertices nrs 1, 2, 
3, nr 1 pointing to direction 0 degrees (the North), nr 2 pointing to 
direction 240 
degrees (approximately SW) and nr 3 pointing to 120 degrees (approximately 
SE).
If seen from above, the turning sense 1-2-3-1 is counter-clockwise. A 
right-handed screw 
will come upward when turned counter-clockwise.
So we may make the convention that this tetrahedron is provided with a 
right-handed 
orientation by listing its vertices 1, 2, 3, 4 exactly in this order.
You will observe that all even permutations of 1, 2, 3, 4 correspond to the 

right-handed 
orientation, and that all odd permutations of 1, 2, 3, 4 to the left-handed 

orientation.
Remark: keep the tetrahedron in place and keep the vertex numbers fixed. 
Just observe what 
screw senses are induced by listing the vertices in different orders.
See Wikipedia: http://en.wikipedia.org/wiki/Orientation_%28mathematics%29
To answer your second question: the simplest of all non-trivial knots, the 
trefoil knot, 
comes in two different forms, which are each other's mirror images. So its 
complement also 
comes in a pair of mirror-image forms. That is why orientation comes into 
play.
See Wikipedia: http://en.wikipedia.org/wiki/Knot_theory
BTW, I would rather like to speak of left-handed and right-handed 
orientations that of 
positive and negative orientations.
Positive and negative suggest that even in mathematics one of the 
two would be 
privileged, which is not the case. One is tempted to think of the 
multiplication 
properties of positive and negative numbers. So I prefer left and 
'right: these terms 
have by themselves nothing to do with numbers.
Johan E. Mebius
===
Subject: Big Bertha Thing coil
Big Bertha Thing coil
Cosmic Ray Series
Possible Real World System Constructs
http://web.onetel.com/~tonylance/coil.html
Access page JPG 9K Image
Astrophysics net ring access site
Newsgroup Reviews including alt.politics
Two drawings of a Gyroscopic Top complete with coil. 
Extract from Introductory Chapter;-
The fact is that the toe is magnetic, and this being the case
it is easy to see that the rolling of the toe on the side
of the metal produces the motion. But Figs. XV. (a) and (b) 
illustrate a top, whose spindle behaves in exactly the same
manner as the toe of the spider top, and yet is in no way magnetic.
The action is purely mechanical, as we shall explain in later pages.
It is an attractive top to watch, especially as it rushes round
the corner when it comes to the end of the coil. 
An Elementary Treatment of the Theory of
Spinning Tops and Gyroscopic Motion.
By Harold Crabtree M.A.
Formerly Scholar of Pembroke College, Cambridge
Assistant Master at Charterhouse
Longmans, Green and Co. 1923
First Edition 1909
Second Edition 1914
New Impression 1923
(C) Copyright Tony Lance 1998
Distribute complete and free of charge to comply. 
Big Bertha Thing fiction
Anything but a fact, changes the face of twentieth century science.
 9. Gyroscopic Top motion along coil masks a weaker field effect.
The Japanese antigravity experiment only worked one way.
With the object spinning one way OK, but the other way failed.
Anything but a fact...
10. One way gravity reduced and other way gravity increased,
    so is bi-polar field efftect.
11. No antigravity, which is not bi-polar, so contradicts
    bi-polar field effect.
can be disrupted by;-
 1. Passive sensor near 2nd slit.
 2. Turning room light on.
Anything but a fact...
12. LCD digital watch, without backlight, near 2nd slit.
13. Thicker wall between slits.
Redshift is both speedometer and tape measure.
According to redshift speedo, the universe has a go faster
stripe painted on the outside.
Anything but a fact...
14. No Big Bang, which needs go faster stripe on the inside
or antigravity.
15. No redshift speedo due to Steady State, which needs
neither to have go faster stripe.
to give different results from same effect. Make the walls
inside and outside the slits the same size and centre of
gravity.
Anything but a fact...
16. Centre of gravity is at slit in one, but not the other.
17. Equal forces act on the slit in one, but not the other.
results from same effect.
Tony Lance
judemarie@bigberthathing.co.uk
NB.(2007) Addendum to Big Bertha Thing strategic (as deleted by Google) 
Full
documentation.
http://www.groupsrv.com/science/post-1881133.html
 max
Assume following scenario:
Example: if a function f(t) is defined:
 f(t)=a_1*cos(t)+a_2*sin(t)
then
 max(f(t))=sqrt(a_1^2+a_2^2)
and if f(t) is required to be such that f(t)<K, then:
 if [a_1,a_2] in R^2, then allowed [a_1,a_2] will be within circle centered 

at [0,0] with radius K.
Do we have anything similar for more complicated cases, e.g.
g(t)=a1*cos(T*t)+a2*sin(T*t)+a3*cos(3*T*t)+a4*sin(3*T*t)+a5*cos(12*T*t)+a6*s

in(12*T*t)
can I use {ai}, i=1,6 to find max(g(t))?
what would be the domain D in R^6 like for [ai] if g(t)<const?
===
Subject: Solution to the integral
I have been trying to solve an integral (basically a result from
Boltzmann-sigmoid fit) that is in this form. Can anybody help in
finding the solution. It looks like this -
 /
 |     exp(bt)
 | ---------------- dt
 |   1+exp(-at)
/
(or integral of [exp(at) dt / (1+exp(-bt))])
where a & b are constants.
===
Subject: Re: Solution to the integral
Mathematica 5.2 gives a result in terms of a hypergeometric function:
In[4]:=
Integrate[Exp[b*t]/(1 + Exp[(-a)*t]), t]
Out[4]=
(E^((a + b)*t)*Hypergeometric2F1[1, 1 + b/a, 2 + b/a, -E^(a*t)])/(a + b)
David
===
Subject: Re: What is this group?
I don't know if this observation has the slightest use for your
purposes, but it seems to me that we have Pauli matrices around.  a=0,
b=1 gives us sigma_1, a=0, b=-i gives us sigma_2.  On the other hand
a=i, b=0 is the matrix i*sigma_3.  As people on the thread have
pointed out, you need to add a condition to ensure that the matrices
are nonsingular if you want a group.
===
Subject: Re: What is this group?
Perhaps that you are talking about SU(1,1), but then you have to add the
hypothesis that |a|^2 - |b|^2 = 1.
Jose Carlos Santos
===
Subject: Re: What is this group?
I definitely cannot do that. It's extremely important to the physics
that the determinant is not 1.
===
Subject: Re: What is this group?
Well, add the hypothesis that |a|^2 - |b|^2 is non-zero (which you
must, if you want to get a group under multiplication); then you
presumably have U(1,1) (remove the S for savings!!!).
Lee Rudolph
===
Subject: Re: What is this group?
kind of unitary group over some kind of field?
===
Subject: Re: What is this group?
...
In general, if G(n) denotes a group of matrices preserving a
positive-definite bilinear form (e.g., if G is O then the matrices
are real and n by n, preserving the form with matrix I_n), then 
G(k,n-k) denotes the group of the same sort preserving a bilinear
form of signature k (or n-k, depending on your convention, which
might depend on whether you're a physicist or not).  In particular,
U(1,1) would be 2-by-2 complex matrices preserving the form with
matrix diag(1,-1), and (as usual) SU(1,1) would be the subgroup
of U(1,1) on which the determinant is 1.
Lee Rudolph
===
Subject: Re: What is this group?
Your matrices do not always have an inverse.  For example,
[a  a  ]
[a* a* ]
does not have an inverse.  Therefore your matrices do not form a
group.  SU(2), which is isomorphic to quaternions, has the form
[ a   b ]
[-b*  a*]
and always has an inverse except for a = b = 0.
- MO
===
Subject: Re: What is this group?
They have an inverse as long as |a| != |b|
And if you had taken a second to check, then you would have discovered
that this is a property that is preserved under multiplication.
It is a group!
===
Subject: Re: What is this group?
Wait... you are saying that he should have taken a second look for
something that YOU DID NOT SPECIFY?
How exactly was he supposed to do that?
Under extra conditions, sure. That was pointed out by several people. 
Were we expected to read your mind?
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Re: What is this group?
            days. My association with the Department is that of an alumnus.
Well, the original matrices do form a group under matrix addition, but
in that case they are simply isomorphic to C x C, which is doubtless
not what the original poster had in mind... 
-- 
It's not denial. I'm just very selective about
 what I accept as reality.
    --- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Why am i obsessed with random numbers?
i invented this simple program that randomly generates 0 or 1 with
equal probability. i have to guess a number between 0..1, for example,
0.6, to predict this number. then the covariance between my guess and
the randomly generated data is calculated. it turns out that i cannot
get the covariance over 0.1, which is statistcally right. but i cannot
for a moment stop to think that i cannot do it. why is that? it's only
one idea fails, i go with another. i also sometimes have two guess,
like, i knew it all along. can someone explain to me, why this happens?
===
Subject: 8-bits mantissa and 5 bits exponent
I did a floating point addition using 8-bits mantissa and 5 bits exponent. 
The answer I obtain for: 
1) 4.5 + 5.0 
01001100 , 00100 
2) -12.25 - 10.5 
10100101, 00001 
3) 7.25 - 14.0 
00001101, 00001 
4) 0.5 + 0.375 
00110000, 00001 
5) 0.2 + 0.0625 
00001100, 00001 
Is my solution above correct.
===
Subject: Re: Never never to late
No sane person, falsly  advised, sit down write for serious 'books',
when they have found the super-conductor formular, no one godfearing.
At least I never did.
When fate is involved you really don't mistake anything, you only do
perfect. If anyone  says different, they are faithful black liars.
 If I become and old man, (take it easy, I naturally don't do after
this)  perhaps I shall concider anyway writing a book?
Love you
Marathon man
===
Subject: Why am i obsessed with random numbers?
I dont compromise when i am attacked. I will make sure that the other
person, initiating the attack or even trying to plan an attack feels
the pain, physical or emotional. However, i can bought, ie, either for
money or for some reward, even a smile for that matter. But plently of
people have trouble understanding it. I can be very complacement when
i am paid to act that way. A lot of people dont understand that. I may
be, in the wrong country, where people have different attitude. How do
i explain that to people?
===
Subject: Why am i obsessed with random numbers?
I dont compromise when i am attacked. I will make sure that the other
person, initiating the attack or even trying to plan an attack feels
the pain, physical or emotional. However, i can bought, ie, either for
money or for some reward, even a smile for that matter. But plently of
people have trouble understanding it. I can be very complacement when
i am paid to act that way. A lot of people dont understand that. I may
be, in the wrong country, where people have different attitude. How do
i explain that to people?
===
Subject: Re: Why am i obsessed with random numbers?
You tell them your mommy is an asshole and that you have distorted thinking 

and want to go back to your homeland. 
===
Subject: fundamentals of thermo
i want solutions to fundamentals of thermodynamics (Moran, Shapiro
6th). please email me at dsu.is.cool@gmail.com
===
Subject: Re: nonlinear, nonhomogenaus ODE
Here's one special case: for f = constant the general solution
can be written as y = -6/a * WeierstrassP(x+C1, f*a/3, C2) where C1 and 
C2 are arbitrary constants.  
In some cases the solution may blow up, going to infinity at a finite
value of x.  Thus for example the solution above has a singularity at 
x=-C1, since (according to Maple) 
WeierstrassP(x,b,c) = x^(-2) + b x^2/20 + c x^4/28 + O(x^6) 
Phase-plane analysis may help determine whether a solution will blow
up or not.  Thus if f and a are positive constants, there is a homoclinic
orbit through the saddle-point equilibrium [y = - sqrt(f/a), dy/dx = 0],
and solutions inside this orbit are bounded and periodic while those 
outside
will blow up.
As far as numerics are concerned, I would expect the standard numerical
ODE solvers of, say, Maple, to work well, at least on the bounded 
solutions.
-- 
Robert Israel              israel@math.MyUniversitysInitials.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada
===
Subject: #11 published examples of mistaken proofs by various authors; 
monograph-book: Correcting the Logical Flaws of Euclid's Infinitude of 
Primes Proof
(# 14) --- quoting from the book MATHEMATICS AND LOGIC , Mark Kac and
Stanislaw M. Ulam 1968 , page 3 ---
The following proof, probably still the simplest, asserts the mere
existence of arbitrarily large primes. Suppose the number were finite;
there would then be a largest prime p. Consider now the number n = p!
+
1 (p! is read p factorial and equals 1x2x3x...xp). This number is
not
divisible by any prime up to p. If there is no prime between p and n;
as we have assumed, then n itself would be a prime, contrary to our
assumption that p is the largest one.
  This remarkably simple and elegant result of Euclid's is one of the
first known proofs by contradiction.
--- end quoting MATHEMATICS AND LOGIC , Kac and Ulam  ---
Kac and Ulam almost have a valid Indirect Method proof. They
reason that n itself would be prime after having conducted a prime-
factor
search. But the real reason that n is prime is because of the
definition of prime
which they missed inserting as their very first step of the proof. The
Indirect Method
pivots on the first step of the proof as the Definition-of-Prime. Kac
and Ulam almost
have a valid Indirect Method but miss it by a mistaken reason.
Secondly, if
they had studied Euclid's own proof, they may have seen that Euclid
did a Direct Method of set cardinality increase.
Again, the need for professors of mathematics to provide both a Direct
and Indirect Method whenever giving a Euclid IP proof is very much
needed.
(# 15) --- quoting Victor J. Katz, A HISTORY OF MATHEMATICS, 2nd
edition,
1998, p 87 ---
The final book on number theory is Book IX. Proposition 20 of
that book shows that there are infinitely many prime numbers.
Proposition IX-20  Prime numbers are more than any assigned
multitude of prime numbers.
In analogy to the proof of proposition VI-1, Euclid has no way
of writing down an arbitrary assigned multitude of primes.
Therefore, he again uses the method of generalizable example.
He picks just three primes, A, B, C, and shows that one can
always find an additional one. To do this, consider the number
N = ABC+1. If N is prime, a prime other than those given has
been found. If N is composite, then it is divisible by a
prime p. Euclid shows that p is distinct from the given primes
A, B, C, because none of these divides N. It follows again
that a new prime p has been found. Euclid presumably assumes
that his readers are convinced that a similar proof will work,
no matter how many primes are originally picked.
--- end quoting Victor J. Katz, A HISTORY OF MATHEMATICS, 2nd edition,
1998, p 87 ---
It is a relief that math professors have started to read
Archimedes Plutonium posts on Internet showing what is wrong and
flawed
in previous renditions given by math professors of Euclid's IP.
However, Mr. Katz did not understand my corrections fully and by
1998, I had not imposed a higher standard that professors provide
both Direct and Indirect concomitantly.
Katz's above is a Direct Method with terminology of
method of generalizable example is that of set cardinality increase.
But Katz does not stick or stay with set cardinality increase with
his last statement.
Katz makes a blunder. His last sentence of Euclid presumably
assumes that his readers are convinced that a similar proof will
work,
points to the fact that Katz believes Euclid gave an Indirect Method.
If Katz does a Direct Method of increasing set cardinality, then he
should
not made a remark of his last sentence which implies Euclid did a
Reductio ad Absurdum. The blunder is this sentence of
no matter how many primes are originally picked is the body of the
Direct Method. For Euclid realized that if any finite set yields a new
prime, then the argument is generalized or elevated to the infinite
set.
Whereas Katz is running a reductio ad absurdum on any finite set of
primes.
Here again, if Katz had been required to write out a full Direct and
also
a full Indirect method proof of Euclid IP, he may have ended up with
two crystal clear and valid proofs instead of one obfuscated and
murky attempt.
(# 16) --- quoting from MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN
TIMES,
1972, Morris Kline, page 80 ---
Proposition 20. Prime numbers are more than any assigned multitude of
prime numbers.
In other words, the number of primes is infinite. Euclid's proof of
this proposition is a classic. He supposes that there is just a finite
number of primes, p1,p2, . . .,pn. He then forms (p1xp2x . . .xpn) +1
and argues that if this new number is a prime, we have a
contradiction,
because this prime is larger than any of the n primes and so we would
have more than n primes. On the other hand, if this new number is
composite it must be divisible (exactly) by a prime. But this prime
divisor is not p1,p2, . . ., or pn because these leave a remainder of
1. Hence there must be some other prime; and again we have a
contradiction of the assumption that there are just the n primes
p1,p2,. . .,pn.
--- end quoting from MATHEMATICAL THOUGHT FROM ANCIENT TO MODERN
TIMES,
1972, Morris Kline, page 80 ---
Kline makes the usual two mistakes. First, Kline erroneously thought
Euclid's IP was indirect proof method. Secondly, Kline
goes ahead and gives an
invalid indirect proof of IP by hunting around for a prime factor.
In the Indirect Method there can be no hunting around for a new prime
because Kline's new number of multiply the lot add 1 is the only
prime candidate in that method and it is automatically and
necessarily prime because the first step of the proof is the
definition
of prime. When you combine the definition and the reductio ad absurdum
step, then the new number Kline forms is necessarily prime.
I do not know where mathematicians erroneously ascribed Euclid with
Indirect
Method. I believe from sketchy sources that it was Gauss
who led astray the mathematics community into believing
the Ancients used the Indirect method frequently when in fact
the Ancients seldom if ever used the method. Euclid may have used
reductio ad absurdum in some subcases of a proof but there is no
evidence that the ancient Greeks built an entire proof on the Indirect
Method:
And this is where perhaps Gauss enters the picture and leads many
astray.
Perhaps it was Gauss in his
century who most clearly used and espouses the mechanics
of the indirect method that would lead other mathematicians
astray.
Kline should have been clued by the fact that reductio ad
absurdum is a Roman term, not a Ancient Greek term, and
so should have investigated.
And although Kline failed to give an accurate
historical account of Euclid's IP and failed to even
give a correct proof of such a simple theorem. Well, I
would still buy his book.
And if Kline was required to have given both a direct and indirect
proof of Euclid
IP, he probably would have made two valid and clear proofs thereof.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
===
Subject: Re: #11 published examples of mistaken proofs by various authors; 
monograph-book: Correcting the Logical Flaws of Euclid's Infinitude of 
Primes Proof
This is a _very_ accurate exposition of Euclid's original approach.
He does. Obviously, you do not understand him correctly.
... is referring to the fact that the proof will not only work for
just _three_ primes (here A, B and C) but for ANY (finite) number of
(given) primes.
Katz: Euclid presumably assumes that his readers are convinced that a
similar proof will work, no matter how many primes are originally
picked.                  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
~~~~~~
It escapes me how a normal person can mis-take this rather clear
statement.
Especially given the preamble: Euclid has no way of writing down an
arbitrary assigned multitude of primes. Therefore, he again uses the
method of generalizable example. He picks just three primes, A, B, C,
                                          ~~~~~~~~~~
and shows that one can always find an additional one.
F.
===
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six members will be reimbursed for their participation as
a List Developer with one dollar each. Your name will move up the list
geometrically so that when your name reaches the #1
position you will be receiving thousands of dollars in CASH!!! What an
opportunity for only $6.00 ($1.00 for each of the first
six people listed above) Send it now, add your own name to the list
and you're in business!
DIRECTIONS FOR HOW TO POST TO NEWSGROUPS
Step 1) You do not need to re-type this entire letter to do your own
posting. Simply put your cursor at the beginning of this
letter and drag your cursor to the bottom of this letter, and select
'copy' from the edit menu. This will copy the entire letter into
the computer's memory.
Step 2) Open a blank 'notepad' file and place your cursor at the top
will paste a copy of the letter into notepad so that you can add your
name to the list.
Step 3) Save your new notepad file as a .txt file. If you want to do
your postings in different settings, you'll always have this file
to go back to.
Step 4) Use Netscape or Internet explorer and try searching for
various newsgroups (on-line forums, message boards, chat sites,
discussions.)
message by
highlighting the text of this letter and selecting paste from the edit
menu. Fill in the Subject, this will be the header that everyone
sees as they scroll through the list of postings in a particular
group, click the post message button. You're done with your first
one! Congratulations...THAT'S IT! All you have to do is jump to
different newsgroups and post away, after you get the hang
of it, it will take about 30 seconds for each newsgroup! **REMEMBER,
THE MORE NEWSGROUPS YOU POST IN,
THE
MORE CASH YOU WILL MAKE!! BUT YOU HAVE TO POST A MINIMUM OF 300**
That's it! You will begin receiving CASH from around the world within
days! You may eventually want to rent a P.O.Box due
to the large amount of mail you will receive. If you wish to stay
anonymous, you can invent a name to use, as long as the
postman will deliver it. **JUST MAKE SURE ALL THE ADDRESSES ARE
CORRECT. **
Now the WHY part: Out of 300 postings, say I receive only 7 replies (a
very low example). So then I made $7.00 with my
name at #6 on the letter. Now, each of the 7 persons who just sent me
$1.00 make the MINIMUM 300 postings, each with
my name at #5 and only 7 persons respond to each of the 7 original 7,
that is another $49.00 for me, now those 49 each make
300 MINIMUM posts with my name at #4 and only 7 replies each, I will
bring in an additional $343.00! Now, those 343
persons turn around and post the MINIMUM 300 with my name at #3 and
only receive 7 replies each, I will make an
additional $2401.00! OK, now here is the fun part, each of those 2401
persons post a MINIMUM 300 letters with my name
at #2 and they each only receive 7 replies, that just made me
$16,807.00!!! Those 16807 persons will all deliver this message
to 300 newsgroups with my name at #1 and if still 7 persons per 300
Newsgroups react I will receive $117,649.00! With an
When your name is no longer on the list, you just take the latest
posting in the newsgroups, and send out another $6.00 to
names on the list, putting your name at number 6 again. And start
posting again. The thing to remember is: do you realize that
thousands of people all over the world are joining the internet and
now!!
So, can you afford $6.00 and see if it really works?? I think
so...People have said, what if the plan is played out and no one
sends you the CASH? So what! What are the chances of that happening
when there are tons of new honest users and new
honest people who are joining the internet and newsgroups everyday and
are willing to give it a try? Estimates are at 20,000 to
50,000 new users, every day, with thousands of those joining the
actual internet.
Keep an eye on the newsgroups and when the cash has stopped coming
(that means your name is no longer on the list), you
just take the latest posting at the newsgroups, send another $6.00 to
the names stated on the list, make your corrections (put
your name at #6) and start posting again.
===
Subject: Help with cholesky factorization please
Hi I am currently writing a matlab program which will recursively
calculate the cholesky factor of a positive definite matrix as part of
a revision assignment. Unfortunately I cannot get into contact with my
lecturer for guidance as he is on holiday. The program is below.
Objective is to fill in XXXX and YYYY
function [l ] = recchol(a)
%
n = max(size(a));
if n==1,
    l = sqrt(XXX);
    return
end
%
v=2:n;
l=zeros(n);
l(1,1) = sqrt(a(1,1));
h      = a(v,1)/l(1,1);
l(v,1) = h;
l(v,v) = recchol(YYY);
I am pretty sure the first XXXX is sqrt(a) since the cholesky
factor of a 1 x 1 pd matrix will be the sqrt of the only term.
Also for the recursive part, I assumed it would take the principal
submatrix as this would be reducing the problem as we now need to
calculate the 2nd column+ which we can represent as a submatrix which
will be added into the a(2,2) index. So I assumed the correct command
to use for YYYY would be a(2:n,2:n) subtracting some more terms using
common formulae for cholesky factor calculations.
===
Subject: About the Name of the Whitehead Group
Does anyone know how the name Whitehead group originated?  I know it was 
named for J. C. Whitehead, but does anyone know who first coined this term 
and how it got accepted?  
Does anyone know what Whitehead called this group?  Dr. Tom Chapman here at 

UK mentioned in our Simple Homotopy Theory class that he does not know.
Jonathan Groves
===
Subject: Number of Roots of Polynomials in R[x], R Has Zero Divisors
It is well-known that if p(x) is a polynomial of degree d in R[x] where R is 

an integral domain, then p(x) has at most d roots (in fact, exactly d roots 

if you count a root of multiplicity m as m roots).  In fact, this is still 
true if R is noncommutative, as long as R has no zero divisors.  
However, when we remove the no zero divisors assumption, then this result is 

no longer true.  For example, let R=Z/(6), and consider the polynomial 
p(x)=x^2+x.  Then p(x) has roots 0, 2, 3, and 5 (mod 6).  Thus, p(x) has 
degree 2 yet has four distinct roots.  
Does anyone know how to tell how many roots an arbitrary p(x) has when R has 

zero divisors?  Of course, if R is small, then it is not too much trouble to 

evaluate p(x) for all elements in R, but this is tedious if R is large, such 

as Z/(500), and cannot be done if R is infinite.  Has such a theorem been 
found?  I guess so, but I have never seen it before.    
Jonathan Groves
===
Subject: Norm equivalence
sorry if this is a repost of an earlier partial message, my editor
messed things up. Anyway, my question is as follows: What conditions
must a norm || || over an infinite vector space V satisfy so that
given a constant c (1 <= c < infinity) there is an inner product norm
||| ||| over the same space V with the property that
1/c*||x|| <= |||x||| <= c*||x|| for all x in V ?
If it matters, we can assume that (V, || ||) is a Banach space.
-Kasimir Blomstedt