mm-213
===
Subject: Re: MTL and MatLab
  I am trying to make Matrix Template Library>
(http://www.osl.iu.edu/research/mtl/) interact with MatLab
6.5. If> anybody has done it, please give me your comments on
what you tnk> about it. I would greatly appreciate it! ,>
IrynaI have used both. What do you mean by having MTL
Ôinteract' with Matlab? Do you want to call 
matlab functions
via the mex interface? Or do want to generate a C++ program
that is callable from matlab?I recently did a project where
results had to be read in Matlab, but the simulation wch
generates them is far too complex to run efficiently as a
Matlab program. I created a class for results that allowed
easy measurments from a simulation, and automates the writing
of a corresponding mat file. If ts is what you are trying to
do, let me know and I can send you the code (I have a version
on the web, but it needs to be updated).Regarding MTL... 
It's
been a long time since any updates came from MTL? In fact he
object oriented numerics websit (oon.org) is looking a bit out
of date. Is there a better reference? Has anytng to replace
blitz or MTL come out ?G.S. [ See
http://www.gotw.ca/resources/clcm.htm for info about ] [
comp.lang.c++.moderated. First time posters: Do ts!
===
Professor at the University of Montana.>Is the polynomial
2*cos(2^k*arccos(x/2)) irreducible for each >positive integer
k? If so, how is it proved?Is it a polynomial? In what?
===
useless? The real> problem is that ts is an *unmoderated*
newsgroup. All posters are> permitted. I have two other main
newsgroup interests. One of them is> christianity. In that
case, I avoid the unmoderated groups> at like a Pharisee
avoids a leper. They make sci.math look like> a candle next to
the space shuttle taking off. It takes a hardworking,> patient
and wise moderator to keeps christians behaving like...uh..>
christians. So in that sense, your half right. But my other
interest is homebrewing, What's your favorite style? I like 
to
brew IPA's. But, so far, my bestbrew is a lagered Bohemian
Pilsner. Smooooth. I'm having a littleproblem with masng ...
poor yield. Math + Homebrew = FUN.> and that group is
UNmoderated,> but the most pleasant and helpful and unßaming
group on all of USENET.> Every newbie who comes along asking
the same dumb question that's> been asked 17 trillion times 
is
welcomed with open arms, given> kind advice and we cheer that
another lost sheep has been converted> from the evils of
Budmilloors and demon megaswill. So your half > wrong. Half
the problem here is that it's unmoderated. The other half 
is>
that it's populated by overly-anal-retentive jerks who tnk
it's> going to be important to point out that I spelled 
you're
your > twice in the preceding paragraphs. There's sometng 
the
same about fanatic legalists in the religious> groups and anal
mathematicians, in that they tnk they've scored> sometng if
they catch you in an error. AHA! You've typed> slander when
you should have typed libel! So what? Therefore> I'm an 
idiot
and he's a savant and now I have to erase s> chalkboards for
m? Hardly. I'm going to have a homebrew and> hang out with 
the
cute ccks wle he re-catalogues s PowerRanger> Collector's
Cards. I _thought_ we had beaten these guys up sufficiently in
the gh> school locker room that they'd be quiet by now. All 
I
did was> explain that if we'd all place our duffel bags
perpendicularly> on the benches, there would be room for all
of us to get dressed> at once. I really don't tnk that was
sufficient reason for> them to put Nair in my bottle of Prell.
Maybe the real problem is that a guy can't e-mail a
===
irreducible for each >positive integer k? If so, how is it
proved? Is it a polynomial? In what? Irreducible over
===
polynomial ?>Is the polynomial 2*cos(2^k*arccos(x/2))
irreducible for each >positive integer k? If so, how is it
proved? Is it a polynomial? In what? Irreducible over what?>
in x, irreducible over QFor example,2*cos(2^4*arccos(x/2)) =
16 14 12 10 8 6 4 2 x - 16 x + 104 x - 352 x + 660 x - 672 x +
===
useless?>so I don't see what the problem is.Well, you do 
need
to be able to find the N key and the>I key. That can be tricky
at first, but you only have to>do it once per session; just
leave your fingers there.Actually that could be a hard task, 
to
locate those two keys.R'ght Ôow, for 
Ô'sta'ce, Ô 
ca''t seem to
===
anyone show me an example of a function (R=real numbers)
f:R-->R>such that for any a,b,c in R, there exists an x in R
===
irreducible polynomial ?Isn't it sometng like Chebychev
===
Assistant Professor at the University of Montana.>Is the
polynomial 2*cos(2^k*arccos(x/2)) irreducible for each
>>positive integer k? If so, how is it proved?>> Is it a
polynomial? In what? Irreducible over what?in x, irreducible
over QFor example,2*cos(2^4*arccos(x/2)) = 16 14 12 10 8 6 4 2
>x - 16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2is
irredicible.Never seen that before; but would it not be
possible to prove itirreducible by using Eisenstein's
Criterion? Looks like the leadingcoefficient is 1, the 
constant
coefficient is 2, and all the otherterms have even 
coefficient.
If the pattern holds for arbitrary k,then there you
===
raf@tiki-lounge.com (Ross A. Finlayson) said:>I'm still 
tnking
that f(x)=1 for irrational x and f(x)=0 for>rational x that f
is everywhere discontinuous. That much is correct.What that
describes is>that between each pair of any two rationals is at
least one>irrational and between any two irrationals is at
least one rational. That part is wrong; it describes no such
tng.?? Perhaps i'm missing sometng.Suppose that A and B are
two subsets of R, mutually disjoint andtheir union is R. 
Define
f to be 0 on A and 1 on B. If f is everywherediscontinuous,
doesn't that imply that between any two elements ofA there 
is
an element of B and vice versa?>What I propose is that given
any>rational that the value greater than it and less than any
other>greater is irrational, There is no such number, as
several different people have shown you.In non-standard
analysis, there might be, however.See Alain Robert's book
about NSA. Rather than beingirrational, it would be
non-standard, though.>Obviously enough then under ts axiom
Adding such an axiom to the standard axioms would yield an>
inconsistent axiom system, and all statements would be
provable. It> would not be an axiom system for the real
numbers.Actually, there is an axiomatic approach of NSA in
wcha few axioms are added to ZF(C), and in wch the above
suggestionmakes sense. The extra axioms are relatively
consistent w.r.t. ZF(C).Again, see Alain Robert's book on
NSA.The other poster might be interested in ts approach
===
Re: Length of Stock> What is the formula for length of stock
required to go around a circumference.>> Eg>> If I wanted to
go around a piece of pipe 3 1/2 in diameter, with 1/4 ßatbar,
how long should I cut the bar. Ts is considering the fact that
I can form it completely round>C = 2 * pi * r = pi * d, where C
= circumference of the circle,>pi = 3.14159..., r = radius, d =
diameter.The diameter of the neutral axis is to be used in ts
formula:http://arcve.metalformingmagazine.com/1999/12/
DieD.pdfAnd note that the outside diameter of a nominal
3.5-inch pipe is not3.5
inches:http://mdmetric.com/tech/pipe0010.htmHTHJoe
===
interesting & thoughtful observation but RE> glosses over one
presumably obvious consideration. let f(x) be the clause size
of a formula> with fewest variables to factor a x-bit> number.
as RE points out, the minimum-variable SAT formula to> factor a
bits each. note> sqrt(2^x) has approx x/2 bits) HOWEVER it is
likely assured that f(x), the> clause-to-bits relationsp,
grows exponentially.I wondered about ts.Generating the clauses
probably requires exponential effort,but the clause to variable
ratio can't be exponential.Given that we have 2N literals, 
the
maximum number of 3-clausesis O(2N^3). For a 100 variable
problem there are C(200,3)possible 3-clauses = 1313400 ~
===
SCP-fBZbrebBAZ-aI+05ErtLcGn0A1vAQ5e3xzn-PtM9iZUVRm18EYArturo
irreducible for each>positive integer k? If so, how is it
proved?ts is up to normalization the n-th (=2^k) chebyshev
polynomial (for
|x|<=2)http://mathworld.wolfram.com/
ChebyshevPolynomialoftheFirstKind.htmlits roots are related to
the n-th roots of unity, and the irreducibility to thespecial
value of p(n)hth (and hope it is not nonsense)klaus Is it a
===
polynomial
===
Subject: Re: Brownian motion approximationCan you spell out
how to do it? I can see that the general theoremfollows easily
if you can prove it where f(t) is a linear function(byscaling,
Markov property, etc.) but how do you show easily that f(t)is
approximated when f is linear? As I mentioned, I can handle
thecase f(t)=0 for all t, because then you can use the
reßectionprincipal. Is B_t - f(t) with f linear a BM with
drift(I have heardthe term before but don't really know what
it is)?> A wle back I posted a question about whether or not> P[sup_{0<=t<=1}|B_t - f(t)| < d] > 0 for all d > 0 and f(t)
continuous>> on [0,1] with f(0) = 0.> >> In other words, does
Brownian motion uniformly approximate any>> continuous
function(with f(0)=0) with positive probability? Someone>>
replied that it does, and ts follows from first proving it for
f(t)>> = 0 for all t and then applying the Cameron- Theorem. I
can do>> it for f(t)=0, but I don't seem to be able to 
find a
reference for the>> Cameron- Theorem, though it seems to be
related to Girsanov's>> Theorem, and maybe even follows from
it. Can someone give me some help>> or lead me to a
reference?> >I don't believe there is any 
significant
difference between the>Cameron--Girsanov theorem, the Girsanov
theorem and the Cameron->theorem. As I understand it the same
theorem was discovered independently>and is now attributed to
all three.> >Rather like the Green-Gauss-Ostrogradsky theorem.
> Ts does not need quoting any complicated theorems. Construct>
a polygonal function h such that |f - h| < d/3 on the
interval,> bound (from below) the probability that |B_t -
h(t)| < d/3 at> all vertices, and bound the probability that B
differs by d/3> from its polygon at those vertices. The Markov
nature of B and> the boundedness and continuity of f enable
===
Calculus QuestionCould someone help me to understand how to
find the minimum distancebetween a surface (say f1(x,y,z)=c1)
and a line (f2(x,y,z)=c2. ibelieve i should be using
===
me an example of a function (R=real numbers) f:R-->R>such that
for any a,b,c in R, there exists an x in R such that a<x<b>and
f(x)=c?Not so that it's continuous for the whole R, but it 
can
be continuouswitn the open interval ]a, b[:
f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]-oo when x=a, oo when
===
polysigned numbers Roger.Wch form of terplex matches my
construction?I do believe we have a subtle difference in our
maths. Certainly yoursis far more generalized. But I would
differ with (a,b,-a-b) beingzero-sized (or just plain zero in
my terms). I can't really handlesometng like (a,b,-a-b) in 
my
representation due to the negatives,but I would compensate for
the negative signs by adding the magnitudeof the sum of a and b
to each ordinate to yield ( Sum(2a,b),sum(a,2b), 0 ). Ts would
yield (in my three-signed representation) *2a * b - a - 2b,
not zero. I assume by zero size you mean equivalentto zero.>
In Terplex, {a,a,a} and {a,b,-a-b} are zero-sized. Multiplying
or> dividing by triples with these characteristics constrains
the result> to a sub-algebra in wch that size is always
zero.What is product( (a,a,a), (1,2,3))?I suppose it is: a +
2aJ + 3aJJ + aJ + 2aJJ + 3a + aJJ +2a +3aJ. = 6a + 6aJ +6aJJ.
{a,b,c} can be written a+ b'J 
+c'J'J, where ÔJ^3=1. a,b,c> are
in a field, and so can be real or complex, etc.; they have 
the>
signs from the field, such as the complex signs {+,i,-,-i} as
well as> the signs ÔJ & ÔJ'J. 
Terplex can also be written in
polar form,> {a+b+c,((a-b)^2+(b-c)^2+(c-a)^2)/2,
ArcTan[2a-b-c,Sqrt(3)(c-b)]},> where the second term is a
squared radius and the trd a polar angle.> On multiplication,
the first two terms multiply and the angles add. Also you have
stated that these terplex values fit in between the> reals and
the complex numbers. The construction I am using produces> the
complex numbers on the three signed stage.> snip You are
re-developing a 3-phase description of the complex plane ->
sometng that I taught to electrical engineering students in
1948! In> Terplex, 1, ÔJ, and ÔJ'J 
are orthogonal directions
that can be> projected onto the complex plane. Projecting them
destroys their> interesting properties.Ts seems to be another
important difference. But here is a conßict.If 1, ÔJ, and 
ÔJ'J
are indeed orthogonal then how can (a,a,a) be zero?If each
direction is unique then (a,a,a) is different than (b,b,b)
andeach have their own unique position. If you agree with ts
then wecertainly are in two different spaces. And it would
seem that ts istrue especially since the superposition with
the complex planedestroys none of the properties of my
system.> Algebras work over fields (e.g. real, complex,
quaternion, octonion)> that have their own signs. Terplex is
just one of the many> conservative algebras that introduce
another set of signs such as ÔJ,> wch are probably better
thought of as directions. Real & complex> numbers work well at
human-scale problems, but other systems are> more appropriate
to the quantum and cosmological scales. I fear that> your
approach does not break out of the complex
strait-jacket.Three-signed numbers are quite equivalent to
complex numbers. Butthere are gher signs wch lead to gher
dimensions. For examplefour-signed math yields a three
dimensional space. Ts four signedmath has a product unlike
anytng that I know of for traditionalRxRxR. Could you comment
on ts?> I don't understand the symbolic system above. They 
are
very long. the> Ô=1' part seems redundant. 
Perhaps sometng got
lost in the font> translation to my macne. I'm not seeing
anytng double struck. Are> the letters s, d, n, o, g, h, p, I,
J, Y, k, l, m important? The Ôd (etc) terms are written that
way to satisfy the Groups.Google> convention that material
should be printable on simple equipment,> wch precludes
double-struck letters. My choice of letters is:-> Ôd 12 dozen;
Ôn 9 nine; Ôo 8 octal; g'7 seventh 
letter> Ôh 6 hex; Ôp 5
penta; Ôi 4 standard nomenclature;> Ôj & 
Ôk 3 & 2 following
Ôi; Ôm &,n 2 minus and negative;> ÔY 
3 symmetry. The -1 terms
completed the definitions. I tried to> make the symbols easily
remembered; the result is messy but mnemonic. Roger
Beresford.Now I get your mnemonics. I probably won't get any
gher than #(fourth sign)for some time to come.I believe that
the closest we can get to an overlap of ourconstructions is
your primals as the field and your terplex algebra asthe
operators to approximate three-signed arithmetic. Why does
===
Conference Topics for DebateComments by Jack Sarfatti on
excerpts from:( to Gary Bekkum for bringing ts paper to my
attention.)ABSTRACTEach approach to the quantum-gravity
problem originates from expertise in one or another area
Jack,I can feel your intense interest to find the mechanism of
gravity and objects but are instead wave structures in a
quantum space. Our perception of their properties was
Ôschaumkommen' of the wave structures. 
(appearances.)I
disagree. I agree with the deBroglie-Bohm-Vigier pilot theory
that from information waves.IT FROM BITmatter cores /zpf < 0
that balance the centrifugal repulsion from quantized rotation
about their centers of mass and from the repulsive self
electric charge.the pilot wave information BIT landscape it is
rolling on in a generalized gradient ßow including the 
fiber
space connections or gauge potentials as in the Bohm-Aharonov
effect that is the Josephson effect in the macro-quantum case.
That is action without reaction for the micro-quantum
approximation with signal locality that applies to not apply
to complex macro-quantum systems. Einstein agreed, but nobody
worked it out.False. It's all worked out in Bohm and 
ley's The
Undivided Universe.Now, it has been worked out. see
QuantumMatter.com andSpaceandMotion.comThe results are
amazing. 1) All the natural laws are found as properties of
the wave structure of the electron.2) Everytng grows out of
only two principles wch are properties of one tng -
space.Awesome. Gravity is the simplest piece of cake. Take a
look. I would love to have your thoughts.I do not know what
you mean. Have you derived the equations for general
relativity from the information wave? That is precisely what I
have done for the giant vacuum pilot wave along with the 
unified
dark energy/matter local field.Any new proposal must be 
couched
in mathematical language and must in suitable limiting cases
yield the battle tested equation of theoretical physics such
asGuv = (8piG/c^4)TuvMaxwell's equationsetc.Otherwise it is
not legitimate physics IMHO.Also there must be contact with
experimental observations both in terms of prediction and
explanation as nicely presented in Deutsch's book The Fabric
of Reality for example in the chapters on proper methodology
in theoretical physics.Ditto for the excess verbal baggage of
less by C. Williams below on the nature of c in E = mc^2. The
hard core of what is bend ts can be found in the book by
Wheeler and Taylor's introductory text on 
Einstein's
relativity. The basic idea of geometrodynamics is the block
universe of 4 dimensions with time and space mixing together
in changes of perspective of uniformly moving observers in the
globally ßat case without gravity to begin with.The issue is
the invariance of the 4D line element ds under the relevant
groups of local frame transformations at a fixed spacetime
event P.Given a local frame of reference with coordinates
x,y,z, tds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2 = dx'^2 + 
dy'^2 +
dz'^2 - c^2dt'^2Here c must be invariant under 
the group as in
the Lorentz transformationsdx' = (1 - (v/c)^2)^-1/2[dx -
vdt]dt' = (1 - (v/c)^2)^-1/2[t - vx/c^2]dy' = 
dydz' = dzfor a
nonaccelerating frame sft at constant velocity v along the
common direction x parallel to x' of the two global inertial
frames S and S'.One cannot describe gravity ts way if one
insists on retarded causality of no teleological future causes
of past effects associated with the ideas of destiny, fate and
purpose.could use Newton's gravity with global special
relativity to produce the three classic tests of GR provided
one introduced the Wheeler-Feynman-Dirac-Hoyle-Narklikar trick
of advanced potentials in addition to retarded potentials. The
fact that Puthoff gets those tests as well with s variable
dielectric vacuum model is no great acevement either because
Einstein's classical geometrodynamics goes beyond those 
tests,
e.g. gravimagnetism and gravity waves and black holes.With
gravity one must use LOCALLY curved spacetime in wch at
spacetime point event Pds^2(P) = guv(P)dx^udx^vwith summation
convention u,v, = 0,1,2,3There are two LOCAL symmetry groups
here.The Poincare group is no good anymore. The translation
subgroup symmetry of the Poincare group is broken by the
locally variable Diff(4) curvature tensor that is the essence
of gravity. The local Lorentz group of invariant tipped light
cone structure is obeyed in the tangent fiber space attached 
to
P.The base space of the tangent bundle obeys the Diff(4) group
of LOCAL general coordinate tensor transformations xu' =
x^u'(x^u) that replaces the translation subgroup of the
globally ßat Poincare group of special relativity.Ts is all
for zero torsion of course.All LOCAL observables must be
tensors or spinors under both groups. A spinor is a square
root of a tensor.Einstein's equivalence principle is
mathematically represented by the tetrad map eu^a(P) from
locally ßat tangent space inertial coordinates a to locally
curved base space non-inertial coordinates u. The a -> a'
transformation is via the 6-parameter Lorentz group of special
relativity. The u -> u' transformation is via the 
4-parameter
Diff(4) group of general relativity.dx^u = eu^a(P)dx^anab =
ea^u(P)eb^v(P)guv(P)Local elimination of the non-tensor
connection field for parallel transport of tensors along world
line paths normally associated with Newtonian gravity
acceleration g for example. These are g-forces or inertial
forces from accelerating non-inertial frames like the surface
of the rotating Earth for example. They are not inhomogeneous
tidal variations in the g-force from the curvature tensor, wch
are never locally eliminated although they are here on Earth
very small of order(scale of measurement) 10^-13 in
centimeters.Where nab is the ßat space-time constant metric
tensor of special relativity and guv(P) is the locally
variable metric wch represents a real gravity field only when
the 4th rank curvature tensor of tidal forces does not vanish.
You can have a variable guv(P) without a real gravity field 
from
using a non-inertial local frame that is accelerating without
tidal forces. Ts is not physically of great interest however.A
LOCAL tensor that vanishes in one LOCAL frame vanishes in ALL
LOCAL frames at fixed point event P for ALL relevant symmetry
groups.If tuv = 0 then tab = 0 and vice versa.There is no such
tng as a gravity force anymore in ts non-Newtonian paradigm of
geometrodynamics. Newton's gravity equations are regained in
the limit of weak curvature and slow speeds compared to c. If
there that minimize their dynamical action in the given metric
field guv(P)Light rays move on null geodesics ds^2 = 0.The
invariance of the speed of light c in global special
relativity is replaced by the above remark!In general there
are gravimagnetic cross terms in the case of nonstationary
metrics and ts complicates what is meant by the speed of
light.For example, if there are no gravimagnetic cross terms
in a simple case with spherical polar coordinates for a
non-geodesic observer subject to spin 1 gauge forces likeds^2
= grr(P)dr^2 - gthetatheta(P)dtheta^2 + gpp(P)dp^2 -
c^2gtt(P)dt^2For a light ray we have0 = grr(P)dr^2 -
gthetatheta(P)r^2dtheta^2 + gpp(P)(rsin(theta))^2dp^2 -
c^2gtt(P)dt^2with the usual convention that the LOCAL metric
field guv(P) is a pure dimensionless number and r(P) is the
Schwarzscld curvature radial coordinate defined such that the
surface area surrounding the center in the static spherically
symmetric spacetime geometry has the Euclidean area 4pir(P)^2
.Consider the null radial geodesic, dtheta = dp = 00 =
grr(P)dr^2 - c^2gtt(P)dt^20 = dR^2 - c^2dT^2wheredR =
grr(P)^1/2drdT = gtt(P)^1/2dtdR and dT are physically measured
space and time intervals for the light ray using meter sticks
and clocks or radars by the non-geodesic observer for small
separations between two lightlike separated events P and P'.
Small means compared to the local radii of spacetime
curvature.For example in the static spherically symmetric
Schwarzscld vacuum metric solution ofRuv = 0for r >
2Gm/c^2gthetatheta = 1gpp = 1grr(P) = (1 - 2Gm/c^2r)^-1gtt(P)
= (1 - 2Gm/c^2r)The black hole event horizon is atgtt(P) = 0dR
= (1 - 2Gm/c^2r)^-1/2drBut the circumference C = 2pirThe change
in C for dr isdC = 2pidrThereforedC/dR = 2pi(1 -
2Gm/c^2r)^1/2--> 0 at the event horizon.The physical radius R
= integral dR is much larger compared to physical
circumference C as it would be in ßat 3D space. If one keeps R
fixed ~ h/mc ~ G*m/c^2, the micro-geon of 
Wheeler's Mass without
mass Charge without charge Spin without spin shrinks to a point
in gh resolution Heisenberg microscope scattering probes with r
~ h/p for momentum transfer p like SLAC deep inelastic electron
scattering off protons.Ts is a semi-classical theory without
quantum electrodynamic vacuum polarization zero point energy
density effects, however the latter were shown by Feynman and
Schwinger to obey the Poincare group in the absence of
gravity. The price for ts is the obscure renormalization, wch
may not be internally mathematically consistent although its
predictions are in remarkable agreement with experiments.
Indeed, the requirement of renormalization with a finite 
number
of fudge factors or epicycles :-) has been a very useful rule
of thumb. Directly micro-quantizing Einstein's general
relativity is not renormalizable, i.e. one needs an infinity 
of
epicycle fudge factors. That tells us we have asked the wrong
question. As A. Wheeler says:The Question is: What is The
Question?Resemblances of Wheeler's remark to 
Cantor's diagonal
argument and Godel's incompleteness theorem of 
self-referential
spontaneous self-organization are not accidental and random.
:-) Wheeler tnks of the universe as a self-excited circuit of
observer-participators.The velocity of light c in ordinary
non-gravitating vacuum with /zpf ~ 0 is directly measured by a
variety of techniques. It is an observable measurable property.
The velocity of light in a medium is c/n where n is the index
of refraction. The physical vacuum also has an index of
refraction n(vac) that is very close to 1 in most situations.
Ts small variation comes primarily from vacuum polarization
zero point ßuctuations of the off mass shell or virtual
electron-positron plasma electrically neutral ionized plasma
inside the vacuum. Ts is vacuum.One must be careful in how to
use both special and general relativity when polarization
effects in real on mass shell media are considered. In the
case of special relativity one should, for example, consult
the text books by Arnold Sommerfeld, Panofsky and Pllips,
Landau and Lifstz. A real on mass shell medium in a sense
spontaneously breaks translational symmetry on scales larger
than the unit cell of the lattice as described in more detail
by P. W. Anderson in s More is different series of papers
collected in A Career in Theoretical Physics (World 
Scientific)
When one includes all dynamical degrees of freedom including
those inside the unit cell of the lattice, global special
relativity is restored to the propagation of light in a real
medium in the usual situation where gravity tidal forces are
forwarded to appropriateparties (bcc's to you and Toby) to
stimulate wider discussion andunderstanding of the important
points you have raised towards re-evaluationand correction of
fundamental errors in scientific conceptualizations sincethe
key departure of Newton's work neutering science by s using
smathematicalizations wch did not incorporate properly s own
determinedreligious faith in the absolute nature of truth as
per s belief in God. Iedited you posts lightly for clearer
reading per common American Englishpunctuation etc., and
changed one word from proof to idea relating tothe Theosopcal
science quote so that your point was not mistaken as thatquote
you cited being some kind of empirical experimental proof of
theyour choice of proof over idea please expand on ts in your
next posttocorrect my misunderstanding and I will forward
appropriately with apologies.Your discussion properly
considered, by those to whom it was sent, shouldalso go a long
way towards restoring, or integrating, etcs into thescientific
discipline of thought since the only way out of
presentconundrums in science is to replace uncertainty about
uncertainty withcertainty about the absolute nature of truth
itself and let the cpsfall where they may in terms of how ts
change in principle modifiesscientific perception 
of the nature
of reality, the various theoryequations etc.I noticed in your
writeups that you reference C as the constant value of
thespeed of light in several places and then in one place you
seem toreference it as the velocity of light. I understand
that in s earlierworks, especially in German, Einstein used
the term velocity and thenover years he too began routinely
using speed apparently due to themathematical convention
dictating that by definition the value of thesquare of the
direction component of the square of a vector property
isdefined identically equal to one.Thus, in trying to
understand, for example, what is the true value of,
say,(10mph-North)^2, the square of the velocity of ten miles
per hour in anortherly direction, the simple (2+2=4) logic of
ordinary math is thrownout the window by ts mathematical
convention per ts dictum and thenormal logical value of the
square of ts vector quantity(100m^2/h^2-North^2) is defined 
for
all intents and purposes as equal to(100m^2/h^2), ie, like
saying because we cannot mentally grasp orunderstand what it
means (North)^2, we define it to be identical to one,ie, no
meaning at all.Wle ts may seem a trivial point for all
purposes witn the realm ofconsidering physical systems
dynamics from the point of view of currentscience since
apparently Galileo's time, ie, the exclusively 
objectivenature
of reality, ie, that observer and observed are separate and
allphysical systems that are real exist independenttly of and
identicallyobservable by all observers, or they are not
considered real by sciencewhen they are not reproducible by
all observers at all times, in the caseof C^2 as the
proportionality constant between the value of the Energy ofany
given system of reality under observation and the value of the
mass ofthat system, and since C itself appears in so many
equations ofelectrodynamics etc., herein apparently lies
another overlooked fundamentalmisconception of scientific
thought compromising the absolute nature oftruth since C
itself is not pegged to the notion of an externalobjective
reality reference frame but is pegged to the identity of
theobserver. That is to say, since C is non-additive and its
square (C^2) isthe proportionality constant between the values
of energy and mass of realphysical systems then ts mathematical
convention seems evidently themain limit of current mathematics
to overcome for a full understanding ofthe relative nature of
reality, ie, the relationsp depicted in Buddsm(and in nduism I
believe) that each person's mind is the creator ofthe 
universe
relative to the identity of that person, ie, the
ego-centricnature of the universe.Wle ts view seems rightly to
folks like Dr. Sarfatti as psycho-babbleit nevertheless is the
view that offers a mathematical handle for astarting point to
conceptualize and integrate into modern physical theoriests
precept that there is a consciousness factor at work between
thenature of the observer and the nature of the system of
reality underobservation, ie, a relationsp of mind between
observer and observedas well acknowledged by quantum physics
experiments over last decadeswhere he describes ts principle
as observersp.Many scientists are wrestling with ts notion of
the relationsp of mindbetween observer and observed, eg,
denoted in O'Leary's books asthe consciousness 
factor and in
Dr. Sarfatti's post quantum physicsof consciousness theory
equations by their complexities that includethe Uncertainty
Principle mathematics and depict the operation ofintent on the
mental field of matter causing a Ôback-action in 
time'reßecting
back via that mental field of matter in a 
Ôcybernetic
feedbackloop of consciousness' of predictible 
Ômoment of
consciousness' durationwch corellates well with latest
experimental results in neuroscience etceven though in
Sarfatti's view, in my words, the observer is transparentie,
there is no accounting for variations between observer
identitiesand corresponding variations in observable systems
dynamics fromobserver to observed (non-reproducibility by all
observers at alltimes of all physical systems dynamics, eg,
psi-phenomena), ie, thedifferences in each observer's 
Ômind'
impacting the physical systemsdynamics of the 
Ôreality' wch
each observer observes.Back-action is not temporal. It is the
direct reaction if IT back on its BIT field. 
IT's BIT field
quantum potential Q now has sources and is not fragile, but
has macro-quantum phase rigidity of wch Andrei Sakharov's
metric elasticity is an example. IT is no longer a test from
in spontaneous self-organization - the participatory universe
as a self-excited circuit. It is not to be confused with the
Wheeler-Feynman advanced potential from future to past. It is
true that when the limits of micro-quantum theory without
back-action hence with signal locality are transcended, then
one gets presponse signal nonlocality wch mimics an advanced
potential effect. Back-action in my sense as quantum action +
post-quantum reaction, and advanced potentials are, of course,
not incompatible in my theory in wch EPR micro-quantum
nonlocality with signal locality are as in Cramer's
transactional generalization of the Wheeler-Feynman classical
advanced potential as first noted by Costa de Beauregard back
in the 1970's. Antony Valentini shows that post-quantum
back-action that pushes the system away from sub-quantal heat
death causes signal nonlocality like what is seen by Dick
Bierman in s mind-matter presponse experiments. Macro-quantum
BEC systems in particular have post-quantum back-action
IMHO.The above is my latest attempt to explain in words what I
have renderedsince 1974 into low level mathematical equations
of a unified fieldtheory (referencing 
field of awareness and the
human mind'sconsciousness orientation function of light as
correction factor toadd back ts omitted true extra value of
the square of the directioncomponent of C^2 wch is more than
(north)^2 because of ts factthat C is non-additive and
therefore as such is fundamental, beingthe only physical not
Ôpegged' to the notion of the exclusively 
objectivenature of
reality but rather Ôpegged' to the 
conceptualization of
the'relative' (or 
Ôego-centric') nature of reality a
conceptualization wchis obvious in human daily life via common
sense in all other realms andis throughout the story of human
thought a fundamental principle asdiscussed in new age
psycho-babble in such terms as we are thecenter of our own
universe and by application of the human mindwe can inßuence
and change the nature of our universe.The above paragraph by
Williams is New Age cargo cult pseudoscience IMHO on a par
with crystal power, orgone, spiral dynamics, monoatomic ghly
deformed nuclei superconducting Ormus powder et-al. It is not
even wrong in Wolfgang Pauli's sense.Bruce DePalma is the 
only
person who has looked at the four equationsof my Tetron Natural
Unified Field Theory and the minimal discussionof their 
meaning
presented to m soon after I met m in May 1979,and s reaction
in subsequent soon talk with me was, , youknow, I understand
your theory. ÔSeeing is believing, right?' was 
sresponse with
a glimmer of humor bend s eyes that made meknow that he had t
the nail on the head with ts response wchzero's in on the
paradoxical relationsp of mind between seeingand believing as
it relates to all levels of human thought includingscience,
religion, spirituality, politics, etc.The above paragraph by
Williams is New Age cargo cult pseudoscience IMHO on a par
with crystal power, orgone, spiral dynamics, monoatomic ghly
deformed nuclei superconducting Ormus powder et-al. It is not
even wrong in Wolfgang Pauli's sense.Exclusive of Dr.
Sarfatti's vigorous ... criticisms,everyone else since has
either said so what? in response, or likeO'Leary have just
ignored and refused to respond to ts theoryof mind, with the
single exception of Dr. Fred Wood who listenedto my first-ever
public lecture on my theory, on September 10, 2001,at my alma
mater California State University at Northridge, and aftermade
a point of telling me that he would tnk seriously about whatI
had said (a prepared written formal read aloud lecture arcved
athttp://groups.yahoo.com/group/gcsc-csun/message/6
)Subsequently, I emailed Dr. Wood and suggested that the next
stepin the application of my tetron thesis is beyond the
mathematicsof my education as a Bachelor of Science in
Chemistry, ie, howto understand the (tensor?) mathematical
relationsp betweents correction factor Tetron -- the
mathematical function appliedto the square of the speed of
light to correct it to its true value ofthe square of the
velocity of light oriented relative to the identityof the
observer, ie, adding back the square of the vectorcomponent of
C to overcome above described mathematicalconvention, ie,
compromise about the nature of light and theabsolute nature of
truth itself in all places where C^2 appears inphysics
equations -- as Tetron applies to C^2, and since manyequations
particularly in electromagnetism contain C, there mustbe a
similar correction factor to apply in such equations wch
isalso the missing link in understanding the consciousness
factoras it relates to these new space energy technologies and
theexpected reconciliation of their presently conßicting
theories ofoperation.The above paragraph by Williams is New
Age cargo cult pseudoscience IMHO on a par with crystal power,
orgone, spiral dynamics, monoatomic ghly deformed nuclei
superconducting Ormus powder et-al. It is not even wrong in
Wolfgang Pauli's sense.Since your theoretical approach well
includes deep backgroundunderstanding on ts Eastern plosophy
concept of the mindas the creator of the universe per Vedic
knowledge you writeand feedback on my above views will help
stimulate productivetnking towards a deeper understanding of
ts consciousnessfactor as it also needs to be formalized in
order to correct themind of science and fully understand what
is going on with allthese various new space energy technology
experiments andthe theories posited to try and understand
these results as wellas how some of the theories (like yours
and Joseph Newman's)have predicted and perhaps even 
empowered
the successfulexperimental results that you have each obtained
by differentconfigurations of rotating magnetic systems with
entirely differenttheoretical underpinnings yet with similar
overunity results.Consciousness IMHO is when one has
post-quantum back-action wch excites the MACRO-QUANTUM BIT BEC
field into self-awareness.Our minds are macroscopic
non-classical information fields of both active and inactive
information in the Bohm-ley sense. Flashes of memory are the
activation of unoccupied basins of attraction of the
MACRO-QUANTUM BIT LANDSCAPE when the IT system point of
sub-microtubular hydrophobically caged electric dipole
Frohlich collective modes roll into that basin.In the torsion
field theory developed in recent decades from theRussian
language-mind views expressed in mathematicallanguage and with
deeper consideration of many documentedpsi-phenomena
experiments in former Soviet Union, goingback to the
1950's-60's apparently, there may also be ts 
ideaof a
relationsp of mind between observer and observed butthe
details of how ts is represented in ts theory are unclearto me
at my level of math education and literature availability.But
it is clear from my personal conversation with Dr.
Spovcourtesy of Dr. Sarfatti's kind invitation for my
informalparticipation with s group in San Francisco one day a
fewyears ago, that torsion field theory also is perplexed by
theresults of DePalma showing variation in gravitational
behaviorbetween spinning and non-spinning ball bearing drop
resultsdiscussed in earlier post.I do not believe DePalma's
claims. Gennady's beliefs in psi are not directly connected
with s torsion math. Bill Page talked about the latter at
Vigier IV.Creon Levit investigated such claims at ISSO
1999-2000 that Williams refers to here. Creon was not
impressed with any of the New Age Free Energy claims promoted
by . But Creon can speak for mself.My hunch here on ts is that
rotating objects too have avector property wch analogous to C
discussed above, isbeing overlooked in its importance as it
relates to the observerbecause every rotating system also may,
similar to C, be seenin the view of its orientation and
rotational properties as relatesto the relative identity of
the observer. I tnk that ts otherapplication of what I talk
about above will not come here untilts issue about C is
resolved, but that there is an importantconnection between the
corrected true value of C^2 as perabove and a deeper
understanding and correction also withtorsion field theory,
although the math involved is way beyondmy education level to
deal with as a language to express theprinciples I have tried
my best above to explain in my style ofCalifornia Chemical
English :-)The above paragraph by Williams is New Age cargo
cult pseudoscience IMHO on a par with crystal power, orgone,
spiral dynamics, monoatomic ghly deformed nuclei
superconducting Ormus powder et-al. It is not even wrong in
Wolfgang Pauli's sense. Crockett Williams
661-822-3309Chartered Life UnderwriterBachelor Science
Chemistrywww.angelfire.com/on/GEAR2000/vision.htmlhttp://
groups.yahoo.com/group/drums-of-peacehttp://groups.yahoo.com/
group/new-energy-solutionshttp://www.josephnewman.comRed Silk
Road Peace March ProjectUSA, California, Japan, Korea, Cna,
Nepal,India, Pakistan, Afghanistan, Iran, Iraq,
Jerusalem,http://groups.yahoo.com/group/silk-road-to-peacehttp
://groups.yahoo.com/group/dharma-walksOne Person Can Make a
Differencewww.kucinich.usRep. Dennis
Kucinichhttp://groups.yahoo.com/group/
Kucinich-for-PresidentSpiritualism, The ghest Form of
PoliticsFor a Culture of Peace & CommunityOur Spiritual Unity
Re-Awakeningwww.leonardpeltier.orgwww.horseforgovernor.comwww.p
rophecykeepers.comhttp://
web.mahatma.org.inwww.peaceinspace.comwww.cesarechavez.orgwww.b
rianwillson.comwww.dharmawalk.orgwww.sathyasai.orgwww.tewari.or
===
minute talk > Well, for (1), all you need is a series. 
ßy's
speed and that time, work out the distance. I expect most
people here have heard it, but I'll pass on the following>
anecdote:> Someone presented Feynman with that problem [bug
ßying between two> approacng vecles], and he of course solved
it very quickly. The other> guy said, would you believe some
people solve it with series! To wch> Feynman responded, 
what's
the other way? BOh Feynman not Neumann ;-) My apologies to both
===
minute talk> > Well, for (1), all you need is a series.> > 
ßy's
speed and that time, work out the distance. I expect most
people here have heard it, but I'll pass on the following>
anecdote:> Someone presented Feynman with that problem [bug
ßying between two> approacng vecles], and he of course solved
it very quickly. The other> guy said, would you believe some
people solve it with series! To wch> Feynman responded, 
what's
the other way? :-) Here (USA), it's almost always (i.e., 
every
time I've ever heard it) told> about Von Neumann mself: he
solves the problem very quickly, exclaims,> Ah! Yes, it is 150
miles! or whatever, and, when the curious onlookers> ask m how
he did it so quickly, he gives a blank look and replies,> I
summed the series. I never heard the algebraic approach
referred to as the Von Neumann> approach before. -ArthurI
meant the sum the series approach is the von Neumann
approach,only to be told that the clever fellow was Feynman
not Neumann. Now I'mutterly confused.Of course 
it's possible
that von Feynmann _didn't_ sum the series andwas pulling s
===
show me an example of a function (R=real numbers) f:R-->R>>such
that for any a,b,c in R, there exists an x in R such that
a<x<b>>and f(x)=c?Not so that it's continuous for the whole 
R,
but it can be continuous>witn the open interval ]a, b[:
f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]-oo when x=a, oo when
x=b and gets all values when between them.Another
example:g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a))f[x_]:=Cos[g[x]]/g
[x]Ts one is continous everywhere except at x=(a+b)/2, and
===
VTZr+6ZbZe8Nfgn2gDuLLEJcmTttvjg7po1SbWvfolpjE+5Ny7fWc1> Just
use a decent news-reader and ignore certain Subject-lines and>
authors.He does use a decent newsreader, and he could ignore
certain Subject-linesand authors with it. Joacm-- Trau niemals
===
GbkCIrZdZeB7XzMoZh-iQSILMHJEkbFUwr+XMlfqVd-92z37xFyVY3
Moderating ts group is not an option. The USENET Powers that
Be> (otherwise knows as the moderators of
news.announce.newgroups) have> decreed that proposals for
changing the moderation status of existing> groups will be not
be accepted. If you want a moderated version of> sci.math then
you'll have to propose a new newsgroup.But 
doesn't
sci.math.research exist? And isn't it moderated? So what 
isall
the fuss about?Joacm-- Trau niemals einem
===
couple questions. Ts is homework, so please post a nudge,not a
solution.1)prove that if f,g continuous, then so are max(f,g)
and min(f,g)After drawing some graphs, ts seems pretty obvious
for the singlepoint a0 -- max(f,g) has 2 cases: it equals to f
or g. Either iscontinuous. However, ts question implies
continuous on R, not justat a single point. Any ideas how to
approach ts?2)Let f be a function with the property that every
point ofdiscontinuity (ie the lim (x->a) f(x) exists, but is
not equal tof(x)) is a removeable discontinuity. Ts implies
lim (y->x)f(y)exists for all x, but f may be discontinuous at
some (even infinitelymany) numbers x.Define g(x) = 
lim (y->x)
f(x). Prove g is continuous.--I don't even know where to 
start
===
one point or another I read parts of these books: * Manfredo
P. do Carmo: Riemannian Geometry> a very nice reading. good
exercises> * R.W. Sharpe : Differential Geometry. Cartan's
Generalization of> Klein's Erlangen Program.> very broad
minded. not allways accurate.> * Kobayas and Numizu>
enciclopedic.> * J. Jost> the analytic view. very dense.> *
Spivak's pentalogy> surpsingly, some parts of it are 
actually
nice.Yes, now it's manufactured as a proper book, 
I'm saving
===
Exponentative closureCan the reals be defined using repeated
exponentative closure?By the exponentative closure F, I define
F/x as the set of all thezeroes of all the polynomial
functions with coeffeicients ANDexponents in F. For example,
the the algebraics are the exponentativeclosure of the
integers. Thus, it can be written A=Z/x. DoesC=A/x? If not
what does A/x equal? Can C be generated byrepeatedly
exponentatively closing the integers a finite number oftimes?
If so, how many? A countable number of times? An
===
question about circular sectors :)> When I run your numbers
backwards, I get L => 1.4142.> Using your first answer where
theta = 90 deg (or Pi[/2 you meant] radians) and radius = 1: L
= 2 (r * sin(theta/2)) = 1.4142> And, your second where theta =
180 deg (Pi radians) and radius = 1/sqrt(2) => 0.7071: L = 2 (r
* sin(theta/2)) = 1.4142Please see my above post again ; L is
===
radius of convergence for one series, I'm trying to 
find the
radius for 2 simliar seriesLet f(z) = sum_k a_k*z^k be a
formal power series with radius ofconvergence 1. Put s_n = a_0
+ ... + a_n and t_n = (s_0 + ... +s_n)/(n+1).Let g(z) = sum_k
s_k*z^k and h(z) = sum_k t_k*z^k.How would I show that the
radius of convergence of h and g are both 1as well? (The lim
===
question>Why would you tnk that? Rudin's texts are known for
their concinnity>and concision, not for detailed
proofs.>concinnity. What a wonderful word! I had never seen
that before. (To save others the trip, here is the first
definition from
http://dictionary.reference.com/search?q=concinnity: 1.
Harmony in the arrangement or interarrangement of parts with
respect to a whole. 2. Studied elegance and facility in style
of expression: He has what one character calls the gifts of
concinnity and concision, that deft swipe with a phrase that
can be so devastating in cldren (Elizabeth Ward). 3. An
instance of harmonious arrangement or studied elegance and
facility.)-- Stephen J. Herschkorn
===
calculation problem> a few weeks ago, i made an observation in
alt.sports.baseball.mn-twins about> what seemed to be a rarity:
all three teams in baseball's american league> clinched 
their
respective divisions on the same date. my exact remark was> i
bet ts has never happened before. several others agreed and
then tried to pin down the probability of ts> happening in any
given season. at ts point the discussion turned a heated>
argument over the probability issue, with widely disparate
answers--we're> talking answers not even in the same 
universe.
worlds apart. anyway, i'm posting ts problem here hoping that 
a
few knowledgeable folks> will be willing to take a crack at it.
wle it helps to have a knowledge> of how competition among
baseball clubs works mathematically, i don't> believe 
thorough
understanding of the sport is required. it seems to be> more of
a math issue than a baseball issue. here's the problem,
specifically: there are three divisions in the american 
league.
every season, each one of> these divisions will be clinched by
one team on a certain date. ts is a> mathematical certainty.
past baseball story tells us that dates before september are
extremely> rare for teams clincng. in fact, the later days of
september, roughly the> 15th through the 28th, are where the
likelihood of teams clincng is the> greatest.> An important
fact that you left out is how the divisions are clinched.There
are 5 teams in each division and 180 games per season
spreadover approximately 200 days. Typically all teams will
play or veryfew will play in a given day. A division is
clinched when one teamhas more wins the (180-losses) of any
other team in the division.It should also be considered that
certain teams have better chances ofwinning than others and
different teams are better against otherteams.If you ignore
that consideration it can be determined by investigatingthe
rows 90-180 of pascal's triangle.I would say that the
probability is about 1%.> i hope i've done an adequate job 
of
spelling out the problem. what are the> approximate odds that
all three divisions will be clinched on the same
===
algebra structureI have a problem in combinatorics (chapter is
generating functions) where f(x) = sum_{x>=0} a_n x^n is in
C[[x]].What is C[[x]]?I have to show that composition of two
f,g in C[[x]] is again in C[[x]] if the constant term of g is
0.Our professor defined ts structure in class but I 
can't find
the definition.Any help on the definition of ts 
structure is
greatly appreciated.Also, what are WEIGHTED generating
===
equation solving software ?I like Mathematica. Never had my
computer crash.Lurch> What is the best software for solving
differential> equations? Is there anytng better than Matlab
--> that at least won't crash and necessitate a reboot> of 
my
computer when I give it a differential equation> it can't
===
polynomial 2*cos(2^k*arccos(x/2)) irreducible for each >
positive integer k? If so, how is it proved? to all! It is now
clear that the answer is yes, and the proof is essentially
trivial: Each polynomial is the square of its predecessor less
2.Ts follows immediately from the identity (cos z)^2 = (1 +
cos(2*z))/2. Now for k=0,1 the polynomials are x and x^2-2 so
it is clear that for each k the leading coeff is 1, the
constant is 2 and all other coeffs are even, so Eisenstein
===
irreducible polynomial ?>>Is the polynomial
2*cos(2^k*arccos(x/2)) irreducible for each >>positive integer
k? If so, how is it proved?>Is it a polynomial? In what?
Irreducible over what?>>in x, irreducible over Q>>For
example,>>2*cos(2^4*arccos(x/2)) =>>16 14 12 10 8 6 4 2 >>x -
16 x + 104 x - 352 x + 660 x - 672 x + 336 x - 64 x + 2>>is
irredicible.>> Never seen that before; but would it not be
possible to prove it> irreducible by using Eisenstein's
Criterion? Looks like the leading> coefficient is 1, the
constant coefficient is 2, and all the other> terms have even
coefficient. If the pattern holds for arbitrary k,> then there
you are.> Right you are. Let p_k(x) = cos(2^{k} *
arccos(x/2)), thenp_1(x) = x^{2} / 2 - 1andp_{k+1}(x) = 2 *
(p_k(x))^2 - 1in x^2, has constant term equal to 2 or -2 and
that 2p_k(1) = -1 and2p_k(2) = 2. As you say, there you are.
You don't even need to
===
satisfiability> ts is an interesting & thoughtful observation
but RE>> glosses over one presumably obvious consideration.>>
let f(x) be the clause size of a formula>> with fewest
variables to factor a x-bit>> number. as RE points out, the
minimum-variable SAT formula to>> factor a x-bit number has
approx x bits.>> (two factors of approx x/2 bits each. note>>
sqrt(2^x) has approx x/2 bits)>> HOWEVER it is likely assured
that f(x), the>> clause-to-bits relationsp, grows
exponentially.I wondered about ts.>Generating the clauses
probably requires exponential effort,>but the clause to
variable ratio can't be exponential.>Given that we have 2N
literals, the maximum number of 3-clauses>is O(2N^3). For a
100 variable problem there are C(200,3)>possible 3-clauses =
1313400 ~ 100^3.Why should all the clauses in ts formula have
at most three literals?You can't necessarily represent a
function in N variables using a 3-CNFformula. The number of
functions that can be represented that way is verysmall by a
counting argument: there are 2^(2^N) functions in N
variables,but only 2^(O(N^3)) possible 3-CNF formulas. What
are the odds thatfactoring is one of those lucky functions for
interesting values of N?-- Daniel A. Jim.8enez
djimenez@cs.utexas.eduAssistant Professor
djimenez@cs.rutgers.eduDepartment of Computer ScienceRutgers
===
material I have read, the terms Borel>Field, Sigma Field, and
Sigma Algebra appear to mean the same,>that is a collection of
sets closed under complementation and>countable unions. Is ts
not correct?>No, as I understand the terms Sigma fields and
sigma algebras are indeed the same tng. The Borel field is the
smallest sigma field containing the topology (i.e., all the
open sets) of a topological space. When refering to R^n, the
usually unstated topology is the usual one.-- Stephen J.
===
smooth function sci.math,I've got an array size 100 of 
integer
values (range: 0-5). Each plot in the array represents a 1ms
time window. I currently have these graphed as stairs in
MatLab (wch looks like a bar graph). I want to make ts
function smooooooth, but i'm not sure of a method to apply 
to
===
Re: Fixed points>Since you have already been given the (an)
answer.....What happens to (0,1) if you pick it up , ßip it
over then set it back>down?>Er, doesn't that leave 1/2 
fixed?--
===
Re: What to tell students in a 10-15 minute talk >>Also, when
are the hour, minute and second hands positioned so that>>they
divide the clockface in three equal sectors?>>The short answer
is never.>>I was aware of the result. The interesting bit is
tnking of>>different mathematical solutions to solving it, wch
leads us to the>>topic of having multiple ways to reach a given
mathematical result. > So if you tnk of the three hands as
rotating unit vectors,> their sum will never be zero. So
here's another problem. Assign> angular velocities to the
three unit vectors so that the mimimum> length of their sum is
as large as possible.There are various answers depending on
what additional restrictionsyou apply. For j = 1,2,3, let Case
j allow up to j hands to have thesame angular velocity. Define
subcase (a) by requiring all hands tomeet at some time, and
subcase (b) by not requiring ts.(Applying no additional
restrictions yields the trivial case 3b.)Solutions:Cases 3a
and 3b: maxmin length is 3. All hands have equal
angularvelocities and initial positions.Cases 2a, 2b, and 1b:
maxmin length is 1. 2a: two hands have the same angular
velocity, and one hand has an angular velocity different from
the other two. 2b: The (2a) solution + another. The second
solution is the same as (2a) except that the first two hands
have opposite initial positions. 1b: The hour and minute hand
start at the same initial position; the second hand, at the
opposite position. The second hand moves twice as fast as the
minute hand in the frame of reference of the hour hand.Case
1a: maxmin length is sqrt((47 - 14 sqrt(7))/27) ~=
0.607346.The second hand moves three times as fast as the
minute hand in theframe of reference of the hour hand.I 
can't
swear by these results: it's easy to slip up when
casesproliferate. Nor have I proven them.I found it
interesting to compute the minimal length for our
standardclocks. It turns out to be 0.0025408119679, and occurs
at2:54:34.56169 and9:05:25.43831.Note: none of the above
results apply to digital clocks.-- | Jim Ferry | Center for
Simulation |++ of Advanced Rockets ||
http://www.uiuc.edu/ph/www/jferry/ ++|
jferry@[delete_ts]uiuc.edu | University of Illinois
===
algebra structure I have a problem in combinatorics (chapter
is generating functions)> where f(x) = sum_{x>=0} a_n x^n is
in C[[x]]. What is C[[x]]? I have to show that composition of
two f,g in C[[x]] is again in C[[x]]> if the constant term of
g is 0. Our professor defined ts structure in class but I 
can't
find the> definition. Any help on the 
definition of ts structure
is greatly appreciated. Also, what are WEIGHTED generating
functions? in advance.C[[x]] are simply formal power series, i
e you do not carefor convergence. Weighted? Just ask m (as you
could dofor the first Q), may be he says: Ôput 
some factors at
thecoefficients'. Hm ... there is no shame to 
ask questions in
===
software ?> I like Mathematica. Never had my computer crash.
I'll check it out.-- Bob Day What is the best software for
solving differential> equations? Is there anytng better than
Matlab --> that at least won't crash and necessitate a 
reboot>
of my computer when I give it a differential equation> it 
can't
===
baseball odds calculation problem>here's the problem,
specifically:there are three divisions in the american league.
every season, each one of>these divisions will be clinched by
one team on a certain date. ts is a>mathematical
certainty.past baseball story tells us that dates before
september are extremely>rare for teams clincng. in fact, the
later days of september, roughly the>15th through the 28th,
are where the likelihood of teams clincng is the>greatest.>You
need to specify the probability distribution of the date of
clincng for each division. For example, assuming the the three
divisions are independent and have the same distribution,
completely supported (a model of your approximation) by {15,
16,..., 28}, then the probability is sum(k=15..28, p(k)^3),
where p(k) is the probability of clincng on the date k. So if
you are saying that each division is equally likely to be
clinched on each day and is independ of the others, the
probability is 14*(1/14)^3 = 1/196 = .0051 (0.51%),
approximately. Or if you assume identical triangular
distributions, p(k) = min(k-14,29-k)/56, the probability
(still assuming independence) is 2/56^3 sum(k=1..7, k^3) =
1/112 = .0089 (0.89%), approximately.Another model would be a
discretized normal over the whole season. Perhaps you want to
make an empirical estimate of the distribution; you may then
do the calculation yourself.-- Stephen J. Herschkorn
===
(generating functions) : algebra structure Visiting Assistant
Professor at the University of Montana.>I have a problem in
combinatorics (chapter is generating functions) >where f(x) =
sum_{x>=0} a_n x^n is in C[[x]].What is C[[x]]?Usually, power
series on nonnegative powers of x. Like polynomials
withcoefficients in C, except they are allowed to have
===
TopicsSome more corrections:The physical radius R = integral
dR is much larger compared to physical circumference C as it
would be in ßat 3D space. If one keeps R fixed ~ h/mc ~
G*m/c^2, the micro-geon of Wheeler's Mass without mass 
Charge
without charge Spin without spin shrinks to a point in gh
resolution Heisenberg microscope scattering probes with r ~
h/p for momentum transfer p like SLAC deep inelastic electron
scattering off protons.Should be:The physical radius R =
integral dR is much larger compared to physical circumference
C as it would be in ßat 3D space. If one keeps R fixed ~
e^2/mc^2 ~ G*m/c^2 (Blackett effect), the micro-geon of
Wheeler's Mass without mass Charge without charge Spin 
without
spin shrinks horizon. We will see later that ts is why
lepto-quarks look like probes with r ~ h/p for momentum
transfer p like SLAC deep inelastic electron scattering off
protons.Note thatG(rho + 3p/c^2) = Gphro(1 + 3w) is replaced
by c^2/zpf in the w = -1 exotic vacuum case that included both
dark energy of negative pressure and dark matter of positive
pressure controlled by the vacuum polarization macro-quantum
coherent local order parameter (0|e+(x)e-(x)|0) completely
missed in the Puthoff-Haisch-Rueda approach to these problems
for the origin of inertia and the origin of gravity from zero
point vacuum quantum ßuctuations in the sense proposed by
Andrei Sakharov in 1967. Sakharov's metric elasticity is a
special case of P.W. Anderson's More is different 
generalized
phase rigidity of the macro-quantum ground state coherence
from spontaneous symmetry breaking.Imagine a dark matter
exotic vacuum core of radius R = e^2/mc^2c^2/zpfR^3/G* ~ mThe
Kerr-Newmann micro-geon picture for spatially extended IT dden
variable is that the inner event horizon is of order classical
electron radius of ~ 1 fermi with outer event horizon out to ~
137 fermi = h/mc.The in-between ergosphere is ionized vacuum
polarization plasma of virtual electron-positron pairs./zpf ~
1/Lp*^2Lp* ~ Lp^2/3(c/Ho)^1/3 ~ 1 fermi (t'Hooft-Susskind
world hologram)m ~ (e^2/c^2)/zpf^1/2 ~ 1 Mev generic
lepto-quark rest ggs field mass from zero point exotic vacuum
energy, hadronic mass ~ 1 GEV from Heisenberg kinetic energy
of confined lepto-quarks as in QCD lite bag model (Frank
Wilczek).G*m^2 ~ e^2 (Blackett effect)G*m^2/hc ~ fine 
structure
constantOne must be careful in how to use both special and
general relativity when polarization effects in real on mass
shell media are considered. In the case of special relativity
one should, for example, consult the text books by Arnold
Sommerfeld, Panofsky and Pllips, Landau and Lifstz. A real on
mass shell medium in a sense spontaneously breaks
translational symmetry on scales larger than the unit cell of
the lattice as described in more detail by P. W. Anderson in s
More is different series of papers collected in A Career in
Theoretical Physics (World Scientific) When one includes all
dynamical degrees of freedom including those inside the unit
cell of the lattice, global special relativity is restored to
the propagation of light in a real medium in the usual
situation where gravity tidal forces are neglile.should beOne
must be careful in how to use both special and general
relativity when polarization effects in real on mass shell
media are considered. In the case of special relativity one
should, for example, consult the text books by Arnold
Sommerfeld, Panofsky and Pllips, Landau and Lifstz. A real on
mass shell medium in a sense spontaneously breaks
translational symmetry on scales smaller than the unit cell of
the lattice as described in more detail by P. W. Anderson in s
More is different series of papers collected in A Career in
Theoretical Physics (World Scientific) When one includes all
dynamical degrees of freedom including those inside the unit
cell of the lattice, global special relativity is restored to
the propagation of light in a real medium in the usual
situation where gravity tidal forces are neglile.Some
corrections:Ditto for the excess verbal baggage of less by C.
Williams below on the nature of c in E = mc^2.should have
beenDitto for the trite excess verbal baggage of less with
more by C. Williams below on the nature of c in E = mc^2.could
use Newton's gravity with global special relativity to 
produce
the three classic tests of GR provided one introduced the
Wheeler-Feynman-Dirac-Hoyle-Narklikar trick of advanced
potentials in addition to retarded potentials. The fact that
Puthoff gets those tests as well with s variable dielectric
vacuum model is no great acevement either because Einstein's
classical geometrodynamics goes beyond those tests, e.g.
gravimagnetism and gravity waves and black holes.The professor
also got the basic black hole effective potential of the
Schwarzscld /zpf = 0 vacuum metric with s advanced potential
method.Interesting, but like the stochastic electrodynamics
approach to EM zero point energy and the semi-classical
attempt by Ed Jaynes not to quantize the EM field, like
Puthoff's PV approach et-al these attempts at the very best
are fragmentary incomplete and fail to ask and answer many of
the really important fundamental questions e.g.What is
consciousness physically?What is the universe made from?We are
===
Express As Single FractionHow do I do ts?Express the following
as a single fraction:4/3ab - 5/6bcand(m^2 + 2)/(m^2 + m) - (m -
===
and you might be the first. after all,I said that it was quite
trivial. seriously, when I was passing though Santa Cruz,I
stopped at Otto-Pagan's office to pursue ts, 
buthe was on s
European half of academia. when I suggested that,changing the
hardware setting on the macne from double-precisionto single,
he pooh-poohed it -- the grad student that I found. anyway, as
I said, monsieur M. had already done it, orat least he made
that inference at Young Hall.correction to what I typed:the
mini-Ms do not appear at *every* magnification, sincethe
rounding-errors are tied to the lengths of the registers,wch
is enough for a few iterations. as your statement impliese.g.
the specification is inherently chaotic,as the term of art
goes, no matter how variously implimented. > that is the
recurrence of mini-bugs or cardioids,> at every level of
magnification, is just an artifact> of the ßoating-point 
ops
(IEEE-755, -855, I tnk).> ts was (really/partially) confirmed
by monsieur M,> when he glroriously begged my (only) technical
question> at a talk for a general audience. > It's quite 
simple
to disprove your claim by setting the rounding> method, wch you
can do in hardware on the Pentium (and many> other processors
as well) to all it's values and seeing what changes,> or
doesn't change, in calculations. FP doubles are good for
several> tens of thousand of iterations, down to an area of
10E-10 or so,> before the precision gives out. The cartoids
are visible several> orders of magnitude above ts.--les ducs
de Buffet;vote NONE OF THE BELOWon Trickier Dick Cheney's
===
consecutive composite integersExamining a table of factors and
primes, I found that for any sequenceof consecutive composite
numbers there is always one integer that hasa prime factor
larger than any other prime factor of any of the
otherintegers. Further, ts prime is not raised to any power.
My questionis: Is ts true for all consecutive composite
sequences and if so, isthere a proof?(Here's an example: 
1500,
1501, ... 1509. The last integer has theprime factor
===
teensy duality the same,for any proof that one hasn't 
actually
read,all the way through (or to the point whereone can see it
through, already) ??> you're saying that the L-wing tng is
just an argument> *about* an actual (R-wing) proof?--les ducs
de Buffet et Schulz (R&D Chair Assoc.Intl.);vote NONE OF THE
BELOWon Trickier Dick Cheney's California
Recall/e-Dereg!http://larouchepub.comhttp://members.tripod.com
===
functions) : algebra structureNNTP-Posting-User:
[DNl+kfRPf6mkEI2haabWc7pkPCvSlQMj]> I have a problem in
combinatorics (chapter is generating functions)> where f(x) =
sum_{x>=0} a_n x^n is in C[[x]]. What is C[[x]]?>Ring of
formal power series. Elements are power series in x
withcoefficients in C, not necessarily convergent.> I have to
show that composition of two f,g in C[[x]] is again in C[[x]]>
if the constant term of g is 0. Our professor defined ts
structure in class but I can't find the> 
definition. Any help on
the definition of ts structure is greatly appreciated.> Also,
what are WEIGHTED generating functions? in
C-113opstall@math.wasngton.eduhttp://www.math.wasngton.edu/~
===
it's just simple numbertheory --skipcodes are that, and they
were apparently usedby (some?) Torah writers/copyists to
ensure accuracy,as with the old CRC in 8-bit communications
programs.I read taht they summed the letters on every 70th,or
skipped to every 70th, or some tng. the computer can be set to
find any messagein any ring of an alphabet, and Drosnin et al
know ts ... ormaybe they can't learn it, not because 
they're
dumb. there was ambiguity in _The Bible Code_ taht he
ignored,like with the variant translations and the fact
thatOld Hebrew has no vowels.repeat, _War and Peace_ or just
the 26 letters in any ordercan be used with the infinite set 
of
co-prime skips,with teh resulting ts being further massagedinto
some m by n array (or what ever). >FAILED.Is ts accurate? And
does ts say sometng about the New Testament>and the belief in
a Christ figure? --les ducs de Buffet et Schulz (R&D Chair
Assoc.Intl.);vote NONE OF THE BELOWon Trickier Dick Cheney's
California
Recall/e-Dereg!http://larouchepub.comhttp://members.tripod.com
===
problem>> here's the problem, specifically:>> 
there are three
divisions in the american league. every season, each >> one
of>> these divisions will be clinched by one team on a certain
date. ts >> is a>> mathematical certainty.>> past baseball
story tells us that dates before september are extremely>>
rare for teams clincng. in fact, the later days of september,
>> roughly the>> 15th through the 28th, are where the
likelihood of teams clincng is >> the>> greatest. You need to
specify the probability distribution of the date of > clincng
for each division. For example, assuming the the three >
divisions are independent and have the same distribution,
completely > supported (a model of your approximation) by {15,
16,..., 28}, then > the probability is sum(k=15..28, p(k)^3),
where p(k) is the > probability of clincng on the date k. So
if you are saying that > each division is equally likely to be
clinched on each day and is > independ of the others, the
probability is 14*(1/14)^3 = 1/196 = > .0051 (0.51%),
approximately. Or if you assume identical triangular >
distributions, p(k) = min(k-14,29-k)/56, the probability
(still > assuming independence) is 2/56^3 sum(k=1..7, k^3) =
1/112 = .0089 > (0.89%), approximately. Another model would be
a discretized normal over the whole season. > Perhaps you want
to make an empirical estimate of the distribution; > you may
then do the calculation yourself.>And as approximation of the
discretized normal, suppose the date less 15 has binomial
distribution with parameters 13 and 1/2. Then the probability
is approximately the much larger 2.76%, by my calculations--
===
Re: Re fermat by Tomaswhat is found at your site is known as
Sopstry. the fact thatFernat's Last Theorem is negative is 
not
problem, just asproofs (of neg or pos statements) by
contradiction are good. of course, three per cent is rational
by definition.Negative statements about numbers are
unverifiable. Take thedefinition of an irrational 
as a number
that is not rational, wherebeing rational means having
periodic decimal expansion. Supposesomeone wants to verify
that %3 is irrational. Since the indirectproof is out he
computes its decimal expansion to the trillionth digitand,
satisfied that no evidence of periodicity looms on the
horizon,declares that %3 is irrational. A miscevous fellow
quickly forms aperiodic decimal whose first period is that 
first
trillion > http://www.users.bigpond.com/pidro/home.htm > --les
ducs de Buffet et Schulz (R&D Chair Assoc.Intl.);vote NONE OF
THE BELOWon Trickier Dick Cheney's California
Recall/e-Dereg!http://larouchepub.comhttp://members.tripod.com
===
me an example of a function (R=real numbers) f:R-->R>such that
for any a,b,c in R, there exists an x in R such that a<x<b>and
f(x)=c?>>Not so that it's continuous for the whole R, but it
can be continuous>>witn the open interval ]a, b[:
>>f[x_]:=Tan[Pi/(b-a)*X-Pi*(a+b)/(2*(b-a))]>>-oo when x=a, oo
when x=b and gets all values when between them.Another
example:g[x_]:=Pi/(b-a)*x-Pi*(a+b)/(2*(b-a))>f[x_]:=Cos[g[x]]/
g[x]Ts one is continous everywhere except at x=(a+b)/2, and
gets all>values between a and b.You are not reading the
question! It says for all a,b,c, soa and b are not constants.
And the question does not say anytng aboutcontinuity.I tnk
there was one correct solution posted. Here is another
possibility.I will define f:R -> [0,1], wch is good enough,
because there are bijectionsfrom [0,1] to R.Let x in R and
look at the decimal expansion of x. Choose m maximalsuch that
x = n . a_1 a_2 ... a_m a_1 a_2 ... a_m ...; that is,the first
m digits after the decimal point repeat.If there is no maximal
m with ts property, then define f(x) = 0.Otherwise, let x = n 
.
a_1 a_2 ... a_m a_1 a_2 ... a_m b_1 b_2 b_3 ...and define f(x)
= 0.b_1 b_3 b_5 ...(The point of taking only the odd b_i is to
enable me to select the even b_iso as to prevent me
accidentally getting a larger value of m in x.)Derek
===
ts? Express the following as a single fraction: 4/3ab - 5/6bc
and (m^2 + 2)/(m^2 + m) - (m - 2)/m > in advance. (1) find a
common denominator. The least common denominator is a good one
to use.(2) convert each fraction to an equivalent fraction, all
having the same (common) denominator found in step 1.(3) Add
(or subtract) numerators and put the result over the common
denominator from step 1.(4) Reduce the result to lowest terms.
Note that if you have used the least common denominator, the
===
baseball odds calculation problem>here's the problem,
specifically:there are three divisions in the american league.
every season, each oneof>these divisions will be clinched by
one team on a certain date. ts isa>mathematical certainty.past
baseball story tells us that dates before september are
extremely>rare for teams clincng. in fact, the later days of
september, roughlythe>15th through the 28th, are where the
likelihood of teams clincng is the>greatest.> You need to
specify the probability distribution of the date of> clincng
for each division. For example, assuming the the three>
divisions are independent and have the same distribution,
completely> supported (a model of your approximation) by {15,
16,..., 28}, then the> probability is sum(k=15..28, p(k)^3),
where p(k) is the probability of> clincng on the date k. So if
you are saying that each division is> equally likely to be
clinched on each day and is independ of the others,> the
probability is 14*(1/14)^3 = 1/196 = .0051 (0.51%),
approximately.> Or if you assume identical triangular
distributions, p(k) => min(k-14,29-k)/56, the probability
(still assuming independence) is> 2/56^3 sum(k=1..7, k^3) =
1/112 = .0089 (0.89%), approximately. Another model would be a
discretized normal over the whole season.> Perhaps you want to
make an empirical estimate of the distribution; you> may then
do the calculation yourself. for the answer. i looked for
clincng dates over the last 27 fullbaseball seasons. in those
seasons, there have been 126 different clinches(4 clinches per
year from 1975-1993 excluding 1981 strike year and then 6and i
was able to find 96 of those dates (they're not 
very easy to
find). idid an estimate based on ts here's the 
distribution of
clinches i found:sept7 1%8 2%9 3%10-11 0%12 1%13 0%14 1%15
1%16 0%17 3%18 2%19 2%20 4%21 3%22 2%23 6%24 4%25 7%26 5%27
7%28 7%29 4%30 6%oct1 3%2 5%3 6%4 3%5 6%6 1%7 1%btw, the
earliest clinch in baseball story, not just the 27 seasons
ilooked at, is apparently september 7th.> --> Stephen J.
===
functioncan anyone show me an example of a function (R=real
numbers) f:R-->R>such that for any a,b,c in R, there exists an
x in R such that a<x<b>and f(x)=c? Let f(x) = 5.> No, let c=6,
===
quantity of matter>Cut< I'm sure that people would stand in
line for blocks to get a signedcopy!! RJ PAs a matter of fact
RJ, I have already written a couple, and can't even 
give'em
away! As long as the gravy train keeps running, nobody wants
===
trying to prove the following theorem:Let P be a polynomial
with real coefficients such that P(x) >=0 forevery real x.
Then, there are polynomials R and S such that P(x) =R^2(x) +
S^2(x) for every complex x.It's easy to see that the degree 
of
P must be even. If r is a realroot of P, then the restriction
of P to the reals has an absoluteminimum at r and, from the
differentiability of P, it follows there'san even number k
such that the k-1 first derivatives of P vanish at rand the
k_th is positive. Therefore, the k-1 derivatives admits
theroot r with multiplicity 1, wch implies P admits r as a
rooth withmultiplicity k. So, we see every real root of P,
when they exist, musthave an even multiplicity (I tnk we could
come to ts sameconclusion a bit faster, considering only the
continuity of polynomialfunctions).Corollary - If all of the
roots of P are real, then, P = Q^2 for somepolynomial Q. So,
for ts particular case the theorem has just beenproved.To
prove the theorem, for the general case, I tried to
usemathematical induction. It didn't work, 
that's why I'm
asking forhelp. What I did is as folows:the previous
paragraph, it's also enough to cover the case ofeven-degree
polynomials with real coefficients and no real roots. 
It'swell
known that every monic trinomial T of the 2nd degree
thatsatisfies such conditions can be written as T(x) = (x-a)^2
+ C^2,where a and C<>0 are real. Therefore, for such
trinomials the theoremholds trivially. Now, suppose there's 
a
natural k such that thetheorem holds for i=1,...k-1 for every
2i-degree polynomial with realcoefficients and no real roots.
If P is a 2k-degree polynomial withthese same properties, then
at some (or several) real r's therestriction of P to the 
reals
attains an absolute minimum m>0. Tsimplies that, for every
real x, the polynomial P-m is non-negative,has one (or
several) real root(s) and (i) P(x) - m =
(x-r_1)^p1*...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the
real roots and thenumbers p_1,...p_n are even. In addition, Q
is monic, has even degree< 2k, has no real root and is
strictly positive on the real line. Bythe induction
assumption, the theorem holds for Q and, in virtue of(i), also
holds for P-m. But now, to complete the induction, itremains to
prove the theorem is good for P, in other words, it remainsto
prove that if the theorem holds for some polynomial P then it
holdsfor P+m for every m>0. That's where I got 
stuck.Actually,
I tnk I chose a very cumbersome way to prove the theorem,there
certainly is a neater proof.Any suggestions are
===
Game-Boards?> I tnk, with Chess, what is considered a capture
might be ambiguous.Each piece has a value and a circle of
control. After a piece moves, it attacks everytng in its
circle. You must stop movement when youenter a hostile circle.
Defenders sum their values. Pawns would get promoted after
===
%-nbwyH<CA'^s-PG9@(a%'R|v~rO=dCg-8R>lYn=yh4r^*
v|!,o}OFN$97k
?cjI1!x?l>5*VZ)c/:of{IPQt>In the material I have read, the
terms Borel>Field, Sigma Field, and Sigma Algebra appear to
mean the same,>that is a collection of sets closed under
complementation and>countable unions. Is ts not correct?> No,
as I understand the terms Sigma fields and sigma algebras are 
>
indeed the same tng. The Borel field is the smallest sigma 
field
> containing the topology (i.e., all the open sets) of a
topological > space. When refering to R^n, the usually
unstated topology is the usual > one.If I recall correctly,
Mr. ez (quoted above by Mr. Herschkorn) is studying out of
Chung's _Course_. Chung uses Borel field, in a 
now old-fasoned
way, as a synonym for sigma-field (aka sigma-algebra). The 
same
usage can be found, for instance, in Doob's classic book on
stochastic processes.penetrating analysis of increasing
families of sigma-fields (aka filtrations). 
Needless to say they
===
pointsTnking with my toes again. : )> -----Original
===
Message-----> Conversation: Fixed points> Subject: Re: Fixed
points Since you have already been given the (an)
answer.....What happens to (0,1) if you pick it up , ßip it
over then > set it back>down? Er, doesn't that leave 1/2 
fixed?
-- > Stephen J. Herschkorn > herschko@rutcor.rutgers.edu>
===
questions. Ts is homework, so please post a nudge,> not a
solution. 1)prove that if f,g continuous, then so are max(f,g)
and min(f,g)> After drawing some graphs, ts seems pretty
obvious for the single> point a0 -- max(f,g) has 2 cases: it
equals to f or g. Either is> continuous. However, ts question
implies continuous on R, not just> at a single point. Any
ideas how to approach ts?Proving a function continuous at any
point proves it continuous at every point. > 2)Let f be a
function with the property that every point of> discontinuity
(ie the lim (x->a) f(x) exists, but is not equal to> f(x)) is
a removeable discontinuity. Ts implies lim (y->x)f(y)> exists
for all x, but f may be discontinuous at some (even 
infinitely>
many) numbers x. Define g(x) = lim (y->x) f(x). Prove g is
continuous. --I don't even know where to start with ts
one.Does you definition of g(x) actually read g(x) = lim 
(y->x)
f(x)or should it be g(x) = lim (y->x) f(y)?In the first case,
g(x) = f(x) at all x, so is not continuous at discontinuities
of f.In the latter case, does g(x) = lim (y -> x) g(y), for
===
future!> Here's 5 peoples posts to verify that mind reading
technology is already> here. When you tnk, you are not silent,
a radar can pick up your thoughts JUST LIKE SPEACH and sound
them out. Because I'm the truman I get it constantly. 
100,000
people in townsville australia know for certain that 100%
clear> mind reading> is possible, all my neighbours listen to
my every thought every day. its the most deous torture
possible to constantly have you're thoughts> played> back to
you audibly and be FORCED to answer truthfully every passing>
remark,> like I do every time I go out. Your voice box gets a
trace stimulus of every thought you tnk, makes a> small> noise
just like speaking, it can be picked up. They can play with
the> timing,> they can hear compressed phonetics of whole
sentences you are about to> tnk, and tell you your thought a
second before you are aware of it. Herc> I CANNOT LIEEEE or
another Jim Carrey, Majestic costarring Laurie Holden> Exbit
A: http://tinyurl.com/fuf8 she looks exactly like Laurie
Holden> Exbit B: http://tinyurl.com/fuf2 government has spied
place them clearly> in between the release dates of The Truman
Show : 1998 : Jim Carrey> Majestic : 2002 : Jim Carrey and
Laurie Holden I'm from Townsville and YOU ARE the Truman!>
http://tinyurl.com/iky5> I was in Townsville over the weekend,
and I heard m.> Very spooky!> http://tinyurl.com/iky8phone
someone in Townsville, half of you must know someone
there,every day I go out people say THERES THE TRUMAN> I'm 
in
Townsville. We're sick of you.> http://tinyurl.com/iky9>
http://tinyurl.com/iky4> You rule Truman!Do you know if the
truman is living in Townsville?> I've been hearing stuff,
yeah> http://tinyurl.com/p0w3> I can remember listening to
Amazing Randi's radio show in the 
Ô60's .> He was aware of the
phenomenon.> It's called Subvocalization and apparently, 
some
people can detect the> acoustial or electro/acoustical energy
from you saying what you're tnking> under your breath 
.That's
how I assume its done, but its done remotely with macnes, a
sateliteI tnk, you can't hear it yourself.Popular Science 
had
a May issue with I was shot by the army's pain beam onthe
cover, wch was a clue to the Ôweather' satelite 
story, that
usetrains of satelites pumping quote 94 GHZ radar beam and
pulsed laser array. Somehow perple claiming to to mind reading
suffer a stunning loss of> accuracy when isolated acoustically
from their victim ( oops..) client. (> oops!), I mean subject.
the Scientific study of ts is slim, but Magick is not an
acceptable> alternative at ts time, when ts is at least in the
hypothesis stage.> Drifting off topic for sci.math??>yes, but
I'm not writing about esp, I'm submitting data 
that requires a
simple statisticalanalysis for 6 months now as to whether it
occurs naturally.I need sci.math to stamp my stat analysis so
other groups won't dismiss it.Its not unlike a 5 mark 
question
from second year stats. H0 the correlation isevident.... H1
Herc is rambling about notng :H0 : ...Spend 10 seconds
checking each tiny url I gave. Why would a man yell sheart out
he's the truman, then get numerous responses like ts :Do you
know if the truman is living in Townsville?> I've been 
hearing
stuff, yeah> http://tinyurl.com/p0w3>Do you want Duggy to post
in sci.math and tell you to check my empiricial datashowing
who I am? He's quite conversant and he'll tell 
you just what
my inner thoughtssound like. I found s James Cook University
email address and he's usedit since before I was in
Townsville.I put the truman verifying posts there to show that
lots of people know thetechnology is already here is complete
form. You can listen to my thoughtstoday if you come to
Townsville.Honestly, media cover up + internet global
communications = apathy.HercNow if you don't mind, I have a
dole form to submit and I have to stand in queuefor half an
hour with 50 people around me all interrogating my thoughts,
thenI'll thaw my last half loaf of frozen bread for lunch 
and
then return into the mysteriesof the internet where the whole
morning of EVERYONE in sight knowingwho I am never happened,
like I have been for 2 years now.and don't set the group
===
need to find an algorithm that can produce a unique
non-predictable 12>> digit (0-9) number for any given 12 digit
number. Ts is to be used to>> create a unique barcode on a
ticket that cannot be predicted. It is not>> required that the
original seed number be computed from the resulting>> barcode,
so some form of one-way hasng function would be acceptable.>>
Any help in ts problem would be appreciated. I've seen wiser
heads than mine recommend a Ruby-Lackov cipher for > ts kind
of tng.> First, you need a random function f(i) that takes a
6-digit decimal > as input and produces a 6-digit decimal
output.> Start by splitting your original 12-digit decimal
number into two > 6-digit parts, A and B. Then perform four
steps:> A=A+f(B) mod 1000000> B=B+f(A) mod 1000000> A=A+f(B)
mod 1000000> B=B+F(A) mod 1000000> Concatenate the final A and
B to form the 12-digit output. The process > is reversible, so
there won't be any duplicates among the output values.> f(i)
should be a good randomizing function, such as the cryptograpc
> hash of the concatenation of a secret key k with the operand
i. Wle you could convert to binary and back, if that's
convenient, > it's not necessary. If you're 
going to do ts in
decimal, you could use > the ASCII decimal digits of A or B as
the input to the hash, and take, > say, the first 32 bits of 
the
hash's output, convert it to decimal, and > use the low-order 
6
digits as the value of f. [I tnk Ruby-Lackov can > tolerate a
small amount of bias in f. If not, I'm sure someone will 
post
> another suggestion.]You would have to be careful in the
selection of your hash function.All standard hash functions
have 2**n different outputs, andI don't know any hash 
function
that produces 10**6 outputs. An example of a bad hash function
would be to take the first 20 bitsmod 1000000 of a standard
cryptograpcal hash, because ts hashfunction is extremely
biased (some values occur twice as often as theothers). When
you use a larger number of bits the bias is reduced.Is it
possible to quantify the maximum allowable bias in
===
Re: JSH your sp has come in!!!!>message> <for scary dream see
previous > Maybe the only point is that I fear James being
overwhelmed byevil.> Hmmm. > I have to ask myself, Why should
I care? James may be the> reincarnation > of Gauss, but is it
really any skin off my nose if he goes> unrecognized? > I've
been worrying about the guy for months (ever since Irealized>
that> he > was not, in fact, a crank, but a genius) and
defending m onts> newsgroup. > My reward? Laughter and bile
from the peanut gallery. And nota> word> from > James mself. A
word of advice to Prof. Connes: Don't wasteyour> time> on > 
.
If he loves being the solitary genius so much,let> m> fight > 
s
own damn battles. He's not worth losing sleep over. Well 
damn
it, I'm losing sleep at least partly because of yourscary >
dream!!! Good writing their Jim Ferry, and I have to give you
credit forthat, > but hey, how many years were you ridiculing
me, and now you expectme > to just go, hey, pal? Show me
you're serious and wade in and respond to some of these >
ant-mathematicians in the current battle. Prove your
sincerity, and um, keep posting any interesting dreamsyou >
have, as I found it interesting puzzling over that one.> >
Exactly! Ts Jim Ferry now tnks you are a genius, but has he>
defended you in your current battle? No!> > I suspect that he
is not *fully* committed to your view ofmathematics.> That is
just a feeling, and I suppose I could be wrong.> > The primary
focus is the odd definition error in core.> > Once 
that's
accepted for what it is, and most importantly FIXED, then 
it's
not about committing to my view of mathematics, but about
showing your commitment to mathematics itself.> > There is ONE
mathematics.> > I, however, have been trying to find out more
about your ObjectRing,> but I feel you are pusng me away.> > I
have spent some effort guiding you along at the Mega Foundation
discussion area. Yes, indeed. But, I expect you remember that I
did ask you a question that I wasunable> to answer. I was using
the notation [a,b,c] to represent an ordered triple ofcomplex>
numbers. And I was wondering if the ordered triple [1,2,8] was
anelement of> the Object Ring. You replied:> == Quit being
lazy!!! You have the definition, figure it out 
foryourself!!!
What amazes me is how often people are willing to ask someone
else todo> their work for them. If you're smart enough, 
answer
your own question. I'm curious to see if you can. 
I've given
the definition for the object ring, so no excuses.> == Well, I
am ashamed to say I still cannot figure it out. Sounds like a
ploy. You have to understand that your record Clive Tooth is
rather long and> involves some rather...sleazy behavior...like
that attack webpage, and> quite a few negative postings over a
period of YEARS.That was just... just... boyish gh spirits,
James. And you joined in thefun by threatening to sue me for
libel!! Ah... happy days...> You don't get the 
benefit of the
doubt from me but have to make the> extra effort yourself, so
quit being lazy!!!Oh... James... You have to admit that I did
help you out, on the Mega board,with sqrt(i) wch you didn't
realize was a complex number. Can't you helpme out just a
little with ts one?And I noticed that you said that Jim Ferry
is on the team! How can I geton the team if you won't help 
me
when I am struggling? By the way, isanybody else on the team?
I tnk that all the team-members should havewell-defined roles
for the up-coming battle.> Could you help me please? It sounds
to me like you tnk you have some angle for even more>
negativity and I've given enough time explaining. Remember
mathematics is a continuing process.> > The object ring is
fascinating in and of itself, so you can't expect me to know
all the answers just because I'm a discoverer.> > After all,
if it were that easy, then math research would have ended long
ago with the first mathematician explaining it all.> > Your
friend.> > Clive> > Time will tell. Yes, as I said, time will
tell.Very true.-- Clive
===
Question> Could someone help me to understand how to find the
minimum distance> between a surface (say f1(x,y,z)=c1) and a
line (f2(x,y,z)=c2. i> believe i should be using gradients.
very much!A single equation, such as f2(x,y,z)=c2, can
describe in a 3d space a surface, possibly a plane, but not a
line.The vector parametric form for a line is g(t) = (u1 +
u2*t, v1+v2*t,w1+w2*t),where (ui,v1,w1) is a point on the line
and (u2,v2,w2) is a vector parallel to the line.If the surface
and line intersect, i.e., the distance between the surface and
the line is zero, then f1(u1+u2*t,v1+v2*t.w1+w2*t) = c1 is true
for some real value of t.If the surface and line do not
intersect and the surface has continuous gradients and no
boundary curves, then the gradient of f2 at any extremal point
(closest to or furthest from the line) must be perpendicular to
===
points<D15ED542E12BD3119FFE00805F6551F010E4CF20@
mailwhqnews.cerner.com>,> -----Original Message----->
[mailto:madrian@pool-151-197-8-253.pl.east.verizon.net]On
===
Behalf Of> Marc> Conversation: Fixed points> Subject: Fixed
points > Suppose f : (0,1) --> (0,1) is continuous. Does f
have to > have a fixed> point? If it was f : [0,1] ---> [0,1]
or f : [0,1] --- (0,1), then yes. Any thoughts? Marc Since you
have already been given the (an) answer..... What happens to
(0,1) if you pick it up , ßip it over then set it back> down?>
Can't you come up with an algebraic formula that describes 
ts>
operation?> Do you mean sometng like f(x) = 1-x, on
===
is a field (not necessarily Borel)>>2.- u is a measure on
F>>3.- G is the minimal Borel Field containing F. >> I really
don't see what sense the terminology >> minimal Borel 
field
makes. Maybe you meant>> that G is the minimal sigma-field
containing F?>That's what I meant. In the material I have
read, the terms Borel>Field, Sigma Field, and Sigma Algebra
appear to mean the same,>that is a collection of sets closed
under complementation and>countable unions. Is ts not
correct?Yes. Some authors, I tnk mostly probabilists rather
than analysts, use Borel field ts way. See e.g. K.L. Chung in 
A
Course in Probability Theory. One trouble with ts terminology
is that it becomes clumsy to talk about the main example of a
Borel field, namely the field of Borel sets. I 
prefer to use
sigma-algebra or sigma-field. >>4.- v is a measure on G.>>5.- 
v
and u agree on F.>>6.- v and u are sigma-finite on F.>>It can 
be
proved that v is the unique extension of u from F to G.
>>Apparently 6.- is a sufficient but not necessary condition
for ts>>uniqueness. Can someone please indicate the necessary
condition and>>outline how the proof would then proceed? Many
>> I doubt that there _is_ a simple necessary and sufficient
condition.Me too. Here, by the way, is an example to show that
without sometng like 6.- you don't have uniqueness in 
general.
Consider the field F of finite unions of intervals 
in the reals,
and the measure u on F such that u(A) = 0 if A is a finite set
and u(A) = infinity otherwise. Of courseit is not 
sigma-finite.
The minimal sigma-field containing F is the 
sigma-field B of
Borel sets. There are lots of extensions of u to B, e.g.
Hausdorff measure of any dimension d with 0 < d < 1.> Can you
please indicate what is the importance of ts>measure extension
subject in terms of further development of measure>and
probability theory? In other words, is understanding of
it>important in terms of understanding new material down the
road? Your>help is always appreciated.My personal opinion is
that it's a rather specialized topic, and it's 
not worth
getting too worked up about, say, the most general form ofthe
uniqueness theorem, unless you run into a case where you
really need it. Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
===
non-Hausdorff homeomorpsm to R^nis there a non-Hausdorff space
===
useless?> But doesn't sci.math.research exist? And 
isn't it
moderated? So what is> all the fuss about?sci.math.research is
oriented towards mathematical research at a fairly gh level.
sci.math isn't (although such topics arise here); 
it's scope
===
propose is that given anyrational that the value greater than
it and less than any othergreater is irrational, There is no
such number, as several different people have shown you. In
non-standard analysis, there might be, however.> See Alain
Robert's book about NSA. Rather than being> irrational, it
would be non-standard, though.I have yet to see any standard
or non-standard model of the reals in wch there is a smallest
positive number. In the various non-standard versions, there
tend to be rather more numbers between any positive number x
and zero, there are all those y such that y/x are ifinitesimal
but positive, then all those z such that z/y is infinitesimal
===
Calculus Question>A single equation, such as f2(x,y,z)=c2, can
describe in a 3d space >a surface, possibly a plane, but not a
line.I could *swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a
===
extension
proofhttp://www.giganews.com/info/dmca.html>>Given:>>1.- F is
a field (not necessarily Borel)>>2.- u is a measure on F>>3.- 
G
is the minimal Borel Field containing F.>> I really don't 
see
what sense the terminology >> minimal Borel field makes. Maybe
you meant>> that G is the minimal sigma-field containing
F?That's what I meant. In the material I have read, the 
terms
Borel>Field, Sigma Field, and Sigma Algebra appear to mean the
same,>that is a collection of sets closed under complementation
and>countable unions. Is ts not correct?If I hadn't read the
other replies I would have simply saidno, ts is not correct -
the last two are the same, but a Borelfield is a special case
of a sigma field (the sigma field generatedby the 
open sets in a
topological space). That's the standardway the terminology 
is
used these days, but I gather thereare people who do use the
term Borel field the way you'vebeen doing - I 
didn't know
that.>>4.- v is a measure on G.>>5.- v and u agree on F.>>6.-
v and u are sigma-finite on F.>>It can be proved that v is the
unique extension of u from F to G. >>Apparently 6.- is a
sufficient but not necessary condition for ts>>uniqueness. Can
someone please indicate the necessary condition and>>outline
how the proof would then proceed? Many >> I doubt that there
_is_ a simple necessary and sufficient condition. Can you
please indicate what is the importance of ts>measure extension
subject in terms of further development of measure>and
probability theory? Well, it happens a lot that the
_existence_ of the extension is used to define measures, by
first defining them on a 
(non-sigma)field... when you do that you
need the uniqueness to know that you've defined 
_a_ measure.>In
other words, is understanding of it>important in terms of
understanding new material down the road? Other people may
have different opinions: If you're just learningmeasure 
theory
my advice would be to skim through ts part asquickly as
possible and concentrate on the stuff coming up thatgets used
in applications of measures, as opposed to constructionsof
measures - if it turns out you get into sometng where ts
isimportant there will be plenty of time to go back to the
===
non-Hausdorff homeomorpsm to R^n>is there a non-Hausdorff
space locally homeomorpc to R^n?Sure tng. Look for a current
thread wch mentionsdoubling points in one of the posts (I tnk
===
Re: Exponentative closure>Can the reals be defined using
repeated exponentative closure?>By the exponentative closure
F, I define F/x as the set of all the>zeroes of all the
polynomial functions with coeffeicients AND>exponents in F.
For example, the the algebraics are the exponentative>closure
of the integers. Thus, it can be written A=Z/x. Does>C=A/x? If
not what does A/x equal? Can C be generated by>repeatedly
exponentatively closing the integers a finite number of>times?
If so, how many? A countable number of times? An
uncountable>number of times?Unless I misunderstand you, F/x is
countable if F is countable. So no,a finite or even a 
countable
number of exponentative closures won'tdo it: the union of
countably many countable sets is countable. I don'tknow 
about
an uncountable number of times.Robert Israel
israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
===
useless? > But doesn't sci.math.research exist? And 
isn't it
moderated? So what is> all the fuss about?sci.math.research
doesn't allow discussions of elementary topics(homework
problems), and other math related tngs like Latex,whether
Maple is better than Mathematica, how easy it is to losea job
by going to the Joint Meetings, whether electronic journalsare
===
newsgroup
useless?http://www.giganews.com/info/dmca.html>[...]Several
people have commented that the filtering systems should>work
fine. Sure they do, if you continuously update them.> 
You're
certainly doing _sometng_ out loud - hard to say whether>>
tnking is the right word, since the problems you're
complaining>> about are so easy to fix. Luckily ts is not a
moderated group,>> so you're free to bounce ideas off us 
just
like JSH is...I haven't been complaining at all. Just 
chatting
with the >original complainer. Everyone knows what the really,
really,>absolutely true problem is, however, and it is the
insufferable>rudeness of several posters. But like you say,
these tngs>are easy to fix.<plonk>See, that 
wasn't all that
===
series, I'm trying to find the radius for 2 
simliar series>Let
f(z) = sum_k a_k*z^k be a formal power series with radius
of>convergence 1. Put s_n = a_0 + ... + a_n and t_n = (s_0 +
... +>s_n)/(n+1).>Let g(z) = sum_k s_k*z^k and h(z) = sum_k
t_k*z^k.>How would I show that the radius of convergence of h
and g are both 1>as well? (The lim sup method doesnt seem to
work).Some nts:Note that for any epsilon > 0, there is C such
that |a_k| < C (1+epsilon)^k for all k. What does that say
about |s_k|?For the other direction, note that a_n = s_n -
s_{n-1}.Robert Israel israel@math.ubc.caDepartment of
Mathematics http://www.math.ubc.ca/~israel University of
===
form for the integral of x^x dx by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h973a2W15776;I've been trying to find a 
generalized (open)
formula for the integral of x^x dx.....does anyone know if
===
analysis: construct ts set ... by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h974dnb19740;Can you construct a set E in [0, 1] s.t. for
every open interval I in [0,1] m(I intersect E) > 0 & m(I
intersect E^c) > 0 m is lebesgue measure E^c is the complement
of E Ts is so tricky! I was tnking sometng with the generalized
Cantor set but everytng I'm trying isn't 
working. Any
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h97D3uP19205; I studied an
alternative set theory wch dissolves Russel's Paradox. In ts
theory, it is possible to get good theorems of ordinal number.
More description about the set theory is available on my home
page. http://boat.zero.ad.jp/~zbi74583/another02.htm I
appreciate any comment about it. --------@ 1.Axiom of Free
Class. @ A. ma [A]A[A]B[A]x[A]y A ma x &@B ma y & A=B -> x=y
Ts axiom means that any Atsumari makes only one class. A. el
x[A]A[A]B ([A]a a el A <-> a el B) -> A=B Ts axiom means that
an Atsumari is decided by its members. A.F [E]x{E]B ([A}a a el
B <-> F(a)) & B ma x Ts is the Schema of comprehention axioms.
Where A,B,C,...are variables for Atsumari, wch means
collection or set or pile in Japanese. a,b,c,...are variables
for Class, [A] means all, [E] means exist, [E]! means exist
only one, ma means makes, el means element, -> means then, <->
means equivalent, V means or, & means and, Atsu is an
abbreviation of Atsumari ~ means negation, {a,b,c} means the
Atsu wch has members a,b,c 2.What does Free Class mean? @ [E]B
a el B & B ma b means a el b in ZF (TR) For example if {a,c} ma
b & {x,y} ma b are true, then a el {a,c} & {a,c} ma b so [E]B a
el B & B ma b it means a el b in ZF. FC ZF {a,c} ma b
corresponds a el b, c el b {x,y} ma b x el b, y el b 3.
Russel's class [A]a a el R <-> ~([E]B a el B & B ma a) The
right side of formula means ~(a el a) in ZF. Ts Atsu R makes
Russel's Class r, so R ma r. Let a=r then, r el R <-> ~([E]B 
r
el B & B ma r) If r el R is true, considering that R ma r is
also true,then [E]B r el B & B ma r is true, then right side
of formula is false. Ts is a contradiction. So, the following
formulas are gotten. ~(r el R), [E]B r el B & B ma r. Ts
conclusion means that the diagonal logic was dissolved. So, it
===
coloring problem by support1.mathforum.org (8.11.6/8.11.6/The
Math Forum, $Revision: 1.9 primary) id h97D48o19230; I studied
a generalized coloring problem. The ordinary coloring problem
is defined as follows. @ @@Place different colors on two
vertices wch are next each other in a plane graph. @@How many
colors are necessary and enough? @ @@And new coloring problem
is the following. @ @@A definition @@D.1. @@Place different
colors on two vertices wch are near each other in a plane
graph. @@How many colors are necessary and enough? @ @@The
term near is defined as follows. @@For different two vertices 
a
and b,@either condition 1. or 2. is filled. @@c.1. a is next 
of
b @@c.2. There are three paths of length 2 between a and b. @
@@c.1 and c.2 is represented as (1,1) and (2,3) respectively.
@ @@If G is a plane graph and @@if edges are added between all
pairs of vertices wch satisfy (2,3)@in G, @@then ts graph is
written as Near_2,3(G) @ @@New problem is also defined as
follows. @ @@Another definition @@D.2. @@What is
chr(Near_2,3(G))? @@@@ where chr(x) means the chromatic number
of x. @ I proved a theorem as follows. [ The@@7 colors theorem.
] @@T.7C.@@7 colors are necessary and enough. More description
about ts problem is available on my HP. @
===
The Octic x^8-x^7+29x^2+29 Revisited by support1.mathforum.org
(8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id
h97D4OF19281; all,By a stroke of luck, I managed to find the
missing piece to thepuzzle of how to solve the resolvent
septic. So here is the complete solution:Given:
x^8-x^7+29x^2+29 =
0Then,x1=(1+(a-b-c-d+e-f-g))/8x2=(1-(a-b-c-d-e+f+g))/8x3=(1-(a
+b-c+d+e-f+g))/8x4=(1+(a+b-c+d-e+f-g))/8x5=(1-(a+b+c-d+e+f-g))
/8x6=(1+(a+b+c-d-e-f+g))/8x7=(1-(a-b+c+d-e-f-g))/8x8=(1+(a-b+c
+d+e+f+g))/8where the 7 variables a,b,c,d,e,f,g are the SQUARE
ROOTS (positive case) of 4v+1 and the v's are the roots of 
the
septic:8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0
(eq.3)with the
solution:v=2(w^11+w^13+w^16+w^18)-2(w+w^12+w^17+w^28)-(w^2+w^5
+w^24+w^27)+(w^3+w^7+w^22+w^26)+(w^4+w^10+w^19+w^25)-(w^8+w^9+
w^20+w^21)where w is ANY complex root of unity <>1 such that
w^29=1.Note: Though there would be 28 such roots, the
properties of theseroots ensure that v will ONLY have 7
distinct values.I found the solution of v in Dave Rusin's
website, and it's by P.Montgomery, though the solution 
wasn't
used in the same way I usedit. Eq.3 wasn't explicitly
mentioned there but when I used theInteger Relations applet
for a particular v, eq.3 popped out. Itlooked familiar and I
realized i saw it before wle trying tosolve the resolvent
(eq.2) of the prior post,
namely:z^7-7z^6-2763z^5-19523z^4+1946979z^3+34928043z^2+
119557031z-3247^2=0(eq.2)and by letting z=4v+1, we get the new
septic:8903+47647v+39672v^2+7192v^3-522v^4-174v^5+v^7 = 0
(eq.3)So there it is. It's so nice to have completion. :)By
the way, do SOME solvable 12-deg polys need an 11-deg
===
Extensions by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id h97D4L919274;> I have
recently extended the Quaternions to larger sets by
requiring>> some (new) group elements to commute. In doing so,
I found ts process>> and its results to be very asthetic. For
one, the law of association is>> regained. However, the
algebra involved is no longer a division algebra,>> i.e. we
may not always follow x = 0 or y = 0 from xy=0 (when x and y
are>> certain elements taken from a linear combination of
group vectors).>...>> Has ts type of tng been done before and
are its conclusions of>> interest?>>It's obvious that there
exist extensions of the quaternions H,>eg H + H (direct
sum),>or algebras of matrices with quaternion 
elements.>>You'd
have to say what properties your extension has>before anyone
could say if it is of interest.>>I am neither refering to the
ring of quaternion matrices nor to the >group>product... have
sent you a copy of ts work.Apologies for not replying to your
email ... lectures have just begun.>My point was that you
seemed surprised to find that >there were algebras extending
the quaternions,>and I noted that ts wasn't too 
surprising.So
the fact that you have constructed such an algebra>could not
be considered interesting in itself -->any interest would have
to lie in the special properties of the algebra.>-- >Timothy
Murphy >e-mail: tim /at/ birdsnest.maths.tcd.ie>(all email
over 80k dispatched to /dev/null)>tel: +353-86-2336090,
+353-1-2842366>s-mail: School of Mathematics, Trinity College,
Dublin 2, IrelandAbsolutely no need to apologize for not having
replied yet; it is good to know you recieved it, as I now asume
to be the case. The group diagram picture I made on page 4 or 5
(I forgot) and its explanation in the text should tell a lot
about the group's properties very quickly. By the way, the
same process of extension (I call it reßection) can be used
again and again to create more group elements, presumably
proceeding to gher dimensions in the process- but Im not quite
sure if ts is somehow equivilant or to the procedure for
extending Clifford Algebras (or, indeed, if my group is
perhaps a Clifford algebra in disguise. Admitted, I don't 
know
enough about Clifford algebras at the momement and am currently
checking ts possibility myself). Note also that many types of
groups can be reßected, but ts does not always give rise to a
new group. For example, the trivial group {1,-1} does not
change after reßection (the complex trivial group {1,-1, i,
===
knight metric? by support1.mathforum.org (8.11.6/8.11.6/The
Math Forum, $Revision: 1.9 primary) id h97E9gZ23741;>Take a
chessboard (with or without infinetely many squares) let the
>distance d((x_1,x_2),(y_1,y_2)) between two squares x and y
of the >chessboard be defined as the minimum number of moves a
knight takes >to reach y from x. >Is d a metric? Not trying to
suggest that ts is some new>question that hasn't been
asked/answered before. Is there a general >formula for
calculating d? More generally, the same question may be >asked
for the other pieces (queen, king, knight, bisp)? Actually,
I>asked myself ts question a few years ago. If I remember back
to the >notes I took, I had sometng like (x_1-y_1, x_2-y_2)=
(even number, >even number), then d(x,y) = even number. If
(x_1-y_1, x_2-y_2)= (even>number, odd number), then d(x,y) =
odd number. Finally, if (x_1-y_1,>x_2-y_2)= (odd number, odd
number), then d(x,y) = even number. In other words, the same
===
Pointless by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id h97E9jb23746;>mathedman
<mathedman@hotmail.cut.com> scribbled the following>on
comp.lang.c:>> Why is ts being discussed in
comp.lang.c???>Obviously because IT IS FULL OF JEWS!!! Jews
are taking over >comp.lang.c because their greedy grubby need
to take over everytng is >finally seeping into the C language.
Next tng you know they will be >trying to rewrite the
standard. The entire reason for my low IQ and >inability to
succeed in life can be attributed to jews. If it wasn't >for
all the damn JEWS in science I wouldn't have to study! They
keep >taking all the women too, being all nice and treating
them with >respect and making me look like a complete ass.
They took all the >jobs, now there is no point even looking
for one. All I can do is sit >around all day filled with self
pity and loatng for the damn JEWS who >did ts to me. I hate my
life and its all the fault of the Jew!> YOU are a total idiot.I
thought he was being sarcastic.-- >/-- Joona Palaste
(palaste@cc.helsinki.fi) ---| Kingpriest of The Flying Lemon
Tree G++ FR FW+ M- #108 D+ ADA N+++|>| <a
href=http://www.helsinki.fi/~palaste>http://www.helsinki.[Capi
talThorn]/~
palaste</a> W++ B OP+ |>----- Finland rules! />It sure is cool
having money and ccks.> - Beavis and Butt-head If he was, then
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h97FaaV30213;> I have recently
extended the Quaternions to larger sets by requiring> some
(new) group elements to commute. In doing so, I found ts
process> and its results to be very asthetic. For one, the law
of association is> regained. However, the algebra involved is
no longer a division algebra,> i.e. we may not always follow x
= 0 or y = 0 from xy=0 (when x and y are> certain elements
taken from a linear combination of group vectors).>>...> Has
ts type of tng been done before and are its conclusions of>
interest?>>It's obvious that there exist extensions of the
quaternions H,>>eg H + H (direct sum),>>or algebras of
matrices with quaternion elements.>>You'd have to say what
properties your extension has>>before anyone could say if it
is of interest.>>I am neither refering to the ring of
quaternion matrices nor to the >group>>product... have sent
you a copy of ts work.>>Apologies for not replying to your
email ... lectures have just begun.>>My point was that you
seemed surprised to find that >>there were algebras extending
the quaternions,>>and I noted that ts wasn't too
surprising.>>So the fact that you have constructed such an
algebra>>could not be considered interesting in itself -->>any
interest would have to lie in the special properties of the
algebra.>>-- >>Timothy Murphy >>e-mail: tim /at/
birdsnest.maths.tcd.ie>>(all email over 80k dispatched to
/dev/null)>>tel: +353-86-2336090, +353-1-2842366>>s-mail:
School of Mathematics, Trinity College, Dublin 2,
IrelandAbsolutely no need to apologize for not having replied
yet; it is >good to know you recieved it, as I now asume to be
the case. The >group diagram picture I made on page 4 or 5 (I
forgot) and its >explanation in the text should tell a lot
about the group's>properties very quickly. By the way, the
same process of extension> (I call it reßection) can be used
again and again to create more> group elements, presumably
proceeding to gher dimensions in the >process- but Im not
quite sure if ts is somehow equivilant or to >the procedure
for extending Clifford Algebras (or, indeed, if my >group is
perhaps a Clifford algebra in disguise. Admitted, I >don't
know enough about Clifford algebras at the momement and am
>currently checking ts possibility myself). Note also that
many >types of groups can be reßected, but ts does not always
give rise >to a new group. For example, the trivial group
{1,-1} does not change> after reßection (the complex trivial
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) with ESMTP id h97G9do32618 id
1A6uP0-0003DA-6dX-AntiAbuse: Ts header was added to track
abuse, please include it with any abuse reportX-AntiAbuse:
Primary Hostname - secure.server-3.comX-AntiAbuse: Original
Domain - mathforum.orgX-AntiAbuse: Originator/Caller UID/GID -
[32107 32107] / [47 12]X-AntiAbuse: Sender Address Domain -
secure.server-3.com<HTML><HEAD><META http-equiv=Content-Type
BGCOLOR=#FFFFFF><CENTER><TABLE BORDERCOLOR=#000040
CELLSPACING=0 CELLPADDING=0 WIDTH=754 BGCOLOR=#0065A8 BORDER=0
VALIGN=TOP><TR> <TD>&nbsp;</TD></TR></TABLE><TABLE
CELLSPACING=0 CELLPADDING=0 WIDTH=754 BGCOLOR=#FFFFFF
BORDER=0><TR> <TD VALIGN=BOTTOM ALIGN=RIGHT> <A
onmouseover=window.status='Home'; return 
true;
onmouseout=window.status=''; return true;
href=http://www.igdnycb.com/Main.html> <IMG ALT=IGDNYC Inc.
SRC=http://www.igdnycb.com/Images/Header1.gif BORDER=0></A>
</TD> <TD VALIGN=BOTTOM ALIGN=LEFT> <IMG
SRC=http://www.igdnycb.com/Images/Header2.gif BORDER=0>
</TD></TR></TABLE><TABLE BORDERCOLOR=#000040 CELLSPACING=0
CELLPADDING=10 WIDTH=754 BGCOLOR=#ffffff BORDER=1
BORDERWIDTH=1><TR> <FORM Name=SendNewsletter
ACTION=http://www.igdnycb.com/cgi-bin/Remove.cgi METHOD=POST>
<TD ALIGN=LEFT WIDTH=620 STYLE=border-width : 0> <P><BR> <FONT
FACE=Verdana, Arial, Helvetica, Sans-Serif COLOR=#000080
SIZE=2> </FONT> <BR><BR> <TABLE BORDER=1 BORDERCOLOR=#FFFFFF
CELLPADDING=0 CELLSPACING=15 ALIGN=LEFT> <TR> <TD ALIGN=LEFT
WIDTH=150 VALIGN=TOP> <A
HREF=http://www.igdnycb.com/News.shtml><FONT FACE=Verdana,
Arial, Helvetica, Sans-Serif COLOR=Blue SIZE=1><U><B>News
Home</B></U></FONT></A> <BR> <HR WIDTH=150> <BR> <HR
WIDTH=150> <A
HREF=http://www.igdnycb.com/Reports_October03_
MarketAnalysis.shtml><FONT FACE=Verdana, Arial, Helvetica,
Sans-Serif COLOR=Blue SIZE=1><U><B>Detailed Statistics and
Graphs</B></U></FONT></A> <BR> <HR WIDTH=150> <FONT
FACE=Verdana COLOR=#000080 SIZE=1><B>Average Prices By
Neighborhood</B> <BR> &nbsp;&nbsp;&nbsp;&nbsp;- <A
HREF=http://www.igdnycb.com/Reports_October03_
MarketAveragePrice_Office.shtml><FONT FACE=Verdana, Arial,
Helvetica, Sans-Serif COLOR=Blue
SIZE=1><U><B>Office</B></U></FONT></A> <BR>
&nbsp;&nbsp;&nbsp;&nbsp;- <A
HREF=http://www.igdnycb.com/Reports_October03_
MarketAveragePrice_Retail.shtml><FONT FACE=Verdana, Arial,
Helvetica, Sans-Serif COLOR=Blue
SIZE=1><U><B>Retail</B></U></FONT></A> <BR>
&nbsp;&nbsp;&nbsp;&nbsp;- <A
HREF=http://www.igdnycb.com/Reports_October03_
MarketAveragePrice_Industrial.shtml><FONT FACE=Verdana, Arial,
Helvetica, Sans-Serif COLOR=Blue
SIZE=1><U><B>Industrial</B></U></FONT></A> <HR WIDTH=150>
<FONT FACE=Verdana COLOR=#000080 SIZE=1><B>IGDNYC In The
News</B> <BR> &nbsp;&nbsp;&nbsp;&nbsp;- <A
HREF=http://search.businessweek.com/search97cgi/s97_cgi?action
=FilterSearch&ServerKey=Primary&filter=bwfilt.hts&
command=
GetMenu&ResultMaxDocs=500&ResultCount=25&resulttemplate=
bwarcv.hts&querytext=Get+Yer+Red-Hot+Office+Towers&x=9&y=2><
FONT FACE=Verdana, Arial, Helvetica COLOR=Blue
SIZE=1><U><B>Business Week</U></B></FONT></A> <BR><BR>
&nbsp;&nbsp;&nbsp;&nbsp;- <A
HREF=http://www.igdnycb.com/DailyVariety.shtml><FONT
FACE=Verdana, Arial, Helvetica COLOR=Blue SIZE=1><U><B>Daily
Variety</U></B></FONT></A> <BR><BR> &nbsp;&nbsp;&nbsp;&nbsp;-
<A HREF=http://www.igdnycb.com/TheNewYorkSun.shtml><FONT
FACE=Verdana, Arial, Helvetica COLOR=Blue SIZE=1><U><B>The New
York Sun</U></B></FONT></A> <BR> <HR WIDTH=150> </TD> <TD
ALIGN=LEFT VALIGN=TOP> <FONT FACE=Verdana, Arial, Helvetica,
Sans-Serif COLOR=#000080 SIZE=2> <UL><BR> As Manhattan office
retail and industrial vacancies continue to decline, landlords
are starting to decrease the concession packages offered to
tenants.... <BR><BR><BR><BR> <UL> <FONT FACE=Verdana, Arial
COLOR=#000080 SIZE=2> <A
HREF=http://www.igdnycb.com/Reports_October03.shtml><FONT
FACE=Verdana, Arial COLOR=Blue
SIZE=2><U><B>here...</B></U></FONT></A> </UL> <BR><BR><BR><BR>
</TD> </TR> </TABLE> </TD></TR><TR> <TD ALIGN=LEFT
STYLE=border-width : 0 COLSPAN=2> <FONT FACE=Verdana, Arial
COLOR=#000000 SIZE=2> To add to our newsletter, please <A
HREF=http://www.igdnycb.com/Add.shtml><FONT FACE=Verdana,
Arial COLOR=Blue SIZE=2><U>Go Here</U></FONT></A> <BR><BR>
<FONT FACE=Verdana, Arial COLOR=#000000 SIZE=2> To remove
<I>post-sci-math@mathforum.org</I> from subsequent
newsletters, please <A
HREF=http://www.igdnycb.com/cgi-bin/Remove.cgi?post-sci-math@
mathforum.org><FONT FACE=Verdana, Arial COLOR=Blue
SIZE=2><U>Go Here<U></FONT></A><BR><BR><img
src=http://www.igdnyc.com/cgi-bin/emailLog.cgi?email=
post-sci-math@mathforum.org height=1 width=1>
</TD></TR></TABLE></FORM><TABLE><TR> <TABLE CELLSPACING=0
CELLPADDING=0 width=754 BGCOLOR=#ffffff BORDER=0> <TBODY> <TR>
<TD ALIGN=middle colSpan=2><FONT face=Arial, Helvetica,
Sans-Serif COLOR=#000080 size=2>1841 Broadway, Suite 200, New
York, NY 10023 &nbsp;&nbsp;&nbsp;&nbsp; Tel: 212.258.2700
&nbsp;&nbsp;&nbsp;&nbsp; Fax: 212.258.3286
&nbsp;&nbsp;&nbsp;&nbsp; <A href=mailto:info.com><FONT
face=Arial, Helvetica, Sans-Serif COLOR=#000800
size=2><B>info.com</B></A> <BR></FONT></FONT></TD></TR> <TR>
<TD ALIGN=middle colSpan=2><FONT face=Arial, Helvetica,
Sans-Serif COLOR=#0065a8 size=2><BR><A
href=http://www.igdnycb.com/Main.html><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>Home</A>
&nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/AboutUs.shtml><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>Why Us?</A>
&nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/OurClients.shtml><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>Our Clients</A>
&nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/UsefulInfo.shtml><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>Useful
Information</A> &nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/Forms.shtml><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>Forms</A>
&nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/News.shtml><FONT face=Arial,
Helvetica, Sans-Serif COLOR=#0065a8 size=2>News</A>
&nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/Testimonials.shtml><FONT
face=Arial, Helvetica, Sans-Serif COLOR=#0065a8
size=2>Testimonials</A> &nbsp;&nbsp;|&nbsp;&nbsp; <A
href=http://www.igdnycb.com/ContactUs.shtml><FONT
COLOR=#0065a8 size=2 CE=Arial, Helvetica, Sans-Serif
FA!>Contact Us</A> <BR></FONT>
===
Subject: Re: The ... spacetime; answer to critic. by
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h97JmJl15824;I know it is a bit
stupid, but 1> how do you prove that a discrete topology is
metrizable?2> X is an set of all positive integers and
T={{},{1,2,3,4...},{2,3,4...},{3,4...},{4...},...} Why is
I. V. Maresin <av95@comtv.ru > As a topological space
spacetime is metrizable, but one does not, > in general, look
at any particular metric (in the topological sense) > on it. >
Why Severian tnks that ts Hausdorf topology on Space-Time is
adequate to its physical sence and geometrical structure? > I
care notng for physical sence (sic). All I was doing was
pointing out that the term metric was being used in two
different ways (like many words in mathematics).>> Do you mean
as two
ways:::::::::::::::::::::::::::::::::::::::::::::::::::::::>
### (pseudo-)Riemann metric -- quadratic form> and> ### metric
(distance)> ?>> Yes, they're different, but the 
first is
applicable only to manifolds.>> Yes, I know that, but the
original poster was confused by these two> distinct concepts
having the same word.> Of course the two concepts are related
ts way. So? The> original poster was talking about one version
of metric> in terms of the definition of the other version -
that's going> to cause confusion, regardless of the fact 
that
the two> are related, so pointing out that they are two
different> tngs seems like a good idea.The original poster. In
the original question. Where he said sometng>about metrics that
applied to one version, then asked about what>he said in re an
instance of the other version.>...And note how these concepts
are related and how it can be applied to S.T.>>seems also a
good idea...> More precisely:>> The standard topology of
Euclidean space is induced by metric.>> You are failing to
specify wch usage of the term metric you are using>
here.>>Directly, I used the metric (distance).>>But the
Euclidean distance is a partial case of Riemann distance,>>so
not important wch I use distance or quadratic form.> The
standard topology of Minkowski space isn't.>> Minkowski 
space
is metrizable.>>mmel! ### Not by Minkowski metric ! ! !He
didn't say it _was_ metrizable by the Minkowski metric.>He
said it was metrizable. It is.You said The standard topology
of Euclidean space is induced >by metric. The standard
topology of Minkowski space isn't.>We assumed that induced 
by
metric meant induced by>some metric.What _did_ you mean by
induced by metric??? If>induced by metric means induced by
Minkowski>metric then the standard topology on euclidean>space
is _not_ induced by metric...>You implicitly defined on 
it>>the
standart (Hausdorf) topology of finite-dimension linear
space>>and said that it's metrizable, as all 
finite-dimension
linear spaces!>>Original question was NOT ABOUT SUCH
TOPOLOGY,>>but IS Minkowski A METRIC on the Space-Time? .>>The
answer
is:>>_________________________________________________________
__________>>[ Minkowski metric on the Space Time ( dx0^2
-dx1^2 -dx2^2 -dx3^2 )>>[ not leads to any metric (distance
function) in classical sence.That's correct. Several people
have explained ts already. Nobody has>disagreed with
ts.>~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
The standard topology of Euclidean space is> the only
(non-trivial) topology invariant to motions (rotation, mirror,
sft).>> Is it? Proof?>>Sorry, in general it isn't. I
mistaken.>>It's true that any non-one-point and non-discrete
invariant topology>>is identical to standard IN ANY BOUNDED
DOMAIN,Say _exactly_ what ts means (and give a nt of the
proof.)Seriously, I cant figure out what it means. Because I
can't>see what it means for a topology on a BOUNDED 
DOMAIN>to
be invariant under sfts, for one tng. Nor under
rotations,>unless it happens that the domain is invariant
under rotations.>but at infinity exist some different
cases...>>Except standard topology, it may be some
COMPACTIFIED topologies.>>One of them is like standard but
accepts only bounded closed subsets>>(or, the same, only open
subsets wch are neighbourhoods of infinity).>>Is there another
cases, I don't know yet.>>, Severian! If had not you sceptic
question,>>I would ignore ts fact a long time...> The standard
topology of Minkowski space is invariant but not unique.>> In
Euclidean case, all smooth maps from R to the Space are
CONTINIOUS> and may represent a point trajectory in Newtonian
mechanics.> In the Space-Time, all smooth maps R->M are
continious (in std. top.),> but not all are physically
allowed.>> physically allowed!>>Ts mean that the 4-speed is
not space-like:>>( d x / d tau )^2 >= 0>>and that we have a
correct time sign:>>( d x0 / d tau ) >= 0> What bullst.>>***
Anecdote:>>enter expression: cos ( pi / 2 )>>Syntax
error!>>enter expression: 1 * 0>>Syntax error!>>enter
expression: 2 + 2>>Syntax error!>>enter expression: .8f.9b.90
.99.8c.89.8c>>.8f.9b.90 is not an argument!>>-->>qq~~~~>/ / >
===
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h97JmW515882;>>Take a chessboard
(with or without infinetely many squares) let the >>distance
d((x_1,x_2),(y_1,y_2)) between two squares x and y of the
>>chessboard be defined as the minimum number of moves a 
knight
takes >>to reach y from x. >>Is d a metric? Not trying to
suggest that ts is some new>>question that hasn't been
asked/answered before. Is there a general >>formula for
calculating d? More generally, the same question may be
>>asked for the other pieces (queen, king, knight, bisp)?
Actually, I>>asked myself ts question a few years ago. If I
remember back to the >>notes I took, I had sometng like
(x_1-y_1, x_2-y_2)= (even number, >>even number), then d(x,y)
= even number. If (x_1-y_1, x_2-y_2)= (even>>number, odd
number), then d(x,y) = odd number. Finally, if
(x_1-y_1,>>x_2-y_2)= (odd number, odd number), then d(x,y) =
even number. In other words, the same rules for adding natural
numbers...C.Dement>Sorry for not quite completing the question
above (even though, >perhaps, obvious):>Is d a metric over the
product space of whole/natural numbers >(corresponding to two
different cases of an infinite chess board)>or the restriction
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision: 1.9 primary) id h97JmTA15867;A grammar would beS ->
BaBB -> BBB -> aBbB -> bBaB -> lambdaI doubt there is a way to
===
Reimann by support1.mathforum.org (8.11.6/8.11.6/The Math
Forum, $Revision: 1.9 primary) id h97KV2u19406;It is possible
prove the Ternary Goldbach Conjecture (TGC) and the Twin Prime
Conjecture (TPC) are true, if the Generalized Riemann
Hypothesis (GRH) is
true.Seehttp://www.ams.org/era/1997-03-15/S1079-6762-97-00031-
0/S1079-6762-97-00031-0.pdffor the proof of GRH-->TGC. Is
there a similar paper for the converse? If the TGC or TPC is
true then, the GRH or (RH) is true. My question is can either
of the following be proved?TGC-->GRH orTPC-->GRH
===
I want a Ôlarge' computably enumerable 
collection of
functions>f_i : N --> N with the following properties:>1. Each
primitive recursive function is included.>2. Each f_i is total
by construction.>3. Given i and n in N, there is a computable
function time: N x N --N wch tells me that the value of f_i(n)
will take at most time(i,n)>to compute by a turing macne or
equivalent.>[ÔTake as long as you want, but PLEASE tell me
when you will be>done!']>4. The function time is computable 
in
ÔAckermann + constant' time,>or at least 
it's behavior is
boundedly nasty in some sense. :)There was a paper 40 years
ago somewhat along these lines. Ifyou want to be able to
predict how long a computation will take,then you have
severely restricted the class of functions. Hereis the
reference: Robert W. Ritce Classes of predictably computable
functions Transactions of the American Mathematical Society,
vol. 106 (1963), pp. 139-173It might help.--Herb
===
Gauss-JordanGauss-Jordan elimination. I know text books are
constantly usingone-way traffic ßow analysis and such, and 
I
really enjoyed tssubject. However, I am taking a technical
writing class and wanted todo a paper on using gauss-jordan
elimination to solve real worldproblems....The problem is I
can't find any resources for my research. Any 
help would be
===
by zero...??> > I'm afraid that I don't 
understand what you
mean You should only be afraid when you *do* understand what
Tapio means. Exactly! :-) Tapio> --> )>Waht is really scary is
I am sure that he's sayingLook guys,The question doesnt make
sense!Zero is really the same tng as a decimal point.Since the
decimal point is a marker, how do you divide by a marker? Why
even post to math,sci with ts kind of stuff?It's time for 
the
bus to Plonk City to leave!It's called the Cerry bus because
===
consecutive composite integers>Examining a table of factors
and primes, I found that for any sequence>of consecutive
composite numbers there is always one integer that has>a prime
factor larger than any other prime factor of any of the
other>integers. Further, ts prime is not raised to any power.
My question>is: Is ts true for all consecutive composite
sequences and if so, is>there a proof?>(Here's an example:
1500, 1501, ... 1509. The last integer has the>prime factor
503.)How does the Further,... apply to ts example: 8, 9?
Robert Israel israel@math.ubc.caDepartment of Mathematics
http://www.math.ubc.ca/~israel University of British Columbia
===
metric?>Take a chessboard (with or without infinetely many
squares) let the >distance d((x_1,x_2),(y_1,y_2)) between two
squares x and y of the >chessboard be defined as the minimum
number of moves a knight takes >to reach y from x. >Is d a
metric? With ts distance, the triangular equation is obviously
true, and that makes it a metric. > Not trying to suggest that
ts is some new>question that hasn't been asked/answered
before. Is there a general >formula for calculating d? More
generally, the same question may be >asked for the other
pieces (queen, king, knight, bisp)? Actually, I>asked myself
ts question a few years ago. If I remember back to the >notes
I took, I had sometng like (x_1-y_1, x_2-y_2)= (even number,
>even number), then d(x,y) = even number. If (x_1-y_1,
x_2-y_2)= (even>number, odd number), then d(x,y) = odd number.
Finally, if (x_1-y_1,>x_2-y_2)= (odd number, odd number), then
d(x,y) = even number. In other >words, the same rules for
===
to satisfiability> Why should all the clauses in ts formula
have at most three literals?> You can't necessarily 
represent
a function in N variables using a 3-CNF> formula. The number
of functions that can be represented that way is very> small
by a counting argument: there are 2^(2^N) functions in N
variables,> but only 2^(O(N^3)) possible 3-CNF formulas. What
are the odds that> factoring is one of those lucky functions
for interesting values of N?You are correct.I can't assume
that the formula can be reduced to 3CNF unlessI am willing to
allow auxiliary variables.One of the references given by
Eppstein:The Propositional Formula Checker HeerHugo, J. F.
Groote and J. P.Warners,
http://ftp.cwi.nl/CWIreports/SEN/SEN-R9905.pdfstates:using
auxiliary variables, any formula can be put into (<=3)CNF
formin linear time.Your counting argument shows why the number
of auxiliary variablestends to grow rapidly. Let N be the
minimum number of variablesin a formula and M be the number of
variables after adding theauxiliary variables.2^(2^N) =
===
Folk Psychology and Social Convention>The peephole you view
your world through is very narrow, Longley.>There are many
here who tnk the following matters are the crux of>building
AI's:How do you recognize what you see? >How do you know how
to move your arm? >How do you choose wch words to say?>How do
you understand what they mean? >How does commonsense reasoning
work? Stop misquoting Lennon & McCartney! how do you recognize
just what you've seen? > i can't tell you but 
i know it's
mine> how do you know how to move your own arm? > i get by
with a little help from my friends> how do you choose wch big
words to say?> lend me your ears and i'll sing you a song> 
how
do you understand what they mean? > i try not to sing out of
key > does commonsense reasoning work?> yes i'm certain that
it happens all the time > oh building AIs with a little help
from my friends> Brilliant, Lisa - I hope the originator of
the quotes I gave can makeuse of your added
into my bed> and sought some rest for limbs with dust
attired,> but there began a trial in my head> to obsess, when
body's overtired:> my thoughts then race, unlike their daily
pace> when they escape and leave me little grace;> for verbal
slings and arrows of bon mots> i am left gawking bereft of
ripostes.> look in darkness on what might have been said> had
i the wits to speak as those admired:> rejoinders, like
galaxies, hang in space> and taunt me in my unrestful repose.>
ts sort of pesky followup is never very far> when you keep
sending crosspostings to here, talk dot bizarre. > .> . i'll
be here at dogberry's fencing school all weekendWhy ts 
thread
was x-posted to the other forums, I don't know - 
butthere's
clearly a lot more humor on t.dot.b than on c.a.p. - bon
mots,rather than non bots [... had i the wits ...]. So, I
===
Re: Numeric one-way hash functionAs usual, I'm missing some 
of
the intermediatepostings, so ts is a reply to all of the
partiesso far.>> I need to find an algorithm that can produce 
a
unique non-predictable 12> digit (0-9) number for any given 12
digit number. Ts is to be used to> create a unique barcode on
a ticket that cannot be predicted. It is not> required that
the original seed number be computed from the resulting>
barcode, so some form of one-way hasng function would be
acceptable.> Any help in ts problem would be appreciated.that
you've thought out your threat model. Yousay that the hash
doesn't need to be reversible,so what happens if I just make
up a 12-digitbarcode and print my own ticket? I have a
funnyfeeling that if you present the actual problemhere, you
might get other suggestions for solvingit or have possible
problems with your proposedsolution identified.For example,
your use of the word ticket makesme tnk authentication. For
that you probablyreally want sometng like a ticket number and
aMAC appended to it. Or sometng.>> I've seen wiser heads 
than
mine recommend a Ruby-Lackov cipher for >> ts kind of tng....
[snipped] ...>> ... [I tnk Ruby-Lackov can >> tolerate a small
amount of bias in f. If not, I'm sure someone will post >>
another suggestion.]Depends what you mean by tolerate. The
securityproofs for Luby-Rackov certainly don't hold up
ifthere's any knowledge at all of the f() functions.Wle on
that subject, I'll also point out thatthe proofs require 
four
*independent* f()functions, not one that is reused. You
cansimulate ts with f_n = hash(key, n, data) for n = 1..4.That
said, for such a small data input, you'dprobably be safe
ignoring those two nits. Problemsare more likely to surface
somewhere else.>You would have to be careful in the selection
of your hash function.>All standard hash functions have 2**n
different outputs, and>I don't know any hash function that
produces 10**6 outputs. An example of a bad hash function
would be to take the first 20 bits>mod 1000000 of a standard
cryptograpcal hash, because ts hash>function is extremely
biased (some values occur twice as often as the>others). When
you use a larger number of bits the bias is reduced.Yes,
there's a regular subject here about how toget unbiased
uniform random in [0..N-1] given apseudorandom bitstream (such
as generated by ahash function), avoiding ts bias thatcreeps in
when you least expect it.Greg.-- Greg Rose232B EC8F 44C6 C853
D68F E107 E6BF CD2F 1081 A37CCrypto Mini-FAQ:
http://www.schlaßy.net/crypto/faq.txtQualcomm Australia:
===
applications for Gauss-Jordan> Gauss-Jordan elimination. I
know text books are constantly using> one-way traffic ßow
analysis and such, and I really enjoyed ts> subject.I tnk you
are asking when might one use numerical methods tosolve large
systems of linear equations.One application is in economic
planning using Leontief input-outputplanning. I actually read
a science fiction novel last week jokedabout ts application.I
have Gauss-Jordan elimination built into my applet for
LinearProgramming mentioned in my sig. LPs have a wide variety
ofapplications.-- Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html To solve Linear Programs: .../LPSolver.htmlr c A
game: .../Keynes.html v s a Whether strength of body or of
mind, or wisdom, or i m p virtue, are found in proportion to
the power or wealth e a e of a man is a question fit perhaps 
to
be discussed by n e . slaves in the hearing of their masters,
but ghly @ r c m unbecoming to reasonable and free men in
===
polynomial problemhttp://www.giganews.com/info/dmca.html>
all,I'm trying to prove the following theorem:Let P be a
polynomial with real coefficients such that P(x) >=0 for>every
real x. Then, there are polynomials R and S such that P(x)
=>R^2(x) + S^2(x) for every complex x.It's easy to see that
the degree of P must be even. If r is a real>root of P, then
the restriction of P to the reals has an absolute>minimum at r
and, from the differentiability of P, it follows there's>an
even number k such that the k-1 first derivatives of P vanish
at r>and the k_th is positive. Therefore, the k-1 derivatives
admits the>root r with multiplicity 1, wch implies P admits r
as a rooth with>multiplicity k. So, we see every real root of
P, when they exist, must>have an even multiplicity (I tnk we
could come to ts same>conclusion a bit faster, considering
only the continuity of polynomial>functions).Corollary - If
all of the roots of P are real, then, P = Q^2 for
some>polynomial Q. So, for ts particular case the theorem has
just been>proved.To prove the theorem, for the general case, I
tried to use>mathematical induction. It didn't work, 
that's why
I'm asking for>help. What I did is as folows:the previous
paragraph, it's also enough to cover the case of>even-degree
polynomials with real coefficients and no real roots. 
It's>well
known that every monic trinomial T of the 2nd degree
that>satisfies such conditions can be written as T(x) = 
(x-a)^2
+ C^2,>where a and C<>0 are real. Therefore, for such
trinomials the theorem>holds trivially. Now, suppose there's 
a
natural k such that the>theorem holds for i=1,...k-1 for every
2i-degree polynomial with real>coefficients and no real roots.
If P is a 2k-degree polynomial with>these same properties,
then at some (or several) real r's the>restriction of P to 
the
reals attains an absolute minimum m>0. Ts>implies that, for
every real x, the polynomial P-m is non-negative,>has one (or
several) real root(s) and (i) P(x) - m =
(x-r_1)^p1>*...(x-r^_n)^p_n * Q(x), where r_1, ..r_n are the
real roots and the>numbers p_1,...p_n are even. In addition, Q
is monic, has even degree>< 2k, has no real root and is
strictly positive on the real line. By>the induction
assumption, the theorem holds for Q and, in virtue of>(i),
also holds for P-m. But now, to complete the induction,
it>remains to prove the theorem is good for P, in other words,
it remains>to prove that if the theorem holds for some
polynomial P then it holds>for P+m for every m>0. That's 
where
I got stuck.Actually, I tnk I chose a very cumbersome way to
prove the theorem,>there certainly is a neater proof.I believe
there is. The real roots all have even order and the
complexroots come in conjugate pairs - ts means there exists a
polynomialF (with complex coefficients) such that P is the
product of F and thecomplex conjugate of F...>Any suggestions
===
can produce a unique non-predictable12>> digit (0-9) number
for any given 12 digit number. Ts is to be used to>> create a
unique barcode on a ticket that cannot be predicted. It is
not>> required that the original seed number be computed from
the resulting>> barcode, so some form of one-way hasng
function would be acceptable.>> Any help in ts problem would
be appreciated.> The simplest way is to encrypt the first
number using AES or> 3DES. You will have to convert the result
from binary to decimal.> Any extra digits can be thrown away.
Change the key variable> regularly and keep the old ones
secret. But shouldn't the bar codes be unique?> Your 
procedure
can generate duplicate bar codes.>With gh grade ciphers like
AES there will be very fewduplicates. If you do not know the
key variable thenthe conversion is unpredictable.The simplest
way to ensure that the bar codes areunique is to add a prime
number to the previousvalue. Lap round when you get to the top
(or a primenumber near the top). Start at a weird
===
Certainly it is possible to formulate the idea of a minimal
5-chromatic>> graph in a way you might find more pleasing. For
instance, we could say>> that it's equivalent to a graph 
that
is 5-chromatic in such a way that>> for every vertex v there
exists a 5-colouring of the graph in wch v>> is the sole
vertex with the colour blue.>The only 5-chroma graph that I am
likely to find pleasing is K5! Fair enough, but there are 
other
minimal 5-chromatic graphs besides K5> even if you aren't
pleased by them :). For example, glue two regular> pentagonal
cones together at the base to get a polyhedron with 7
vertices,> and form the natural adjacency graph on those
vertices (of course, we get> a planar graph). Then add one
more edge joining the apex vertices. The resulting graph is
5-chromatic, but removing any vertex, no matter> wch one,
always gives a graph that is 4-chromatic (and also planar,> if
I'm not mistaken). Personally I find it just as 
pleasing as K5
:).> The 5-chroma graph is non-planar and therefore, cannot be
===
Re: Minimal Graph, Four Color Theorem > No, you are confused
because you read badly. Nobody has been arguing that> the four
color theorem is false. Nobody!> I will concede that you are
not arguing against the FCT if you willconcede that I have no
===
points of Qare the limit points of Q exactly R?well, I know ts
is true, and I can prove it. the reason i ask thoughis because
i doubt myself on ts.let (xn) be a sequence in the set of all
rationals 0<=p<=1. Further,suppose xn is an enumeration of the
rationals between 0 and 1. i tnkthat all rationals between 0
and 1 are cluster points of xn, mainlybecause all are limit
points. i could also work it from the definitionsince a
rational lies between any two rationals, but i starteddoubting
myself for some reason.can someone tell me if I am wrong about
===
group. i'm extremely confused as to how to change the system
of ode's into amatrix version.let's say you 
have the the
lorenz system. { x'[t] = - a*y[t] - a*z[t], 
y'[t] = r*x[t] +
y[t] - x[t]*z[t] , z'[t] = x[t]*y[t]- b*z[t]}where a, r, b,
are constants, how do i represent the above as a matrix
differential equationssystem?and then find the eigenvalues and
===
another question about circular sectors :)Oops!> When I run
your numbers backwards, I get L => 1.4142.> Using your first
answer where theta = 90 deg (or Pi[/2 you meant]radians) and
radius = 1: L = 2 (r * sin(theta/2)) = 1.4142> And, your
second where theta = 180 deg (Pi radians) and radius
=1/sqrt(2) => 0.7071: L = 2 (r * sin(theta/2)) = 1.4142>
Please see my above post again ; L is the SEMI- chord, not
===
single equation, such as f2(x,y,z)=c2, can describe in a 3d
space>a surface, possibly a plane, but not a line. I could
*swear* that {(x,y,z) in R^3 : x^2+y^2=0} was a line,> last
time I looked. Lee RudolphTs actually happened.Many years ago
a problem similar to ts came up in a CalcIII class I
wasteacng.I mentioned that situations like ts are called
Degenerate caseA girl raised her hand and saidBut that's 
what
===
differential equation.> group. i'm extremely confused as to
how to change the system of ode's into a> matrix version.
let's say you have the the lorenz system. { 
x'[t] = - a*y[t] -
a*z[t],> y'[t] = r*x[t] + y[t] - x[t]*z[t] ,> 
z'[t] =
x[t]*y[t]- b*z[t]} where a, r, b, are constants, how do i
represent the above as a matrix differential equations>
system? and then find the eigenvalues and so on. You want to
write [x y z]^T as a column vector: [ x ] X = [ y ] [ z ]and
then express the system of DEs as follows: X' = AXwhere A is
some 3x3 matrix.Since the ith row of the product AX is the
product of the ithrow of the matrix A with the single column
of X, and since theindividual equations are sums of (sometng)
times (x,y,z) invarious combinations, what you try to do is to
populate thematrix A with suitable coefficients that make that
happen.For instance, x' = -a y - a zcan be written as 
follows:
[ x ] x' = [0 -a -a] [ y ] [ z ]Note how the 
coefficient of x
(namely, 0) got put into thex position, the coefficient of y
(that is, -a), and thecoefficient of z (again -a), were placed
into the y and zpositions, respectively, of the row vector.The
equations for y' and z' are not remarkably 
different,except for
the fact that you can't use constant 
coefficients,but then ts
isn't a DE with constant coefficients.Once you 
have each DE
written in ts form: x_i = R_i*Xwith a row vector R_i, then put
them together to build thematrix A, mentioned earlier,
===
The Bible Code> it's not even wrong. it's just 
simple
numbertheory --> skipcodes are that, and they were apparently
used> by (some?) Torah writers/copyists to ensure accuracy,>
as with the old CRC in 8-bit communications programs.> I read
taht they summed the letters on every 70th,> or skipped to
every 70th, or some tng.> the computer can be set to find any
message> in any ring of an alphabet, and Drosnin et al know ts
... or> maybe they can't learn it, not because 
they're dumb.>
there was ambiguity in _The Bible Code_ taht he ignored,> like
with the variant translations and the fact that> Old Hebrew has
no vowels. repeat, _War and Peace_ or just the 26 letters in
any order> can be used with the infinite set of co-prime
skips,> with teh resulting ts being further massaged> into
some m by n array (or what ever).FAILED.> >Is ts accurate? And
does ts say sometng about the New Testamentand the belief in a
Christ figure? --les ducs de Buffet et Schulz (R&D Chair
Assoc.Intl.);> vote NONE OF THE BELOW> on Trickier Dick
Cheney's California Recall/e-Dereg!> http://larouchepub.com>
http://members.tripod.com/~american_almanac/A defective theory
when applied does neither prove or disprove a hypothesis.The
Code stuff has been applied successfully to Moby Dick to
predictmoderns events with Astounding accuracy. (NOT!)rj
===
matterThe inertia of a slowly moving object; body or mass of
material substancewill increase, or decrease; in proportion to
an increase, or decrease in itsspeed, will it not?Can ts be
plotted as a component of its motion? Maybe called its
momentum,or impetus?If so, what would ts component look like
===
matrix differential equation. ... stuff deleted ...Some stuff,
with a badly-spaced equation:> For instance, x' = -a y - a z
can be written as follows: [ x ]> x' = [0 -a -a] [ y ]> [ z 
]>
I just have to fix the spacing here. Won't be a 
minute. [ x ] x'
= [0 -a -a] [ y ] [ z ] Let's see whether that works.> 
Dale.>
===
QuestionI would greatly appreciate some comments upon the
correctness of theassertion about the following equation (1)
under the given conditions. y = (a^m + b^m)(a^m - b^m)
(1)Conditions: (y, a, b ) = 1; m is a non-integer > 0; prime p
> 3.Assertion: If y is a p-th power then both a^m + b^m and a^m
===
will not as easy to lie in future! and don't set the group
header on me>my mistake, cutting a group is fine, I jumped to
the conclusion you set the forward,I often find myself posting
just to alt.kibble once day i'll go there and 
it'll beall
===
easy to lie in future! and don't set the group header on me>
my mistake, cutting a group is fine, I jumped to the 
conclusion
you set the forward,> I often find myself posting just to
alt.kibble once day i'll go there and it'll 
be> all posts from
me.>dole office took nearly 2 hours btw, had to murmur a 
sermon
to everyone to keep themquiet, then next month you can all
tune into it word for word on everyone loves raymondor nic
cages new prison scene.Hercaka the very poor star who gets
===
Riemann-Zeta Hypothesis possibly be explained to a math
majorwho has only just begun to study Real Analysis? What is
thesignificance of zero solutions lying on a critical line? 
And
whatsubstantial mathematical theorems are dependent on a
hypothesis thathas yet to be proven?I feel like a twelve year
old leafing through Wiles' proof of 
Fermat'sLast
===
there any algorithm to calculate the deep holes of leech
lattice? Yes, it's not hard to find one that 
calculates all
holes of any lattice.Could you please tell me the method? I
probably don't need all the deep holesbut I wanna 
find some
deepholes. a lot! Any software can do that? I tried MAGMA but
MAGMA gave no result I'm sure that on a lattice as 
complicated
as Leech any known> algorithm would be totally intractible. I
read SPLAG however I am not clear about it Pity. I wanna have
the whole collection of leech lattice deep holes is the best
account you're going to find. Finding> the Leech 
lattice holes
was a major piece of research. Although> finding holes is a
computable problem, just running some> general-purpose
===
Factorial/Exponential Identity, Infinitylim n->oo ((sum n)^n -
sum(n^n)) / n!^2 = 1(sum n)^n - sum(n^n) = n! ^2(sum n)^n =
sum(n^n) + n! ^2((n^2+n)/2)^n = sum(n^n) + n! ^2(n^2+n)^n =
2^n ( sum (n^n) + n! ^2 )n^n (n+1)^n = 2^n ( sum (n^n) + n! ^2
)The sum of the integers from {1, 2, ..., n ,...} to the 
n'th
power,(1+2+...+n+...)^n is greater than the sum of the numbers
of the n'thpower {1^n, 2^n, 3^n, ..., n^n, ...}, (1^n + 2^n 
+
... + n^n + ... ). Their difference is the square of the
factorial of n: 1^2 2^2 ...(n-1)^2 n^2 .... That is to say,
lim n->oo ((sum n)^n - sum(n^n)) /n!^2 = 1.Stirling's
approximation for n! is as so:lim n->oo ( n! e^n ) / ( n^n
sqrt(2pi n) ) = 1Thusly:lim n->oo ( (sum n)^n - sum(n^n) )
e^2n / ( n^2n 2 pi n) ) = 1lim n->oo ( (n+1)^n - ( sum(n^n) /
n^n ) ) e^2n / (2^n n^n 2pi n ) =1lim n->oo ( (n+1)^n -
(sum(n^n)/n^n) ) e^2n / ( 2^(n+1) n^(n+1) pi) =1As described
earlier in ts thread, other equations describe n! inthe limit.
Euler lived in the 1700's. Euler did some amazing work.I 
wonder
if there is a closed for sum(n^n). There are closed formsfor
sum(n^x), they are defined in terms of Bernoulli polynomials
thecoefficients of wch are generated by recurrence 
relation.For
small values of n, sum(n^n):n = 1, sum (n^n) = 1n = 2, sum
(n^n) = 5n = 3, sum (n^n) = 36n = 4, sum (n^n) = 354n = 5, sum
(n^n) = 4425With (sum n)^n:n = 1, (sum n)^n = 1n = 2, (sum n)^n
= 9n = 3, (sum n)^n = 216n = 4, (sum n)^n = 10000n = 5, (sum
n)^n = 759375With n! ^2:n = 1, n! ^2 = 1n = 2, n! ^2 = 4n = 3,
n! ^2 = 36n = 4, n! ^2 = 576n = 5, n! ^2 = 14400It appears that
as n diverges that (sum n)^n - sum (n^n) ~= (sum n)^n,yet it is
of course less than (sum n)^n, and it appears to be n! ^2.I
guess I should start calculating ((sum n)^n - sum (n^n)) /
n!^2 forsmall and increasing values of n and see if it holds
true that ittends towards unity.I read Richmond and 
Merlini's
paper, as mentioned earlier, aboutgeneralizations of Stirling
cycle numbers [ n x ] to complexarguments where n-x is an
integer, and don't understand it but itlooks interesting. 
I'm
more interested currently in finding closedform solutions for 
[
n+1 n-x+1 ], or rather |s(n+1, n-x+1)|, toevaluate
(n+1)(n+2)...(n+n), in the strange case of the
===
= To Sum Of Some Divisors Take an n-by-n-grid, n>= 3.> > Place
the integers 2 to (n^2 +1) into the grid, one DISTINCT integer
per grid-square, so that:> > If s(k,j) = a grid-square (ie. an
element of an n-by-n matrix), then (for all k and j where n >=
k >= 3 and n >= j >= 1) s(k,j) = (any divisor >= 2 of
s(k-1,j)) + (any divisor >= 2 of s(k-2,j)),> and> (for all k
and j where n >= k >= 1 and n >= j >= 3) s(k,j) =(any divisor
>= 2 of s(k,j-1)) + (any divisor >= 2 of s(k,j-2))[...] an n=3
example is: 5 3 8 2 6 4 7 9 10> Is there an n=4 example?? There
appear to be 2 basic solutions and of course their transposes:>
5 2 7 9> 3 12 6 15> 8 4 10 14> 11 16 13 17 5 3 8 11> 2 12 4 16>
7 6 10 13> 9 15 14 17 11 2 13 15> 3 12 6 9> 14 4 16 8> 5 10 7
17 11 3 14 5> 2 12 4 10> 13 6 16 7> 15 9 8 17 wch a bit
tediously could be extended to 5x5 or 6x6> but not much
further because it uses nested for loops,> one level per cell,
rather than a recursive approach. > The program takes about 1
second to exhaust the 4x4 case.> -jiwAhh...So there ARE
solutions after all!I was neglecting the likelyhood of bigger
integers, such as the 11 and12, being in the upper-left,
===
>The only minimal counter-example to the FCT is K5! No, K5 is
NOT a counterexample to the Four Color Theorem, because the> 4
color theorem states that any ->planar<- graph can be colored
with> at most 4 colors in such a way that no two adjacent
vertices share the> same color.The conjecture that there
exists a 5-chroma graph may be recolored to>4-chroma is false.
There is no such conjecture. Let H be any subgraph of G, where
G has n vertices and H has n-1>vertices. Then, the description
of H seems to imply that the deletion>of Ôany' 
vertex from G
will make c(H)<=4. Ts is true if G is a minimal counterexample
for the 4 Color Theorem.But ts interpretation is generally
false and is valid only for>n=5!!! The triple exclamation
points make you look like a raving loon. So> start by removing
them.Point taken, . Could you explain why?> Then note that the
original argument started by ->assuming<- that the> FCT is
->false<-, from wch we deduce that if ts is the case, then>
among them, there is one with the least number of vertices.
Call n the> number of vertices of ts HYPOTHETICAL
counterexample. Then, by the> definition of n, any graph with
fewer than n vertices must be> 4-colorable. In particular, if
you took ts HYPOTHETICAL example G,> and removed one vertex,
then the resulting graph would be 4-colorable. What exactly
are you having trouble understanding about the above>
argument? Try to answer without using a ->single<- exclamation
point.> I understand the argument perfectly. I have given the
problem somethought and I have concluded that HYPOTHETICAL G
is impossible. Nograph meets all three criteria; ie, G is
===
Antidiagonal, Infinity >What I propose is that given
anyrational that the value greater than it and less than any
othergreater is irrational,> > There is no such number, as
several different people have shown you. In non-standard
analysis, there might be, however.> See Alain Robert's book
about NSA. Rather than being> irrational, it would be
non-standard, though. I have yet to see any standard or
non-standard model of the reals in > wch there is a smallest
positive number. In the various non-standard versions, there
tend to be rather more > numbers between any positive number x
and zero, there are all those > y such that y/x are 
ifinitesimal
but positive, then all those z such > that z/y is 
infinitesimal
but positive, and so on ad infinitum.Between any two odd
integers is an even integer, between any two evenintegers is
an odd integer. The density in their union of either isone
half.Here I equate density with measure in the unit interval.
I don't care if you ignore gravity, it won't 
do you much good,
I'mhere only concerned with considering a model where the
rationals andirrationals alternate in the reals.If there are
more irrationals than rationals and rationals andirrationals
are disjoint and distinct, then, where they are eachtotally
ordered, then there necessarily would be irrationals with
norationals between them. Yet, there are not.I'm trying to 
tnk
of a function between the unit interval's realsand 
irrationals.
The claim is that one exists because the rationalsmap onto the
integers and the integers don't map to the reals, thusthat 
the
irrationals map onto the reals else the reals would be aunion
of two sets that don't map onto the reals. Yet, a
constructionexplicitly mapping each element of the irrationals
to each element ofthe reals is not given. I'm also still
looking for a mapping betweenR[0,1)^N and R[0,1).I like to tnk
that the rationals and irrationals alternate and thatthe
function f(x)=x+iota maps Q[0,1) onto P(0,1), and f(x)=x-iota
mapsQ(0,1] to P(0,1).Then again I tnk the impulse function
evaluates to half infinityat zero, and consider the Gamma
function on negative integers to havevalues of various finite
multiples of a scalar infinity.Now I'm looking 
at the post
about mapping R <-> P. I don'timmediately grasp vector space
over a field and linearlyindependent. You have the sequence b
being a sequence of reals eachlinearly independent over Q, and
a set C of reals of {b_0, b_1, ...}linearly independent over Q,
with the initial sequence element b_0being a rational. RQ=P,
you claim that R injects into P byf(b_n)=b_{n+1} and f(c)=c.
Why do you have braces around n+1 insteadof parentheses? Then
you have F(c)=c, for c in C. I tnk you meanthat c in C is not
an element of the sequence b. Then you say toextend that to
all of R by linearity over Q. So you claim a functionf(r)=p
for r in R and p in P to be defined for all reals. 
What's rfor
f(r)=pi? What's p for f(2)? Why f and F, presumably a
sft-keyerror?http://mathworld.wolfram.com/
LinearlyIndependent.htmlhttp://mathworld.wolfram.com/
VectorSpace.htmlhttp://www.wikipedia.org/wiki/Vector_space:A
set V is a vector space over a field F, if given an
operationvector addition defined in V, denoted v+w for all v, 
w
in v, and anoperation scalar multiplication in V, denoted a*v
for all v in V and ain F, the following 10 properties hold for
all a, b, in F and u, v,and w in V:1. v+w E V2. u+(v+w) =
(u+v)+w3. v+0 = v4. v-v = 05. v+w = w+v6. a*v E V7. a*(b*v) =
(a*b)*v8. 1*v=v9. a*(v+w) = a*v + a*w10. (a+b)*v = a*v +
b*vThose each hold for V = R and F = Q. Properties 1 through 5
indicate that V is an abelian group undervector addition. The
intersection of all subspaces containing a given set of
vectorsis called their span; if no vector can be removed
without diminisngthe span, the set is called linearly
independent.So you say each element of the sequence represents
a set of vectors ora set of a vector, I'm not sure wch, and
that it is linearlyindependent over Q because removing that
vector from the set ofvectors would diminish the span of the
intersection of the subspacesof the vector space.Please neaten
that up provide a more self-contained explanation. Alsoexplain.
Wle you're at it show a bijection between R^N and R.Some 
talk
here is about the nosntandard treatment of the reals:
thehyperreals. One tng to note is that *R, the hyperreals, as
a setcontains the same elements as R, the reals. It's just a
different wayto consider them.About the uniform probability
distributions over intervals of reals,that's not about 
making
some new definition of what a probabilitydistribution is. 
It's
about applying the characteristics of aprobability
distribution to an infinite population. We were talkingabout
the probability of an infinite binary seqence having one
elementbeing on, the rest off. That probability is expressed
as n/2^n, as ndiverges to infinity. The probability of any
possible sequence isequal to 2^n/2^n, in the limit: one. So
anyways out of those npossible sequences with one on bit and
the rest off bits, each isequally probable. The probability of
each among all possible infinitebinary sequences is 
being1/2^n,
the probability of each among allinfinite binary sequences 
with
one on bit is 1/n. So a theoretical(read: thought experiment)
method to generate an element of N is toonce again ßip
infinitely many coins. At ts point it's a crazy, 
orrather,
unconventional thought experiment in that the first coin
tosssays whether it is oo/2 or greater or less than oo/2.
Assume it's along sequence of zeros, then it would be saying
about whether theresult is greater than or equal to oo/4,
oo/8, oo/16, etcetera. Without a method to generate a sample
from a uniform distribution overthe natural numbers, it's
still that the probability of selecting anyis 1/|N|.Of course
that's ludicrous but at the same time it allows us 
toconsider
the realm of thought in concern of ts issue and to thentalk
about the probability of selecting a given element of the
naturalintegers assuming a uniform probability distribution
over theintegers. At least we seem to have some agreement that
a uniformprobability distribution over an interval of the reals
exists, and asimple method to sample an element of an interval
of the reals exists.infinitesimals, it talks about
1-infinitesimals, 2-infinitesimals,etcetera, 
n-infinitesimals,
===
Divisibility Of A Derivative By...(Calculus /#-Theory)Ts is a
slight generalization of the theorem in thePrime-Derivative
Puzzle
rnum=21&prev=from mid August.Let q and r be any positive
integers.Let, for all x where -1 < x < 1, f(x) =
(1-x)^((1-x)^(-q)) *(1+x)^((1+x)^(-r))In ascii-art mode: f(x)
= -q -r (1-x) (1+x)(1-x) *(1+x)Then:GCD(q+r ,m)always
dividesthe (m+1)th derivative of f(x) at x = 0.(Ts derivative,
and all derivatives, of f(x), at x =0, areintegers.),Leroy
===
have been arguing the question, &#8220;Two coins were ßipped,
and> at least one is a head. What are the chances for two
heads?&#8221;, in> sci.math, for some time. I argue that the
correct answer is 1/2. Unambiguously!!! Our question, as
written, has correct answer 1/2. Dr. Holt should> concur.> To
answer 1/3, you must assume a slightly different question.
When we> are willing to assume a different questions, we can
get different> answers. Eldon Moritz Apparently there is
sometng missing from the exposition wch was not> included in
the (extensive) argument I snipped. Ts is a standard logic>
problem, wch yields to the following: 1) There are exactly
four possible outcomes for ßipping two coins, HH,> HT, TH,
TT.> 2) For Ôfair' coins, any of the four 
outcomes is equally
likely.> 3) There are 3 outcomes with Ôat least one 
head'.> 4)
If we choose the universe of discourse as the set of trials
with Ôat> least one head', the occurrence of two 
heads will
happen 1 out of 3 times. In short, if you run the experiment
100 times, 75 results (should) have> Ôat least one 
head'. 25
(should) have two heads. 25/75 = 1/3. Are you discussing a
different problem?> There are four equally likely outcomes for
the toss. Prior to thestatement at least one is a head.After
the statement, there are three left, they are no longer
equallylikely. It is more likely to get the 
Ôheads' statement
with HH, thanwith HT, or with TH.That is, assuming that the
statement is true. As I showed earlier,assume it false and you
assume a different question.Eldon> --> There are two tngs you
must never attempt to prove: the unprovable --> and the
obvious.> --> Democracy: The triumph of popularity over
===
be defined using repeated exponentative closure?>By the
exponentative closure F, I define F/x as the set of all
the>zeroes of all the polynomial functions with coeffeicients
AND>exponents in F. For example, the the algebraics are the
exponentative>closure of the integers. Thus, it can be written
A=Z/x. Does>C=A/x? If not what does A/x equal? Can C be
generated by>repeatedly exponentatively closing the integers a
finite number of>times? If so, how many? A countable number of
times? An uncountable>number of times? Unless I misunderstand
you, F/x is countable if F is countable. So no,> a finite or
even a countable number of exponentative closures won't> do
it: the union of countably many countable sets is countable. I
don't> know about an uncountable number of times. Robert 
Israel
israel@math.ubc.ca> Department of Mathematics
http://www.math.ubc.ca/~israel > University of British
Columbia > Vancouver, BC, Canada V6T 1Z2A further question
along those lines is what is is the set, E, ofnumbers
generated by repeated sums of products of rational powers
ofrationals? It is a subset of the algebraics, but does it
form afield? What can be said about the set of sums and
products ofelements of E to the power of elements of
===
numbers in a decreasing sequence.Say, take 2 numbers from
{1,2,3} and arrange as {2,1}. What is thenumber of possible
===
be explained to a math major> who has only just begun to study
Real Analysis? What is the> significance of zero solutions 
lying
on a critical line? And what> substantial mathematical theorems
are dependent on a hypothesis that> has yet to be proven? I
feel like a twelve year old leafing through 
Wiles' proof of
Fermat's> Last Theorem.Interestingly, there are no fewer 
than
four recent pop bookson the Riemann Hypothesis. (Probably
because of itsappearance on the Clay Mathematics Institute
list of problems.)One sees: Derbysre, Prime Obsession du
Sautoy, The Music of the Primes Sabbagh, The Riemann
Hypothesis Edwards, Riemann's Zeta 
FunctionI've read only
Derbysre and du Sautoy, anddespite the fact that Derbysre is a
journalist and columnist,he illustrates (not proves) more math
than du Sautoy,who teaches math at Oxford (though he is a
journalisttoo). Both of these have a (fairly consistent)
accountof the story, but Debysre spends more time ingiving a
taste of complex analysis, wle du Sautoyis broader in s
approach to the story parts, especiallymore recent stuff.
===
ts?> Express the following as a single fraction: 4/3ab -
5/6bc> (m^2 + 2)/(m^2 + m) - (m - 2)/m>You do it the same way
you do it for fractions in arithematic.The general formula is
derived thus a/b + r/s = as/bs + br/bs = (as +
===
out of y numbers in a decreasing sequence.> Say, take 2
numbers from {1,2,3} and arrange as {2,1}. What is the> number
of possible combinations ?y choose x (assuming the y numbers
===
you please explain these integrators differece?1.
1/(1-z^(-1))2. z^(-1)/(1-z^(-1))3. Tz^(-1)/(1-z^(-1)) a
===
Identity) A propos, here are cites from Pierre Bourdieu's>
_Language & Symbolic Power_ (the titles are mine).> Enjoy!
Censorsp There Oughta Be A Law Cool-Hand Luke > The Social
Conditions for the Effectiveness of Ritual Discourse Heretical
Discourse> The *Skeptron*The skeptron is passed to the orator
before he begins s speechso that he may speak with authority
(......). It is an attributeof the person who brings a
message, a sacred personage whosemission is to transmit the
message of authority.E. Benveniste, in Indo-European Language
and
Society<http://www.angelfire.com/ca/sanmateoissues/Qualia3.html

>If, as Austin observes, there are utterances whose role is
not onlyto Ôdescribe a state of affairs or state some 
fact',
but also to'execute an action', ts is because 
the power of
words resides inthe fact that they are not pronounced on
behalf of the person whois only the Ôcarrier' of 
these words:
the authorized spokespersonis only able to use words to act on
other agents and, through theiraction, on tngs themselves,
because s speech concentrates witnit the accumulated symbolic
capital of the group wch has delegatedm and of wch he is the
*authorized representative*.The laws of social physics are
only apparently independent of thelaws of physics, and the
power wch certain *slogans* have to secureefforts from others
without expending efforts themselves--wch is thevery aim of
magical action--is rooted in the capital wch the grouphas
accumulated through its effort and whose effective use
issubordinated to a whole set of conditions, those wch define
the*rituals of social magic*. Most of the conditions that have
to befulfilled in order for a performative utterance to 
succeed
come downto the question of the appropriateness of the
speaker--or better still,s social function--and of the
discourse he utters. A performativeutterance is destined to
fail each time that it is not pronouncedby a person who has
the Ôpower' to pronounce it, or, more 
generally,each time that
the Ôparticular persons and circumstances in a 
givencase' are
not Ôappropriate for the invocation of the particularspeaker
invoked'; in short, each time that the speaker does not
havethe authority to emit the words that he utters. But
perhaps themost important tng to remember is that the success
of these operationsof social magic--comprised by *acts of
authority*, or, what amounts tothe same, *authorized acts*--is
dependent on the combination of asystematic set of
interdependent conditions wch constitute socialrituals.It is
clear that all the efforts to find, in the
specificallylinguistic logic of different forms of
argumentation, rhetoric andstyle, the source of their symbolic
efficacy are destined to failas long as they do not establish
the relationsp between theproperties of discourses, the
properties of the person who pro-nounces them and the
properties of the institution wch authorizesm to pronounce
them. The limits (and the interest) of Austin'sattempt to
define performative utterances lie in the fact that hedoes not
exactly do what he tnks he is doing, and ts preventsm from
following it through to the end. Believing that he
wascontributing to the plosophy of language, he was in fact
workingout a theory of a particular class of symbolic
expressions, of wchthe discourse of authority is only the
paradigmatic form, and whosespecific efficacy 
stems from the
fact that they seem to possess *inthemselves* the source of a
power wch in reality resides in theinstitutiional conditions
of their production and reception.The specificity of the
discourse of authority (e.g. a lecture, asermon, etc.)
consists in the fact that it is not enough for itto be
*understood* (in certain cases it may even fail to
beunderstood without losing its power), and that it exercises
itsspecific effect only when it is *recognized* as such.
Tsrecognition, whether accompanied by understanding or
not--isgranted, in the manner of sometng taken for granted,
onlyunder certain conditions, namely, those wch define
legitimateusage; namely, it must be uttered by the person
legitimatelylicensed to do so, the holder of the *skeptron*,
known andrecognized as being able and enabled to produce ts
particularclass of discourse: a priest, a teacher, a poet,
etc.; it mustbe uttered in a legitimate situation, that is, in
front of legitimatereceivers (one cannot read a piece of
Dadaist poetry at a Cabinetmeeting); finally, it must be
enunciated according to the leg-itimate forms (syntactic,
phonetic, etc.) What one might callthe *liturgical*
conditions, namely, the set of prescriptionswch govern the
*form* of the public manifestation of authority,like
ceremonial etiquette, the code of gestures and
officiallyprescribed rites, are clearly only an *element*,
albeit the mostvisible one, in a system of conditions of wch
the most importantand indispensable are those wch produce the
disposition towardsrecognition in the sense of misrecognition
and belief, that is, thedelegation of authority wch confers
its authority on authorizeddiscourse. By focusing exclusively
on the formal conditions forthe effectiveness of ritual, one
overlooks the fact that theritual conditions that must be
fulfilled in order for ritual tofunction and for the sacrament
to be both *valid* and *effective*are never sufficient as long
as the conditions wch produce therecognition of ts ritual are
not met: the language of authoritynever governs without the
collaboration of those that it governs,without the help of the
social mechanisms capable of producing tscomplicity, based on
misrecognition, wch is the basis of allauthority. In order to
gauge the magnitude in Austin's and otherstrictly formalist
analyses of symbolic systems, it suffices toshow that the
language of authority is only the limiting case ofthe
legitimate language, whose authority does not reside, as
theracism of social class would have it, in the set of
prosodic andarticulatory variations wch define distinguished
pronunciation,or in the complexity of the syntax or the
richness of the vocabulary,in other words in the intrinsic
properties of discourse itself, butrather in the social
conditions of production and reproduction ofthe distribution
<sic: more likely, Ôdistinction'> between 
theclasses of
knowledge and recognition of the legitimate language.(Pierre
===
Re: The Riemann Hypothesis> Xevious <cptriker@aol.com> wonders
in message > Can the Riemann-Zeta Hypothesis possibly be
explained to a math major> who has only just begun to study
Real Analysis? What is the> significance of zero solutions
lying on a critical line? And what> substantial mathematical
theorems are dependent on a hypothesis that> has yet to be
proven? Interestingly, there are no fewer than four recent pop
books> on the Riemann Hypothesis. (Probably because of its>
appearance on the Clay Mathematics Institute list of
problems.) One sees:> Derbysre, Prime Obsession> du Sautoy,
The Music of the Primes> Sabbagh, The Riemann Hypothesis>
Edwards, Riemann's Zeta FunctionAnother book that deserves
mention here is Julian Havil's book, Gamma. 
It's about Euler's
constant, gamma, but it gets around to some useful material on
the zeta function. I tnk all of these books are reviewed on
the MAA website so you can see another opinion before you jump
in. Executive summary: the zeta function can be thougnt of as a
generating function for the primes. Anytng we learn about the
zeros of zeta has immediate implications for the distribution
of the primes. Just about any quantitative question about
prime numbers, or tngs that depend on prime numbers like
divisors, can be answered more precisely the more we know
===
Question> I would greatly appreciate some comments upon the
correctness of the> assertion about the following equation (1)
under the given conditions. y = (a^m + b^m)(a^m - b^m) (1)
Conditions: (y, a, b ) = 1; m is a non-integer > 0; prime p >
3. Assertion: If y is a p-th power then both a^m + b^m and a^m
- b^m> separately be p-th power.Huh? If m is a non-integer then
it seems unlikely to me that a^m + b^m will be an integer. In
===
limit points of Q> are the limit points of Q exactly R? well, I
know ts is true, and I can prove it. the reason i ask though>
is because i doubt myself on ts. let (xn) be a sequence in the
set of all rationals 0<=p<=1. Further,> suppose xn is an
enumeration of the rationals between 0 and 1. i tnk> that all
rationals between 0 and 1 are cluster points of xn, mainly>
because all are limit points. i could also work it from the
definition> since a rational lies between any two rationals,
but i started> doubting myself for some reason. can someone
tell me if I am wrong about anytng in the above?Yes: you are
===
imply Reimann> It is possible prove the Ternary Goldbach
Conjecture (TGC) and the Twin Prime > Conjecture (TPC) are
true, if the Generalized Riemann Hypothesis (GRH) is >
true.GRH implies twin primes? News to me. > Is there a similar
paper for the converse? If the TGC or TPC is true then, > the
===
Fixed points> Suppose f : (0,1) --> (0,1) is continuous. Does
f have to have a fixed> point? If it was f : [0,1] ---> [0,1]
or f : [0,1] ---> (0,1), then yes. Any thoughts? MarcDoesn't
Brouwer have sometng to say on ts? I don't
===
prove the following theorem: Let P be a polynomial with real
coefficients such that P(x) >=0 for> every real x. Then, there
are polynomials R and S such that P(x) => R^2(x) + S^2(x) for
every complex x.Show P factors over the reals as a product of
irreducible quadratics times a product of squares of linear
polynomials. Show that an irreducible quadratic is a sum of 2
squares. Show that a product of two sums of two squares is a
===
construct ts set ...> Can you construct a set E in [0, 1] s.t.
for every open interval I in [0,1] > m(I intersect E) > 0 & m(I
intersect E^c) > 0 m is lebesgue measure > E^c is the
complement of E take away fat disjoint Cantor sets K1 and J1.
Now from B2 (K1 U J1) take away fat disjoint Cantor sets K2
and J2. Continue, and set E = ... (If ts is a homework problem
and you use ts nt, be sure to give credit to
===
Why do you take so much trouble to expose such a reasoner as>
Mr. Smith? I answer as a deceased friend of mine used to
answer> on like occasions - A man's capacity is no measure 
of
s power> to do miscef. Mr. Smith has untiring energy, wch does
> sometng; self-evident honesty of conviction, wch does more;>
and a long purse, wch does most of all. He has made at least>
ten publications, full of figures few readers can critize. A
great> many people are staggered to ts extend, that they
imagine there> must be the indefinite sometng in the 
mysterious
all ts.> They are brought to the point of suspicion that the
mathematicians> ought not to treat all ts with such
undisguised contempt,> at least.> -- A Budget of Paradoxes,
structured social space, a field of forces, a 
forcefield. It
contains people who dominate and others who are
dominated.Constant, permanent relationsps of inequality
operate inside tsspace, wch at the same time becomes a space
in wch the variousactors struggle for the transformation or
preservation of the field. All the individuals in ts universe
bring to the competition allthe relative power at their
disposal. It is ts power that definestheir position in the 
field
and, as a result, their strategies.Economic competition between
networks or newspapers for viewers,readers, or for marketshare,
takes place concretely in the formof a contest between
journalists. Ts contest has its own,specific stakes - the
scoop, the Ôexclusive', professionalreputations, 
and so on. Ts
kind of competition is neitherexperienced nor thought of as a
struggle purely for economicgain, even though it remains
subject to pressures derivingfrom the position the news medium
itself occupies witn alarger set of economic and symbolic power
relations. Today,invisible but objective relations connect
people and partieswho may never meet - nevertheless, in
everytng these entitiesdo, they are led, consciously or
unconsciously, to take into accountthe same pressures and
effects, because they belong to the same world!On Television,
by Pierre Bourdieu
===
Subject: Re: billard mechanics I posted sometng like ts in
sci.physics but people seem to be mainly> with their heads in
the stars there..> I have programmed, ages ago, ts billiard
mechanics engine, using very> basic physical laws on momentum
collision. It works in the sense that it> creates :> 1. good
collisions between two balls that each have a specific
velocity> vector, and> 2. it never draws balls over each
other,> although I'm not satisfied with the 
amount of
calculation needed for those> two tngs (it involves solving a
quadratic equation and then trigonometry).> But, as you may be
aware, in snooker or pool one starts with the red balls> (and
the pink) toucng each other.> Ts kind of ruins ts whole nice
model since one can no longer use an> O(n^2) algorithm to scan
the balls for possible collisions and then work> them,
since..well it becomes a mess. When the wte ball ts the pack
of> red balls, I get a nice 4-dimensional representation of
chaos, I tnk,> wch isn't my intention. As you know, when 
balls
touch each other then the> energy of the first ball gets
transferred to the last. Most of it, anyway.> In fact I tnk it
calls for an entirely new strategy. Does anybody have any>
ideas how I could :> - devise a collision strategy that works
on singular collisions as well> as on multiple simultaneous
collisions> or> - treat these chained collisions separately
(ts would involve> sorting the toucng balls with respect to
the ball(s) that will t them,> since a computer can't know 
in
what direction the force gets transferred - meaning, ts is
probably impossible to accomplish.Correct. A triangle of balls
is statically indeterminant, that is, theyare just like a
triangular pyramid of balls in static equilibrium. Thereare
not enough constraints to specify the forces on all the balls.
I would probably do best to treat the table of balls as a table
of static> balls, with force fields on them, instead of each
ball having its vx and vy,> but really, it's not that 
easy..>
If anybody has any ideas or experience with ts, I'd be very
grateful,I have toyed with ts type of problem over the years,
and I tnk thebest tng is just to never let them touch. With
double precision youcan leave dither in the 8th place or so,
and use your single collisionalgorithm. Lew Mammel,
===