mm-270
Subject: Re: Cantor Paradox :-)
> I was under the impression that there was a maximum line
length
>> allowed in emails, around 80 characters I think.
>> There is no such limit.
>> Oh yeah? Prove it. Cite some RFC that says there is no
limit.
>> Just kidding. for the information. I wasn't too sure on that
>> point.
 It's an oft-quoted rule of netiquette that you should limit
line
> lengths to 80 characters. Some people even say 60 characters.
 I think that the RFC I quoted gives that rule, but it
applies only to
> the body, not to the headers (evidently).
There actually is a quote, in rfc822.
Each header field may be represented on exactly one line con-
sisting of the name of the field and its body, and terminated
by a CRLF; this is what the parser sees. For readability, the
field-body portion of long header fields may be folded onto
multiple lines of the actual field. Long is commonly inter-
preted to mean greater than 65 or 72 characters. The former
length serves as a limit, when the message is to be viewed on
most simple terminals which use simple display software; how-
ever, the limit is not imposed by this standard.
And I think also the successors impose no limit.
I understand now that Forte Agent (which David uses) indeed
has a bug.
It is based on what sendmail does do (insert a CRLF followed
by a
tab), but not on what the standard allows.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam,
nederland,
+31205924131
home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
Subject: Re: Eulerian path in infinite graph
>>Unfortunately, I'm having a hard time with one of the
exercises, which
>>asks for the reader to show that the infinite square grid is
an Eulerian
>>graph by showing an explicit two-way Eulerian path (i.e.,
one path that
>>covers every edges of the graph and that extends in both
directions).
>>Where can I find a hint for this excercise?
> Hint: spiral.
. I had already shown that the infinite grid was Hamiltonian by
using exactly this technique of showing a two-way spiral, but
this
particular two-way spiral left some edges uncovered.
> It's not especially difficult if you just start drawing some
pictures.
> Just make sure both ends of the path are always on the
boundary of the
> region you've already covered.
Ok, thank you very much. I will try it again with the hint of
the
paragraph above.
BTW, this is a particularly nice exercise, since any finite
grid
(sufficiently large) is clearly not Eulerian (many vertices of
odd
degree), but the infinite graph is Eulerian. Nice.
again, Rog.8erio Brito.
--
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
=-=-=-=
Rog.8erio Brito - rbrito@ime.usp.br -
http://www.ime.usp.br/~rbrito
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
=-=-=-=
Subject: Re: New Knowledge
First you need to get interested in the problem - to enjoy
playing
with it. Then you make a guess which would explain how stuff
works
behind the scene. Then you work out the consequencess of your
guess.
You check them out, how they hold. Your new explanation should
fit
everything what is known, it should not be unnecesserily
complicated,
it should not be mushy (adjustable by ad hoc additions
designed to
make it fit - or contain untestable propositions) and should
predict
some new testable stuff.
The hardest part is not fooling yourself. If you can do that,
you
won't fool others.
> How, when and why does *New Knowledge* come to humankind?
Subject: Re: New Knowledge
charset=Windows-1252
N:dlzc D:aol T:com (dlzc) <N: dlzc1 D:cox T:net@nospam.comHow,
when and why does *New Knowledge* come to humankind?
How: Deductive reasoning applied to observation. Divine
inspiration.
> When: Often, when serendipity strikes; however, chance
favors the
> prepared mind.
> Why: Human desire for knowledge, occasionally purely for its
own sake
> but more often goal-oriented.
> Not very imaginitive answers. What are you driving at?
> Steve Turner
What am *I* driving at..... well, like I said:
> How, when and why does *New Knowledge* come to humankind
> In all the answers you have seen posted thus far, you can
detect
> a certain lost unease, a groping in the dark....after their
laughs have
> died down..... Steve, if I'd knew the real, manifest and
practical
> answer to this, then I would not be posting
here.....ahahahahaha...
> The frontier of new thought is even scarier and closer then
one
> normally does admit to: Really new thoughts/ideas do come
very
> seldom. 99.9999999....% of what we think and say is parroting
> of/about old and ancient stuff........ Now, let's see
whether anybody
> will come up with a *new* thought or idea that is truly
original.
Just a little niggle...
New Thought =/= New Knowledge
> New Idea =/= New Knowledge
So I guess the question is, how would you define knowledge,
new,
and
> the domain for whom this might be new knowledge.
Tesla walked some strange ground, and even though his
> philosophy/belief/reasoning were apparently two steps removed
from
> reality, they seemed to parallel reality pretty well.
I guess I'll see where your definitions line up, before I add
more.
> David A. Smith
>
ahahahaha........Smitty, make your own, be my guest
or be independent. I don't care, Smitty, except that
you should produce/express a novel unheard thought,
and instead of parakeeting precambrian arguments for
argument's sake.........ahahahaha......take care Smitty,
ahahahaha...........ahahahanson
Subject: Re: New Knowledge
Dear hanson:
> N:dlzc D:aol T:com (dlzc) <N: dlzc1 D:cox T:net@nospam.com>
...
> What am *I* driving at..... well, like I said:
> How, when and why does *New Knowledge* come to humankind
> In all the answers you have seen posted thus far, you can
detect
> a certain lost unease, a groping in the dark....after their
laughs
have
> died down..... Steve, if I'd knew the real, manifest and
practical
> answer to this, then I would not be posting
here.....ahahahahaha...
> The frontier of new thought is even scarier and closer then
one
> normally does admit to: Really new thoughts/ideas do come
very
> seldom. 99.9999999....% of what we think and say is parroting
> of/about old and ancient stuff........ Now, let's see
whether anybody
> will come up with a *new* thought or idea that is truly
original.
Just a little niggle...
New Thought =/= New Knowledge
> New Idea =/= New Knowledge
So I guess the question is, how would you define knowledge,
new,
and
> the domain for whom this might be new knowledge.
Tesla walked some strange ground, and even though his
> philosophy/belief/reasoning were apparently two steps
removed
from
> reality, they seemed to parallel reality pretty well.
I guess I'll see where your definitions line up, before I add
more.
> ahahahaha........Smitty, make your own, be my guest
> or be independent. I don't care, Smitty, except that
> you should produce/express a novel unheard thought,
> and instead of parakeeting precambrian arguments for
> argument's sake.........ahahahaha......take care Smitty,
A thought or idea that can be expressed in words, is already
known. It
is constructed from known building blocks. Even the language
of music
is
written in building blocks we already know.
Knowledge is new to each person. A baby gains the knowledge of
how to pee,
how to walk, how to talk, how to manipulate his/her
environment.
New knowledge comes from chaos. New knowledge comes from
walking
new
ground. New knowledge is more than thought or idea, as it
entails
*testing*.
I suggest you study the works of Buddha (pre-cambrian indeed!)
if you want
to know more about new knowledge.
David A. Smith
Subject: Re: Why are there 63360 inches in a mile?
>The book has a website:
>http://www.measuringamerica.com/home.php
> ...
>> But why didn't they just define a rod to be 16 feet,
instead of 16.5???
>> I don't know - maybe they were just dumb.
>> If anyone knows, I'd be interested.
> ...> http://math.ucr.edu/home/baez/inches.html
But there actually are some people so ignorant that they seem
> to think rods are 16 rather than 16.5 feet; see for example
> http://www.merrimackcanoes.com/lingo.htm (et al) which says
> ROD: A way of measuring distance when portaging a canoe.
> Most people accept that a rod is 16 feet, or one canoe
length.
> Of course it is trivial to remember there are 320 rods per
mile,
> so no right-thinking person has any excuse whatsoever to not
> know the actual length of a rod.
One current and ongoing use of rods in day-to-day measurement
> is (as hinted in the quote) to measure canoe portages. For
> example, distances are given in rods in the table
> http://www.rook.org/earl/bwca/portages/portage1.html
> which lists portages in the BWCA mostly on routes between
> Trout Lake at the west end and Burntside Lake at the east.
> -jiw
A rod is a quarter of a chain (five and a half yards ans,
just to add a little decimalization into this thread...
There are 10 square chains to an acre!
Jack Fearnley
Subject: Re: Ln typo or....?
>I have a problem that requires a proof that ln(a) + ln(b) =
ln(a + b)
> Then you're in deep doo-doo, unless for example a = b = 2.
>....which makes no sense to me because ln(a) + ln(b) is not
equal
>to ln(a + b).
> Right. The proof you sketch is actually not so hard if you
> switch to proving the right thing: ln(ab) = ln(a) + ln(b).
> Must have been a typo.
Yes, but it isn't so obvious to me that the sum of 2 areas
under the
curve of y = 1/x is the same as another single area, minus any
circular
reasoning. Say for ln(ab) = ln(a) + ln(b) we take ln(a) as the
area from
x = 1 to 2 and ln(b) as the area from 1 to 3, this is the same
as ln 2 +
ln 3 = ln 6. The area from x = 1 to 6 isn't obviously the same
as the
sum of the two smaller areas. What am I missing?
Phil Holman
Subject: is LU and LDU matrix factorization unique? or under
what condition
are they unique?
Dear all,
Here is my question: is LU and LDU matrix factorization unique?
(here for LDU we follow the definition that L is UNIT lower
triangular
matrix, D is diagonal, U is UNIT upper triangular matrix)
How to prove that they are unique?
If they are not unique, then under what condition do they
become unique?
a lot,
Joenyim
Subject: Re: is LU and LDU matrix factorization unique? or
under what
condition are they unique?
Content-transfer-encoding: 8bit
> Dear all,
Here is my question: is LU and LDU matrix factorization unique?
(here for LDU we follow the definition that L is UNIT lower
triangular
> matrix, D is diagonal, U is UNIT upper triangular matrix)
How to prove that they are unique?
Yes. If L_1*D_1*U_1 = L_2*D_2*U_2 then, with all the proper
hypotheses,
L_1 = L_2, D_1 = D_2 and U_1 = U_2.
The proof goes something like this:
U_1*U_2^(-1) = D_1^(-1)*L_1^(-1)*L_2*D_2.
The LHS is upper triangular with unit diagonal. The RHS is
lower
triangular. So, LHS = I. Etc.
--
Subject: Schedule-writing
This seems like somehow it should be a problem in graph theory.
Not sure :P
I am looking for an algorithm to help write a schedule: a group
of people, divided into 3 shifts, to cover several positions
which
must be manned 24/7, giving people fairly regular weekends
(but not
necessarily on the weekend proper)... and the algorithm should
be
flexible enough to handle both one-time requests for days off
as well
as regular requests (like every other thursday). Also it
should (over
long periods of time) work everyone close to an equal amount,
regardless of above mentioned requests.
Any hints? Any specific algorithms I should look up? This is of
course no homework. I am infinitely in your debt for any help
you can
give.
T
Subject: Re: New Randomness Test
>> 1. PRNG-generated sequences.
>> 2. Decimal (or any other base) expansions of normal
numbers, like pi
et
>> al.
>> 3. Strings obtained from natural phenomena, like
radioactive decay.
The P of the first stands for pseudo, and many of these are
cyclical. I
> would say the cyclical ones would not qualify.
If we design a [P]RNG to actually output a nonpatterned,
noncyclical,
> normal series, however, I suspect it would use something
from category
2.
> (I put the Pseudo in brackets, because some would say it
qualifies as a
> random series even though it is deterministic.)
Am I overlooking something, or have you actually listed only
two
genuine
> categories?
> It all boils down to how one defines random in the first
place -
> something you seem to have focused on, directly or
indirectly, from the
> start of this thread. I.e. I think that this discussion is
pretty much
one
> of form, not of substance.
> If you want to drop PRNGs on the grounds that
> they are cyclical and deterministic, that's fine by me. If
you insist in
> including the decimal expansion of pi, on the grounds that
it is not
> cyclical, although deterministic, that's fine too.
Being deterministic would disqualify a series as random by SOME
definitions.
I suspect that cyclicality would disqualify a series as random
by MOST if
not
ALL knowledgable definitions which aren't qualified with
pseudo.
> Still, I am curious as to why you have no problem with the
deterministic
> character of pi, while objecting to the cyclical character
of PRNGs. To
> me, they are both objectionable or acceptable.
If one is using a number series to deal cards for online
poker, or to
generate
keys for encryption, anything which is cyclical may introduce
detectable
artifacts
and be subject to successful attack. Sometimes this problem
can be
overcome.
At other times, people select something like the pi series to
mutate a key
or algorithm because of the higher level of confidence this
disorder
allows.
Deterministic origin or not, the pi series shares certain
characterstics
with
true random numbers which may be put to good use.
As Carbuncle Al pointed out, pi series is the champ in freedom
from some
subtle
kinds of seeming bias.
Servo
Subject: Re: Super Cosmos World Picture
> The World Picture of Super Cosmos
> http://focus.aps.org/
> Highlights of what subjectively struck me as worth reporting
> 1. WMAP ( http://map.gsfc.nasa.gov/m_mm.html ) of CMB
(Cosmic Microwave
> Black Body Radiation) in thermal equilibrium in slow enough
adiabatic
> expansion of the Hubble spherical horizon of our past light
cone. That
> is, think of our 3D space as effectively bounded by a
slightly distorted
> spherical wave front that is c/H(t) away from us in our
past. That is,
> it took light approximately 1/H(t) seconds to reach us from
that initial
> wave front. Note, I use the word approximately.
You know I was thinking about this very thing, and how as the
universe
expands.
If you could stand on it, a flat rubber sheet of it for
instance, it would
appear to
be round, and you would see ships slowly disappear over the
horizon.
So the universe is older than previously thought eh? That is
no big
surprise really, when the Hubble only studies light, and light
bends under the influence of gravity. So as the universe
expands,
the point where we can no longer see any light, would be
light speed * distance of the visible horizon.
And that is what anyone would suspect, is the age of the
almost static universe.
And why inside a large Dyson sphere, the horizon does not
appear
to go up. It looks level, or again, curved slightly down.
If you could see more than 10 miles or so, through the
atmosphere
that is.
> This Hubble sphere is one of an infinity of Level 1 parallel
IT
> Universes on a single 3D spatially flat post-inflationary
bubble,
> which is the 3D version of Euclid's globally flat infinite
plane without
> boundary. There are a further Level IIinfinity of these
bubbles (or
> infinite 3D rubber sheets expanding homogeneously and
isotropically on
> the largest of scales only) in a Cosmic Champagne Cocktail
as it were
> perhaps from the Qabalistic Vineyard of one Rashi des Troyes
(1040 -
> Parallel Universes for the Level I and Level II
classification of the
> chaotic inflation scenario with the Weak Anthropic Principle
and
> Darwinian Natural Selection on the cosmic scale.
> Angular resolution of WMAP space probe is 40 x better than
COBE space
probe.
> Polarization of CMB is measured from last scattering of
light from the
> time of 'recombination' when stable hydrogen atoms formed
for the first
> time ~ 380,000 light years from the Big Bang, and also from
ionization
> of atoms in space by UV photons from the first stars forming
only 200
> million light years from the Big Bang! The age of the
Universe ~ c/Ho is
> 13.7 billion light years. This is the radius of the Hubble
spherical
> horizon back to the Big Bang. Therefore, the Holographic
Thermodynamic
> Entropy S of our Universe now is of order (10^28)^2/(10)^-26
~ 10^82
BITS.
> Atomic matter is about 4% of the large-scale structure of
the Universe,
> i.e. Omega(Atomic Matter) ~ 0.04.
> Exotic vacuum with gravitating positive zero point pressure
is about 23%
> of the large-scale structure of the Universe, i.e.
Omega(Dark Matter) ~
> 0.23.
> I have made a theoretical inference here that dark matter is
not made
> symmetry partners. That is, dark matter detectors, in
principle, will
> never click with The Right Stuff, if they click at all it
will be
with
> false positives, with Fool's Gold like Enron Stock After The
Fall.
> Exotic vacuum with anti-gravitating negative zero point
pressure is
> about 73% of the large-scale structure of the Universe, i.e.
Omega(Dark
> Matter) ~ 0.73.
> Note that scale-dependence is crucial. You can imagine, in
the sense of
> adaptive windowed wavelet transforms not the usual Fourier
transforms
> of time and space, a power spectrum of different kinds of
stuff
> plotted against scale of resolution. The above percentages
are only
> measured at the largest scales ~ 10^26 to 10^28 cm in a
coarse
> graining sense where the FRW metric approximation applies.
> Homework Problem 1
> Note that the Fourier transform of the temporal
autocorrelation function
> of some disturbance is its power spectrum. What is the
> autocorrelation function corresponding to the adaptive
windowed wavelet
> power spectrum where scale is plotted instead of
frequency/wave
> numbers?
> 2. SDSS of 200,000 galaxies + WMAP above give
> ho ~ 0.7 +- 0.04
> Omega(Dark Matter + Atomic Matter) = 0.3 +- 0.04
> Upper limit on rest mass of the electron neutrino ~ 0.6 ev
> http://arxiv.org/abs/astro-ph/0310723
> SDSS 3D conformal scale-dependent wavelet-transform? map of
past
light
> cone from scale of Earth's core ->Solar System -> Milky Way
Galaxy - Clusters of Galaxies -> CMB last light.
> We have produced a new conformal map of the universe
illustrating
> recent discoveries, ranging from Kuiper belt objects in the
Solar
> system, to the galaxies and quasars from the Sloan Digital
Sky Survey.
> This map projection, based on the logarithm map of the
complex plane,
> preserves shapes locally, and yet is able to display the
entire range of
> astronomical scales from the Earth's neighborhood to the
cosmic
> microwave background. The conformal nature of the
projection, preserving
> shapes locally, may be of particular use for analyzing large
scale
> structure. Prominent in the map is a Sloan Great Wall of
galaxies 1.37
> billion light years long, 80% longer than the Great Wall
discovered by
> Geller and Huchra and therefore the largest observed
structure in the
> universe.
> http://arxiv.org/abs/astro-ph/0310571
> 3. Speed of Gravity?
> What Tom Van Flandern alleges in
> http://www.ldolphin.org/vanFlandern/gravityspeed.html about
superluminal
> near field of gravity is controversial as is what he says
about GPS in
> http://www.metaresearch.org/ But it is not my purpose here
to dig into
> those issues. On the other hand his reference to William
Walker's work
> is sound http://arxiv.org/abs/physics/0001063
> Near-field Analysis of Superluminally Propagating
Electromagnetic and
> Gravitational Fields
> A near-field analysis based on Maxwells equations is
presented which
> indicates that the fields generated by both an electric and
a magnetic
> dipole or quadrapole, and also the gravitational waves
generated by a
> quadrapole mass source propagate superluminally in the
nearfield of the
> source and reduce to the speed of light as the waves
propagate into the
> farfield. Both the phase speed and the group speed are shown
to be
> superluminal in the nearfield of these systems. Although the
information
> speed is shown to differ from group speed in the nearfield
of these
> systems, provided the noise of the signal is small and the
modulation
> method is known, the information can be extracted in a time
period much
> smaller than the wave propagation time, thereby making the
information
> speed only slightly less than the superluminal group speed.
It is shown
> that relativity theory indicates that these superluminal
signals can be
> reflected off of a moving frame causing the information to
arrive before
> the signal was transmitted (i.e. backward in time). It is
unknown if
> these signals can be used to change the past.
> http://arxiv.org/abs/gr-qc/0304090
> A mainstream experiment using gravity lensing by Jupiter of
an aligned
> distant quasar's light give a speed of gravity 1.06 +-0.21
the speed of
> light http://arxiv.org/abs/astro-ph/0302294
> However Clifford Will disagrees with that claim in
> We calculate the delay in the propagation of a light signal
past a
> massive body that moves with speed v, under the assumption
that the
> speed of propagation of the gravitational interaction c_g
differs from
> that of light. Using the post-Newtonian approximation, we
consider an
> expansion in powers of v/c beyond the leading ``Shapiro''
time delay
> effect, while working to first order only in Gm/c^2, and
show that the
> altered propagation speed of the gravitational signal has no
effect
> whatsoever on the time delay to first order in v/c beyond
the leading
> term, although it will have an effect to second and higher
order. We
> show that the only other possible effects of an altered
speed c_g at
> this order arise from a modification of the parametrized
post-Newtonian
> (PPN) coefficient alpha_1 of the metric from the value zero
predicted
> by general relativity. Current solar-system measurements
already provide
> tight bounds on such a modification. We conclude that recent
> measurements of the propagation of radio signals past
Jupiter are
> sensitive to alpha_1, but are not directly sensitive to the
speed of
> propagation of gravity.
> http://arxiv.org/abs/astro-ph/0301145
> 4. Gravity Waves? LIGO a giant strain gauge to measure
ripples in
> space-time curvature. Two spiraling in neutron stars emit
gravity waves
> that change distance between two mirrors 4km apart by less
than one
> thousanth of the diameter of a proton! And they say that
people who see
> UFOs are crazy? :-) Well Kip Thorne at Cal Tech thinks this
can be
> measured and many millions of dollars are being spent to do
so. Let's
> hope they do better than the hot fusion boys. :-) Nothing
exciting to
> 5. The Big Rip Doomsday
> The dark energy data from type 1a supernovae requires -1/3 >
w =
> (pressure of dark energy)/(energy density of dark energy)
> In my theory, dark energy and dark matter are both w = -1
zero point
> energy density of exotic vacuum regions that are positive
and negative
> respectively, hence negative and positive quantum pressures
> respectively. Exotic vacuum with w = -1 and positive
pressure will be
> mistaken for w ~ 0 CDM
>
http://web.mit.edu/afs/athena.mit.edu/user/r/e/redingtn/www/
netadv/specr/012/
012.html
> The Big Rip destroying future Universe only happens if -1 >
w i.e.
> phantom energy. There is no Big Rip in my theory where w =
-1 on the
nose.
> Appendix 1. Digression on the Universe as a Hologram - one
particular
> model that is probably false.
> The pixel resolution of the Hubble surface deteriorates as
the Universe
> 3D space expands and accelerates. The quantum of area or
cosmic pixel
> size is
> Lp*^2 ~ Lp^4/3(c/H(t))^2/3
> 3D space is the holographic image of the expanding
accelerating Hubble
> 2D surface.
> The number of Bekenstein BITS in the Universe then gives the
Entropy of
> the Universe S(t) at Cosmic Time t to be
> S(t) ~ k(c/H(t))^2Lp*^-2 ~ k(c/H(t)Lp)^4/3
> This explains the Second Law of Thermodynamics Arrow of Time
as tied
> to the expansion of the Universe. That's The Good News.
There is also
> some Bad News. That is, this particular model may be false.
I do not
> know as yet.
> k = Boltzmann's constant, Lp^2 = hG(Newton)/c^3 ~ 10^-66
cm^2,
> c/H(today) ~ 10^28 cm, Lp*(today) ~ 1 Gev, i.e. 1 fermi
(10^-13 cm)
> Witten's alpha' today ~ 1/(1Gev)^2 = Universal
Geometrodynamic Regge
> Slope of hadronic resonances of the strong force ~ (string
tension)^-1
> All of this is testable and this model may be false.
> Rest mass of lepto-quarks m ~ (e/c)^2Lp*^-1 ~ Mev
> Therefore, e/m ~ (c^2/e)Lp*
> IF e/m is to be a constant of the Universe's evolution in
global FRW
> large-scale cosmic time ~ h/k(Absolute Temperature of the
CMB), then
> (c^2/e)^2Lp*^2 is constant
> That is,
> (c4/e^2)hG*/c^3 = (hc/e^2)G* is constant in cosmic time
though not in
scale!
> Note, that the only way to keep H(t) constant would be
> H(t) = R(t)^-1dR(t)/dt = Ho
> R(t) = Roe^Hot
> i.e. a state of continuous inflation.
> If we exclude this, then the e/m ratios of the lepto-quarks
must be
> increasing if this particular version of the world hologram
idea were
true.
Subject: Re: 0 and natural numbers
I think a formal definition would require that you first
construct
> the
> set corresponding to the equivalence relation using the
axioms
of
> set
> theory, and then prove that it has the required properties --
> reflexive, symmetric, transitive, and so on. But, as I said
in
my
> previous posting, you may not have to explicity define this
> equivalence relation to construct the set of integers.
The Schaum's book developed everything relating to equivalence
> relations
> before getting to the various types of numbers. (And I wish i
could
> remember where I put that book.)
Besides, don't you have to establish that your z is a
partition?
It
> seems that this would require the same steps as
demonstrating the
> properties of an equivalence relation.
I'm not sure. But using my program, I have managed to show:
ALL(a):ALL(b):[a e n & b e n => EXIST(c):[c e z & (a,b) e c]
> & ALL(c1):ALL(c2):[c1 e z & c2 e z & (a,b) e c1 & (a,b) e c2
= c1=c2]]
In other words, for every pair of natural numbers a,b, there
exists
a
> unique
> integer c such that (a,b) e c. I think this will allow me to
construct
> integer addition and multiplication based on addition and
multiplication
> for
> the natural numbers, which I have already established.
Again, this
> project
> is very much a work in progress.
Dan
Free download of DC Proof 1.0 at http://www.dcproof.com
> Then you are partitioning NxN and calling the the sets of
that
partition
> your integers.
But the sets of any partition are equivalence classes of a
unique
> equivalence relation, and vice versa.
So how does your construct differ from the common equivalence
class
> construct of the integers?
The underlying set is the same, if I understand you correctly,
but I
never
> explicitly make use of an equivalence of ordered pairs of
natural
numbers,
> as such. Informally, I have
0 = {(1,1), (2,2), (3,3)... }
> +1 = {(2,1), (3,2), (4,3)... }
> +2 = {(3,1), (4,2), (5,3)... }
> -1 = {(1,2), (2,3), (3,4)... }
> -2 = {(1,3), (2,4), (3,5)... }
Maybe this is sounding all rather vague, and I am not
expressing myself
> well. I am, of course, leaving out a LOT of detail. I hope
to clarify
things
> when I post my proofs at my website in the not too distant
future.
Dan
Free download of DC Proof 1.0 at http://www.dcproof.com
If you make use of the equivalence classes, then your
construction is
> essentially no different from the more common one.
I suppose that's one way to look at it. I believe there are
subtle
differences in my approach, however.
Dan
Free download of DC Proof 1.0 at http://www.dcproof.com
Subject: Re: Multiplying Dice, then adding
> Here is a dice game which I post more of as a question than
as a
> game-idea.
> You have 2 dice and 2 players.
combination, on each face next to each of the six numbers on
that die.
> (One sign per face.)
And they do this secretly before play begins.
> Next, the dice are cast together on each round, and then the
*product*
> P of the 2 dice's values (including the signs) is taken.
> Finally, on each round, the product (which can be positive or
> negative) is added to a running-total.
After a fixed number of rounds, player 1 wins if the
running-total is
> positive, player 2 wins if the running-total is negative.
> (A total of 0 is a tie.)
> What would be a good strategy for either player in choosing
whether
> to put a + or a - on any particular face of his/her die?
Leroy
> Quet
> As noted already in a reply, this is definitely more of a
> psychological game than a mathematical game, as far as
strategy.
Since it's just about the same game as a random walk,
which math-tarts call science, it's no more psychological than
than telephones are psychological.
Or since most psychologist are more familiar with
Mel Gibson than they are science:
It's obviously why Mel's got money, and most
mathematicians still work for Mao or one
his subcontractors in beautiful downtown Cairo.
Subject: Re: No such thing as a random number?
You have a proof that not all arbitratry digit sequences are
equally
>probable in the decimal expansion of pi?
He means the Schoenfeld Pi. It's much nicer but you can't use
it
> without paying him royalties.
Well, but since only mathematicians
are actually stupid enough to use
patented random numbers, you
can tell Schoenfeld that the
universe is still relativistically
invariant to moron though.
Subject: Re: Fun with axioms
> But then of course there's the ultimate reality check
> for mathematics: physics.
I believe you have that backwards. When physics is done
> ignoring
> the
> theoretical
> coherence of mathematics you will find that a great many
> resources
> have
> been
> unnecessarily directed toward an infinite regress of
> triangles.
> Galileo.
> Galvani
> Volta
> Joule
> Hooke
> Faraday
> Would you wish me to enlarge the list further?
Unlike you, Franz, I am not in the habit of denying the
brilliance
> of
> *great*
> men and women with substantive ideas.
Reread and try to understand your own words before spouting.
> Practice on the following quotation from yourself, for
instance:
When physics is done ignoring the theoretical coherence of
> mathematics
> you
> will find that a great many resources have been unnecessarily
> directed
> toward an infinite regress of triangles.
> Sure Franz.
for agreeing with me.
> I have erased the irrelevant straw man you raised.
Ahh... relevance.
Let's see, that goes back to utility and Bentham.
So, following Guenther...
Have you gangraped anyone lately Franz?
> Mitch, That remark characterises yourself quite precisely.
You win Franz. I am nowhere near as skilled at character
assassination as
someone like you.
Subject: Re: Fun with axioms
>The neat thing about building a math with axioms, is that you
can
>>first build one, then go back and change just one axioms
slightly.
>>Go through the exercise of building the math again, and see
the
>>differences between the first and the second build.
Assuming the axioms are consistent. This is not
>automatic.
> Of course. That's a part of the learning experience.
..................
good mathematicians are quite fussy about what structures
defined
>via axioms they are willing to bother studying. If you write
down
>an arbitrary list of axioms and ask a mathematician to study
it's
>consequences, they'll probably look at you funny.
>Oh, no, no, no. The whole point is to do the work yourself,
>>NOT ask a mathematician.
As someone who has consulted on mathematics and statistics,
>many of my clients could not possibly have done so.
> That's why you're here :-). I'm simply talking about a hobby
> that is self-gratifying, can be done anywhere at anytime, and
> doesn't need a gazillion dollars of infrastructure before
> getting started. A half-dozen project notebooks, pencil, and
> a very big eraser is all the equipment needed. A compass
> and straight-edge is useful for geometry.
When one formulates a real world problem in mathematical
>terms, so that one can get further results, one must make
>are theorems, these are the consequences of the assumptions.
>If the consequences do not have acceptable real world
>properties, the assumptions must be changed, or it must
>be recognized that what the client wants is impossible.
> Yep.
..............
But then of course there's the ultimate reality check
>for mathematics: physics.
The Feynman integral is an example of contradictory
>assumptions. What is it?
> I'm looking forward to this discussion.
Indeed, this has me curious too. I have downloaded some
papers, but I doubt
it
is enough to sort out the details.
:-)
mitch
Subject: Re: Fun with axioms
Have you gangraped anyone lately Franz?
> All by himself? I guess he's just too much man.
Oh no. I just figure he is part of that kind of crowd.
Subject: Hints for self taught topology?
I am undertaking the quest to teach myself topology. Text I'm
using right now is Munkres' Topology: a first course, but this
is a
library book (interlibrary loan) so I may or may not be able
to obtain
my own copy and might have to settle for something less highly
recommended. I'm no stranger to self teaching, I've self
taught myself
galois theory and just about everything leading up to that, as
well as
the standard calculus package and diffeq. But I *am* a
stranger to
topology and it strikes me as exceedingly subtle (which is a
good
thing:) and I look forward to mastering it.
Are there any common pitfalls, etc. that I need to be ready
for,
or anything else I should know for this mission?
Sincerely,
SA
Subject: Re: Hints for self taught topology?
> I am undertaking the quest to teach myself topology. Text I'm
> using right now is Munkres' Topology: a first course, but
this is a
> library book (interlibrary loan) so I may or may not be able
to obtain
> my own copy and might have to settle for something less
highly
> recommended. I'm no stranger to self teaching, I've self
taught myself
> galois theory and just about everything leading up to that,
as well as
> the standard calculus package and diffeq. But I *am* a
stranger to
> topology and it strikes me as exceedingly subtle (which is a
good
> thing:) and I look forward to mastering it.
> Are there any common pitfalls, etc. that I need to be ready
for,
> or anything else I should know for this mission?
http://at.yorku.ca/topology/educ.htm
Subject: Re: Is this correct?
Content-transfer-encoding: 8bit
> I have completed this proof and I was wondering if there
needs to be
> anything added or changed to it. Thank you very much for you
help.
Question:
> An affine map is a function f : R -> R of the form
> f(x) = ax + b
> where a and b are fixed real numbers with a /= 0. Prove that
Aff, the
> set
> of all affine maps, is a group under the operation of
composition.
> (Note:
> Don't forget to show that Aff is closed under composition.)
Proof:
> If g : R -> R has the form g(x) =cx+d, where c /= 0, then
fg(x)=f(g(x))=f(cx+d)=a(cx+d)+b=acx+(ad+b).
Since ac /= 0, the composite fg is an affline map. The identity
> function
> 1 : R -> R is an affline map (set a=1 and b=0), while the
inverse of f
> is easliy seen to be x=> (a^-1)x - (a^-1)b.
Again thank you for your help.
Looks good to me
--
Subject: Re: Die Cantor Die
> If mathematics, like religion, lived in the private domain,
> then the mathematicians, like the religionists, would be
fully
> justified in closing their ears to objections from outsiders.
> But, of course, mathematics lives in the public domain.
And therefore mathematicians are obliged to listen to
unmathematical
twaddle about mathematics?