mm-377
>>Let S be the real numbers (0,1)
>>Since it is a set of real numbers it is partially ordered
and every
>chain is
>>obviously bounded by 1
>>Yet (0,1) does not have a maximal element.
>>What am I missing here?
>The chain (0,1), for example, does *not* have a least upper
bound *in
S.*
>More relevant to a question about Zorn's lemma is the fact
that (0,1)
>does not have an upper bound in S.
>I don't know if this is relevant to the original poster's
reason for
>confusion, but in Kaplansky's _Set Theory and Metric Spaces_
(1972), he
>defines Zorn's Lemma as
>Let L be a partially ordered set in which every chain has an
upper bound.
>Then L contains a maximal element.
>Our teacher instructed us to write in the word nonempty before
partially
>ordered, and in L after upper bound. To be fair, Kaplansky
notes
in
>his remarks on the lemma that the upper bound must be in L.
Your teacher's right about nonempty - the statement above is
false
unless we insert those words.
But, while it's true that the upper bound must be in L, that
doesn't
mean that the statement above needs to be modified! Saying
Let L be a partially ordered set in which every chain has an
upper
bound _says_ that the upper bound is in L. The partially
ordered
set is L, period; something outside L cannot be an upper bound
for anything, _in_ the particular partial order in question.
(To be fair to your teacher<g>, the confusion in the OP shows
that emphasizing that the upper bound must be in L is
probably a good idea pedagogically. But there's no problem
with the statement above - if one reads that and doesn't
realize that the upper bound must be in L then one is
simply not reading it correctly.)
===
Subject: Re: Zorn's Lemma Question
>I don't know if this is relevant to the original poster's
reason for
>confusion, but in Kaplansky's _Set Theory and Metric Spaces_
(1972),
>he defines Zorn's Lemma as
>Let L be a partially ordered set in which every chain has an
upper
>bound. Then L contains a maximal element.
>Our teacher instructed us to write in the word nonempty before
partially ordered, and in L after upper bound. To be fair,
>Kaplansky notes in his remarks on the lemma that the upper
bound
>must be in L.
>Did your teacher explain why he asked you to deface your
books? As
>there is no nonempty before chain, it's not needed before
partially ordered set either; if the partially ordered set is
empty,
>then the empty chain has no upper bound in L.
Drat, I got that wrong just now.
>As for in L, I think
>the in which takes care of that; the pronoun which refers to
L.
>Putting in which at the front tells us that the chain and the
upper
>bound are both in L.
===
Subject: Re: ellipse from 4 points
It could be that 3 points define a circle, 3 define a number
of ellipses,
but 4 points define one ellipse(?), and 5 points covers the
general
solution. I can visualize 3 points and their circle, then
stretching the
circle to include the 4th point, did I stretch it the right
way into an
ellipse? If so then one can go from general circle equn back
and forth to
general ellipse equn.
The general conic form includes rotation and offset from
origin, so
distinguishing which type of conic section it is form the
general equation,
is not obvious. so assume a coordinate system for the ellipse
that is
simple
and see if any 4 points can be mapped onto it.
If so, one could assume the xy axis inline with the ellipse
foci to
simplify
coordinates
x^2/a^2+y^2/b^2=1
remap the points to the new coordinate system
that is one approach, but it seems to be light enough
variables (dof) to
cover the 4 points.
one to sleep on!
'''''''''''''''''''''''''''''
'''''''''''''''''''''''''''''
> In sci.math,
> <m@hotmail.com
> I have the position of 4 points on a 2D plane. The points are
> unequally spaced.
> Is there anyway I could fit an ellipse (or any other
circular shape)
> to these points (it has to pass through the 4 points)?
> FYI, this is for an image processing algorithm. I have tried
to use
> Hermite Interpolation, but I can't seem to find a way to get
the
> tangent values at each of the 4 points so the curve looks
like an
> ellipse/circle.
> Any help would be appreciated greatly.
> Well, the most straightforward (and hardest!) method would
be to
> take the general equation of a conic section:
> Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
> and come up with 8 equations in 6 unknowns. Not exactly the
prettiest.
> It gets weirder as one should be able to use 5 factors:
> ax^2 + bxy + cy^2 + dx + ey + 1 = 0
> (where a = A/F etc.)
> but one requires 3 points to define a circle, so I'm
obviously
> missing something degrees-of-freedom wise.
> --
> #191, ewill3@earthlink.net
> It's still legal to go .sigless.
===
Subject: Re: Difficulty of calculus vs. discrete math
> I have taken college courses both in calculus and in discrete
> mathematics. What surprised me was the difficulties that
other
> students were having with discrete mathematics. For some
reason, they
> found calculus much easier.
> The discrete math was baby stuff: formal logic, divisibility,
> combinatorics, and the like.
> Why would one find calculus easy and discrete math difficult?
Given the prominence of calculus within the undergraduate
curriculum,
perhaps high schools and students alike spend more time
preparing for
those calculus classes already during the high school period.
If this is really the case, those students would be
preconditioned
for calculus.
===
Subject: Interactive Proof Writing Tutorial (Freeware)
Students,
My new proof writing program, DC Proof 1.0, can help you learn
the basics
of
logic and mathematical proof. Included with the free download
is a
self-study tutorial with exercises (plus hints and full
solutions). You can
also participate in a forum just for DC Proof users, hosted by
Yahoo!
Groups. And you can have your proofs published at the DC Proof
Online
website! No need to worry about making mistakes -- DC Proof
checks every
line of your proof as you enter it.
Visit DC Proof Online now for your FREE download at:
http://www.dcproof.com
Also available at the Tucows download site:
http://www.tucows.com
===
Subject: Re: Difficulty of calculus vs. discrete math
> The discrete math was baby stuff: formal logic, divisibility,
> combinatorics, and the like.
Perhaps it was more abstract===
Subject: Re: Zorn's Lemma Question
>Let S be the real numbers (0,1)
>Since it is a set of real numbers it is partially ordered and
every
>chain is
>obviously bounded by 1
>Yet (0,1) does not have a maximal element.
>What am I missing here?
>>The chain (0,1), for example, does *not* have a least upper
bound *in
S.*
>> More relevant to a question about Zorn's lemma is the fact
that (0,1)
>> does not have an upper bound in S.
>I don't know if this is relevant to the original poster's
reason for
>confusion, but in Kaplansky's _Set Theory and Metric Spaces_
(1972), he
>defines Zorn's Lemma as
>Let L be a partially ordered set in which every chain has an
upper bound.
>Then L contains a maximal element.
>Our teacher instructed us to write in the word nonempty before
partially
>ordered, and in L after upper bound. To be fair, Kaplansky
notes in
>his remarks on the lemma that the upper bound must be in L.
>Your teacher's right about nonempty - the statement above is
false
>unless we insert those words.
In case you missed Fred Galvin's post (it doesn't appear to me
to
be posted as a direct reply to your post): What I said here was
wrong. Wrong, wrong, wrong - the lemma is correct exactly
as stated.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
I would suggest to use the COMMON quantifiers
(Ax) and (Ex)
instead of the extremely uncommon and rather irritating
ALL(x): and EXISTS(x):
A quantifier is not a function, mind that! (In addition this
would
*improve* readability [of complex statements] a good deal!)
Same for the connective or, the symbole here definitely SHOULD
BE
v
and NOT
|.
Moreover imho it would be a good idea to use the simple symbols
-> and <-
instead of the more complicated and less clear
=> and <=>.
Note that there are approaches in logic where the latter
symbols are
used either at meta-level or to express some semantical
relation(s).
Again, to use idiomatic notation is NOT user-friendly, but a
rather
bad idea from a pedagogical point of view.
Ever considered that there might be A WORLD beyond your
solipsistic
approach?
F.
--
I do tend to feel Hughes & Cresswell is a more authoritative
source than you. (D. Ullrich)
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Again, to use idiomatic notation is NOT user-friendly, but a
rather
> bad idea from a pedagogical point of view.
> Ever considered that there might be A WORLD beyond your
solipsistic
> approach?
Ah, right, the use of
( and )
instead of
[ and ]
would also be helpful, imho.
Is there a *reason* for the latter notation??? (I mean beyond
just
trying to be different?)
F.
P.S.
This way,
P <=> ~Q <=> ~[P <=> Q]
would read
(P <-> ~Q) <-> ~(P <-> Q).
Note that additional brackets are rather helpful here to
differentiate
between the two cases
(A <-> B) <-> C
and
A <-> (B <-> C).
--
I do tend to feel Hughes & Cresswell is a more authoritative
source than you. (D. Ullrich)
===
Subject: re:Interesting problem
If the sets are only required to be closed under addition,
it's easy
to construct such A and B using the Hamel basis.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
Thank you for your well-meaning suggestions. Given the
practical
limitations
of the standard keyboard, I have chosen the notation to be (1)
as close as
possible to that used in mathematics classrooms and textbooks,
and (2) as
easy to learn as possible. We have been over this before, and
I'm sorry if
it bothers you.
> <dchris@allstream.net> tried to advertise his program ,
again.
> I would suggest to use the COMMON quantifiers
> (Ax) and (Ex)
> instead of the extremely uncommon and rather irritating
> ALL(x): and EXISTS(x):
> A quantifier is not a function, mind that! (In addition this
would
> *improve* readability [of complex statements] a good deal!)
> Same for the connective or, the symbole here definitely
SHOULD BE
> v
> and NOT
> |.
> Moreover imho it would be a good idea to use the simple
symbols
> -> and <-
> instead of the more complicated and less clear
> => and <=>.
> Note that there are approaches in logic where the latter
symbols are
> used either at meta-level or to express some semantical
relation(s).
> Again, to use idiomatic notation is NOT user-friendly, but a
rather
> bad idea from a pedagogical point of view.
> Ever considered that there might be A WORLD beyond your
solipsistic
> approach?
> F.
I do tend to feel Hughes & Cresswell is a more authoritative
> source than you. (D. Ullrich)
===
Subject: Re: Zorn's Lemma Question
Adjunct Assistant Professor at the University of Montana.
>Let S be the real numbers (0,1)
>Since it is a set of real numbers it is partially ordered and
every
>chain is
>obviously bounded by 1
>Yet (0,1) does not have a maximal element.
>What am I missing here?
>>The chain (0,1), for example, does *not* have a least upper
bound *in
S.*
>> More relevant to a question about Zorn's lemma is the fact
that (0,1)
>> does not have an upper bound in S.
>I don't know if this is relevant to the original poster's
reason for
>confusion, but in Kaplansky's _Set Theory and Metric Spaces_
(1972), he
>defines Zorn's Lemma as
>Let L be a partially ordered set in which every chain has an
upper bound.
>Then L contains a maximal element.
>Our teacher instructed us to write in the word nonempty before
partially
>ordered, and in L after upper bound. To be fair, Kaplansky
notes in
>his remarks on the lemma that the upper bound must be in L.
>Your teacher's right about nonempty - the statement above is
false
>unless we insert those words.
How so?
If L is empty, then it is false that every chain has an upper
bound:
the empty chain does not have an upper bound, and the
statement is
true by vacuoity. If L is nonempty, then the empty chain is
certainly
bounded: by any element of L; and any nonempty chain is
bounded,
giving the usual statement with nonempty in it.
Or did I miss something?
========
===
Subject: Re: Zorn's Lemma Question
Adjunct Assistant Professor at the University of Montana.
>Your teacher's right about nonempty - the statement above is
false
>unless we insert those words.
>How so?
Oops; never mind. I've seen your response now. My newsfeeder
is acting
up. No new messages all day Wed., then no new messages
overnight last
night, and they are now trickling in at odd intervals...
========
===
Subject: Re: ellipse from 4 points
> Well, the most straightforward (and hardest!) method would
be to
> take the general equation of a conic section:
> Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
> and come up with 8 equations in 6 unknowns. Not exactly the
prettiest.
4 equations. Each point gives you one equation. Of course, the
equations are homogeneous: (2A,2B,2C,2D,2E,2F) is as good as
(A,B,C,D,E,F).
> It gets weirder as one should be able to use 5 factors:
> ax^2 + bxy + cy^2 + dx + ey + 1 = 0
> (where a = A/F etc.)
> but one requires 3 points to define a circle, so I'm
obviously
> missing something degrees-of-freedom wise.
A circle requires a = c and b = 0. Those are the two degrees
of freedom
it loses relative to a more general conic, such as an ellipse
(which
requires 5 points to define it).
iel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Thank you for your well-meaning suggestions. Given the
practical
limitations
> of the standard keyboard, I have chosen the notation to be
(1) as close
as
> possible to that used in mathematics classrooms and
textbooks, and (2) as
> easy to learn as possible. We have been over this before,
and I'm sorry
if
> it bothers you.
How about making it setable?
It might also be a good idea to decouple the displayed
notation from the
keyboard limitations.
===
Subject: re:Interesting problem
Using this I can show that for every composite n, the positive
reals
can be split into n sets closed under addition and
multiplication.
Unfortunately 2 is prime.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: Vf8cy-ZegeMoV3FIwVmRCoxRsx6CADCP2znm9NlalLgfzBocddBnwy
> Thank you for your well-meaning suggestions. Given the
practical
limitations
> of the standard keyboard...
Huh?! This is in no way related to my /well-meant
suggestions/. :-)
> ...I have chosen the notation to be (1) as close as possible
to that used
in
> mathematics classrooms and textbooks, and (2) as easy to
learn as
possible.
Look, , you may repeat your mantra as long as you want, but
it's
still a lie.
Neither in mathematical classrooms nor in ANY textbook *I*
know your
idiosyncratic notation will be found.
It's just idiotic to claim otherwise.
Where on earth (in which mathematical or logical textbook)
have you ever
seen the notation
EXISTS(x): or ALL(x):
for quantifier?! :-) You are just lying that's all. (Right,
some CS
professor m a y have u s e d it in ONE of your classes, so
what?!)
Same for or: it's completely idiotic to claim that anyone
actually
would use
|
for the logical connective or. Hello Sir, this is not BNF, ok?
(And,
btw, concerning BNF I would rather interpret it as some sort
of xor.)
In standard logical textbooks | denotes the Sheffer stroke.
http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Sheffer+stroke
http://en.wikipedia.org/wiki/Sheffer_stroke
Seems that you are unable to change your erroneous views even
when told
otherwise. Well, that's just fine. But *I* certainly will not
support
such a nonsensical approach (by not making up my mind).
Are you sure that you are the right person to present a
teaching tool?
F.
===
Subject: v'' + A v' + B v = 0 for vectors
I've learned about first order diff.eqs. in vectors. You can
diagonalize
and
solve then convert back to get the original variables. But is
there a
general
method for solving second order equations like
v'' + Av' + Bv = 0
?
in Philly
PS: how much worse does it get if there is a f(t) on the RHS?
===
Subject: Re: v'' + A v' + B v = 0 for vectors
> I've learned about first order diff.eqs. in vectors. You can
diagonalize
and
> solve then convert back to get the original variables. But
is there a
general
> method for solving second order equations like
> v'' + Av' + Bv = 0
> ?
I would say, convert to first order. Twice as many variables,
but you already know the method for it.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Note that additional brackets are rather helpful here to
differentiate
> between the two cases
> (A <-> B) <-> C
> and
> A <-> (B <-> C).
But we do NOT NEED it in case of & and v, since
(A & B) & C is equiv. with A & (B & C)
and
(A v B) v C is equiv. with A v (B v C).
Hence the software may interpret an expression
A & B & C
in either way: (A & B) & C or A & (B & C).
A.
===
Subject: Re: This Week's Finds in Mathematical Physics (Week
202)
> ...
> |T| = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...
> Lo and behold! The coefficient of x^n is the number of
binary trees
> with n leaves!
> There's also another approach where we work directly with the
> structure types themselves, instead of taking generating
functions.
> This is harder because we can't subtract structure types, or
divide
> them by 2, or take square roots of them - at least, not
without
> stretching the rules of this game. All we can do is use the
> isomorphism
> T = X + T^2
> and the basic rules of category theory. It's not as
efficient, but it's
> illuminating. It's also incredibly simple: we just keep
sticking in
X + T^2 wherever we see T on the right-hand side, over and over
again.
> Like this:
> T = X + T^2
> T = X + (X + T^2)^2
> T = X + (X + (X + T^2)^2)^2
> and so on. You might not think we're getting anywhere, but if
> you stop at the nth stage and expand out what we've got,
you'll
> get the first n terms of the Taylor expansion we had before!
> At least, you will if you count stages and terms correctly.
A mind boggling thing occured to me: the partition function of
rooted
planar
binary trees f(x) has the property that f(x) converges to
infinity if
and only
if x lies outside the Mandelbrot set!
Squark
[If you wish to contact me by e-mail,
use Trvbsl_Ovdmfbstup@fyjuf.dpn
where I replaced each letter by the next alphabetically]
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
G. Frege says...
><dchris@allstream.net> tried to advertise his program , again.
>I would suggest to use the COMMON quantifiers
> (Ax) and (Ex)
>instead of the extremely uncommon and rather irritating
> ALL(x): and EXISTS(x):
I didn't know that there was any standard way to write
first-order logic using ASCII characters. The standard
in math textbooks is to use upside-down A for for all
and a backwards E for there existsThe same comment for or,
implies, and: there are standard
ways to write these things in textbooks, using special
symbols, but I didn't know that there were any standards for
writing them using ASCII.
I actually don't see why it matters much. It takes about
4 lines to describe your notation, and from then on, I
don't see why anyone would be confused.
--
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: bcwXAkZeweJHpEy1j8FDe+fc7w-XSB7HnBp+CTqJSJcQuRjPZyRoQ7
> I would suggest to use the COMMON quantifiers
> (Ax) and (Ex)
> instead of the extremely uncommon and rather irritating
> ALL(x): and EXISTS(x):
> I didn't know that there was any standard way to write
> first-order logic using ASCII characters.
Huh? Standard? Standard?! What the hell are you talking about?!
> in math textbooks is to use upside-down A for for all
> and a backwards E for there exists
Right. But since we cannot do t h a t in ASCII we have to use
the
normal A and normal E as BEST approximation, won't you think
so?
Now G. Greene claimed that this means we would HAVE TO write
Ax and Ex.
Well, ok, that's certainly a reasonable point of view. But in
the
mentioned context m a y b e writing
(Ax) and (Ex)
is a better choice. (But I don't know for sure.)
> The same comment for or
Nonsense. Why talking bull?
Either you write
v
or you write
/,
there's certainly NO OTHER COMMON way to write the logical
connective
or as a SYMBOL (trying to resemble the USUAL symbol used in
print).
> implies
I would propose either
-
or
=>.
But the latter has some serious disadvantages, imho.
> I actually don't see...
Right. That's obvious. So why talk?
> It takes about 4 lines to describe your notation, and from
then on,
> I don't see why anyone would be confused.
Right. That's just what I'm trying to do. :-)
F.
Reference:
http://planetmath.org/encyclopedia/LogicalLanguage.html
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Note that additional brackets are rather helpful here to
differentiate
> between the two cases
> (A <-> B) <-> C
> and
> A <-> (B <-> C).
> But we do NOT NEED it in case of & and v, since
> (A & B) & C is equiv. with A & (B & C)
> and
> (A v B) v C is equiv. with A v (B v C).
> Hence the software may interpret an expression
> A & B & C
> in either way: (A & B) & C or A & (B & C).
A & B & C is interpreted as (A & B) & C and is not seen as
equivalent to A
&
(B & C). The associativity of & requires a simple proof using
what I call
the Split and Join rules.
===
Subject: Science Without Math? (model-free common sense
steering)
Neural networks are model-free estimators, in that they do not
require
an
in-depth understanding of the phenomena they are modeling.
http://www.arcon.com/arconneu.html
THE MATHEMATICS OF CROSSING THE STREET:
You are at the curb deciding, Should I
cross the street? Well, it depends.
AT THE CURB
The walk light is on, but you see a
truck approaching fast. How fast?
There is no exact number. Instead, there are an infinite
number of
possibilities - from 1kph to over 100kph and everything in
between. You
don't have a radar gun, so instead you watch the truck for a
second or two,
and sum its speed up in two words very fast. That is good
enough.
Your senses have told you the truck is coming very fast, but
you need more
information before you can decide whether or not to risk
crossing. How
far
down the street is the truck? Is it slowing down? Again, there
are no
exact numbers, so you sum up the situation - close, not slowing
quickly
enoughSomehow your brain adds fast + close + not slowing
quickly
enough, and
warns you instantly that the risk is high. It is purely
cognitive
process.
It involves a complex combination of sensory information and
experience.
...Since there are no exact numbers in this story, the
mathematical version
must be told with fuzzy numbers...
But, the process is still not quite over. Should I wait or
cross? You
have
to make the decision. Risk tolerance leads to different spins
and endings.
If you walk with a cane, you reason, The risk is high, so I'll
wait.
You
watch as the truck runs the red light. If you are a jogger,
impatient to
cross, you disregard the evidence, step into the intersection,
and jump
back
just in time to save your life.
http://www.decyde.com/crossingthestreet.html
Fuzzy logic works the way that humans think as opposed to the
way that
computers typically work. For example, consider the task of
driving a car.
You notice that the stoplight ahead is
red and the car ahead is braking. Your
mind might go through the thought process,
I see that I need to stop. The roads are
wet because it's raining and there is a
car only a short distance in front of me.
Therefore I need to apply a significant
pressure on the brake pedal.
This is all subconscious (in general), but that's the way we
think - in
fuzzy terms. Do our brains compute the precise distance to the
car ahead of
us and the exact coefficient of friction between our tires and
the road,
and
then use a Kalman filter to derive the optimal pressure which
should be
applied to the brakes? Of course not. We use common-sense
rules and they
seem to work pretty well. On the other hand, when we do
finally get around
to pressing the brake pedal there is some exact force that we
apply, say
1.326 pounds. So although we think in fuzzy, noncrisp ways,
our final
actions are crisp. The process of translating the results of
fuzzy
reasoning
to a crisp, nonfuzzy action is called defuzzification.
http://www.innovatia.com/software/papers/fuzzy.htm
...In particularly vast networks in fast moving environments,
the split
second it takes to traverse the circuit is greater than the
time it takes
for the situation to change. In reaction, the last node tends
to compensate
by ordering a large correction. But this also is delayed by
the long
journey
across many nodes, so that it arrives missing its moving mark,
birthing yet
another gratuitous correction.
The same effect causes student drivers
to zigzag down the road, as each late
large correction of the steering wheel
overreacts to the last late overcorrection.
Until the student driver learns to tighten
the feedback loop to smaller, quicker
corrections, he cannot help but swerve down
the highway hunting (in vain) for the center.
This then is the bane of the simple auto-circuit. It is liable
to
flutter
or chatter, that is, to nervously oscillate from one
overreaction to
another, hunting for its rest. There are a thousand tricks to
defeat this
tendency of overcompensation, one trick each for the thousand
advance
circuits that have been invented.
http://www.kk.org/outofcontrol/ch7-c.html
Fuzzy systems are based on
storage of common-sense rules.
For example, a fuzzy Army-ant robot controller might have the
fuzzy
association if load is heavy, then signal for help longer.
Fuzzy
phenomena
admit degrees: some loads are heavier than others; some signal
durations
are
longer then others.
A single association (heavy,longer)
encodes all combinations...
Fuzzy systems reason with
parallel associative inference.
A fuzzy system reasons with multivalued sets, instead of true
or false
propositions, and it may adaptively modify its fuzzy
associations from
representative numerical samples.
http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html
Wired: What is fuzzy logic and why do critics call it the
cocaine of
science?
Kosko: Fuzzy logic is Spock's worst nightmare - a way of doing
science
without math. It's a new branch of machine intelligence that
tries to make
computers think the way people think and not the other way
around. You
don't
write equations for how to wash clothes. Instead you load a
chip with vague
rules like if the wash water is dirty, add more soap, and if
very
dirty,
add a lot more. All wash water is dirty and not dirty - to
some degree.
It's just common sense. But it breaks the old either/or logic
of Aristotle.
That offends some scientists, who would like us to think and
talk like
off/on switches. But they still haven't produced a statement
of fact like
the sky is blue or E=mc^2 that is 100 percent true or 100
percent
false.
Fact ain't math. You can never get the science right to more
than a few
decimal places. That's one reason we find chaos when we look
at things up
close...
...Fuzzy systems are universal computers. I proved that as a
theorem - the
fuzzy approximation theorem. In theory, you can replace every
book on
physics or economics with equivalent books that have fuzzy
systems where
the
equations used to be. Fuzzy systems are model-free estimators.
You
don't
have to guess at equations to build a bridge from inputs to
outputs. Fuzzy
rules build that bridge for you. There is math behind the
rules, but you
don't need to know it to program a fuzzy system. You can
program it in
English. If the air is cool, turn the AC down a little. But
the math is
not fuzzy. That's why you can capture fuzzy logic in a digital
chip.
Most of the first fuzzy systems were in control - as in
adjusting a camera
lens or backing up a trailer truck to a loading dock. Now
we're applying
fuzzy systems to wireless communications and multimedia. The
fuzzy rules
can
randomly spread signals over a wide bandwidth or teach an
intelligent
agent the kind of houses or sunsets you prefer. The math says
we can apply
them anywhere. In practice, it may not be so easy.
http://www.wired.com/wired/archive/3.02/kosko_pr.html
Fuzzy logic is a superset of conventional(Boolean) logic that
has been
extended to handle the concept of partial truth- truth values
between
completely true and completely false. As its name suggests, it
is
the
logic underlying modes of reasoning which are approximate
rather than
exact.
The importance of fuzzy logic derives
from the fact that most modes of human
reasoning and especially common_sense
reasoning are approximate in nature.
Boolean vs. Fuzzy: 300 years B.C., the Greek philosopher,
Aristotle came up
with binary logic(0,1), which is now the principle foundation
of
Mathematics. It came down to one law: A or not-A, either this
or not this.
For example, a typical rose is either red or not red. It
cannot be red and
not red. Every statement or sentence is true or false or has
the truth
value
1 or 0. This is Aristotle's law of bivalence and was
philosophically
correct
for over two thousand years.
Two centuries before Aristotle, Buddha, had the belief which
contradicted
the black-and-white world of worlds, which went beyond the
bivalent cocoon
and see the world as it is, filled with contradictions, with
things and not
things. He stated that a rose, could be to a certain degree
completely red,
but at the same time could also be at a certain degree not
red. Meaning
that
it can be red and not red at the same time.
Conventional(Boolean) logic states that a glass can be full or
not full of
water. However, suppose one were to fill the glass only
halfway. Then the
glass can be half-full and half-not-full. Clearly, this
disprove's
Aristotle's law of bivalence. This concept of certain degree or
multivalence
is the fundamental concept which propelled Zader Lofti of
University
Berkely
in the 1960's to introduce fuzzy logic. The essential
characteristics of
fuzzy logic founded by him are as follows.
In 1965, Lofti Zadeh formally developed multivalued set
theory, and
introduced the term fuzzy into the technical literature.
Nowadays, the
recent emergence of fuzzy commercial products, as well as new
theory, has
generated a new interest in multivalued systems. Yet already
engineers have
successfully applied fuzzy systems in many commercial areas :
intelligent
subways automation, emergency breakers, cement mixers, Kanji
characters
recognition, control air conditioners, automatic washing
machines, guide of
robot-arm manipulators, and so on.
Fuzzy systems store banks of fuzzy associations or
common-sense rules
such
as IF traffic is heavy in this direction, THEN keep the light
green
longer
that might be articulated by an human expert. Some traffic
configuration
are
heavier that others and some green-light duration are longer
than others,
so
that, the single fuzzy association (HEAVY, LONGER) encodes all
these
combinations. That is to say, fuzzy systems directly encode
structured
knowledge but in a numerical framework : by entering the fuzzy
association
(HEAVY, LONGER) as a single entry in a rule database we are
defining an
input-output transformation.
http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
Fuzzy Logic is a computational paradigm capable of modelling
the own
uncertainness of human beings. Fuzzy reasoning is nothing else
than a Fuzzy
Logic-based formalism for encoding human knowledge or common
sense in a
numerical framework. Indeed, the mathematical concepts on
which Fuzzy Logic
is supported are very easy to understand. In a Fuzzy
Controller, human
experience is codified by means of linguistic if-then rules,
which compute
control actions upon given conditions. Fuzzy Logic has been
applied to
problems that are difficult to solve mathematically. One of
its main
advantages lies in the fact that it offers a straightforward
methodology
for
modelling and controlling non-linear systems, which are
difficult to face
by
means of conventional techniques.
http://www.wkap.nl/prod/b/1-4020-7359-3
Fuzzy logic models itself on the pattern of human reasoning in
its use of
approximate information and uncertainty to generate decisions.
It was
designed (during late 1980s and early 1990s) to mathematically
represent
vagueness and develop tools for dealing with imprecision
inherent in
several
problems. Normally, in digital computers one uses the binary
logic
where
the digital signal has two discrete levels : low (logic zero)
or high
(logic
one); nothing in-between. Fuzzy systems use soft linguistic
variables (e.g.
hot, tall, slow, light, heavy, dry, small, positive, ...etc.)
and a range
of
their weightage (or truth) values, called membership
functions, in the
interval (0, 1), enabling the new computers to make human-like
decisions.
Since human beings tend to use words rather than numbers to
describe
behaviour patterns, fuzzy controls avoid the conventional
rigidity of
computers and allow them to use parameters based on common
sense.
http://www.tribuneindia.com/2002/20021024/science.htm
Fuzzy logic best summed up by common sense
Computer Corner
John Boyd
Fuzzy logic was introduced to the world 27 years ago by
Professor
Lotfi Zadeh in his Fuzzy Sets paper published in Information
Control magazine, though it is only recently that we've seen it
applied across a broad range of products.
Some readers have asked for more explanation on fuzzy logic, so
here's an attempt to defuzzify the subject a little further.
Simply put, fuzzy logic is aimed at enhancing our prissy
computer
technology with a touch of common sense.
One problem with the conventional digital computer is that it
is
such a scrupulously either-or beast. It cannot be easily coaxed
to handle approximations or vague notions like young, a lot and
probably.
Yet most of us rely on such terms daily because we happen to be
humans dealing with other humans, not robots building cars.
It's an easy matter to arbitrarily program a computer so it
designates everyone falling into the age-range 0f 15 to 18
as being a youth. Such a precise category has come to be called
a crisp set since the emergence of fuzzy logic.
Yet we all know some 14-year-olds can look older than some
late-developers turning 20. Such exceptions, however, cannot
be accounted for in conventional computing. Or at least not
without an inordinate amount of additional programming and
expense.
As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied
when I asked him if we couldn't just continue using
conventional
programming and technology for controlling new products,
instead
of fuzzy, Well, we could. And you could probably swim across
the
Pacific if you got enough support from enough people. But ...
To overcome this problem, Zadeh was inspired to develop his
fuzzy
theory and the math to go with it that could be used to create
fuzzy sets based on imprecise natural language.
Each member in a fuzzy set (such as the youths and others
considered
in the above example) is assigned one of a continuous range of
values
(called the membership value) between zero and one.
Whereas in the above crisp set a 13-year-old going on 14 would
still
have to be considered a minor and thus be designated as zero in
binary logic, fuzzy logic could assign him a membership value
of
say 0.1. Likewise, an immature 20-year-old who would normally
fall
outside our either-or crisp-set range could be assigned a
membership
value of 0.9 depending upon the criteria we use to measure
youth.
Working out just what criteria to use, what values should be
assigned
each member and deciding what rules are necessary to govern the
relationships between members is the key to successfully
applying
fuzzy control in products.
In some applications, determining the optimum rules has become
so
complex, some manufacturers have resorted to employing the aid
of
neural networks, which may be stretching a good thing too far,
given
fuzzy logic's original purpose to get round complexity.
Still, the flexibility in herent in fuzzy is clearly useful in
dealing with approximate calculations, such as about 100It can
be used in artificial intelligence to provide us with an
almost true answer. It can also infer a common-sense result
even
when the data is not precise.
Our handwritten 5 in 250 would be treated as 5, not the letter
S,
for instance, in Sony's fuzzy-based Palmtop computer.
While we have all seen fuzzy logic-based products from the
likes of
Matsua, Sanyo and Hitachi, one unlikely company that has made
fuzzy technology a central part of its business strategy is
Omron
Corp.
It began its research into fuzzy logic in 1984 and has since
applied
for over 700 patents. This puts it in the forefront of fuzzy
applications in areas like factory and industry control, as
well as
in medical equipment.
In 1989, Omron also signed on lotfi Zadeh as a senior advisor.
Earlier this year at the Business Show in Harumi, Omron
demonstrated
its fuzzy workstation. Omron manufactures both standard
Motorola
68040-based and 88000 reduced-instruction or RISC-based
workstations
that can be fitted with a fuzzy inference board, turning them
into
the world's first fuzzy workstations.
Omron claims such a RISC-based workstation can achieve 4 ion
operations per second, an incredible speed if they haven't
fuzzed
on the number. Fuzzy logic is used in the workstations to store
and retrieve fuzzy information and make inferences.
Ranging in price from Y2.5 million to almost Y4 million (a US
dollar
is about 120 Yens -FM), these machines are not the kind of
products
you will find down in Akihabara. (a section of Tokyo famous
for its
quantity and variety of electronic goods -FM) Rather, they are
typically aimed at value-added resellers in niche markets, and
engineers who want to develop fuzzy applications, fuzzy
databases
and expert systems, as well as fuzzy inference systems.
However, the entrepreneurs among you may be interested in
Omron's
FB-30AT fuzzy inference board for the IBM PC and compatible
wares.
It features a 24 MHz FP-3000 fuzzy chip capable of processing
up
to 128 rules, with five antecedents and 2 consequents. Training
software and a compiler is also available.
Omron has also produced a fine little booklet on fuzzy called
Clearly Fuzzy that I dipped into when writing this column.
Tadashi Katsuno, at Omron's public relations section, tells me
he still has a limited number of copies left that he will send
to the first readers of Computer Corner who write to him with
contact information.
The address is Omron Corp., International Public Relations
Section,
Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105.
--------------------------------------------------------------
------
- Farzin Mokhtarian
farzin@apollo3.ntt.jp
http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/
areas/fuzzy/doc/in
tro/j_times.tgz
http://www.ece.utep.edu/research/webfuzzy/about.html
http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS
_NOE_ASI99/ASI
'99_ViharosMonostori.htm
http://www.bjarne.ca/pmflp.pdf
http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htm
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Thank you for your well-meaning suggestions. Given the
practical
limitations
> of the standard keyboard...
> Huh?! This is in no way related to my /well-meant
suggestions/. :-)
> ...I have chosen the notation to be (1) as close as possible
to that
used in
> mathematics classrooms and textbooks, and (2) as easy to
learn as
possible.
> Look, , you may repeat your mantra as long as you want, but
it's
> still a lie.
> Neither in mathematical classrooms nor in ANY textbook *I*
know your
> idiosyncratic notation will be found.
> It's just idiotic to claim otherwise.
For some reason, you seem to be taking this whole notation
thing rather
personally. Aren't you getting a little carried away with this
talk of
lies and idiotic claims? This NOT a matter of life and death,
after
all!
> Where on earth (in which mathematical or logical textbook)
have you ever
> seen the notation
> EXISTS(x): or ALL(x):
Here we go again!
In every undergraduate alegebra and calculus textbook I have
seen, they use
the expressions like there exists and for all in defintions,
theorems,
etc. Given this fact, which is more meaningful for typical
undergraduate:
EXIST(x) or (Ex) or a backwards E? Keep in mind that variable
and
predicate names may be any length in my notation. Ex would be
interpreted
as logical proposition or predicate. Should I disallow all
propositions
beginning with E or A? Also, the user need not even type in
EXIST( ): -- there is a control-key shortcut Ctrl-e to insert
this
string,
Ctrl-a for the universal quantifier.
> for quantifier?! :-) You are just lying that's all. (Right,
some CS
> professor m a y have u s e d it in ONE of your classes, so
what?!)
> Same for or: it's completely idiotic to claim that anyone
actually
> would use
> |
> for the logical connective or. Hello Sir, this is not BNF,
ok? (And,
> btw, concerning BNF I would rather interpret it as some sort
of xor.)
> In standard logical textbooks | denotes the Sheffer stroke.
> http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Sheffer+stroke
> http://en.wikipedia.org/wiki/Sheffer_stroke
I guess you would like to see v used. With the longer names,
however,
something like AvB would be interpreted as logical proposition
or
predicate.
Should I also dissallow every proposition name with a v in it?
Of
course
not. So, some other symbol was needed. I have seen | used in
some
programming languages as a logical OR-operator, so it seemed a
natural
choice.
> Seems that you are unable to change your erroneous views
even when told
> otherwise.
How can a notation be erroneous? It is simply a convention.
Well, that's just fine. But *I* certainly will not support
> such a nonsensical approach (by not making up my mind).
> Are you sure that you are the right person to present a
teaching
tool?
Do you have anything better? I will let the user be the judge.
Christensen
Visit DC Proof Online at http://www.dcproof.com -- Free
Download
===
Subject: re:Interesting problem
Actually I can't. Never mind.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
G. Frege says...
>I would suggest to use the COMMON quantifiers
> (Ax) and (Ex)
>instead of the extremely uncommon and rather irritating
> ALL(x): and EXISTS(x):
>I didn't know that there was any standard way to write
>first-order logic using ASCII characters.
>Huh? Standard? Standard?! What the hell are you talking
about?!
Basically, I was trying to say in a polite way that you are
being
a jerk. But nevermind polite: You are being a jerk.
--
===
Subject: Re: Science Without Math? (model-free common sense
steering)
> Neural networks are model-free estimators, in that they do
not
require
an
> in-depth understanding of the phenomena they are modeling.
> http://www.arcon.com/arconneu.html
> THE MATHEMATICS OF CROSSING THE STREET:
> You are at the curb deciding, Should I
> cross the street? Well, it depends.
> AT THE CURB
> The walk light is on, but you see a
> truck approaching fast. How fast?
> There is no exact number. Instead, there are an infinite
number of
> possibilities - from 1kph to over 100kph and everything in
between. You
> don't have a radar gun, so instead you watch the truck for a
second or
two,
> and sum its speed up in two words very fast. That is good
enough.
> Your senses have told you the truck is coming very fast, but
you need
more
> information before you can decide whether or not to risk
crossing. How
far
> down the street is the truck? Is it slowing down? Again,
there are
no
> exact numbers, so you sum up the situation - close, not
slowing
quickly
> enough> Somehow your brain adds fast + close + not slowing
quickly
enough,
and
> warns you instantly that the risk is high. It is purely
cognitive
process.
> It involves a complex combination of sensory information and
experience.
> ...Since there are no exact numbers in this story, the
mathematical
version
> must be told with fuzzy numbers...
> But, the process is still not quite over. Should I wait or
cross? You
have
> to make the decision. Risk tolerance leads to different
spins and
endings.
> If you walk with a cane, you reason, The risk is high, so
I'll wait.
You
> watch as the truck runs the red light. If you are a jogger,
impatient to
> cross, you disregard the evidence, step into the
intersection, and jump
back
> just in time to save your life.
> http://www.decyde.com/crossingthestreet.html
> Fuzzy logic works the way that humans think as opposed to
the way that
> computers typically work. For example, consider the task of
driving a
car.
> You notice that the stoplight ahead is
> red and the car ahead is braking. Your
> mind might go through the thought process,
> I see that I need to stop. The roads are
> wet because it's raining and there is a
> car only a short distance in front of me.
> Therefore I need to apply a significant
> pressure on the brake pedal.
> This is all subconscious (in general), but that's the way we
think - in
> fuzzy terms. Do our brains compute the precise distance to
the car ahead
of
> us and the exact coefficient of friction between our tires
and the road,
and
> then use a Kalman filter to derive the optimal pressure
which should be
> applied to the brakes? Of course not. We use common-sense
rules and they
> seem to work pretty well. On the other hand, when we do
finally get
around
> to pressing the brake pedal there is some exact force that
we apply, say
> 1.326 pounds. So although we think in fuzzy, noncrisp ways,
our final
> actions are crisp. The process of translating the results of
fuzzy
reasoning
> to a crisp, nonfuzzy action is called defuzzification.
> http://www.innovatia.com/software/papers/fuzzy.htm
> ...In particularly vast networks in fast moving
environments, the split
> second it takes to traverse the circuit is greater than the
time it takes
> for the situation to change. In reaction, the last node
tends to
compensate
> by ordering a large correction. But this also is delayed by
the long
journey
> across many nodes, so that it arrives missing its moving
mark, birthing
yet
> another gratuitous correction.
> The same effect causes student drivers
> to zigzag down the road, as each late
> large correction of the steering wheel
> overreacts to the last late overcorrection.
> Until the student driver learns to tighten
> the feedback loop to smaller, quicker
> corrections, he cannot help but swerve down
> the highway hunting (in vain) for the center.
> This then is the bane of the simple auto-circuit. It is
liable to
flutter
> or chatter, that is, to nervously oscillate from one
overreaction to
> another, hunting for its rest. There are a thousand tricks
to defeat this
> tendency of overcompensation, one trick each for the
thousand advance
> circuits that have been invented.
> http://www.kk.org/outofcontrol/ch7-c.html
> Fuzzy systems are based on
> storage of common-sense rules.
> For example, a fuzzy Army-ant robot controller might have
the fuzzy
> association if load is heavy, then signal for help longer.
Fuzzy
phenomena
> admit degrees: some loads are heavier than others; some
signal durations
are
> longer then others.
> A single association (heavy,longer)
> encodes all combinations...
> Fuzzy systems reason with
> parallel associative inference.
> A fuzzy system reasons with multivalued sets, instead of
true or false
> propositions, and it may adaptively modify its fuzzy
associations from
> representative numerical samples.
> http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html
> Wired: What is fuzzy logic and why do critics call it the
cocaine of
> science?
> Kosko: Fuzzy logic is Spock's worst nightmare - a way of
doing science
> without math. It's a new branch of machine intelligence that
tries to
make
> computers think the way people think and not the other way
around. You
don't
> write equations for how to wash clothes. Instead you load a
chip with
vague
> rules like if the wash water is dirty, add more soap, and if
very
dirty,
> add a lot more. All wash water is dirty and not dirty - to
some degree.
> It's just common sense. But it breaks the old either/or
logic of
Aristotle.
> That offends some scientists, who would like us to think and
talk like
> off/on switches. But they still haven't produced a statement
of fact like
the sky is blue or E=mc^2 that is 100 percent true or 100
percent
false.
> Fact ain't math. You can never get the science right to more
than a few
> decimal places. That's one reason we find chaos when we look
at things up
> close...
> ...Fuzzy systems are universal computers. I proved that as a
theorem -
the
> fuzzy approximation theorem. In theory, you can replace
every book on
> physics or economics with equivalent books that have fuzzy
systems where
the
> equations used to be. Fuzzy systems are model-free
estimators. You
don't
> have to guess at equations to build a bridge from inputs to
outputs.
Fuzzy
> rules build that bridge for you. There is math behind the
rules, but you
> don't need to know it to program a fuzzy system. You can
program it in
> English. If the air is cool, turn the AC down a little. But
the math
is
> not fuzzy. That's why you can capture fuzzy logic in a
digital chip.
> Most of the first fuzzy systems were in control - as in
adjusting a
camera
> lens or backing up a trailer truck to a loading dock. Now
we're applying
> fuzzy systems to wireless communications and multimedia. The
fuzzy rules
can
randomly spread signals over a wide bandwidth or teach an
intelligent
> agent the kind of houses or sunsets you prefer. The math
says we can
apply
> them anywhere. In practice, it may not be so easy.
> http://www.wired.com/wired/archive/3.02/kosko_pr.html
> Fuzzy logic is a superset of conventional(Boolean) logic
that has been
> extended to handle the concept of partial truth- truth
values between
completely true and completely false. As its name suggests, it
is
the
> logic underlying modes of reasoning which are approximate
rather than
exact.
> The importance of fuzzy logic derives
> from the fact that most modes of human
> reasoning and especially common_sense
> reasoning are approximate in nature.
> Boolean vs. Fuzzy: 300 years B.C., the Greek philosopher,
Aristotle came
up
> with binary logic(0,1), which is now the principle
foundation of
> Mathematics. It came down to one law: A or not-A, either
this or not
this.
> For example, a typical rose is either red or not red. It
cannot be red
and
> not red. Every statement or sentence is true or false or has
the truth
value
> 1 or 0. This is Aristotle's law of bivalence and was
philosophically
correct
> for over two thousand years.
> Two centuries before Aristotle, Buddha, had the belief which
contradicted
> the black-and-white world of worlds, which went beyond the
bivalent
cocoon
> and see the world as it is, filled with contradictions, with
things and
not
> things. He stated that a rose, could be to a certain degree
completely
red,
> but at the same time could also be at a certain degree not
red. Meaning
that
> it can be red and not red at the same time.
> Conventional(Boolean) logic states that a glass can be full
or not full
of
> water. However, suppose one were to fill the glass only
halfway. Then the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
multivalence
> is the fundamental concept which propelled Zader Lofti of
University
Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics of
> fuzzy logic founded by him are as follows.
> In 1965, Lofti Zadeh formally developed multivalued set
theory, and
> introduced the term fuzzy into the technical literature.
Nowadays, the
> recent emergence of fuzzy commercial products, as well as
new theory, has
> generated a new interest in multivalued systems. Yet already
engineers
have
> successfully applied fuzzy systems in many commercial areas
: intelligent
> subways automation, emergency breakers, cement mixers, Kanji
characters
> recognition, control air conditioners, automatic washing
machines, guide
of
> robot-arm manipulators, and so on.
> Fuzzy systems store banks of fuzzy associations or
common-sense rules
such
> as IF traffic is heavy in this direction, THEN keep the
light green
longer
> that might be articulated by an human expert. Some traffic
configuration
are
> heavier that others and some green-light duration are longer
than others,
so
> that, the single fuzzy association (HEAVY, LONGER) encodes
all these
> combinations. That is to say, fuzzy systems directly encode
structured
> knowledge but in a numerical framework : by entering the
fuzzy
association
> (HEAVY, LONGER) as a single entry in a rule database we are
defining an
> input-output transformation.
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
> Fuzzy Logic is a computational paradigm capable of modelling
the own
> uncertainness of human beings. Fuzzy reasoning is nothing
else than a
Fuzzy
> Logic-based formalism for encoding human knowledge or common
sense in a
> numerical framework. Indeed, the mathematical concepts on
which Fuzzy
Logic
> is supported are very easy to understand. In a Fuzzy
Controller, human
> experience is codified by means of linguistic if-then rules,
which
compute
> control actions upon given conditions. Fuzzy Logic has been
applied to
> problems that are difficult to solve mathematically. One of
its main
> advantages lies in the fact that it offers a straightforward
methodology
for
> modelling and controlling non-linear systems, which are
difficult to face
by
> means of conventional techniques.
> http://www.wkap.nl/prod/b/1-4020-7359-3
> Fuzzy logic models itself on the pattern of human reasoning
in its use of
> approximate information and uncertainty to generate
decisions. It was
> designed (during late 1980s and early 1990s) to
mathematically represent
> vagueness and develop tools for dealing with imprecision
inherent in
several
> problems. Normally, in digital computers one uses the binary
logic
where
> the digital signal has two discrete levels : low (logic
zero) or high
(logic
> one); nothing in-between. Fuzzy systems use soft linguistic
variables
(e.g.
> hot, tall, slow, light, heavy, dry, small, positive,
...etc.) and a range
of
> their weightage (or truth) values, called membership
functions, in the
> interval (0, 1), enabling the new computers to make
human-like decisions.
> Since human beings tend to use words rather than numbers to
describe
> behaviour patterns, fuzzy controls avoid the conventional
rigidity of
> computers and allow them to use parameters based on common
sense.
> http://www.tribuneindia.com/2002/20021024/science.htm
> Fuzzy logic best summed up by common sense
> Computer Corner
> John Boyd
> Fuzzy logic was introduced to the world 27 years ago by
Professor
> Lotfi Zadeh in his Fuzzy Sets paper published in Information
> Control magazine, though it is only recently that we've seen
it
> applied across a broad range of products.
> Some readers have asked for more explanation on fuzzy logic,
so
> here's an attempt to defuzzify the subject a little further.
> Simply put, fuzzy logic is aimed at enhancing our prissy
computer
> technology with a touch of common sense.
> One problem with the conventional digital computer is that
it is
> such a scrupulously either-or beast. It cannot be easily
coaxed
> to handle approximations or vague notions like young, a lot
and
> probably.
> Yet most of us rely on such terms daily because we happen to
be
> humans dealing with other humans, not robots building cars.
> It's an easy matter to arbitrarily program a computer so it
> designates everyone falling into the age-range 0f 15 to 18
> as being a youth. Such a precise category has come to be
called
> a crisp set since the emergence of fuzzy logic.
> Yet we all know some 14-year-olds can look older than some
> late-developers turning 20. Such exceptions, however, cannot
> be accounted for in conventional computing. Or at least not
> without an inordinate amount of additional programming and
> expense.
> As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied
> when I asked him if we couldn't just continue using
conventional
> programming and technology for controlling new products,
instead
> of fuzzy, Well, we could. And you could probably swim across
the
> Pacific if you got enough support from enough people. But ...
> To overcome this problem, Zadeh was inspired to develop his
fuzzy
> theory and the math to go with it that could be used to
create
> fuzzy sets based on imprecise natural language.
> Each member in a fuzzy set (such as the youths and others
considered
> in the above example) is assigned one of a continuous range
of values
> (called the membership value) between zero and one.
> Whereas in the above crisp set a 13-year-old going on 14
would still
> have to be considered a minor and thus be designated as zero
in
> binary logic, fuzzy logic could assign him a membership
value of
> say 0.1. Likewise, an immature 20-year-old who would
normally fall
> outside our either-or crisp-set range could be assigned a
membership
> value of 0.9 depending upon the criteria we use to measure
youth.
> Working out just what criteria to use, what values should be
assigned
> each member and deciding what rules are necessary to govern
the
> relationships between members is the key to successfully
applying
> fuzzy control in products.
> In some applications, determining the optimum rules has
become so
> complex, some manufacturers have resorted to employing the
aid of
> neural networks, which may be stretching a good thing too
far, given
> fuzzy logic's original purpose to get round complexity.
> Still, the flexibility in herent in fuzzy is clearly useful
in
> dealing with approximate calculations, such as about 100> It
can be used in artificial intelligence to provide us with an
almost true answer. It can also infer a common-sense result
even
> when the data is not precise.
> Our handwritten 5 in 250 would be treated as 5, not the
letter S,
> for instance, in Sony's fuzzy-based Palmtop computer.
> While we have all seen fuzzy logic-based products from the
likes of
> Matsua, Sanyo and Hitachi, one unlikely company that has made
> fuzzy technology a central part of its business strategy is
Omron
> Corp.
> It began its research into fuzzy logic in 1984 and has since
applied
> for over 700 patents. This puts it in the forefront of fuzzy
> applications in areas like factory and industry control, as
well as
> in medical equipment.
> In 1989, Omron also signed on lotfi Zadeh as a senior
advisor.
> Earlier this year at the Business Show in Harumi, Omron
demonstrated
> its fuzzy workstation. Omron manufactures both standard
Motorola
> 68040-based and 88000 reduced-instruction or RISC-based
workstations
> that can be fitted with a fuzzy inference board, turning
them into
> the world's first fuzzy workstations.
> Omron claims such a RISC-based workstation can achieve 4 ion
> operations per second, an incredible speed if they haven't
fuzzed
> on the number. Fuzzy logic is used in the workstations to
store
> and retrieve fuzzy information and make inferences.
> Ranging in price from Y2.5 million to almost Y4 million (a
US dollar
> is about 120 Yens -FM), these machines are not the kind of
products
> you will find down in Akihabara. (a section of Tokyo famous
for its
> quantity and variety of electronic goods -FM) Rather, they
are
> typically aimed at value-added resellers in niche markets,
and
> engineers who want to develop fuzzy applications, fuzzy
databases
> and expert systems, as well as fuzzy inference systems.
> However, the entrepreneurs among you may be interested in
Omron's
> FB-30AT fuzzy inference board for the IBM PC and compatible
wares.
> It features a 24 MHz FP-3000 fuzzy chip capable of
processing up
> to 128 rules, with five antecedents and 2 consequents.
Training
> software and a compiler is also available.
> Omron has also produced a fine little booklet on fuzzy called
Clearly Fuzzy that I dipped into when writing this column.
> Tadashi Katsuno, at Omron's public relations section, tells
me
> he still has a limited number of copies left that he will
send
> to the first readers of Computer Corner who write to him with
> contact information.
> The address is Omron Corp., International Public Relations
Section,
> Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105.
>
--------------------------------------------------------------
------
> - Farzin Mokhtarian
> farzin@apollo3.ntt.jp
http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/
areas/fuzzy/doc/in
tro/j_times.tgz
> http://www.ece.utep.edu/research/webfuzzy/about.html
http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS
_NOE_ASI99/ASI
'99_ViharosMonostori.htm
> http://www.bjarne.ca/pmflp.pdf
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
>
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htm
All unconscious processes.
===
Subject: Re: 'erf' function in C
> The attraction of the method for evaluating
> cPhi(x)=1-Phi(x)
> through the intermediary
> R(x)=cPhi(x)/phi(x)
> is that R(x) has an easily developed Taylor
> series. Thus an easy way to evaluate R(z+h),
> given a known value a=R(z), is through
[stuff deleted]
> Some of my early implementations of the method
> (one of which on newsgroups several years ago),
> kept a table of R(z) for 128 or 64 values z,
> and the series needed only a few steps to terminate.
> Subsequent exchanges with interested parties led
> to consideration of 8 then 4 and finally one value,
> z=0, for which evaluation of the odd terms in the
> Taylor series can be avoided, thanks to the
> observation of Daly.
already found the earlier versions of your algorithm
[specifically, the
ones with 8 and 128 pre-calculated values for R(z)] using
Google Groups.
Are you familiar with the algorithm developed by W.D. Cody?
See this
link for a C implementation with a stated accuracy of 18
significant
decimal digits
http://tigger.smu.edu.sg/software/mnp-stuff/stat.ubc.ca/
pnorms2.c
===
Subject: Re: v'' + A v' + B v = 0 for vectors
>I've learned about first order diff.eqs. in vectors. You can
diagonalize
and
>solve then convert back to get the original variables. But is
there a
general
>method for solving second order equations like
>v'' + Av' + Bv = 0
>?
>I would say, convert to first order. Twice as many variables,
>but you already know the method for it.
Actually it can be done directly: v = a exp(rt) is a solution
(where
a is a constant vector and r is a complex constant)
if r^2 a + r A a + B a = 0. So you want to find the values of r
for which det(r^2 I + r A + B) = 0, and then take a basis of
the null
space of r^2 I + r A + B for those values of r. If that's
enough
to give you a complete set of solutions, then good - otherwise
you
have to consider solutions of the form (at+b) exp(rt) etc.
For the inhomogeneous equation v'' + A v' + B v = b exp(rt)
you try v = a exp(rt), and find (if r^2 I + r A + B is
invertible) a = (r^2 I + r A + B)^(-1) b.
Department of Mathematics http://www.math.ubc.ca/~israel
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
G. Frege says...
>in math textbooks is to use upside-down A for for all
>and a backwards E for there exists
>Right. But since we cannot do t h a t in ASCII we have to use
the
normal A and normal E as BEST approximation, won't you think
so?
No, I don't think so, at all. The difference is that an
upside-down
A can never be mistaken for the name of a variable or function
symbol,
but A by itself certain can be. In contrast, All and Exists is
pretty unambiguous.
Another difference between ASCII and math textbooks is that
in math, you can use a different character for just about
every function symbol or variable. In contrast, ASCII only
provides 26 possible function and variable names. To get more,
you use multiple-character names. But if you allow
multiple-characters,
then how do you distinguish the quantified expression
Ax
from the name Ax (which might be, for example, your set of
axioms)?
I disagree completely with your suggestions. I think that the
notation that is using would be preferable to the notation
you are suggesting.
And I'm not just saying that because I think you're a jerk.
--
===
>For some reason, you seem to be taking this whole notation
thing rather
>personally. Aren't you getting a little carried away with
this talk of
lies and idiotic claims? This NOT a matter of life and death,
after
all!
Exactly. I don't think that there is anything wrong with your
notation.
If it irritates some people, well, they probably would be
irritated no
matter what you did.
--
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
>> in math textbooks is to use upside-down A for for all
>> and a backwards E for there existsRight. But since we
cannot do t h a t in ASCII we have to use the
normal A and normal E as BEST approximation, won't you think
so?
> No, I don't think so, at all. The difference is that an
upside-down
> A can never be mistaken for the name of a variable or
function symbol,
> but A by itself certain can be.
Completely agree with you! That's why I (personally) would
prefer to
write
(Ax) (Ex)
instead of
Ax Ex
(-the variant G. Greene would prefer).
This way confusion is successfully avoided, imho.
> In contrast, All and Exists is pretty unambiguous.
That's right. BUT readability isn't that good in formulas with
many
quantifiers. Moreover. (Ax) and (Ex) STILL resembles standard
symbols
used in print quite well. Actually Smullyan used them (with
upside-down
A and E) in his famous book about First-Order Logic> Another
difference between ASCII and math textbooks is that
> in math, you can use a different character for just about
> every function symbol or variable. In contrast, ASCII only
> provides 26 possible function and variable names. To get
more,
> you use multiple-character names.
Well, one might also use indices.
> But if you allow multiple-characters,
> then how do you distinguish the quantified expression
> Ax
> from the name Ax (which might be, for example, your set of
axioms)?
Right, that is indeed a problem, but again. Remember that I
/proposed/
to use: (Ax) and (Ex), leading to
(Ax)(x e Ax -> ...).
Is this really confusing? I don't think so. And right, why not
use
UPPERCASE letters in such cases:
(Ax)(x e AX -> ...)
Now compare the following two formulas:
- 's idiosyncratic notation:
EXISTS(x):(EXISTS(z):(z e x | x = 0) & ALL(w):(w e x <=> w e u
& Phi)
- A very COMMON notation in sci.logic (you are just too big a
fool to
realize that):
(Ex)(Ez)(z e x v x = 0) & (Aw)(w e x <-> w e u & Phi)
or even simpler:
ExEz(z e x v x = 0) & Aw(w e x <-> w e u & Phi)
> I disagree completely with your suggestions.
So what? What's your problem man, just pretending to be an
asshole, or
what?
> I think that the notation that is using would be preferable
to
> the notation you are suggesting.
F.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Christensen says...
For some reason, you seem to be taking this whole notation
thing rather
>personally. Aren't you getting a little carried away with
this talk of
lies and idiotic claims? This NOT a matter of life and death,
after all!
Exactly. I don't think that there is anything wrong with your
notation.
>If it irritates some people, well, they probably would be
irritated no
>matter what you did.
Not mean to take side at all about lies and idiotic claims;
there
does exist a degree of
(natural) language invariance in mathematics. Years ago, I had
encountered & learnt the up-side-own A
and inverted E in a highschool text book, before knowing what
exists
- which is not in my native langauge -
means.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> I would suggest to use the COMMON quantifiers
> (Ax) and (Ex)
> instead of the extremely uncommon and rather irritating
> ALL(x): and EXISTS(x):
>> I didn't know that there was any standard way to write
>> first-order logic using ASCII characters.
>Huh? Standard? Standard?! What the hell are you talking
about?!
> Basically, I was trying to say in a polite way that you are
being
> a jerk. But nevermind polite: You are being a jerk.
Oh right, *I* am a jerk because you ing asshole full of are
Well, ok. *plonk*
===
Subject: Re: Bayesian Class and Math/Stat Teaching Techniques
> One of my avocations is evaluating the work of bozos
> like you in lawsuits.
> Why do think I'm a bozo?
> It looks like he just felt the need to call -somebody- a
bozo.
> You just happened to get in the way.
> It makes a man feel so good -- BOZO, BOZO, BOZO!
> There, I said it! I feel better already.
> -- Robert Doctor of Bozology Dodier
Glad to help. :-)
There are lies, damned lies, and quotes from literary icons.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> Right, that is indeed a problem, but again. Remember that I
/proposed/
> to use: (Ax) and (Ex), leading to
> (Ax)(x e Ax -> ...).
> Is this really confusing? I don't think so.
If I encountered such a formula I'd stare at it for a while,
and then
understanding would dawn: it's written using an old notation
for
universal quantification in which for all x is rendered as
(x)...
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
Wovon man nicht sprechen kann, daruber muss man schweigen
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: rqizCcZvZeDc85a89bfLocYoGEogaxMJCYc1miK7N5cb8QEwJxGwsP
> there does exist a degree of (natural) language invariance
in mathematics.
> Years ago, I had encountered & learnt the up-side-own A and
inverted
E
> in a highschool text book, before knowing what exists -
which is not
in
> my native language - means.
Right. Up-side-own A and inverted E (without brackets) is more
or
less /standard/ these days.
On the other hand, the /Encyclop. Britannica/ (1999) used
(Ax) and (Ex)
again. :-) [Of course with up-side-own A and inverted E.]
But since there's really NO realistic situation where normal A
and
normal E in brackets could produce any confusion, imho it's
rather
*natural* to use t h e m as substitutes for up-side-own A and
inverted
E in ASCII contexts. No?
The *advantage* simply would be that ASCII formulas and
printed formulas
would look as similar as (reasonable) possible. That actually
might be
helpful for a beginner, imho.
F.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: Z6JbosZAgeAKFkS2o87rfeKZuLGk0NZJ3yZNkZgqohIJcoHBgi5sQy
> Right, that is indeed a problem, but again. Remember that I
/proposed/
> to use: (Ax) and (Ex), leading to
> (Ax)(x e Ax -> ...).
> Is this really confusing? I don't think so.
> If I encountered such a formula I'd stare at it for a while,
and then
> understanding would dawn: it's written using an old notation
for
> universal quantification in which for all x is rendered as
(x)...
Right. How about
Ax(x e AX -> ...) ?
Better now? ;-)
F.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
there does exist a degree of (natural) language invariance in
mathematics.
>Years ago, I had encountered & learnt the up-side-own A and
inverted
E
>in a highschool text book, before knowing what exists - which
is not
in
>my native language - means.
Right. Up-side-own A and inverted E (without brackets) is
more or
>less /standard/ these days.
>On the other hand, the /Encyclop. Britannica/ (1999) used
> (Ax) and (Ex)
>again. :-) [Of course with up-side-own A and inverted E.]
>But since there's really NO realistic situation where normal
A and
>normal E in brackets could produce any confusion, imho it's
rather
>*natural* to use t h e m as substitutes for up-side-own A and
inverted
E in ASCII contexts. No?
Agree. That is, I was just a little bit opposing the use of
exists,
forall: in complex expressions,
they tend to make the expressions longer and I for one would
rather
avoid typing them.
>The *advantage* simply would be that ASCII formulas and
printed formulas
>would look as similar as (reasonable) possible. That actually
might be
>helpful for a beginner, imho.
>F.
>
===
Subject: Re: ellipse from 4 points
> Well, the most straightforward (and hardest!) method would
be to
> take the general equation of a conic section:
> Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
> and come up with 8 equations in 6 unknowns. Not exactly the
prettiest.
Ummm.... Plugging in the values of 4 points gives 4 equations
in 6
unknowns, not 8 equations. So really not so bad to solve. And
since
the equations are homogeneous, you'll generally just get a
one-parameter set of solutions. In vector space terms, the set
of
solutions has dimension one. And if you start by translating
the
points so that one of them is (0,0), then one of the equations
is just
F=0.
And I beg to disagree, IMHO this is actually quite pretty. It
is the
beginning of a large story concerning the spaces that
parametrize
collections of curves with certain properties. This leads to
problems
in intersection theory in algebraic geometry, with lots of
beautiful
theorems and also many still unsolved problems.
For example, if you take a similar problem for cubics, you get
a 9
dimensional space of curves, and forcing them to go through a
certain
collection of points leads to a similar linear algebra
problem. Cubic
curves are elliptic curves, which have been in the news in
recent
years (e.g., Wiles' proof of Fermat's last theorem,
cryptography).
JS
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: TFO3E2ZYoeHkCjCwBD4F0sln8d02Qu2mi4b5UTIDofIVxCF7DKtXwu
>But since there's really NO realistic situation where normal
A and
>normal E in brackets could produce any confusion, imho it's
rather
>*natural* to use t h e m as substitutes for up-side-own A and
inverted
E in ASCII contexts. No?
> Agree. That is, I was just a little bit opposing the use of
exists,
forall: in complex expressions, they tend to make the
expressions
> longer [...]
Right, here's a quote of one of my other posts in this thread:
---------------------------------------------------------------
Now compare the following formulas:
- 's idiosyncratic notation:
EXISTS(x):(EXISTS(z):(z e x | x = 0) & ALL(w):(w e x <=> w e u
& Phi)
- a very COMMON notation in sci.logic:
(Ex)(Ez)(z e x v x = 0) & (Aw)(w e x <-> w e u & Phi)
or the even simpler:
ExEz(z e x v x = 0) & Aw(w e x <-> w e u & Phi)
---------------------------------------------------------------
F.
I disagree completely with your suggestions. (Daryl McCullough)
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: T8WIcEZfgeRujd5qtXiid2CCE1PRG3kowODS5VGc8x2xcjsOrZ9M0j
> I don't think that there is anything wrong with your
notation.
Well, let's face it Daryl, that's just an idiotic claim.
For example, it's extremely misleading to use | instead of the
connective vWho the did EVER write
p | q
when he MEANT
p v q or p / q or p or q
???
Hint: The | usually is called the Sheffer stroke, and it is
used to
denote the NAND truth-function.
http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Sheffer+stroke
http://en.wikipedia.org/wiki/Sheffer_stroke
Don't think, look! (L. Wittgenstein)
F.
===
Subject: Re: Science Without Math? (model-free common sense
steering)
> Conventional(Boolean) logic states that a glass can be full
or not full
of
> water. However, suppose one were to fill the glass only
halfway. Then the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
multivalence
> is the fundamental concept which propelled Zader Lofti of
University
Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics of
> fuzzy logic founded by him are as follows.
I don't buy this. A glass can be either [half-full] or
not-[half-full],
not
both. half-not-full is not equivalent to not-[half-full]. The
'not'
shouldn't be stuck in the middle of the statement, but
appended to the
front
of it. ie: A glass with 50% water is half-not-full, but to say
its
not-[half-full] is incorrect. I understand the concept of non
boolean
logics, and the benefit of fuzzy systems in various
applications, but this
is a pretty bad example IMO. Especially the part about it
'clearly'
disproving Aristotle's law of bivalence.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: boID4iZpQepiX1yjoCS7gAEu+4yG4KGVKkfQhalPZjSb5W0UsldHsx
> It might also be a good idea to decouple the displayed
notation from the
> keyboard limitations.
Well, actually, he already did that (in at least one case):
e (the element-relation)
is represented by an epsilon. That's certainly a good idea. On
the other
hand, how to get ASCII-Text from that...?
Idea...: There's a menu which allows to type in the ASCII
characters
that are used to represent that graphical symbol.
Symbol ASCII characters
Ex Ex
Ax Ax
: :
^^^
upside-down
E and A.
This way the software could produce proofs as ASCII text, but
still
use common symbols for normal display.
F.
===
Subject: Coin-Flipping Machine
Story from NPR:
http://www.npr.org/display_pages/features/feature_1697475.html
Apparently, coin-toss-outcomes are more a function of
human-unpredictability than of the coin's unpredictabilty.
So, perhaps it is more accurate to talk about, rather than a
biased
coin, a biased HUMAN-BEING!...
;)
(What!?!...Human-beings are BIASED!?...Huh???..)
(I apologize if this link has been mentioned recently on
sci.math
already.)
Leroy Quet
===
Subject: Congruence Involving 6 & Some Sums-of-Sums
Let, for each nonnegative integer k,
a(6k) = a(6k +1) = 1;
a(6k +2) = a(6k +5) = 0;
a(6k +3) =a(6k +4) = -1.
Let A(0,m) = a(m);
and for all positive integers n, and for nonnegative integers
m,
A(n,m) = sum{k=0 to m} A(n-1,k);
Then, for q and r = any nonnegative integers:
m!*(A(q,m+1) -binomial(q+m,q-1))
is congruent to
m!*A(r,m) (-1)^m (mod {m+q+r}).
So, more specifically, from the above we get:
For ODD m,
(m-1)!*A(r,m)
is congruent to
(r+m)!/(m r!) (mod {m+2r+1}).
For EVEN m,
(m-1)!*A(r,m)
is congruent to
(r+m-2)!/(m (r-2)!) (mod {m+2r-2}).
(Someone might enjoy confirming the above congruences...)
I wonder if any of these congruences have any interesting
number-theory implications...
Leroy
Quet
===
Subject: Re: Difficulty of calculus vs. discrete math
This might just be echoing James's post, and another one too,
but
there are a few slogans one comes across in learning maths.
Two that
stuck in my mind were that the easier a problem is to state
the harder
it is to solve (above a certain level), and there are reasons
the
definitions are hard and the theorems easy. Perhaps it is this
observation alluded to that in calculus there are many
problems all
having the same solution, yet in discrete maths it is often
that each
has a different method for solving it, if one even exists at
all. A
personal favourite would be the if there are six people at a
party
there are 3 mutual friends or 3 people who don't know each
other, easy
proof, generalize it and it becomes imposssibly difficult to
solve.
Perhaps Erdos's observation (of another problem) that maths
isn't
ready for these things yet is applicable.
===
Subject: min area to flip 2 hinged rods
I thought of this problem but have no idea how to tackle it.
2 rigid rods of unit length are hinged to each other.
Initially they are
parallel to each other, much like a closed pair of divider.
This divider is placed on a piece of paper. What is the
minimum area of
the
paper which allow the divider to be opened such that the angle
between the
two legs extend from 0 to 360 degrees. The divider is to touch
the paper
and no part of the divider is to extend beyond the paper
throughout the
whole process.
What area of mathematics deal with such problem ?
===
Subject: some complex integration questions
I have a few quick questions about integration of complex
functions.
first, how does one integrate over a curve that crosses a
branch cut?
is it possible? if c(t)=2e^it, t in [-pi,pi], then can
integration of
1/(z^2-1)=1/2(1/(z-1)+1/(z+1)) be done as usual? what care do
i have
to take when doing this?
and lastly, is integration of complex functions ever done over
regions
in the plane instead only over curves?
thanks
===
Subject: Re: Science Without Math? (model-free common sense
steering)
: Neural networks are model-free estimators, in that they do
not
require
an
: in-depth understanding of the phenomena they are modeling.
:
: http://www.arcon.com/arconneu.html
:
: THE MATHEMATICS OF CROSSING THE STREET:
:
: There is no exact number. Instead, there are an infinite
number of
: possibilities - from 1kph to over 100kph and everything in
between. You
: don't have a radar gun, so instead you watch the truck for a
second or
two,
: and sum its speed up in two words very fast. That is good
enough.
Phew.
: Your senses have told you the truck is coming very fast, but
you need
more
: information before you can decide whether or not to risk
crossing. How
far
: down the street is the truck? Is it slowing down? Again,
there are
no
: exact numbers, so you sum up the situation - close, not
slowing
quickly
: enough:
: Somehow your brain adds fast + close + not slowing quickly
enough,
and
: warns you instantly that the risk is high. It is purely
cognitive
process.
: It involves a complex combination of sensory information and
experience.
:
: ...Since there are no exact numbers in this story, the
mathematical
version
: must be told with fuzzy numbers...
:
: But, the process is still not quite over. Should I wait or
cross? You
have
: to make the decision. Risk tolerance leads to different
spins and
endings.
: If you walk with a cane, you reason, The risk is high, so
I'll wait.
You
: watch as the truck runs the red light. If you are a jogger,
impatient to
: cross, you disregard the evidence, step into the
intersection, and jump
back
: just in time to save your life.
$#!! Where was my head? =)
: http://www.decyde.com/crossingthestreet.html
:
: Fuzzy logic works the way that humans think as opposed to
the way that
: computers typically work. For example, consider the task of
driving a
car.
:
: You notice that the stoplight ahead is
: red and the car ahead is braking. Your
: mind might go through the thought process,
:
: I see that I need to stop. The roads are
: wet because it's raining and there is a
: car only a short distance in front of me.
:
: Therefore I need to apply a significant
: pressure on the brake pedal.
:
: This is all subconscious (in general), but that's the way we
think - in
: fuzzy terms. Do our brains compute the precise distance to
the car ahead
of
: us and the exact coefficient of friction between our tires
and the road,
and
: then use a Kalman filter to derive the optimal pressure
which should be
: applied to the brakes? Of course not. We use common-sense
rules and they
: seem to work pretty well. On the other hand, when we do
finally get
around
: to pressing the brake pedal there is some exact force that
we apply, say
: 1.326 pounds. So although we think in fuzzy, noncrisp ways,
our final
: actions are crisp. The process of translating the results of
fuzzy
reasoning
: to a crisp, nonfuzzy action is called defuzzification.
:
: http://www.innovatia.com/software/papers/fuzzy.htm
:
: ...In particularly vast networks in fast moving
environments, the split
: second it takes to traverse the circuit is greater than the
time it takes
: for the situation to change. In reaction, the last node
tends to
compensate
: by ordering a large correction. But this also is delayed by
the long
journey
: across many nodes, so that it arrives missing its moving
mark, birthing
yet
: another gratuitous correction.
:
: The same effect causes student drivers
: to zigzag down the road, as each late
: large correction of the steering wheel
: overreacts to the last late overcorrection.
:
: Until the student driver learns to tighten
: the feedback loop to smaller, quicker
: corrections, he cannot help but swerve down
: the highway hunting (in vain) for the center.
:
: This then is the bane of the simple auto-circuit. It is
liable to
flutter
: or chatter, that is, to nervously oscillate from one
overreaction to
: another, hunting for its rest. There are a thousand tricks
to defeat this
: tendency of overcompensation, one trick each for the
thousand advance
: circuits that have been invented.
:
: http://www.kk.org/outofcontrol/ch7-c.html
:
: Fuzzy systems are based on
: storage of common-sense rules.
:
: For example, a fuzzy Army-ant robot controller might have
the fuzzy
: association if load is heavy, then signal for help longer.
Fuzzy
phenomena
: admit degrees: some loads are heavier than others; some
signal durations
are
: longer then others.
:
: A single association (heavy,longer)
: encodes all combinations...
:
: Fuzzy systems reason with
: parallel associative inference.
:
: A fuzzy system reasons with multivalued sets, instead of
true or false
: propositions, and it may adaptively modify its fuzzy
associations from
: representative numerical samples.
:
: http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html
:
: Wired: What is fuzzy logic and why do critics call it the
cocaine of
: science?
:
: Kosko: Fuzzy logic is Spock's worst nightmare - a way of
doing science
: without math. It's a new branch of machine intelligence that
tries to
make
: computers think the way people think and not the other way
around. You
don't
: write equations for how to wash clothes. Instead you load a
chip with
vague
: rules like if the wash water is dirty, add more soap, and if
very
dirty,
: add a lot more. All wash water is dirty and not dirty - to
some degree.
: It's just common sense. But it breaks the old either/or
logic of
Aristotle.
: That offends some scientists, who would like us to think and
talk like
: off/on switches. But they still haven't produced a statement
of fact like
: the sky is blue or E=mc^2 that is 100 percent true or 100
percent
false.
: Fact ain't math. You can never get the science right to more
than a few
: decimal places. That's one reason we find chaos when we look
at things up
: close...
:
: ...Fuzzy systems are universal computers. I proved that as a
theorem -
the
: fuzzy approximation theorem. In theory, you can replace
every book on
: physics or economics with equivalent books that have fuzzy
systems where
the
: equations used to be. Fuzzy systems are model-free
estimators. You
don't
: have to guess at equations to build a bridge from inputs to
outputs.
Fuzzy
: rules build that bridge for you. There is math behind the
rules, but you
: don't need to know it to program a fuzzy system. You can
program it in
: English. If the air is cool, turn the AC down a little. But
the math
is
: not fuzzy. That's why you can capture fuzzy logic in a
digital chip.
:
: Most of the first fuzzy systems were in control - as in
adjusting a
camera
: lens or backing up a trailer truck to a loading dock. Now
we're applying
: fuzzy systems to wireless communications and multimedia. The
fuzzy rules
can
: randomly spread signals over a wide bandwidth or teach an
intelligent
: agent the kind of houses or sunsets you prefer. The math
says we can
apply
: them anywhere. In practice, it may not be so easy.
:
: http://www.wired.com/wired/archive/3.02/kosko_pr.html
:
: Fuzzy logic is a superset of conventional(Boolean) logic
that has been
: extended to handle the concept of partial truth- truth
values between
: completely true and completely false. As its name suggests,
it is
the
: logic underlying modes of reasoning which are approximate
rather than
exact.
:
: The importance of fuzzy logic derives
: from the fact that most modes of human
: reasoning and especially common_sense
: reasoning are approximate in nature.
:
: Boolean vs. Fuzzy: 300 years B.C., the Greek philosopher,
Aristotle came
up
: with binary logic(0,1), which is now the principle
foundation of
: Mathematics. It came down to one law: A or not-A, either
this or not
this.
: For example, a typical rose is either red or not red. It
cannot be red
and
: not red. Every statement or sentence is true or false or has
the truth
value
: 1 or 0. This is Aristotle's law of bivalence and was
philosophically
correct
: for over two thousand years.
:
: Two centuries before Aristotle, Buddha, had the belief which
contradicted
: the black-and-white world of worlds, which went beyond the
bivalent
cocoon
: and see the world as it is, filled with contradictions, with
things and
not
: things. He stated that a rose, could be to a certain degree
completely
red,
: but at the same time could also be at a certain degree not
red. Meaning
that
: it can be red and not red at the same time.
:
: Conventional(Boolean) logic states that a glass can be full
or not full
of
: water. However, suppose one were to fill the glass only
halfway. Then the
: glass can be half-full and half-not-full. Clearly, this
disprove's
: Aristotle's law of bivalence. This concept of certain degree
or
multivalence
: is the fundamental concept which propelled Zader Lofti of
University
Berkely
: in the 1960's to introduce fuzzy logic. The essential
characteristics of
: fuzzy logic founded by him are as follows.
:
:
: In 1965, Lofti Zadeh formally developed multivalued set
theory, and
: introduced the term fuzzy into the technical literature.
Nowadays, the
: recent emergence of fuzzy commercial products, as well as
new theory, has
: generated a new interest in multivalued systems. Yet already
engineers
have
: successfully applied fuzzy systems in many commercial areas
: intelligent
: subways automation, emergency breakers, cement mixers, Kanji
characters
: recognition, control air conditioners, automatic washing
machines, guide
of
: robot-arm manipulators, and so on.
:
: Fuzzy systems store banks of fuzzy associations or
common-sense rules
such
: as IF traffic is heavy in this direction, THEN keep the
light green
longer
: that might be articulated by an human expert. Some traffic
configuration
are
: heavier that others and some green-light duration are longer
than others,
so
: that, the single fuzzy association (HEAVY, LONGER) encodes
all these
: combinations. That is to say, fuzzy systems directly encode
structured
: knowledge but in a numerical framework : by entering the
fuzzy
association
: (HEAVY, LONGER) as a single entry in a rule database we are
defining an
: input-output transformation.
:
: http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
:
: Fuzzy Logic is a computational paradigm capable of modelling
the own
: uncertainness of human beings. Fuzzy reasoning is nothing
else than a
Fuzzy
: Logic-based formalism for encoding human knowledge or common
sense in a
: numerical framework. Indeed, the mathematical concepts on
which Fuzzy
Logic
: is supported are very easy to understand. In a Fuzzy
Controller, human
: experience is codified by means of linguistic if-then rules,
which
compute
: control actions upon given conditions. Fuzzy Logic has been
applied to
: problems that are difficult to solve mathematically. One of
its main
: advantages lies in the fact that it offers a straightforward
methodology
for
: modelling and controlling non-linear systems, which are
difficult to face
by
: means of conventional techniques.
:
: http://www.wkap.nl/prod/b/1-4020-7359-3
:
: Fuzzy logic models itself on the pattern of human reasoning
in its use of
: approximate information and uncertainty to generate
decisions. It was
: designed (during late 1980s and early 1990s) to
mathematically represent
: vagueness and develop tools for dealing with imprecision
inherent in
several
: problems. Normally, in digital computers one uses the binary
logic
where
: the digital signal has two discrete levels : low (logic
zero) or high
(logic
: one); nothing in-between. Fuzzy systems use soft linguistic
variables
(e.g.
: hot, tall, slow, light, heavy, dry, small, positive,
...etc.) and a range
of
: their weightage (or truth) values, called membership
functions, in the
: interval (0, 1), enabling the new computers to make
human-like decisions.
: Since human beings tend to use words rather than numbers to
describe
: behaviour patterns, fuzzy controls avoid the conventional
rigidity of
: computers and allow them to use parameters based on common
sense.
:
: http://www.tribuneindia.com/2002/20021024/science.htm
:
:
: Fuzzy logic best summed up by common sense
:
: Computer Corner
: John Boyd
:
: Fuzzy logic was introduced to the world 27 years ago by
Professor
: Lotfi Zadeh in his Fuzzy Sets paper published in Information
: Control magazine, though it is only recently that we've seen
it
: applied across a broad range of products.
:
: Some readers have asked for more explanation on fuzzy logic,
so
: here's an attempt to defuzzify the subject a little further.
:
: Simply put, fuzzy logic is aimed at enhancing our prissy
computer
: technology with a touch of common sense.
:
: One problem with the conventional digital computer is that
it is
: such a scrupulously either-or beast. It cannot be easily
coaxed
: to handle approximations or vague notions like young, a lot
and
: probably.
:
: Yet most of us rely on such terms daily because we happen to
be
: humans dealing with other humans, not robots building cars.
:
: It's an easy matter to arbitrarily program a computer so it
: designates everyone falling into the age-range 0f 15 to 18
: as being a youth. Such a precise category has come to be
called
: a crisp set since the emergence of fuzzy logic.
:
: Yet we all know some 14-year-olds can look older than some
: late-developers turning 20. Such exceptions, however, cannot
: be accounted for in conventional computing. Or at least not
: without an inordinate amount of additional programming and
: expense.
:
: As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied
: when I asked him if we couldn't just continue using
conventional
: programming and technology for controlling new products,
instead
: of fuzzy, Well, we could. And you could probably swim across
the
: Pacific if you got enough support from enough people. But ...
:
: To overcome this problem, Zadeh was inspired to develop his
fuzzy
: theory and the math to go with it that could be used to
create
: fuzzy sets based on imprecise natural language.
:
: Each member in a fuzzy set (such as the youths and others
considered
: in the above example) is assigned one of a continuous range
of values
: (called the membership value) between zero and one.
:
: Whereas in the above crisp set a 13-year-old going on 14
would still
: have to be considered a minor and thus be designated as zero
in
: binary logic, fuzzy logic could assign him a membership
value of
: say 0.1. Likewise, an immature 20-year-old who would
normally fall
: outside our either-or crisp-set range could be assigned a
membership
: value of 0.9 depending upon the criteria we use to measure
youth.
:
: Working out just what criteria to use, what values should be
assigned
: each member and deciding what rules are necessary to govern
the
: relationships between members is the key to successfully
applying
: fuzzy control in products.
:
: In some applications, determining the optimum rules has
become so
: complex, some manufacturers have resorted to employing the
aid of
: neural networks, which may be stretching a good thing too
far, given
: fuzzy logic's original purpose to get round complexity.
:
: Still, the flexibility in herent in fuzzy is clearly useful
in
: dealing with approximate calculations, such as about 100:
: It can be used in artificial intelligence to provide us with
an
: almost true answer. It can also infer a common-sense result
even
: when the data is not precise.
:
: Our handwritten 5 in 250 would be treated as 5, not the
letter S,
: for instance, in Sony's fuzzy-based Palmtop computer.
:
: While we have all seen fuzzy logic-based products from the
likes of
: Matsua, Sanyo and Hitachi, one unlikely company that has made
: fuzzy technology a central part of its business strategy is
Omron
: Corp.
:
: It began its research into fuzzy logic in 1984 and has since
applied
: for over 700 patents. This puts it in the forefront of fuzzy
: applications in areas like factory and industry control, as
well as
: in medical equipment.
:
: In 1989, Omron also signed on lotfi Zadeh as a senior
advisor.
:
: Earlier this year at the Business Show in Harumi, Omron
demonstrated
: its fuzzy workstation. Omron manufactures both standard
Motorola
: 68040-based and 88000 reduced-instruction or RISC-based
workstations
: that can be fitted with a fuzzy inference board, turning
them into
: the world's first fuzzy workstations.
:
: Omron claims such a RISC-based workstation can achieve 4 ion
: operations per second, an incredible speed if they haven't
fuzzed
: on the number. Fuzzy logic is used in the workstations to
store
: and retrieve fuzzy information and make inferences.
:
: Ranging in price from Y2.5 million to almost Y4 million (a
US dollar
: is about 120 Yens -FM), these machines are not the kind of
products
: you will find down in Akihabara. (a section of Tokyo famous
for its
: quantity and variety of electronic goods -FM) Rather, they
are
: typically aimed at value-added resellers in niche markets,
and
: engineers who want to develop fuzzy applications, fuzzy
databases
: and expert systems, as well as fuzzy inference systems.
:
: However, the entrepreneurs among you may be interested in
Omron's
: FB-30AT fuzzy inference board for the IBM PC and compatible
wares.
: It features a 24 MHz FP-3000 fuzzy chip capable of
processing up
: to 128 rules, with five antecedents and 2 consequents.
Training
: software and a compiler is also available.
:
: Omron has also produced a fine little booklet on fuzzy called
: Clearly Fuzzy that I dipped into when writing this column.
:
: Tadashi Katsuno, at Omron's public relations section, tells
me
: he still has a limited number of copies left that he will
send
: to the first readers of Computer Corner who write to him with
: contact information.
:
: The address is Omron Corp., International Public Relations
Section,
: Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105.
:
--------------------------------------------------------------
------
:
:
http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/
areas/fuzzy/doc/in
tro/j_times.tgz
:
: http://www.ece.utep.edu/research/webfuzzy/about.html
:
http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS
_NOE_ASI99/ASI
'99_ViharosMonostori.htm
: http://www.bjarne.ca/pmflp.pdf
: http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
:
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htm
Sometimes I really believe that people give many names to the
same thing
when it is something we really appreciate. There is model in
there, but it
is prior to numbers. It is topological, connective. It looks
like:
a -> b
and introduces a fundamental asymmetry we can play with.
Automata build up digraphs of these upon which they evolve
through state
transitions. Abstraction neural nets are often lattices,
though you toss
in
a loop or ring or whatever (like in nematodes) and you have
recurrence.
All of them say the same thing: you are passing information
through a
structure of transformations.
I took a class in Control Theory once in my undergrad years,
and the
professor used to often suggest books to me by Norbert Wiener
and others in
the cybernetics community or Bucky's synergetics. When I could
understand
dynamics better, I was able to get into Rene Thom and
catastrophes through
another suggestion when he was teaching a non-linear diffy-que
class.
The discrete becomes continuous.
And the logics, all of them, so very useful. The dynamics they
play
calculating away the universe.
Information
thinking...
thinking...
thinking...
===-=-=-=-=-
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
> I don't think that there is anything wrong with your
notation.
> Well, let's face it Daryl, that's just an idiotic claim.
> For example, it's extremely misleading to use | instead of
the
> connective v> Who the did EVER write
> p | q
> when he MEANT
> p v q or p / q or p or q
> ???
Somebody whose been taking CompSci for too many years?
Honestly, I've done it before, particularly when I've been
working on
programming and then go to take a break. Of course, my scratch
work
contains
all sorts of nonstandard symbols... :-/
That said... well... I'll stay out of this discussion ;-)
> Hint: The | usually is called the Sheffer stroke, and it is
used to
> denote the NAND truth-function.
> http://www.swif.uniba.it/lei/foldop/foldoc.cgi?Sheffer+stroke
> http://en.wikipedia.org/wiki/Sheffer_stroke
Don't think, look! (L. Wittgenstein)
> F.
Jonathan Christensen
---
Outgoing mail is certified Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
===
Subject: Re: min area to flip 2 hinged rods
> I thought of this problem but have no idea how to tackle it.
> 2 rigid rods of unit length are hinged to each other.
Initially they are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
> What area of mathematics deal with such problem ?
A circle eith radius equal to the length of one rod (or the
longer rod, if
they are not the same)? Unless I've understood it incorrectly,
it's pretty
simple. One rod lies on a radius of the circle, with the hinge
at the
center. Then the other rod is swung out around and sweeps over
the entire
area of the circle before it gets to 360 degrees, at which
point it is back
where it started.
But... maybe I did understand it wrong.
Jonathan
---
Outgoing mail is certified Virus Free.
Checked by AVG anti-virus system (http://www.grisoft.com).
===
Subject: Re: min area to flip 2 hinged rods
> I thought of this problem but have no idea how to tackle it.
> 2 rigid rods of unit length are hinged to each other.
Initially they are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
> the paper which allow the divider to be opened such that the
angle
between
> the
> two legs extend from 0 to 360 degrees. The divider is to
touch the paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
Since no part of the divider is to extend beyond the paper, we
assume
that
it is laid down on the paper, not stood up on one point. We
further assume
that the whole of (the underside of) the divider must be in
contact with
the paper throughout any rotation operation, and that the
paper does not
move.
With these fairly simple assumptions in place, it's hard to
see how the
answer will be anything other than pi square units (area of a
circle of
radius one unit).
> What area of mathematics deal with such problem ?
Geometry.
Richard Heathfield : binary@eton.powernet.co.uk
Usenet is a strange place. - Dennis M Ritchie, 29 July 1999.
C FAQ: http://www.eskimo.com/~scs/C-faq/top.html
K&R answers, C books, etc: http://users.powernet.co.uk/eton
===
Subject: Differentiate many variables
I have a system consisting of 6 atoms. They all interact with
one another
(bonds and van der waals forces).
I have the equations for the interactions between atoms.
Each atom is in 3D space. I can alter the x,y,z for each atom.
I will have a function that depends on these 3*6 variables.
Constructing
this function should be fairly easy.
(For ease of work, I presume that I should fix one of the
atoms at the
origin).
I want to minimise this function (analytically).
So, How do I differentiate this 18 variable function?
I do have a degree in comp + math but it was a long time ago,
and I don't
even know what this would be called. Multivariate
differentiation? Partial
differentiation?
Anyone know how to do this (or have any links)
Also, I've got some tools that I'll be using once I got the
method straight
in my head: Mathematica, mathcad and matlab.
Anyone got any idea how I would use one of the packages to
help me?
(preferably mathematica, as I heard that the strongest
symbolically)
===
Subject: Re: min area to flip 2 hinged rods
the area of the smaller unhinged end of the two rods
(area-wise) plus
the larger unhinged end's area, plus a fudge factor (I'm
supplying no
math because you didn't specify the shape of the paper). You
didn't
specify that the rods were laying on the paper.
> I thought of this problem but have no idea how to tackle it.
> 2 rigid rods of unit length are hinged to each other.
Initially they are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
> What area of mathematics deal with such problem ?
===
Subject: Re: min area to flip 2 hinged rods
> I thought of this problem but have no idea how to tackle it.
> 2 rigid rods of unit length are hinged to each other.
Initially they
are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area
of
> the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the
paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
> What area of mathematics deal with such problem ?
> A circle eith radius equal to the length of one rod (or the
longer rod,
if
> they are not the same)? Unless I've understood it
incorrectly, it's
pretty
> simple. One rod lies on a radius of the circle, with the
hinge at the
> center. Then the other rod is swung out around and sweeps
over the entire
> area of the circle before it gets to 360 degrees, at which
point it is
back
> where it started.
> But... maybe I did understand it wrong.
> Jonathan
It can certainly be done in a circle of that radius, but it
can also be
done in a half circle of the same radius if the hinge is
allowed to
change positions,
If r is the length of each segment, it can be done within the
curve
|x|^s + |y|^s = r^s, for s = 1, where the circle would be s =
2.
I think it can be done with s = 1/2.
It may well be that the minimal s for which it can be done
gives the
desired minimal enclosed area curve.
On the other hand, there are closed curves enclosing
arbitrarily small
areas in which a line segment may be rotated through two right
angles.
===
Subject: Re: min area to flip 2 hinged rods
> I thought of this problem but have no idea how to tackle it.
> 2 rigid rods of unit length are hinged to each other.
Initially they
are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area
of
> the paper which allow the divider to be opened such that the
angle
between
> the
> two legs extend from 0 to 360 degrees. The divider is to
touch the
paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
> Since no part of the divider is to extend beyond the paper,
we assume
that
> it is laid down on the paper, not stood up on one point. We
further assume
> that the whole of (the underside of) the divider must be in
contact with
> the paper throughout any rotation operation, and that the
paper does not
> move.
> With these fairly simple assumptions in place, it's hard to
see how the
> answer will be anything other than pi square units (area of
a circle of
> radius one unit).
I don't think so. Imagine you pull the ends of the divider away
from each other until the divider is straight, then nudge the
hinge past
vertical and push the ends back together. The envelope should
be a sort
of diamond with in-bowed sides, and it should fit inside a
circle of
unit radius with plenty of room to spare.
----j7y
**
jere7my tho?rpe / 734-769-0913 There is no spoon. SPOON!
There
> j7y@liws.org <<< is no spoon. SPOON! There is
no
invert liws to reply via email spoon. SPOON! -- The Tick vs.
Neo
===
Subject: Re: Interesting problem
>I'm still not convinced that a non-empty A can exist...
> Neither am I. But I'm also not convinced that it can't exist.
> Note that if you only had to worry about addition, not
multiplication,
> there would be examples: e.g. take an additive but nonlinear
function
> f on the reals (f(x+y) = f(x)+f(y) for all x,y in R, but not
f(x) = cx.
> Such exotic beasts are consequences of the Axiom of Choice
(e.g.
> using a Hamel basis for the reals over the rationals). Then
> take A = {x > 0: f(x) >= 0} and B = {x > 0: f(x) < 0}.
However,
> these won't satisfy the multiplicative requirements.
Here's a simpler version: Let U be a finite extension of the
algebraic
numbers, and U+ = U n R+. Does there exist A,B meeting the
requirements: U+ = A u B, A n B = {}, A and B closed under +,
*?
I _think_ the answer is yes, but I don't quite have a proof
yet...
Cheers - Chas
(n == intersction, u == union)
===
Subject: Re: some complex integration questions
> I have a few quick questions about integration of complex
functions.
> first, how does one integrate over a curve that crosses a
branch cut?
> is it possible? if c(t)=2e^it, t in [-pi,pi], then can
integration of
> 1/(z^2-1)=1/2(1/(z-1)+1/(z+1)) be done as usual? what care
do i have
> to take when doing this?
I do not understand your question. Your decomposition of
1/(z^2 - 1) is
almost correct; it should be 1/(z^2 - 1) = 1/2(1/(z - 1) -
1/(z + 1)).
So, all that is left for you to do is to calculate the
integrals of
1/(z - 1) and 1/(z + 1) over your curve. For this, you use
Cauchy's
formula or the definition of index. There are no branch cuts
here.
Jose Carlos Santos
===
Subject: Re: min area to flip 2 hinged rods
> With these fairly simple assumptions in place, it's hard to
see how the
> answer will be anything other than pi square units (area of
a circle of
> radius one unit).
> I don't think so. Imagine you pull the ends of the divider
away
> from each other until the divider is straight, then nudge
the hinge past
> vertical and push the ends back together. The envelope
should be a sort
> of diamond with in-bowed sides, and it should fit inside a
circle of
> unit radius with plenty of room to spare.
The shape I, Mister Math Whiz, was trying to describe is
apparently
an astroid. I believe its area should be 3/8 pi square units,
which
is less than half the area of the unit circle.
No idea if that's a minimum, though.
----j7y
**
jere7my tho?rpe / 734-769-0913 There is no spoon. SPOON!
There
> j7y@liws.org <<< is no spoon. SPOON! There is
no
invert liws to reply via email spoon. SPOON! -- The Tick vs.
Neo
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
|: > Oh, did you miss the thread on modality in language as
well?
|:
|: What, Andrew Patterson's nonsense? That was (a) independent
|: of your posturing and (b) obviously received largely with
|: indifference.
|
|You see, that is the great evil laying at the heart of all
your anger with
|my post. There are people out there besides you, Brian, with
many varied
|interests. As long as we stay on topic to a particular group,
everyone
|should have the right to post their interests and questions.
The usenet
|does not follow office politics. Newcomers have all the same
rights as
|those who have been posting for years.
It's not a newcomers-versus-oldtimers issue. Newcomers and
oldtimers are
not essentially different on this issue.
It's a basic issue of shared resources. Bandwidth is very
cheap indeed,
and like with most resources, there's no simple maximum
available amount
of it. But like with most resources, the environment degrades
as people
use more and more of it beyond a certain point.
Nearly every communication channel that's free for the sender
is being
overused in an unpleasant way. Now and then I get woken up at
night by
people honking their horns to get people's attention. As I
pick up my
newspaper in the morning, I find that junk mail has been added
to it by
sticking it under the rubber band, in addition to the usual
advertising
inside it. Or sometimes it's attached to the door knob,
despite our
no-soliciting policy. Outside, egocentric youngsters have
spraypainted
their nicknames on buildings. At work, we've had a guy trying
to sell
Kincaid paintings from office to office, who didn't want to
leave.
Then there's the guy who thinks shouting to the receptionist
at the
other end of the office is easier than using the intercom. :-)
Our mailserver at work is pretty good at filtering out spam,
but once in
awhile somebody decides to pummel it with mail addressed to a
large number
of common names @ our domain, hoping that at least some of
them will be
somebody's user name. When I go to the bathroom, fairly often
someone has
attached advertisements for some work-from-home scheme to the
wall of the
stall. When I then leave work, I routinely find that
advertising has been
put under the windshield wiper. Back at home, every so often
somebody
repeats a message saying they've been assigned to review my
mortgage
and need to talk to me. And of course the usual messages from
Nigeria
and so on when I check my email.
I'm not claiming that what you're doing is as obnoxious as
those things,
but it shares with them a certain selfish tency. How can we
tell that
it's selfish? Just honestly perform the thought-experiment of
imagining
what effect it would have on the netnews-reading experience if
everybody
were as casual about massive crossposting as you have been. It
doesn't
take too much imagination; overly crossposted threads are not
so rare,
and most of us have some idea how poorly they tend to work.
A posting does not become on topic to a group just because it
contains
some paragraph that would be on topic on its own. If messages,
with such
weak ties to the groups they're initially posted to, initiate
threads of
any length, thread-drift usually causes them rather soon to
become
entirely off-topic except in one or two of the original
groups. If people
were really on their toes, they would stop crossposting at
that point, but
quite often that doesn't work. Sci.math has had prolonged
discussions of
all kinds of hot-button topics inflicted on it. It seems that
typically
once a group such as talk.abortion gets involved, you can
assume it will
take months for the noise-to-signal ratio to go back down to
normal again.
I had a partizan on one side of the abortion flame war tell me
point blank
that he considered his extended criticism of someone's
character to be
on topic in sci.math, because his enemy happened to be a
mathematician.
One difference between a newcomer and an oldtimer, I suppose,
is that if
you are familiar with the way usenet used to be, you can see
how much less
pleasant it is now, due to the proliferation of noise.
|: > Expense?
|:
|: Yes, expense. You are, for example, directly responsible
|: for cluttering sci.lang with off-topic mathematics and
|: complaints about the lack of physical content in your posts
|: from sci.physics.
|
|One collection of people upset that I am posting mathematics,
another
upset
|that there is not enough. Maybe it is these two groups that
should be
|arguing between each other and not be including me at all,
I'd be a heck of a lot more sympathetic, if I didn't see you
doing such
disengenuous things. Here you probably don't realize how
obvious it is
that you're playing dumb in order to sound innocent. It would
take some
amazing degree of confusion for a person to suppose that
people on
sci.math complaining that they're having too many
nonmathematical
postings inflicted on them, and people in sci.lang complaining
that
they're getting too many mathematical postings inflicted on
them, are
disagreeing *with each other*.
All this would also fail to be objectionable if there really
wasn't any
nicer way for you to call attention to your attempted
interdisciplinary
discussion than to crosspost so much of it. You could easily
have chosen
one place as home for the discussion, and posted only the bits
actually
relevant to various other groups, with a reminder of where the
whole
big thing was available. Doing it the way you did it instead
serves only
as an attention-getting move, an attempt to draw the attention
of people
(for example) reading sci.physics, but not interested enough
in what you
have to say to consider it worth checking out your home page
or wherever
you housed the rest of the discussion. You could just as well
do it in
a polite way, but you don't bother to.
|but I think the
|more prudent action would be for those who don't find content
they are
|interested in to just skip my threads.
This is the standard excuse. Just delete the email, just give
a few
seconds of my time to the salesman, just wait a bit for the
noise to go
away, just toss out the junk mail, and so on, and so on, and
so on, and
so on, and so on, and so on. It's true that this is normally
the best way
of handling junk, but it's disengenuous coming from one of the
sources of
the junk, because it disregards the responsibility of the
sender to have
some shred of self-restraint. It just ignores how much
cheesier life has
become on account of so many communication channels now being
half-filled
with stuff that's there ONLY because the sender considers
their interests
in getting attention more important than the receiver's
preferences in
what to pay attention to.
[...]
|You start attacks, mister Scott, and
|that puts you in error here.
Massive crossposting is an attack.
Keith Ramsay
===
Subject: re:Interesting problem
If by finite extension you mean extension by one element (or
finitely
many elements) then it's possible.
If F+ can be split into two closed sets (under mult. and add.)
then if
we add x transcendental over F to obtain the field F(x) which
consist
of rational functions of x over F, we can consider those
functions
with leading coefficients in A or B.
Doesn't quite work if x is algebraic over F.
===
Subject: e^i(pi) = -1 revisited with Doubly Infinites Re:
infinite rightward
strings tacked-on to p-adics serves as Orthogonality and makes
Doubly-Infinites the points of Lobachevskian Geometry
I would have liked to post this followup to a post I made
several
hours earlier tonight but Google has not yet correlated that
post to
the newsgroup board and so I post this one as a non-sequitor
followup.
I remember the last time I visited e^i(pi) = -1 was in the
early 1990s
after learning that the p-adics of the 5-adics have a number
that is
truly i. And I remember Karl Heuer trying to help me see if
that
equation can be broken down into its number parts rather than a
symbol-equation. So we had -1 as a Real, and we had e and pi
as Reals
and we had i as a 5-adic. But then no progress.
Well, tonight I have something new and different that I did
not have
in the early 1990s. Tonight I have the idea that the Reals
were a fake
or fictional set and that the only true numbers are either
p-adics or
Doubly-Infinites.
So let me see if progress now can be achieved with the equation
e^i(pi) = -1.
I prefer 10-adics because all of us are so used to seeing the
decimal
system but that my hinder us because i is in the 5-adics. So
can we
re-arrange the 5-adic i into a 10-adic?
So in the equation e^i(pi) = -1, let us mark out what numbers
we have.
Since Reals are a fictional set then e and pi are
Doubly-Infinites.
That leaves i and -1 and they are thus p-adics. In 10-adics -1
is
easily replaced as ....999999. But can we use the 5-adic i in
10-adics? If we can us the 5-adic i or transform it into a
10-adic i
then it is easy to transfer straight across the Doubly
Infinite for e
and pi as .....00002.71...... and .....0003.14..... If we can,
would
leave the question has how one multiplies p-adic with
Doubly-Infinites
to achieve an end result of .....9999999 which is
representative of
-1.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
===
Subject: Re: min area to flip 2 hinged rods
> 2 rigid rods of unit length are hinged to each other.
Initially they are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
Unless I'm horribly mistaken, it's just a circle of diameter 1
(or a
semicircle of diameter 1, if you count refolding back the way
you
started for the part about opening from 0 to 360), but I can't
prove
it offhand. For what I do know for a fact, I can state the
following:
the area of the minimal figure is between 1/2 (big enough to
hold both
rods when at right angles) and pi (circle, a working upper
bound),
must have two perpendicular chords of length at least 1 and a
chord of
length 2. Indeed, you must be able to fit ANY isoceles
triangle with
two sides of length 1 into the figure somehow.
Consider fixing one rod and swinging the other around; then
you trace
this circle out, so this is an upper bound. Now the only think
you
can do differently is not to fix the rod while swinging the
other arm
out; while I'm not going to do this for you, consider two
continuous
parametric paths through space p1(t) and p2(t) which represent
the
positions of each end of the previously fixed rod. The only
condition
on these is that the distance between them remains 1. Also, the
second rod rotates to be at an angle of t with the first two;
call its
location q(t). Then consider the area of the union of triangles
p1(t), p2(t), q(t) for all t from 0 degrees to 360 degrees,
which you
should be able to do with a clever path integral, find the
minimum
over all paths (which I assume is pi, given by p1, p2
constant) and
you'd have a rigorous proof of the fact. For one thing, the
area of
the ribbon traced out by the rod (p1, p2) would have to be
rather
small to avoid exceeding pi on its own; it is possible that a
solution
involving oscillation over a fixed, small area could exist, but
===
Subject: Re: Coin-Flipping Machine
> Story from NPR:
>
http://www.npr.org/display_pages/features/feature_1697475.html
> Apparently, coin-toss-outcomes are more a function of
> human-unpredictability than of the coin's unpredictabilty.
> So, perhaps it is more accurate to talk about, rather than a
biased
> coin, a biased HUMAN-BEING!...
> ;)
> (What!?!...Human-beings are BIASED!?...Huh???..)
> (I apologize if this link has been mentioned recently on
sci.math
> already.)
> Leroy Quet
A coin, when flipped the same way, will come up with the same
outcome
We now have *proof* for the vague philosophical concept of
determinism! I'm going to go publicly pronounce the death of
free
will; brb.
(sarcasm, if you're too stupid to realize)
===
Subject: Re: Coin-Flipping Machine
> Story from NPR:
>
http://www.npr.org/display_pages/features/feature_1697475.html
> Apparently, coin-toss-outcomes are more a function of
> human-unpredictability than of the coin's unpredictabilty.
> So, perhaps it is more accurate to talk about, rather than a
biased
> coin, a biased HUMAN-BEING!...
> ;)
> (What!?!...Human-beings are BIASED!?...Huh???..)
> (I apologize if this link has been mentioned recently on
sci.math
> already.)
> Leroy Quet
By the way, in my previous post, by you I don't mean Leroy of
course. He's cool. I'm talking to all the trolls who crosspost
all
this junk about philosophy and the beginnings of the universe
and the
nature of learning. I *swear* we don't care; just go away.
===
Subject: Re: min area to flip 2 hinged rods
>I thought of this problem but have no idea how to tackle it.
>2 rigid rods of unit length are hinged to each other.
Initially they
>are parallel to each other, much like a closed pair of
divider.
>This divider is placed on a piece of paper. What is the
minimum area
>of the paper which allow the divider to be opened such that
the angle
>between the
>two legs extend from 0 to 360 degrees. The divider is to
touch the
>paper and no part of the divider is to extend beyond the paper
>throughout the whole process.
>Since no part of the divider is to extend beyond the paper, we
assume
>that it is laid down on the paper, not stood up on one point.
We further
>assume that the whole of (the underside of) the divider must
be in
>contact with the paper throughout any rotation operation, and
that the
>paper does not move.
>With these fairly simple assumptions in place, it's hard to
see how the
>answer will be anything other than pi square units (area of a
circle of
>radius one unit).
> I don't think so. Imagine you pull the ends of the divider
away
> from each other until the divider is straight, then nudge
the hinge past
> vertical and push the ends back together. The envelope
should be a sort
> of diamond with in-bowed sides, and it should fit inside a
circle of
> unit radius with plenty of room to spare.
Ah, so the paper /is/ moving! (Or rather, the paper and the
divider hinge
move relative to each other -- and therefore one of my
assumptions is
incorrect.) Yes, that certainly makes the problem more
interesting. :-)
Being no geometer, the most I can salvage out of this is to
say that pi
constitutes an upper limit on the minimum area of paper
required.
Richard Heathfield : binary@eton.powernet.co.uk
Usenet is a strange place. - Dennis M Ritchie, 29 July 1999.
C FAQ: http://www.eskimo.com/~scs/C-faq/top.html
K&R answers, C books, etc: http://users.powernet.co.uk/eton
===
Subject: Re: Minimally simple finite groups?
[...]
>Is it covered in Carter's Simple Groups of Lie Type? How about
>the Atlas? (Not that a copy of that can be had for love nor
>money, as far as I've been able to discover. :-(
Thank you! It never occurred to me that they'd have something
not
available at amazon.com.
===
Subject: f continuous
f:R-->R surjective with the property
( for any x(n) real sequence f(x(n)) converge => x(n) converge
)
Prove that f is continous
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
X-ID: EYH6DMZewexTNOnivyQKy-r55WLAcUmIBNqPCJas38vrzjN8L40eo8
> Imho, it's extremely misleading to use | instead of the
> connective v
> Who [...] did EVER write
> p | q
> when he MEANT
> p v q or p / q or p or q
> ???
> Somebody whose been taking CompSci for too many years?
Right. Actually, I KNOW that. ;-) But surely there's a world
beyond the
CS department, isn't it? :-)
Indeed, OTTER for example uses | to denote disjunction in its
ASCII
/input files/. But frankly, I would hardly call OTTER a
teaching tool.
(And there's a certain difference between an ASCII input file
of a
highly sophisticated automatic proving software and an
interactive proof
writer intended as a teaching tool. ;-)
> Honestly, I've done it before, particularly when I've been
working on
> programming and then go to take a break. Of course, my
scratch work
> contains all sorts of nonstandard symbols... :-/
Right. And I guess that's rather normal. Why should anyone
care?
For example, in a *personal* (e-mail) communication *I*
recently
proposed to use | for abbreviating xor. :-) Rationale: It's
just
simpler than >-< or _v or xor, and esthetically more appealing.
With other words, we are free to do that in _certain_ contexts.
- This way, Either P or ~P would be expressed with P | ~P.
Well,
right, but *I* certainly would n e v e r consider to adopt | to
express or. ;-)
F.
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
<95ou30tv1r3s3dqi07dlad0od54ulm5jl6@4ax.com
<SBK%b.344$Xy3.1121@tor-nn1.netcom.ca
<4m1v3094a4ea2vetiakqr2l2ipps2ahmh9@4ax.com
<1sO%b.360$Xy3.1137@tor-nn1.netcom.ca>
<c1oe0i01d7n@drn.newsguy.com
<f9nv30d5uqu20vm2odjvljeu771n07q1n7@4ax.com
<c1p9ni$1l3arr$1@ID-221186.news.uni-berlin.de
>Who the did EVER write
>p | q
>when he MEANT
>p v q or p / q or p or q
>???
Well, OTTER seems to do so IIRC, irritating me no end...
> Somebody whose been taking CompSci for too many years?
Wouldn't cs people rather use + (from switching theory, where |
also means NAND)?
Note that the | notation used in some computing languages
doesn't really denote a logical operator but a truth-evaluation
function for which it's usually assumed that the right side
isn't evaluated if the left side returns true.
regards
Stephan
===
Subject: re:Interesting problem
Then again, it's impossible to split a field into two such
sets.
===
Subject: Re: Number Theory Problem!
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i1SD7en03941;
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) with ESMTP id i1S8mwi16155
by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 $, proapp) id i1S8mwq00509;
><pre
>Does anyone know how to prove the statement:
>Let p be a prime and let a,b be integers such that a^p is
congruent to b^p
modulo p.
>Prove that a^p is congruent to b^p mudulo p^2.
>Also, If n>4 and n is composite, prove that (n-1) is
congruent to 0 modulo
n.
>Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of every
integer n.
>I would appreciate any suggestion or solutions on how to
prove the above
problems/statement!.
>Ferdinand
>Please send to my e-mail sirferdz23i@yahoo.com
===
Subject: Re: Number Theory Problem!
Ferdinand Balmes <sirferdz23i@yahoo.com> escribi.97:
>Also, If n>4 and n is composite, prove that (n-1) is
congruent to 0
>modulo n.
(n - 1) =/= 0 (mod n) always that n > 1!!
>Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of
>every integer n.
1/5n^5 + 1/3n^3 + 7/15n = n(3n^4 + 5n^2 + 7)/15
Study if n(3n^4 + 5n^2 + 7) is always multiple of 3 and 5.
Consider first n
= 0, +/- 1 (mod 3), and second n = 0, +/-1 and +/-2 (mod 5)
Saludos,
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Coin-Flipping Machine
>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>http://www.npr.org/display_pages/features/feature_1697475.html
>Apparently, coin-toss-outcomes are more a function of
>human-unpredictability than of the coin's unpredictabilty.
What a waste of time. I could have told him that. Oh, but then
he
wouldn't have gotten a grant to study a non-problem.
>So, perhaps it is more accurate to talk about, rather than a
biased
>coin, a biased HUMAN-BEING!...
>;)
towards heads). Of course, coins don't have a symmetrical mass
distribution, so I could have told him that also. And I didn't
see
any reference to the control test of an unstamped coin blank.
And is he conducting the tests in a vacuum chamber at a
controlled
temperature? I wonder how much of my tax money he fleeced out
of the government for this stupid project?
>(What!?!...Human-beings are BIASED!?...Huh???..)
Not biased, RANDOM. Unless he puts a lot of effort into it, the
human cannot reproduce the same conditions of the flip from
one try to the next. This uncertainty is what makes the flips
fair. If you knew exactly what the initial conditions were, you
would know exactly what the outcome would be. But you cannot
know this information, so you cannot predict the outcome.
>(I apologize if this link has been mentioned recently on
sci.math
>already.)
>Leroy Quet
--
===
Subject: Re: Number Theory Problem!
Ferdinand Balmes <sirferdz23i@yahoo.com> a .8ecrit s le
message de
>Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of every
integer n.
24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
===
Subject: Re: Differentiate many variables
> I have a system consisting of 6 atoms. They all interact
with one another
> (bonds and van der waals forces).
> I have the equations for the interactions between atoms.
> Each atom is in 3D space. I can alter the x,y,z for each
atom.
> I will have a function that depends on these 3*6 variables.
Constructing
> this function should be fairly easy.
> (For ease of work, I presume that I should fix one of the
atoms at the
> origin).
> I want to minimise this function (analytically).
> So, How do I differentiate this 18 variable function?
> I do have a degree in comp + math but it was a long time
ago, and I don't
> even know what this would be called. Multivariate
differentiation?
Partial
> differentiation?
Same way you differentiate any other function of multiple
variables:
you find the gradient. The gradient of a function f is a
vector whose
i-th component (component in the direction of the coordinate
x_i) has
magnitude @f/@x_i. It points in the direction of steepest
ascent for
f -- in this case, a direction in 18-space.
If you back down the gradient direction, you may find a local
minimum,
you may find the global minimum, or you may find there is no
minimum.
And by the way, you are implicitely ignoring a possible
orientation of
each atom, and treating them as points. This may be adequate,
I'm not
sure.
===
Subject: Re: f continuous
> f:R-->R surjective with the property
> ( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
> Prove that f is continous
Apply the theorem that in a first countable topological space,
functions which preserve limits are continuous.
===
Subject: Re: f continuous
> f:R-->R surjective with the property
> ( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
> Prove that f is continous
> Apply the theorem that in a first countable topological
space,
> functions which preserve limits are continuous.
But here f doesn't preserve limits, but instead does
something in reverse of that.
===
Subject: Logic question #2
Hey guys,
Here is question #2. I haven't the slightest idea with this
one. If
--------------------------------------------------------------
--------------
-
The rest of this (except for the Roman numerals and the
letters to
select your choice) is in a simple code.
(i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg.
(A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F)
vla
(ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh
irxuwk.
(A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh
--------------------------------------------------------------
--------------
--
===
Subject: Re: Number Theory Problem!
>Also, If n>4 and n is composite, prove that (n-1) is
congruent to 0 modulo
n.
You probably mean prove that (n-1)! is divisible by n.
Hint: either n = p^2 with p > 2, or n = ab with 1 < a < b < n.
John Robertson
===
Subject: Re: Science Without Math? (model-free common sense
steering)
Isn't this simply the difference between intuitive reasoning
and
mathematical reasoning? I realize many here would say that all
intuitive
thinking can be reduced to mathematics (albiet impossibly
complicated), but
isn't the real differece is one of quality, not quantity? It
is the
difference between an analogue computer and a digital one.
Certianly the
way a human reasons is closest to the former, not the latter.
Why do we have this need to measure human reasoning abilities
in terms of
computers anyway? Even to the extent of coming up with the
term fuzzy
logic to describe the human's comparatively weak reasoning
abilities as
opposed to the computer, which over the last 20 years or so
seems to have
become the accepted standard that all things--even those
things human--are
measured by.
Scott
> Neural networks are model-free estimators, in that they do
not
require
an
> in-depth understanding of the phenomena they are modeling.
> http://www.arcon.com/arconneu.html
> THE MATHEMATICS OF CROSSING THE STREET:
> You are at the curb deciding, Should I
> cross the street? Well, it depends.
> AT THE CURB
> The walk light is on, but you see a
> truck approaching fast. How fast?
> There is no exact number. Instead, there are an infinite
number of
> possibilities - from 1kph to over 100kph and everything in
between. You
> don't have a radar gun, so instead you watch the truck for a
second or
two,
> and sum its speed up in two words very fast. That is good
enough.
> Your senses have told you the truck is coming very fast, but
you need
more
> information before you can decide whether or not to risk
crossing. How
far
> down the street is the truck? Is it slowing down? Again,
there are
no
> exact numbers, so you sum up the situation - close, not
slowing
quickly
> enough> Somehow your brain adds fast + close + not slowing
quickly
enough,
and
> warns you instantly that the risk is high. It is purely
cognitive
process.
> It involves a complex combination of sensory information and
experience.
> ...Since there are no exact numbers in this story, the
mathematical
version
> must be told with fuzzy numbers...
> But, the process is still not quite over. Should I wait or
cross? You
have
> to make the decision. Risk tolerance leads to different
spins and
endings.
> If you walk with a cane, you reason, The risk is high, so
I'll wait.
You
> watch as the truck runs the red light. If you are a jogger,
impatient to
> cross, you disregard the evidence, step into the
intersection, and jump
back
> just in time to save your life.
> http://www.decyde.com/crossingthestreet.html
> Fuzzy logic works the way that humans think as opposed to
the way that
> computers typically work. For example, consider the task of
driving a
car.
> You notice that the stoplight ahead is
> red and the car ahead is braking. Your
> mind might go through the thought process,
> I see that I need to stop. The roads are
> wet because it's raining and there is a
> car only a short distance in front of me.
> Therefore I need to apply a significant
> pressure on the brake pedal.
> This is all subconscious (in general), but that's the way we
think - in
> fuzzy terms. Do our brains compute the precise distance to
the car ahead
of
> us and the exact coefficient of friction between our tires
and the road,
and
> then use a Kalman filter to derive the optimal pressure
which should be
> applied to the brakes? Of course not. We use common-sense
rules and they
> seem to work pretty well. On the other hand, when we do
finally get
around
> to pressing the brake pedal there is some exact force that
we apply, say
> 1.326 pounds. So although we think in fuzzy, noncrisp ways,
our final
> actions are crisp. The process of translating the results of
fuzzy
reasoning
> to a crisp, nonfuzzy action is called defuzzification.
> http://www.innovatia.com/software/papers/fuzzy.htm
> ...In particularly vast networks in fast moving
environments, the split
> second it takes to traverse the circuit is greater than the
time it takes
> for the situation to change. In reaction, the last node
tends to
compensate
> by ordering a large correction. But this also is delayed by
the long
journey
> across many nodes, so that it arrives missing its moving
mark, birthing
yet
> another gratuitous correction.
> The same effect causes student drivers
> to zigzag down the road, as each late
> large correction of the steering wheel
> overreacts to the last late overcorrection.
> Until the student driver learns to tighten
> the feedback loop to smaller, quicker
> corrections, he cannot help but swerve down
> the highway hunting (in vain) for the center.
> This then is the bane of the simple auto-circuit. It is
liable to
flutter
> or chatter, that is, to nervously oscillate from one
overreaction to
> another, hunting for its rest. There are a thousand tricks
to defeat this
> tendency of overcompensation, one trick each for the
thousand advance
> circuits that have been invented.
> http://www.kk.org/outofcontrol/ch7-c.html
> Fuzzy systems are based on
> storage of common-sense rules.
> For example, a fuzzy Army-ant robot controller might have
the fuzzy
> association if load is heavy, then signal for help longer.
Fuzzy
phenomena
> admit degrees: some loads are heavier than others; some
signal durations
are
> longer then others.
> A single association (heavy,longer)
> encodes all combinations...
> Fuzzy systems reason with
> parallel associative inference.
> A fuzzy system reasons with multivalued sets, instead of
true or false
> propositions, and it may adaptively modify its fuzzy
associations from
> representative numerical samples.
> http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html
> Wired: What is fuzzy logic and why do critics call it the
cocaine of
> science?
> Kosko: Fuzzy logic is Spock's worst nightmare - a way of
doing science
> without math. It's a new branch of machine intelligence that
tries to
make
> computers think the way people think and not the other way
around. You
don't
> write equations for how to wash clothes. Instead you load a
chip with
vague
> rules like if the wash water is dirty, add more soap, and if
very
dirty,
> add a lot more. All wash water is dirty and not dirty - to
some degree.
> It's just common sense. But it breaks the old either/or
logic of
Aristotle.
> That offends some scientists, who would like us to think and
talk like
> off/on switches. But they still haven't produced a statement
of fact like
the sky is blue or E=mc^2 that is 100 percent true or 100
percent
false.
> Fact ain't math. You can never get the science right to more
than a few
> decimal places. That's one reason we find chaos when we look
at things up
> close...
> ...Fuzzy systems are universal computers. I proved that as a
theorem -
the
> fuzzy approximation theorem. In theory, you can replace
every book on
> physics or economics with equivalent books that have fuzzy
systems where
the
> equations used to be. Fuzzy systems are model-free
estimators. You
don't
> have to guess at equations to build a bridge from inputs to
outputs.
Fuzzy
> rules build that bridge for you. There is math behind the
rules, but you
> don't need to know it to program a fuzzy system. You can
program it in
> English. If the air is cool, turn the AC down a little. But
the math
is
> not fuzzy. That's why you can capture fuzzy logic in a
digital chip.
> Most of the first fuzzy systems were in control - as in
adjusting a
camera
> lens or backing up a trailer truck to a loading dock. Now
we're applying
> fuzzy systems to wireless communications and multimedia. The
fuzzy rules
can
randomly spread signals over a wide bandwidth or teach an
intelligent
> agent the kind of houses or sunsets you prefer. The math
says we can
apply
> them anywhere. In practice, it may not be so easy.
> http://www.wired.com/wired/archive/3.02/kosko_pr.html
> Fuzzy logic is a superset of conventional(Boolean) logic
that has been
> extended to handle the concept of partial truth- truth
values between
completely true and completely false. As its name suggests, it
is
the
> logic underlying modes of reasoning which are approximate
rather than
exact.
> The importance of fuzzy logic derives
> from the fact that most modes of human
> reasoning and especially common_sense
> reasoning are approximate in nature.
> Boolean vs. Fuzzy: 300 years B.C., the Greek philosopher,
Aristotle came
up
> with binary logic(0,1), which is now the principle
foundation of
> Mathematics. It came down to one law: A or not-A, either
this or not
this.
> For example, a typical rose is either red or not red. It
cannot be red
and
> not red. Every statement or sentence is true or false or has
the truth
value
> 1 or 0. This is Aristotle's law of bivalence and was
philosophically
correct
> for over two thousand years.
> Two centuries before Aristotle, Buddha, had the belief which
contradicted
> the black-and-white world of worlds, which went beyond the
bivalent
cocoon
> and see the world as it is, filled with contradictions, with
things and
not
> things. He stated that a rose, could be to a certain degree
completely
red,
> but at the same time could also be at a certain degree not
red. Meaning
that
> it can be red and not red at the same time.
> Conventional(Boolean) logic states that a glass can be full
or not full
of
> water. However, suppose one were to fill the glass only
halfway. Then the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
multivalence
> is the fundamental concept which propelled Zader Lofti of
University
Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics of
> fuzzy logic founded by him are as follows.
> In 1965, Lofti Zadeh formally developed multivalued set
theory, and
> introduced the term fuzzy into the technical literature.
Nowadays, the
> recent emergence of fuzzy commercial products, as well as
new theory, has
> generated a new interest in multivalued systems. Yet already
engineers
have
> successfully applied fuzzy systems in many commercial areas
: intelligent
> subways automation, emergency breakers, cement mixers, Kanji
characters
> recognition, control air conditioners, automatic washing
machines, guide
of
> robot-arm manipulators, and so on.
> Fuzzy systems store banks of fuzzy associations or
common-sense rules
such
> as IF traffic is heavy in this direction, THEN keep the
light green
longer
> that might be articulated by an human expert. Some traffic
configuration
are
> heavier that others and some green-light duration are longer
than others,
so
> that, the single fuzzy association (HEAVY, LONGER) encodes
all these
> combinations. That is to say, fuzzy systems directly encode
structured
> knowledge but in a numerical framework : by entering the
fuzzy
association
> (HEAVY, LONGER) as a single entry in a rule database we are
defining an
> input-output transformation.
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
> Fuzzy Logic is a computational paradigm capable of modelling
the own
> uncertainness of human beings. Fuzzy reasoning is nothing
else than a
Fuzzy
> Logic-based formalism for encoding human knowledge or common
sense in a
> numerical framework. Indeed, the mathematical concepts on
which Fuzzy
Logic
> is supported are very easy to understand. In a Fuzzy
Controller, human
> experience is codified by means of linguistic if-then rules,
which
compute
> control actions upon given conditions. Fuzzy Logic has been
applied to
> problems that are difficult to solve mathematically. One of
its main
> advantages lies in the fact that it offers a straightforward
methodology
for
> modelling and controlling non-linear systems, which are
difficult to face
by
> means of conventional techniques.
> http://www.wkap.nl/prod/b/1-4020-7359-3
> Fuzzy logic models itself on the pattern of human reasoning
in its use of
> approximate information and uncertainty to generate
decisions. It was
> designed (during late 1980s and early 1990s) to
mathematically represent
> vagueness and develop tools for dealing with imprecision
inherent in
several
> problems. Normally, in digital computers one uses the binary
logic
where
> the digital signal has two discrete levels : low (logic
zero) or high
(logic
> one); nothing in-between. Fuzzy systems use soft linguistic
variables
(e.g.
> hot, tall, slow, light, heavy, dry, small, positive,
...etc.) and a range
of
> their weightage (or truth) values, called membership
functions, in the
> interval (0, 1), enabling the new computers to make
human-like decisions.
> Since human beings tend to use words rather than numbers to
describe
> behaviour patterns, fuzzy controls avoid the conventional
rigidity of
> computers and allow them to use parameters based on common
sense.
> http://www.tribuneindia.com/2002/20021024/science.htm
> Fuzzy logic best summed up by common sense
> Computer Corner
> John Boyd
> Fuzzy logic was introduced to the world 27 years ago by
Professor
> Lotfi Zadeh in his Fuzzy Sets paper published in Information
> Control magazine, though it is only recently that we've seen
it
> applied across a broad range of products.
> Some readers have asked for more explanation on fuzzy logic,
so
> here's an attempt to defuzzify the subject a little further.
> Simply put, fuzzy logic is aimed at enhancing our prissy
computer
> technology with a touch of common sense.
> One problem with the conventional digital computer is that
it is
> such a scrupulously either-or beast. It cannot be easily
coaxed
> to handle approximations or vague notions like young, a lot
and
> probably.
> Yet most of us rely on such terms daily because we happen to
be
> humans dealing with other humans, not robots building cars.
> It's an easy matter to arbitrarily program a computer so it
> designates everyone falling into the age-range 0f 15 to 18
> as being a youth. Such a precise category has come to be
called
> a crisp set since the emergence of fuzzy logic.
> Yet we all know some 14-year-olds can look older than some
> late-developers turning 20. Such exceptions, however, cannot
> be accounted for in conventional computing. Or at least not
> without an inordinate amount of additional programming and
> expense.
> As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied
> when I asked him if we couldn't just continue using
conventional
> programming and technology for controlling new products,
instead
> of fuzzy, Well, we could. And you could probably swim across
the
> Pacific if you got enough support from enough people. But ...
> To overcome this problem, Zadeh was inspired to develop his
fuzzy
> theory and the math to go with it that could be used to
create
> fuzzy sets based on imprecise natural language.
> Each member in a fuzzy set (such as the youths and others
considered
> in the above example) is assigned one of a continuous range
of values
> (called the membership value) between zero and one.
> Whereas in the above crisp set a 13-year-old going on 14
would still
> have to be considered a minor and thus be designated as zero
in
> binary logic, fuzzy logic could assign him a membership
value of
> say 0.1. Likewise, an immature 20-year-old who would
normally fall
> outside our either-or crisp-set range could be assigned a
membership
> value of 0.9 depending upon the criteria we use to measure
youth.
> Working out just what criteria to use, what values should be
assigned
> each member and deciding what rules are necessary to govern
the
> relationships between members is the key to successfully
applying
> fuzzy control in products.
> In some applications, determining the optimum rules has
become so
> complex, some manufacturers have resorted to employing the
aid of
> neural networks, which may be stretching a good thing too
far, given
> fuzzy logic's original purpose to get round complexity.
> Still, the flexibility in herent in fuzzy is clearly useful
in
> dealing with approximate calculations, such as about 100> It
can be used in artificial intelligence to provide us with an
almost true answer. It can also infer a common-sense result
even
> when the data is not precise.
> Our handwritten 5 in 250 would be treated as 5, not the
letter S,
> for instance, in Sony's fuzzy-based Palmtop computer.
> While we have all seen fuzzy logic-based products from the
likes of
> Matsua, Sanyo and Hitachi, one unlikely company that has made
> fuzzy technology a central part of its business strategy is
Omron
> Corp.
> It began its research into fuzzy logic in 1984 and has since
applied
> for over 700 patents. This puts it in the forefront of fuzzy
> applications in areas like factory and industry control, as
well as
> in medical equipment.
> In 1989, Omron also signed on lotfi Zadeh as a senior
advisor.
> Earlier this year at the Business Show in Harumi, Omron
demonstrated
> its fuzzy workstation. Omron manufactures both standard
Motorola
> 68040-based and 88000 reduced-instruction or RISC-based
workstations
> that can be fitted with a fuzzy inference board, turning
them into
> the world's first fuzzy workstations.
> Omron claims such a RISC-based workstation can achieve 4 ion
> operations per second, an incredible speed if they haven't
fuzzed
> on the number. Fuzzy logic is used in the workstations to
store
> and retrieve fuzzy information and make inferences.
> Ranging in price from Y2.5 million to almost Y4 million (a
US dollar
> is about 120 Yens -FM), these machines are not the kind of
products
> you will find down in Akihabara. (a section of Tokyo famous
for its
> quantity and variety of electronic goods -FM) Rather, they
are
> typically aimed at value-added resellers in niche markets,
and
> engineers who want to develop fuzzy applications, fuzzy
databases
> and expert systems, as well as fuzzy inference systems.
> However, the entrepreneurs among you may be interested in
Omron's
> FB-30AT fuzzy inference board for the IBM PC and compatible
wares.
> It features a 24 MHz FP-3000 fuzzy chip capable of
processing up
> to 128 rules, with five antecedents and 2 consequents.
Training
> software and a compiler is also available.
> Omron has also produced a fine little booklet on fuzzy called
Clearly Fuzzy that I dipped into when writing this column.
> Tadashi Katsuno, at Omron's public relations section, tells
me
> he still has a limited number of copies left that he will
send
> to the first readers of Computer Corner who write to him with
> contact information.
> The address is Omron Corp., International Public Relations
Section,
> Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105.
>
--------------------------------------------------------------
------
> - Farzin Mokhtarian
> farzin@apollo3.ntt.jp
http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/
areas/fuzzy/doc/in
tro/j_times.tgz
> http://www.ece.utep.edu/research/webfuzzy/about.html
http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS
_NOE_ASI99/ASI
'99_ViharosMonostori.htm
> http://www.bjarne.ca/pmflp.pdf
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
>
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htm
===
Subject: inverse galois problem
hello
I'm looking for help.
i study inverse galois problem for abelian group
i know that for every abelian group G we can found K in order
to
Gal(KQ)=G
i search a polynome P whose verify Gal(Q[x]/P(x))=Z/7Z
i know only Z/7Z is a quotient of (Z/29Z)*
sorry for my bad english and thank you for your help
===
Subject: Re: Logic question #2
> Hey guys,
> Here is question #2. I haven't the slightest idea with this
one. If
>
--------------------------------------------------------------
------------
---
> The rest of this (except for the Roman numerals and the
letters to
> select your choice) is in a simple code.
> (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg.
There are six choices. Tick off the one after the second.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F)
one two three four five
> vla
> six
> (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh
irxuwk.
There are now five choices. Tick off the one after the fourth.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh
one two three four five
>
--------------------------------------------------------------
------------
----
The puzzle seems too simple, unless the point was to solve the
cipher.
===
Subject: Re: f continuous
> Apply the theorem that in a first countable topological
space,
What say this theorem ?
===
Subject: a little - big problem
solving it.
Let U be an open set in R^n.
Consider
f in C^{infty}(U, R)
such that
exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 .
Then f belongs to I^k_{x_0}(U, R) .
--------------------------------------------------------------
--------------
--
I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U, R)
k-times with itself and I_{x_0}(U, R) denotes the ideal in
C^{infty} (U,R)
of function vaniscing at p.
In other words, f belongs to I^k_{x_0}(U, R) if and only if f
is of the form:
f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k +
+...+
+ h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k
where m is a natural number , say m=1 or m >1 ,
h_i belongs to C^{infty}(U, R)
and
g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R.
Note that for m=0 , the proposition above is false,
consider for example k=2 and f(x,y)=x^2+y^2.
Then, exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1}
= lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0
,on the other hand, it's impossible to write f as
f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R)
My ask is:
Is the proposition above true with m=1 or m>1 ?
Tern
===
Subject: Scimitar Theorem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i1SGVTv23566;
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) with ESMTP id i1SDq9i08995
by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 $, proapp) id i1SDq9u09663;
Let y = f(x) be the shape of a (zero-width, finite, planar)
scimitar
blade, and also of its scabbard. The blade remains in contact
with the
scabbard for any amount of blade withdrawal. Prove that y must
be a
circular arc.
phil
===
Subject: Re: a little - big problem
Excuse me, there's a litlle mistake on the previous message.
..............
,on the other hand, it's impossible to write f as
f = h_0 g_01 * g_02 with g_01 and g_02 in vanishing at (0,0)
..............
===
Subject: Re: Differentiate many variables
> I have a system consisting of 6 atoms. They all interact
with one another
> (bonds and van der waals forces).
> I have the equations for the interactions between atoms.
> Each atom is in 3D space. I can alter the x,y,z for each
atom.
> I will have a function that depends on these 3*6 variables.
Constructing
> this function should be fairly easy.
> (For ease of work, I presume that I should fix one of the
atoms at the
> origin).
> I want to minimise this function (analytically).
> So, How do I differentiate this 18 variable function?
> I do have a degree in comp + math but it was a long time
ago, and I don't
> even know what this would be called. Multivariate
differentiation?
Partial
> differentiation?
> Anyone know how to do this (or have any links)
> Also, I've got some tools that I'll be using once I got the
method
straight
> in my head: Mathematica, mathcad and matlab.
> Anyone got any idea how I would use one of the packages to
help me?
> (preferably mathematica, as I heard that the strongest
symbolically)
>
Hi :
(Just trying to guide you to the right track but be aware that
my
knowledge of these subjects is small).
Supposing (just for economy of writing) that your function F
depended
only on the positions of 2 atoms (instead of 6) you should
first
define for Mathematica your function of those 3*2 variables.
For
example with:
F[x1_,y1_,z1_,x2_,y2_,z2_]:= (your analytical expression of
the 6
variables)
Next remember that the increase dF of a function F when their
variables are increased respectively by
dx1,dy1,dz1,dx2,dy2,dz2 is
called the total differential of the function and is given by:
dF = (@F/@x1)dx1 + (@F/@y1)dy1 + ... + (@F/@z2)dz2 [1]
where @F/@xi means the partial derivative of the function F
respect
to the variable xi.
Mathematica implements total differentials with the operator
Dt[function], so if you want to calculate the total
differential of
your F, you can input in Mathematica:
Dt[F[x1,y1,z1,x2,y2,z2]]
and you will obtain an expression with the pertinent partial
derivatives of the expression [1] already calculated and the
differential increments dx1, dy1,... of the variables written
in the
form Dt[x1], Dt[y1],..
you are trying to solve a big problem (called in Physics a
many-body
problem) that AFAIK can only be solved analytically in very few
circumstances. So I can't guess how are you now going to
minimise your
function F.
Best regards
Carlos L
===
Subject: Re: Scimitar Theorem
> Let y = f(x) be the shape of a (zero-width, finite, planar)
scimitar
> blade, and also of its scabbard. The blade remains in
contact with the
> scabbard for any amount of blade withdrawal. Prove that y
must be a
> circular arc.
Need a clearer definition of in contact with===
Subject: Re: ellipse from 4 points
> I have the position of 4 points on a 2D plane. The points are
> unequally spaced.
> Is there anyway I could fit an ellipse (or any other
circular shape)
> to these points (it has to pass through the 4 points)?
Was thinking about this a bit more, and it seems to lead to
all sorts
of interesting questions in real algebraic geometry.
(Admittedly may
not be very relevant to the OP's practical question.)
To set it up, note that each 5-tuple of points
P=(P1,P2,P3,P4,P5) in
(R^2)^5 determines a conic C_P in the plane. Well, need to
discard
some 5-tuples that are degenerate, but most 5-tuples give a
unique
conic. So we can divide (R^2)^5 into four distinct pieces:
E = points P for which C_P is an ellipse
R = points P for which C_P is a parabola
H = points P for which C_P is an hyperbola
D = points P for which C_P is degenerate (e.g., consists of
two lines,
or for which there is more than one conic through the five
points)
E and H are open sets, while R and D are lower dimensional.
Now look at the projection map, say onto the first four
coordinates:
F : (R^2)^5 --> (R^2)^4
F(P1,P2,P3,P4,P5) = (P1,P2,P3,P4)
Question: Is F(E) all of (R^2)4?
Of course, this is just a fancy way of asking if every four
points in
R^2 can be placed onto an ellipse. But it sets things up in a
natural
way to be generalized. If F(E) is not all of (R^2)^4, it would
be
interesting to describe what it looks like. Ditto for F(H), or
course.
And what does F(R) look like?
Finally, I'll mention that since the order of the points is
irrelevant, one should be really taking P as a point in
(R^2)^5/S_5,
that is, mod out by the symmetric group. Offhand I don't know
what
this quotient space looks like, but it's undoubtedly known. A
similar
problem that's often assigned as an exercise in algebra or
algebraic
geometry classes is to describe the quotient space C^n/S_n
(here C is
the complex numbers). The answer is that this quotient is
isomorphic
to C^n. The proof uses the fundamental theorem of algebra and
the fact
that every symmetric polynomial is itself a polymomial in the
elementary symmetric polynomials.
JS
===
Subject: Re: ellipse from 4 points
> Was thinking about this a bit more, and it seems to lead to
all sorts
> of interesting questions in real algebraic geometry.
(Admittedly may
> not be very relevant to the OP's practical question.)
> To set it up, note that each 5-tuple of points
P=(P1,P2,P3,P4,P5) in
> (R^2)^5 determines a conic C_P in the plane. Well, need to
discard
> some 5-tuples that are degenerate, but most 5-tuples give a
unique
> conic. So we can divide (R^2)^5 into four distinct pieces:
> E = points P for which C_P is an ellipse
> R = points P for which C_P is a parabola
> H = points P for which C_P is an hyperbola
> D = points P for which C_P is degenerate (e.g., consists of
two lines,
> or for which there is more than one conic through the five
points)
D = points P for which at least 3 points in the 5-tuple are
collinear.
> E and H are open sets, while R and D are lower dimensional.
> Now look at the projection map, say onto the first four
coordinates:
> F : (R^2)^5 --> (R^2)^4
> F(P1,P2,P3,P4,P5) = (P1,P2,P3,P4)
> Question: Is F(E) all of (R^2)4?
> Of course, this is just a fancy way of asking if every four
points in
> R^2 can be placed onto an ellipse. But it sets things up in
a natural
> way to be generalized. If F(E) is not all of (R^2)^4, it
would be
> interesting to describe what it looks like. Ditto for F(H),
or course.
> And what does F(R) look like?
The 4-tuples in F(E) should consist of convex quadrilaterals.
The
4-tuples in F(R) should be convex quadrilaterals other than
parallelograms. (Note that a degenerate parabola consists of
two
parallel lines.) F(H) should simply be all quadrilaterals
(i.e., no 3
points collinear).
iel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: Bayesian Class and Math/Stat Teaching Techniques
boundary=------------050400060200000302000001
--------------------------------------------------------------
-------
But Radford, most word problems address situations far from the
studemt's potential field of application. They are seen to be
as
little more relevant that the more abstract problems. In
secondary
school a common trig problem involves a tree, a river, and the
tree's
shadow leading to determination of the tree's height. Neat
problem but
not too much related to Fourier transforms. IMHO few stats
instructors,
at least at the lower academic levels, have in fact analyzed
and
interpreted much real data tied to real world problems. Anyhow
a
blanket condemnation of the student reeks of the abdication of
responsibility or the attempted teaching of students unready or
unqualified for college.
department was just different. They liked theory and they
liked formulas.
>They liked elegant solutions and proofs, even if they were
irrelevant to
>application. I sensed a certain disdain for word problems and
real
world
>analogies and explanations to help the students conceptualize
the theory
>because real math students don't need those crutches.
My experience, and that of other math/stat instructors whom
I've
>talked to, is quite the opposite. It's the STUDENTS who don't
like
>word problems, and resist applications (eg, to physics),
because to do
>them they have to actually understand the mathematical
material (and
>even some physics!), rather than just applying formulas
without really
>knowing what they're doing. This may not be true of real math
>students, however, who ought to be able to do the word
problems (but
>who may find the standard ones to be too easy to be
interesting).
> Radford Neal
>-------------------------------------------------------------
--------------
-
>Radford M. Neal
radford@cs.utoronto.ca
>Dept. of Statistics and Dept. of Computer Science
radford@utstat.utoronto.ca
>University of Toronto
http://www.cs.utoronto.ca/~radford
>-------------------------------------------------------------
--------------
-
>
===
Subject: Re: f continuous
>f:R-->R surjective with the property
>( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
>Prove that f is continous
What about the function f(x) = x except when x = 0 or 1 where
f(0) = 1 and f(1) = 0?
--Lynn
===
Subject: Re: Number Theory Problem!
Ferdinand Balmes <sirferdz23i@yahoo.com> escribi.97:
><pre
>Does anyone know how to prove the statement:
>Let p be a prime and let a,b be integers such that a^p is
congruent
>to b^p modulo p.
>Prove that a^p is congruent to b^p mudulo p^2.
a^p = a (mod p) if p is prime. Then a^p = b^p (mod p) ===> a =
b (mod p)
==
b = a + k*p
b^p = (a + k*p)^p = a^p + p*a^(p-1)k*p + ... + k^p*p^p = a^p +
p^2*M ==
b^p = a^p (mod p^2)
Saludos,
===
Subject: Re: f continuous
> f:R-->R surjective with the property
> ( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
> Prove that f is continous
It is clear from the condition that f is injective so it is a
bijection. Set g = f^-1. Then g is also a bijection and the
given
condition implies g is continuous. So it is enough to show a
continuous bijection R->R has continuous inverse. This is easy
- use
the fact that a continuous injection R->R is monotone.
===
Subject: Re: f continuous
>What about the function f(x) = x except when x = 0 or 1 where
>f(0) = 1 and f(1) = 0?
>--Lynn
Woops, posted that too quickly. Never mind.
--Lynn
===
===
Subject: Re: inverse galois problem
> hello
> I'm looking for help.
> i study inverse galois problem for abelian group
> i know that for every abelian group G we can found K in
order to
> Gal(KQ)=G i search a polynome P whose verify
Gal(Q[x]/P(x))=Z/7Z
> i know only Z/7Z is a quotient of (Z/29Z)*
Let z be a primitive 29-th root of unity.
Take a number a with muliplicative order 7 mod 29. Any 4-th
power will
do as long as it's not 1 mod 29. Then
let w = z + z^a + z^{a^2} + ... + z^{a^6}.
Find the minimum polynomial of w.
(you could express w^7 as an integer linear combination of 1,
w, ..., w^6
or you could do something smarter).
That's your P.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: inverse galois problem
>hello
>I'm looking for help.
>i study inverse galois problem for abelian group
>i know that for every abelian group G we can found K in order
to
>Gal(KQ)=G i search a polynome P whose verify
Gal(Q[x]/P(x))=Z/7Z
>i know only Z/7Z is a quotient of (Z/29Z)*
> Let z be a primitive 29-th root of unity.
> Take a number a with muliplicative order 7 mod 29. Any 4-th
power will
> do as long as it's not 1 mod 29. Then
> let w = z + z^a + z^{a^2} + ... + z^{a^6}.
> Find the minimum polynomial of w.
> (you could express w^7 as an integer linear combination of
1, w, ..., w^6
> or you could do something smarter).
> That's your P.
Doh! I'm getting my Gaussian periods mixed up :-(
Swap 4 and 7 in the above so ....
Take a number a with muliplicative order 4 mod 29. Any 7-th
power of
a quadratic nonresidue will do. Then let w = z + z^a + z^{a^2}
+ z^{a^3}.
Well we can take a = 12, and get w = z + z^{12} + z^{-1} +
z^{-12}
= cos(2pi/29) + cos(24pi/29) when choosing the obvious z.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Number Theory Problem!
Ferdinand Balmes <sirferdz23i@yahoo.com> a .8ecrit s le message
de
>Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of every
> integer n.
> 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
That's nice! Is it true that every polynomial with rational
coefficients f(n) such that f(n) is an integer for every
interger n is
a linear combination of binomial coefficients C(n,k)?
Mark Sapir
===
Subject: Re: Give me that old time ontology: (was: the
anticlassicalist }{
i:
|
|> Why don't you state one single mind bogling fact
|
|Classical propositional calculus has non-Boolean
|models! Related to this and even more surprising
|is that classical propositional calculus can be
|modeled by a non-Boolean lattice [Reference 6], a
|fact apparently overlooked for over 100 years!
|Common intuition is that classical propositional
|calculus and Boolean algebra go hand-in-hand.
|Lattice O6 is a counterexample that shows this
|intuition is false.
Let me say first that I do find this amusing and worthy of
note. To
me, however, it falls somewhat short of mind-boggling. Being
amazing
is harder than it seems.
My first reaction was that it would have to depend on what
exactly one
meant by modeled. For example, since there's a translation of
classical
logic into intuitionist logic, any model (however you define
it) of
intuitionist logic will in a less direct or meaningful way
constitute
some kind of model of classical logic. So
. 1
|
. p
|
. 0
which is a Heyting algebra (where ~~p=1) could count as a
model in a
*weak* sense, where one has redefined the connectives to be
something
other than their normal meaning in a Heyting algebra. For
instance,
the classical x or y is defined as ~(~x & ~y), so that p or p
evaluates
to 1. Pretty soon, though, you notice it's a little silly to
keep
pretending to distinguish between elements with the same
double negation,
and when you identify them, you have a Boolean algebra again.
In the referenced example,
. 1
/
~q . . p
| |
~p . . q
/
. 0
X->Y is defined to mean ~X or Y, rather than being the minimal
element Z
with the property that X&Z <= Y. So in particular, it's
possible to have
X<->Y (i.e. (X->Y)&(Y->X)) evaluate to 1 even though X and Y
are distinct
elements of the lattice.
The example should serve as a reminder, then, that the sense
in which a
Boolean algebra incarnates classical logic depends on assuming
something
that ensures that equivalent elements (in the sense of X<->Y
evaluating
to 1) are equal, as elements of the algebra. Since p<->q
evaluates to 1,
we'd need to identify p with q, and ~p with ~q, which leaves
us with the
Boolean algebra
. 1
/
~p . . p=q
/
. 0
having the property that the same expressions evaluate to 1 in
it as
evaluate to 1 in the given six-element example.
I'd categorize this as cute rather than mind bogglingKeith
Ramsay
===
Subject: what is the z-transform of sinc function?
Can anybody tell me what is the z-transform of sinc function
and what
is
its region of convergence?
-Joenyim
===
Subject: Re: min area to flip 2 hinged rods
...
> 2 rigid rods of unit length are hinged to each other.
Initially they are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
I assume these hypothetical rods have no thickness, and move
only
within the plane formed by the top flat surface of the paper. I
think around pi/3 should be enough area. Suppose the hinge C
is at
the origin and the rod tips A and B are at (1,0) to start.
While
raising A, slide C to the right past (1,0) until AC is
vertical.
I think area used so far is around pi/8, and pi/2 radians have
been
swept. Now pull B down while C goes up and left a little ways
and
A swings through part of the area already used, so area used
so far
is about pi/4, and swept angle is pi. Now push C to the right,
etc.
> What area of mathematics deal with such problem ?
Computational Geometry
-jiw
===
Subject: What are Bessel functions?
What are Bessel functions? What do they do?
What is their purpose?
===
Subject: how to clean up derivative and integral Re: cracks in
Euclidean
Geometry and why Reals are fake
The biggest evidence of cracks in the Real Numbers as a system
of
numbers is the fact of the plethora of integration and
differentiation
measures. We have Riemann integral and we have Lebesgue
integral and
Stieltjes integral and Radon integral and 50 other integrals.
Then we
have 50 or more types of differentiation. Messy, you say. That
is only
the starting of all the gaps and holes in Real Analysis. The
entire
subject of Real Analysis is gap ridden, hole ridden and
overall sloppy
and messy. The reason being is that the Real Numbers are a
nonexistant
entity. They are fakes. Just as the NaturalNumbers equals
Finite-Integers are nonexistant and a fake set.
What is real is the P-adics and the Doubly-Infinites. The
trouble with
both the Finite-Integers and the Reals is that no number
exists which
is not an infinite string. The P-adics are infinite strings
leftward
and the Doubly Infinites are leftward and rightward infinite.
The reason Number Theory had more unsolved problems
accumulating since
2,000 years ago is because they had no infinite componentry to
their
numbers. So Number Theory was even more lousy messed up than
REal
Analysis. At least REal Analysis had some infinite strings
such as
3.3333..... whereas finite-integers never had infinite strings
and so
the accumulation of unsolved problems in Number Theory was huge
compared with the messy gaps and holes in REal Analysis in
things such
as differentiation and integrals. Once that Integrals and
Derivatives
are given over to Doubly Infinites will all of the holes and
gaps in
REal Analysis begin to disappear and vanish.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
(www.iw.net/~a_plutonium) website of the science of AP under
revision
what used to be my old science website
www.newphys.se/elektromagnum/physics/LudwigPlutonium from
years 1993
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
Welcome to my filter.
===
Subject: Re: What are Bessel functions?
>What are Bessel functions? What do they do?
>What is their purpose?
Try
http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
for
starters. Google gives many more references. The first place I
ever
saw them as a student was as solutions to the heat equation in
a
circular rod.
--Lynn
===
Subject: Re: What are Bessel functions?
> What are Bessel functions? What do they do?
> What is their purpose?
Expand exp(z(t - 1/t)/2) in a Laurent series in powers of t.
The
coefficient of t^n is J_n(z) the Bessel _coefficient_ of order
n
(n integral).
J_n was studied by Euler when investigating the vibration of
drum
skins. Lagrange also studied them. Bessel studied them in
connection
with planetary motion. The Bernoulli's studied them--iel in
connection with swinging chains.
J_n satisfies Bessel's equation:
d^2y/dz^2 + (dy/dz)/z + (1 - (n/z)^2)*y = 0
n does not need to be an integer in the DE. The general
solution is the
Bessel _function_:
J_n(z) = 1/(2*pi*i)*(z/2)^n *
int_{-infty}^0 t^{-n-1} exp(t - z^2/(4*t)) dt
= sum_{r=0}^infty [ (-1)^r * z^{n + 2*r}
/(2^{n + 2*r} r! Gamma(n + r + 1)) ]
One can expand arbitrary functions in a J_n series (n
integral) rather
like a Fourier expansion.
I don't know if they are in fashion now. You may wish to read
G N
Watson Theory of Bessel Functions CUP and older books on
calculus/analysis.
G.C.
===
Subject: Finding out base of a number
I need help with the following problem:
749 in base 11 equals 279 in base b. What is base b?
===
Subject: Re: Finding out base of a number
> I need help with the following problem:
> 749 in base 11 equals 279 in base b. What is base b?
In base 10
7*11^2 + 4*11 + 9 = 2*b^2 + 7*b + 9
2*b^2 + 7*b - 11*(7*11 + 4) = 0
Can you solve quadratics?
G.C.
===
Subject: Re: e^i(pi) = -1 revisited with Doubly Infinites Re:
infinite
rightward strings tacked-on to p-adics serves as Orthogonality
and makes
Doubly-Infinites the points of Lobachevskian Geometry
> I would have liked to post this followup to a post I made
several
> hours earlier tonight but Google has not yet correlated that
post to
> the newsgroup board and so I post this one as a non-sequitor
followup.
> I remember the last time I visited e^i(pi) = -1 was in the
early 1990s
> after learning that the p-adics of the 5-adics have a number
that is
> truly i. And I remember Karl Heuer trying to help me see if
that
> equation can be broken down into its number parts rather
than a
> symbol-equation. So we had -1 as a Real, and we had e and pi
as Reals
> and we had i as a 5-adic. But then no progress.
> Well, tonight I have something new and different that I did
not have
> in the early 1990s. Tonight I have the idea that the Reals
were a fake
> or fictional set and that the only true numbers are either
p-adics or
> Doubly-Infinites.
> So let me see if progress now can be achieved with the
equation
> e^i(pi) = -1.
The progress has already been done.
Since Euler not only proved that the equation
is true, but he also proved that you need Geometric a
proof to prove it. Goedel proofs are insufficient.
You need a pair of duel functions to prove that it's true.
So that the consistency relationship:
Jerk Mod Wanker == 0 is also preserved
throughout the prove. Otherwise you
consistently get the retarded answer:
e^log(-pi) is bounded from above.
===
Subject: Non-linear congruence in Z
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i1SLJsl15916;
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) with ESMTP id i1SKeVi12787
by proapp.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 $, proapp) id i1SKeVv25810;
How can I solve the following non-linear congruence in Z?
Does there exist x,y from Z such that:
y(x^2-y)=7 (mod 17)
I don't think it's wise to try out all combinations for x,y
from {0,1,...16}.
Any good reference for such equations?
Hardy-Wright doesn't seem to discuss this type of congruences.
Cron
===
Subject: Proposition for Euclidean geometry
Does anyone know a Godel proposition for Euclidian space?
JS
===
Subject: Class vs. Dimension Equations
When I talk about the dimension equation of a finite group, I
mean the
one expressing the order of the group as the sum of the
squares of the
dimensions of the irreducible representations of the group
over the
field of complex numbers.
My question is: Is the dimension equation directly computable
(without
knowing the group) from the class equation or vice versa?
---- David
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
> Welcome to my filter.
Most understandable.
I would not call the Reals fake, although there is a
philosophical
question about whether they're invented or discovered, and I
don't know
about cracks in Euclidean Geometry, but there are cracks in the
real number line.
Do I get to meet your filter? :-)
George
PS Good luck with D'Inverno.
===
Subject: Re: min area to flip 2 hinged rods
> ...
> 2 rigid rods of unit length are hinged to each other.
Initially they
are
> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area of
the
> paper which allow the divider to be opened such that the
angle between
the
> two legs extend from 0 to 360 degrees. The divider is to
touch the
paper
> and no part of the divider is to extend beyond the paper
throughout the
> whole process.
There is a related problem about passing a ladder of length L
around a
right angled corner from a corridor of width A to another of
width B.
Assuming the ladder is kept horizontal and that its thickness
is
negligible, it transpires that if L^(2/3) <= A^(2/3) + B^2/3),
the
desired passage of the ladder is just possible.
Thus we may simplify the problem of the hinged rod by
providing a much
smaller definite upper bound on the area that has been
proposed yet.
Assuming each rod to be of length 1, the boundary curve need
be no
larger that (x^2)^(1/3) + (y^2)^(1/3) = 1
Since this bounds an area of 3*pi/32 in each quadrant, the
total area
required will be less than 3*pi/8 ~ 1.7881 square units.
Since the rods are not constrained by the axes but only by the
curve
itself, and not, as was the ladder, constrained by the walls,
a slightly
smaller boundary is possible, possibly one of the similar form
|x|^u +
|y|^u = 1 with u < 2/3.
Can anyone come up with an area less that 3*pi/8?
===
Subject: Re: Alternative ways to solve a quadratic equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) with ESMTP id i1SLrYi19049
You can approximate the roots by using the trace function of a
graphing
calculator.
--
Take out the trash to reply
===
Subject: Re: min area to flip 2 hinged rods
> With these fairly simple assumptions in place, it's hard to
see how
the
> answer will be anything other than pi square units (area of
a circle
of
> radius one unit).
> I don't think so. Imagine you pull the ends of the divider
away
> from each other until the divider is straight, then nudge
the hinge past
> vertical and push the ends back together. The envelope
should be a sort
> of diamond with in-bowed sides, and it should fit inside a
circle of
> unit radius with plenty of room to spare.
> The shape I, Mister Math Whiz, was trying to describe is
apparently
> an astroid. I believe its area should be 3/8 pi square
units, which
> is less than half the area of the unit circle.
> No idea if that's a minimum, though.
> ----j7y
And if you are allowed to roll the divider over, while it still
touches the paper, you can cut the 3/8 pi in half by having
only half
of an astroid, plus a little more for the rolling operation.
Unfold the divider so it is straight, roll it over, and fold
it back
within the same half of the astroid you openned it within.
3/16 pi +epsilon, perhaps.
Leroy Quet
===
Subject: Re: Congruence Involving 6 & Some Sums-of-Sums
> Let, for each nonnegative integer k,
> a(6k) = a(6k +1) = 1;
> a(6k +2) = a(6k +5) = 0;
> a(6k +3) =a(6k +4) = -1.
> Let A(0,m) = a(m);
> and for all positive integers n, and for nonnegative
integers m,
> A(n,m) = sum{k=0 to m} A(n-1,k);
> Then, for q and r = any nonnegative integers:
> m!*(A(q,m+1) -binomial(q+m,q-1))
> is congruent to
> m!*A(r,m) (-1)^m (mod {m+q+r}).
> So, more specifically, from the above we get:
> For ODD m,
> (m-1)!*A(r,m)
> is congruent to
> (r+m)!/(m r!) (mod {m+2r+1}).
> For EVEN m,
> (m-1)!*A(r,m)
> is congruent to
> (r+m-2)!/(m (r-2)!) (mod {m+2r-2}).
> (Someone might enjoy confirming the above congruences...)
> I wonder if any of these congruences have any interesting
> number-theory implications...
> Leroy
> Quet
I must point out that
A(n,m) =
sum{k>=0} (binomial(m+n-1-6k,n-1) +binomial(m+n-2-6k,n-1)
-binomial(m+n-4-6k,n-1) -binomial(m+n-5-6k,n-1)),
where
binomial(q,j) = q!/(j!(q-j)!)
if q >=0, as before,
but, here,
binomial(q,j) = 0
for q < 0.
Leroy Quet
===
Subject: Re: a little - big problem
i think it's a problem in taylor expansion:
you can show that all partial derivatives up to the (k-1)th
order vanish at
x_0,
next, look at the k-th order taylor residue of f around x_0 -
it's an expression exactly of the desired type (i.e., belongs
to I^k)
> solving it.
> Let U be an open set in R^n.
> Consider
> f in C^{infty}(U, R)
> such that
> exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 .
> Then f belongs to I^k_{x_0}(U, R) .
>
--------------------------------------------------------------
------------
--
> I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U,
R)
> k-times with itself and I_{x_0}(U, R) denotes the ideal in
C^{infty}
(U,R)
> of function vaniscing at p.
> In other words, f belongs to I^k_{x_0}(U, R) if and only if f
> is of the form:
> f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k +
> +...+
> + h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k
> where m is a natural number , say m=1 or m >1 ,
> h_i belongs to C^{infty}(U, R)
> and
> g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R.
> Note that for m=0 , the proposition above is false,
> consider for example k=2 and f(x,y)=x^2+y^2.
> Then, exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1}
> = lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0
> ,on the other hand, it's impossible to write f as
> f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R)
> My ask is:
> Is the proposition above true with m=1 or m>1 ?
> Tern
===
Subject: Re: Finding out base of a number
>I need help with the following problem:
>749 in base 11 equals 279 in base b. What is base b?
If you are looking for an integer b perhaps you have copied the
problem incorrectly?
--Lynn
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
Selections from the website, where he proves that he is a
supergenius:
Archimedes Plutonium (my true legal name)
Below in chemistry I have a circular periodic table [...] God
is 231Pu
and the best bible is the best most up-to-date physics
textbook.
If the Brain Locus theory is correct, then through a single
atom in the
brain can all the thinking and thoughts be conducted.
I make this biological speculation that the source of my
supergenius
is that there is a Pu atom located in my brain, the focus of
my mind.
The brain is a parabolic reflecting telescope which has one
atom as
the center focus.
They walk among us. A scary thing indeed.
Servo
===
Subject: Prime factors of number near googolplexplex
Hello folks,
In (probably) the first page on Internet about numbers near
googolplexplex, that is, 10^googolplex, I put the prime
factors I
found using trial division of numbers in the range
googolplexplex-999
to googolplexplex+999.
The URL is: http://www.alpertron.com.ar/GOOGOLP.HTM
Dario Alejandro Alpern
Buenos Aires - Argentina
http://www.alpertron.com.ar/ENGLISH.HTM
===
Subject: Re: Logic question #2
> Hey guys,
> Here is question #2. I haven't the slightest idea with this
one. If
>
--------------------------------------------------------------
------------
> ---
> The rest of this (except for the Roman numerals and the
letters to
> select your choice) is in a simple code.
> (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg.
> There are six choices. Tick off the one after the second.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F)
> one two three four five
> vla
> six
 (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh
irxuwk.
> There are now five choices. Tick off the one after the
fourth.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh
> one two three four five
>
--------------------------------------------------------------
------------
> ----
> The puzzle seems too simple, unless the point was to solve
the cipher.
Richard,
How did you decipher the code? Did you just look at it and
know? Tell
me please. Or am I just stupid, or slow, or maybe both? LOL.
Anyway,
thanks a lot. Question # 3 is on it's way.
Heidi
===
Subject: Re: Finding out base of a number
| Vikram Hegde asked:
| I need help with the following problem:
|
| 749 in base 11 equals 279 in base b. What is base b?
If you meant:
749 in base 11 equals 297 in base b...
then b would be nineteen. ________________________Gerard S.
===
Subject: Re: What are Bessel functions?
> What are Bessel functions? What do they do?
Bessel functions (among other things) describe vibration modes
of disk
(or anulus). (Recall how sin and cos describe vibrations of a
string.) Bessel functions come from a Sturm-Liouville set of
equations and hence enjoy many nice properties such as being
othongonal (wrt to the inner produce natural for disk/anulus
shaped
spaces).
> What is their purpose?
They are a nice orthogonal set of functions which span a
Hilbert
space. You can make Bessel transforms which act similarly to
Fourier
transforms.
They often appear when you deal with round or circular things.
===
Subject: Re: Differentiate many variables
> I have a system consisting of 6 atoms. They all interact
with one another
> (bonds and van der waals forces).
> I have the equations for the interactions between atoms.
> Each atom is in 3D space. I can alter the x,y,z for each
atom.
> I will have a function that depends on these 3*6 variables.
Constructing
> this function should be fairly easy.
> (For ease of work, I presume that I should fix one of the
atoms at the
> origin).
> I want to minimise this function (analytically).
> So, How do I differentiate this 18 variable function?
> I do have a degree in comp + math but it was a long time
ago, and I don't
> even know what this would be called. Multivariate
differentiation?
Partial
> differentiation?
> Anyone know how to do this (or have any links)
> Also, I've got some tools that I'll be using once I got the
method
straight
> in my head: Mathematica, mathcad and matlab.
> Anyone got any idea how I would use one of the packages to
help me?
> (preferably mathematica, as I heard that the strongest
symbolically)
>
There is the 3 body ( n- body) problem based on gravitational
forces
between any two bodies. The final solution leads to a number of
singular Lagrange points. E.g.,browse
Microsoft PowerPoint - CS395TF02-lect2.ppt
===
Subject: Re: Prime factors of number near googolplexplex
> Hello folks,
> In (probably) the first page on Internet about numbers near
> googolplexplex, that is, 10^googolplex, I put the prime
factors I
> found using trial division of numbers in the range
googolplexplex-999
> to googolplexplex+999.
> The URL is: http://www.alpertron.com.ar/GOOGOLP.HTM
 Dario Alejandro Alpern
> Buenos Aires - Argentina
> http://www.alpertron.com.ar/ENGLISH.HTM
Very impressive. How'd you do that? I would think that
factoring numbers
that large would not be possible on any computing equipment or
software
available today.
===
Subject: Re: Non-linear congruence in Z
Cron <anonymous@mathforum.org> escribi.97:
> How can I solve the following non-linear congruence in Z?
> Does there exist x,y from Z such that:
> y(x^2-y)=7 (mod 17)
> I don't think it's wise to try out all combinations for x,y
> from {0,1,...16}.
> Any good reference for such equations?
> Hardy-Wright doesn't seem to discuss this type of
congruences.
10 for X=0 to 16
20 for Y=0 to 16
30 if Y*(X^2-Y)@17=7 then print X,Y
40 next Y
50 next X
x y
== ==
3 10
3 16
5 1
5 7
12 1
12 7
14 10
14 16
If you want to do it by hand, let y = 0, 1, ... ,16 and solve
the quadratic
congruence for x.
===
Subject: Re: Logic question #2
> Hey guys,
> Here is question #2. I haven't the slightest idea with this
one. If
>
--------------------------------------------------------------
------------
> ---
> The rest of this (except for the Roman numerals and the
letters to
> select your choice) is in a simple code.
> (i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg.
> There are six choices. Tick off the one after the second.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh (F)
> one two three four five
> vla
> six
> (ii) Wkhuh duh qrz ilyh fkrlfhv. Wlfn rii wkh rqh diwhu wkh
irxuwk.
> There are now five choices. Tick off the one after the
fourth.
> (A) rqh (B) wzr (C) wkuhh (D) irxu (E) ilyh
> one two three four five
>
--------------------------------------------------------------
------------
> ----
> The puzzle seems too simple, unless the point was to solve
the cipher.
> Richard,
> How did you decipher the code? Did you just look at it and
know? Tell
> me please. Or am I just stupid, or slow, or maybe both? LOL.
Anyway,
> thanks a lot. Question # 3 is on it's way.
In simple English, the triad the is just about the most
common. Start
by
trying to find a fit for that.
===
Subject: Logic question #3
Hello again guys,
--------------------------------------------------------------
--------------
-
On the planet Fingal live the Fingas. Fingas look human,
except that
they have no thumbs, and the number of fingers on each hand
may be
different. This gives them their surnames, so that a Finga
with three
fingers on his left hand and seven on his right might be named
Joseph
3-7.
There is a wake on Fingal, to mourn the death of Finnegan, a
Finga.
There is the beautiful ceremony of the touching of the
fingers, when
the whole population forms a chain, a Finga touching one
neighbour to
the right, finger to finger. Last night before Finnegan died,
he was
in the chain, and the chain formed a complete circle. How
beautiful!
Now Finnah 6-9 begins a chain, touching Joa 9-11, and so on,
and
forming one circle with some of the Fingers. Fella 9-6 begins a
separate chain, forming a second circle with the rest of the
Fingas.
What is Finnegan's name?
(A) Finnegan 6-9 (B) Finnegan 9-6 (C) Finnegan 9-9 (D)
Finnegan 11-9 (E) Finnegan 11-6 (F) Finnegan 6-11
--------------------------------------------------------------
--------------
--
===
Subject: eigenvalue must be defined only for square matrix?
I think the concept of eigenvalue can be defined also for
rectangular
nonsquare matrix...
but the equation |A-lamda*I|=0 only defined for square
matrix...
what's the problem?
Anybody clarify my confusion?
===
Subject: UK universities
Can anyone recomment UK universities or colleges with a
mathematics
department which is strong in (some of) the following fields:
- (differential) topology
- set theory
- logics
- differential geometry
- it has kind to connections to theoretical physics
(cosmology, field
theory)
R.M.
===
Subject: why the gradient of a multi-variate function is
column vector?
Why the first differential of a multi-variate function is row
vector and
the
gradient vector is defined to be the transpose of that first
differential
and be column vector?
I don't understand. can anybody clarify for me?
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
> |: > Oh, did you miss the thread on modality in language as
well?
> |:
> |: What, Andrew Patterson's nonsense? That was (a)
independent
> |: of your posturing and (b) obviously received largely with
> |: indifference.
> |
> |You see, that is the great evil laying at the heart of all
your anger
with
> |my post. There are people out there besides you, Brian,
with many
varied
> |interests. As long as we stay on topic to a particular
group, everyone
> |should have the right to post their interests and
questions. The usenet
> |does not follow office politics. Newcomers have all the
same rights as
> |those who have been posting for years.
> It's not a newcomers-versus-oldtimers issue. Newcomers and
oldtimers are
> not essentially different on this issue.
> It's a basic issue of shared resources. Bandwidth is very
cheap indeed,
> and like with most resources, there's no simple maximum
available amount
> of it. But like with most resources, the environment
degrades as people
> use more and more of it beyond a certain point.
But, with scientists the message is always the same.
It doesn't matter what *bandwidth* costs, since
we are not paying for bandwidth. We are paying
for information.
As the same goes for water. It doesn't matter
what water costs. Since we are not paying for water.
We are paying for ice.
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
X-SessionID: LWa0c-13332-Y4-33183@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: 9d6c2097 456c6e33 d9ba6f9c f4ba4829 7cd95c08
>The biggest evidence of cracks in the Real Numbers as a
system of
>numbers is the fact of the plethora of integration and
differentiation
>measures.
>Welcome to my filter.
You're a quick learner. Congrats.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
===
Subject: Re: Finding out base of a number
You could expand both numbers:
749 in base 11 equals 7 * 11^2 + 4 * 11 + 9
279 in base b equals 2 * b^2 + 7 * b + 9
Setting these equal
7 * 11^2 + 4 * 11 + 9 = 2 * b^2 + 7 * b + 9
Or
900 = 2b^2 + 7b + 9.
This yields the quadratic
2b^2 + 7b - 891 = 0
So
b = ( -7 + - sqrt(49 - 4 * 2 * (-891)) ) / 4
Unfortunately, this is not a whole number. So it would seem
that
something is not right.
I could, of course, be wrong.
Brian
>I need help with the following problem:
>749 in base 11 equals 279 in base b. What is base b?
===
Subject: Re: eigenvalue must be defined only for square matrix?
> I think the concept of eigenvalue can be defined also for
rectangular
> nonsquare matrix...
How?
A x = lambda x, where x is a column vector would seem to
require A
to be square.
===
Subject: Re: Non-linear congruence in Z
> How can I solve the following non-linear congruence in Z?
> Does there exist x,y from Z such that:
> y(x^2-y)=7 (mod 17)
> I don't think it's wise to try out all combinations for x,y
> from {0,1,...16}.
> Any good reference for such equations?
> Hardy-Wright doesn't seem to discuss this type of
congruences.
 Cron
Rewrite as
x^2 = 7/y + y (mod 17).
Now you can plug in the 17 possible values for y and use
quadratic
reciprocity to check if the righthand side is a square. And if
you
don't want to find the reciprocal of each y mod 17, you can
multiply
both sides by y^2 and let z=xy, so then you have to solve
z^2 = 7y + y^3 (mod 17).
Of course, you'll need to disregard the solution y=z=0, since
that
doesn't give a valid (x,y).
Using more theory, one can observe that the equation z^2 = 7y
+ y^3
mod 17 is a (nonsingular) elliptic curve over the field with 17
elements. By the Hasse-Weil theorem, an elliptic curve mod p
has at
least (sqrt(p)-1)^2 points. So for p=17, there are at least 10
points.
Now one of those points is at infinity, and one of them is
(0,0),
and there may be up to two other points with z=0. (I haven't
checked,
is -7 a square mod 17?) Anyway, this leaves at least six
points with z
not equal to 0.
So without having plugged in any numbers, we can conclude that
your
original equation has at least 6 different solutions (x,y) mod
17. Of
course, it may have more than that. If you want an upper bound,
without doing any computations, use the upper bound of
(sqrt(p)+1)^2
from the Hasse-Weil theorem. However, if you want to actually
find a
solution, your best bet is probably to just plug in some y
values to
find one with 7y+y^3 equal to a square mod 17. Since you have
approximately a 50-50 of chance winning for each y that you
try, it
won't take too long.
> Any good reference for such equations?
There are lots of books that discuss elliptic curves over
finite
fields. At the risk of being accused of shameless
selfpromotion,
there's a chapter that discusses it in very elementary terms
in A
Friendly Introduction to Number Theory, there's a treatment at
an
advanced undergraduate level in Rational Points on Elliptic
Curves,
and there's a proof of the Hasse-Weil theorem in Chapter V of
The
Arithmetic of Elliptic Curves, which is a basic graduate text.
Also,
since elliptic curves over finite fields are some important
for modern
cryptography, you'll generally find a chapter or two about
them in any
basic cryptography textbook.
Joe Silverman
===
Subject: Re: what is the z-transform of sinc function?
> Can anybody tell me what is the z-transform of sinc function
and what
is
> its region of convergence?
i thought originally that it's a homework problem, but i
wonder if it is
since i have never thought of using the sinc() function in the
discrete-time
world. without scaling the independent variable, then, for
discrete-time,
n,
{ 1 n=0
x[n] = sinc(n) = sin(pi*n)/(pi*n) = {
{ 0 n!=0
so the Z transform of it is the same as for the discrete
impulse function
(sometimes called the Kronecker Delta)
{ 1 n=0
d[n] = {
{ 0 n!=0
and that is
Z{d[n]} = 1 for all z
now if you were to scale the input a little:
x[n] = sinc(a*n) = sin(pi*a*n)/(pi*a*n)
for some real, a, then that's a little harder and i don't know
what the
answer is right off the bat. come to think of it, i don't even
know what
the Laplace Transform of the sinc() function is since i do not
know how to
analytically extend the rect() function to use complex
arguments.
FWIW
r b-j
===
Subject: Re: Logic question #3
===
>Subject: Logic question #3
>Message-id: <505be127.0402281649.cf31114@posting.google.com
>Hello again guys,
>-------------------------------------------------------------
--------------
--
>On the planet Fingal live the Fingas. Fingas look human,
except that
>they have no thumbs, and the number of fingers on each hand
may be
>different. This gives them their surnames, so that a Finga
with three
>fingers on his left hand and seven on his right might be
named Joseph
>3-7.
>There is a wake on Fingal, to mourn the death of Finnegan, a
Finga.
>There is the beautiful ceremony of the touching of the
fingers, when
>the whole population forms a chain, a Finga touching one
neighbour to
>the right, finger to finger. Last night before Finnegan died,
he was
>in the chain, and the chain formed a complete circle. How
beautiful!
>Now Finnah 6-9 begins a chain, touching Joa 9-11, and so on,
and
>forming one circle with some of the Fingers. Fella 9-6 begins
a
>separate chain, forming a second circle with the rest of the
Fingas.
>What is Finnegan's name?
>(A) Finnegan 6-9 (B) Finnegan 9-6 (C) Finnegan 9-9 (D)
>Finnegan 11-9 (E) Finnegan 11-6 (F) Finnegan 6-11
Nicely stated, Tameika. I'm sure James Joyce knew...
Marvin
Marvin Sebourn
osugeography@aol.com
===
===
Subject: Re: some complex integration questions
> I have a few quick questions about integration of complex
functions.
> first, how does one integrate over a curve that crosses a
branch cut?
> is it possible? if c(t)=2e^it, t in [-pi,pi], then can
integration of
> 1/(z^2-1)=1/2(1/(z-1)+1/(z+1)) be done as usual? what care
do i have
> to take when doing this?
> I do not understand your question. Your decomposition of
1/(z^2 - 1) is
> almost correct; it should be 1/(z^2 - 1) = 1/2(1/(z - 1) -
1/(z + 1)).
> So, all that is left for you to do is to calculate the
integrals of
> 1/(z - 1) and 1/(z + 1) over your curve. For this, you use
Cauchy's
> formula or the definition of index. There are no branch cuts
here.
 Jose Carlos Santos
I'm sorry, i do see now that i said the question in a weird
way. what
i mean is that the primitive of 1/(z+1) is Log(z+1), but the
curve C
crosses the branch cut of Log, and is discontinuous there.
does that
not matter? I could be confused, and I did understand how to
apply
cauchy, but I just figured it wouldnt be applicable because of
the
===
Subject: y = x^x
I've looked into the function y=x^x and am a bit confused
about its
nature... with x>0 everything is predictable, but with x<=0 I
am not
sure what its graph would look like.
I realize that many negative x values will produce imaginary
numbers,
but what does the rest of the graph look like? Does it just
approach
y=0? And how exactly are arbitrary powers calculated like that?
===
Subject: Re: y = x^x
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
> I realize that many negative x values will produce imaginary
numbers,
> but what does the rest of the graph look like? Does it just
approach
> y=0? And how exactly are arbitrary powers calculated like
that?
I believe it looks like a U shape as a negative times a
negative is a
positive number. Hope this helps.
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
: |: > Oh, did you miss the thread on modality in language as
well?
: |:
: |: What, Andrew Patterson's nonsense? That was (a)
independent
: |: of your posturing and (b) obviously received largely with
: |: indifference.
: |
: |You see, that is the great evil laying at the heart of all
your anger
with
: |my post. There are people out there besides you, Brian,
with many
varied
: |interests. As long as we stay on topic to a particular
group, everyone
: |should have the right to post their interests and
questions. The usenet
: |does not follow office politics. Newcomers have all the
same rights as
: |those who have been posting for years.
:
: It's not a newcomers-versus-oldtimers issue. Newcomers and
oldtimers are
: not essentially different on this issue.
:
: It's a basic issue of shared resources. Bandwidth is very
cheap indeed,
: and like with most resources, there's no simple maximum
available amount
: of it. But like with most resources, the environment
degrades as people
: use more and more of it beyond a certain point.
:
: Nearly every communication channel that's free for the
sender is being
: overused in an unpleasant way. Now and then I get woken up
at night by
: people honking their horns to get people's attention. As I
pick up my
: newspaper in the morning, I find that junk mail has been
added to it by
: sticking it under the rubber band, in addition to the usual
advertising
: inside it. Or sometimes it's attached to the door knob,
despite our
: no-soliciting policy. Outside, egocentric youngsters have
spraypainted
: their nicknames on buildings. At work, we've had a guy
trying to sell
: Kincaid paintings from office to office, who didn't want to
leave.
: Then there's the guy who thinks shouting to the receptionist
at the
: other end of the office is easier than using the intercom.
:-)
:
: Our mailserver at work is pretty good at filtering out spam,
but once in
: awhile somebody decides to pummel it with mail addressed to
a large
number
: of common names @ our domain, hoping that at least some of
them will
be
: somebody's user name. When I go to the bathroom, fairly
often someone has
: attached advertisements for some work-from-home scheme to
the wall of the
: stall. When I then leave work, I routinely find that
advertising has been
: put under the windshield wiper. Back at home, every so often
somebody
: repeats a message saying they've been assigned to review my
mortgage
: and need to talk to me. And of course the usual messages
from Nigeria
: and so on when I check my email.
:
: I'm not claiming that what you're doing is as obnoxious as
those things,
: but it shares with them a certain selfish tency. How can we
tell that
: it's selfish? Just honestly perform the thought-experiment
of imagining
: what effect it would have on the netnews-reading experience
if everybody
: were as casual about massive crossposting as you have been.
It doesn't
: take too much imagination; overly crossposted threads are
not so rare,
: and most of us have some idea how poorly they tend to work.
The difference I have been arguing between _all_ of the things
you list and
what I have done is topicality. The way (the only way) that
usenet works
to
be useful to its users is through the filter of topicality. On
the usenet,
I can find postings on categories of information without
having to sift
through advertisement requesting that I enlarge body parts I
don't even
possess or get hooked on whatever pharmaceuticals are
fashionably addictive
this week. Which is what all of your listed annoyances do, and
I have not.
Yes, free speech is selfish. Its the ultimate in selfish, that
I have the
right to believe and evangelise _my_ beliefs. Even though I
strongly
dislike Ayn Rand these days, that is one of the romantic
beliefs which I
was
given by her which I do not think I will ever give up or have
shame in.
those found around the politics newsgroups can be quite
topical to all of
the groups mentioned, though there is a tendency in some of
the posters to
veer away from the political topics and into social
evaluations that may
have better topicality elsewhere. There are some posters like
Archimedes
Plutonium or James Harris whose topics (which have included
weight loss and
conspiracies) have often not been appropriate to any of the
groups posted
to. Jack Sarfatti likes to post his UFO tales along with
zero-point energy
drives, fables of espionage, and self-inflicted persecutions
that may
have
only pieces of modern physics applicable to the groups posted
to (very
rarely appropriate to sci.math, however), and although I enjoy
the
ramblings
myself (and think that much that is written shows an intense
interest in
certain leading-edge topics only reorganised into his ongoing
fictional
life), again topicality bites him. I see some around the
philosophical
newsgroups who usually show good prescience for cross-posting
with
topicality, but every now and then someone comes along
completely oblivious
to topical distinctions.
In other words, it works sometimes and sometimes it doesn't.
There is
nothing inherently evil about the concept, which is why the
usenet
structure
allows it.
Try to focus on why it is I posted in the first place and what
my topic
was.
I had a question (with some other questions decorating it)
about the
importance of nonclassical logic in the research communities,
in particular
of Heyting algebras, and whether the communities using them
felt that this
indicated that education in their particular fields should be
adjusted. In
particular, I was hoping for stories by users of Heyting
algebras about
whether there had been difficulties in adjusting to using a
model that
didn't possess the boolean structure taught in most logic
prerequisites and
how they had even learned of such structures if they had not
been taught
them in school (or had they?). This I wanted to take as the
starting point
for exploring why boolean algebras have been given such a
prominence in
education despite their not being applicable to large numbers
of sentences,
both in natural languages and the sciences.
were fairly fundamental models in their fields that possesed
these
structures in a quest to elicit responses from these
communities. I
checked
topicality concerns in all groups I wanted to post to, read
the FAQs again
(I have read them in the past for many of the groups just for
their
educational value), and saw that this type of question was
quite topical.
I
checked the google archives and remember from my monitoring of
the groups
several of the issues explored previously, including
constructivism on
sci.math with very good contributions from yourself KRamsay,
quantum logic
on the physics newsgroups, sci.logic covering many of the
bases, and even
(despite the objections) mathematical discussion on linguistic
models in
sci.lang. This topic is often already a part of computer
science curricula
due to its foundational importance in relation to the lambda
calculus, and
again I could list the cognitive research as well. So I knew
when I posted
(after much consideration, I must stress) just how topical my
post was to
all communities, and I was driven to try to find those who
might have input
on my question.
Even sci.lang has this in its FAQ:
1. What is sci.lang for?
Discussion of the scientific or historical study of human
language(s).
Note the sci. prefix. The main concern here is with _facts_ and
theories accounting for them.
For advice on English usage, see alt.usage.english or
misc.writing.
For casual chatter about other languages see
soc.culture.<whatever>.
Discussion of or in Greek or Latin is available in
sci.classics.
The sci.lang.translation newsgroup focusses on translation and
issues of
concern to translators and interpreters.
The comp.ai.nat-lang newsgroup focusses on natural language
processing
by computers.
Which shows quite clearly that not only am I topical (theories
accounting
for the facts of human languages -- in particular models of
natural
language
and its logic) but that quite a number of topics participated
in by those
who took it upon themselves to throw insults my way are
actually suggested
in the FAQ to be placed elsewhere!
And I must stress that I didn't come here in order to propose
some vague
connection between the various theories even though I point to
some of the
work in that direction. I came asking questions about what the
professionals using the mathematical structures saw as being
important in
structuring the information (with educational applications my
goal). I
have
no theory of everything (yet!) as others have attacked me for
implying, nor
has it ever been my goal to propose such in these threads. I
am not even
asking for help with my homework or selling any products.
: A posting does not become on topic to a group just because it
contains
: some paragraph that would be on topic on its own. If
messages, with such
: weak ties to the groups they're initially posted to,
initiate threads of
: any length, thread-drift usually causes them rather soon to
become
: entirely off-topic except in one or two of the original
groups. If people
: were really on their toes, they would stop crossposting at
that point,
but
: quite often that doesn't work. Sci.math has had prolonged
discussions of
: all kinds of hot-button topics inflicted on it. It seems
that typically
: once a group such as talk.abortion gets involved, you can
assume it will
: take months for the noise-to-signal ratio to go back down to
normal
again.
: I had a partizan on one side of the abortion flame war tell
me point
blank
: that he considered his extended criticism of someone's
character to be
: on topic in sci.math, because his enemy happened to be a
mathematician.
Later below you accuse me of being disingenuous, yet you lay
all of these
implicit package deals on me as if I was the cause or in some
way similar
to
any of these things you mention. I did not veer topicality in
these
threads; that was initiated by others with some kind of
annoyance reaction
they can't help themselves sharing. Even you, KRamsay, whose
opinion I was
hoping to hear, has not contributed much to the actual topic
of these
threads, though you have been eager to point out your opinions
in other
only
loosely related areas.
My posts have much more than some paragraph of topicality; the
entire
topic of my post is topical to all the groups. The examples I
listed cover
many fields, focusing a paragraph or two on specifics to
illustrate my
point. But that's all those are for: illustrations, not topic
covering.
The topic covering is in my actual questions.
: One difference between a newcomer and an oldtimer, I
suppose, is that if
: you are familiar with the way usenet used to be, you can see
how much
less
: pleasant it is now, due to the proliferation of noise.
I've seen it. I've watched the newsgroups for a long time. I
know that
Google was responsible for a huge upsurge in popularity, and
before that I
watched in 93-94 the beginnings of growth due to the internet
and the ease
of downloading newsreaders for numerous systems. I have
watched not just
quantity of posts go up, but the professional quality of the
posts go down
as younger and younger participants began submitting.
Quite a lot of that I see as a good thing. The usenet, in my
opinion, is
becoming (along with other message discussion board formats)
an integral
part of education. Students are working out with those willing
to help
them
the various struggles they encounter, and others can monitor
such
discussions and learn along. I have certainly learned quite a
bit reading
various newsgroups over the years.
As for the concerns of decreasing professional quality and
discussion of
research topics, these symptoms are indicators for the need of
more refined
topicalities and the need to create new groups. The recent
creation of
string theory and discrete model groups in physics shows that
this is
occurring without any major problems. Sci.math would benefit
as well from
such plans, and if I ever get the server farm I was promised
from my work
(now seems unlikely) I would certainly look into proposing a
few groups of
my own.
: |: > Expense?
: |:
: |: Yes, expense. You are, for example, directly responsible
: |: for cluttering sci.lang with off-topic mathematics and
: |: complaints about the lack of physical content in your
posts
: |: from sci.physics.
: |
: |One collection of people upset that I am posting
mathematics, another
upset
: |that there is not enough. Maybe it is these two groups that
should be
: |arguing between each other and not be including me at all,
:
: I'd be a heck of a lot more sympathetic, if I didn't see you
doing such
: disengenuous things. Here you probably don't realize how
obvious it is
: that you're playing dumb in order to sound innocent. It
would take some
: amazing degree of confusion for a person to suppose that
people on
: sci.math complaining that they're having too many
nonmathematical
: postings inflicted on them, and people in sci.lang
complaining that
: they're getting too many mathematical postings inflicted on
them, are
: disagreeing *with each other*.
I'm not playing dumb at all here; I am accusing the poster of
playing dumb.
The fact that a post has mathematics in it does not make it
off topic for
sci.lang. In fact, from the FAQ quote I post above, I would
think it would
be eminently topical. I intentionally removed most
mathematical formalism
from my post and relegated the particular discussion of
symbolics to the
extensive list of references I posted in my more focus on the
constructive
less mathematical background. This openned me up to criticism
from those
in
the mathematical and physics communities who wanted more meat,
and I
tried
to answer their concerns as well. I didn't come here with
these posts to
teach the topics I mentioned; I was only asking about the
relevance of the
logical structure to reasoning in the models and what this
implied for
education. I was forced to add an amount of teaching to my
posts because
of
the attacks I received.
: All this would also fail to be objectionable if there really
wasn't any
: nicer way for you to call attention to your attempted
interdisciplinary
: discussion than to crosspost so much of it. You could easily
have
chosen
: one place as home for the discussion, and posted only the
bits
actually
: relevant to various other groups, with a reminder of where
the whole
: big thing was available. Doing it the way you did it instead
serves only
: as an attention-getting move, an attempt to draw the
attention of people
: (for example) reading sci.physics, but not interested enough
in what you
: have to say to consider it worth checking out your home page
or wherever
: you housed the rest of the discussion. You could just as
well do it in
: a polite way, but you don't bother to.
Almost every other day I see someone post anger to someones
post because it
did not make use of the cross-posting facilities of usenet and
thus didn't
keep responses together. I absolutely admit that I tried the
approach that
I felt would be most likely to get response from the
admittedly small
communities I was targeting, but that is the entire point of
usenet
cross-posting (which was built into the structure of this
information
source -- ie. _intended_). The only criteria I can be accused
of is lack
of
topicality, and that case has _not_ been made.
: |but I think the
: |more prudent action would be for those who don't find
content they are
: |interested in to just skip my threads.
:
: This is the standard excuse. Just delete the email, just
give a few
: seconds of my time to the salesman, just wait a bit for the
noise to go
: away, just toss out the junk mail, and so on, and so on, and
so on, and
: so on, and so on, and so on. It's true that this is normally
the best way
: of handling junk, but it's disengenuous coming from one of
the sources of
: the junk, because it disregards the responsibility of the
sender to have
: some shred of self-restraint. It just ignores how much
cheesier life has
: become on account of so many communication channels now
being half-filled
: with stuff that's there ONLY because the sender considers
their interests
: in getting attention more important than the receiver's
preferences in
: what to pay attention to.
Its not an excuse; it is a use of technology. Again you cast
your implicit
associations on me despite the singular issue of topicality
being the
important distinction. That you find my questions junk is
unfortunate,
because I had hoped for some better type of response to you.
Even you
admitted elsewhere that constructive reasoning often has a
different
character to classical reasoning, which is fundamentally my
point about
education, but you don't want to go into detail with me
because you still
harbor anger at alot of the other spammers out there and I am
the target
you
have chosen to take it out on.
: [...]
: |You start attacks, mister Scott, and
: |that puts you in error here.
:
: Massive crossposting is an attack.
Here is a useful usenet maxim:
Sombunall (some but not all) usenet crossposts are legitimate
uses of the
technology.
Here's another one:
Sombunall usenet crossposters are cranks.
===-=-=-=-=-
===
Subject: easy algebra problem.....
show that all a in Q(sqrt(2),sqrt(3)) is constructible.
------------------------------
i know that Q, sqrt(2), sqrt(3) is constructible.
but i can't explain logically.
help me......please....
===
Subject: Re: y = x^x
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
Look at
<http://www.peda.com/grafeq/gallery/rogue/xx_exponential.html>.
HTH
David
===
Subject: Re: Finding out base of a number
>I need help with the following problem:
>749 in base 11 equals 279 in base b. What is base b?
> If you are looking for an integer b perhaps you have copied
the
> problem incorrectly?
> --Lynn
When a = 11, b must be a solution to the equation
7*a^2 + 4*a + 9 = 2*b^2 + 7*b + 9
Unfortunately neither solution for b is rational, so the
solutions are
unlikely to be used as a base.
===
Subject: Re: y = x^x
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
> I realize that many negative x values will produce imaginary
numbers,
> but what does the rest of the graph look like? Does it just
approach
> y=0? And how exactly are arbitrary powers calculated like
that?
I don't think it will produce imaginary numbers for negative x
since -n^-n
is just 1/-n^n.
Will look similar to exponential for x = 1 or greater, but is
U-shaped as y
goes negative in the range x=0 to x=1 since it passes through
1.0 at both
those points.
I couldn't visualize it, other than it should 'oscillate'
positive and
negative, for negative x so plotted it in Mathcad. Looks
similar to a
damped sinusoid running backward for negative x, oscillating
around zero.
KeithK
===
Subject: cantor's theorem
Here is an interesting argument that Cantor's proof that there
are
uncountably infinite reals is wrong. I found it pointed to on
crank.net of all places, but I could not find any hole in the
reasoning.
http://maxpages.com/cantoriswrong
Perhaps someone else can, if there is any hole. Basically it
is saying
that the real number constructed out of the diagonal of the
countably
infinite list may still be in the list, even though there is no
element of the list that has the exact same digits as this
number.
They note that that .01111111....=.10000000000..... to make
this
conclusion.
Craig
===
Subject: Re: y = x^x
David, I saw the reference you provided after I posted the
Mathcad results
for negative x. Mathcad assumed rational x. Do you think that
plot was
correct for rational x or was it erroneously combining the two
separate
curves shown in your reference into a single oscillating curve?
KeithK
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
> Look at
<http://www.peda.com/grafeq/gallery/rogue/xx_exponential.html>.
> HTH
> David
===
Subject: Re: John Michael Osbourne - December 3rd 1948
> His given names contain 11
>letters (5th prime). His first name adds to 47 (5+5+5th
prime),
How long is his pecker?
===
Subject: Points on a Line, Infinity
Having a conversation, two-way communication, is the same
whether it's
a pair of lovers, semaphores or broadband, or two prisoners
tapping
through the wall: there is room for mutual understanding, and
the
lack of it, in the lack there being both misunderstanding and
ununderstanding, partial understanding, and even mutual
understanding
through mutual errors in understanding.
One concept I try to understand and present is that of the
points of
the real line as a contiguous sequence. The reals have the
quality of
being dense in themselves, between any two distinct or definite
reals there are infinitely many more as endpoints of
subintervals of
the interval between them, iteratively redividing.
This is a rather large jump from the previous paragraph, but
one thing
I consider is how if the points are contiguous that
consecutive points
are in a way one-sided. These are points some finitely many in
succession from a definite point, but infinitely far in
succession
from any next definite point. They exist because the set of
reals
consists only of points and the set of reals comprise all
elements of
the continuous real number line: the set of all reals contains
only
points, every real.
If we consider these consecutive points after a given point
only
infinitesimally distant, then we have a consideration that in
many
systems the infinitesimal is only zero. Besides that, where in
those
models the infinitesimal being zero is conducive to simplified
expression of those models, in other models an infinitesimal
has the
property of being non-zero yet less than any definite positive
value,
that is to say less than any positive ratio of integers, yet
positive
and non-zero.
Back to the one-sidedness, it is similar to considerations of
those
functions we have described about functions continuous at only
a
point, or set of points, in the image of their codomain, their
range.
Here, one-sided is an unfortunate term, although in some ways
it
expresses the quality of the point. In this context I want to
disquality anything except a point. Consider the n-D Euclidean
vector
basis, any line that intersects the point reaches only and
exactly
that one point, not so for any collection of points. It's not
one-sided to consider the intervening points of endpoints of an
interval one-sidedIn one way it leads me back to the
consideration of the sampling
frequency of the rationals being the frequency of the reals
among the
reals, being continuous necessarily the most dense of sets of
elements
on the line where the hyperreals contain only reals.
Many of these infinital (infinitary) relations exist mutually
and some
only in their particular context.
So then, of the one-sidedness of the infinitesimal that
directly
follows in indefinite magnitude a given point, it is about
that the
only way to define that point is in terms of that ordered
relation. If
we consider two functions that asymptotically approach the
same real
value from different directions, from the positive or negative
side,
those aren't necessarily the previous and next points on the
line
but rather parametric expressions of points on that line.
The radius of the unit n-sphere, as n->oo, is one, unity, a
scalar
value.
There are ways to extend infinitesimal analysis to analyze
variously
these collections of points on lines. Yet, where that is true,
two
methods of analysis can also lead to different results. I
think when
that is so that they are different results, but they are also
the same
result.
Consider the unit square, with area equal to one, and a square
with a
side of each interval between integers x and x+1 for natural x.
Divide the unit in half vertically, place the latter half in
the
second square. Divide each of those in half, placing those
quarters
in the next two squares. Repeat. Allow the division to
complete,
infinitely many times to where only point-width rectangles
remain.
The integration of f(x)=1 over the naturals evaluates to two,
on the
last division when only a point width is left the area
doubles, the
cleavage leaves each strip as a point-width.
Ignore the innuendo, the area of the unit square is one. It is.
Placing two or more points in consecutive and continuous order
changes
one of the points.
Ross F.
===
Subject: Re: min area to flip 2 hinged rods
>And if you are allowed to roll the divider over, while it
still
>touches the paper, you can cut the 3/8 pi in half by having
only half
>of an astroid, plus a little more for the rolling operation.
>Unfold the divider so it is straight, roll it over, and fold
it back
>within the same half of the astroid you openned it within.
>3/16 pi +epsilon, perhaps.
>Leroy Quet
And if you allow it to slip as you roll it over you don't need
that
little bit more.
Geoff G
===
Subject: Re: easy algebra problem.....
> show that all a in Q(sqrt(2),sqrt(3)) is constructible.
> ------------------------------
> i know that Q, sqrt(2), sqrt(3) is constructible.
> but i can't explain logically.
Q(sqrt(2),sqrt(3)) is an extension of Q(sqrt(2)) of degree 2
and Q(sqrt(2)) is also an extension of Q of degree 2.
Therefore,
every element of Q(sqrt(2),sqrt(3)) is constructible.
More generelly, whenever you have a tower of subfields of the
field of complex numbers
Q = K_0 in K_1 in K_2 ... K_n
such that each K_i is an extension of K_{i - 1} of degree 2,
then
every element of K_n is constructible.
Jose Carlos Santos
===
Subject: Re: Proposition for Euclidean geometry
John Schoenfeld
> Does anyone know a Godel proposition for Euclidian space?
Not directly, but obviously the fact one can construct
Cartesian
coordinates
means it contains arithmetic so the theorem applies.
===
Subject: Re: cantor's theorem
> Here is an interesting argument that Cantor's proof that
there are
> uncountably infinite reals is wrong. I found it pointed to on
> crank.net of all places, but I could not find any hole in the
> reasoning.
> http://maxpages.com/cantoriswrong
> Perhaps someone else can, if there is any hole. Basically it
is saying
> that the real number constructed out of the diagonal of the
countably
> infinite list may still be in the list, even though there is
no
> element of the list that has the exact same digits as this
number.
> They note that that .01111111....=.10000000000..... to make
this
> conclusion.
This comes up every once in a while; it shows that a bad choice
of how to swap digits is possible, but it does NOT show that
there
is no good way. In fact, the problem is easy to avoid, and
Cantor
was careful enough to avoid it.
An easy way is to swap (decimal) digits is as follows: for
numbers
with two decimal representations (.x0000..., .y999...), chose
the one
with the 9's. Then if the n-th digit of the n-th item is a 5,
put a
4 in that place in the number under construction, else put a 5.
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
Archimedes Plutonium
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures. We have Riemann integral and we have Lebesgue
integral and
> Stieltjes integral and Radon integral and 50 other
integrals. Then we
> have 50 or more types of differentiation. Messy, you say.
That is only
> the starting of all the gaps and holes in Real Analysis. The
entire
> subject of Real Analysis is gap ridden, hole ridden and
overall sloppy
> and messy. The reason being is that the Real Numbers are a
nonexistant
> entity. They are fakes. Just as the NaturalNumbers equals
> Finite-Integers are nonexistant and a fake set.
Your biggest problem is you do not know what the LUB axiom is
and its
implication of completeness.
===
Subject: Re: Number Theory Problem!
Mark Sapir <m..v..s@mail.ru> a .8ecrit s le message de
Ferdinand Balmes <sirferdz23i@yahoo.com> a .8ecrit s le message
de
Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of
every
> integer n.
 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
> That's nice! Is it true that every polynomial with rational
> coefficients f(n) such that f(n) is an integer for every
interger n is
> a linear combination of binomial coefficients C(n,k)?
> Mark Sapir
Yes
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
George Jones
> Most understandable.
> I would not call the Reals fake, although there is a
philosophical
> question about whether they're invented or discovered, and I
don't know
> about cracks in Euclidean Geometry, but there are cracks in
the
> real number line.
Having studied analysis at graduate school and non standard
analysis I
would
be very interested in knowing what they are.
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
|> It's a basic issue of shared resources. Bandwidth is very
cheap indeed,
|> and like with most resources, there's no simple maximum
available amount
|> of it. But like with most resources, the environment
degrades as people
|> use more and more of it beyond a certain point.
|
| But, with scientists the message is always the same.
| It doesn't matter what *bandwidth* costs, since
| we are not paying for bandwidth. We are paying
| for information.
We're not paying in money; we're paying in time. The more
junk, the longer it takes to wade through it.
Keith Ramsay
===
Subject: Re: y = x^x
>I've looked into the function y=x^x and am a bit confused
about its
>nature... with x>0 everything is predictable, but with x<=0 I
am not
>sure what its graph would look like.
>I realize that many negative x values will produce imaginary
numbers,
>but what does the rest of the graph look like? Does it just
approach
>y=0? And how exactly are arbitrary powers calculated like
that?
By definition, x^x = exp(x ln(x)). Now ln(x) is a multivalued
function
in general, so we have to be careful about what branches we
are using.
Let's say we allow any branch. When will x^x have a real
value, when
x is a negative real?
What we need is Im(x ln(x)) = n pi for n an integer. Now
Im(ln(x)) = k pi where k is an odd integer, so Im(x ln(x)) = x
k pi.
Thus we need x = n/k where n is an integer and k an odd
integer.
In fact, for any negative rational of the form x = -a/b where
a and b are positive integers and b is odd, we take
ln(x) = ln(a/b) + b i pi,
x ln(x) = -a/b ln(a/b) - a i pi
x^x = exp(x ln(x)) = exp(-a i pi) exp(-a/b ln(a/b))
= (-1)^a |x|^x
Since rationals of the form odd/odd are dense in the reals, as
are
rationals of the form even/odd, the graph will look like the
graph
of |x|^x together with the graph of -|x|^x for x < 0.
Department of Mathematics http://www.math.ubc.ca/~israel
===
Subject: Re: easy algebra problem.....
> show that all a in Q(sqrt(2),sqrt(3)) is constructible.
> ------------------------------
> i know that Q, sqrt(2), sqrt(3) is constructible.
> but i can't explain logically.
> help me......please....
Do you mean geometrically constructible?
If so, then sqrt(n+1) gives the length of the hypoteneuse
of a right tringle with sides 1 and sqrt(n). sqrt(1) is
easy, and now this allows sqrt(2), that allows sqrt(3), ...
Also, 1,2,3, ... are easy, and adding lengths is easy.
Yoyu can use similar triangles to multiply or divide
segment lengths a.b is just a segment having the same
ratio of lenght to b that a has to 1, for instance.
Now you can multiply and divide ... and that's all the
field operations.
You can also get square roots with ruler and compass;
to get sqrt(a) you need segments of length a and 1, and
a circle of radius (1+a)/2 ...
===
Subject: Re: Proposition for Euclidean geometry
> Does anyone know a Godel proposition for Euclidian space?
> JS
The first-order theory of geometry (what can be said
in terms of colinearity points, coincidence of lines,
betweenness, and congruence) is decidable.
===
Subject: Re: Differentiate many variables
continued..
http://www.google.com/search?sourceid=navclient&q=Microsoft+
PowerPoint+%2D+C
S395TF02%2Dlect2%2Eppt
to link to Microsoft PowerPoint - CS395TF02-lect2.ppt
===
Subject: Re: cantor's theorem
> Here is an interesting argument that Cantor's proof that
there are
> uncountably infinite reals is wrong. I found it pointed to on
> crank.net of all places, but I could not find any hole in the
> reasoning.
> http://maxpages.com/cantoriswrong
> Perhaps someone else can, if there is any hole. Basically it
is saying
> that the real number constructed out of the diagonal of the
countably
> infinite list may still be in the list, even though there is
no
> element of the list that has the exact same digits as this
number.
> They note that that .01111111....=.10000000000..... to make
this
> conclusion.
> Craig
This is old stuff and is easy to refute by altering the rule
by which
the nonmember of the list is constructed.
If you insist on using base two, one does it in terms of pairs
of
digits, counting from the radix point,
If the n'th pair of binary digits in the n'th number in your
list is not
01 make the n'th pair of your construction 01, otherwise make
it 01.
Since it is now impossible that the constructed number shall
be one of
those with dual representation, as well as being different
from every
number in the list, the list cannot contain every real number.
Note that for numeral bases of 4 or larger, one need only use
one digit
at a time, instead of two, to avoid creating a real number
with two
representations.
===
Subject: Re: Difficulty of calculus vs. discrete math
> I have taken college courses both in calculus and in discrete
> mathematics. What surprised me was the difficulties that
other
> students were having with discrete mathematics. For some
reason, they
> found calculus much easier.
> The discrete math was baby stuff: formal logic, divisibility,
> combinatorics, and the like.
> Why would one find calculus easy and discrete math difficult?
Here are a couple of ideas, based on my own experience:
First, you need to be aware that discrete math is quite
different from
calculus. There are certain things that are analogous between
them, but
for the most part the techniques and ideas are different.
Second, consider the preparation that students receive. Many
of the
ideas taught in algebra classes are introduced with (in
retrospect) the
purpose of preparing students for calculus. Students are shown
how to
do various manipulations of expressions which will be commonly
used in
calculus.
Third, the baby stuff you mention is, for the most part, a
collection
of concepts that are totally foreign to students when they
first see it.
Formal logic is barely touched on in public education.
Combinatorics
requires problem solving skills just to see which tools apply
for a
given problem. Calculus and algebra are often taught in a
cookbook
fashion that does not promote understanding of underlying
principles.
Many problems in discrete math are unsolvable for someone who
does not
understand the corresponding underlying principles.
Finally, an example. How do you know to use a permutation to
count the
number of ways to deal a card to each of 5 people, but a
combination to
count the number of way sto deal a 5-card poker hand? Many
students
don't see the distinction.
Will Twentyman
email: wtwentyman at copper dot net
===
Subject: Re: Differentiate many variables
continued..
http://www.google.com/search?sourceid=navclient&q=Microsoft+
PowerPoint+%2D+C
S395TF02%2Dlect2%2Eppt
to link to Microsoft PowerPoint - CS395TF02-lect2.ppt
===
Subject: Re: Number Theory Problem!
En el mensaje:c1s5hd$1lphhs$1@ID-143665.news.uni-berlin.de,
panh <inconnu@wanadoo.fr> escribi.97:
Mark Sapir <m..v..s@mail.ru> a .8ecrit s le message de
>Ferdinand Balmes <sirferdz23i@yahoo.com> a .8ecrit s le
message
> Another is, Prove that 1/5n^5 + 1/3n^3 + 7/15n is an integer
of
>> every
>> integer n.
> 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
>That's nice! Is it true that every polynomial with rational
>coefficients f(n) such that f(n) is an integer for every
interger n
>is a linear combination of binomial coefficients C(n,k)?
> Yes
It is obvious because {C(n, k) | 0 <= k <= m} is a basis of
the vectorial
space of polynomials in n with degree less or equal than m.
But the
coefficients of that lineal combination must be necessary
integers?
(By the way, as solution of the OP question is slightly
onerous ...)
===
Subject: Re: cantor's theorem
> Here is an interesting argument that Cantor's proof that
there are
> uncountably infinite reals is wrong. I found it pointed to on
> crank.net of all places, but I could not find any hole in the
> reasoning.
> http://maxpages.com/cantoriswrong
> Perhaps someone else can, if there is any hole. Basically it
is saying
> that the real number constructed out of the diagonal of the
countably
> infinite list may still be in the list, even though there is
no
> element of the list that has the exact same digits as this
number.
> They note that that .01111111....=.10000000000..... to make
this
> conclusion.
> Craig
> This is old stuff and is easy to refute by altering the rule
by which
> the nonmember of the list is constructed.
> If you insist on using base two, one does it in terms of
pairs of
> digits, counting from the radix point,
Alternatively, since a number has at most two representations,
you
could expand the list to include all representations of every
number
on the list, and then diagonalize.
===
Subject: Re: e^i(pi) = -1 revisited with Doubly Infinites Re:
infinite
rightward strings tacked-on to p-adics serves as Orthogonality
and makes
Doubly-Infinites the points of Lobachevskian Geometry
> The progress has already been done.
> Since Euler not only proved that the equation
> is true, but he also proved that you need Geometric a
> proof to prove it. Goedel proofs are insufficient.
> You need a pair of duel functions to prove that it's true.
Duel functions?
That was Galois, not Godel :-)
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: a little - big problem
> i think it's a problem in taylor expansion:
> you can show that all partial derivatives up to the (k-1)th
order vanish
at
> x_0,
> next, look at the k-th order taylor residue of f around x_0 -
> it's an expression exactly of the desired type (i.e.,
belongs to I^k)
Thank you professor for having answered me.
I need a clarification:
So,
since exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1}
= lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0
we are able to conclude that x^2+y^2 belongs to I^2_0(R^2,R) ?
I say yes. What do you say?
Tern
===
Subject: Re: min area to flip 2 hinged rods
>> ...
>> 2 rigid rods of unit length are hinged to each other.
Initially they
are
>> parallel to each other, much like a closed pair of divider.
> This divider is placed on a piece of paper. What is the
minimum area
of the
>> paper which allow the divider to be opened such that the
angle
between the
>> two legs extend from 0 to 360 degrees. The divider is to
touch the
paper
>> and no part of the divider is to extend beyond the paper
throughout
the
>> whole process.
>There is a related problem about passing a ladder of length L
around a
>right angled corner from a corridor of width A to another of
width B.
>Assuming the ladder is kept horizontal and that its thickness
is
>negligible, it transpires that if L^(2/3) <= A^(2/3) +
B^2/3), the
>desired passage of the ladder is just possible.
>Thus we may simplify the problem of the hinged rod by
providing a much
>smaller definite upper bound on the area that has been
proposed yet.
>Assuming each rod to be of length 1, the boundary curve need
be no
>larger that (x^2)^(1/3) + (y^2)^(1/3) = 1
>Since this bounds an area of 3*pi/32 in each quadrant, the
total area
>required will be less than 3*pi/8 ~ 1.7881 square units.
>Since the rods are not constrained by the axes but only by
the curve
>itself, and not, as was the ladder, constrained by the walls,
a slightly
>smaller boundary is possible, possibly one of the similar
form |x|^u +
>|y|^u = 1 with u < 2/3.
>Can anyone come up with an area less that 3*pi/8?
> Yes!
> I can't compute the area exactly (maybe someone else can
help?)
> but it is somewhere between sqrt(3)/4 = .43 and 3/4 = .75.
I'm
> guessing it's about .56, which is less than half of 3pi/8.
> Here is the construction of the paper. Start with the origin
O.
> Then place three segments with endpoints O and A = (1,0), O
and
> B = (-1/2, sqrt(3)/2), and O and C = (-1/2, -sqrt(3)/2), so
that
> there is an angle of 2pi/3 between each pair of segments. Now
> play the ladder game on each pair. For example, start with a
> 'ladder' of length 1 with endpoints O and A. Allow the
endpoint
> that is at O to move along the segment towards B, while
keeping
> the other endpoint on the x-axis. The area that this ladder
(and
> the three others) sweeps out is the entire paper. I've found
a
> parameterization of the curve, but I can't find a direct
formula.
> This is why I can't give an exact area. The estimates are
easy
> to see, however.
> Now, here is how we can actually get this divider to go
through
> 360 degrees. Let the ends of the divider be R,S,T, with S in
the
> middle. These points will stay on the three lines (O-A, O-B,
O-C)
> the whole time. Start with
> R,T at O, S at B.
> R->A and T->C while S->O. ... (1)
> Let R,S stay on the x-axis as S->A and T->O. (so R->2A) ....
(2)
> T->B as S->O (so R->A). ... (3)
> R->O and T->O while S->C. ... (4)
> I thought about the ladder problem when I saw this, too. I
think
> this construction may be improved (because of some
asymmetry),
> but I don't know the best way to proceed. I'd be happy if
anyone
> could find a direct formula for the curve!
Ingenious (and a good bit of ASCIIsation). But we can
immediately do
better than this: by not constraining the ends of the rods to
be on
the star lines, the emaciated triangle shape (sorry, don't know
fancy -oid name) can be made a bit more emaciated.
I think that the optimum shape is not radially symmetric: in
the
directions of B and C, the star does not need to extend beyond
those
points, but along the x-axis to the right, an _extremely_
emaciated
arm that extends out to a radius of 2-m, where m is the radius
of the
inscribed circle in the star would be better than a cusp that
stopped
suddenly at A. Finding an optimisation analytically looks
hideous, but
perhaps someone can do a geometric simulation.
Brian Chandler
----------------
Jigsaw puzzles from Japan
http://imaginatorium.org/shop/
imaginatorium@despammed.com
===
Subject: Re: a little - big problem
> i think it's a problem in taylor expansion:
> you can show that all partial derivatives up to the (k-1)th
order vanish
at
> x_0,
> next, look at the k-th order taylor residue of f around x_0 -
> it's an expression exactly of the desired type (i.e.,
belongs to I^k)
Thank you professor for having answered me.
I need a clarification:
So,
since exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1}
= lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0
we are able to conclude that x^2+y^2 belongs to I^2_0(R^2,R) ?
I say yes. What do you say?
Tern
===
Subject: Re: Number Theory Problem!
>panh <inconnu@wanadoo.fr> escribi.97:
Mark Sapir <m..v..s@mail.ru> a .8ecrit s le message de
>> That's nice! Is it true that every polynomial with rational
>> coefficients f(n) such that f(n) is an integer for every
interger n
>> is a linear combination of binomial coefficients C(n,k)?
>Yes
>It is obvious because {C(n, k) | 0 <= k <= m} is a basis of
the vectorial
>space of polynomials in n with degree less or equal than m.
But the
>coefficients of that lineal combination must be necessary
integers?
Yes, just write down the first m+1 values, take mth-order
finite
differences,
and you can easily read off those coefficients (which are of
course
integers).
-- Erick
===
Subject: Re: Points on a Line, Infinity
> Having a conversation, two-way communication, is the same
whether it's
> a pair of lovers, semaphores or broadband, or two prisoners
tapping
> through the wall: there is room for mutual understanding,
and the
> lack of it, in the lack there being both misunderstanding and
> ununderstanding, partial understanding, and even mutual
understanding
> through mutual errors in understanding.
> One concept I try to understand and present is that of the
points of
> the real line as a contiguous sequence. The reals have the
quality of
> being dense in themselves, between any two distinct or
definite
> reals there are infinitely many more as endpoints of
subintervals of
> the interval between them, iteratively redividing.
> This is a rather large jump from the previous paragraph, but
one thing
> I consider is how if the points are contiguous that
consecutive points
> are in a way one-sided. These are points some finitely many
in
> succession from a definite point, but infinitely far in
succession
> from any next definite point. They exist because the set of
reals
> consists only of points and the set of reals comprise all
elements of
> the continuous real number line: the set of all reals
contains only
> points, every real.
> If we consider these consecutive points after a given point
only
> infinitesimally distant, then we have a consideration that
in many
> systems the infinitesimal is only zero. Besides that, where
in those
> models the infinitesimal being zero is conducive to
simplified
> expression of those models, in other models an infinitesimal
has the
> property of being non-zero yet less than any definite
positive value,
> that is to say less than any positive ratio of integers, yet
positive
> and non-zero.
> Back to the one-sidedness, it is similar to considerations
of those
> functions we have described about functions continuous at
only a
> point, or set of points, in the image of their codomain,
their range.
> Here, one-sided is an unfortunate term, although in some
ways it
> expresses the quality of the point. In this context I want to
> disquality anything except a point. Consider the n-D
Euclidean vector
> basis, any line that intersects the point reaches only and
exactly
> that one point, not so for any collection of points. It's not
> one-sided to consider the intervening points of endpoints of
an
> interval one-sided
> In one way it leads me back to the consideration of the
sampling
> frequency of the rationals being the frequency of the reals
among the
> reals, being continuous necessarily the most dense of sets
of elements
> on the line where the hyperreals contain only reals.
> Many of these infinital (infinitary) relations exist
mutually and some
> only in their particular context.
> So then, of the one-sidedness of the infinitesimal that
directly
> follows in indefinite magnitude a given point, it is about
that the
> only way to define that point is in terms of that ordered
relation. If
> we consider two functions that asymptotically approach the
same real
> value from different directions, from the positive or
negative side,
> those aren't necessarily the previous and next points on the
line
> but rather parametric expressions of points on that line.
> The radius of the unit n-sphere, as n->oo, is one, unity, a
scalar
> value.
> There are ways to extend infinitesimal analysis to analyze
variously
> these collections of points on lines. Yet, where that is
true, two
> methods of analysis can also lead to different results. I
think when
> that is so that they are different results, but they are
also the same
> result.
> Consider the unit square, with area equal to one, and a
square with a
> side of each interval between integers x and x+1 for natural
x.
> Divide the unit in half vertically, place the latter half in
the
> second square. Divide each of those in half, placing those
quarters
> in the next two squares. Repeat. Allow the division to
complete,
> infinitely many times to where only point-width rectangles
remain.
> The integration of f(x)=1 over the naturals evaluates to
two, on the
> last division when only a point width is left the area
doubles, the
> cleavage leaves each strip as a point-width.
> Ignore the innuendo, the area of the unit square is one. It
is.
> Placing two or more points in consecutive and continuous
order changes
> one of the points.
Always nice to see you here, Ross. I see that the buzzphrase
generator
is working very well lately.
===
Subject: Re: SymbMath.com: web-based computer algebra system
> Would you please show these results?
> Ah Dr Huang, thanks for addressing my concerns.
Hey I tried some of the functions and they gave results
inconsistent
> with matlab. Do you think I should tell them to fix their
program?
> www.SymbMath.com
===
Subject: Re: Data analysis software
> It plots and analyses any x-y data for peak location, peak
> height,
> peak
> width, semi-derivative, derivative, integral, semi-integral,
> convolution,
> deconvolution, curve fitting, and separating overlapped peaks
> and
> background.
>
www.chemSoftware.com
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures. We have Riemann integral and we have Lebesgue
integral and
> Stieltjes integral and Radon integral and 50 other
integrals. Then we
> have 50 or more types of differentiation. Messy, you say.
That is only
> the starting of all the gaps and holes in Real Analysis. The
entire
> subject of Real Analysis is gap ridden, hole ridden and
overall sloppy
> and messy. The reason being is that the Real Numbers are a
nonexistant
> entity. They are fakes. Just as the NaturalNumbers equals
> Finite-Integers are nonexistant and a fake set.
The real numbers are just as fictitious as rational numbers
and the
positive
integers. One can construct the ring of rational integers from
the positive
integers. One can then construct the field of rational numbers
from the
rational integers. The topological completion of the rational
number field
is the real number field. Either by adding limit points of
Cauch sequences
or Dedikind Cuts (the equivalent) the set of real numbers we
know and love
can be constructed.
All mathematical abstractions are fictions.
===
Subject: Re: Number Theory Problem!
> That's nice! Is it true that every polynomial with rational
> coefficients f(n) such that f(n) is an integer for every
interger n
> is a linear combination of binomial coefficients C(n,k)?
>> Yes
>It is obvious because {C(n, k) | 0 <= k <= m} is a basis of
the
>vectorial space of polynomials in n with degree less or equal
than
>m. But the coefficients of that lineal combination must be
necessary
>integers?
> Yes, just write down the first m+1 values, take mth-order
finite
> differences, and you can easily read off those coefficients
(which
> are of course integers).
Of course! I don't know in what I was thinking ...
===
Subject: Re: Non-linear congruence in Z
> Does there exist x,y from Z such that:
> y(x^2-y) = 7 (mod 17)
> I don't think it's wise to try out all combinations for x,y
It can be solved by hand in a couple minutes as follows.
Y^2 - xx Y + 7 has discriminant D = x^4+6 (mod 17)
To check when D is square, tabulate x^4, by squaring x^2
+-x | 0 1 2 3 4 5 6 7 8 (mod 17)
----+---------------------------
x^2 | 0 1 4 -8 -1 8 2 -2 -4 (square prior row)
|
x^4 | 0 1 -1 -4 1 -4 4 4 -1 (square prior row)
D = x^4+6 is square when 6 + row2 intersects row1
Tis true only when x^4 = -4, so +-x = 3,5 and D = 2
So we've 8 solutions:
quadratic formula
x xx Y = (xx+-6)/2 = (-b+-sqrt(D))/2, b = -xx, D=2, sqrt(D)=6
--- -- -------------
+-3 -8 -1, -7
+-5 8 1, 7
===
Subject: re:cantor's theorem
It won't work with the decimal number system, will it?
===
Subject: Re: f continuous
moubinool.omarjee
> f:R-->R surjective with the property
> ( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
> Prove that f is continous
I think f is a homeomorphism. We know f is injective, for
if f(A)=f(B) and A<>B, we can let
x(2n) = A
x(2n+1) = B
and get a contradiction of hypothesis. So f is bijective. If we
can show that f is monotone, we will be home.
===
Subject: Galois group
I have to determine the structure and the elements of a Galois
group
relatively to the splitting field of the polynomial
f(x)=x^3-2. This
polinomial is in Q[x] so I have to find an extension. The
roots of f(x) are
2^(1/3)
2^(1/3)(-1/2 + i (1/2)3^(1/2))
2^(1/3)(-1/2 - i (1/2)3^(1/2)),
so the splitting field should be K=Q(2^(1/3),i 3^(1/2)) and
|Q(2^(1/3),i 3^(1/2)):Q(2^(1/3))|*| Q(2^(1/3)):Q| = 6.
Then |Gal(f(x)/Q)|=6 too.
My question is:
which are the automorphisms of the Galois group? (Show me the
automorphisms
and _how they work_ if you could).
TIA
===
Subject: Re: y = x^x
> By definition, x^x = exp(x ln(x)).
If you want a continuous answer, choose the principal
logarithm.
Is continuity an advantage?
The disadvantage is the it is non-real even for cases like
(-1/3)^(-1/3) .
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
> Welcome to my filter.
> Most understandable.
> I would not call the Reals fake, although there is a
philosophical
> question about whether they're invented or discovered,
All structural possibilities are already inherent in the rule
set,
hence all inventions -- both physical and purely logical -- are
equally deserving of the title discovery. Which subset of
discoveries we feel are further deserving of the subtitle
invention
depends partially on the method of search. Debating whether
the reals
are an invention _or_ a discovery misses this point: they
certainly
are a discovery, the question remains to what extent they are
an
invention, which calls for analyzing what we want to mean by
invention rather than a rush to truth value.
Now ... kind sir ... having paid my philosophical dues for the
week, I
wonder if I may pry your brain for some baby questions in
tensorial
calculus?
:-)
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
> Selections from the website, where he proves that he is a
supergenius:
Archimedes Plutonium (my true legal name)
Below in chemistry I have a circular periodic table [...] God
is 231Pu
> and the best bible is the best most up-to-date physics
textbook.
If the Brain Locus theory is correct, then through a single
atom in the
> brain can all the thinking and thoughts be conducted.
I make this biological speculation that the source of my
supergenius
> is that there is a Pu atom located in my brain, the focus of
my mind.
> The brain is a parabolic reflecting telescope which has one
atom as
> the center focus.
> They walk among us. A scary thing indeed.
I don't know .... I very much like:
the best bible is the best most up-to-date physics textbook
===
Subject: Math That Continues To Be Too Advanced For Mainstream
Economists
In the given example, the wage rate is multiplicatively
separable,
and so has no effect on the choice of technique and thus on
the amount
of labor chosen.
this involves the ceteris paribus assumption that is basic to
the
concept of demand curves (this example is not really one of
supply
and demand but rather isolated factor demand for labor). A
demand curve is a relationship between price and quantity
demanded.
If something else changes, such as prices of related goods in
demand
(here the discount rate would be an example), this shifts the
curve.
To be charitable to the original poster, his confusion
probably stems
from his notion that a change by the firm in question in the
number of
workers it hires would change the discount rate for the
economy.
I have explained in this thread how the linked post is not
related
to simple supply and demand curve analysis of labor markets
and only a
misunderstanding of the supply and demand approach would make
anyone
think so, such as with the misuse in the context of tools like
IRR.
I tired to explain that his linked page doesn't really
critique the
competitive labor market supply and demand model, where
quantities are
related to prices, and other factors shift said curves. It
evidently
didn't take, probably owing to confusion on Mr. Vienneau's
part on the
difference between partial and general equilbrium. It's a
common
intermediate mistake.
1.0 INTRODUCTION
This post considers a competitive firm facing a given
technology and
given prices. Two levels of the wage are considered; other
prices
modeled below remain at one given level. Yet the
cost-minimizing firm
chooses to adopt a less labor-intensive technique at the lower
wage.
So I consider the thought experiment that mainstream economists
have claimed yields a non upward-sloping labor demand curve.
Yet, I do
not obtain such a curve. This result raises some questions:
under what
special-case conditions will cost-minizing competitive firms
prefer to
hire more workers when the wage is at a lower level? What are
the
assumptions needed to derive well-behaved factor input curves?
How
are such curves derived?
Mainstream intermediate microeconomics is mostly nonsense, as
has
been demonstrated in the literature long ago.
2.0 THE SETTING OF THE PROBLEM
Consider a widget-producing firm. The managers of the firm
know of
two techniques for producing widgets, Alpha and Beta. Table 1
shows the
amount of labor inputs required to produce a widget for each
technique.
The interesting qualitative properties of this example depend
merely on
the difference in labor inputs between the techniques each
year. Thus,
if every input of labor in both techniques was increased by
the same
amount (e.g., 25 person-years), the results of this example
would look
much the same. One instance discussed in the literature is a
choice
between using a plot of land for either grazing or mining;
mining
requires a high initial expenditure and a costly cleanup phase
at the
end of the use of the land. Other empirical examples have been
discussed
in the literature, usually in the context of environmental
economics or
geography.
TABLE 1: INPUTS OF LABOR NEEDED PER WIDGET PRODUCED
YEAR
BEFORE
OUTPUT ALPHA TECHNIQUE BETA TECHNIQUE
0 33 person-years 0 person-years
1 0 person-years 52 person-years
2 20 person-years 0 person-years
The managers of the firm face three choices for the use of
revenue
from previously-produced widgets:
(1) Buy a bond.
(2) Produce widgets with the Alpha technique.
(3) Produce widgets with the Beta technique.
The managers, in choosing among these alternatives, take
prices as given.
Let p dollars per widget represent the price that widgets sell
for, and
let w dollars per person-year represent the wage. Let i = 25%
be the
interest rate earned on a bond. That is, if one spends $100 at
the
start of the year on bonds, one will receive interest payments
at the
end of the year of $25 in perpetuity (or until one decides to
sell the
bond). Only one level for the price of bonds and of widgets is
considered
below.
3.0 QUANTITY FLOWS
This analysis is intended to establish the labor-intensity of
the
choosen technique at various levels of the wage. My first step
is to
determine the labor-intensity of each technique. For this
purpose,
consider steady-state quantity flows. Table 2 shows the labor
inputs
required to produce each year an output of one widget with the
Alpha
technique. As is easily seen, the labor-intensity of the Alpha
technique is 53 Person-Years Per Widget. Table 3 shows the
labor
inputs required to produce a constant flow of one widget per
year
with the Beta technique. The labor-intensity of the Beta
technique is
52 Person-Year Per Widget.
TABLE 2: LABOR INPUTS FOR ALPHA PER WIDGET PRODUCED
...33
0 33
20 0 33
20 0 33
20 0 33
20 0 33
20 0 33
20 0...
20
TABLE 3: LABOR INPUTS FOR BETA PER WIDGET PRODUCED
...0
52 0
0 52 0
0 52 0
0 52 0
0 52 0
0 52 0
0 52...
4.0 THE COST-MINIMIZING TECHNIQUE
4.1 HIGH WAGES
For a given price of widgets, p, consider the wage:
w/p = 4/257 ~ 0.0156 Widgets Per Person-Year
Recall that the interest rate on bonds is i = 25%. Suppose the
rate of
returns the firm uses for present value calculations, r, is
also
equal to 25%. Table 4 shows the cost of producing widgets with
each
of the two techniques with this price system.
TABLE 4: WIDGET PRODUCTION COSTS FOR INITIAL WAGE, r = 25%
TECHNIQUE COST PER WIDGET PRODUCED
Alpha 33 (4/257) + 20 (4/257) (1 + 1/4)^2 = 1
Beta 52 (4/257) (1 + 1/4) = 260/257 > 1
Table 4 shows that the Internal Rate of Return in producing
widgets
with the Alpha technique is 25%; the revenue gained from
selling widgets
is equal to their cost of production under Alpha at this rate
of
return. Furthermore, the cost of producing widgets under the
Beta
technique at this rate of return exceeds the revenue gained.
Thus, the
firm would not want to produce widgets with the Beta technique
at this
wage.
Suppose one wanted to speak of the opportunity cost of using
capitalWhat rate of return would the managers of this firm be
giving up for
every dollar invested in the Beta technique? The best rate the
firm can
obtain is 25%, and this 25% rate is gained in either buying
bonds or in
producing widgets with the Alpha technique.
4.2 LOW WAGES
Now consider the firm's choices at a lower wage, namely:
w/p = 1/91 ~ 0.0110 Widgets per person-year
The firm still needs to do present value arithmetic to
determine
which option is best. Suppose the firm uses the interest
obtainable
on bonds, 25%, for such calculations. Table 5 results.
TABLE 5: WIDGET PRODUCTION COSTS FOR LOW WAGE, r = 25%
TECHNIQUE COST PER WIDGET PRODUCED
Alpha 33 (1/91) + 20 (1/91) (1 + 1/4)^2 = 257/364 ~ 0.7060
Beta 52 (1/91) (1 + 1/4) = 5/7 ~ 0.7143
In the calculations shown in Table 5, the Alpha technique is
cheaper
than Beta. Since the cost of producing a widget with the Alpha
technique,
at a 25% interest rate, is cheaper than the price of widget,
the rate
of return in using the Alpha technique is better than that
obtained
from buying a bond. Evidently the opportunity cost of capital
for
this firm under these circumstances is not 25%, but some
higher rate
of return.
What is the best rate of return obtainable by the firm under
these
conditions? One might hypothesize that it is the internal rate
of return
obtainable in producing widgets with the Alpha technique.
Table 6 shows
costs of the two techniques under this hypothesis, that is,
with a rate
of return of approximately 70%.
TABLE 6: WIDGET PRODUCTION COSTS FOR LOW WAGE,
r = 100 [Sqrt( 29/10) - 1] %
TECHNIQUE COST PER WIDGET PRODUCED
Alpha 33 (1/91) + 20 (1/91) (29/10) = 1
Beta 52 (1/91) Sqrt(29/10) = (4/7) Sqrt(29/10) < 1
The analysis of the choice of technique is not complete with
Table 6.
Note that the cost of producing a widget with the Beta
technique is less
than unity. That is, the internal rate of return in using the
Beta
technique exceeds the rate of return used in drawing up the
table. If
the managers of this firm were to purchase a bond, the
otherwise best
rate of return they could obtain - that is, the opportunity
cost of
capital - would not be r = 70% used in drawing up the table.
It would
be whatever rate is obtainable in using the Beta technique.
The internal rate of return in producing widgets with the Beta
technique is 75%, as shown in Table 7. Note that the cost of
producing
widgets with the Alpha technique at this discount rate exceeds
the
price obtained. Clearly, a cost-minimizing firm would not
produce
widgets with Alpha technique at this wage.
TABLE 7: WIDGET PRODUCTION COSTS FOR LOW WAGE, r = 75%
TECHNIQUE COST PER WIDGET PRODUCED
Alpha 33 (1/91) + 20 (1/91) (1 + 3/4)^2 = 29/28 > 1
Beta 52 (1/91) (1 + 3/4) = 1
5.0 CONCLUSION
Note that in the above analysis, the price of bonds and the
interest
rate obtainable on bonds has been assumed constant. Of modeled
prices,
only the wage has been considered to be at different levels.
If the
opportunity cost of capital is taken to be the best rate of
return
that the firm has available, it clearly varies with the wage.
That is,
the discount rate this firm uses in its internal accounting is
dependent on the wage, even though the rate used elsewhere by
agents
in the economy may differ. In other words, the discount rate
for the
economy, whatever that may be, is not shown to vary with the
wage
merely by the above analysis. It is true that other markets
outside
the labor market are implicitly shown to be thrown out of
equilibrium
by a shift in the wage. But, since the above analysis is one of
partial equilibrium, these disequilibrium forces are ignored
in the
analysis.
Anyway, Table 8 summarizes this analysis. One sees that a
cost-minimizing competitive firm may want to employ less
workers,
given the level of output, at a lower wage. If the supply of
workers were to increase, say because of increased immigration,
the labor market might only be able to come into equilibrium
(in
this partial equilibrium analysis) with more workers employed
at
a higher wage.
TABLE 8: THE COST-MINIMIZING TECHNIQUE
WAGE TECHNIQUE LABOR-INTENSITY
(p/91) Widgets Per Person-Year Beta 52 Person-Years Per Widget
(4 p/257) Widgets Per Person-Year Alpha 53 Person-Years Per
Widget
An analysis of the cost-minimizing technique for the entire
possible
range of wages is provided for this example at:
<http://www.dreamscape.com/rvien/Sraffa4/Sraffa4.html
So much for the theory that wages and employment are
determined by the
intersection of well-behaved supply and demand curves in the
labor
market.
Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html
To solve Linear Programs: .../LPSolver.html
r 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 highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
===
Subject: re:Proposition for Euclidean geometry
Depends on what sort of proposition you're looking for.
===
Subject: re:Prime factors of number near googolplexplex
He didn't factor those numbers, just guessed some of their
small
factor.
===
Subject: Re: eigenvalue must be defined only for square matrix?
> I think the concept of eigenvalue can be defined also for
rectangular
> nonsquare matrix...
> How?
> A x = lambda x, where x is a column vector would seem to
require A
> to be square
I think singular values are a generalization of eigenvalues to
rectangular matrices. Apparently the singular value sigma
satisfies
the following equations for some vectors u and v:
A v = sigma u
A' u = sigma v
See, for example,
http://www.mathworks.com/access/helpdesk/help/techdoc/math_
anal/mat_l26a.
shtml
Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html
To solve Linear Programs: .../LPSolver.html
r 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 highly
@ r c m unbecoming to reasonable and free men in search of
d o the truth. -- Rousseau
===
Subject: Re: Galois group
I am going to let K = Q( 2^(1/3), (-1 + i*sqrt(3))/2). It's
the same
as what you have, but it will be easier for me when it comes
to the
automorphisms. In fact, let
r = 2^(1/3)
and
p = (-1 + i*sqrt(3))/2)
Let f: K --> K be an automorphism. What does it do with r and
p?
f(r) = r, rp, or rp^2
f(p) = p or p^2.
This gives a total of 6 possibilities.
Then, we can define automorphism f, g: K --> K by the
following:
f(r) = rp and f(p) = p
g(r) = r and g(p) = p^2
Extend these linearly to all of K. With these, you should see
that
f^3 = g^2 = 1, so that the Galois group is S_3.
Hope this helps,
Brian
>I have to determine the structure and the elements of a
Galois group
>relatively to the splitting field of the polynomial
f(x)=x^3-2. This
>polinomial is in Q[x] so I have to find an extension. The
roots of f(x)
are
>2^(1/3)
>2^(1/3)(-1/2 + i (1/2)3^(1/2))
>2^(1/3)(-1/2 - i (1/2)3^(1/2)),
>so the splitting field should be K=Q(2^(1/3),i 3^(1/2)) and
>|Q(2^(1/3),i 3^(1/2)):Q(2^(1/3))|*| Q(2^(1/3)):Q| = 6.
>Then |Gal(f(x)/Q)|=6 too.
>My question is:
>which are the automorphisms of the Galois group? (Show me the
automorphisms
>and _how they work_ if you could).
>TIA
===
Subject: re: hang Them By Their Own G-Strings?
PS
to say that if Brian Greene had devoted himself to research,
instead of
writing popular books, he would not have accomplished great
things.
Money, money is very tempting.
I replied: Two and a half million dollar advances cannot be
sneezed at.
Brian is laughing on his way to the bank and he can get away
on NOVA
with talking about parallel universes, extra dimensions, time
travel to
the past and he can even talk to ETs on the telephone and
not be called
crazy. Go figure. (GRIN)
http://stardrive.org/cartoon/spectra.html
===
Subject: re:Galois group
The group must be isomorphic to a transitive subgroup of S3
(the
symmetry group of 3 elements) and contains the complex
conjugation
which is of order two. Thus it's isomorphic to the complete
permutation group of the three roots of f.
===
Subject: Re: UK universities
R.M. <raimuehl@web.de> schrieb im Newsbeitrag
> Can anyone recomment UK universities or colleges with a
mathematics
> department which is strong in (some of) the following fields:
> - (differential) topology
> - set theory
> - logics
> - differential geometry
> - it has kind to connections to theoretical physics
(cosmology, field
> theory)
Oxford would be my first choice in differential geometry and
topology.
===
Subject: Upside-Down Functions Have Some Remarkable Properties
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
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$Revision:
1.9 $, proapp) id i1T76UP17686;
An Upside-Down Function (UDF) is defined here as a function
which
has a non-trivial multiplicative inverse, where non-trivial
means
that the multiplicative inverse has a recognizable
mathematical use
other than as an index comparing two of the same types of
quant-
ities. A function may be UDF restricted to some domain or
range,
and it may be UDF based on current mathematical knowledge, and
occurrence of something in an intermediate step of computation
is
not regarded as a non-trivial use.
For example, 1 - x is UDF because 1 + x + x^2 + ... = 1/(1 - x)
for /x/ < 1 as the geoemtric series. Similarly, exp(x) is UDF
because exp(x)exp(-x) = exp(x)/exp(x) = 1 and exp(-x) is
exponential decay with rate 1 in radioactivity. However,
log(x) is not UDF, since its multiplicative inverse 1/(log x)
doesn't have any recognizable non-trivial mathematical use.
UDFs are disguised by the usual emphasis on Composition of
Functions in Category Theory, but they play several interesting
roles:
A. In mathematical physics, 1 - x occurs in the Special Theory
of Relativity (SR) as sqrt(1 - v^2/c^2) if we set x = v^2/c^2,
and it occurs in the Heisenberg Uncertainty Principle (HUP) if
we write the latter as DEL(x)DEL(p) > 1 and write y = DEL(x)DEL
(p) so that 1 - y < 0.
B. The expression 1 - x is the special case of 1 + y - x which
crosses fuzzy multivalued logic (implication), probability-
statistics (probable influence), and geometry and its topology
intersection (proximity functions), when y = 0. In other words,
in probable influence P(A-->B) = 1 + P(AB) - P(A), the
probability P(AB) = 0, which typically means that A and B are
disjoint up to probability zero.
C. Both of the main UDFs 1 - x and exp(-x) are special cases of
the Riccati Differential Equation growth-expansion-contraction
solution dy/dt = A(t) + B(t)y + C(t)y^2 with x = t, and
therefore
are essentially quite different from Curvilinear Motion in one
direction at a time or other types of change of state.
Therefore, their occurrence in item B above is somewhat
paradoxical.
D. The fact that log(x) is not UDF raises the question of how
fundamental it is (compared with exp(x) or exp(-x) types) as
a measure or part of a measure of informationE. If we write 1
- x = 1/[1 + x + x^2 + ... ], for /x/ < 1, and
if we assign physical units of some sort to x (or biological or
psychological or whatever units), then if 1 is regarded as a
dimensional constant in the same units, the right-hand-side of
this equation seems to be an upside-down unit scale object
and also a generalization of a multiplicative inverse term in
a Laurent (Complex) Series.
Osher Doctorow
===
Subject: When the term you differentiate does not become zero
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
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I first came across this method in the film Stand and Deliver,
you may have
seen it.
Anyway, I had a look at this method, and as a trick for simple
by parts
integrals it works fine provided that the one you
differentiate becomes zero.
As I realised in lectures this is not always the case so I
considered a way
to cheat the cheat method, and found a little gem.
Consider the indefinite integral of xlnx dx
using lnx as a priority for the differential it can be quickly
seen that it
doesn't revert to zero.
by parts in can be shown that the answer is (1/2 x^2 lnx) -
(1/4 x^2) + C
Using the table;
sign d/dx integral dx
+ lnx x
- 1/x x^2/2
+ -x^-2 x^3/6
The trick is now to take the integral of the product in the
last row.
Giving; 1/2 x^2 lnx - 1/6 x^3 x^-1 - 1/6 int(x^-2 x^3)dx
this then comes out as above once simplified.
The trick is knowing when to stop!!!
Try some simple ones and you will get the hang of it.
Take care
(Elec Eng)
===
Subject: Re: Number Theory Problem!
> 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
> That's nice! Is it true that every polynomial with rational
> coefficients f(n) such that f(n) is an integer for every
interger n is
> a linear combination of binomial coefficients C(n,k)?
> Mark Sapir
> Yes
For those who (like me) wanted a reference here it is (copied
from
http://www.math.tamu.edu/~harold.boas/courses/math696/
Maple-functions.html):
G. P.97lya proved [.86ber ganzwertige ganze Funktionen, Rend.
Circ. Mat.
Palermo 40 (1915) 1-16] that the polynomials that take integer
values
at all integer points are precisely the linear combinations
with
integral coefficients of the binomial polynomials of the form
x(x-1)...(x-k+1)/k!, where k=0, 1, .... See Manjul Bhargava,
The
factorial function and generalizations, American Mathematical
Monthly
107 (2000), number 9 (November), 783-799.
Mark Sapir
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
: | I've written
: |stories. Fiction. I like the idea of a fiction lying itself
into
reality,
: |much like the mythology of galathaea. But also like a
scientific model,
: |which never knows itself to be true but seeks justification.
:
: Do you actually have a scientific model, or just a sketch?
Yes. I've actually tried to mention it in several places now
and even
began
and v. But I've left alot to the literature so far because I
have not had
do not have literature access. Again I stress that none of
this is my
theory; it is stuff already well accepted in the literature,
though maybe
not as well known as I would like.
What I have been talking about is the categorial analysis of
logic. My
model is merely an ontology interpretted in terms of the
ell-Benabou
language, building a Kripke model structure inside of a
category through
the
use of the Joyal interpretation. I am using that to build a
scientific
model by attaching an observable ontology with which to
evaluate
experimental correlation.
This is explained in some detail in books like Saunders
MacLane and Ieke
Moerdijk's Sheaves in geometry and logic: a first introduction
to topos
theory (particularly chapter VI, sections 5 and 6), Robert
Goldlatt's
Topoi: the categorial analysis of logic (chapters 8 and 11),
and its
application to the scientific models as explained in, for
example, John
Casti's Reality rules i: picturing the world in mathematics -
the
fundamentals -ell-Benabou allows us to use a collection of
objects and their
transformations (much like the conceptual maps one might build
for natural
language or mathematical objects and their categories) as the
basic types
of
the ontology. A term rho of a type X is a morphism rho:U -> X
whose
codomain is X. Each variable of type X is a term of type X
whose
interpretation is the identity morphism on X. Using a
subobject classifier
Omega (or its generalisations) in the ontology allows one to
build
mereological notions which include the formulas of the
ontology.
Terms can combine through composition. Elements are defined as
relationships between terms of X and terms of the power object
Omega^X.
Quantifiers involve constructions on the power object of type
X and the
power object of the terminal type (which is just the subobject
classifier).
Validity in the theory is identified with those terms of Omega
that factor
through the morphism from 1 to Omega named true. One can derive
notions
of forcing as well and describe the collection process. Using
the common
bracket notation of collection (as found, for instance, in set
theory), we
can define U ||- phi(alpha) for some generalised element alpha
: U -> X
with
image(alpha) an element of Sub(X), if and only if image(alpha)
<=
{x|phi(x)}, ie. alpha factors through {x|phi(x)} as in:
{x|phi(x)} ---------------> 1
% / /
. | |
. | | true
. | |
. V V
U ------> X -------------------> Omega
alpha phi(x)
You build an observable language through a selection of
morphisms with
domain the subobject classifier and codomain a measurement
object (the real
numbers, a probability measured interval, a discrete detector,
etc.).
These
can then be subjected to correlation and other test analyses.
Not only do
we have this formal system as an expression of a generalised
logic (which,
as for example in the quantum case, can be significantly
generalised beyond
topoi), but we also seem to have a lot of indicators from
quite a number of
different directions that a very similar structure can be
interpreted on
cognitive models, ie. that we may employ a very similar method
of mereology
to construct our conceptual spaces and the logics we employ in
our natural
communications.
I identify these models with lies because I believe theorising
models is a
uniquely _creative_ task, where there is a process of pattern
recognition
not guaranteed to be accurate which picks out real world
correlations and
attempts fuzzy significance tests connected with some model
construction
game. In other words, the models are never guaranteed to be
anything more
than fictions, particularly when ontology is not covered by
observable
epistemology, but often these fictions are useful. We can
even, for
example, define preorders of model correlations with
experiments.
: |: | > I know many models whose Heyting structure is far
more simplistic
: |: | > than the corresponding Boolean embedding.
: |: |
: |: | Can you name them? Heyting algebras are always
infinite, afaik.
: |:
: |: Note that simplistic means excessively simplified: |:
: |: Boolean algebras are a special case of Heyting algebras,
and there are
: |: plenty of finite Heyting algebras even excluding finite
boolean
algebras.
: |
: |Yes, additional structure can make these potential
infinities collapse.
:
: Eh? It's not a matter of infinities collapsinginto a
skeletal category by taking the quotient through the
isomorphism
equivalence. Adding, for instance, the formula
forall p, we have p / ~(p)
into a Heyting structure causes a collapse of what may have
previously been
a potential infinity of truth values if appropriate
existential formulas
held.
: |I wanted to stress that both algebras are finitely
generated, however,
:
: Both of what algebras? The Heyting algebras you describe as
simpler than
: the Boolean algebras they embed in? I would guess that a
finitely
: generated Heyting algebra need not embed in any finitely
generated
boolean
: algebra.
Finitely generated in the term calculus as formulas defining
the existence
and axiomatisation of the universe of the ontology. In other
words, a
finite list of axioms. This is applying to the general objects
of Heyting
and Boolean algebras, and not to specific instances which
carry more axioms
/ definitions with their structure. Its a computational (or
proof theory)
notion of finiteness I am stressing here.
: |and it
: |is only through the potential application of axioms over an
infinity
that
: |any infinities arise.
:
: I don't think the phrase application of axioms over an
infinity
carries
: any specific meaning.
It does computationally or proof theoretically. What I am
saying is that
one arrives only at potential notions of infinity through, for
example,
application of the sequent calculus to determine apartness
between objects
for an infinte class of elements of the ontology.
: |This, I feel, is why the notion of potential
: |infinity is so stressed in concstructive circles, because
much of the
: |distinction in concepts only occurs over this realm, the
finite sharing
most
: |properties with the boolean.
:
: The notion of potential infinity is not often stressed by
: constructivists, in my experience. It's stressed by some
people talking
: *about* constructivism, in an attempt to characterize the
difference
: between constructivism and classical philosophy of
mathematics.
:
: It sounds like you mean to say that finite Heyting algebras
share some
: natural properties with Boolean algebras. Perhaps you'd like
to state
: some?
Finitely presented Heyting algebras are co-heyting, thus
semi-boolean.
Finite collections have constructive properties that are
naturally boolean
through proofs by exhaustion (ie. is-an-element-of is
bivalent, etc.).
[...]
: |Heyting algebras are much more general than
: |merely models for constructive mathematics. They also
underly a huge
number
: |of models in the sciences, which has been why I have listed
so many of
them.
: |In fact, with the theory of causal sets, we can attach a
natural Heyting
: |structure to many theories that share the underlying causal
structure.
:
: There are plenty of partial orderings you could cite, but it
doesn't add
: up to much of a motivation to think Heyting with them.
I think it does. I think that the fact that the category of
digraphs is a
topos (and thus admits a natural Heyting structure to
propositions) is very
important to reasoning about such structures. I think this is
all pointing
to a type of universality not shared by the boolean. Many of
the papers I
reference make a point about the model being Heyting and work
on a
semantical interpretation of formulas in the model, which is
pretty much
all
there is to thinking Heyting: |: Part of what makes the
situation confusing is that the ratio between
: |: ordinary mathematics done constructively and
metamathematics about
: |: it is much lower than the ratio between ordinary
mathematics done not
: |: bothering with constructivity and the metamathematics of
that. It's to
: |: the point that Mathematical Reviews places constructive
mathematics
under
: |: the 03 (logic) category. You might, for example, wonder
whether
such
: |: things as Martin-Lof's type theories count as
counterexamples to my
claim
: |: above. It's possible that somebody out there has been
actually doing
: |: mathematics in them, but not as far as I know.
: |
: |The Heyting structure extends to untyped lambda calculi as
well, through
: |Curry-Howard, and CS students regularly study computability
and
mathematical
: |frameworks. Obviously, turing-completeness is often a
requirement for
any
: |new language proposed, and discovering turing-completeness
in c++'s
template
: |metaprogramming mechanism was a crucial step to modern
generative
: |programming paradigms, so much of the focus is on the
cut-elimination
: |operations and similar reduction theorems. However,
illustrating
reasoning
: |in terms of the logical structure is not as well taught. I
liken this
to
: |the fact that the logical reasoning in quantum mechanics is
rarely
taught
in
: |terms of orthomodular lattices, though doing so obviously
prevents a lot
of
: |the conceptual difficulties associated with quantum
mechanics.
:
: Apparently you have some liking for theoretical computer
science.
:
: You might want to have a look at Mackey's book on quantum
mechanics in
: which he presents it in terms of axioms about questions,
which amount
: to the closed subspaces of a Hilbert space.
:
: I'm afraid using orthomodular lattices doesn't really keep
people from
: one slit, or through both at the same time?, nor should it.
As Feynman
: once said, all the weirdness boils down to just that one
simple case. If
: you can make coherent sense of that situation, then the rest
falls into
: place. I don't doubt that orthomodular lattices are a useful
concept, but
: I don't think it has much to do with the conceptual
difficulties people
: tend to have with quantum mechanics.
I think your fears are misfounded. That is exactly what you
see quantum
logic explaining. Why is it that the sentence the electron
passes
through
slit 1 and hits the screen at position x gives a gaussian
distribution
for
a particular idealisation of slit, the sentence the electron
passes
through
slit 2 and hits the screen at position x gives another
gaussian, but the
sentence the electron hits the screen at position x without
mention of
a
detection at either slit 1 or 2 gives the characteristic
interference
pattern? That is the type of question which is eminently the
domain of
quantum logic.
: |: If you were to count as schools of classical mathematics
all the
: |: different nonconstructive formal systems in which one
could do
: |: mathematics, the number would hugely exceed the number of
constructive
: |: formal systems. Even if you were to restrict yourself
just to
classical
: |: theories in which _some_ mathematics has actually been
done, you can
find
: |: set theorists who've taken as their starting points
initial
assumptions
: |: of varying strengths.
: |
: |I believe this narrows the applicability of the logical
structure far
too
: |much.
:
: Please remember that this was all in response to the remark
that if you
: use a constructive foundation, you suffer from a wealth of
alternative
: constructivisms. Obviously applying a logic foundationally
is not the
: only way to apply it, or else logicians would have little to
talk about.
:
: There's a world of difference between saying you have some
kind of cute,
: simple algebraic structure that appears often, and saying
that you have
: a fundamental concept.
:
: |There is some type of universality in Heyting structures
not shared
: |by the Boolean that allows it to model propositions of a
huge variety.
Many
: |of the early foundationalists saw this and did much
research in the area
: |(Kleene, Tarski, etc.). With the semantical identification
with S4, one
: |finds a deep identification with notions of possible worlds,
descriptions
of
: |necessity, and the basic modality of science,
computability, and proof.
: |Certainly, as well, proof theory today is highly influenced
by its
Heyting
: |structure, and I am looking for why this is not so in more
fields that
: |implicitly have the structure hiding away in their analyses.
:
: Best of luck with your investigation. But it seems to me
that you're
: being a little too glib with the attribution of deep
identification:
: I once saw a book that argued for the existence of God on
the basis that
: the golden ratio (sqrt(5)+1)/2 occurred in a supernatural
profusion. I
: think they failed to appreciate the extent to which this
occurred simply
: because it's a root of a very simple equation, x^2=x+1.
:
: In your examples, you mention a number of cases where
there's an obvious
: partial ordering in the background, and the Heyting algebra
structure
: comes from it. Sure, this is commonplace, but where does
that really
: leave us? For a concept to be really fruitful or deep, it
has to be
: more than just frequent in occurrence. If you err too far on
the side of
: generality, you wind up with many examples, but not being
able to add
: much content to any of them.
:
: On the other hand, if you managed to uncover a good
constructive theorem
: or a few, you might really have something.
I think thats one of the great points about the categorial
approach. It
shows in particular that a particular method of building a
model for a
particular pantheon of objects illustrates mereological and
geometric
relationships. When the pantheon admits the basic mereological
structure
of
a topos, one automatically has a Heyting structure for the
model. That, I
feel, is pretty fundamental. That such a structure is related
with
evolutionary, causal models as well as the intimate relation to
computation,
I think there becomes enough established that it might just be
important to
restructure teaching a bit to introduce these notions.
But I am looking for the opinion of others here, and am
interested in
objections. Do you think that the notion of a subobject
classifier,
exponentiation, and the definition of a topos are not
fundamentally useful
notions?
: [...]
: |: Nevertheless, I'd just as soon not have someone trying to
get people
: |: interested in it in the manner Galathaea has been trying
to.
: |:
: |: The excessive cross-posting is a bad sign. The fact alone
that one has
: |: had to try to justify it almost always means that one has
gone too
far.
: |: And excessive cross-posting usually means that someone
feels entitled
to
: |: grab attention at others' expense.
: |
: |Do you disagree with any of the points I have made in the
towards a
: |constructive education or more focus... posts? Or do you
believe
that
I
: |have in some other way violated the constraints of the
groups'
topicalities?
:
: I consider them only very poorly topical in the majority of
the groups to
: which you posted them. I think perhaps you are rather
generous in general
: in attributing mutual relevance to ideas you like, for one
thing.
:
: Remember that the intended purpose of these rules is to make
it easier
for
: people to read what they want to and not be bothered with
what they don't
: want to. I think you've approached this in a manner much
like the typical
: excessive crossposter does. They usually hugely overestimate
the interest
: that their particular message will have for the readership
of the groups
: they're posting to (or don't care).
I never overestimated the interest. I've been quite upfront
about this. I
hoped that a small number of readers would be interested
enough in my
questions to participate. But the low numbers I predicted have
nothing to
with topicality. My post was quite topical to the groups,
despite your
repeated assertions otherwise. Just like when one sees a post
on, say,
cobordism in sci.math, one doesn't get upset at the number of
regular
readers of the group that wouldn't be interested or have
covered the
particular material. Topicality is not measured the way you
want to
measure
it, which is fortunate because your method is subjective,
evolving, and
easy
to throw around hand waving with.
[...]
: |What I have tried to find is people in all of these groups
from several
: |directions (which I have worked hard to detail), to see
what that
: |communities ideas are concerning the education proposal,
because it is a
: |fractured and disparate community which I felt might share
a common
goal.
:
: Often the best approach is to make a brief mention in
various groups that
: you intend to start a discussion somewhere else, and then
leave those
: groups alone. If anybody actually is keenly interested in
reading it,
they
: will find it easy to subscribe to the group in which the
discussion is
now
: taking place.
No. That's the go away response of someone who really has
control
problems. Seriously, these groups are here for the discussion.
There is
absolutely no reason for me to go elsewhere.
: |I have seen many pleas against the cross posting. None of
them have
been
: |very convincing in my opinion, since I have made it quite
clear the
points
: |of topicality I want to discuss.
:
: Yes, what *you* want to discuss. If they don't have much
interest, they
: get to keep deleting it as postings generated from other
groups keep
: showing up.
Its times like this I really start to understand a lot of the
posts from
Jeff Relf over in sci.physics about news reader software. Keep
deleting
sounds like one is in dire need of better software. Even the
mathforums
web
archive for those who don't have an nntp service with their
access provider
does a good job of keeping a thread together logically.
In other words, I think if you have every new posting being
tossed out as
something new to ignore, it is likely a problem with old
technology and I
am
sure that there are freely downloadable solutions for most
major operating
systems.
: |Often these have been from people who
: |admitted they were unfamiliar with the actual work in the
topic they
were
: |attempting to defend, and usually they were uniterested in
making any
effort
: |to learn about it. All of my main posts have worked to make
this
absolutely
: |clear.
: |
:
: Massive crossposting has been dubbed velveeta to distinguish
it from
: spam proper. Anything that spreads a message around more
than ten times
: the number of places where it really belongs, though, falls
into a common
: category, however you want to term it.
Again the sly little really belongs coloring your statement...
: |: I would generally advise against being a self-proclaimed
liar, even if
: |: this is meant in a humorous way (which I don't know).
: |
: |Its just a fact that many psychologists have verified that
most people
: |(percentages close to unity) lie in their life. I've done
it.
:
: Calling someone who lies to only the usual degree a liar
without more
: context is a little bit like calling someone a tennis player
because
: they have, at some point, played tennis. I am not a
badminton player: I am someone who's at some time played
badminton.
:
: If there's nothing more to it than this, then I'd have to
say it's
: a bit of not especially amusing whimsy.
Except for the deeper notion of language as a lie, the
attachment of
meaning
to utterance strings or visual symbolics through creative,
metaphorical
relationships to real life structures, and similar connections
between
fiction and lies, there really is not much more to it. I'm not
the one who
keeps bringing up my signature.
: |I write it in my signature to annoy those who cannot get
over it. Its
an
: |annoyance they will have to carry with them until they
forget my
signature
: |(rather transitory unless they keep reading my threads), or
until they
: |accept at a much more fundamental level the metaphor and
fiction that
: |underlies their entire perception of the world and methods
of modeling
it.
:
: Count on their either (a) ignoring it, or (b) deciding
you're sort of an
: annoying person but not otherwise bothering with it.
Except to keep bringing it up, like several have so far.
[...]
: | and so I have
: |had to skim huge pantheons of objects to become familiar
with the
various
: |territories. However, my learning model includes going
deeper and
deeper
: |into the topics I feel need most exploring to understand
the structural
: |questions I want to answer. For example, I bohminised
Witten's cubic
: |bosonic string model during my analysis of extensions of
realist
ontologies
: |of quantum mechanics in order to demonstrate to myself that
some of my
: |notions concerning the isomorphism of Bohm and its relation
to
quantisation
: |could carry over to some modern theories. I have done
original research
in
: |the study of functors from the category of Poisson
manifolds to the
category
: |of Hilbert spaces. I have derived results on the
combinatorial
enumeration
: |of certain magma types. Many would not consider these types
of
calculations
: |to be that of the skimmer type, and the classification is
often used
in
a
: |derogatory way.
:
: I mention the source partly to indicate that it isn't meant
as
: necessarily derogatory. Her point was to help people realize
that they
: might be more one way or the other, and that they should
quit trying to
: apply a style that doesn't really suit them, and instead
develop whatever
: their own appropriate style is. On the axis between
believing in one
right
: way to do everything, and believing we each should do our
own thing,
: this author certainly would fall well toward the liberal end.
:
: I mention it as a way of indicating that I also accept such
differences
: as naturally present, and don't intend to count it against
someone if
: they merely have a different style from mine.
:
: On the other hand, your postings I've seen so far have been
far from
: dense in content. I suspect you think they are, because you
count as
: contentual remarks that are only suggestive.
What is it about my topic that I am supposed to go into more
dense
content??? I didn't come here with the goal of teaching
everything about
the relationships I mention, I came to have a conversation
with those who
already understood the concepts. I am asking about education
here. I only
I received a lot of questions and realised that some education
of my own
was
necessary. But there are a ton of books already written about
the topics
to the large set of theory.
However, I have had quite detailed posts on other topics in my
usenet
career. Whether it be the hypergeometric relations I have
posted in
sci.math or any of quite a large number of posts in
sci.physics.research or
elsewhere, I have my deeper side. In my real life I have to
write
technical white papers concerning cryptographic techniques and
other
technological reports for my company's IP all the time.
Your implication is indeed doing that very thing passive
aggressively you
deny you are doing, making a derogatory implication about my
capabilities
to
perform.
[...]
: |: A lot of the discussion I've stayed out of just because
there doesn't
seem
: |: to be all that much content in it. Let's please knock it
off with the
: |: massive cross-posting and deal more patiently with the
various topics
one
: |: at a time.
: |
: |Most of the lack of content has been from those spamming
their own
: |newsgroups, not asking for intelligent discussion, just
spamming with
: |insults and the like.
:
: I was talking about your postings, not theirs.
:
: Please don't stretch the term spamming beyond all sense.
Spamming
: implies wide distribution.
:
: It's often a convenient excuse by massive crossposters that
they're
: not to blame for the resulting flood of irritation.
Spamming of newsgroups implies a lack of topicality. That is
what I am
defending.
: |I am always eager to go into more depth as time
: |permits me, and I have been struggling to give myself more
and more time
as
: |the questions turn more and more to a technical nature.
: |
: |: If someone wants to chip in on the mathematical side of
constructivism,
: |: try helping me satisfy some of my curiosity. I've had the
question of
the
: |: degree to which the Jor-Holder theorem is constructive on
the back
: |: burner for a long time. It's easy to see that the fact
that any two
: |: decomposition series have a common refinement is
constructive. But
then
: |: given two decompositions with simple quotients, it's not
clear to me
that
: |: we should be able to get isomorphisms between them in
some order. We
can
: |: get a common refinement where not all the quotients are
nontrivial,
but
: |: we have no way in general to determine whether a quotient
group is
: |: trivial. On the other hand, I haven't thought of a good
counterexample,
: |: either.
: |
: |So are you questioning the second isomorphism theorem of
groups as not
being
: |constructive?
:
: The second isomorphism theorem says that if H and K are
subgroups of a
: group, with K normal, then HK/K is isomorphic to H/(H^K).
That's the one
: used in the theorem that any two decompositions (whether
with simple
: quotients or not) there's a common refinement. Those are
constructive.
: Given an element of HK/K, it's represented by an element of
HK, and the
: factor in H is a representative of the corresponding element
of H/(H^K).
: Converly an element in H representing a class in H/(H^K)
represents the
: class in HK/K that maps to it. There's no problem there.
:
: |Although I haven't explored this before, I never noticed
anything hiding
in
: |there that wasn't extendible to constructive definitions of
groups or
made
: |use of bivalence. I thought it was a simple application of
intersections
: |and joins, but now you have me intrigued. Would you like to
expand on
this?
:
: Take the special case of a group with two decomposition
series of length
: 2: suppose I have a group G with two simple normal subgroups
N1 and N2
: with simple quotients G/N1 and G/N2. I can write a common
refinement of
: the two decompositions either as 1<N1^N2<N1<N1N2<G or
1<N1^N2<N2<N1N2<G
: using the fact that N1N2/N1 is isomorphic to N1/(N1^N2) and
N1N2/N2 is
: isomorphic to N2/(N1^N2).
:
: Classically, one says that two of these quotients are
trivial; either
: N1=N2 or N1^N2=1. In the latter case, G/N1 is isomorphic to
N2 and G/N2
: is isomorphic to N1, and G is a product of N1 and N2.
However, I don't
: know whether there's a method of determining which is the
case.
:
: Finite examples don't help, since it's always possible in
principle
: to determine whether finite groups are isomorphic or not. I
need simple
: infinite groups that are quotients of subgroups of a common
group, and
: are hard to distinguish up to isomorphism.
:
: Remember the strength of the definitions I have. I'm
assuming a fair bit
: of constructive content in my premises.
So basically it comes to the question of whether one can solve
the
isomorphism problem for particular groups being studied? For
example, if
we
have a finite presentation for both infinite simple groups,
then something
like the elementary Tietze transformations can be used to
investigate
whether there is an equivalent presentation for both groups.
That could be
used to prove isomorphism if you were lucky enough to be given
finite
presentations, but not disprove it. At least it gives a
preliminary result
on one class of infinite simple groups, though, and that is
always
progress.
I do see where the difficulty lies here. I doubt that the fact
that
simplicity is a Markov property is gonna interfere, but you
still need a
method for the isomorphism problem. The alternate procedure,
proving that
two groups are not isomorphic (and thus constructively
deciding for the
other alternative isomorphism or set of isomorphisms if the
dichotomy is
established constructively), may be easier in specific cases,
but then you
have to rely on specific structure information and couldn't
build a general
procedure.
: |: I'm interested in a strong version of the concept of
simple group
(with
: |: apartness). I'm willing to assume that simplicity holds
for the
quotient
: |: groups in the following form. If x and y are elements of
the group,
and
: |: x<>1, then y can be expressed as a product of conjugates
of x and
x^{-1}.
: |: I think that's a pretty strong assumption, but not crazy.
I'm not sure
: |: for instance whether it holds (constructively, of course)
for the
: |: classical simple Lie groups.
: |
: |Usually, I find it is more natural to approach problems
like this in the
: |opposite direction when looking for constructive deductions.
:
: After reading the following, I wonder in what sense you
think this is
: opposite to my approach. A sort of generators-and-relations
approach
: to constructing examples is often good. I think it makes
sense also to
: try examples based on infinite alternating groups and the
like.
I think I made a mistake when I first replied. Rereading this
certainly
does give me the impression that I just recapitulated what you
had written,
not giving an opposite view in any way.
[...]
: |I think this approach is very basic to the logic that I
would desire
being
: |taught more. This is the computational approach so inherent
to
: |construction, that you build the structures you desire to
study through
: |finitely axiomatising the definitions and deduce
constructive
consequences,
: |which does oppose the infinite axiomatics underlying certain
classical
: |constructions. It is very much the difference between
bottom-up and
: |top-down approaches.
:
: I'm not convinced there's a serious contrast. I think
serious researchers
: in these areas do combinations of what you would call top
down and bottom
: up.
[...]
The contrast I am trying to point out is the difference in
proof methods
that are constructive versus those that are nonconstructive.
Classically,
you have both options so I agree that serious researchers
often use both.
I
just believe they should be focused on separately in
education, because
they
really seem to involve different methodologies. I sincerely
believe that
separating and focusing on direct and indirect proofs is very
important to
delineating, and clarifying, approaches in the toolkit of the
practicing
mathematician.
===-=-=-=-=-
===
Subject: Re: Number Theory Problem!
En el mensaje:aa8e2a84.0402290627.462b3c2e@posting.google.com,
Mark Sapir <m..v..s@mail.ru> escribi.97:
> 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 + 1/3n^3 +
7/15n
>> That's nice! Is it true that every polynomial with rational
>> coefficients f(n) such that f(n) is an integer for every
interger n
>> is a linear combination of binomial coefficients C(n,k)?
>> Mark Sapir
>Yes
> For those who (like me) wanted a reference here it is
(copied from
http://www.math.tamu.edu/~harold.boas/courses/math696/
Maple-functions.html):
> G. P.97lya proved [.86ber ganzwertige ganze Funktionen,
Rend. Circ.
Mat.
> Palermo 40 (1915) 1-16] that the polynomials that take
integer values
> at all integer points are precisely the linear combinations
with
> integral coefficients of the binomial polynomials of the form
> x(x-1)...(x-k+1)/k!, where k=0, 1, .... See Manjul Bhargava,
The
> factorial function and generalizations, American
Mathematical Monthly
> 107 (2000), number 9 (November), 783-799.
But it is a consequence of the finite difference formula, no?
===
Subject: Re: cantor's theorem
> Here is an interesting argument that Cantor's proof that
there are
> uncountably infinite reals is wrong. I found it pointed to on
> crank.net of all places, but I could not find any hole in the
> reasoning.
> http://maxpages.com/cantoriswrong
> Perhaps someone else can, if there is any hole. Basically it
is saying
> that the real number constructed out of the diagonal of the
countably
> infinite list may still be in the list, even though there is
no
> element of the list that has the exact same digits as this
number.
> They note that that .01111111....=.10000000000..... to make
this
> conclusion.
Hint: there's little point in trying to understand a so-called
refutation of Cantor's proof until you've understood the proof
itself.
G.C.
===
Subject: Tableau height
I have a bced tableau with N nodes (i, j).
2
/
3 9
/ /
5 12 10
/ / /
7 14 13 11
/ /
8 16
Where 2 is the root (0,0)
and 16 is (4, 1).
The root is numbered (0, 0) and child nodes are numbered
(i + 1, j) and (i, j + 1).
The height of the tableau is:
max {i | (i, j) is a leaf }
Where a leaf is a childless node. That is, height is the
maximum value of i
in a leaf node.
The height of the attached tableau is thus 4.
I need to find a expression for the height of the tableau in
terms of N.
N seems to be related to the telescoping sum
1 + 2 + 3 + 4...h...h+1
where h = height.
I am stuck at this point.
Any suggestions would be gratefully received.
Christensen
===
Subject: Re: Pigeonhole Speed-Up
> Consider the following inference, based on the Pigeonhole
Principle:
> (i) There are at least m objects
> (ii) There are exactly n A's
> (iii) The f of each object is an A
> Therefore,
> (iv) There are two distinct objects a, b such that f(a) =
f(b).
<snip
> Problem 2 (Philosophical):
> We all know that PHP(m,n) is true, if m>n. The problem is
*how* we know
that
> it's always true.
> In particular, consider an application of this. E.g.,
> (i) There are at least one ion people
> (ii) There are exactly 180 countries
> (iii) Each person is in exactly one country,
> So,
> (iv) There are at least two distinct people in the same
country.
> Presumably everyone agrees that (iv) follows logically from
(i),(ii),(iii).
> But in order to justify this belief, one must appeal to
genuine
mathematics,
> as one cannot actually perform the maths-free computation
feasibly. If
the
> speed-up of such proofs goes as fast as I suspect it does,
then it's
> possible that a first-order proof of (iv) from
(i),(ii),(iii) may have
more
> However, informal mathematical reasoning, about a *set* of
people, and a
> *set* of countries, and their cardinal numbers, and a
*function* from
people
> to countries, provides a quick proof of (iv) from
(i),(ii),(iii). In
other
> words, there seems to be a problem for a very strict
formalist about
> mathematics: they cannot accept the above reasoning, since
it directly
> appeals to numbers, sets and functions.
> --- Jeff
I'm not sure if a formalist would accept this reasoning -
can't be
sure since I'm not a formalist! Anyway, I think the formalist
could
say that, even though the proof might be too big for the
universe, we
are still capable of reasoning that there *is* a proof and how
the
proof would go. This would pose the question why the formalist
prefers reasoning about proofs more than numbers, but this is
already
a standard problem, and presumably he has an answer ready.
===
Subject: Re: y = x^x
> David, I saw the reference you provided after I posted the
Mathcad
> results for negative x. Mathcad assumed rational x.
That seems strange to me, but then I don't know Mathcad. (But
assuming
that x is rational _does_ sound like the sort of thing a cad
would do! ;-)
> Do you think that
> plot was correct for rational x or was it erroneously
combining the two
> separate curves shown in your reference into a single
oscillating curve?
If it shows a smoothly oscillating curve, then it's wrong. For
negative x,
the graph _jumps_ back and forth -- and so certainly is not
smooth --
between |x|^x and its negative. See the explanation by Robert
Israel.
GrafEq is the only program known to me which gives the graph
correctly for
negative x. For other programs, I suggest that, for x < 0, you
graph both
|x|^x and -|x|^x, and then visualize how the graph of x^x
jumps back and
forth between them, depending on whether x is of the form
even/odd or
odd/odd.
David
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
> Look at
<http://www.peda.com/grafeq/gallery/rogue/xx_exponential.html>.
===
Subject: How to solve complexvalue optimization in MATEMATICA ?
Hi all,I wonder if somebody would like to help me with an
optimization
problem in Matematica.
It is an electronics problem where I have
so called S-parameters(S11,S12,S21,S22) which are
characterizing an
amplifier system (transistor circuit) and are complex valued.
I want to maximize a function
Gt=
[(1-|R_s|^2)/|1-R_in*R_s|^2]*S21*[(1-|R_l|^2)/|1-S22*R_l|^2]
Constraints: R_in= S11 + S12*S21*R_l/(1-S22*R_l) < 1
|R_l| < 1 AND outside circle with radius r_l at point
C_l
|R_s| < 1 AND outside circle with radius r_s at point
C_s
r_l and r_s are real valued functions of the S-parameters.
C_l and C_s are complex valued functions of the S-parameters.
Hoping for some help...
Kindest regards,
Lasse Karagiannis
===
Subject: Re: Finding out base of a number
> | Vikram Hegde asked:
> | I need help with the following problem:
> |
> | 749 in base 11 equals 279 in base b. What is base b?
> If you meant:
> 749 in base 11 equals 297 in base b...
> then b would be nineteen. ________________________Gerard S.
Yeah, I made a mistake. I mean 297, not 279.
===
Subject: Re: Galois group
> I have to determine the structure and the elements of a
Galois group
> relatively to the splitting field of the polynomial
f(x)=x^3-2. This
> polinomial is in Q[x] so I have to find an extension. The
roots of f(x)
are
> 2^(1/3)
> 2^(1/3)(-1/2 + i (1/2)3^(1/2))
> 2^(1/3)(-1/2 - i (1/2)3^(1/2)),
> so the splitting field should be K=Q(2^(1/3),i 3^(1/2)) and
> |Q(2^(1/3),i 3^(1/2)):Q(2^(1/3))|*| Q(2^(1/3)):Q| = 6.
> Then |Gal(f(x)/Q)|=6 too.
> My question is:
> which are the automorphisms of the Galois group? (Show me the
automorphisms
> and _how they work_ if you could).
> TIA
It's easier to let a = 2^(1/3) and r = (-1+3^(1/2))/2 and
think of the
splitting field as being generated by a and r, i.e., K=Q(a,r).
The
Galois group is the symmetric group on three letters, given by
all
possible ways of permuting the three roots of your polynomial.
The
Galois group is generated by the automorphisms s and t defined
by:
s(a) = ra, s(r) = r
t(a) = a, t(r) = r^2
Then s^3 = e, t^2 = e (since r^3=1), and it's easy enough to
check
that
st = ts^2.
Thus:
(ts^2)(a) = t(s(s(a))) = t(s(ra)) = t(r^2a) = ra
(st)(a) = s(t(a)) = s(a) = ra
(ts^2)(r) = t(s(s(r))) = t(s(r)) = t(r) = r^2
(st)(r) = s(t(r)) = s(r^2) = r^2
The group generated by s and t with relations s^3 = t^2 = e and
st=ts^2 is the symmetric group S_3.
Joe Silverman
===
Subject: Re: y = x^x
> I've looked into the function y=x^x and am a bit confused
about its
> nature... with x>0 everything is predictable, but with x<=0
I am not
> sure what its graph would look like.
> I realize that many negative x values will produce imaginary
numbers,
> but what does the rest of the graph look like? Does it just
approach
> y=0? And how exactly are arbitrary powers calculated like
that?
> I believe it looks like a U shape as a negative times a
negative is a
> positive number. Hope this helps.
I don't think so.
===
Subject: Difficult Analysis Problem
I am facing the following strange problem :
Given f : I -> R , with the property that f composed with f so
f *f
and f*f*f are infinity many times differentiable, is f
infinitely time
differentiable or not?
Note: * means composed
If not, can you give me a counter example?
===
Subject: Re: Pigeonhole Speed-Up
>Consider the following inference, based on the Pigeonhole
Principle:
> (i) There are at least m objects
> (ii) There are exactly n A's
> (iii) The f of each object is an A
>Therefore,
> (iv) There are two distinct objects a, b such that f(a) =
f(b).
><snip
>Problem 2 (Philosophical):
>We all know that PHP(m,n) is true, if m>n. The problem is
*how* we know
that
>it's always true.
>In particular, consider an application of this. E.g.,
> (i) There are at least one ion people
> (ii) There are exactly 180 countries
> (iii) Each person is in exactly one country,
>So,
> (iv) There are at least two distinct people in the same
country.
>Presumably everyone agrees that (iv) follows logically from
(i),(ii),(iii).
>But in order to justify this belief, one must appeal to
genuine
mathematics,
>as one cannot actually perform the maths-free computation
feasibly. If
the
>speed-up of such proofs goes as fast as I suspect it does,
then it's
>possible that a first-order proof of (iv) from (i),(ii),(iii)
may have
more
>However, informal mathematical reasoning, about a *set* of
people, and a
>*set* of countries, and their cardinal numbers, and a
*function* from
people
>to countries, provides a quick proof of (iv) from
(i),(ii),(iii). In
other
>words, there seems to be a problem for a very strict
formalist about
>mathematics: they cannot accept the above reasoning, since it
directly
>appeals to numbers, sets and functions.
>--- Jeff
>I'm not sure if a formalist would accept this reasoning -
can't be
>sure since I'm not a formalist! Anyway, I think the formalist
could
>say that, even though the proof might be too big for the
universe, we
>are still capable of reasoning that there *is* a proof and
how the
>proof would go.
In a sense, this proof must be an abstract entity of some
sort. This
appears
to violate the basic idea behind very strict formalism. It
couldn't be a
mental entity, since clearly we cannot construct a proof
larger than the
physical universe. So, we appear to be forced to admit
arbitrarily large
finite structures (proofs, etc.) into our ontology, although
as abstract
entities. If the formalist says they're happy with this, then
good.
> This would pose the question why the formalist
>prefers reasoning about proofs more than numbers, but this is
already
>a standard problem, and presumably he has an answer ready.
The technical problem is related to a number of central issues
in
complexity
theory.
The philosophical problem occurs only for a very strict kind
of formalist
or
nominalist who doesn't accept the arbitrarily large finite,
but thinks that
the natural numbers either don't exist period, or that small
finite numbers
exist because of concrete physical exemplifications, but
somehow run out
because there aren't enough actual collections of physical
things.
In a sense, the intended structure for syntax is isomorphic to
the
arithmetic structure (as exploited in Goedel's incompleteness
theorems), so
talking about expressions, arbitrary concatenations thereof,
and arbitrary
substitutions but rejecting talk of arbitrarily large finite
numbers, with
addition and multiplication, doesn't make coherent sense.
A reasonable person who understands the Pigeonhole Principle
will accept
that the formula PHP(m, n) is valid, if m > n. So, the
people-countries
example PHP(10^9, 180) is valid, even though a direct
verification is
impossible---the proof would be fantastically huge.
Nonetheless, its
ordinary mathematical proof is straightforward.(Actually, if
one insists
upon expanding out the cardinality statement There are at
least 10^9
Fs,
the formula PHP(10^9, 180) cannot even be written down. I
calculate that
cardinality statements have O(n^3) number of symbols, so it
has O(10^{27})
symbols. So, quite aside from verifying its validity, one
cannot in
practice
even express it.)
Although we can see that (iv) is a logical consequence of
(i),(ii),(iii)
but
it seems that we don't see this fact by logical
inference---e.g., by
applying modus ponens and various logical rules of inference
zillions of
times. Rather our more abstract (combinatorial) mathematical
understanding
of finite sets, their cardinalities and functions between them
tells us
that
(iv) is a logical consequence of (i),(ii) and (iii), and such
understanding
is incompatible with strict finitism, etc.
--- Jeff
===
Subject: Re: Science Without Math? (model-free common sense
steering)
> Neural networks are model-free estimators, in that they do
not
require
> an
> in-depth understanding of the phenomena they are modeling.
> http://www.arcon.com/arconneu.html
> THE MATHEMATICS OF CROSSING THE STREET:
> You are at the curb deciding, Should I
> cross the street? Well, it depends.
> AT THE CURB
> The walk light is on, but you see a
> truck approaching fast. How fast?
> There is no exact number. Instead, there are an infinite
number of
> possibilities - from 1kph to over 100kph and everything in
between. You
> don't have a radar gun, so instead you watch the truck for a
second or
> two,
> and sum its speed up in two words very fast. That is good
enough.
> Your senses have told you the truck is coming very fast, but
you need
more
> information before you can decide whether or not to risk
crossing.
How
> far
> down the street is the truck? Is it slowing down? Again,
there are
no
> exact numbers, so you sum up the situation - close, not
slowing
quickly
> enough
> Somehow your brain adds fast + close + not slowing quickly
enough,
> and
> warns you instantly that the risk is high. It is purely
cognitive
> process.
> It involves a complex combination of sensory information and
experience.
> ...Since there are no exact numbers in this story, the
mathematical
> version
> must be told with fuzzy numbers...
> But, the process is still not quite over. Should I wait or
cross?
You
> have
> to make the decision. Risk tolerance leads to different
spins and
endings.
> If you walk with a cane, you reason, The risk is high, so
I'll
wait.
You
> watch as the truck runs the red light. If you are a jogger,
impatient
to
> cross, you disregard the evidence, step into the
intersection, and jump
> back
> just in time to save your life.
> http://www.decyde.com/crossingthestreet.html
> Fuzzy logic works the way that humans think as opposed to
the way that
> computers typically work. For example, consider the task of
driving a
car.
> You notice that the stoplight ahead is
> red and the car ahead is braking. Your
> mind might go through the thought process,
> I see that I need to stop. The roads are
> wet because it's raining and there is a
> car only a short distance in front of me.
> Therefore I need to apply a significant
> pressure on the brake pedal.
> This is all subconscious (in general), but that's the way we
think - in
> fuzzy terms. Do our brains compute the precise distance to
the car
ahead
> of
> us and the exact coefficient of friction between our tires
and the
road,
> and
> then use a Kalman filter to derive the optimal pressure
which should be
> applied to the brakes? Of course not. We use common-sense
rules and
they
> seem to work pretty well. On the other hand, when we do
finally get
around
> to pressing the brake pedal there is some exact force that
we apply,
say
> 1.326 pounds. So although we think in fuzzy, noncrisp ways,
our final
> actions are crisp. The process of translating the results of
fuzzy
> reasoning
> to a crisp, nonfuzzy action is called defuzzification.
> http://www.innovatia.com/software/papers/fuzzy.htm
> ...In particularly vast networks in fast moving
environments, the split
> second it takes to traverse the circuit is greater than the
time it
takes
> for the situation to change. In reaction, the last node
tends to
> compensate
> by ordering a large correction. But this also is delayed by
the long
> journey
> across many nodes, so that it arrives missing its moving
mark, birthing
> yet
> another gratuitous correction.
> The same effect causes student drivers
> to zigzag down the road, as each late
> large correction of the steering wheel
> overreacts to the last late overcorrection.
> Until the student driver learns to tighten
> the feedback loop to smaller, quicker
> corrections, he cannot help but swerve down
> the highway hunting (in vain) for the center.
> This then is the bane of the simple auto-circuit. It is
liable to
flutter
> or chatter, that is, to nervously oscillate from one
overreaction
to
> another, hunting for its rest. There are a thousand tricks
to defeat
this
> tendency of overcompensation, one trick each for the
thousand advance
> circuits that have been invented.
> http://www.kk.org/outofcontrol/ch7-c.html
> Fuzzy systems are based on
> storage of common-sense rules.
> For example, a fuzzy Army-ant robot controller might have
the fuzzy
> association if load is heavy, then signal for help longer.
Fuzzy
> phenomena
> admit degrees: some loads are heavier than others; some
signal
durations
> are
> longer then others.
> A single association (heavy,longer)
> encodes all combinations...
> Fuzzy systems reason with
> parallel associative inference.
> A fuzzy system reasons with multivalued sets, instead of
true or false
> propositions, and it may adaptively modify its fuzzy
associations from
> representative numerical samples.
> http://www-2.cs.cmu.edu/~unsal/thesis/thesisch2.html
> Wired: What is fuzzy logic and why do critics call it the
cocaine of
> science?
> Kosko: Fuzzy logic is Spock's worst nightmare - a way of
doing science
> without math. It's a new branch of machine intelligence that
tries to
make
> computers think the way people think and not the other way
around. You
> don't
> write equations for how to wash clothes. Instead you load a
chip with
> vague
> rules like if the wash water is dirty, add more soap, and if
very
> dirty,
> add a lot more. All wash water is dirty and not dirty - to
some
degree.
> It's just common sense. But it breaks the old either/or
logic of
> Aristotle.
> That offends some scientists, who would like us to think and
talk like
> off/on switches. But they still haven't produced a statement
of fact
like
the sky is blue or E=mc^2 that is 100 percent true or 100
percent
> false.
> Fact ain't math. You can never get the science right to more
than a few
> decimal places. That's one reason we find chaos when we look
at things
up
> close...
> ...Fuzzy systems are universal computers. I proved that as a
theorem -
the
> fuzzy approximation theorem. In theory, you can replace
every book on
> physics or economics with equivalent books that have fuzzy
systems
where
> the
> equations used to be. Fuzzy systems are model-free
estimators. You
don't
> have to guess at equations to build a bridge from inputs to
outputs.
Fuzzy
> rules build that bridge for you. There is math behind the
rules, but
you
> don't need to know it to program a fuzzy system. You can
program it in
> English. If the air is cool, turn the AC down a little. But
the
math
is
> not fuzzy. That's why you can capture fuzzy logic in a
digital chip.
> Most of the first fuzzy systems were in control - as in
adjusting a
camera
> lens or backing up a trailer truck to a loading dock. Now
we're
applying
> fuzzy systems to wireless communications and multimedia. The
fuzzy
rules
> can
randomly spread signals over a wide bandwidth or teach an
intelligent
> agent the kind of houses or sunsets you prefer. The math
says we can
apply
> them anywhere. In practice, it may not be so easy.
> http://www.wired.com/wired/archive/3.02/kosko_pr.html
> Fuzzy logic is a superset of conventional(Boolean) logic
that has been
> extended to handle the concept of partial truth- truth
values between
completely true and completely false. As its name suggests, it
is
the
> logic underlying modes of reasoning which are approximate
rather than
> exact.
> The importance of fuzzy logic derives
> from the fact that most modes of human
> reasoning and especially common_sense
> reasoning are approximate in nature.
> Boolean vs. Fuzzy: 300 years B.C., the Greek philosopher,
Aristotle
came
> up
> with binary logic(0,1), which is now the principle
foundation of
> Mathematics. It came down to one law: A or not-A, either
this or not
this.
> For example, a typical rose is either red or not red. It
cannot be red
and
> not red. Every statement or sentence is true or false or has
the truth
> value
> 1 or 0. This is Aristotle's law of bivalence and was
philosophically
> correct
> for over two thousand years.
> Two centuries before Aristotle, Buddha, had the belief which
contradicted
> the black-and-white world of worlds, which went beyond the
bivalent
cocoon
> and see the world as it is, filled with contradictions, with
things and
> not
> things. He stated that a rose, could be to a certain degree
completely
> red,
> but at the same time could also be at a certain degree not
red. Meaning
> that
> it can be red and not red at the same time.
> Conventional(Boolean) logic states that a glass can be full
or not full
of
> water. However, suppose one were to fill the glass only
halfway. Then
the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
> multivalence
> is the fundamental concept which propelled Zader Lofti of
University
> Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics
of
> fuzzy logic founded by him are as follows.
> In 1965, Lofti Zadeh formally developed multivalued set
theory, and
> introduced the term fuzzy into the technical literature.
Nowadays, the
> recent emergence of fuzzy commercial products, as well as
new theory,
has
> generated a new interest in multivalued systems. Yet already
engineers
> have
> successfully applied fuzzy systems in many commercial areas :
intelligent
> subways automation, emergency breakers, cement mixers, Kanji
characters
> recognition, control air conditioners, automatic washing
machines,
guide
> of
> robot-arm manipulators, and so on.
> Fuzzy systems store banks of fuzzy associations or
common-sense
rules
> such
> as IF traffic is heavy in this direction, THEN keep the
light green
> longer
> that might be articulated by an human expert. Some traffic
configuration
> are
> heavier that others and some green-light duration are longer
than
others,
> so
> that, the single fuzzy association (HEAVY, LONGER) encodes
all these
> combinations. That is to say, fuzzy systems directly encode
structured
> knowledge but in a numerical framework : by entering the
fuzzy
association
> (HEAVY, LONGER) as a single entry in a rule database we are
defining an
> input-output transformation.
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
> Fuzzy Logic is a computational paradigm capable of modelling
the own
> uncertainness of human beings. Fuzzy reasoning is nothing
else than a
> Fuzzy
> Logic-based formalism for encoding human knowledge or common
sense in a
> numerical framework. Indeed, the mathematical concepts on
which Fuzzy
> Logic
> is supported are very easy to understand. In a Fuzzy
Controller, human
> experience is codified by means of linguistic if-then rules,
which
compute
> control actions upon given conditions. Fuzzy Logic has been
applied to
> problems that are difficult to solve mathematically. One of
its main
> advantages lies in the fact that it offers a straightforward
methodology
> for
> modelling and controlling non-linear systems, which are
difficult to
face
> by
> means of conventional techniques.
> http://www.wkap.nl/prod/b/1-4020-7359-3
> Fuzzy logic models itself on the pattern of human reasoning
in its use
of
> approximate information and uncertainty to generate
decisions. It was
> designed (during late 1980s and early 1990s) to
mathematically
represent
> vagueness and develop tools for dealing with imprecision
inherent in
> several
> problems. Normally, in digital computers one uses the binary
logic
where
> the digital signal has two discrete levels : low (logic
zero) or high
> (logic
> one); nothing in-between. Fuzzy systems use soft linguistic
variables
> (e.g.
> hot, tall, slow, light, heavy, dry, small, positive,
...etc.) and a
range
> of
> their weightage (or truth) values, called membership
functions, in the
> interval (0, 1), enabling the new computers to make
human-like
decisions.
> Since human beings tend to use words rather than numbers to
describe
> behaviour patterns, fuzzy controls avoid the conventional
rigidity of
> computers and allow them to use parameters based on common
sense.
> http://www.tribuneindia.com/2002/20021024/science.htm
> Fuzzy logic best summed up by common sense
> Computer Corner
> John Boyd
> Fuzzy logic was introduced to the world 27 years ago by
Professor
> Lotfi Zadeh in his Fuzzy Sets paper published in Information
> Control magazine, though it is only recently that we've seen
it
> applied across a broad range of products.
> Some readers have asked for more explanation on fuzzy logic,
so
> here's an attempt to defuzzify the subject a little further.
> Simply put, fuzzy logic is aimed at enhancing our prissy
computer
> technology with a touch of common sense.
> One problem with the conventional digital computer is that
it is
> such a scrupulously either-or beast. It cannot be easily
coaxed
> to handle approximations or vague notions like young, a lot
and
> probably.
> Yet most of us rely on such terms daily because we happen to
be
> humans dealing with other humans, not robots building cars.
> It's an easy matter to arbitrarily program a computer so it
> designates everyone falling into the age-range 0f 15 to 18
> as being a youth. Such a precise category has come to be
called
> a crisp set since the emergence of fuzzy logic.
> Yet we all know some 14-year-olds can look older than some
> late-developers turning 20. Such exceptions, however, cannot
> be accounted for in conventional computing. Or at least not
> without an inordinate amount of additional programming and
> expense.
> As Tetsuya Yamada, a senior engineer at Hitachi Ltd., replied
> when I asked him if we couldn't just continue using
conventional
> programming and technology for controlling new products,
instead
> of fuzzy, Well, we could. And you could probably swim across
the
> Pacific if you got enough support from enough people. But ...
> To overcome this problem, Zadeh was inspired to develop his
fuzzy
> theory and the math to go with it that could be used to
create
> fuzzy sets based on imprecise natural language.
> Each member in a fuzzy set (such as the youths and others
considered
> in the above example) is assigned one of a continuous range
of values
> (called the membership value) between zero and one.
> Whereas in the above crisp set a 13-year-old going on 14
would still
> have to be considered a minor and thus be designated as zero
in
> binary logic, fuzzy logic could assign him a membership
value of
> say 0.1. Likewise, an immature 20-year-old who would
normally fall
> outside our either-or crisp-set range could be assigned a
membership
> value of 0.9 depending upon the criteria we use to measure
youth.
> Working out just what criteria to use, what values should be
assigned
> each member and deciding what rules are necessary to govern
the
> relationships between members is the key to successfully
applying
> fuzzy control in products.
> In some applications, determining the optimum rules has
become so
> complex, some manufacturers have resorted to employing the
aid of
> neural networks, which may be stretching a good thing too
far, given
> fuzzy logic's original purpose to get round complexity.
> Still, the flexibility in herent in fuzzy is clearly useful
in
> dealing with approximate calculations, such as about 100
> It can be used in artificial intelligence to provide us with
an
almost true answer. It can also infer a common-sense result
even
> when the data is not precise.
> Our handwritten 5 in 250 would be treated as 5, not the
letter
S,
> for instance, in Sony's fuzzy-based Palmtop computer.
> While we have all seen fuzzy logic-based products from the
likes of
> Matsua, Sanyo and Hitachi, one unlikely company that has made
> fuzzy technology a central part of its business strategy is
Omron
> Corp.
> It began its research into fuzzy logic in 1984 and has since
applied
> for over 700 patents. This puts it in the forefront of fuzzy
> applications in areas like factory and industry control, as
well as
> in medical equipment.
> In 1989, Omron also signed on lotfi Zadeh as a senior
advisor.
> Earlier this year at the Business Show in Harumi, Omron
demonstrated
> its fuzzy workstation. Omron manufactures both standard
Motorola
> 68040-based and 88000 reduced-instruction or RISC-based
workstations
> that can be fitted with a fuzzy inference board, turning
them into
> the world's first fuzzy workstations.
> Omron claims such a RISC-based workstation can achieve 4 ion
> operations per second, an incredible speed if they haven't
fuzzed
> on the number. Fuzzy logic is used in the workstations to
store
> and retrieve fuzzy information and make inferences.
> Ranging in price from Y2.5 million to almost Y4 million (a
US dollar
> is about 120 Yens -FM), these machines are not the kind of
products
> you will find down in Akihabara. (a section of Tokyo famous
for its
> quantity and variety of electronic goods -FM) Rather, they
are
> typically aimed at value-added resellers in niche markets,
and
> engineers who want to develop fuzzy applications, fuzzy
databases
> and expert systems, as well as fuzzy inference systems.
> However, the entrepreneurs among you may be interested in
Omron's
> FB-30AT fuzzy inference board for the IBM PC and compatible
wares.
> It features a 24 MHz FP-3000 fuzzy chip capable of
processing up
> to 128 rules, with five antecedents and 2 consequents.
Training
> software and a compiler is also available.
> Omron has also produced a fine little booklet on fuzzy called
Clearly Fuzzy that I dipped into when writing this column.
> Tadashi Katsuno, at Omron's public relations section, tells
me
> he still has a limited number of copies left that he will
send
> to the first readers of Computer Corner who write to him with
> contact information.
> The address is Omron Corp., International Public Relations
Section,
> Omron Tokyo Bld., 3-4-10 Toranomon, Minato Ward, Tokyo 105.
>
--------------------------------------------------------------
------
> - Farzin Mokhtarian
> farzin@apollo3.ntt.jp
http://www-cgi.cs.cmu.edu/afs/cs/project/ai-repository/ai/
areas/fuzzy/doc/in
> tro/j_times.tgz
> http://www.ece.utep.edu/research/webfuzzy/about.html
http://www.sztaki.hu/~viharos/homepage/Publications/1999_ICIMS
_NOE_ASI99/ASI
> '99_ViharosMonostori.htm
> http://www.bjarne.ca/pmflp.pdf
> http://www.etse.urv.es/~aoller/fuzzy/fuzzy_logic.htm
>
http://www-pablo.cs.uiuc.edu/Project/PPFS/PPFSII/
FuzzyLogicControl.htm
> All unconscious processes.
Right, but which maths will survive and which will die after
the cerebral
code is figured out?
===
Subject: Catenary curves question
I'm editing a puzzle book which asks:
What will happen to the shape of a catenary curve if you tie
and suspend
two weights at equal horizontal distances apart?
I can't find the answer anywhere on the Internet - does anyone
know?
David
===
Subject: Re: cantor's theorem
> Here is an interesting argument that Cantor's proof that
there are
> uncountably infinite reals is wrong. I found it pointed to on
> crank.net of all places, but I could not find any hole in the
> reasoning.
> http://maxpages.com/cantoriswrong
> Perhaps someone else can, if there is any hole. Basically it
is saying
> that the real number constructed out of the diagonal of the
countably
> infinite list may still be in the list, even though there is
no
> element of the list that has the exact same digits as this
number.
> They note that that .01111111....=.10000000000..... to make
this
> conclusion.
> This comes up every once in a while; it shows that a bad
choice
> of how to swap digits is possible, but it does NOT show that
there
> is no good way. In fact, the problem is easy to avoid, and
Cantor
> was careful enough to avoid it.
> An easy way is to swap (decimal) digits is as follows: for
numbers
> with two decimal representations (.x0000..., .y999...),
chose the one
> with the 9's. Then if the n-th digit of the n-th item is a
5, put a
> 4 in that place in the number under construction, else put a
5.
Well, I'm glad that there is an answer and we don't have to
reinvent
set theory. Thank you. Isn't it ironic that in Cantor's time,
he was
considered the renegade mathematician by many and now his
stuff is
well accpepted and anyone who opposes it is considered
renegade?
Craig
===
Subject: Re: Science Without Math? (model-free common sense
steering)
> Conventional(Boolean) logic states that a glass can be full
or not full
of
> water. However, suppose one were to fill the glass only
halfway. Then
the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
> multivalence
> is the fundamental concept which propelled Zader Lofti of
University
> Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics
of
> fuzzy logic founded by him are as follows.
> I don't buy this. A glass can be either [half-full] or
not-[half-full],
not
> both. half-not-full is not equivalent to not-[half-full].
The 'not'
> shouldn't be stuck in the middle of the statement, but
appended to the
front
> of it. ie: A glass with 50% water is half-not-full, but to
say its
> not-[half-full] is incorrect. I understand the concept of
non boolean
> logics, and the benefit of fuzzy systems in various
applications, but
this
> is a pretty bad example IMO. Especially the part about it
'clearly'
> disproving Aristotle's law of bivalence.
A = water
A = ~A in this 1/2 example
>
===
Subject: Re: Difficult Analysis Problem
>I am facing the following strange problem :
>Given f : I -> R , with the property that f composed with f
so f *f
>and f*f*f are infinity many times differentiable, is f
infinitely time
>differentiable or not?
>Note: * means composed
>If not, can you give me a counter example?
Counterexample:
Let f:R->R be defined by f(x)=1 if x<0 and f(x)=0 else.
Then f*f=f*f*f=...=0, but f is obviously not at all
differentiable.
HTH!
Benedikt Plitt
beplitt@math.uni-muenster.de
===
Subject: Re: Upside-Down Functions Have Some Remarkable
Properties
> An Upside-Down Function (UDF) is defined here as a function
which
> has a non-trivial multiplicative inverse, where non-trivial
means
> that the multiplicative inverse has a recognizable
mathematical use
> other than as an index comparing two of the same types of
quant-
> ities. A function may be UDF restricted to some domain or
range,
> and it may be UDF based on current mathematical knowledge,
and
> occurrence of something in an intermediate step of
computation is
> not regarded as a non-trivial use.
> For example, 1 - x is UDF because 1 + x + x^2 + ... = 1/(1 -
x)
> for /x/ < 1 as the geoemtric series. Similarly, exp(x) is UDF
> because exp(x)exp(-x) = exp(x)/exp(x) = 1 and exp(-x) is
> exponential decay with rate 1 in radioactivity. However,
> log(x) is not UDF, since its multiplicative inverse 1/(log x)
> doesn't have any recognizable non-trivial mathematical use.
Yes it does. Its integral
integral dx/log x
is used as an approximation to pi(x) in analytic number theory.
> B. The expression 1 - x is the special case of 1 + y - x
which
> crosses fuzzy multivalued logic (implication), probability-
> statistics (probable influence), and geometry and its
topology
> intersection (proximity functions), when y = 0. In other
words,
> in probable influence P(A-->B) = 1 + P(AB) - P(A), the
> probability P(AB) = 0, which typically means that A and B are
> disjoint up to probability zero.
We never did find out what A -> B meant did we :-(
> D. The fact that log(x) is not UDF raises the question of how
There's no fact here. It's your arbitrary decision, based on
your
own prejudices that this has no non-trivial use
> E. If we write 1 - x = 1/[1 + x + x^2 + ... ], for /x/ < 1,
and
> if we assign physical units of some sort to x (or biological
or
> psychological or whatever units), then if 1 is regarded as a
> dimensional constant in the same units, the right-hand-side
of
> this equation seems to be an upside-down unit scale object
> and also a generalization of a multiplicative inverse term in
> a Laurent (Complex) Series.
Like er, yeah man.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: cantor's theorem
> Well, I'm glad that there is an answer and we don't have to
reinvent
> set theory. Thank you. Isn't it ironic that in Cantor's
time, he was
> considered the renegade mathematician by many and now his
stuff is
> well accpepted and anyone who opposes it is considered
renegade?
Although those sci.maths posters who oppose Cantor may consider
themselves renegades and outlaws, the rest of us consider them
idiots and imbeciles.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Pigeonhole Speed-Up
> The philosophical problem occurs only for a very strict kind
of formalist
> or nominalist
Many thanks for that elegant and definitive refutation of
nominalism.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: confusion in a rule like Pascal's triangle
of Chemical Education (American Chemical Society), about a
rule like
Pascal's triangle for predicting the relative intensities of
nuclear
magnetic resonance spectra signals.The standard method is to
write the
set of nuclear spins(I) as a polynomial and then raise the
polynomial
to the number of equivalent nuclei splitting the signal, and
expanding
the multinomial. The coefficents in the expansion of
multinomial give
the intensities of the siganls.
multinomial ,one can easily generate a table, the construction
of
which is causing some confusion to me. The exact wordings of
the
author are One can generate the table by writing down a row of
1's,
2I+1 (I=nuclear spin) in number: additional rows are then
constructed
by taking the sum of of 2I+1 entries in the row above and to
the left
of the entry desired(non existent entry being assigned values
of
zero), until the row number is equal to equivalent nuclei , n,
in the
moeity under consideration.
The example of which is given as: If I= 2,
No. of nuclein I=2
1: 1 1 1 1
2: 1 2 3 4 5 4 3 2 1
3: 1 3 6 10 15 18 19 15 10 6 3 1
4: 1 4 10 20 35 52 68 80 85 80 68 52 35 20 10 4 1
The rule given by the author for constructing the table like
the one
given above seems to work upto a certain column, say for n=2,
where
the rule seems to follow till sixth column. How are we getting
4, 3,
2 and 1 in the second row? Same for the rest of the rows.
Somone in sci.chem suggested that I write the rows as a
symmetrical
pyramid, but still that is not clear because for n=3 and n=4
there 12
and 17 terms respectively.The author has not mentioned making
symmetrical pyramid, and has given the table in the same form
which I
have reproduced. I am still unable to construct a complete
table
like this one by the rules given by the author. Can somebody
point
out how the author has constructed the table or what point I am
missing?
===
Subject: Re: Galois group
JHS <jhsntru@yahoo.com> ha scritto nel messaggio
> I have to determine the structure and the elements of a
Galois group
> relatively to the splitting field of the polynomial
f(x)=x^3-2. This
> polinomial is in Q[x] so I have to find an extension. The
roots of f(x)
are
> 2^(1/3)
> 2^(1/3)(-1/2 + i (1/2)3^(1/2))
> 2^(1/3)(-1/2 - i (1/2)3^(1/2)),
> so the splitting field should be K=Q(2^(1/3),i 3^(1/2)) and
> |Q(2^(1/3),i 3^(1/2)):Q(2^(1/3))|*| Q(2^(1/3)):Q| = 6.
> Then |Gal(f(x)/Q)|=6 too.
> My question is:
> which are the automorphisms of the Galois group? (Show me the
automorphisms
> and _how they work_ if you could).
> TIA
> It's easier to let a = 2^(1/3) and r = (-1+3^(1/2))/2 and
think of the
> splitting field as being generated by a and r, i.e.,
K=Q(a,r). The
> Galois group is the symmetric group on three letters, given
by all
> possible ways of permuting the three roots of your
polynomial. The
> Galois group is generated by the automorphisms s and t
defined by:
> s(a) = ra, s(r) = r
> t(a) = a, t(r) = r^2
> Then s^3 = e, t^2 = e (since r^3=1), and it's easy enough to
check
> that
Why t^2=e?
t^2(r)=t(r^2)=r and t^2(a)=t(a)=a that is not e, isn't it?
===
Subject: Re: y = x^x
> I believe it looks like a U shape as a negative times a
negative is a
> positive number. Hope this helps.
> I don't think so.
Yes, spot the big mistake of the year. Lol, sorry. I thought
it was X*X for
some reason. It's like an S shape if I'm correct.
Easiest way to find out is get a bit of graph paper, plot a
conservative
scale and start putting numbers into x and start plotting them.
===
Subject: Re: confusion in a rule like Pascal's triangle
> No. of nuclein I=2
> 1: 1 1 1 1
Should this be 1 1 1 1 1 ?
> 2: 1 2 3 4 5 4 3 2 1
> 3: 1 3 6 10 15 18 19 15 10 6 3 1
Should this be 1 3 6 10 15 18 19 18 15 10 6 3 1 ?
> 4: 1 4 10 20 35 52 68 80 85 80 68 52 35 20 10 4 1
Modulo my corrections these are the coefficients of the
polynomials
(1 + x + x^2 + x^3 + x^4)^n for n = 1, 2, 3 and 4.
We get entry k in row n by adding entries k-4, k-3, k-2, k-1
and k
in row n-1 (if any of these are absent, suppose they are zero).
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Upside-Down Functions Have Some Remarkable
Properties
> An Upside-Down Function (UDF) is defined here as a function
which
> has a non-trivial multiplicative inverse, where non-trivial
means
> that the multiplicative inverse has a recognizable
mathematical use
> other than as an index comparing two of the same types of
quant-
> ities. A function may be UDF restricted to some domain or
range,
> and it may be UDF based on current mathematical knowledge,
and
> occurrence of something in an intermediate step of
computation is
> not regarded as a non-trivial use.
> For example, 1 - x is UDF because 1 + x + x^2 + ... = 1/(1 -
x)
> for /x/ < 1 as the geoemtric series. Similarly, exp(x) is UDF
> because exp(x)exp(-x) = exp(x)/exp(x) = 1 and exp(-x) is
> exponential decay with rate 1 in radioactivity. However,
> log(x) is not UDF, since its multiplicative inverse 1/(log x)
> doesn't have any recognizable non-trivial mathematical use.
Yes it does. Its integral
integral dx/log x
is used as an approximation to pi(x) in analytic number theory.
> B. The expression 1 - x is the special case of 1 + y - x
which
> crosses fuzzy multivalued logic (implication), probability-
> statistics (probable influence), and geometry and its
topology
> intersection (proximity functions), when y = 0. In other
words,
> in probable influence P(A-->B) = 1 + P(AB) - P(A), the
> probability P(AB) = 0, which typically means that A and B are
> disjoint up to probability zero.
We never did find out what A -> B meant did we :-(
> D. The fact that log(x) is not UDF raises the question of how
There's no fact here. It's your arbitrary decision, based on
your
own prejudices that this has no non-trivial use
> E. If we write 1 - x = 1/[1 + x + x^2 + ... ], for /x/ < 1,
and
> if we assign physical units of some sort to x (or biological
or
> psychological or whatever units), then if 1 is regarded as a
> dimensional constant in the same units, the right-hand-side
of
> this equation seems to be an upside-down unit scale object
> and also a generalization of a multiplicative inverse term in
> a Laurent (Complex) Series.
Like er, yeah man.
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
Lacan, Jacques, 79, 91-92; mistakes his penis for a square
root, 88-9
Francis Wheen, _How Mumbo-Jumbo Conquered the World_
===
Subject: Re: Hang them by their own G-strings.?
Nothing
Hey Jack-o, men wear T-straps or banana hammocks. Girls and
transvestites wear G-strings. Now you know twice as much as
you did
two sentences ago.
--
Uncle Al
http://www.mazepath.com/uncleal/qz.pdf
http://www.mazepath.com/uncleal/eotvos.htm
(Do something naughty to physics)
===
Subject: Re: cantor's theorem
> It won't work with the decimal number system, will it?
 ----------------------------------------------------------
> ** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **
> ----------------------------------------------------------
>
Sure. If the nth digit of the nth number is 2 use 3, otherwise
use 2.
===
Subject: multiplication and exponentiation with
Doubly-Infinites to P-adics
Re: making geometrical sense of e^i(pi) = -1 Re: e^i(pi) = -1
revisited
(snipped)
> Back in 1994, Dominique Bernardi, Theorie des Nombres
Universite
> Pierre et Marie
> Curie 4 place Jussieu - F75005 Paris dob@ccr.jussieu.fr,
> bernardi@mathp7.jussieu.fr was kind enough to post the
5-adic i and i.
> I am hoping by luck that I can get the equation e^i(pi) = -1
the
> easiest means possible where I end up with ......333333 x
......3 [sic]
> which of course is
> .....999999 in 10-adics and which resembles -1.
> So I need to look at pi and e to many place values and where
I replace
> i with one of these two 5-adics. I am hoping that I need not
adjust
> the 5-adics i because 10-adics is based on 2,5 prime.
> So if I take pi to say some 20 digits and multiply by i,
what number
> results?
> Then this result, call it x, is the exponent of e. And if I
take e to
> 20 digits
> with exponent x, the question I want is does it come out to a
> beautiful:
> .....999999999
> 33231021412240312040104032030331303024331122040204
> 13204114144413133414410311104224243403300234244140
> 00424243131240230101323111334132240141323322314123
> 21413141441403332212002214433021104311210431204321
> 11434140213402312410420004234221031320231214002203
> 21333413042344124233211100442012420011310044412431
> 22031433201410124424000213333432410434233221123404
> 42143230100410420334203424100032234444224314211134
> 30043114414204130142242310240033430142334143134044
> 34124000314134442112203220440401423331244432340112
> 30222012001411114001311402311204222201440332220204
> 03441013402041421431114122232231314404431040203313
> 42342124033202411203314310333210434302444430231004
> 41341230110400344411141422114221231410120410221024
> 30333101301443120201230101434024414211021132214302
> 21211203143222020224422142231041301241413141242020
> 04212233022222432322013342021301231201224210143322
> 21213423032204132404340412414113141420113322404240
> 44020201313204214343121333110312204303121122130321
> 23031303003041112232442230011423340133234013240123
> 33010304231042132034024440210223413124213230442241
> 23111031402100320211233344002432024433134400032013
> 22413011243034320020444231111012034010211223321040
> 02301214344034024110241020344412210000220130233310
> 14401330030240314302202134204411014302110301310400
> 10320444130310002332241224004043021113200012104332
> 14222432443033330443133042133240222243004112224240
> 41003431042403023013330322212213130040013404241131
> 02102320411242033241130134111234010142000014213440
> 03103214334044100033303022330223213034324034223420
> 40021240434020423422001224022343112442244124411030
> 23233241301222424220022302213403143203031303202424
> 40232211422222012122431102423143213243220234301122
> 13420133102441441142042231342412101314412021132344
I made a mistake by saying .....3333333 X .....333333 is
.....99999
when I am hoping to easily find .....333333 X ....000003 =
....9999
Which brings up the question of how is multiplication defined
between
P-adics and Doubly-Infinites. Here I am hoping again that the
equation
e^i(pi)=-1 will do all the work and show us how multiplication
and
exponentiation is defined instead of humans having to define
it. I
doubt that there is any other mathematical equation up to the
task of
defining multiplication and exponentiation between P-adics and
Doubly
Infinites.
pi is 3.14159265.... if my memory is correct
e = 2.71828... if memory is correct
But I contend these are not REals because the Reals are a fake
set
especially going out long distance in the Reals. So pi and e
are
Doubly-Infinites.
pi = .......00003.14159265.......
e = ........00002.71828.........
5-adic i = .........303243121.2
5-adic i = .........30422141201323.3
I said that P-adics are all positive and that Doubly-Infinites
are all
negative because of their infinite string rightwards serves as
an
Orthogonality so that P-adics are curved in space with convex
curvature and Doubly-Infinites are concave bent in space.
I am hoping that the equation e^i(pi)=-1 defines naturally what
multiplication and exponentiation between p-adics and
doubly-infinites
should be.
Since I do not see it naturally as of this moment I can only
make a
stab at it.
I would think that multiplying i by pi of:
........30422141201323.3 X .......00003.14159265....... =
...3969.914159....
I am thinking that the infinite rightward string in pi is
untouched or
can be untouched and that the only significant portion to
multiply is
the leftward portion of a Doubly-Infinite because the rightward
portion is mostly orthogonality. I am thinking that whenever a
p-adic
is multiplied with a Doubly-Infinite that the endresult is
always a
Doubly-Infinite but I could be wrong on that presumption. Much
of this
is presumption because this whole field of knowledge has never
been
explored and I am the first other than perhaps Karl Heuer who
made a
remark circa 1994 I wonder what these are (Doubly-Infinites)?
And none of this is easy, for if it were easy then I am rather
certain
that in the 20th century a few mathematicians, even perhaps
Kurt
Hensel would have stumbled onto the Euler Equation e^i(pi)=-1
where a
doubly-infinites and p-adics solve that algebraic equation. I
still
hope that some of this exploration will come easily instead of
every
step of the way massively hard and difficult.
What I am hoping for now is to get the Euler equation
e^i(pi)=-1 with
a e and pi
of doubly-infinites and a i and ...999999. as p-adics to where
the
side of the equation: e^i(pi) after replacing the e and the i
and
the pi with perhaps 10-adics or 5-adics or 7-adics or 22-adics
or
19-adics so that they combine to yield the sought for number of
....999999. If I can find such then multiplication and
exponentiation
are self defined because the Euler equation e^i(pi)=-1 forces
it.
I wish Abian were here to help.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
(www.iw.net/~a_plutonium) website of the science of AP under
revision
what used to be my old science website
www.newphys.se/elektromagnum/physics/LudwigPlutonium from
years 1993
===
Subject: Re: Prime factors of number near googolplexplex
> He didn't factor those numbers, just guessed some of their
small
> factor.
Well, he might have factored googleplexplex itself, since it
factors
into the form 2^n*5^n, and all he would have to do is work out
the value
of n.
===
Subject: Differential Geometry Book
I am taking a first course in Differential Geometry at the
graduate
level, and we are using Do Carmo's Differential
Geometry of Curves and Surfaces. While a thourough
and often used text, it is a bit difficult to read at some
times.
Does anyone know of a book that could be used to supplement Do
Carmo?
I would prefer something that does not assume too much
knowledge on
the part of the student... Maybe even something that could be
used
by undergraduates.
And, of course, cheaper is good, too. :lol:
Merkinball
===
Subject: Re: Upside-Down Functions Have Some Remarkable
Properties
> An Upside-Down Function (UDF) is defined here as a function
which
> has a non-trivial multiplicative inverse, where non-trivial
means
> that the multiplicative inverse has a recognizable
mathematical use
> other than as an index comparing two of the same types of
quant-
> ities. A function may be UDF restricted to some domain or
range,
> and it may be UDF based on current mathematical knowledge,
and
> occurrence of something in an intermediate step of
computation is
> not regarded as a non-trivial use.
Sec(x), csc(x) and cot(x) spring to mind, as do their
hyperbolic
analogues.
===
Subject: Re: When the term you differentiate does not become
zero
> I first came across this method in the film Stand and
Deliver, you may
have
> seen it.
> Anyway, I had a look at this method, and as a trick for
simple by parts
> integrals it works fine provided that the one you
differentiate becomes
zero.
> As I realised in lectures this is not always the case so I
considered a
way
> to cheat the cheat method, and found a little gem.
> Consider the indefinite integral of xlnx dx
> using lnx as a priority for the differential it can be
quickly seen that
it
> doesn't revert to zero.
> by parts in can be shown that the answer is (1/2 x^2 lnx) -
(1/4 x^2) + C
> Using the table;
> sign d/dx integral dx
> + lnx x
> - 1/x x^2/2
> + -x^-2 x^3/6
> The trick is now to take the integral of the product in the
last row.
> Giving; 1/2 x^2 lnx - 1/6 x^3 x^-1 - 1/6 int(x^-2 x^3)dx
> this then comes out as above once simplified.
> The trick is knowing when to stop!!!
> Try some simple ones and you will get the hang of it.
> Take care
(Elec Eng)
The usual way to do integral ln(x) dx is by itegration by
parts with
integral ln(x) 1 dx = integral u dv
with u = ln(x) and dv = 1 dx.
===
Subject: Re: Catenary curves question
> I'm editing a puzzle book which asks:
What will happen to the shape of a catenary curve if you tie
and
suspend
> two weights at equal horizontal distances apart?
> I can't find the answer anywhere on the Internet - does
anyone know?
> David
Assuming a perfectly flexible cable hanging between two fixed
endpoints
of equal elevation and close enough together so that the
direction of
gravitational acceleration may be regarded as constant, the
shape of the
cable will be a symmetric section of a catenary curve.
Hanging weights from the cable at any points between the ends
will
increase the tension along the entire cable, so you will have
the cable
shaped piecewise into parts of a flatter catenary, as if
sections of the
catenary have been excised where the weights hang and the
remainders
reattatched.
The lengths of each (metaphorically) excised section will be a
length of
cable equal in weight to the weight being hung at that point
on the
cable.
Note that this is the length of cable excised and not the
horizontal
extent of the cable excised.
Hope this helps.
===
Subject: Re: Proposition for Euclidean geometry
>Depends on what sort of proposition you're looking for.
Let's just start with dinner and a movie, OK?
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> Your biggest problem is you do not know what the LUB axiom
is and its
> implication of completeness.
>
I once had a friend in Melbourne Australia next to Burwood
Heights
School by the name of John Hobba. But I would guess there are
many
Hobba in Australia. I was a maths teacher and John was a
literature
teacher.
Anyway, I am sure I am rusty on calculus. Run LUB axiom by me
again.
The focus of attention of thoughts on the Cracks in the REals
should
broadly look at derivative and integral and I remember in my
College
days the plethora of integration. When you have a lot of a
thing
spells mess and spells trouble. Another mess is the idea of
continuous
everywhere but nowhere differentiable.
, the reason the NaturalNumbers= FiniteIntegers is flawed is
because all numbers must have an infinity-component and so
whenever a
NumberTheory tries to prove something for all FiniteIntegers
such as
FLT, perfect-numbers, Goldbach, twin primes, thousands of Erdos
problems etc etc, the reason they fail is because the set of
FiniteIntegers is a fake set and doomed to failure for any
problem
asking about all FiniteIntegersSince REals still have that
finite-componentry on the leftward
string then Reals are another ill-defined set. But the REals
should
not be a flagrant violator of theorems that FiniteIntegers are
and
that the fakeness of Reals should be somewhat muted and
dampened
whereas Number Theory run amok. So the cracks in Reals is
softened and
have to peer more closely to see the cracks in Reals whereas in
Numbertheory the cracks are huge.
I know of only one REal Analysis theorem that shows the cracks
of
Reals and it is obviously the Riemann Hypothesis for it tries
to reach
out to infinity the 1/2 Real line. Contrasting NumberTheory
where the
Erdos problems number in the thousands where infinity is
attempted to
be grappled with. If Erdos replaced FiniteIntegers with
P-adics, after
coming to realize the FiniteIntegers are a fictional set, then
all the
Erdos problems will vanish almost overnight.
If you replace the Riemann Hypothesis of the Reals with Doubly
Infinites and the NaturalNumbers with P-adics then RH is solved
overnight.
, what I am trying to say above is that the cracks in
Numbertheory
are obvious because so many theorems grapple with the Naturals
out at
infinity and thus unsolved whereas theorems in Real Analysis
generally
stick with the REals that are 1 trillion or less. Few theorems
in REal
Analysis ask for the Reals at infinity unlike Number theory.
Riemann
Hypothesis is one of the few theorems that puts the Reals all
the way
out to infinity into play.
Perhaps the NP conjecture is another Real Analysis problem
that puts
the Reals out at infinity into play.
So, that leaves the derivative and the integral as areas of
REal
Analysis where the REals are questioned at infinity and thus
dredges
up the messyness of the REals when put to the test of infinity.
So, , please remind me again about LUB.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
(www.iw.net/~a_plutonium) website of the science of AP under
revision
what used to be my old science website
www.newphys.se/elektromagnum/physics/LudwigPlutonium from
years 1993
===
Subject: 6 numbers and Brian Greene's new book
Notes for a book review to appear in Super Cosmos.
Sarfatti Commentary 1
Sir Martin's agent is John Brockman who has the monopoly on
pop physics
media and who is good at maximizing the profits. Hence there
are no
equations in the book. A few equations in the back in an
Appendix would
have been nice for more scientifically literate readers, but
that would
spoil the corporate bottom line - a pity.
What are the six numbers that control the Universe we live in?
The idea
is that there are an infinity of infinity of parallel
universes limited
by the Hubble horizons of their past light cones from any
observation
point that reach back to the Big Bangs creating each one of
them in
quantum vacuum phase transitions. The Weak Anthropic Principle
(WAP) of
generalized Darwinian natural selection applies. There is no
mystery to
the fine tuning of these 6 control parameters. Intelligent
Design is
not needed. God is not needed -- at least not for this aspect
of The
Problem.
1. N = 10^36 = e^2/Gmp^2 ~ square of the (e/m) ratio for
elementary
e = electric charge on the proton of mass mp, where G is
Newton's
parameter for gravity at large scales. Note I use variable
parameter
rather than constant since spin 1 gauge force coupling
strengths run
and there is no reason that spin 2 coupling of gravity should
also not
run, i.e. be scale dependent.
This number measures the strength of the electrical forces
that hold
atoms together divided by the force of gravity between them.
If N had a
few less zeros only a short-lived miniature universe could
exist ... p.2
If G is a scale-dependent variable, one must not violate the
kinds of
constraints discussed in this book. For example, Hal Puthoff's
idea for
metric engineering the shape of space-time requires changing
the e/m
ratios - not a good idea and not at all necessary to achieve
the mission
objective of practical free-float warp drive through
traversable
wormhole star gates making Star Trek Real.
2. Another number E = 0.007 defines how firmly atomic nuclei
bind
together ... if E were 0.006 or 0.008, we could not exist.
3. The cosmic number Omega measures the amount of stuff in the
universe. Stuff comes in two distinct forms real and virtual
because of Heisenberg's quantum uncertainty principle. For
example,
suppose a quantum of stuff has energy E at time t with physical
fluctuations delta(E) and delta(t.) If this quantum is virtual
then
its fluctuations obey
delta(E) delta (t) < h = Planck's quantum of dynamical action
In contrast, if the quantum of stuff is real then its
fluctuations obey
delta(E) delta(t) > h = Planck's quantum of dynamical action
That is, the product of conjugate pairs of properties of the
stuff
wave coherent phase, do not have enough action if they are
virtual --
meaning they are inside the vacuum. Real stuff is outside the
vacuum.
Another way to look at this virtual/real distinction is that
real quanta
outside the vacuum have their energy E and momentum p yoked
together by
Einstein's famous equation E = Mc^2 where M^2 = (p/c)^2 + m^2.
M is the
total mass and m is the rest mass. Real photons, forming the
far
field, have zero m, but non-zero M. Real photons also only
have two
transverse states of spin-polarization and they propagate
energy and
momentum through space on null geodesics that lie on light
cones.
Virtual quanta inside the vacuum do not obey this law. They
can have
any energy E and any momentum p at the same time (crudely
speaking)
unlike the same quanta in their real phase of existence
outside the
vacuum where there is a unique function connecting energy to
momentum.
The near induction fields in the electrical equipment we are
surrounded
by are entirely made out of virtual photons with an extra
longitudinal
state of spin-polarization. These near induction
electromagnetic fields
do not lie on the light cones. Whether or not a given quantum
is real or
virtual depends on whether or not its conjugate pairs of
fluctuations or
uncertainties exceed a critical threshold h of action.
4. Einstein's Cosmological Constant / for anti-gravitating
dark zero
point quantum exotic vacuum fluctuation energy is next. Sir
Martin does
not pin down the physical nature of this number as precisely
as I have
just done. Indeed, my way of interpreting this observed number
is a
definite prediction of my original theory and I could be wrong
in the
sense of Sir Karl Popper's falsifiabilty, which any good
physics
theory must obey. It is not yet clear if the physics of string
theory
that is described by Brian Greene in his two popular books is
Good
Physics or Bogus Physics in Martin Gardner's sense, or is
Cargo Cult
physics in Richard Feynman's sense. The issue is whether or
not the
picture of physical reality given by M Theory as described by
Greene,
Ed Witten and other members of what Doctor Faustus called The
String
Mafia is falsifiable physics or unfalsifiable cultism
masquerading as
physics. String theory is definitely seductively beautiful
mathematics,
but unless one is a Platonist in the sense of Max Tegmark's
Level IV
definite falsifiable predictions, like my theory does, it is
not Good
Physics. Note that Good Physics can be Wrong Physics. The
problem with
string theory or M theory is that it is not yet decided if it
is not
even wrong in Wolfgang Pauli's sense. Therefore, the public
should not
be hoodwinked and fooled into thinking that Brian Greene's
story is
anything more than wild speculation at this point in our
understanding.
It may turn out to be true, i.e. not falsified by facts but it
should not be itself considered as proven or, more precisely,
battle-tested the way Newton's mechanics, Maxwell's
electrodynamics,
thermodynamics, quantum theory and Einstein's relativity
(special and
general) have been. Double standards should not be allowed.
The same
criteria applied by skeptics to flying saucers and the
paranormal need
to be objectively applied to string theory as presented by
Brian
Greene to the public. Note that Brian, in his NOVA show, does
talk about
time travel to the past and to parallel universes next door in
hyperspace with extra dimensions and has a scene where he
talks to an
extra-terrestrial on the telephone. All of that is in my book
Destiny
Matrix published in 2002 at least one year before the NOVA
show was
presented. In fact there is a double standard in Physics Today
backed by
large multi-national corporate funding that is creating an
artificial
reality not unlike the artificial reality of WMD in Iraq as
shown by
Noam Chomsky on C-Span http://www.zmag.org/chomsky/index.cfm .
The
scientific process, like the political process, has been
corrupted with
fiction pretending to be fact and we must speak out. It's nigh
time for
media on PBS and elsewhere. Of course, in a complex world with
incomplete information it is not easy to distinguish fact from
fiction
on many important issues which require trust in the people
managing and
brokering the knowledge. What I say about string theory also
applies to
loop quantum gravity of spin foams of the John Baez crowd.
Also an
interesting approach, but also at this point wildly
speculative without
seamless integration into known theories like Einstein's GR
last I
checked. I could be wrong about the last remark. I mean, is
there a
clear derivation of Einstein's GR field equation from the
quantized area
and volume operators of loop quantum gravity in an appropriate
limiting
case?
5. Is Q ~ 10^-5 the seeds of WMAP. More on that later.
6. D the dimensions of the universe at large scales.
to be continued
===
Subject: Help Reading Erdos
I'm Italian and I would really like to know wheter it is
possible to get some writing of Erdos, or if they are not
available to the public (or if they are not available only to
the public in forein countries - :-( - ). Note: who does
publish
them? Where does he publish them?
I tried to post on sci.math.research but it did not work,
anyway
if someone would inform me on this subject I'll appreciate it.
Michele
===
Subject: Re: This Week's Finds in Mathematical Physics (Week
203)
> Also available at http://math.ucr.edu/home/baez/week203.html
> This Week's Finds in Mathematical Physics - Week 203
> John Baez
> Topologists know David Joyce as the inventor of the quandle
- an
> algebraic structure that captures most of the information in
a knot.
<http://www.math.uga.edu/~szwang/colloquium/abst-Carter.html
<http://adela.karlin.mff.cuni.cz/lat/~stanovsk/math/survey.pdf
Kewl... but does encompass handedness?
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
Quis custodiet ipsos custodes? The Net!
===
Subject: Re: ellipse from 4 points
Hello Everyone,
First of all, thank you for all your replies. In most of my
cases the
points which I start off with our a feasible set of 4 points.
That is,
I don't have a case where in one of the 4 points is the
centroid of a
triangle formed by the other 3 points.
I have written a simple Matlab program (below) which tries to
draw
an ellipse from any 4 points based on Ken's example. It is
interesting
to see the different shapes you get by varying the value of t.
The
program is very crude, but it gives you a good insight.
Ken, I get the value of 't' using the condition ab - h^2 >0, by
solving the quadratic you basically get 2 roots i.e t1 and t2.
By taking t = (t1 + t2) / 2; I am able to get an ellipse for
most of
my cases.
Does this mean that there is lower limit and upper limit for t?
Again, thank you all for your help.
Rgds,
> ....
> I have the position of 4 points on a 2D plane....
> (The usual kind of plane. :-)
> Is there any way I could fit an ellipse ....
> to pass through the 4 points?....
> Yes. I'll outline a method (pure algebra, no calculus), then
go
> through an example.
> In general, 5 points in a plane determine a unique conic
(ellipse
> or hyperbola or whatever). Specifying only 4 points allows
you an
> infinite family called a pencil of conics. If you know the
equations of
> any two members of the pencil, in the form S = 0 and S' = 0,
then
> all the others have equations S + tS' = 0 where t can be any
real
> number. For the two basic conics it's a good idea to use
line-pairs
> (degenerate conics), because their equations are easy to
find.
> For a simple example I'll take the 4 given points to be
(0,0),
> (1,0), (0,1), (3,1).
> Sort them into two pairs, say {(0,0), (1,0)} and {(0,1),
> (3,1)}. The line through the first pair is y = 0, and the
line through
> the second pair is y - 1 = 0, so these two lines together
make up the
> degenerate conic y(y - 1) = 0.
> Similarly, the line through {(0,0), (0,1)} and the line
through
> {(1,0), (3,1)} make up the line-pair x(x - 2y - 1) = 0.
Similarly,
> the line through {(0,0), (3,1)} and the line through {(1,0),
(0,1)}
> make up the line-pair (x - 3y)(x + y - 1) = 0.
> I've calculated all three line-pairs to show you the idea,
but you
> need only two of them. I'll choose the two simplest.
> x(x - 2y - 1) = 0 can be the conic S = 0, and
> y(y - 1) = 0 can be the conic S' = 0.
> Then all other conics through the four points have equations
> x(x - 2y - 1) + ty(y - 1) = 0 for various real numbers t.
> Multiplying out gives
> x^2 - 2xy + t(y^2) - x - ty = 0. (*)
> Various values of t give conics of various kinds, but you're
> interested in ellipses only. The condition for a conic
> a(x^2) + 2hxy + b(y^2) + 2gx + 2fy + c = 0
> to be an ellipse is ab - h^2 > 0.
> In our example (*), a = 1, b = t, h = -1, so
> ab - h^2 = t - 1. Hence every real number t > 1 makes (*) the
> equation of an ellipse through the original 4 points.
> In cases which interest you, you should be able to use the
same
> method to find infinitely many ellipses through your four
points. Among
> all those you can choose which you like best, perhaps by
imposing some
> other condition on it.
>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%MATLAB File
%%%Draw Ellipse from 4 unequally spaced points
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
%Set position of the 4 points
x1=85;
y1=107;
x2=105;
y2=122;
x3=88;
y3=147;
x4=67;
y4=134;
%Get min and max, x and y values for interpolating
x_array=[x1 x2 x3 x4];
y_array=[y1 y2 y3 y4];
lower_x=min(x_array);
higher_x=max(x_array);
lower_y=min(y_array);
higher_y=max(y_array);
%Plots the 4 points
plot(x1,y1,'ro')
hold on;
plot(x2,y2,'bo')
hold on;
plot(x3,y3,'yo')
hold on;
plot(x4,y4,'go')
hold on;
%%Pair 1 - pt1 and pt2
if ((x2-x1)==0)
m1=y2-y1;
c1=0;
else
m1 = (y2-y1) / (x2-x1);
c1= y1 - m1*x1;
end
%%Pair 2 - pt3 and pt4
if ((x4-x3)==0)
m2=y4-y3;
c2=0;
else
m2 = (y4-y3) / (x4-x3);
c2= y3 - m2*x3;
end
%%Pair 3 - pt1 and pt3
if ((x3-x1)==0)
m3=y3-y1;
c3=0;
else
m3 = (y3-y1) / (x3-x1);
c3= y1 - m3*x1;
end
%%Pair 4 - pt2 and pt4
if ((x4-x2)==0)
m4=y4-y2;
c4=0;
else
m4 = (y4-y2) / (x4-x2);
c4= y2 - m4*x2;
end
%%%%%%%%%Set t value
aa1 = (4*m3*m4 - m3*m3 - m4*m4 - 2*m3*m4) / 4;
bb1 = (4*m1*m2 + 4*m3*m4 - 2*m1*m3 - 2*m2*m3 - 2*m1*m4 -
2*m2*m4) / 4;
cc1 = (4*m1*m2 - m1*m1 - m2*m2 - 2*m1*m2) / 4;
pp1=[aa1 bb1 cc1];
rr1=roots(pp1);
t = (rr1(1,1) + rr1(2,1)) / 2.0;
%%%%Interpolate , find y from x
for x=lower_x:higher_x
a = 1 + t;
b = -(x*m1) -(x*m2) - (t*x*m3) - (t*x*m4) - c1 -c2 -t*c3 -
t*c4;
c = (x*x)*(m1*m2 + t*m3*m4) + x*(m1*c2 + m2*c1 + t*m3*c4 +
t*m4*c3) +
(c1*c2) + (t*c3*c4);
p = [a b c];
r=roots(p);
yy1=r(1,1);
yy2=r(2,1);
hold on;
plot(x,yy1,'k.')
hold on;
plot(x,yy2,'k.')
end
%%%%Interpolate , find x from y
for y=lower_y:higher_y
a = (m1*m2) + (t*m3*m4);
b = -(y*m1) -(y*m2) - (t*y*m3) - (t*y*m4) + (m1*c2) + (m2*c1) +
(t*m3*c4) + (t*m4*c3);
c = (1+t)*(y*y) + y*(-c1-c2-t*c3-t*c4) + (c1*c2) + (t*c3*c4);
p = [a b c];
r=roots(p);
xx1=r(1,1);
xx2=r(2,1);
hold on;
plot(xx1,y,'k.')
hold on;
plot(xx2,y,'k.')
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%END
===
Subject: d1 cos (ax + b) + d2 cos(cx + d) = k ; x = ?
Greetings,
how can I solve
d1 cos (ax + b) + d2 cos(cx + d) = k
( and sin (ax + b) + sin(cx + d) = k )
?
I need to express x (through some arccoss & arcsins & ...).
Cannot be iterative.
The best would be the solution for quadratic arguments
[ ...cos (ax^2 + bx + c)... ], even better generally for
polynomical
argument and the very best some hints for cos f(x) + cos g(x)
= k, if
it's possible. But I'll be happy with linear case.
-----
REASONS:
Actually, what I'm trying to do is find a numerical solution
for
collision of the peak of moving bones and parametric surface
segment
(for movement upon such surface).
In 2D (bone vs parametric curve), angles between n bones are
described
by a1(t), a2(t), ... an(t); time t and surface parameter u in
<0, 1>.
Lengths of bones are d1, ... dn. 1 rotational degree of
freedom, no
translational. I wanted to solve system of equations
forward_kinematics_x(t) = surface_x(u, t)
forward_kinematics_y(t) = surface_y(u, t)
equal to
d1 cos a1(t) + d2 cos [a1(t) + a2(t)] ... + dn cos [a1(t) +
an(t)] =
p3 u^3 + p2 u^2 + p1 u + p0
d1 sin a1(t) + d2 sin [a1(t) + a2(t)] ... + dn sin [a1(t) +
an(t)] =
q3 u^3 + q2 u^2 + q1 u + q0
Having the exact time of collision, I won't need to be
bothered by
per-frame collision tests, I'll have I have I can choose
I'm willing to do any sacrifices on generality, like linear
ai(t)
functions and amount of bones n limited to 3 or 2, if it can't
be
avoided. :(
Thank you very much for any kind of hint, link or opinion about
solvability
Marek
===
Subject: Re: e^i(pi) = -1 revisited with Doubly Infinites Re:
infinite
rightward strings tacked-on to p-adics serves as Orthogonality
and makes
Doubly-Infinites the points of Lobachevskian Geometry
> The progress has already been done.
> Since Euler not only proved that the equation
> is true, but he also proved that you need Geometric a
> proof to prove it. Goedel proofs are insufficient.
> You need a pair of duel functions to prove that it's true.
> Duel functions?
> That was Galois, not Godel :-)
You are mistaken. Galois proved some
distorted thing about *Groups* and
*quadratic* equations. He didn't
prove anything about polynomials.
Other than the well-known fact that
mathematicians merely exist for
Communist job security reasons
rather than logic.
===
Subject: Re: min area to flip 2 hinged rods
> Assuming each rod to be of length 1, the boundary curve need
be no
> larger that (x^2)^(1/3) + (y^2)^(1/3) = 1
> Since this bounds an area of 3*pi/32 in each quadrant, the
total area
> required will be less than 3*pi/8 ~ 1.7881 square units.
...
> Can anyone come up with an area less that 3*pi/8?
I think I can improve on the astroid solution, but I'm not
sure by
how much. First off, here's the astroid, in case anyone
doesn't know
what shape we're talking about:
http://www-gap.dcs.st-and.ac.uk/~history/Curves/Astroid.html
The original idea is that we start with the hinge at (-1,0) and
both tips at (0,0), then pull the tips apart until they are at
(0,1) and
(0,-1) and the hinge is at (0,0), then close them again with
the hinge
moving toward (1,0). That sweeps out an astroid with area 3/8
pi, as
pictured.
Instead, start with the tips at (1,0) and the _hinge_ at (0,0).
Constrain the tips to run along the righthand curve of the
astroid until
they reach (0,1) and (0,-1), then move them together along the
Y axis
until they meet, with the hinge moving toward (1,0). (I.e., the
closing is the same as in the original solution.) This should
_nearly_ fit into the right half of the astroid, plus a small,
curvy
corner sticking out into the negative X region (caused by the
hinge
poking out as the dividers are opened, then receding as they
close).
As I say, I haven't calculated the total area, but it should be
only a little more than half the area of the astroid.
(I'm assuming that flipping the dividers is illegal, since that
question is functionally equivalent to opening the dividers to
180
degrees, which is easier and less interesting, imho.)
----j7y
**
jere7my tho?rpe / 734-769-0913 There is no spoon. SPOON!
There
> j7y@liws.org <<< is no spoon. SPOON! There is
no
invert liws to reply via email spoon. SPOON! -- The Tick vs.
Neo
===
Subject: Re: Class vs. Dimension Equations
>When I talk about the dimension equation of a finite group, I
mean the
>one expressing the order of the group as the sum of the
squares of the
>dimensions of the irreducible representations of the group
over the
>field of complex numbers.
>My question is: Is the dimension equation directly computable
(without
>knowing the group) from the class equation or vice versa?
>---- David
You don't say what you mean by the class equation.
Are you asking whether there exist two finite groups having
conjugacy
classes
with the same sizes, but having different degrees of their
irreducible
complex characters?
The answer to that is yes. There are many examples of order
128.
One example, in the notation of GAP or Magma is
SmallGroup(128,36) and
SmallGroup(128,734). They both have conjugacy classes with
sizes:
[1,1,1,1,2,2,2,2,2,2,4,4,4,4,8,8,8,8,8,8,8,8,8,8,8,8]
The irreducible character degrees of SmallGroup(128,36) are
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,4,4,4,4,4,4]
whereas those of SmallGroup(128,734) are
[1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4]
There are examples of order 64 of two groups with the same
character
degrees and different class sizes. For example
SmallGroup(64,4) and
SmallGroup(64,257).
Derek Holt.
===
Subject: Re: d1 cos (ax + b) + d2 cos(cx + d) = k ; x = ?
En el mensaje:2f7b6a6f.0402291244.2a1e584d@posting.google.com,
Magh <8agh@st.fmph.uniba.sk> escribi.97:
> Greetings,
> how can I solve
> d1 cos (ax + b) + d2 cos(cx + d) = k
> ( and sin (ax + b) + sin(cx + d) = k )
Differentiating the last with respect to x,
a*cos(a*x + b) + c*cos(c*x+d) = 0
d1*cos(a*x + b) + d2*cos(c*x + d) = k
cos(ax + b) = c*k/(c*d1 - a*d2)
cos(cx + d) = -a*k/(c*d1- a*d2)
x = (acos(c*k/(c*d1 - a*d2)) - b)/a
x = (acos(-a*k/(c*d1- a*d2)) - d)/c
If it has sense or no, you would must know ...
Saludos,
===
Subject: Re: how to clean up derivative and integral Re:
cracks in Euclidean
Geometry and why Reals are fake
> The biggest evidence of cracks in the Real Numbers as a
system of
> numbers is the fact of the plethora of integration and
differentiation
> measures.
> Selections from the website, where he proves that he is a
supergenius:
Archimedes Plutonium (my true legal name)
Below in chemistry I have a circular periodic table [...] God
is
231Pu
> and the best bible is the best most up-to-date physics
textbook.
If the Brain Locus theory is correct, then through a single
atom in
the
> brain can all the thinking and thoughts be conducted.
I make this biological speculation that the source of my
supergenius
> is that there is a Pu atom located in my brain, the focus of
my mind.
> The brain is a parabolic reflecting telescope which has one
atom as
> the center focus.
> They walk among us. A scary thing indeed.
> I don't know .... I very much like:
the best bible is the best most up-to-date physics textbook
Perhaps, if you plan to worship the universe. I don't share
Einstein and
Spinoza's deism, and would be loathe to analogize science with
religion.
Servo
===
Subject: Re: Help Reading Erdos
> I'm Italian and I would really like to know wheter it is
> possible to get some writing of Erdos, or if they are not
> available to the public (or if they are not available only to
> the public in forein countries - :-( - ). Note: who does
publish
> them? Where does he publish them?
> I tried to post on sci.math.research but it did not work,
anyway
> if someone would inform me on this subject I'll appreciate
it.
There is a pdf here
http://www.zblmath.fiz-karlsruhe.de/MATH/general/erdos/
index.htm
G.C.
===
Subject: Re: Catenary curves question
> I'm editing a puzzle book which asks:
What will happen to the shape of a catenary curve if you tie
and
suspend
> two weights at equal horizontal distances apart?
Parabola--sorry can't remember a reference.
> I can't find the answer anywhere on the Internet - does
anyone know?
> David
G.C.
===
Subject: Re: Differential Geometry Book
> I am taking a first course in Differential Geometry at the
graduate
> level, and we are using Do Carmo's Differential
> Geometry of Curves and Surfaces. While a thourough
> and often used text, it is a bit difficult to read at some
times.
> Does anyone know of a book that could be used to supplement
Do Carmo?
> I would prefer something that does not assume too much
knowledge on
> the part of the student... Maybe even something that could
be used
> by undergraduates.
> And, of course, cheaper is good, too. :lol:
Barrett O'Neil Elementary Differential Geometry Academic Press.
$77.95 from Amazon.
G.C.
===
Subject: Re: ellipse from 4 points
> ....
> Then all other conics through the four points have equations
> x(x - 2y - 1) + ty(y - 1) = 0 for various real numbers t.
> Multiplying out gives
> x^2 - 2xy + t(y^2) - x - ty = 0. (*)
> Various values of t give conics of various kinds, but you're
> interested in ellipses only. The condition for a conic
> a(x^2) + 2hxy + b(y^2) + 2gx + 2fy + c = 0
> to be an ellipse is ab - h^2 > 0.
> In our example (*), a = 1, b = t, h = -1, so
> ab - h^2 = t - 1. Hence every real number t > 1 makes (*) the
> equation of an ellipse through the original 4 points....
> ....
> Ken, I get the value of 't' using the condition ab - h^2 >0,
by
> solving the quadratic you basically get 2 roots i.e t1 and
t2.
> By taking t = (t1 + t2) / 2; I am able to get an ellipse for
most of
> my cases.
> Does this mean that there is lower limit and upper limit for
t?....
In my very simple example the inequality ab - h^2 > 0 came out
linear in t, but usually it's quadratic. To solve a quadratic
inequality l(t^2) + mt + n > 0, first solve the corresponding
equation
l(t^2) + mt + n = 0 for its roots t_1 and t_2. If you think
about
the graph of a quadratic, you'll see that it's positive
outside the
interval (t_1, t_2) if l > 0, but inside that interval if l <
0.
Using my suggested method starting from line-pairs will give
you l < 0,
so the values of t giving you ellipses are all the real numbers
between t_1 and t_2 as you said. Your (t_1 + t_2)/2 is just one
special case.
===
Subject: Re: Logic question #2
> Hey guys,
> Here is question #2. I haven't the slightest idea with this
one. If
> The puzzle seems too simple, unless the point was to solve
the cipher.
> Richard,
> How did you decipher the code?
I can't speak for Richard, but this is how I deciphered it:
>The rest of this (except for the Roman numerals and the
letters to
>select your choice) is in a simple code.
>(i) Wkhuh duh vla fkrlfhv. Wlfn rii wkh rqh diwhu wkh vhfrqg.
First realize that the word the is the most common 3-letter
word.
The 3-letter sequence wkh appears twice in these sentences, so
it is
probably the word THE. So w=T, k=H,and h=E and the sentences
become:
(w=T k=H h=E)
>(i) THEuE duh vla fkrlfEv. Tlfn rii THE rqE diTEu THE vEfrqg.
Now the first word is either THERE or THESE, so u=R or u=S. If
u=S, then the second word, dSE, could only be USE so d=U.
However,
if u=S, then it is dRE and could only be ARE so d=A. The
sentence
can now be one of the following:
(u=S d=U)
>(i) THESE USE vla fkrlfEv. Tlfn rii THE rqE UiTES THE vEfrqg.
(u=R d=A)
>(i) THERE ARE vla fkrlfEv. Tlfn rii THE rqE AiTER THE vEfrqg.
Now for u=S and d=U the third from the last word, UiTES, makes
no
sense, but for u=R and d=A AiTER which must be either AFTER or
ALTER
so i=F or i=L and we have:
(i=F)
>(i) THERE ARE vla fkrlfEv. Tlfn rFF THE rqE AFTER THE vEfrqg.
(i=L)
>(i) THERE ARE vla fkrlfEv. Tlfn rLL THE rqE ALTER THE vEfrqg.
The problem with i=L is the word rLL, which can't be ALL
because A
is represented by d, not r. It could only be ILL, which doesn't
make much sense in the context of the surrounding words, so we
proceed
with i=L.
For i=L, the word rFF can only be OFF so r=O:
(r=O)
>(i) THERE ARE vla fkOlfEv. Tlfn OFF THE OqE AFTER THE vEfOqg.
OqE must be ONE, so q=N:
(q=N)
>(i) THERE ARE vla fkOlfEv. Tlfn OFF THE ONE AFTER THE vEfONg.
At this point, the sentences get tricky - but lets look at the
multiple choices after we plug in these letters:
>(A) ONE (B) TzO (C) THREE (D) FOxR (E) FlyE (F)vla
It immediately becomes obvious that they are ONE TWO THREE
FOUR FIVE
and (probably) SIX, so...
(z=W x=U l=I y=V v=S a=X)
>(i) THERE ARE SIX fHOIfES. TIfn OFF THE ONE AFTER THE SEfONg.
fHOIfES look like CHOICES, so f=C:
(f=C)
>(i) THERE ARE SIX CHOICES. TICn OFF THE ONE AFTER THE SECONg.
Finally, it appears obvious that TICn must be TICK and SECONg
must
be SECOND, so...
(n=K g=D)
>(i) THERE ARE SIX CHOICES. TICK OFF THE ONE AFTER THE SECOND.
>(A) ONE (B) TWO (C) THREE (D) FOUR (E) FIVE (F) SIX
The second part becomes:
>(ii) THERE ARE NOW FIVE CHOICES. TICK OFF THE ONE AFTER THE
FOURTH
>(A) ONE (B) TWO (C) THREE (D) FOUR (E) FIVE
Assuming one of the three-letter words is THE or that the most
frequently occuring letter is E is the easiest way to decipher
things
like this.
Squishua
===
Subject: Which topology text would be better for self-study?
I checked out a couple of topology books to try to learn from
on my own.
One
is Munkres Topology and the other is Introduction to Topology
by Crump
Baker
http://www.amazon.com/exec/obidos/tg/detail/-/0697059723/qid=
1078093279/sr
=1-8/ref=sr_1_8/102-3578915-2424159?v=glance&s=books
I know that the Munkres book is popular but I'm not sure if it
would be okay
to
learn with on my own. The book by Baker covers point-set
topology only and
that's all I am trying to learn. THe chapters are Preliminary
Topics,
Topological Spaces, Subspaces and Continuity, Product Spaces,
Connectedness,
Compactness, Separation Properties, and Metric Spaces.
If there is a better book for me, I wouldn't mind buying it.
===
Subject: Re: Differential Geometry Book
There is a Springer Undergraduate text: Elementary Topics in
Differential Geometry, by J.A. Thorpe.
Amazon has it for $52.95 new, and about $25 used.
Here is the link.
http://www.amazon.com/exec/obidos/search-handle-form/103-
5011499-5100656
Brian
>I am taking a first course in Differential Geometry at the
graduate
>level, and we are using Do Carmo's Differential
>Geometry of Curves and Surfaces. While a thourough
>and often used text, it is a bit difficult to read at some
times.
>Does anyone know of a book that could be used to supplement
Do Carmo?
>I would prefer something that does not assume too much
knowledge on
>the part of the student... Maybe even something that could be
used
>by undergraduates.
>And, of course, cheaper is good, too. :lol:
>Merkinball
>
===
Subject: Re: Help Reading Erdos
> I'm Italian and I would really like to know wheter it is
> possible to get some writing of Erdos, or if they are not
> available to the public (or if they are not available only to
> the public in forein countries - :-( - ). Note: who does
publish
> them? Where does he publish them?
mathematical research, which will be available in university
libraries in Italy & elsewhere. As he published well over 1000
look unless you know what you're trying to find.
He also published several books, and maybe it will be easier to
find his work in one of these. Again, a university library
would
be the place to go.
of numbers. Translated from the second Hungarian edition by
Barry
Guiduli. Undergraduate Texts in Mathematics. Springer-Verlag,
New York,
MR1601954 (99b:05031) Chung, Fan; Graham, Ron Erd.9as on
graphs. His
legacy of unsolved problems. A K Peters, Ltd., Wellesley, MA,
1998.
xiv+142 pp. ISBN: 1-56881-079-2; 1-56881-111-X (Reviewer: R.
H. Schelp)
05Cxx
MR1425199 (97f:00033) The mathematics of Paul Erd.9as. II.
Edited by
Ronald L. Graham and Jaroslav Nev setv ril. Algorithms and
Combinatorics, 14. Springer-Verlag, Berlin, 1997. xvi+577 pp.
ISBN:
3-540-61031-6 00B30 (03-06 05-06 52C10)
[23] MR1425172 (97f:00032) The mathematics of Paul Erd.9as. I.
Edited by
Ronald L. Graham and Jaroslav Nev setv ril. Algorithms and
Combinatorics, 13. Springer-Verlag, Berlin, 1997. xvi+399 pp.
ISBN:
3-540-61032-4 00B30 (05-06 11-06)
MR1003606 (90g:11081) Erd.9as, P.; Gruber, P. M.; Hammer, J.
Lattice
points. Pitman Monographs and Surveys in Pure and Applied
Mathematics,
39. Longman Scientific & Technical, Harlow; copublished in the
United
States with John Wiley & Sons, Inc., New York, 1989. viii+184
pp. ISBN:
0-582-01478-6 (Reviewer: T. H. Jackson) 11Hxx (05Bxx 11P21
52A43)
MR0795592 (87g:04002) Erd.9as, Paul; Hajnal, Andr.87s;
M.87t.8e, Attila;
Rado,
Richard Combinatorial set theory: partition relations for
cardinals.
Studies in Logic and the Foundations of Mathematics, 106.
North-Holland
Publishing Co., Amsterdam, 1984. 347 pp. ISBN: 0-444-86157-2
(Reviewer:
Carlos A. Di Prisco) 04-02 (03-02 03E05 04A20)
MR0592420 (82j:10001) Erd.9as, P.; Graham, R. L. Old and new
problems and
results in combinatorial number theory. Monographies de
L'Enseignement
Math.8ematique [Monographs of L'Enseignement Math.8ematique],
28.
Universit.8e
de Gen.8fve, L'Enseignement Math.8ematique, Geneva, 1980. 128
pp.
(Reviewer:
L. C. Eggan) 10-02 (05-02)
MR0382007 (52 #2895) Erd.9as, Paul; Spencer, Joel
Probabilistic methods in
combinatorics. Probability and Mathematical Statistics, Vol.
17.
Academic Press [A subsidiary of Harcourt Brace Jovanovich,
Publishers],
New York-London, 1974. 106 pp. (Reviewer: P. E. O'Neil) 05-02
MR0532670 (58 #27144) Erd.9as, Paul The art of counting:
Selected
writings. Edited by Joel Spencer and with a dedication by
Richard Rado.
Mathematicians of Our Time, Vol. 5. The MIT Press, Cambridge,
Mass.-London, 1973. xxiii+742 pp. 01A75
===
Subject: Re: ellipse from 4 points
ETAtAhUAtax2xMURmfpmYEtF/18FsDt9AgwCFBBWtuUS/
S6fBDG1ZShybWEYnd7v
You cannot uniquely define an ellipse from just four points.
You need
additional information such as the orientation of the axes.
With FIVE
points an ellipse is uniquely defined if it exists, but it may
not exist
for an arbitrary collection of five points.
--OL
===
Subject: Re: Interactive Proof Writing Tutorial (Freeware)
: G. Frege says...
:
: ><dchris@allstream.net> tried to advertise his program ,
again.
:
:
: >I would suggest to use the COMMON quantifiers
:
: > (Ax) and (Ex)
The fact that something is common doesn't make it defensible.
Idiocy is common. Or, as Harlan Ellison put it,
The two most common things in the universe are hydrogen
and stupidity : >instead of the extremely uncommon and rather
irritating
:
: > ALL(x): and EXISTS(x):
Well, of COURSE *that's* irritating.
That will fall of its own weight; it's unparseable.
: I didn't know that there was any standard way to write
: first-order logic using ASCII characters. The standard
: in math textbooks is to use upside-down A for for all
: and a backwards E for there exists : The same comment for
or, implies, and: there are standard
: ways to write these things in textbooks, using special
: symbols, but I didn't know that there were any standards for
: writing them using ASCII.
Since Ascii is what we've got, it behooves US to SET
some standards. (Ax) and (Ex) are stupid because the
parentheses
are redunt; they don't teach anything. EVERYBODY CAN ALREADY
SEE
that whatever string of lower-case letters immediately follows
the
upper-case quantifier is intended to list the bound variables.
Where you NEED bracketing is around the MATRIX/scope of the
SENTENCE
being quantified over. This notation deprives you of that.
Correct is Axyz(phi(x,y,z)) or some variation on that theme.
(Ax) was born ridiculous.
---
The history of our nation has demonstrated that separate is
seldom, if
ever, equal.
--- Supreme Judicial Court of Massachusetts, adv.Sen.#2175
===
Subject: Re: the anticlassicalist }{ ii: the spectre continues
> |> It's a basic issue of shared resources. Bandwidth is very
cheap
indeed,
> |> and like with most resources, there's no simple maximum
available
amount
> |> of it. But like with most resources, the environment
degrades as
people
> |> use more and more of it beyond a certain point.
> |
> | But, with scientists the message is always the same.
> | It doesn't matter what *bandwidth* costs, since
> | we are not paying for bandwidth. We are paying
> | for information.
> We're not paying in money; we're paying in time. The more
> junk, the longer it takes to wade through it.
Scientists do not pay in time.
They pay in *clocks*, just like
their moron leader Einstone did.
And since Goedel payed with a
proof rather than a head
clock, we're not really
interfering with your job,
we're simply just up to our tricks,
racking up the points in the
Engineers vs. The Morons Game.
> Keith Ramsay
===
Subject: Re: Finding out base of a number
ETAtAhQjLBdOvqgP2UPW3QEWhMeeI2ViWAIVAIdsYod5B/
gcQBmm1PFKGCkZvkp5
7*11^2+4*11+9 = 2*b^2+7b+9. Solve the quadratic quatin for an
appropriate root b.
Hey wait, why am I not getting a rational root!? You sure it's
not 297
base b?
--OL
===
Subject: Proof help is needed
Could someone please help me get started with this proof. It
would
Prove that if G is a group of order 25, then either G is
cyclic or
g^5=1 for all g E G.
Again thank you.
===
Subject: Re: Logic question #3
> Hello again guys,
>
--------------------------------------------------------------
---------------
> On the planet Fingal live the Fingas. Fingas look human,
except that
> they have no thumbs, and the number of fingers on each hand
may be
> different. This gives them their surnames, so that a Finga
with three
> fingers on his left hand and seven on his right might be
named Joseph
> 3-7.
> There is a wake on Fingal, to mourn the death of Finnegan, a
Finga.
> There is the beautiful ceremony of the touching of the
fingers, when
> the whole population forms a chain, a Finga touching one
neighbour to
> the right, finger to finger. Last night before Finnegan
died, he was
> in the chain, and the chain formed a complete circle. How
beautiful!
> Now Finnah 6-9 begins a chain, touching Joa 9-11, and so on,
and
> forming one circle with some of the Fingers. Fella 9-6
begins a
> separate chain, forming a second circle with the rest of the
Fingas.
> What is Finnegan's name?
> (A) Finnegan 6-9 (B) Finnegan 9-6 (C) Finnegan 9-9 (D)
> Finnegan 11-9 (E) Finnegan 11-6 (F) Finnegan 6-11
>
--------------------------------------------------------------
---------------
-
Anyone?? Please help!!
===
Subject: Re: y = x^x
x^x = exp(x*lnx)
> I believe it looks like a U shape as a negative times a
negative is a
> positive number. Hope this helps.
> I don't think so.
> Yes, spot the big mistake of the year. Lol, sorry. I thought
it was X*X
for
> some reason. It's like an S shape if I'm correct.
> Easiest way to find out is get a bit of graph paper, plot a
conservative
> scale and start putting numbers into x and start plotting
them.
===
Subject: Re: some complex integration questions
ETAtAhUAx4HY9SChFojUrejSswJtd4pN528CFHug8Erl9FmSpK2AzDvoRDwphty
J
Your original funciton has no branch cuts but the indefinite
integral
does. That is perfectly OK, in fact much desired. If the
branch cut is
generated by the integration process as opposed to being
nherent to the
function, you're supposed to plow right through it and into a
different
branch of the definite integral. You DO NOT jump back to the
original
branch at the discontinuity, for the intergal is supposed to be
continuous even if it is path-dependent. That's how a pole
generates a
residual value when you inegrate around it!
--OL
===
Subject: Re: Number Theory Problem!
ETAsAhQg0gXywob4LO+yEWurvjlNe0+
1zwIUfB3BUb42uFqBF4LCurclyZIbtFg=
For the first problem:
Certainly a^p == p and b^p == b mod p (using == to mean
congruent). So
a == b mod p. Thus a = b+kp for some integer k. Then a^p =
(b+kp)^p.
Apply the Binomial Theorem to the right side, subtract off the
common
term b^p from both sides, and observe that all the terms in the
difference are multiples of p^2.
--OL
===
===
Subject: Re: Proof help is needed
Adjunct Assistant Professor at the University of Montana.
>Could someone please help me get started with this proof. It
would
> Prove that if G is a group of order 25, then either G is
cyclic or
>g^5=1 for all g E G.
The order of an element must divide the order of the group.
What are
the possibilities? And what happens if there is one element
which is
not of order 1 or 5? What can you say about the cyclic
subgroup it
generates?
========
===
Subject: Logic question #4
Hey,
Here is question #4. I realize this is more a math question
than a
logic question. I figured part (i) to be 21. If you didn't get
that,
let me know.
--------------------------------------------------------------
--------------
--
Direct communication lines are being made between pairs of
people in
an office of nine: three managers, their three secretaries,
and three
technicians. The three managers are connected to each other.
Each
manager is also connected to his own secretary. The
secretaries are
connected to each other and to each technician. The
technicians are
also connected to each other.
(i) How many lines are there?
(A) 18 (B) 42 (C) 21 (D) 9 (E) 36 (F) 27
(ii) What is the greatest number of lines that can
simultaneously go
bad and still allow a message to be passed on from any one
person to
any other?
(A) 18 (B) 18 (C) 21 (D) 9 (E) 16 (F) 3
(iii) A quartette is a group of four persons each of whom is
connected
by a direct line. How many quartettes are there?
(A) 3 (B) 6 (C) 9 (D) 15 (E) 21 (F) 27
--------------------------------------------------------------
--------------
-
===
Subject: Re: What are Bessel functions?
ETAsAhQ7ydisCv+LANVl4le8Us6yPacSIgIUEk8O3Tt2Ddx5ZYo2TteF7ER+
iyI=
Bessel functions are named for a mathematician who described
the
properties of series solutions o the following second-order
differntial
equation:
x^2 y + xy' + (x^2 - v^2)y = 0
They have many applications, perhaps the simplest being the
role they
play in separation of variables when you solve partial
differential
equations in cylindrical or spherical coordinates. Any good
math text
should give ou an introduction to these things.
--OL
===
Subject: Re: a little - big problem
> Let U be an open set in R^n.
> Consider
> f in C^{infty}(U, R)
> such that
> exists lim_{x -> x_0} f(x) / ||x-x_0||^{k-1} = 0 .
> Then f belongs to I^k_{x_0}(U, R) .
>
--------------------------------------------------------------
--------------
> I^k_{x_0}(U, R) denotes the product of the ideal I_{x_0}(U,
R)
> k-times with itself and I_{x_0}(U, R) denotes the ideal in
C^{infty}
(U,R)
> of function vaniscing at p.
> In other words, f belongs to I^k_{x_0}(U, R) if and only if f
> is of the form:
> f = h_0 g_01 * g_02 * .. * g_0{k-1}*g_0k +
> +...+
> + h_m g_m1 * g_02 * .. * g_m{k-1}*g_0k
> where m is a natural number , say m=1 or m >1,
> h_i belongs to C^{infty}(U, R)
> and
> g_ij belongs to I_{x_0}(U,R) <==> g_ij(x_0)= 0 in R.
> Note that for m=0 , the proposition above is false,
> consider for example k=2 and f(x,y)=x^2+y^2.
> Then, exists lim_{(x,y) -> x_0} (x^2+y^2) / ||(x,y)||^{2-1}
> = lim_{(x,y) -> x_0} (x^2+y^2)^{1/2}=0
> ,on the other hand, it's impossible to write f as
> f = h_0 g_01 * g_02 with g_01 and g_02 in C^{infty}(U, R)
> My ask is:
> Is the proposition above true with m=1 or m>1 ?
At the end you talk about various values of m, yet there is no
m in the
proposition you stated. Furthermore, I don't think you want to
say
I^k_{x_0}(U, R) is the product of the ideal I_{x_0}(U, R) k
times with
itself.
I'm going to use different notation and try to simplify
things. Set x_0 =
0. Let U be an open ball centered at 0 (I think everything
will extend
easily to any open set). All functions mentioned will belong to
C^{infty}(U, R) without further mention. I^1 is the ideal
consisting of all
f vanishing at 0. For k > 1, I^k is the ideal generated by
k-fold products
f1*f2*...*fk, where each fj belongs to I^1. In other words,
I^k is the set
of all finite sums of such k-fold products. (Note: your h
functions are
not needed. Why?) For fixed m in {1,2,...} (why do you start
with m = 0?)
let I^k,m consist of all functions that are the sum of m
functions of the
form f1*f2*...*fk, where each fj belongs to I^1.
Finally, let J^k be the ideal of functions satisfying the
limit condition
partial derivatives of f of order < k at 0 vanish.
Proposition: J^k = I^k.
Proof sketch: This is true when k = 1, and for all k we have
I^k contained
in J^k. For the other inclusion, assume the result for k-1.
Let f be in
J^k. Then
f(x) = sum(j=1,n) x_j*int_[0,1] Djf(tx) dt,
where Dj is the partial derivative with respect to the jth
coordinate
variable. But each Djf is in J^(k-1). So the integral, which
is a C^oo
function of x, is also in J^(k-1), hence in I^(k-1) by the
induction
hypothesis. This shows f is in I^k.////
As far as the I^k,m go, no value of m will work for all n. In
particular,
|x|^2 is not in I^2,m if m < n. Proof: Suppose m < n and
|x|^2 = sum (j=1,m) fj*gj
where the f's and g's are in I^1. Then we must have
(1) |x|^2 = sum (j=1,m) L(fj)*L(gj),
where L(f) denotes the linear part of the Taylor expansion of
f at 0. Let
Sj be the zero set of L(fj). Then Sj is a linear subspace of
R^n of
dimension at least n-1. Because m < n, a simple dimension
counting argument
shows the intersection of all the Sj's contains a
one-dimensional line
through the origin. But then the sum on the right of (1)
vanishes on this
line, contradicting (1).
===
===
Subject: Re: Alternative ways to solve a quadratic equation
ETAtAhUAjMtlPW9w1+DoG1R+h2eeKmSrIlYCFA6hLaoqgy2Q8+
m2bI0CQ7pWxafi
Would this count?
Take the quadratic formula and ratioalize he NUMERATOR. Thus
x = 2c/(-b (+/-) sqrt(b^2-4ac))
This has advantages over the standard formula in some computer
applications and I have actually used it myself.
Responding to an earlier post, Newton's Method can be used to
exactly
pinpoint a rational root of ANY polynomial equation with
integer
coefficients. Recall from the Rational Root Theorem that a
rational
root times the leading coefficient must be an integer. If a
Newton
approximant times the leading coefficient looks like it might
be close
to an integer, try the corresponding rational solution.
--OL
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Coin-Flipping Machine
>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>http://www.npr.org/display_pages/features/feature_1697475.html
>Apparently, coin-toss-outcomes are more a function of
>human-unpredictability than of the coin's unpredictabilty.
> What a waste of time. I could have told him that. Oh, but
then he
> wouldn't have gotten a grant to study a non-problem.
Write again when you've contributed one-hundredth as much to
mathematics
as Persi has.
===
Subject: Re: Number Theory Problem!
> En el
mensaje:aa8e2a84.0402290627.462b3c2e@posting.google.com,
> Mark Sapir <m..v..s@mail.ru> escribi.97:
> 24C(n,5)+48C(n,4)+32C(n,3)+8C(n,2)+C(n,1)=1/5n^5 +
1/3n^3 + 7/15n
> That's nice! Is it true that every polynomial with rational
> coefficients f(n) such that f(n) is an integer for every
interger n
> is a linear combination of binomial coefficients C(n,k)?
> Mark Sapir
> Yes
> For those who (like me) wanted a reference here it is
(copied from
>
http://www.math.tamu.edu/~harold.boas/courses/math696/
Maple-functions.html):
> G. P.97lya proved [.86ber ganzwertige ganze Funktionen,
Rend. Circ.
Mat.
> Palermo 40 (1915) 1-16] that the polynomials that take
integer values
> at all integer points are precisely the linear combinations
with
> integral coefficients of the binomial polynomials of the form
> x(x-1)...(x-k+1)/k!, where k=0, 1, .... See Manjul Bhargava,
The
> factorial function and generalizations, American
Mathematical Monthly
> 107 (2000), number 9 (November), 783-799.
> But it is a consequence of the finite difference formula, no?
Yes, of course. So the true reference should be to Newton.
Mark Sapir
===
Subject: Re: Which topology text would be better for
self-study?
> I checked out a couple of topology books to try to learn
from on my own.
One
> is Munkres Topology and the other is Introduction to
Topology by Crump
Baker
>
http://www.amazon.com/exec/obidos/tg/detail/-/0697059723/qid=
1078093279/sr
> =1-8/ref=sr_1_8/102-3578915-2424159?v=glance&s=books
> I know that the Munkres book is popular but I'm not sure if
it would be
okay to
> learn with on my own. The book by Baker covers point-set
topology only
and
> that's all I am trying to learn. THe chapters are
Preliminary Topics,
> Topological Spaces, Subspaces and Continuity, Product Spaces,
Connectedness,
> Compactness, Separation Properties, and Metric Spaces.
> If there is a better book for me, I wouldn't mind buying it.
I don't know about better for you, but Dover has reprinted
Hocking and Young in an inexpensive paperback. Roughly the
first half is devoted to point-set topology, up to homotopy.
A good companion for any of these is Steen's Counter-examples
in Topology, also available as a Dover reprint. I'd almost
call this a *necessary* companion, regardless of what other
book you choose.
If you're studying point-set topology as background for
analysis, Kelley's General Topology is very good. It's
in Springer's GTM series and more expensive than the Dover
books, but it's probably in your library (most likely in an
older hardcover edition, not from Springer).
===
Subject: Re: eigenvalue must be defined only for square matrix?
...
> A x = lambda x, where x is a column vector would seem to
require A
> to be square
> I think singular values are a generalization of eigenvalues
to
> rectangular matrices. Apparently the singular value sigma
satisfies
> the following equations for some vectors u and v:
> A v = sigma u
> A' u = sigma v
And this means:
A'A v = sigma^2 v
and
AA' u = sigma^2 u
So a short definition of singular value might be:
the singular values of a matrix A are the positive square
roots of the
matrix AA' or A'A. (We have to be a bit careful for different
sizes of
these two matrices, however, the larger has only additional
singular
values with value 0.)
Note that the singular values of a square matrix are not
necessarily the
eigenvalues of that same matrix. There may even be no relation
at all.
For instance, look at:
( 1 1 )
( 0 1 )
The eigenvalues are 1 and 1, the singular values are [sqrt(5)
+- 1]/2,
quite a difference. So singular values are *not* a
generalisation of
eigenvalues. They do something different, see any text about
numerical
algebra.
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: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
===
>Subject: Coin-Flipping Machine
>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>http://www.npr.org/display_pages/features/feature_1697475.html
>Apparently, coin-toss-outcomes are more a function of
>human-unpredictability than of the coin's unpredictabilty.
>What a waste of time. I could have told him that. Oh, but
then he
>wouldn't have gotten a grant to study a non-problem.
So?
>Write again when you've contributed one-hundredth as much to
mathematics
>as Persi has.
Touched a nerve did I? Having a great body of work entitles
him to such
stupidity?
Sorry, I'm not an acedemic. If that's how the system works,
then fine. I
didn't
know
I was supposed to close one eye and look the other way.
Write again when you can explain how this work is important to
mathematical
knowledge.
>--
--
===
Subject: Re: John Michael Osbourne - December 3rd 1948
<darylsk@shaw.ca> : stumbled drunkenly into the group and
rudely
farted the following noxious cloud:
>207 Ozzy 3 12 48 338/28 +2998
>John 47 Michael 51 Osbourne 109
>flipped and fell on him, resulting in several broken bones.
<<<snipped a lot of psycho crap from Shawn pedophile-boy
Kabatoff>>
The following (courtesy of Waxy.org) is sort of an unofficial
FAQ
explaining the psychotic nonsense posted to Usenet by Shawn
Daryl
Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia and
meWARNING: Read below before even thinking about responding to
this
twit.
http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643
Usenet has the tendency to provide a public forum for those
who would
normally be scribbling in a closet. For example, take Daryl
Shawn
Kabatoff. For the last few years, he's methodically gathered
statistics from various sources, ranging from local newspaper
obituary pages to the food court of the Saskatoon Midtown
Plaza mall.
With all the raw data he's collected, he's attempting to prove
daily
that our full names are in mathematical harmony with our
birthdays.
His rants normally focus on a single individual he's met or
read
about, starting with calculations related to their birthdate
and full
names, blending in whatever other personal information about
their
family members, spouses, birthplace, and career he's been able
to
zealotry, and personal torment. I've never seen anything like
it.
With all the prime numbers, Fibonacci sequences and biblical
references, it's like reading the notebooks of Maximillian
Cohen and
John Nash combined. Unsurprisingly, several posts unfold to
reveal a
history of painful mental illness. If you have some time, take
a look.
I've detailed his posting history and a several sample posts
below.
Usenet Posting History:
January 27, 1999 to July 5, 2000 as Catsco@home.com
December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca
Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca
January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com
(original
posts have been
removed from Google Groups archive)
April 26, 2002 to Present as dar_kabatoff@hotmail.com
Selected Posts:
Tessa Lynne Smith
Dastageer Sakhizai and Helen Smith
Brett David Maki
Andrew Meredith Cotton
Kathryn Lee Hipperson
Amanda Dawn Newton
Mona Marie Etcheverry
Tony Peter Nuspl
Lisa Charlene McMillan
Grant Allyn Wood
Comments
scarier still is that saskatoon is my hometown, though not my
current
residence. and every single place he's mentioned in his posts
(most
notably nervous harold's and the roastary) were either places
i've
been (as it's a small city of 200K) or hangouts, ie. the two
places
mentioned. chances are i could email some friends back home
and find
out if they know of him, they (my friends that is) being of the
broadway-centred slacker ilk. myself, too, until i got out of
there.
eh, anyways. thought it odd to see all this. midtown mall. i
ate my
meals there, whilst waiting several days in line for star wars
episode
one, at the theatre across the street.
posted by andy raad on May 22, 2002 06:20 PM
Fascinating. It's like he's trying to take chaos and bind it
into
whatever rules he can find, religious, logical and otherwise.
Numbers
and math have a reliable pattern, something that can always be
proven
to true or false. People and religion do not. It reminds me of
Darren
Aronofsky's movie Pi. It's the story of an paraniod genius who
is
trying to find a pattern in Pi. A group that takes interest in
his
work is convinced that the existence of Pi, a number whose
existence
can be proven but no quantified, is proof of the existence of
God.
Kabatoff's hunt for patterns in something as random as name
selection
is a way to reconcile his deeply logical thought process with
his
conflicting religious views.
Exactly. I probably shouldn't have, but I e-mailed Daryl
yesterday,
asking him if he'd be willing to create a numerological
analysis for
me. I also asked him if he had seen either Pi or A Beautiful
Mind, and
what he thought of them. If he replies, I'll be sure to post
it.
I baked many pumpkin pies for Shawn (he likes pumpkin pies). I
rubbed
pumpkin pie all over my breasts for him, and my breasts turned
orange.
I am a pumpkin for Shawn.
posted by Trisha Blondie on July 24, 2002 10:41 PM
Um, that's swell. So, you're in love with him?
Shawn once went to a funeral for a Jehovah Witness that shot
himself
and the lemon tarts were very bad, they were not only sour but
were
rubbery as well. Shawn said that the guy was some kind of
Jehovah
Witness prophet, he saw in advance that the lemon tarts at his
funeral
were to be very very bad, and so he shot himself. Shawn said
that he
never ate pumpkin pie at a funeral but would like to some day.
Shawn
likes pumpkin pie and so I have been practicing to make very
good
pumpkin pies.
posted by Trisha Blondie on July 25, 2002 02:49 PM
Shawn said that the lemon tarts were sour, bitter and rubbery.
I don't think this guy takes notes. I think he has Total
Recall, and
it has driven him insane...
Oh... I almost forgot... I didnt spend thousands of dollars a
day
tormenting Daryl... We got a deal on tormenting that fiscal
year, it
only came to about 37cents a day....
Mr. Kabatoff attempts to portray himself as a victim, but in
fact he
is a violent predatory pedophile who is well known to his
local law
enforcement. In his post to multiple newsgroups with the
subject
Collecting Mail For The Coming Anti-Christ, he encourages
mothers to
send him photos of their naked daughters. Mr Kabatoff
explains, I
personally did not want photographs being mailed to (the coming
Ant-Christ) that were of underage children unless the parent
was
signing consent. He is banned from virtually all the shopping
malls
in his community because he stalks young people and sexually
harasses
them. He has an extensive arrest record which includes sexual
molestation charges. He's been hospitalized in mental
institutions
about his contact with young girls in many posts. Search
newsgroup
archives for posts by him containing the word nubile. As part
of his
harrassment, he provides personal details in a public forum,
such as
the real names of real children, in these and other posts.
About one
wanted her and her sister dead.
http://groups.google.com/groups?q=Daryl+Shawn+Kabatoff+dead+or
+in+my+bed&hl=
en&lr=&ie=UTF-8&selm=asqm35%24tjq5j%241%40ID-136124.
news.dfncis.de&rnu
He not only curses children and prays for their death in his
posts, he
also enjoys attending the funerals of young people: And so,
since
nubile sweeties are found in greatest abunce at the funerals of
high school students, then it is the funerals of high school
students
that make the very very best funerals, especially if there is
food...
I stuff my face (and my pockets) with all the good food and
look at
all the pretty nubile sweeties and have the time of my life..
.http://groups.google.com/groups?q=Daryl+Shawn+Kabatoff+nubile
+sex&hl=en&l
r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039%
40twister.socal.rr.
com&rnum=1
Many of his posts are sent to alt.teens.advice. However, he
liberally
spams, floods and crossposts his off-topic threatening and
offensive
missives to countless newsgroups. Some people HAVE problems
and some
folks ARE problems. Don't dismiss Mr. Kabatoff as a harmless
nut. When
he sends these posts to any newgroup, please help by reporting
him to
I knew of him when I was attending the University of
Saskatchewan.
He'd hang out in the Arts computer lab and all you'd see is
screens of
numbers racing by on his laptop. I have an original copy of his
Collecting Mail for the Coming Anti-Christ pamphlet, and have
seen
him be hauled away by campus security on more than one
occasion. My
friends and I refer to him as Crazy Number ManI've been
posting to (and about) Shawn for over two years with big
gaps in between. He has seen Pi and didn't like it and didn't
think it
resembled him at all. (Wrong, it fits him to a tee) He doesn't
have
total recall and has stated that he travels with a lap top to
notate
items. Also, he uses cut n' paste a lot if you read all the way
through his ramblings. He is anti-social as shown by his angry
statements towards those who, by his own admission, have been
kind
(but not kind enough) to him. Still, he's intelligent and
seems to be
able to take a joke on occassion. That's where I came in.
ALOHA
Reply to group
(Unsolicited e-mail is deleted from the server unread
if it comes from anyone not already in my addressbook.
I'll never even see it)
===
Subject: Re: John Michael Osbourne - December 3rd 1948
> <darylsk@shaw.ca> : stumbled drunkenly into the group and
rudely
> farted the following noxious cloud:
>207 Ozzy 3 12 48 338/28 +2998
>John 47 Michael 51 Osbourne 109
>flipped and fell on him, resulting in several broken bones.
> <<<snipped a lot of psycho crap from Shawn pedophile-boy
Kabatoff> The following (courtesy of Waxy.org) is sort of an
unofficial FAQ
> explaining the psychotic nonsense posted to Usenet by Shawn
Daryl
> Kabatoff AKA Dar, AKA Probababbilities. And now AKA marcia
and
> me
> WARNING: Read below before even thinking about responding to
this
> twit.
> http://www.waxy.org/archive/2002/05/21/dar_kaba.shtml#000643
> Usenet has the tendency to provide a public forum for those
who would
> normally be scribbling in a closet. For example, take Daryl
Shawn
> Kabatoff. For the last few years, he's methodically gathered
statistics from various sources, ranging from local newspaper
> obituary pages to the food court of the Saskatoon Midtown
Plaza mall.
> With all the raw data he's collected, he's attempting to
prove daily
> that our full names are in mathematical harmony with our
birthdays.
> His rants normally focus on a single individual he's met or
read
> about, starting with calculations related to their birthdate
and full
> names, blending in whatever other personal information about
their
> family members, spouses, birthplace, and career he's been
able to
> zealotry, and personal torment. I've never seen anything
like it.
> With all the prime numbers, Fibonacci sequences and biblical
> references, it's like reading the notebooks of Maximillian
Cohen and
> John Nash combined. Unsurprisingly, several posts unfold to
reveal a
> history of painful mental illness. If you have some time,
take a look.
> I've detailed his posting history and a several sample posts
below.
> Usenet Posting History:
> January 27, 1999 to July 5, 2000 as Catsco@home.com
> December 9, 2000 to May 4, 2001 as s.kabatoff@sk.sympatico.ca
> Oct 30, 2001 to Oct 31, 2001 as kabatoff@the.link.ca
> January 20, 2002 to April 17, 2002 as s_kabatoff@hotmail.com
(original
> posts have been
> removed from Google Groups archive)
> April 26, 2002 to Present as dar_kabatoff@hotmail.com
> Selected Posts:
> Tessa Lynne Smith
> Dastageer Sakhizai and Helen Smith
> Brett David Maki
> Andrew Meredith Cotton
> Kathryn Lee Hipperson
> Amanda Dawn Newton
> Mona Marie Etcheverry
> Tony Peter Nuspl
> Lisa Charlene McMillan
> Grant Allyn Wood
> Comments
> scarier still is that saskatoon is my hometown, though not
my current
> residence. and every single place he's mentioned in his
posts (most
> notably nervous harold's and the roastary) were either
places i've
> been (as it's a small city of 200K) or hangouts, ie. the two
places
> mentioned. chances are i could email some friends back home
and find
> out if they know of him, they (my friends that is) being of
the
> broadway-centred slacker ilk. myself, too, until i got out
of there.
> eh, anyways. thought it odd to see all this. midtown mall. i
ate my
> meals there, whilst waiting several days in line for star
wars episode
> one, at the theatre across the street.
> posted by andy raad on May 22, 2002 06:20 PM
> Fascinating. It's like he's trying to take chaos and bind it
into
> whatever rules he can find, religious, logical and
otherwise. Numbers
> and math have a reliable pattern, something that can always
be proven
> to true or false. People and religion do not. It reminds me
of Darren
> Aronofsky's movie Pi. It's the story of an paraniod genius
who is
> trying to find a pattern in Pi. A group that takes interest
in his
> work is convinced that the existence of Pi, a number whose
existence
> can be proven but no quantified, is proof of the existence
of God.
> Kabatoff's hunt for patterns in something as random as name
selection
> is a way to reconcile his deeply logical thought process
with his
> conflicting religious views.
> Exactly. I probably shouldn't have, but I e-mailed Daryl
yesterday,
> asking him if he'd be willing to create a numerological
analysis for
> me. I also asked him if he had seen either Pi or A Beautiful
Mind, and
> what he thought of them. If he replies, I'll be sure to post
it.
> I baked many pumpkin pies for Shawn (he likes pumpkin pies).
I rubbed
> pumpkin pie all over my breasts for him, and my breasts
turned orange.
> I am a pumpkin for Shawn.
> posted by Trisha Blondie on July 24, 2002 10:41 PM
> Um, that's swell. So, you're in love with him?
> Shawn once went to a funeral for a Jehovah Witness that shot
himself
> and the lemon tarts were very bad, they were not only sour
but were
> rubbery as well. Shawn said that the guy was some kind of
Jehovah
> Witness prophet, he saw in advance that the lemon tarts at
his funeral
> were to be very very bad, and so he shot himself. Shawn said
that he
> never ate pumpkin pie at a funeral but would like to some
day. Shawn
> likes pumpkin pie and so I have been practicing to make very
good
> pumpkin pies.
> posted by Trisha Blondie on July 25, 2002 02:49 PM
> Shawn said that the lemon tarts were sour, bitter and
rubbery.
> I don't think this guy takes notes. I think he has Total
Recall, and
> it has driven him insane...
> Oh... I almost forgot... I didnt spend thousands of dollars
a day
> tormenting Daryl... We got a deal on tormenting that fiscal
year, it
> only came to about 37cents a day....
> Mr. Kabatoff attempts to portray himself as a victim, but in
fact he
> is a violent predatory pedophile who is well known to his
local law
> enforcement. In his post to multiple newsgroups with the
subject
Collecting Mail For The Coming Anti-Christ, he encourages
mothers to
> send him photos of their naked daughters. Mr Kabatoff
explains, I
> personally did not want photographs being mailed to (the
coming
> Ant-Christ) that were of underage children unless the parent
was
> signing consent. He is banned from virtually all the
shopping malls
> in his community because he stalks young people and sexually
harasses
> them. He has an extensive arrest record which includes sexual
> molestation charges. He's been hospitalized in mental
institutions
> about his contact with young girls in many posts. Search
newsgroup
> archives for posts by him containing the word nubile. As
part of his
> harrassment, he provides personal details in a public forum,
such as
> the real names of real children, in these and other posts.
About one
> wanted her and her sister dead.
>
http://groups.google.com/groups?q=Daryl+Shawn+Kabatoff+dead+or
+in+my+bed&hl=e
n&lr=&ie=UTF-8&selm=asqm35%24tjq5j%241%40ID-136124.
news.dfncis.de&rnu
> He not only curses children and prays for their death in his
posts, he
> also enjoys attending the funerals of young people: And so,
since
> nubile sweeties are found in greatest abunce at the funerals
of
> high school students, then it is the funerals of high school
students
> that make the very very best funerals, especially if there
is food...
> I stuff my face (and my pockets) with all the good food and
look at
> all the pretty nubile sweeties and have the time of my life..
>
.http://groups.google.com/groups?q=Daryl+Shawn+Kabatoff+nubile
+sex&hl=en&l
>
r=&ie=UTF-8&scoring=d&selm=LfXN8.63042%24R53.25142039%
40twister.socal.rr.
> com&rnum=1
> Many of his posts are sent to alt.teens.advice. However, he
liberally
> spams, floods and crossposts his off-topic threatening and
offensive
> missives to countless newsgroups. Some people HAVE problems
and some
> folks ARE problems. Don't dismiss Mr. Kabatoff as a harmless
nut. When
> he sends these posts to any newgroup, please help by
reporting him to
> I knew of him when I was attending the University of
Saskatchewan.
> He'd hang out in the Arts computer lab and all you'd see is
screens of
> numbers racing by on his laptop. I have an original copy of
his
Collecting Mail for the Coming Anti-Christ pamphlet, and have
seen
> him be hauled away by campus security on more than one
occasion. My
> friends and I refer to him as Crazy Number Man
> I've been posting to (and about) Shawn for over two years
with big
> gaps in between. He has seen Pi and didn't like it and
didn't think it
> resembled him at all. (Wrong, it fits him to a tee) He
doesn't have
> total recall and has stated that he travels with a lap top
to notate
> items. Also, he uses cut n' paste a lot if you read all the
way
> through his ramblings. He is anti-social as shown by his
angry
> statements towards those who, by his own admission, have
been kind
> (but not kind enough) to him. Still, he's intelligent and
seems to be
> able to take a joke on occassion. That's where I came in.
> ALOHA
> Reply to group
> (Unsolicited e-mail is deleted from the server unread
> if it comes from anyone not already in my addressbook.
> I'll never even see it)
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
===
>>Subject: Coin-Flipping Machine
>>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>>http://www.npr.org/display_pages/features/feature_1697475.
html
>Apparently, coin-toss-outcomes are more a function of
>>human-unpredictability than of the coin's unpredictabilty.
> What a waste of time. I could have told him that. Oh, but
then he
>> wouldn't have gotten a grant to study a non-problem.
> So?
>Write again when you've contributed one-hundredth as much to
mathematics
>as Persi has.
> Touched a nerve did I? Having a great body of work entitles
him to
> such stupidity? Sorry, I'm not an acedemic. If that's how
the system
> works, then fine. I didn't know I was supposed to close one
eye and
> look the other way.
> Write again when you can explain how this work is important
to
mathematical
> knowledge.
You've accused Diaconis of wasting time, studying a
non-problem,
and stupidity. You haven't given anything remotely resembling
justification for these accusations. So the only thing you've
given me to go on is your reputation versus his. With all due
respect, I know enough about him & his work to be pretty
confident
that whatever he's studying he's got a bloody good reason for
it.
I also suggest that three or four paragraphs on an NPR website
is not a sufficient basis on which to make adverse judgements
of
a mathematical research program.
===
Subject: Re: Alternative ways to solve a quadratic equation
> Would this count?
> Take the quadratic formula and ratioalize he NUMERATOR. Thus
> x = 2c/(-b (+/-) sqrt(b^2-4ac))
Which is what I suggested in this thread on 25 February.
But maybe that post hasn't reached your site yet.
===
Subject: re:Which topology text would be better for self-study?
This Dover book is cheap and easy to read, making it very
suitable to
self-study. It only covers point-set
topology, though. It has nothing from homology theory or
algebraic
topology at all.
http://www.amazon.com/exec/obidos/tg/detail/-/0486663523/qid=
1078107793/sr=1
-1/ref=sr_1_1/002-0705903-6106436?v=glance&s=books
===
Subject: Quarter circle function
x^2 + y^2 = r^2
For a radius of one:
f(x)=sqrt(1-x^2) gives a quarter circle centered at (0,0).
What is the function for a quarter circle centered at (1,1)?
===
Subject: Re: Catenary curves question
> I'm editing a puzzle book which asks:
What will happen to the shape of a catenary curve if you tie
and
suspend
> two weights at equal horizontal distances apart?
> I can't find the answer anywhere on the Internet - does
anyone know?
> David
A V- like shape of catenary assumes a more U- like shape. It
is an
exercise in Engineering Mechanics of suspended catenary bridge
building application. To answer the question quantitatively, an
iteration is needed. The following may be suggested to the
interested
students.
If the original catenary y=a Cosh [x/a] is suspended between
two
points of given distance 2 X apart at same height/y-
coordinate, it is
the symmetrical case. Arc length is (from lowest point to
fixed point
) L= a Sinh [ X/a] (verify this). Find a by iteration such as
Newton-Raphson.
Weights are now tied introducing taut end portions arc length
H below
original fixed suspension points hanging vertically straight
down. A
shallower catenary A Cosh [x/A] of U shape in between the new
tied
points is formed. Sketch for yourself how the catenaries of
parameters
a and A are placed together. ( A > a) .
As the catenary arc lengths are same before and after the tie
procedure, equate them:
L = H + A Sinh [X/A] ; this transcendental equation can be
solved
for the new A value by iteration. Finding a is strictly not
needed,
but plotting both profiles together is a satisfying exercise.
Code
and sketch the two cases together in Mathematica, Matlab or
Maple.
This serves also as simulation exercise.
The solution is beautifully amenable to verification by direct
physics
experiment. I suggest you take, for example a flexible hanging
Venetian blinds (vertical mover with small hinged beads in it)
that
make it flexible. The three given constants may be chosen,
say, as X=
5 inches, (half-) string length = L = 8 inches, H=1 inch in
order to
find A.
After doing all this, then if still interested, proceed to the
unsymmetrical case where the original suspension points are
not at the
same height.
[A side remark, not relevant at this stage of learning. The
situation
here is however not an isoperimetric problem in variational
calculus,
because between the two portions for second case we have
introduced a
new slope discontinuity.]
Hope this helps.
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
===
>>Subject: Coin-Flipping Machine
>>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>>Story from NPR:
>>http://www.npr.org/display_pages/features/feature_1697475.
html
>>Apparently, coin-toss-outcomes are more a function of
>>human-unpredictability than of the coin's unpredictabilty.
> What a waste of time. I could have told him that. Oh, but
then he
>> wouldn't have gotten a grant to study a non-problem.
>So?
>Write again when you've contributed one-hundredth as much to
mathematics
>as Persi has.
>Touched a nerve did I? Having a great body of work entitles
him to
>such stupidity? Sorry, I'm not an acedemic. If that's how the
system
>works, then fine. I didn't know I was supposed to close one
eye and
>look the other way.
>Write again when you can explain how this work is important to
mathematical
>knowledge.
>You've accused Diaconis of wasting time, studying a
non-problem,
>and stupidity. You haven't given anything remotely resembling
>justification for these accusations.
Huh? Didn't you read the URL in question? What more
justification
do I need?
>So the only thing you've
>given me to go on is your reputation versus his.
I have no reputation, so I'll come out second best in any such
comparison. But aren't such comparisons a type of logical
fallacy?
Shouldn't _any_ criticism be based on merit, not reputation?
Are James Harris and John Correy actually right about
mathemeticians? And all this time I thought they were cranks.
>With all due
>respect, I know enough about him & his work to be pretty
confident
>that whatever he's studying he's got a bloody good reason for
it.
In other words, you cannot explain how this research is
important.
And yet, you support it without anything remotely resembling
justification. That makes you no better than me, eh?
>I also suggest that three or four paragraphs on an NPR website
>is not a sufficient basis on which to make adverse judgements
of
>a mathematical research program.
Well, who's fault is that? Mine? Sorry, but I call a spade a
spade.
If it looks stupid, I'll say so. Go ahead, prove me wrong.
Point out how
the coin-flipping machine was an insignificant element of a
much more
important work that was distorted be the media. I won't mind.
I don't
post from an .edu domain. On the other hand, I won't be
embarrassed
if it turns out I'm wrong. Can you say the same?
>--
--
===
Subject: Req. help on On Formally Undecidable Propositions...
Greetings,
I just have a question about one symbol used in Godel's paper,
On
Formally Undecidable Propositions of Principia Mathematica and
Related
SystemsThe paper I'm referring to is the web-based version at:
http://home.ddc.net/ygg/etext/godel/godel3.htm
My question refers to this paragraph:
The following formulae (I-V) are called axioms (they are set
out with
the help of the customarily defined abbreviations: .,
<non-ASCII
symbols removed>,
and subject to the usual conventions about omission of
brackets):
What is the meaning of the . symbol?
What are the usual conventions about omission of brackets?
I tried searching for basic set-theory notation but could not
find
an answer to my question through that search. Hopefully someone
here can help me out.
Cheers,
Chris
===
Subject: Re: Prime factors of number near googolplexplex
> He didn't factor those numbers, just guessed some of their
small
> factor.
I'm curious how one guesses small factors like 3898556974549,
2000665038247,
1301687275127, 2017970156617, or 2184279173669?
===
Subject: differential equation - difficult
I would very much like to solve:
D_1/2[F(t),t] = -F(t)*Log(F(t)) [1]
where D_1/2[f,t] := d^(1/2)f(t)/dt^(1/2) is the fractional 1/2
order
differential operator (sometimes known as the
semi-derivitive). i.e. [1]
is
a nonlinear fractional order differential equation.
One possible way i thought of was to invert the equation,
literealy, i.e.
D_1/2[t,F] = -1/(F(t)*Log(F(t)))
and then reverse the operator by taking D_-1/2 on both sides -
t = -D_-1/2 [ 1/(F*Log(F)) , F] = -1/Sqrt(pi) * Integral (0,F)
1/(g*Log(g))
*1/Sqrt(F-g) dg (by the Rieman-Liouville defintiion of
fractional
integration)
but there are problems with this: 1. i dont know if i can
simply invert the
equation when the operator is of fractional order...and 2) i
cant do the
above integral anyway...so its probably doesnt get me anywhere
even if it
is
allowed. I fear there is little hope for an explicit solution
of [1].
another way of framing this problem is to Solve the following
nonlinear
weakly singular integral equation (of Abel type: i say of
because it is
like the Abel integral equation however it is worse since the
unkown is on
both sides of the equality)
F(t) = -1/Sqrt(pi) * Integral(0,t) F(k)*Log(F(k))/Sqrt(t-k) dk
cheers
moth
===
Subject: Re: Quarter circle function
> x^2 + y^2 = r^2
> For a radius of one:
> f(x)=sqrt(1-x^2) gives a quarter circle centered at (0,0).
> What is the function for a quarter circle centered at (1,1)?
The equation of a whole circle of radius r centered at (1,1) is
(x-1)^2 + (y-1)^2 = r^2
The upper semi-circle is given by
y = sqrt(r^2 - (x-1)^2) + 1, for 1-r <= x <= 1+r
The upper right quadrant is given by that same equation with x
restricted to the interval 1 <= x <= r+1.
===
Subject: re:Which topology text would be better for self-study?
I'll take a look at those books. If it helps at all, I'm not
the best at
math
so I was hoping for as gentle and explanatory an introduction
as possible,
if
that makes any sense. At this level, I know it's next to
imopossible to
have a
text with solutions but I really do like it.
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Coin-Flipping Machine
>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>http://www.npr.org/display_pages/features/feature_1697475.html
>Apparently, coin-toss-outcomes are more a function of
>human-unpredictability than of the coin's unpredictabilty.
> What a waste of time. I could have told him that. Oh, but
then he
> wouldn't have gotten a grant to study a non-problem.
>So, perhaps it is more accurate to talk about, rather than a
biased
>coin, a biased HUMAN-BEING!...
>;)
> towards heads). Of course, coins don't have a symmetrical
mass
> distribution, so I could have told him that also. And I
didn't see
> any reference to the control test of an unstamped coin blank.
> And is he conducting the tests in a vacuum chamber at a
controlled
> temperature? I wonder how much of my tax money he fleeced out
> of the government for this stupid project?
The answer to how much tax money he fleeced out of the
government would
be none. The origin of the coin flipping machine came from a
comment he
made in a lecture sometime in the Fall of 1995, in which he
said that
when he was growing up he saw such a machine advertised in
scientific
toys catalog. He got curious, called up the catalog to see if
they had
ever had such a thing, and was told no. He mentionned this to
someone
in the physics department, and a couple days later some guys
in a
physics lab had built it out of spare parts lying around the
lab. No
grant applications, no spending money, just some geeks playing
around
and having fun.
I skipped the talk, but am told that
the recent result, which presumably the NPR report is
referring to, says
roughly that if you flip a coin with an initial velocity that
is a
random variable with a density, then in the limit as the
height of the
flip tends to infinity, the probability of coming up in the
original
orientation is greater than 1/2. Likewise, given flipping to
some
initial height, the limiting probability of the original
orientation is
greater than 1/2 as the angular velocity tends to infinity.
Again, I
have neither read the paper nor attended a talk about it, so
the actual
result might be different.
--
To respond, change dot to a dot, and com to net
===
Subject: The Triangle: A One-Page Primer
Summary: The Triangle: A One-Page Primer
Message-Id: <slrnc45e2v.9fh.cjf@syn12.vpn.linuxforce.net
My compilation, The Triangle: A One-Page Primer is available at
http://www.CJFearnley.com/triangle.primer.pdf
Please let me know of any corrections and especially if you
find the
content and style useful.
The lesson of history is that our firmest convictions are not
to be
asserted dogmatically; in fact they should be most suspect;
they mark
not our conquests but our limitations and our bounds. --
Morris Kline
Christopher J. Fearnley | Linux/Internet/Web Consulting
Chris@CJFearnley.com | Design Science Revolutionary
http://www.CJFearnley.com | Explorer in Universe
Dare to be Na.95ve -- Bucky Fuller
===
Subject: Inverse Primes in Base Factorial
This is something unusual I observed about the factorial base.
The representation of 1/p, p prime,
will have p-1 places and the last
coefficient is equal to p-1.
1/2 = 1/2! = .1
1/3 = 2/3! = .02
1/5 = 1/3! + 4/5! = .0104
1/7 = 3/4! + 2/5! + 6/7! = .003206
...
Does anyone know why 1/p has this pattern?
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
>You've accused Diaconis of wasting time, studying a
non-problem,
>and stupidity. You haven't given anything remotely resembling
>justification for these accusations.
> Huh? Didn't you read the URL in question? What more
justification
> do I need?
I did read the page. Nothing I saw there justifies accusing
Diaconis
of wasting time, studying a non-problem, or stupidity.
Evidently,
you disagree. Perhaps you could be a little more forthcoming,
as
it's clear that I need convincing.
>So the only thing you've
>given me to go on is your reputation versus his.
> I have no reputation, so I'll come out second best in any
such
> comparison. But aren't such comparisons a type of logical
fallacy?
> Shouldn't _any_ criticism be based on merit, not reputation?
> Are James Harris and John Correy actually right about
> mathemeticians? And all this time I thought they were cranks.
When you make a criticism based on merit, I'll read it, and
judge
it on its merits. So far you've only made accusations which
you say
are founded in the NPR piece and which I say are not. Since I
haven't read Diaconis' write-up, all I have to go on is your
unfounded accusations versus his body of work. It makes perfect
sense under the circumstances to give Diaconis the benefit of
the
doubt and to assume - until you give me some good reason to
think
otherwise - that he has not been wasting his time & is not
being
stupid.
What's so hard to understand about that?
>With all due
>respect, I know enough about him & his work to be pretty
confident
>that whatever he's studying he's got a bloody good reason for
it.
> In other words, you cannot explain how this research is
important.
> And yet, you support it without anything remotely resembling
> justification. That makes you no better than me, eh?
I don't recall saying that I am better than you. It appears
that I
am distinctly inferior to you, in that you are able to find
evidence
of Diaconis' stupidity where I can't.
Diaconis has apparently found that If a coin starts out heads,
it ends
up heads when caught more often than it does tails. Is that
important?
I don't know. I find it interesting; if I know Diaconis at
all, he
won't stop there, but will continue until he has found a model
to
explain why the coin ends up heads more often than tails, and
that
model will have some pretty interesting mathematics in it.
You are saying that you already knew the coin would come up
heads
more often? You have a model which explains why this should be?
Show your work! When someone announces a proof that all the
non-trivial
zeros of the zeta function are on the critical line, are you
going
to say that you could have told her that, too? Will you expect
people
to believe you?
>I also suggest that three or four paragraphs on an NPR website
>is not a sufficient basis on which to make adverse judgements
of
>a mathematical research program.
> Well, who's fault is that? Mine? Sorry, but I call a spade a
spade.
Whose fault is what? That NPR condensed a 30-page paper
[warning - I
made that number up] into a few paragraphs? No, that's not
your fault.
That you chose to judge a 30-page paper by what you got out of
- or
read into - those paragraphs? Well, of course that's your
fault - whose
fault other than yours could it be?
> If it looks stupid, I'll say so. Go ahead, prove me wrong.
Point out how
> the coin-flipping machine was an insignificant element of a
much more
> important work that was distorted be the media. I won't mind.
At the risk of oversimplifying things to a ridiculous degree,
let's say
there are two possibilities here: the research is worthless,
or it's
worthwhile. Given nothing else whatsoever to go on, we might
take the
position that the two outcomes are equally likely. You say
there is
something else to go on; the NPR summary is enough to convince
you the
research is worthless. I don't see that, but I do see
Diaconis' track
record as enough to convince me that there's probably a lot
more here
than what's in the NPR report.
> I don't post from an .edu domain.
I have no idea how that observation advances this discussion.
> On the other hand, I won't be embarrassed
> if it turns out I'm wrong. Can you say the same?
Yes: I, too, won't be embarrassed if it turns out that you are
wrong.
Seriously, though, I've been wrong many many many many times
in this
newsgroup, I have sometimes been embarrassed at how wrong I've
been,
I have tried to come clean when I have been wrong, and I have
tried
to learn something from the experience. What's your point?
===
Subject: Re: This Week's Finds in Mathematical Physics (Week
203)
> Also available at http://math.ucr.edu/home/baez/week203.html
> This Week's Finds in Mathematical Physics - Week 203
> John Baez
> Last week I posed this puzzle: to find a Golden ObjectHi
John Baez and all...
Converting a Black Hole to a Golden Hole.
I came across the Golden Number studying
BH's. I hypothesized the conventional GR equation
g_00 = 1 - 2m /r
would *look* like
g_00 = 1 - 2M*sqrt(g_00) /r
where m=M*sqrt(g_00).
The reason for this conjecture is found in the
red shift. Suppose a photons energy has a fixed
ratio to mass m when emitted by the gravitating
mass as measured by an observer at the surface.
The idea being, the surface observer calculates
the gravitating mass and Energy E, and then measures
the emitted photon to be energy e, and figures the
ratio of Energy/energy to be
ratio = E/e (invariant)
The photon itself is red shifted as it tends to infinity,
yet the ratio is preserved due to it's invariance, ie
it's in units of Energy/energy.
The observer at infinity receives the red shifted
photon and using the invariant ratio determines
the gravitating mass at that location to be
mass = Mass*sqrt(g_00).
Hence the observer at infinity will determine
g_00 = 1-2M*sqrt(g_00) /r = 1-2m /r,
So using the invariant ratio E/e every observer
will be able to agree to the value of g_00 at
every r using the shifting frequency of photons
as they change in altitude.
Every observer can set 2M/r=1 to determine a
relative *event horizon*. To do this, algebraically
set,
g_00 = X^2 = 1 - X and rearrange to,
X^2 + X -1 =0
Using the quadratic equation , X is,
sqrt(5) -1
X = ------------- if I'm not mistaken.
2
This is also known as the Golden Number G in
J Baez's notation, where,
G = sqrt(g_00) at the common event horizon
defined by 2M/r =1, for all altitudes rG is invariant for a
stationary observer at any r
hence a Golden Hole. ((I conditioned stationary
for CYA, the effects of motion aren't germain)).
There is a complex *hidden* caveat in my conjecture,
because it assumed gravitational field intensity (from
a Newtonian perspective and system of reference)
reduced in proportion to the red shift, to preserve
the invariant ratio.
Indeed this is how I used the same value of r
that r2 = x2 + y2 + z2 is unperturbed by gravitation,
and coordiante light speed variations that are only real
and accountable phenomena in curved spacetime in a
GR field.
In defense, the r in g_00= 1-2m/r is usually defined
that way, and is used that way in the conjecture.
Ken S. Tucker
===
Subject: ?? categoricity or naive felicity ??
: |
: |> Why don't you state one single mind bogling fact
: |
: |Classical propositional calculus has non-Boolean
: |models! Related to this and even more surprising
: |is that classical propositional calculus can be
: |modeled by a non-Boolean lattice [Reference 6], a
: |fact apparently overlooked for over 100 years!
: |Common intuition is that classical propositional
: |calculus and Boolean algebra go hand-in-hand.
: |Lattice O6 is a counterexample that shows this
: |intuition is false.
:
: Let me say first that I do find this amusing and worthy of
note. To
: me, however, it falls somewhat short of mind-boggling. Being
amazing
: is harder than it seems.
:
: My first reaction was that it would have to depend on what
exactly one
: meant by modeled. For example, since there's a translation of
classical
: logic into intuitionist logic, any model (however you define
it) of
: intuitionist logic will in a less direct or meaningful way
constitute
: some kind of model of classical logic. So
:
: . 1
: |
: . p
: |
: . 0
:
: which is a Heyting algebra (where ~~p=1) could count as a
model in a
: *weak* sense, where one has redefined the connectives to be
something
: other than their normal meaning in a Heyting algebra. For
instance,
: the classical x or y is defined as ~(~x & ~y), so that p or
p evaluates
: to 1. Pretty soon, though, you notice it's a little silly to
keep
: pretending to distinguish between elements with the same
double negation,
: and when you identify them, you have a Boolean algebra again.
present is not making a very useful distinction unless there
is something
in
even stronger claim that the models they present may be more
faithful to
the
semantics of the logics in a certain sense. Although I still
need to read
on categoricity I need to clear, I think that is an amazing
possibility.
: In the referenced example,
:
: . 1
: /
: ~q . . p
: | |
: ~p . . q
: /
: . 0
:
: X->Y is defined to mean ~X or Y, rather than being the
minimal element Z
: with the property that X&Z <= Y. So in particular, it's
possible to have
: X<->Y (i.e. (X->Y)&(Y->X)) evaluate to 1 even though X and Y
are distinct
: elements of the lattice.
The paper discusses a little the difference between the
equivalence
relations and identity, which is a widely established field of
research
that
in my opinion would be more well known if people had to deal
with such
problems by learning more nonclassical logics (its a real
world problem in
pattern recognition). There is a theorem (2.2) which covers
which cases
equivalence works as identity, and then the paper shows an
example where
the
four specific implications being explored did not fit
equivalence with
identity.
That example was O6, the lattice you show. It was saying the
same thing
you
are. That's the only place in the paper where that lattice is
used.
Are you sure you're not the skimmer? =)
: The example should serve as a reminder, then, that the sense
in which a
: Boolean algebra incarnates classical logic depends on
assuming something
: that ensures that equivalent elements (in the sense of X<->Y
evaluating
: to 1) are equal, as elements of the algebra. Since p<->q
evaluates to 1,
: we'd need to identify p with q, and ~p with ~q, which leaves
us with the
: Boolean algebra
:
: . 1
: /
: ~p . . p=q
: /
: . 0
:
: having the property that the same expressions evaluate to 1
in it as
: evaluate to 1 in the given six-element example.
:
: I'd categorize this as cute rather than mind bogglingOk,
Keith, you have to give me some, here. You just pulled out one
example
you made up and then showed an example demonstrating a theorem
that
explained something about usefulness of certain notions over
certain types
of structures which they used to illustrate the same point as
you. You
have
just cleaning up some foundational points there, particularly
the one you
question here.
confused with category theory). The uneasiness being presented
in the
the fact that their are various weakenings of the axiom system
which also
model the logics. There is a degree of unnecessary definition
which
overly restricts the standard models, and that even at the
propositional
level they are not categorical.
It's something I didn't know. 's post surprised me.
And this notion that when you use the weaker model, you more
faithfully
represent the properties of the logic in the properties of the
lattice,
that
has me intrigued (and has me tracking down papers and planning
my next
library trip). Because that make an explicit claim about
faithfulness I
have only looked at from other, less accurate positions.
You see, you call me a skimmer, Keith, even though from some
of your
responses it seemed you had not noticed some of the basic
positions I have
already mentioned, and you add all of these little package
deals that you
try to ambivalently offer as mere possibilities or indirect
associations
of my not understanding in detail what I am talking about. I
am much too
obsessive to start a post with only a half-assed notion of
what I was
talking about, but you, and quite a number of others,
_instead_of_answering_the_questions_of_my_post_
proceeded to push and prod me to see how much of a bs'er I
was. When I
tire
one of the pack out by answering all of their questions, they
back out for
a
while and someone else comes in. This is what I meant about
the male alpha
problem, which you believe is some simplistic explanation.
It's an obvious explanation, Keith. There is a difference.
===-=-=-=-=-
===
Subject: Re: ?? categoricity or naive felicity ??
: That example was O6, the lattice you show. It was saying the
same thing
you
: are. That's the only place in the paper where that property
of the
lattice
: is used. ^^^^^^^^^^^^^^^
Just to clear up a weak sentence.
===-=-=-=-=-
===
Subject: Re: Coin-Flipping Machine
> Story from NPR:
>
http://www.npr.org/display_pages/features/feature_1697475.html
A somewhat more detailed story is at
===
Subject: Re: Logic question #2
> so we proceed with i=L.
> For i=L, the word rFF can only be OFF
Oops. I meant i=F.
Squishua
===
Subject: Re: Hang them by their own G-strings.?
In sci.math, Uncle Al
<UncleAl0@hate.spam.net
<4042391C.79B41B5F@hate.spam.net>:
> Nothing
> Hey Jack-o, men wear T-straps or banana hammocks. Girls and
> transvestites wear G-strings. Now you know twice as much as
you did
> two sentences ago.
Just to confuse things even further, the President's cadre of
Secret Service are occasionally called G-men...
[.sigsnip]
#191, ewill3@earthlink.net -- I'm not confused, though. :-)
It's still legal to go .sigless.
===
Subject: Re: Coin-Flipping Machine
> towards heads).
says. I think what it says is that the coin has a slight
tendency
to come up whichever way it started - e.g., if it's tails when
you
flip it, then it has a slight tendency to be tails when you
catch
it.
Could you have told him that?
===
Subject: Re: Give me that old time ontology: (was: the
anticlassicalist }{
i:
> |
> |> Why don't you state one single mind bogling fact
> |
> |Classical propositional calculus has non-Boolean
> |models! Related to this and even more surprising
> |is that classical propositional calculus can be
> |modeled by a non-Boolean lattice [Reference 6], a
> |fact apparently overlooked for over 100 years!
> |Common intuition is that classical propositional
> |calculus and Boolean algebra go hand-in-hand.
> |Lattice O6 is a counterexample that shows this
> |intuition is false.
> Let me say first that I do find this amusing and worthy of
note. To
> me, however, it falls somewhat short of mind-boggling. Being
amazing
> is harder than it seems.
[...]
> I'd categorize this as cute rather than mind bogglingHmm...
Perhaps you could look at Lemma 4 in
http://citeseer.nj.nec.com/feigelson97forbidden.html
and explain how the failure of logical equivalence and mutual
exclusion
relate
to this non-classical structure.
The one thing consistent concerning arrogant assholes is that
they have a
lot
to say about things they haven't thought about.
:-)
===
Subject: Re: Class vs. Dimension Equations
>When I talk about the dimension equation of a finite group, I
mean the
>one expressing the order of the group as the sum of the
squares of the
>dimensions of the irreducible representations of the group
over the
>field of complex numbers.
>My question is: Is the dimension equation directly computable
(without
>knowing the group) from the class equation or vice versa?
>---- David
> You don't say what you mean by the class equation.
You surmised correctly, from what you posted. Good work and
thank you
for informing me that this term isn't so standard. Forgive me
for
having grown up (in part) on Artin's blue algebra book.
===
Subject: Re: Logic question #4
> Here is question #4. I realize this is more a math question
than a
> logic question. I figured part (i) to be 21. If you didn't
get that,
> let me know.
> Direct communication lines are being made between pairs of
people in
> an office of nine: three managers, their three secretaries,
and three
> technicians. The three managers are connected to each other.
Each
> manager is also connected to his own secretary. The
secretaries are
> connected to each other and to each technician. The
technicians are
> also connected to each other.
> (i) How many lines are there?
Who cares when the real problem is what's wrong with management
that it has no need to communicate directly with technicians?
NASA lost two shuttles just because managment refused to
listen to
enginers.
(C) 21
> (ii) What is the greatest number of lines that can
simultaneously go
> bad and still allow a message to be passed on from any one
person to
> any other?
(F) 3
> (iii) A quartette is a group of four persons each of whom is
connected
> by a direct line. How many quartettes are there?
No managers are in any quartets.
(D) 15
===
Subject: Re: Inverse Primes in Base Factorial
> This is something unusual I observed about the factorial
base.
> The representation of 1/p, p prime,
> will have p-1 places and the last
> coefficient is equal to p-1.
> 1/2 = 1/2! = .1
> 1/3 = 2/3! = .02
> 1/5 = 1/3! + 4/5! = .0104
> 1/7 = 3/4! + 2/5! + 6/7! = .003206
> ...
> Does anyone know why 1/p has this pattern?
It is Wilson's Theorem that says
(p-1)! = -1 (mod p).
===
Subject: Re: ?? categoricity or naive felicity ??
Oh... wait, I see know. I couldn't get to the web site the
first time I
tried, so I didn't realise you were pulling your info off of
it. You might
not have even read the paper.
That is just a postulate I'm tossing around, because it makes
your
statements more understandable (though still off the mark). I'm
postulating
that you are one of those skimmer types that likes to go
around calling
everyone else skimmers to take the attention off of themselves.
===-=-=-=-=-
===
Subject: re:Which topology text would be better for self-study?
> I'll take a look at those books. If it helps at all, I'm not
the best at
math
> so I was hoping for as gentle and explanatory an
introduction as possible,
if
> that makes any sense. At this level, I know it's next to
imopossible to
have a
> text with solutions but I really do like it.
The exercises are there for you to learn. If you get stuck,
show us your
work and where the snafu.
Here's some on line stuff
http://at.yorku.ca/topology/educ.htm
http://at.yorku.ca/i/a/a/b/23.dir/index.htm
===
Subject: Re: Differential Geometry Book
Merkinball
> I am taking a first course in Differential Geometry at the
graduate
> level, and we are using Do Carmo's Differential
> Geometry of Curves and Surfaces. While a thourough
> and often used text, it is a bit difficult to read at some
times.
> Does anyone know of a book that could be used to supplement
Do Carmo?
The Schaum series has one:
http://www.amazon.com/exec/obidos/tg/detail/-/0070379858/104-
0047037-2675155
?v=glance
It's still in the 1969 edition, without the newer formalism,
e.g. on
differential forms. And it's only about curves and surfaces,
not
higher-dimensional spaces. But it's quite thorough and
detailed as far it
goes, with lots of exercises. US$15.95 according to the above
site.
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
===
>Subject: Coin-Flipping Machine
>Message-id: <b4be2fdf.0402271711.d80d369@posting.google.com
>Story from NPR:
>http://www.npr.org/display_pages/features/feature_1697475.html
>Apparently, coin-toss-outcomes are more a function of
>human-unpredictability than of the coin's unpredictabilty.
>What a waste of time. I could have told him that. Oh, but
then he
>wouldn't have gotten a grant to study a non-problem.
>So, perhaps it is more accurate to talk about, rather than a
biased
>coin, a biased HUMAN-BEING!...
>;)
>towards heads). Of course, coins don't have a symmetrical mass
>distribution, so I could have told him that also. And I
didn't see
>any reference to the control test of an unstamped coin blank.
>And is he conducting the tests in a vacuum chamber at a
controlled
>temperature? I wonder how much of my tax money he fleeced out
>of the government for this stupid project?
>The answer to how much tax money he fleeced out of the
government would
>be none. The origin of the coin flipping machine came from a
comment he
>made in a lecture sometime in the Fall of 1995, in which he
said that
>when he was growing up he saw such a machine advertised in
scientific
>toys catalog. He got curious, called up the catalog to see if
they had
>ever had such a thing, and was told no. He mentionned this to
someone
>in the physics department, and a couple days later some guys
in a
>physics lab had built it out of spare parts lying around the
lab. No
>grant applications, no spending money, just some geeks
playing around
>and having fun.
Ok.
>I skipped the talk, but am told that
>the recent result, which presumably the NPR report is
referring to, says
>roughly that if you flip a coin with an initial velocity that
is a
>random variable with a density, then in the limit as the
height of the
>flip tends to infinity, the probability of coming up in the
original
>orientation is greater than 1/2. Likewise, given flipping to
some
>initial height, the limiting probability of the original
orientation is
>greater than 1/2 as the angular velocity tends to infinity.
<quote
Diaconis, now at Stanford University, found that if a coin is
launched
exactly
the same way, it lands exactly the same way.
</quote
How am I supposed to infer that exactly the same way means
...an initial velocity that is a random variable with a
densityIf a coin is launched exactly the same way, why
_wouldn't_ the
path be the same?
>Again, I
>have neither read the paper nor attended a talk about it, so
the actual
>result might be different.
>--
>--
>To respond, change dot to a dot, and com to net
--
===
Subject: Re: d1 cos (ax + b) + d2 cos(cx + d) = k ; x = ?
<c1tk6v$1lp5g7$1@ID-137122.news.uni-berlin.de
> En el
mensaje:2f7b6a6f.0402291244.2a1e584d@posting.google.com,
> Magh <8agh@st.fmph.uniba.sk> escribi:
> how can I solve
> d1 cos (ax + b) + d2 cos(cx + d) = k
> ( and sin (ax + b) + sin(cx + d) = k )
Differentiating the last twice by x
- a^2 sin(ax + b) - c^2 sin(cx + d) = 0
sin(cx + d) = -(a^2 / c^2) sin(ax + b)
substituting
(1 - a^2 / c^2) sin(ax + b) = k
x = [-b + arcsin k/(1 - a^2/c^2)]/a
Does the answer check out?
Can that approach be justified?
> Differentiating the last with respect to x,
> a*cos(a*x + b) + c*cos(c*x+d) = 0
> d1*cos(a*x + b) + d2*cos(c*x + d) = k
> cos(ax + b) = c*k/(c*d1 - a*d2)
> cos(cx + d) = -a*k/(c*d1- a*d2)
> x = (acos(c*k/(c*d1 - a*d2)) - b)/a
> x = (acos(-a*k/(c*d1- a*d2)) - d)/c
Cute, did he have two problems or one?
===
Subject: Re: f continuous
Larry Hammick
moubinool.omarjee
> f:R-->R surjective with the property
> ( for any x(n) real sequence f(x(n)) converge => x(n)
converge )
> Prove that f is continous
> I think f is a homeomorphism. We know f is injective, for
> if f(A)=f(B) and A<>B, we can let
> x(2n) = A
> x(2n+1) = B
> and get a contradiction of hypothesis. So f is bijective. If
we
> can show that f is monotone, we will be home.
Or if we can show that the inverse of f is continuous. --LH
===
Subject: Re: Coin-Flipping Machine
> How am I supposed to infer that exactly the same way means
...an initial velocity that is a random variable with a
density> If a coin is launched exactly the same way, why
_wouldn't_ the
> path be the same?
Oh, I don't know. Air fluctuations? Different gravities? If
you can't
possibly think of ways the path could be different, then
you're not
thinking very hard.
On the other hand, if the question was if *everything* was the
same, why
wouldn't the path be the same, then your point would be valid.
But that's
not the question asked.
===
Subject: Re: Inverse Primes in Base Factorial
> This is something unusual I observed about the factorial
base.
> The representation of 1/p, p prime,
> will have p-1 places and the last
> coefficient is equal to p-1.
> 1/2 = 1/2! = .1
> 1/3 = 2/3! = .02
> 1/5 = 1/3! + 4/5! = .0104
> 1/7 = 3/4! + 2/5! + 6/7! = .003206
> ...
> Does anyone know why 1/p has this pattern?
Yes.
Note that a factorial base representation of any fraction with
k-1
places corresponds to a fraction (probably not in lowest
terms) with a
denominator of k! . Conversely, any fraction with a
denominator of k!
can be directly converted to a factorial base representation
with k-1
places. When p is prime, 1/p can not be written as a fraction
with
(p-1)! as the denominator, so any factorial representation
must have
more than p-2 places. On the other hand, 1/p = (p-1)!/p!, so
it can be
written with p-1 places.
As has already been pointed out, Wilson's Theorem covers the
other
point. Each term except for the last becomes a multiple of
1/(p-1)! =
p/p! . The sum of all the terms must be (p-1)!/p! . The last
numerator
must therefore be the remainder when (p-1)! is divided by p.
iel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
>towards heads).
>says.
Allow me to quote the the headline:
<quote
The Not So Random Coin Toss
Mathematicians Say Slight but Real Bias Toward Heads
</quote
>I think what it says is that the coin has a slight tendency
>to come up whichever way it started - e.g., if it's tails
when you
>flip it, then it has a slight tendency to be tails when you
catch
>it.
Well, it doesn't say that specifically, it only
says that heads tends to heads. But earlier
is says that if flipped exactly the same way
it lands exactly the same way.
>Could you have told him that?
Told him that Newtonian mechanics are valid?
Yeah, I think I could have told him that.
>--
--
===
Subject: Re: Scimitar Theorem
> Let y = f(x) be the shape of a (zero-width, finite, planar)
scimitar
> blade, and also of its scabbard. The blade remains in
contact with the
> scabbard for any amount of blade withdrawal. Prove that y
must be a
> circular arc.
> phil
It is well-known (see e.g. Differential Geometry by Lipschutz
in
Schaum'
outline series) that curvature and torsion (as functions of
arc length)
uniquely specify a 3D-curve (up to position and orientation).
Obviously for
the unsheathing of the sword to succeed as described (with
idealistic
model)
both curvature and torsion must be constant.
The uniqueness result above then leaves us with a task of
constructing
curves
with specified constant curvature and torsion. These are
achieved by
a straight line (zero curvature), a circular arc (constant
curvature, no
torsion) and a helix. As you asked for a planar curve, the
helix doesn't
count.
In other words: even in 3D the only alternatives would be the
dagger,
the (circular) scimitar and a corkscrew (left-handed or
right-handed).
Cheers,
Jyrki Lahtonen, Turku, Finland
===
Subject: is there a reason why to make mix numbers improper
when adding?
Is there a reason why to make mix numbers improper when adding?
It seems when subtracting and adding, adding a subtracting the
whole
numbers and fraction parts should be sufficient? what'ch think
===
Subject: Re: Prime factors of number near googolplexplex
>He didn't factor those numbers, just guessed some of their
small
>factor.
> I'm curious how one guesses small factors like 3898556974549,
> 2000665038247, 1301687275127, 2017970156617, or
2184279173669?
The trick is to find the modular residue of 10^(10^(10^100))
modulo n, for
n
small, which is not so hard (as, say, here it is modulo 7 :
(10^a) mod
7=[1,3,2,6,4,5,1,...] so you need to find a mod 6. But (10^b)
mod
6=[1,4,4,4....], so your answer is 4. For higher n, you may
have to use the
method 3 or 4 times, though...)
===
Subject: where to get latex help
Sorry to post in the wrong group, but that's the thing...Now
that
comp.tex.text is gone, who can I ask for help with LaTeX? TIA.
Peace
===
Subject: Re: Science Without Math? (model-free common sense
steering)
message
> Conventional(Boolean) logic states that a glass can be full
or not
full
> of
> water. However, suppose one were to fill the glass only
halfway. Then
> the
> glass can be half-full and half-not-full. Clearly, this
disprove's
> Aristotle's law of bivalence. This concept of certain degree
or
> multivalence
> is the fundamental concept which propelled Zader Lofti of
University
> Berkely
> in the 1960's to introduce fuzzy logic. The essential
characteristics
of
> fuzzy logic founded by him are as follows.
> I don't buy this. A glass can be either [half-full] or
not-[half-full],
> not
> both. half-not-full is not equivalent to not-[half-full].
The 'not'
> shouldn't be stuck in the middle of the statement, but
appended to the
> front
> of it. ie: A glass with 50% water is half-not-full, but to
say its
> not-[half-full] is incorrect. I understand the concept of
non boolean
> logics, and the benefit of fuzzy systems in various
applications, but
this
> is a pretty bad example IMO. Especially the part about it
'clearly'
> disproving Aristotle's law of bivalence.
> A = water
> A = ~A in this 1/2 example
<quote
Conventional(Boolean) logic states that a glass can be full or
not full of
water. However, ~suppose one were to fill the glass only
halfway~. Then the
glass can be ~half-full~ and *half-not-full*. Clearly, this
disprove's
Aristotle's law of bivalence.
</quote
a glass can be
A = full, or
not full = ~A
of water. However, suppose one were to fill the glass only
halfway. Then
the glass can be half-full and *half-not-full*.
half-A = half-full
~(half-A) = not (half full)
which is not equal to half-not-full (see *).
If your statement is half-full (see ~), then the boolean
negation of
this
is: not-half-full and is not equal to half-not-full
===
>Subject: Re: Coin-Flipping Machine
>You've accused Diaconis of wasting time, studying a
non-problem,
>and stupidity. You haven't given anything remotely resembling
>justification for these accusations.
>Huh? Didn't you read the URL in question? What more
justification
>do I need?
>I did read the page. Nothing I saw there justifies accusing
Diaconis
>of wasting time, studying a non-problem, or stupidity.
Evidently,
>you disagree. Perhaps you could be a little more forthcoming,
as
>it's clear that I need convincing.
Didn't you ever take a physics class? Doesn't
if a coin is launched exactly the same way,
it lands exactly the same way.
strike you as being a stupid waste of time?
Aren't you supposed to already know this?
Isn't this a non-problem?
>So the only thing you've
>given me to go on is your reputation versus his.
>I have no reputation, so I'll come out second best in any such
>comparison. But aren't such comparisons a type of logical
fallacy?
>Shouldn't _any_ criticism be based on merit, not reputation?
>Are James Harris and John Correy actually right about
>mathemeticians? And all this time I thought they were cranks.
>When you make a criticism based on merit, I'll read it, and
judge
>it on its merits. So far you've only made accusations which
you say
>are founded in the NPR piece and which I say are not. Since I
>haven't read Diaconis' write-up, all I have to go on is your
>unfounded accusations versus his body of work. It makes
perfect
>sense under the circumstances to give Diaconis the benefit of
the
>doubt and to assume - until you give me some good reason to
think
>otherwise - that he has not been wasting his time & is not
being
>stupid.
>What's so hard to understand about that?
Ok, you don't see anything wrong with the NPR piece, in which
case your position is understandable. Wrong, but
understandable.
>With all due
>respect, I know enough about him & his work to be pretty
confident
>that whatever he's studying he's got a bloody good reason for
it.
>In other words, you cannot explain how this research is
important.
>And yet, you support it without anything remotely resembling
>justification. That makes you no better than me, eh?
>I don't recall saying that I am better than you.
No, but you said Diaconis was better than me.
Specifically, you told me to write back when I
have contributed as much to mathematics as
Diaconis has. I guess I just assumed that that
intellectual snobbery extends to yourself.
>It appears that I
>am distinctly inferior to you, in that you are able to find
evidence
>of Diaconis' stupidity where I can't.
>Diaconis has apparently found that If a coin starts out
heads, it ends
>up heads when caught more often than it does tails. Is that
important?
Only if it contradicts what one would expect from
physics. If all parameters are held constant (as would
be expected in a mechanical coin flipper), why would
you expect the outcome to be random?
>I don't know. I find it interesting; if I know Diaconis at
all, he
>won't stop there, but will continue until he has found a
model to
>explain why the coin ends up heads more often than tails, and
that
>model will have some pretty interesting mathematics in it.
Fine. Maybe he should not let NPR report
on his work then.
>You are saying that you already knew the coin would come up
heads
>more often?
I didn't say that. I said that I already knew that
if a coin is launched exactly the same way,
it lands exactly the same way.
>You have a model which explains why this should be?
>Show your work! When someone announces a proof that all the
non-trivial
>zeros of the zeta function are on the critical line, are you
going
>to say that you could have told her that, too?
>Will you expect people
>to believe you?
>I also suggest that three or four paragraphs on an NPR website
>is not a sufficient basis on which to make adverse judgements
of
>a mathematical research program.
>Well, who's fault is that? Mine? Sorry, but I call a spade a
spade.
>Whose fault is what? That NPR condensed a 30-page paper
[warning - I
>made that number up] into a few paragraphs? No, that's not
your fault.
>That you chose to judge a 30-page paper by what you got out
of - or
>read into - those paragraphs? Well, of course that's your
fault - whose
>fault other than yours could it be?
NPR's? Duh.
>If it looks stupid, I'll say so. Go ahead, prove me wrong.
Point out how
>the coin-flipping machine was an insignificant element of a
much more
>important work that was distorted be the media. I won't mind.
>At the risk of oversimplifying things to a ridiculous degree,
let's say
>there are two possibilities here: the research is worthless,
or it's
>worthwhile. Given nothing else whatsoever to go on, we might
take the
>position that the two outcomes are equally likely. You say
there is
>something else to go on; the NPR summary is enough to
convince you the
>research is worthless. I don't see that, but I do see
Diaconis' track
>record as enough to convince me that there's probably a lot
more here
>than what's in the NPR report.
Then maybe you should criticize NPR for
trivializing Diaconis' work. If NPR made his
work appear stupid, then that's what I'm
going to comment on. By the way, what
color are the Emporer's new clothes?
>I don't post from an .edu domain.
>I have no idea how that observation advances this discussion.
Let's see, you told me that unless I am a peer
of Diaconis', I have no right to comment.
Now where would I get the notion that not
being affiliated with an educational institution
doesn't give me the right to comment either?
>On the other hand, I won't be embarrassed
>if it turns out I'm wrong. Can you say the same?
>Yes: I, too, won't be embarrassed if it turns out that you
are wrong.
>Seriously, though, I've been wrong many many many many times
in this
>newsgroup, I have sometimes been embarrassed at how wrong
I've been,
>I have tried to come clean when I have been wrong, and I have
tried
>to learn something from the experience. What's your point?
You brought up the whole issue of reputation.
I thought it was important to you.
>--
--
===
Subject: Re: what is the z-transform of sinc function?
> Can anybody tell me what is the z-transform of sinc function
and
> what is its region of convergence?
> -Joenyim
Does this help?
sin(x) = x - (x^3)/3! + (x^5)/5! - .... (converges everywhere)
so sinx/x evaluates to:
1 - (x^2)/3! + (x^4)/5! - ...
(still infinite converges everywhere, even at x=0)
with BZT applied as f(z) = f(s) where s=(z-1)/(z+1) this gives:
1- ((z-1)/(z+1))^2/3! + ((z-1)/(z+1))^4/5! - ....
I'm not sure if this converges at z=-1.
I guess, it should, but I'm not able to see it from the
formula.
===
Subject: Re: Coin-Flipping Machine
===
>Subject: Re: Coin-Flipping Machine
>Message-id: <LnA0c.154220$uV3.703871@attbi_s51
>How am I supposed to infer that exactly the same way means
...an initial velocity that is a random variable with a
density>If a coin is launched exactly the same way, why
_wouldn't_ the
>path be the same?
>Oh, I don't know. Air fluctuations? Different gravities? If
you can't
>possibly think of ways the path could be different, then
you're not
>thinking very hard.
Then you didn't see my comment about an unstamped
coin blank in a vacuum chamber?
>On the other hand, if the question was if *everything* was
the same, why
>wouldn't the path be the same, then your point would be valid.
And would you then agree that if my point is valid,
then the coin-flipping machine is pointless and stupid?
>But that's not the question asked.
Then the answer there is a slight bias towards heads
is meaningless, isn't it?
>
--
===
Subject: Re: Req. help on On Formally Undecidable
Propositions...
> My question refers to this paragraph:
The following formulae (I-V) are called axioms (they are set
out with
> the help of the customarily defined abbreviations: .,
<non-ASCII
> symbols removed>,
> and subject to the usual conventions about omission of
brackets):
> What is the meaning of the . symbol?
it means and - tfff in truthtable talk.... i assume the choice
of
that character is meant to indicate multiplication, which, in
terms of
lattice theory, works somewhat like truthtable-and...
> What are the usual conventions about omission of brackets?
could be any combination of several things all dealing with
how lax
you can be, and still maintain an unambiguous order of
operations....
(a) leaving off outermost parentheses - (a->b).c rather than
((a->b).c)
(b) Using an order of operations rather than (most)
parentheses. The
typical order I've used is as follows, in decreasing order of
binding-strength:
(1) negation
(2) and, or (equal strength)
(3) conditional, biconditional (equal strength)
for example, ~a->~a^~b would be parsed as (~a) -> ((~a) ^
(~b)) -
note tho, i'm using the previous convention...
i may have 1, 2, and 3 mixed up somewhat... can't recall
exactly....
cdj
===
Subject: a natural diameter within each specific n-adic Re:
how we turn
e^i(pi) = -1 into c = (pi)d
What would really be nice if it was all so very easy. Here are
a lot
of ifs and an easy path, but is it true??
Given the Universe is an atom of 231Pu which has 22 subshells
in 7
shells of which 19 subshells are occupied. That gives in
Rational form
a Collapsed wavefunction for the rational form of pi and e,
22/7 and
19/7 respectively.
Suppose if there exists only two types of numbers, either a
p-adic/n-adic or a doubly-infinite. Then ....99999 is the
largest
number in 10-adics. And pi is the doubly-infinite of
3.14159..... and
e is the doubly-infinite 2.71828.....
Suppose if the 10-adics circumscribe a huge circle starting at
0 going
to 1 then 2 and on around to the last number of ....999999.
Then what
is the diameter of this 10-adic circle? Suppose it is the
10-adic
......3333333. In 6-adics the diameter would be .....22222222.
In
12-adics it would be a diameter of ...44444.
Where each n-adic/p-adic forms a convex curve that ends up as
a closed
circular object. Each n-adic/p-adic has a largest number and
forms a
circular object and thus each adic has a natural built in
diameter.
Now using the formula C = pi(d) on 10-adics we have
....99999999 = pi
X ....3333.
Suppose pi is the doubly-infinite of 3.14159..... and suppose
that
multiplication of a doubly infinite by a adic ends up being a
doubly-infinite.
Suppose that when you multiply a doubly-infinite by a adic you
only
multiply the
leftward strings with one another.
Thus our pi(d) becomes 3.14159.... X ....33333.
The endresult is ....99999999999.14159....
Not exactly ....99999999999 which is a adic whereas
....999999.14159... is a doubly infinite.
Perhaps some relief can be provided by saying that pi and e
exists in
each adic of the closest or most nearby rational form such as
22/7 and
19/7 are for 10-adics. In that manner, we can deal with Eulers
equation of e^i(pi)= -1 and with the C = pi(d) all with adics.
Archimedes Plutonium
whole entire Universe is just one big atom where dots
of the electron-dot-cloud are galaxies
(www.iw.net/~a_plutonium) website of the science of AP under
revision
what used to be my old science website
www.newphys.se/elektromagnum/physics/LudwigPlutonium from
years 1993
===
Subject: Re: Coin-Flipping Machine
> And would you then agree that if my point is valid,
> then the coin-flipping machine is pointless and stupid?
Sure. And cows would fly, and volcanos would shoot out
marshmallow creme,
too.
If your point is valid, then a lot of false things follow.