11992
Subject: Re: Determining Limits of an Arbitrarily Rotated
Ellispe
> Anyone have the derivation for the equations to determine
the X and
> Y-axis limits (mins and maxes) of an arbitrarily rotated
ellipse in an
> X-Y plane? Currently given is the axis-alligned equation of
the
> ellipse and the angle of rotation to the X-axis. Conversly,
the
> generalized ellipse equation can be derived, too.
> Thanks in advance.
> A paper which derives the equations for determining these
values for a
> generalized ellipse can be found at
> http://www.crbond.com/papers/ellipse.pdf .
> This paper was written to present a method for scan
converting ellipses
at
> arbitrary angles.
> There are two things you must never attempt to prove: the
unprovable --
> and the obvious.
>
Perfecto! :o) Thanks so much, C
Subject: More glossary items (was: Extending Python Syntax
with @)
Originator: claird@lairds.com (Cameron Laird)
>months. I took five years of calculus in college, and I still
don't
>see the connection between lambda functions and calculus.
They're both about getting results.
Serious. What you learned as calculus was earlier called
the analytic calculus or the differential and integral
calculus. It's about getting results (very roughly speaking)
that have to do with smooth properties (of shapes and motions,
let's say).
Lambda is entirely different--except that it's also a calculus,
that is, largely expressible algorithmically. Lambda happens
to be about computations.
There are other calculi--quaternion calculus and vector
calculus are two not-too-uncommonly-heard ones. Lambda and
the one you learned in college courses are much the most often
used in such references.
--
Business: http://www.Phaseit.net
Subject: Re: More glossary items
> There are other calculi--quaternion calculus and vector
> calculus are two not-too-uncommonly-heard ones.
Indeed, in mathematics the word calculus really just
means a method of calculating something. It's actually
the Latin word for stone, presumably dating back to
the time when people used stones as counters when
doing arithmetic.
--
http://www.cosc.canterbury.ac.nz/~greg
Subject: Root to One
This attempts to solve polynomials.
Eqn of plane,
a[1]x[1]+a[2]x[2]+a[3]x[3]+a[4]x[4]+a[5]x[5]+a[6]x[6]+a[7]x[7]
+a[0]=0
N=(a[1],a[2],a[3],a[4],a[5],a[6],a[7]) normal
T=(t,t^2,t^3,t^4,t^5,t^6,t^7) space curve
N*T + a[0]=0 7th degree polynomial or
a[1]t+a[2]t^2+a[3]t^3+a[4]t^4+a[5]t^5+a[6]t^6+a[7]t^7+a[0]=0
T=(-a[0]/|N|^2)N This is shown by
N*T = (-a[0]/|N|^2)N*N
decoding,
t =(-a[0]a[1])/|N|^2 = x[1]
t^2=(-a[0]a[2])/|N|^2 = x[2]
t^3=(-a[0]a[3])/|N|^2 = x[3]
t^4=(-a[0]a[4])/|N|^2 = x[4]
t^5=(-a[0]a[5])/|N|^2 = x[5]
t^6=(-a[0]a[6])/|N|^2 = x[6]
t^7=(-a[0]a[7])/|N|^2 = x[7]
substituting these in the equation below,
t=
2{
2{
2{
2{
2{
2{x[7]}^(6/7)-x[6]
}^(5/6)-x[5]
}^(4/5)-x[4]
}^(3/4)-x[3]
}^(2/3)-x[2]
}^(1/2)-x[1]
solve,
x^3+2x^2-3x+1=0
a[1]=-3
a[2]= 2
a[3]= 1
a[0]= 1
|N|^2 = 9+4+1=15
t = -a[0]a[1]/|N|^2 = 3/15 = x[1]
t^2= -a[0]a[2]/|N|^2 = -2/15 = x[2]
t^3= -a[0]a[3]/|N|^2 = -1/15 = x[3]
t = 2{2{x[3]}^(2/3)-x[2]}^(1/2)-x[1]
x = 2{2{-1/15}^(2/3)-(-2/15)}^(1/2)-3/15
x = .540+/-i(.183) , -3.079 roots from calculator
Angle x subtends known arclength A and chord B
(B/A)(x/2)=sin(x/2) trigonometric representation
sin(x/2)=x/2-(x/2)^3/3!+(x/2)^5/5!-... Maclaurin sine
B/A=1-(x/2)^2/3!+(x/2)^4/5!-... cancel x/2
t=x/2 x=2t
a[odd]=0
a[0] =1-B/A
a[2] = -1/3!
a[4] = 1/5!
a[6] = -1/7!
a[8] = 1/9!
a[10]= -1/11!
a[12]= 1/13!
a[14]= -1/15!
|N|^2 = (-1/3!)^2+(1/5!)^2+(-1/7!)^2+...+(-1/15!)^2
x[2] =-a[0]a[ 2]/|N|^2= -(1-B/A)(-1/3! )/|N|^2
x[4] =-a[0]a[ 4]/|N|^2= -(1-B/A)( 1/5! )/|N|^2
x[6] =-a[0]a[ 6]/|N|^2= -(1-B/A)(-1/7! )/|N|^2
.
.
x[14]=-a[0]a[14]/|N|^2= -(1-B/A)(-1/15!)/|N|^2
(x/2)^2 =
2{
2{
2{
2{
2{
2{x[14]}^(12/14)-x[12]
}^(10/12)-x[10]
}^( 8/10)-x[ 8]
}^( 6/8 )-x[ 6]
}^( 4/6 )-x[ 4]
}^( 2/4 )-x[ 2]
For small angles, B=A and x=0
Subject: Roots to Polynomials
Eqn of plane,
a[1]x[1]+a[2]x[2]+a[3]x[3]+a[4]x[4]+a[5]x[5]+a[6]x[6]+a[7]x[7]
+a[0]=0
N=(a[1],a[2],a[3],a[4],a[5],a[6],a[7]) normal
T=(t,t^2,t^3,t^4,t^5,t^6,t^7) space curve
N*T + a[0]=0 7th degree polynomial or
a[1]t+a[2]t^2+a[3]t^3+a[4]t^4+a[5]t^5+a[6]t^6+a[7]t^7+a[0]=0
T=(-a[0]/|N|^2)N This is shown by
N*T = (-a[0]/|N|^2)N*N
decoding,
t =(-a[0]a[1])/|N|^2 = x[1]
t^2=(-a[0]a[2])/|N|^2 = x[2]
t^3=(-a[0]a[3])/|N|^2 = x[3]
t^4=(-a[0]a[4])/|N|^2 = x[4]
t^5=(-a[0]a[5])/|N|^2 = x[5]
t^6=(-a[0]a[6])/|N|^2 = x[6]
t^7=(-a[0]a[7])/|N|^2 = x[7]
substituting these in the equation below,
t=
2{
2{
2{
2{
2{
2{x[7]}^(6/7)-x[6]
}^(5/6)-x[5]
}^(4/5)-x[4]
}^(3/4)-x[3]
}^(2/3)-x[2]
}^(1/2)-x[1]
solve,
x^3+2x^2-3x+1=0
a[1]=-3
a[2]= 2
a[3]= 1
a[0]= 1
|N|^2 = 9+4+1=15
t = -a[0]a[1]/|N|^2 = 3/15 = x[1]
t^2= -a[0]a[2]/|N|^2 = -2/15 = x[2]
t^3= -a[0]a[3]/|N|^2 = -1/15 = x[3]
t = 2{2{x[3]}^(2/3)-x[2]}^(1/2)-x[1]
x = 2{2{-1/15}^(2/3)-(-2/15)}^(1/2)-3/15
x = .540+/-i(.183) , -3.079 roots from calculator
Angle x subtends known arclength A and chord B
(B/A)(x/2)=sin(x/2) trigonometric representation
sin(x/2)=x/2-(x/2)^3/3!+(x/2)^5/5!-... Maclaurin sine
B/A=1-(x/2)^2/3!+(x/2)^4/5!-... cancel x/2
t=x/2 x=2t
a[odd]=0
a[0] =1-B/A
a[2] = -1/3!
a[4] = 1/5!
a[6] = -1/7!
a[8] = 1/9!
a[10]= -1/11!
a[12]= 1/13!
a[14]= -1/15!
|N|^2 = (-1/3!)^2+(1/5!)^2+(-1/7!)^2+...+(-1/15!)^2
x[2] =-a[0]a[ 2]/|N|^2= -(1-B/A)(-1/3! )/|N|^2
x[4] =-a[0]a[ 4]/|N|^2= -(1-B/A)( 1/5! )/|N|^2
x[6] =-a[0]a[ 6]/|N|^2= -(1-B/A)(-1/7! )/|N|^2
.
.
x[14]=-a[0]a[14]/|N|^2= -(1-B/A)(-1/15!)/|N|^2
(x/2)^2 =
2{
2{
2{
2{
2{
2{x[14]}^(12/14)-x[12]
}^(10/12)-x[10]
}^( 8/10)-x[ 8]
}^( 6/8 )-x[ 6]
}^( 4/6 )-x[ 4]
}^( 2/4 )-x[ 2]
Subject: Re: Sum, Over Some Prime Multiples, Is Always m
> Let P be any subset of the primes.
> (example: P contains all primes of even-index,
> or those congruent to 1 (mod 6).)
> Let A be the set of positive integers: including 1 and every
positive
> integer which is a multiple of only the primes in P, and A
contains
> no member which is divisible by a prime not in P.
> Let B be the set of positive integers: including 1 and every
positive
> integer which is a multiple of only the primes *not* in P,
and B
> contains no member which is divisible by a prime *in* P.
> (Yes, there are positive integers in neither set, as long as
P does
> not contain every prime.)
> Let g(x) be the number of distinct elements of A which are
<= x,
> for x = a positive real.
> So then, for m = any positive integer:
> ---
> / g(m/k)
> ---
> k=elements of B, k <= m
> always = m.
> In linear-mode:
> sum{k=elements of B, k<=m} g(m/k)
> always equals m.
> (Right?)
>
A consequence (which probably is trivially shown by a more
direct
method) of the above:
If, for a given P (and A),
limit{m->oo} g(m)/m = x
exists and is finite nonzero, then
sum{k=elements of B} 1/k
( = product{p=primes not in P} 1/(1 -1/p) )
converges to 1/x.
And so, from this we see that
sum{p=primes not in P} 1/p converges
if the limit of g(m)/m approaches a limit which is finite
nonzero.
But...I am being VERY nonrigorous above, and could very well
be wrong.
Have I posted truth anyway?
Subject: Re: Sum, Over Some Prime Multiples, Is Always m
> Let P be any subset of the primes.
> (example: P contains all primes of even-index,
> or those congruent to 1 (mod 6).)
> Let A be the set of positive integers: including 1 and every
positive
> integer which is a multiple of only the primes in P, and A
contains
> no member which is divisible by a prime not in P.
> Let B be the set of positive integers: including 1 and every
positive
> integer which is a multiple of only the primes *not* in P,
and B
> contains no member which is divisible by a prime *in* P.
> (Yes, there are positive integers in neither set, as long as
P does
> not contain every prime.)
> Let g(x) be the number of distinct elements of A which are
<= x,
> for x = a positive real.
> So then, for m = any positive integer:
> ---
> / g(m/k)
> ---
> k=elements of B, k <= m
> always = m.
> In linear-mode:
> sum{k=elements of B, k<=m} g(m/k)
> always equals m.
> (Right?)
>
> A consequence (which probably is trivially shown by a more
direct
> method) of the above:
> If, for a given P (and A),
> limit{m->oo} g(m)/m = x
> exists and is finite nonzero, then
> sum{k=elements of B} 1/k
> ( = product{p=primes not in P} 1/(1 -1/p) )
>
> converges to 1/x.
> And so, from this we see that
> sum{p=primes not in P} 1/p converges
> if the limit of g(m)/m approaches a limit which is finite
nonzero.
> But...I am being VERY nonrigorous above, and could very well
be wrong.
> Have I posted truth anyway?
>
So, unrigorously again,
If sum{p=primes in P} 1/p
AND
sum{p=primes not in P} 1/p
both diverge, then:
the percentage of all positive integers neither in A nor in B
approaches
100%.
In other words, if j(m) = number of positive integers <= m
which are
not in A nor in B (ie. number of positive integers <= m which
are
divisible
by *both* primes in P and in not-P),
then limit{m->oo} j(m)/m = 1
IF the 2 prime-reciprocal series above *both* diverge.
Even though this result can be shown, shown if true, without
the use
of the result in this thread's original post, it is directly
related
to this result.
Subject: Re: Determining the Equation of a Shadow Cast onto a
Surface
<462e18e1.0403111517.71dd612@posting.google.com
>> Given a flat circular disk (high-gain antenna), a planar
surface (a
>> solar array), and an arbitrary vector to a point light
source (the
>> Sun), I would like to determine the equation of the
resultant shadow
>> (which would be an ellipse) that would be cast onto the
plane.
>Oops, I posted this in the wrong place.
>As additional information, the solar array is a group of
maybe 10,000
>individual solar cells, and this analysis is to determine the
power
>output reduction over the entire array if a portion of those
cells are
>shadowed from the Sun by the high-gain antenna. So, certain
>assumptions can simplify this analysis. Fringes can be
ignored (a few
>cells won't degrade the power output significantly). The
Sun's disk
>size can be ignored.
Is this for visualization or for engineering?
The reduction in power is going to depend on the sine of the
angle of the
disk with respect to the sun's rays and percentage of the
shadow which
falls
on the array, multiplied by the area of the disk. I don't
think you
necessarily need to know the shape (equation) of the shadow on
the array
except to calculate the percentage which falls on the array.
(Of course,
if
the array is not perpendicular to the direction of the sun,
the output will
further be reduced by the sine of that angle...)
>Given knowledge of the physical characteristics of this
spacecraft
>(geometries and sizes), locations and orientations of the
articulable
>solar array and high-gain antenna, and solar system
ephemerous, I was
>planning on using an orthoganal projection approach to model
the
>shadow cast on the array.
Orthagonal projection of the disk onto the array will only
happen when the
array is perpendicular to the sun's rays. Is that always going
to be the
case?
>Yeah, I realize the shadow might project off to a parabola
with some
>points of the disk at infinity if the Sun is low enough (I
belive I
>could use some sort of projective transformation).
Assuming the sun is larger than the disk, I don't think you
need to worry
about that!
The above may not (yet) represent the opinions of my employer.
Subject: Re: Determining the Equation of a Shadow Cast onto a
Surface
>> Given a flat circular disk (high-gain antenna), a planar
surface (a
>> solar array), and an arbitrary vector to a point light
source (the
>> Sun), I would like to determine the equation of the
resultant shadow
>> (which would be an ellipse) that would be cast onto the
plane.
>Oops, I posted this in the wrong place.
>As additional information, the solar array is a group of
maybe 10,000
>individual solar cells, and this analysis is to determine the
power
>output reduction over the entire array if a portion of those
cells are
>shadowed from the Sun by the high-gain antenna. So, certain
>assumptions can simplify this analysis. Fringes can be
ignored (a few
>cells won't degrade the power output significantly). The
Sun's disk
>size can be ignored.
> Is this for visualization or for engineering?
Enginnering
> The reduction in power is going to depend on the sine of the
angle of the
> disk with respect to the sun's rays and percentage of the
shadow which
falls
> on the array, multiplied by the area of the disk. I don't
think you
> necessarily need to know the shape (equation) of the shadow
on the array
> except to calculate the percentage which falls on the array.
(Of course,
if
> the array is not perpendicular to the direction of the sun,
the output
will
> further be reduced by the sine of that angle...)
It's a little more complicated than this. The individual solar
cells
are grouped in what are called strings of 16 cells. The input
to
the power prediction software is as follows:
1) The number of strings with 0 cells in shadow
2) The number of strings with 1 cell in shadow
.
.
.
17) The number of strings with all 16 cells in shadow
18) The incedence angle between the solar array normal and the
Sun
vector
Thus, my algorithm first isolates a bounding square on the
array in
which the shadow is cast (the minimum and maximum limits of the
shadow ellipse in solar array coordinates), and then
determines which
cells in that square are in the elliptical shadow.
The arrays will be normal to the Sun during most of the ops,
however,
a constraint is that we do not loose contact, so the HGA
pointing at
the Earth is given higher priority. This could result in some
off-pointing of the arrays.
>Given knowledge of the physical characteristics of this
spacecraft
>(geometries and sizes), locations and orientations of the
articulable
>solar array and high-gain antenna, and solar system
ephemerous, I was
>planning on using an orthoganal projection approach to model
the
>shadow cast on the array.
> Orthagonal projection of the disk onto the array will only
happen when
the
> array is perpendicular to the sun's rays. Is that always
going to be the
> case?
>Yeah, I realize the shadow might project off to a parabola
with some
>points of the disk at infinity if the Sun is low enough (I
belive I
>could use some sort of projective transformation).
> Assuming the sun is larger than the disk, I don't think you
need to worry
> about that!
>
> The above may not (yet) represent the opinions of my
employer.
Subject: Re: Question about Ox Metrics & Monte Carlo
Simulations
>I have a programming assignment I need to work on that
involves the Ox
>Metrics programming environment. Is anyone in this group
familiar with
>programming in Ox for the purpose of Monte Carlo simulations
(for an
>econometrics assignment)? If so, what expierience have you
had with
>this or do you have any recommendations for doing Monte Carlo
>simulations in general?
Monte Carlo simulations model many cases using random input
variables in
order to generate expected statistics. For this to work, it's
important
that:
* Your random input variables have a distribution which
matches reality
* Your model is accurate
* You run a large number of cases (tip: Do 2 or more runs with
different
random seeds. If the results are different, increase the
number of cases
per run. As the number of cases increases, the results should
converge to
the same set of values for any random seed.)
Sorry I know nothing about Ox.
The above may not (yet) represent the opinions of my employer.
Subject: Re: Want ro receive A+ paper?
whatever happened to bullying people into doing your homework?
Subject: Could someone explain this...
I am trying to break my way into Calculus, but I need some
help in
understanding limits, (which I know I need to know to move
on...)
Could someone please explain how to calculate a limit at
Infinity?
Subject: Re: Could someone explain this...
> I am trying to break my way into Calculus, but I need some
help in
> understanding limits, (which I know I need to know to move
on...)
> Could someone please explain how to calculate a limit at
Infinity?
Intuitively, the limit as a function f goes to infinity is a
number L
that f gets arbitrarily close to as the domain of the function
gets
arbitrarily large.
For example, f(x) = x. As x gets very large, f(x) gets very
large, so we
say the limit is infinity.
Another example: f(x) = 1/x. As x gets very large, f(x) gets
very close
to 0, so the limit is 0.
Perhaps if you post some of the specific problems you're having
difficulty with, we can offer more specific help. Are you
working from a
book?
Subject: Re: Could someone explain this...
>> I am trying to break my way into Calculus, but I need some
help in
>> understanding limits, (which I know I need to know to move
on...)
>
>> Could someone please explain how to calculate a limit at
Infinity?
>Intuitively, the limit as a function f goes to infinity is a
number L
>that f gets arbitrarily close to as the domain of the
function gets
>arbitrarily large.
>For example, f(x) = x. As x gets very large, f(x) gets very
large, so we
>say the limit is infinity.
>Another example: f(x) = 1/x. As x gets very large, f(x) gets
very close
>to 0, so the limit is 0.
>Perhaps if you post some of the specific problems you're
having
>difficulty with, we can offer more specific help. Are you
working from a
>book?
In calculus, I'd also expect l'Hopital's rule to be covered.
You have to
understand derivatives before you can use it.
If that value of a function is indeterminate, that is 0/0 or
infinity/infinity, you can take the derivative (differential)
of the
numerator and denominator. For example, if f(x) = 2x / (x+1),
if you plug
in x = oo (oo is the closest thing to the sideways 8 infinity
sign on my
keyboard) you get oo/oo. But if you take the derivative of the
top and
bottom, it's 2/1 = 2.
The above may not (yet) represent the opinions of my employer.
Subject: Re: Could someone explain this...
I'm trying to teach myself...
I can work derivatives and integrals... but I know that I have
to work with
limits to understand some of the more complicated theroms.
Subject: Plotting Hyperelliptic Curves!!!
How do you plot Hyperelliptic Curves? I tried using Maple,
Mathematica
using Implicit Plot but doesn't give me the exact curve I
need. Is
there any other way?
Thank in advance for any help.
Subject: Re: Plotting Hyperelliptic Curves!!!
> How do you plot Hyperelliptic Curves? I tried using Maple,
Mathematica
> using Implicit Plot but doesn't give me the exact curve I
need. Is
> there any other way?
Jacobi functions? what is the function being integrated and
its arguments ?
Subject: Re: Plotting Hyperelliptic Curves!!!
> How do you plot Hyperelliptic Curves? I tried using Maple,
Mathematica
> using Implicit Plot but doesn't give me the exact curve I
need. Is
> there any other way?
> Thank in advance for any help.
>
may be you will find this link to a Function renderer
useful.
It allows you to input any function while producing an
excellent
graphic output. Remember: it is cardware! (You must send a
postcard to
the programmer.)
Eckard
Subject: Re: Cantor Paradox
>Nathan's list is the list of all real numbers definable by a
finite
>string. He must specify definable in what language, say L.
>Then the issue is whether the diagonal number for Nathan's
list can
>be defined in L. Its definition contains such expressions as
the
>n-th digit of the n-th number defined in L and it is not
clear that
>these can be defined in L.
>Since a contradiction results from the supposition that the
diagonal
>number can be defined in L, Nathan's paradox actually works
as a
>reductio ad absurdum to show that it cannot be defined in L.
> Well, Nathan evidently thinks it IS clear that the diagonal
can be
> defined in L. Let L be English. He evidently thinks he has
defined
> in English exactly how we go about constructing the diagonal.
Wow, sounds like Nathan has rediscovered Richard's Paradox
(1905). Not
bad for an 11-year-old, if he really did think of himself.
Subject: Re: Cantor Paradox
Discussion, linux)
> Wow, sounds like Nathan has rediscovered Richard's Paradox
(1905). Not
> bad for an 11-year-old, if he really did think of himself.
He's not an 11-year-old. Whether he's ever read Richard's
Paradox or
not, he clearly has the background to choose the issues
carefully
here.
(I hadn't seen the paradox before, so I appreciate having a
name
attached to it.)
--
Contrariwise, continued Tweedledee, if it was so, it might be,
and
if it were so, it would be; but as it isn't, it ain't. That's
logic!
-- Lewis Carroll
Subject: Re: Cantor Paradox
>Nathan's list is the list of all real numbers definable by a
finite
>string. He must specify definable in what language, say L.
>Then the issue is whether the diagonal number for Nathan's
list can
>be defined in L. Its definition contains such expressions as
the
>n-th digit of the n-th number defined in L and it is not
clear that
>these can be defined in L.
>Since a contradiction results from the supposition that the
diagonal
>number can be defined in L, Nathan's paradox actually works
as a
>reductio ad absurdum to show that it cannot be defined in L.
> Well, Nathan evidently thinks it IS clear that the diagonal
can be
> defined in L. Let L be English. He evidently thinks he has
defined
> in English exactly how we go about constructing the diagonal.
Well, he clearly can't have.
It's not particularly clear to me that the n-th digit of the
n-th
real number defined by a finite string in English is
well-defined at
all. It's not clear to me that English is that well-defined.
If English IS that well-defined, then clearly the n-th digit
of the
n-th real number defined by a finite string in English can't
be a
string with a definite meaning in English. The logic is
inescapable.
> The distinction between paradox and reductio ad absurdum can
be
> controversial. A mathematical argument reaching an absurd
conclusion
> calls into question the underlying assumptions of the
argument, but
> there can be disagreements about which underlying assumption
is
> faulty. If there is only one possible candidate, the
argument is a
> reductio ad absurdum. But if it not clear that there is only
one
> possible candidate then, until we figure things out, we
should use
> the word paradox.
> Nathan evidently does not believe that an ability to
describe his
> diagonal in L is obviously the sole possible candidate for
> faultiness, and so he evidently finds it appropriate to call
into
> question other propositions upon which his argument is based.
He's arguing against Cantor's theorem. That's a theorem of
ZFC. If he
wants to argue that the propositions on which Cantor's theorem
is
based lead to contradiction, he should find a contradiction in
ZFC.
I'm not aware of any definition of English in ZFC. We could
let L be
the language of second-order arithmetic with abstraction
terms. Then
the reason Nathan's argument to a contradiction doesn't go
through in
ZFC is that L can't define the n-th digit of the n-th number
defined
in L. Nathan hasn't found a contradiction in ZFC. He hasn't
given us
any reason to doubt Cantor's theorem.
> Perhaps Nathan thinks that diagonalization in general is
suspect.
> Perhaps he thinks there is some problem in the concept of
infinity or
> of denumerability. Perhaps he thinks there is some much more
deeply
> rooted proposition whose faultiness he has uncovered.
Perhaps he
> even thinks there is paradox at the heart of the mathematical
> enterprise.
Yes, but he hasn't given us any reason to believe that,
because he
hasn't found a contradiction in ZFC.
> Danny Purvis
Subject: Re: Cantor Paradox
>Even more, Nathans list, if it can exist at all, is
already an
>infinite string and it must be given before the diagonal can
be
>constructed.
>
>Which means that any diagonal requires an infinite string, the
>list, for its definition.
> I'm having trouble seeing this point. Why would this same
objection
> not apply to the standard argument showing the reals are not
> denumerable? It seems to me that Nathan can construct his
diagonal
> as far out as he wants to, one step at a time, just the way
the
> diagonal in the standard argument is constructed, only
assuming the
> availability of an algorithm for translating strings to real
numbers.
> Danny Purvis
> The diagonal is supposed to, according to Nathan, to require
only
> finitely many characters for its definition, but the list on
which it is
> defined must contain infinitely many. If the diagonal cannot
exist
> without an infinite list, then it requires infinitely many
characters
> for its definition and cannot be one of Nathan's finite
string numbers.
> A standard real need not be definable by a finite string of
characters,
> in fact it is necessary that most of them not be so
definable.
You are proving your point by assuming what you are trying to
prove.
We don't have to just consider the diagonal proof.
Cantor's first proof, published in 1874, is a finite
description of a real
number that can not be in the set of all definable numbers.
Saying that the diagonal number is not definable is self
defeating.
If the missing real can't be defined, how can you prove whether
it is or is not in Nathan's list?
If the missing number is definable, it can't be in Nathan's
list,
proving the definable reals are uncountable.
Russell
- 2 many 2 count
Subject: Re: Cantor Paradox
>
> >Even more, Nathans list, if it can exist at all, is
already an
> >infinite string and it must be given before the diagonal
can be
> >constructed.
>
> >Which means that any diagonal requires an infinite string,
the
> >list, for its definition.
>
>
> I'm having trouble seeing this point. Why would this same
objection
> not apply to the standard argument showing the reals are not
> denumerable? It seems to me that Nathan can construct his
diagonal
> as far out as he wants to, one step at a time, just the way
the
> diagonal in the standard argument is constructed, only
assuming the
> availability of an algorithm for translating strings to real
numbers.
>
> Danny Purvis
>
> The diagonal is supposed to, according to Nathan, to require
only
> finitely many characters for its definition, but the list on
which it
is
> defined must contain infinitely many. If the diagonal cannot
exist
> without an infinite list, then it requires infinitely many
characters
> for its definition and cannot be one of Nathan's finite
string numbers.
> A standard real need not be definable by a finite string of
characters,
> in fact it is necessary that most of them not be so
definable.
> You are proving your point by assuming what you are trying
to prove.
> We don't have to just consider the diagonal proof.
> Cantor's first proof, published in 1874, is a finite
description of a
real
> number that can not be in the set of all definable numbers.
> Saying that the diagonal number is not definable is self
defeating.
> If the missing real can't be defined, how can you prove
whether
> it is or is not in Nathan's list?
The claim is that the diagonal number is not definable *in L*,
where
Nathan's list is the list of numbers definable in L.
It may be definable in a metalanguage, more powerful than L.
We need to use such a language to define Nathan's list in the
first
place.
> If the missing number is definable, it can't be in Nathan's
list,
> proving the definable reals are uncountable.
>
> - 2 many 2 count
Subject: Re: Cantor Paradox
> Saying that the diagonal number is not definable is self
defeating.
> If the missing real can't be defined, how can you prove
whether
> it is or is not in Nathan's list?
> The claim is that the diagonal number is not definable *in
L*, where
> Nathan's list is the list of numbers definable in L.
> It may be definable in a metalanguage, more powerful than L.
> We need to use such a language to define Nathan's list in
the first
> place.
Nathan's set of definable real numbers is just a list of real
numbers.
We can use any language we want to define these real numbers.
Why should we assume Cantor's missing number exists if
we can't define this number in some language?
I don't see how a metalanguage enters into this.
Any language that can be used to define Cantor's missing number
can also be used to define Nathan's list.
> If the missing number is definable, it can't be in Nathan's
list,
> proving the definable reals are uncountable.
Claiming the diagonal number is only definable in a
metalanguage begs the question of whether this
missing number even exists.
Does this mean Cantor's number is a meta-real number?
Russell
- the universe is one dimensional
Subject: Re: Cantor Paradox
> I don't see how a metalanguage enters into this.
> Any language that can be used to define Cantor's missing
number
> can also be used to define Nathan's list.
Not necessarily. If it requires a priory existence of an
infinite list
to define the diagonal, and Nathan's numbers must be definable
by finite
strings, then that language in which the diagonal is defined
CANNOT be
used for the numbers in Nathan's list.
> If the missing number is definable, it can't be in Nathan's
list,
> proving the definable reals are uncountable.
> Claiming the diagonal number is only definable in a
> metalanguage begs the question of whether this
> missing number even exists.
How so?
> Does this mean Cantor's number is a meta-real number?
Since Cantor identifies irrational numbers with infinite
strings ( a
radix point and infinitely many digit characters), none of
Nathan's crap
is apposite.
Subject: Re: Cantor Paradox
> Saying that the diagonal number is not definable is self
defeating.
> If the missing real can't be defined, how can you prove
whether
> it is or is not in Nathan's list?
>
> The claim is that the diagonal number is not definable *in
L*, where
> Nathan's list is the list of numbers definable in L.
> It may be definable in a metalanguage, more powerful than L.
> We need to use such a language to define Nathan's list in
the first
> place.
> Nathan's set of definable real numbers is just a list of
real numbers.
> We can use any language we want to define these real numbers.
Yes, but if the list is the list of all real numbers definable
in a
language L, you will have to use a language more powerful than
L to
define the list, a language in which you can talk about the
semantics
of L. That's how the metalanguage enters.
> Why should we assume Cantor's missing number exists if
> we can't define this number in some language?
I told you that we could define it in a language, but it can't
be L.
> I don't see how a metalanguage enters into this.
> Any language that can be used to define Cantor's missing
number
> can also be used to define Nathan's list.
> If the missing number is definable, it can't be in Nathan's
list,
> proving the definable reals are uncountable.
> Claiming the diagonal number is only definable in a
> metalanguage begs the question of whether this
> missing number even exists.
It exists as long as the list exists. The list exists as long
as the
language L has a well-defined semantics. In which case there
will be a
metalanguage for talking about the semantics of L.
> Does this mean Cantor's number is a meta-real number?
>
> - the universe is one dimensional
Subject: Re: Cantor Paradox
> Saying that the diagonal number is not definable is self
defeating.
>> If the missing real can't be defined, how can you prove
whether
>> it is or is not in Nathan's list?
>>
>> The claim is that the diagonal number is not definable *in
L*, where
>> Nathan's list is the list of numbers definable in L.
>> It may be definable in a metalanguage, more powerful than L.
>> We need to use such a language to define Nathan's list in
the first
>> place.
> Nathan's set of definable real numbers is just a list of real
> numbers. We can use any language we want to define these real
> numbers. Why should we assume Cantor's missing number exists
if
> we can't define this number in some language?
Nathan's set is the set
{ x in R | exists formula P such that |= P(x) & E!y P(y) }
How do you define this set in the object language without
being able
to represent |=?
Russell, I'm not trying to be insulting here, but if you don't
understand that last question, you're not likely to understand
the
issues here. If you want to pursue this conversation and you
really
don't know what I mean by that last question, then you'll need
some
background reading.
> I don't see how a metalanguage enters into this.
> Any language that can be used to define Cantor's missing
number
> can also be used to define Nathan's list.
>> If the missing number is definable, it can't be in Nathan's
list,
>> proving the definable reals are uncountable.
> Claiming the diagonal number is only definable in a
> metalanguage begs the question of whether this
> missing number even exists.
> Does this mean Cantor's number is a meta-real number?
This nonsense question confirms that you simply don't
understand
object language/metalanguage issues. (Please note: I only
claim a
superficial understanding myself.)
--
Customers have come to SCO asking what they can do to respect
and
help protect the rights of the SCO intellectual property in
Linux.
SCO has created the Intellectual Property License for Linux in
response to these customers needs. -- SCO responds to needs.
Subject: Re: Cantor Paradox
>
> >Even more, Nathans list, if it can exist at all, is
already an
> >infinite string and it must be given before the diagonal
can be
> >constructed.
>
> >Which means that any diagonal requires an infinite string,
the
> >list, for its definition.
>
>
> I'm having trouble seeing this point. Why would this same
objection
> not apply to the standard argument showing the reals are not
> denumerable? It seems to me that Nathan can construct his
diagonal
> as far out as he wants to, one step at a time, just the way
the
> diagonal in the standard argument is constructed, only
assuming the
> availability of an algorithm for translating strings to real
numbers.
>
> Danny Purvis
>
> The diagonal is supposed to, according to Nathan, to require
only
> finitely many characters for its definition, but the list on
which it
is
> defined must contain infinitely many. If the diagonal cannot
exist
> without an infinite list, then it requires infinitely many
characters
> for its definition and cannot be one of Nathan's finite
string numbers.
> A standard real need not be definable by a finite string of
characters,
> in fact it is necessary that most of them not be so
definable.
> You are proving your point by assuming what you are trying
to prove.
> We don't have to just consider the diagonal proof.
> Cantor's first proof, published in 1874, is a finite
description of a
real
> number that can not be in the set of all definable numbers.
Cantor's first proof does not contain a definition of such a
number but
merely a proof that at least one such number has to exist if
the reals
are to have the LUB property.
Subject: Re: area of a circle
In-reply-to: Ignacio Larrosa Ca.96estro
En el mensaje:404F0FEA.8010603@free.invalid,
>philippe 92 escribi.97:
>> [...] [Pizza theorem]
>>> I came up with a pretty short proof based on the fact that
when you
>>> have two intersecting chords, opposing arcs on the circle
sum to
>>> twice the angle between the chords. Since the chords are
>>> perpindicular, we have that the opposing arcs must be
supplementary.
>>> Sliding these arcs and chords together, we see that the
chords form
>>> a right triangle with the diameter of the circle as the
hypotenuse.
>>> See
>>> http://www.whim.org/nebula/math/images/perpchord.gif
>> [...]
>> Fine !
>> I had forgotten this general property of two intersecting
chords.
>> This improves the Pizza Theorem :
>> Not only the areas are equal, but also the portions of the
edge (where
>> usually there is no toping...).
>Certainly, Rob Johnson's proof is much more easy and pretty.
>That didn't seems true for me is that is said in MathWorld:
>If a circular pizza is divided into 8, 12, 16, ...slices by
making cuts
at
>equal angles from an arbitrary point, then the sums of the
areas of
>alternate slices are equal.
>(http://mathworld.wolfram.com/PizzaTheorem.html)
>If the number n of slices is 4 multiple, but not 8 multiple,
it is not
>true, or at least, it isn't supported by the mentionated
proofs. If n =
12,
>by example, you can get 3 bunchs of 4 slices with equal sum
of areas, but
>not 2 bunch of six alternate slices verifying that.
The correct version of the theorem is givent in paper57.pdf.
You are
correct that the version given in PizzaTheorem.html on
MathWorld is not
correct for 8n+4 pieces. For 4n pieces and n people, each
person needs
to take every nth piece, thereby getting 4 pieces each and
getting the
same amount of pizza.
>your first message,
>http://www.maths.unsw.edu.au/~mikeh/webpapers/paper57.pdf.
>(the 'crust' in the corollary)
Rob Johnson Find a configuration of five circles, where each pair
> intersects under the same angle @
> (cos @12=(r1^2+r2^2-d12^2)/(2r1r2) - complex @ are
> allowed, r=0 or oo (points and lines) too.)
> * What is the angle between a point and a circle? (0
degrees, 180
degrees?)
> * What would it be good for to allow lines? -
> Can't we transform lines into circles
> by application of an appropriate angle-preserving map?
http://www.channel1.com/users/bobwb/synergetics/photos/
x6girdle6.html
--Give Earth a Trickier Dick Cheeny -- out of office, after
GIGA years.
http://www.benfranklinbooks.com/
http://www.rand.org/publications/randreview/issues/rr.12.00/
http://members.tripod.com/~american_almanac
Subject: Re: 5 circles
schrieb Brian Quincy Hutchings :
> sounds like Leibniz's *pentagrammum mirificum*,
> or what ever it's called, five geodesics on sphere.
Are you thinking of Gauss's Pentagramma Mirificum???
http://134.76.163.65/servlet/digbib?template=view.html&id=
136942&startpage=1
12&endpage=124&image-path=http://134.76.176.141/cgi-bin/
letgifsfly.cgi&image-
subpath=/3580&image-subpath=3580&pagenumber=112&imageset-id=
3580
But this pentagram consists of five circles (great circles
even),
for which the i-th circle intersects circles (i-1) and (i+1)
*only*
at a 90 degree angle (for i=1,2,3,4,5),
but circles 1 and 3, for example, do meet at a different angle.
Am I missing something? -
I thought, Hauke asked for five circles such that *all*
of the 20 pairwise intersections angles were equal!?
[ ... ]
> I didn't get the notation in the equation;
cosine of alpha-sub(1,2) equals radius-sub(1) to the second
power....
d-sub(1,2) is the distance between the centers?
Looks O.K. - Given two circles with midpoints M1, M2 and radii
r1, r2
which intersect at a point I under an angle alpha,
the distance from I to M1 is r1,
the distance from I to M2 is r2,
and the angle at I is either alpha or 180 degree minus alpha
(depending on how you define the intersection angle of two
circles),
so that by the theorem of cosines the square of the distance
between M1 and M2 is given by
d12^2 = r1^2+r2^2- 2*r1*r2*cos(alpha)
>> Find a configuration of five circles, where each pair
>> intersects under the same angle @
>> (cos @12=(r1^2+r2^2-d12^2)/(2r1r2) - complex @ are
>> allowed, r=0 or oo (points and lines) too.)
Sorry, but the links you appended were not particularly useful
to me.
Thomas
---
Ich hab Dich leider nicht verstanden, aber sag das bitte nicht
nochmal.
Heinz Rudolph Kunze
Subject: Re: 5 circles
I formulated it this way because the usual determinant
approach gives the unique value @=120 deg...but you
can't draw it, some circles will always have radii
being NOT positive real.
--
Hauke Reddmann <:-EX8 fc3a501@uni-hamburg.de
als man ankam wollte man werden, die geschichte schreiben,
die doofen sollen sterben, der plan als man damals nach
hamburg kam
(Kettcar)
Subject: Re: 5 circles
> I formulated it this way because the usual determinant
> approach gives the unique value @=120 deg...but you
> can't draw it, some circles will always have radii
> being NOT positive real.
I see! - So you were using distance geometric methods, too.
Are you interested in the Maple worksheet with the
calculations?
Subject: Re: The Dirac Delta?
>>(It's actually the integral of the function 1 with respect
to the
>>delta function which equals 1. Saying that the integral of
>>the delta function is 1 would imply that it's Lebesgue
integrable,
>>which it certainly is not.)
> Bing! I was waiting for someone to say this.
> I remember my first real QM class, and the dirac function was
> introduced. It was the first time I had to seriously work on
> understanding it. Finally the instructor understood what I
wanted in
> my persistent questions in his office (not to challenge the
physics,
> but to understand how to make the math rigorous) and he said
we never
> use the delta function except to multiply it with some other
function
> and take an integral. Still not rigorous, but with that
proviso my
> brain clicked.
> Once more, I do wish physicists would stop using physicist
> pseudo-math. But still, there are many things I wish for,
and this is
> hardly the only one I won't get.
> Can we agree to stop calling the dirac delta function a
function
> now?
> Thomas
Dirac's Delta is a functional (member of the dual space of a
proper
function space - e.g. infinitely derivable function with
compact
support). This function space is isomorphic to a sub space of
the dual
space. Some of the results in the dual space have ready match
in the
function space, but some no. E.g., heaviside function is a real
function. Dirac's delta is not (look at the common physicist
definition
- zero all over except in the origin with integral one), just
work out
lebesgue integral from the definition. However, other concepts
are
transferable to the real functions - e.g., convolution with
the delta
function is defined.
This is similar situation to extending the real field into the
complex
one, working some stuff there, and getting back results for
the reals.
Try any book on functional analysis (the classic one being by
Dunford &
Schwartz).
Subject: generalization of stokes' and divergence theorems?
Stokes theorem says SS[ curl F dS ] over S = S[ F dr ] over C,
and the
divergence theorem says SSS[ div F dV] over E = SS[ F dS ]
over S.
Is there a theorem that says, SSSS[ something(F) dH ] over H =
SSS[ F dV]
over V. Wouldn't V have to have an orientation? It seems that
in R^3 we
couldn't have that, but in R^4 I'd think you could define
something like
that.
Is there a general something where I can write:
SS...n[ something(F) dA ] over A = SS...(n-1)[ F dB] over B
or something similar?
Thanks,
Jeremy
Subject: Re: generalization of stokes' and divergence theorems?
> Stokes theorem says SS[ curl F dS ] over S = S[ F dr ] over
C, and the
> divergence theorem says SSS[ div F dV] over E = SS[ F dS ]
over S.
> Is there a theorem that says, SSSS[ something(F) dH ] over H
= SSS[ F dV]
> over V. Wouldn't V have to have an orientation? It seems
that in R^3 we
> couldn't have that, but in R^4 I'd think you could define
something like
> that.
> Is there a general something where I can write:
> SS...n[ something(F) dA ] over A = SS...(n-1)[ F dB] over B
> or something similar?
http://en.wikipedia.org/wiki/Stokes'_theorem
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: Re: generalization of stokes' and divergence theorems?
The theorems you cite (Green, Stikes, Grad, Div, Curl, etc.
are from vector
field theory in 3d. The ones that you wish to obtain require an
understanding
of differential geometry on arbitrary manifolds.
Check out Michael Spivak's work or that of Harley Flanders
Subject: Re: generalization of stokes' and divergence theorems?
> The theorems you cite (Green, Stikes, Grad, Div, Curl, etc.
are from
> vector
> field theory in 3d. The ones that you wish to obtain require
an
> understanding of differential geometry on arbitrary
manifolds.
The term differential *geometry* could be misleading; perhaps
calculas on
manifolds fits best.
--
I'm on warm milk and laxatives
Cherry-flavored antacids
reverse my forename for mail! - saibot
Subject: Re: generalization of stokes' and divergence theorems?
>Check out Michael Spivak's work or that of Harley Flanders
I liked Spivak's _Calculus on Manifolds_.
Keith Ramsay
Subject: Help with equation
Can anyone tell me what the mathematical equation is for the
following
statement:
The whole is worth more than the sum of its parts.
Thanks in advance,
xillion
Subject: Re: Help with equation
In sci.math, xillion
:
> Can anyone tell me what the mathematical equation is
> for the following statement:
> The whole is worth more than the sum of its parts.
> Thanks in advance,
> xillion
The equation? This is far too vague a specification.
However, one could be slightly arbitrary; the following
equation is a more or less literal transilation of
the statement:
S = sum(i,+oo) s_i > S_n = sum(i=1,n) s_i
(in more traditional math texts 'sum' would be a large sigma).
and this is true for any convergent series where each term is
positive.
For example,
S = sum(i=1,+oo) (2^(-i)) = 1
and
S_n = sum(i=1,n) (2^(-i)) = 1 - 2^(-n)
as can easily be verified. But this is one of an infinite
number
of examples; another one, for instance, is
e^x = sum(i=0,+oo) x^i/i! > sum(i=0,n) x^i/i!
for x > 0.
Another example, which does not converge, is
+oo = sum(i=1,+oo) 1/i > sum(i=1,n) 1/i
Unfortunately, the whole is not necessarily an infinite series,
its parts is not necessarily its terms, and worth does not
refer to numeric value. As a philosophical statement, it's
quite profound. As a mathematical statement, it's far too
vague.
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
Subject: Re: Help with equation
Gestault=True
> Can anyone tell me what the mathematical equation is for the
following
statement:
> The whole is worth more than the sum of its parts.
> Thanks in advance,
> xillion
Subject: Re: Help with equation
> Can anyone tell me what the mathematical equation is for the
following
> statement:
> The whole is worth more than the sum of its parts.
> Thanks in advance,
> xillion
What makes you think there is one?
Subject: Re: Random pi digits and Mahler's theorem
charset=Windows-1252
[typo correction]
Extending the previous results (well-known, no doubt) ...
Theorem.
Let X be uniformly distributed on interval (a, b), where a and
b
are integers, and let m be a real number > 1. Then
pr( exist integers p,q > 1, |X - p/q| <= 1/q^m ) <=
2(Zeta(m)-1)
where Zeta is the Riemann Zeta function.
Proof (sketch):
pr( exist integers p,q > 1, |X - p/q| <= 1/q^m )
<= sum( pr(|X - p/q| <= 1/q^m, integers p,q > 1 )
(subadditivity)
<= sum( 2(1/q^m)/(b-a) + (q(b-a)-1)(2/q^m)/(b-a), q > 1) [*]
<= sum( 2/q^(m-1), q > 1 )
<= 2 (Zeta(m-1) - 1)
where [*] follows by noting the cases summarized in the
following
table, each case corresponding to the joint occurrence of the
conditions indicated in the first two columns of a row:
contrib. to sum
p/q - 1/q^m p/q + 1/q^m cases #cases (per case)
--------------------------------------------------------------
---
<= a > a p = aq 1 (1/q^m)/(b-a)
< b >= b p = bq 1 (1/q^m)/(b-a)
> a < b aq < p < bq q(b-a)-1 (2/q^m)/(b-a)
other other other 0
--------------------------------------------------------------
---
QED.
Some examples:
m 2 ( Zeta(m-1) - 1)
-----------------------
4 0.40411...
5 0.16464...
6 0.73855... 10^-1
7 0.34686... 10^-1
8 0.16698... 10^-1
9 0.81547... 10^-2
10 0.40167... 10^-2
15 0.12249... 10^-3
20 0.38164... 10^-5
30 0.37253... 10^-8
40 0.36379... 10^-11
42 0.90949... 10^-12
--r.e.s.
Subject: Re: Random pi digits and Mahler's theorem
charset=Windows-1252
Extending the previous results (well-known, no doubt) ...
Theorem.
Let X be uniformly distributed on interval (a, b), where a and
b
are integers, and let m be a real number > 1. Then
pr( exist integers p,q > 1, |X - p/q| <= 1/q^m ) <=
2(Zeta(m)-1)
where Zeta is the Riemann Zeta function.
Proof (sketch):
pr( exist integers p,q > 1, |X - p/q| <= 1/q^m )
<= sum( pr(|X - p/q| <= 1/q^m, integers p,q > 1 )
(subadditivity)
<= sum( 2(1/q^m)/(b-a) + (q(b-a)-1)(2/q^m)/(b-a), q > 1) [*]
<= sum( 2/q^(m-1), q > 1 )
<= 2 (Zeta(m) - 1)
where [*] follows by noting the cases summarized in the
following
table, each case corresponding to the joint occurrence of the
conditions indicated in the first two columns of a row:
contrib. to sum
p/q - 1/q^m p/q + 1/q^m cases #cases (per case)
--------------------------------------------------------------
---
<= a > a p = aq 1 (1/q^m)/(b-a)
< b >= b p = bq 1 (1/q^m)/(b-a)
> a < b aq < p < bq q(b-a)-1 (2/q^m)/(b-a)
other other other 0
--------------------------------------------------------------
---
QED.
Some examples:
m 2 ( Zeta(m) - 1 )
----------------------
4 0.40411...
5 0.16464...
6 0.73855... 10^-1
7 0.34686... 10^-1
8 0.16698... 10^-1
9 0.81547... 10^-2
10 0.40167... 10^-2
15 0.12249... 10^-3
20 0.38164... 10^-5
30 0.37253... 10^-8
40 0.36379... 10^-11
42 0.90949... 10^-12
--r.e.s.
Subject: (clarification) chapter 2 in math faq
There have been many threads on are the natural numbers in the
integers...
and I understand the usual responses to these threads
(essentially what you
said below).
But my original question was about the domains of the
operations on the
sets
Z,Q,... , not the sets.
What type of ignoring goes on here? Is + just domain-less when
it
doesn't
matter?
Ie, in elementary number theory, frequently there is division,
multiplication by sqrt(2), etc. Obviously we can't always work
over only
(Z,+_Z,*_Z). Is it understood that there is a constant
switching of
number system _operations_, or something else?
Thank you,
Khintchine
> the math faq talks about construction of the number systems.
in 2.2.8
> it mentions that the naturals are not a subset of the
integers, etc. it
> defines some embeddings so they can be considered to be
subsets.
> so it's unambiguous to talk about the number 3. what about
the
expression
do people generally imagine little Z,Q,R,C's floating around
under the
+
> signs?
Fishfry said,
> No, people just ignore these fine points and imagine the
integers being
> a subset of the rationals being a subset of the reals. The
only time the
> natural embeddings are important is if you are discussing the
> construction of the number systems.
Subject: stochastic group S(2,4)
How to prove it is isomorphic to alternating group of order 12?
Subject: Re: stochastic group S(2,4)
> How to prove it is isomorphic to alternating group of order
12?
Um, what is it?
--
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: stochastic group S(2,4)
>> How to prove it is isomorphic to alternating group of order
12?
>Um, what is it?
Never heard of it. S(2,4) is sometimes used for symplectic
group Sp(2,4),
but that is the same as SL(2,4) and is isomorphic to
alternating group of
order 60.
It's not SO^+(2,4) or SO^-(2,4) either - they have orders 6
and 10.
Derek Holt.
Subject: Re: Proofs
>I would like to know how I may approach if someone asks me to
prove a
>wrong statement. For example if someone asks me the statement
'prove
>that 101 is an even integer'.
> In this case I would suggest that you prove that the
statement is
false.
> In your case, this is rather obvious, e.g. you could assume
that there
> is an integer k with 2*k=101, but then 50 Anyway, this statement is obviously false, so you could just
say that
> and that's all.
> If someone asked me to prove that 101 is an even integer, I
would first
> mention to that person that he must not be dealing with
decimal numbers,
> because in base ten, 101 is not even. Then I would prove to
him that 101
> in an odd base is even.
Indicating that context is everything. This could be a quite
serious
question on a Friday afternoon (especially at Coslo's or
whatever the local
equivalent is), or in a problem solving class, or just
somewhere where
clear statement of assumptions is important (or even critical)
(are you
listening, NASA: a billion dollars here, a billion dollars
there, and
pretty soon you're talking real money).
Jon Miller
Subject: Re: tensors
>I'm understanding tensors are things who obey certain
transformation
>properties and algebraic properties and everything, but I
don't understand
>what they physically/geometrically mean.
Have a look at http://math.ucr.edu/home/baez/gr/gr.html
Keith Ramsay
Subject: Re: tensors
> I'm understanding tensors are things who obey certain
transformation
> properties and algebraic properties and everything, but I
don't
understand
> what they physically/geometrically mean. What is the physical
significance
> of a covariant, contravariant, or mixed tensor of a specific
order?
I.e.,
> what's the physical difference between A^ij, A_ij, and A^i_j?
Would 'columns of columns', 'rows of rows', and 'rows of
columns' satify
the
question or are you looking for some physical orientation
explanation like,
whats the significant diferrence between 'left handed' and
'right handed'
viewpoints? There's a prevailing viewpoint under which 'column
vectors'
transform covariantly, but if you find linearly independent
functionals in
that
space, they form a dual space of contravariant dual row
vectors. The two
spaces
are images of the same 'physcal' phenomenon, its just
convention that
begins
with seeing it one way first.
> I'm also
> confused as to why we write a dot product of vectors in
tensor notation
as
> a_i b^i, why not a_i b_i, or a^i b^i? Is there a difference
between a^i
b_i
> and a_i b^i? I guess I'm not really understanding the
difference between
> covariant, contravariant, and mixed, other than that the
differences in
how
> they transform.
This is just for consistency. Group operations are dependent
on left or
right
action. Those transforms don't let you jiggle them around
willynilly
Say for a 2x3 matrix, the row dimension is 2 and column
dimension is 3,
and
that limits what you can do with the rows versus column
operations.
Hope this helped a bit.
MK
Subject: Re: tensors
Schaum's Outline on Vector Vnalysis will help you keep track
of your
Christoffel Symbols when understanding and manipulating
tensors.
A good book on Differential Geometry will help. Try Dirk
Struik (MIT),
Dover
Publ.
Subject: Re: simulating bivariate exponential
>I am looking for a way of simulating from the bivariate
exponential
distribution given by Marshall & Olkin:
>F(x,y) = exp{-[Micro]1x-[Micro]2y-([Micro]3+d)max(x,y)}
>in order to simulate a trial with surrogate and primary
endpoint failure
times.
Is that the tail distibution (P{X>x, Y>x})? The unnormalized
density?
It certainly cannot be the CDF.
How about generating one variate according to the marginal and
the other
according to the conditional distribution?
--
Stephen J. Herschkorn herschko@rutcor.rutgers.edu
Subject: Re: Math Shirts (again)
> Hey, I just wanted to update (see:
>
http://groups.google.com/groups?dq=&start=25&hl=en&lr=&ie=UTF-
8&oe=UTF-8&gro
>
up=sci.math&selm=4bc08be9.0312282107.564f1695%
40posting.google.com
> )
> Got a new url going, now it's simply
http://www.mathshirts.com and I'm
> adding new designs as I think of them
> Comments, critiques, congratulations and cognac are all
welcomed and
> invited.
> I have seen studnets wearing shirts bearing Maxwell's
equaions. Maybe
> something of that level of tech-geeky-inside-jokiness would
be more
> attractive.
Yeah, I've seen them too .. and that's the problem, I don't
want to
just make designs that are already out there, if you want
that, go for
it, but in all honesty, I'm making these designs for myself,
and I
figure there are going to be some people out there with similar
tastes. I feel like there's nothing besides inside jokes,
equations on
shirts and elementary school teacher designs.
Subject: Re: Math Shirts (again)
> I have seen students wearing shirts bearing Maxwell's
equations. May be
> something of that level of tech-geeky-inside-jokiness would
be more
> attractive.
It is nevertheless likeable, this one may be ?
http://www.scienceteecher.com/astronomy_all.htm
Mental images have spurred many a mathematician, physicist
and chemist during their creative persuits....
Among the equations and images I consider T-Shirt display
worthy :
and the negatively curved vortex as model for hyperbolic
geometry.
F= m a and the falling apple and the rocket.
e ^ [ 2 Pi n i] = 1 Euler's equation
Euler-Bernoulli Equation of Variational calculus
_
Action Integral: Max _/ (T-V)dt
D N A molecule with cross bands of Crick/Watson
Zhukowski Airfoil, W= z+ a^2/z, and the Kitty Hawk WB aircraft
Fractal images Mandelbrot / Julia anmd FeigenBaum diagram
Platonic Solids and the mixed surface Icosidodecahedron with
its
great circles.
Pyramids of Egypt.
Schroedinger's Equation
Subject: Re: David Ullrich's parachute
<$D$zGFA64NUAFwMT@meden.demon.co.uk
Discussion, linux)
>>> Working quickly, Mr. Ullrich used his sand-making device
to add
>>> an infinite amount of new sand to his backpack. At
infinity, the
>>> Earthward forces and the Moonward forces balanced
eachother perfectly,
>>> and our friend remained suspended in mid-air until the Air
Force could
>>> rescue him.
>>I'm not a physicist, so you'll have to help me here. Does he
achieve
>>equilibrium before or after the black hole forms?
> After I think, as the perfect equilibrium is approached as
the mass
> tends to infinity.
> However, he may be hit by the Earth, moving at high
velocity, before the
> black hole is formed, but this depends on exactly how fast
the sand is
> produced.
> But, if the infinite amount of sand is produced
instantaneously
> equilibrium is obtained and the black hole created at the
same time.
I think that Nathan has taught us an important lesson. If we
allow
infinite mass in a very limited space in our physical models
(and also
decide to suspend the conservation of matter/energy), then
counterintuitive things happen.
Obviously, we must reject Cantor.
(Ignoring, for the moment, the idea of limits tending to
infinity
predating Cantor and the means of comparing two such limits by
their
limit of their ratios.)
Nathan, not your best effort. You're trying to hard. Go play
soccer
and take your mind off it for an afternoon.
--
My proofs are out there.
-- James S. Harris
Subject: need help on prob and stat~ Thanks
1) There are k taxis in Vancouver, numbered consecutively 1 to
k. Suppose
that the taxi that brings you from the airport is number 123.
(a) Devise a suitable model for this situation. Give an
unbiased estimate
for
k.
(b) A friend claims to have heard that k 1000. Do the observed
data
provide grounds to reject the claim at the 5% signi cance
level? (Hint:
under
the claim, what is the probability your taxi number would be
this small?)
(c) Suppose your next taxi is number 200. Answer (a) and (b)
again using
all the data now available. (Hint: in (a), consider the sum of
the two taxi
numbers).
2) Let X1,X2.... be iid with mean and variance. Let Sn^2 be
the sample
variance of X1,...Xn. Show lim n-> infinity Sn^2= var with
probability 1.
Subject: Gaston Julia's Sets
I took a the escape theory of the series for the Gaston
Julia's set
and remind the reader to learn why it is an escape number to
define
the set.
Gaston's Differentail- defined using the escape number
from a Gaston Julia's Set website
**************************************************************
******
The pioneering work of the French mathematician Gaston Julia,
published in 1918 when Julia was 25 years old, was essentially
forgotten by the mathematical world until the 1980's, when
computers
made possible the visualization of his creation. Julia's idea
was to
observe the behavior of the orbit of a complex number (see
Vignette 6
and Vignette 10) under iteration of a function f . That is,
begin
with a complex number z0 , visualized as a point in the plane,
and
apply f to z0 . The resulting value is fed back into the
function f
to obtain a new complex number z1. This in turn is fed back
into f to
obtain z2, and so on. The resulting sequence of complex
numbers {z0 ,
z1 , z2 , ...} is called the orbit of z0 under f .
As an example, consider the function . Starting with z0 = 0, we
obtain the following orbit:
The terms in this sequence remain bounded -- that is, they
never get
very far from 0, the origin of the complex plane.
On the other hand, for the same function f, the orbits of other
starting points may move arbitrarily far away from 0. As an
example,
if we start with z0 = 2, we get the following orbit:
We will refer to a complex number z0 as a prisoner if its
orbit under
f is bounded, and an escapee if the orbit is unbounded -- that
is,
terms in the orbit get arbitrarily far away from 0. The set of
all
prisoners for a given function f is called its prisoner set,
and the
set of all escapees is called the escape set.
In the above examples, we used complex numbers z0 for which the
imaginary part is 0, just to simplify the calculations. But the
analogous calculations can be done for any complex value of z0
, using
the multiplication and addition of complex numbers as
discussed in
Vignette 10. For example, for the function , the orbit of is
In this example, we see that is in the prisoner set for ,
because its
orbit is bounded.
Julia Sets
Julia was interested in the properties, for various functions,
of the
prisoner set and the escape set -- and also what is now called
the
Julia set for the function. The Julia set is defined to be the
boundary between the prisoner set and the escape set. The
Julia set
for f consists of the points In order to determine these sets
experimentally, it is necessary to compute the orbits of a
great many
starting points z0 , since we must know the orbit in order to
determine whether the starting point is a prisoner or an
escapee. And
if we had to do the calculation of orbits such as the last one
above
purely by hand -- for several thousand starting values -- we
would
quickly lose interest in the project!
That's where mathematical theory comes into play. Even without
actually doing all of the tedious calculations, a number of
things can
be proved about prisoner sets, escape sets and Julia sets. But
rather
than looking at the theory, we will instead consider the fact
that,
starting in the 1980's, the availability of computers led
people back
to the idea of a visual exploration of Julia sets -- with
incredibly
beautiful results. After all, a computer could be programmed
to do
the calculation of an orbit of a starting value of z0 -- and
would
not even object if asked to do a similar calculation for
thousands or
even millions of such similar values!
For simplicity, let's consider only functions of the form ,
where c is
some fixed complex number. It can be proven that if the orbit
of a
starting value z0 ever leaves the circle of radius 2 centered
at the
origin, then the rest of the terms in the orbit get
successively
farther from 0, and in fact the orbit is unbounded. Such a
starting
value z0 would therefore be an element of the escape set for f
.
Armed with this simple mathematical theorem and a good bit of
computer
power, we are ready to visualize the prisoner set and escape
set for a
given function of the form . The Julia set can then be
pictured as
the boundary between the two sets.
**************************************************************
*************
Subject: Re: Gaston Julia's Sets
> Julia was interested in the properties, for various
functions, of the
> prisoner set and the escape set -- and also what is now
called the
> Julia set for the function. The Julia set is defined to be
the
> boundary between the prisoner set and the escape set.
These are oversimplifications intended for non-mathematicians.
For correct versions, see (for example) Devaney's book
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
Subject: Re: question about topology
Rup
> Sorry, why does H_k(R^n,R^n{x})=H_k(S_n-1)?
H_k(R^n,R^n{x})=H_k(Rn{x})
Just another excison.
(with a little unimportant problem for k=0 I don't want to
discuss)
R^n{x} deformation retract to S_n-1. So they have same
homotopy type
and same homology type. It is this way of thinking that make
homotopy
and holology so powerfull.
If you are interested in the topics. Read the chapter about
homology in
Hatcher's book Algebraic Topology. Just a few hours to
understand the
stuff. Other chapters or other books like the great book of
Spanier are
more difficult. It can be foung online.
Subject: New source of the math!!!
I have come across a source providing good support to those
experiencing difficulties with doing homework in maths
including
arithmetic(http://www.bymath.com/studyguide/ari/form1.htm),
geometry
(http://www.bymath.com/studyguide/geo/pro/pro1/pro1.htm),
algebra
(http://www.bymath.com/studyguide/alg/pro/pro1/pro1.htm),
functions
and graphics
(http://www.bymath.com/studyguide/fun/pro/pro.htm),
principles of
analysis(http://www.bymath.com/studyguide/ana/pro/pro1/pro1.
htm),
all this is also supported by a good deal of examples and
illustrations.
Subject: Re: Die Petry, die
>> It clearly makes sense to talk about
>> natural numbers in relation to computation. Sets of natural
>> numbers, sets of sets of natural numbers, sets of sets of
sets
>> of natural numbers and so on... these are the familiar kinds
>> of sets by which set theorists model most ordinary
mathematics.
>> At what point does it become meaningless? I don't see that
>> it does. Is the complaint only about higher set theory?
> Over the past 15 years, in both newsgroup discussions and
> in private email discussions, I have expended a great deal
> of energy trying to explain to you the idea of observability
> and the motivations for it. It is clear to me that all that
> energy was wasted.
Yes.
--
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: Trapezoidal Rule & Simpson's Rule - Comparison
> I'm looking for a simple and obvious example where the
Trapezoidal Rule
> would give a better approximation to a given definite
integral than
> Simpson's Rule. Assume the number of subintervals used is
the same for
>both approximations.
For f:[A,B]-->R integrable on [A,B] let us denote
I(f)-INTEGRAL_{x=a to x=b}f(x) dx , -infty < A =< a < b =< B <
infty
,
Consider the quadrature formulas
b-a
(1) I(f)= ----- ( f(a)+ f(b) ) + RT(f)
2
b-a a+b
(2) I(f)= -----( f(a) +4*f(-----) + f(b) ) +RS(f)
6 2
where RT(f)=RT(f;a,b) , RS(f)=RS(f;a,b) are the remainders in
trapezoidal formula respectively in Simpson (Kepler)
quadrature.
Observe that
2*(b-a) f(a)+f(b) a+b
RT(f)-RS(f)= -------( --------- - f(-----) ) .
3 2 2
Therefore we have following assertion :
Proposition. The inequality
============
RT(f;a,b) <= RS(f;a,b) for all [a,b] , subset in [A,B] ,
if and only if f:[A,B]---> is a concave function.
=====
In fact, If f is in C^{2}[a,b], m => I'm looking for a simple and obvious example where the
Trapezoidal Rule
>> would give a better approximation to a given definite
integral than
>> Simpson's Rule. Assume the number of subintervals used is
the same for
both
>> approximations.
>Let f(x) = x^(2/5) on the interval [-1,1] and use one
subinterval.
I assume you mean two subintervals, i.e. you evaluate f at the
three
points -1, 0 and 1, with Trap = 1/2 f(-1) + f(0) + 1/2 f(1) and
Simpson = 1/3 f(-1) + 4/3 f(0) + 1/3 f(1).
A singularity is not necessary, of course: try cos(pi x) which
the trapezoidal rule (for these points) happens to get exactly
right
but Simpson doesn't. And this generalizes easily to give
examples
with more subintervals.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
Subject: re:Die Cantor Die
Godel gave a big blow to all those who advocated Petry's views
at his
time.
Of course the examples of undecidable statements he gave
aren't very
important in applications (unless you need the consistency of
arithmetics to sleep at night), but later more natural
examples have
been found, some of them about simple games like the hydra
game.
Few believe now that the Riemann Hypothesis is unprovable in
ZF, but
is it provable in finite arithmetics? It sure is useful to
derive
properties of finite arithmetics that are important in
applications.
Some results in operator theory that motivated some quantum
mechanical
models probably lie outside the scope of finite arithmetics
too.
Examples of genuinely undecidable statements of finite
arithmetics are
found all the time, but more importantly, many useful results
are
obtained from ZF that no one knows how to prove (or sometimes
even
formulate) by finite methods.
Petry's program would require that we abandon the concept of
proof and
focus on just using the results we believe to be true.
But historically it was always the search for proves that
driven
mathematical investigations. So the results will cease
arriving and
we'll have no new mathematics to apply.
Posted Via Usenet.com Premium Usenet Newsgroup Services
----------------------------------------------------------
** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **
----------------------------------------------------------
http://www.usenet.com
Subject: Re: Die Cantor Die
> And, to repeat myself because so many of my detractors
> never seem to catch on, formal systems themselves are
> objects that live in the world of mathematics as I
> envision it.
Except that calculations in them are arbitrarily excluded
from Petry's subjective objective world of computation.
--
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: Die Cantor Die
> The basics ideas I stress are: (1) mathematics can be
> treated as a science
Well there we are. That's a fundamental blunder.
--
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: Die Cantor Die
> We don't lose any of our capacity to understand
> the reality we live in if we believe that the purpose of all
> abstract thinking is to relate the phenomena we observe in
> the real world to phenomena we observe in the world of
> computation.
> Why would I think the purpose of all abstract thinking is to
relate
> observable phenomena to computational phenomena?
> The idea is that the mind can be understood by analogy with
> artificial intelligence software running on a computer (the
> brain).
Shorly after electricity was discovered many people thought
the mind could be understood as electrical interactions.
Frog legs could be made to twitch with a spark, etc.
Nowadays, we know that thinking does involve
electrical impulses, but few people would say that
the mind can be understood by anology with electricity.
Every generation thinks thinking can be explained by the
latest cool idea. Certainly, the mind does have a computational
element, but there is a lot more to the human mind than
computation. I know. I've written AI programs and they
don't think like humans. This is probably a good thing.
I wouldn't want a human trying to control a large chemical
plant
or trying to land the Space Shuttle from orbit.
Russell
- Zeno was right. Motion is impossible.
Subject: Re: Die Cantor Die
> Certainly, the mind does have a computational
> element, but there is a lot more to the human mind than
> computation.
Well, Penrose would agree with you. But I've never seen
an argument for your view that I find remotely convincing.
Presumably the workings of the brain obey the laws of
quantum mechanics, which in turn could be modelled in a
large computer. So maybe you're saying the mind is more
than the workings of the brain.
I will not continue this line of discussion.
Subject: Re: Die Cantor Die
> Mathematics has ALWAYS been an exquisite game, with its own
> rules and in its own world. Wanna play? Accept the rules.
> In deed, that's seems to be the way the pure mathematicians
> view it. To everyone else - those who actually apply it -
> mathematics is more than just a game. There is an underlying
> reality to it, which is computation.
> It's not the computation, it's the utility, that everyone
else is
> interested in.
> But without the pure mathematicians to create the stuff that
these
> non-mathematicians are so eager to profit from, it wouldn't
be there for
> them at all.
Many advances in pure mathematics were motivated by practical
concerns. The Egyptians figured out how to take the roots of
numbers so they could build better pyramids.
Newton developed calculus to describe the motion of planets.
I think pure and applied mathematics complement each other.
Russell
- 2 many 2 count
Subject: Re: Die Cantor Die
> I think pure and applied mathematics complement each other.
>
> - 2 many 2 count
And as long as neither tries to inhibit what the other does,
it shall
remain so.
Subject: Re: Die Cantor Die
>>Which of the axioms do you object to?
> The objection is to the logic as much as the axioms.
My first reaction on reading this statement is to suspect that
you have
a disagreement with the workings of formal mathematical logic.
If so,
you may find that you have a lot of work to do to get anyone
to join you
in changing the foundations of mathematics. If I am
misunderstanding
your meaning, then I have no clue what you are trying to say.
> Certainly the usual constructivist objections to the law
> of the excluded middle and the axiom of choice are
> things that I accept but do not stress. (They are
> not fundamental objections)
Ok.
> The basics ideas I stress are: (1) mathematics can be
> treated as a science which requires statements to
> have falsifiable implications
Falsifiable against what? The best you can do is say that a
certain
section of mathematics is not a model of reality. I view it as
the job
of physicists, chemists, etc to determine which mathematical
models best
represent reality.
> (2)when we observe
> infinitary objects, we are observing approximations,
> and the logic we use must then deal with uncertainty
There is already a branch of mathematics that deals with these
uncertainties without the need to change the foundations of
mathematics.
> (3) infinity is a useful fiction, but emphatically, it
> is a fiction. When we create this fiction, we must be
> careful that every object living in the world of the
> infinite has corresponding observable approximations.
Infinity means not finite. If you refuse to accept that there
is
anything that is not finite, I ask you: how many integers are
there? An
upper bound will be sufficient, but it must be finite.
> And, to repeat myself because so many of my detractors
> never seem to catch on, formal systems themselves are
> objects that live in the world of mathematics as I
> envision it.
What makes you think they aren't already?
> Those who apply mathematics would have no trouble
> living with these requirements.
Unless those requirements destroyed the foundations of
mathematics and
made it impossible to do anything with computations knowing
that they
worked as expected. This is why I suggest you start doing the
work,
rather than telling us about what you would like to see done.
Be more
than a visionary, help build your vision.
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
> The basics ideas I stress are: (1) mathematics can be
> treated as a science which requires statements to
> have falsifiable implications
> Falsifiable against what?
The world of computation.
Let me give a simple example. Goldbach's Conjecture makes
a prediction that every even number >2 is the sum of two
primes. That is falsifiable in the sense that if we were to
pick some even number, and verify by computation that there
is no pair of primes (each less than our number) whose sum
is equal to our number, we would have falsified the conjecture.
> The best you can do is say that a certain
> section of mathematics is not a model of reality.
You're not following my arguments. Maybe it's my fault.
> This is why I suggest you start doing the work,
> rather than telling us about what you would like to see
done. Be more
> than a visionary, help build your vision.
Good advice is always welcome. Thanks.
Subject: Re: Die Cantor Die
>>>The basics ideas I stress are: (1) mathematics can be
>>>treated as a science which requires statements to
>>>have falsifiable implications
>>Falsifiable against what?
> The world of computation.
Isn't mathematics the mechanism of computation? I am concerned
that you
may enter a circular loop here. Mathematics can be treated as
a science
which requires statements to have falsifiable implications
against the
world of computation which is done in mathematics. What
happens if the
computation you are testing a new idea against is an incorrect
computation? What if one of your estimates is off?
> Let me give a simple example. Goldbach's Conjecture makes
> a prediction that every even number >2 is the sum of two
> primes. That is falsifiable in the sense that if we were to
> pick some even number, and verify by computation that there
> is no pair of primes (each less than our number) whose sum
> is equal to our number, we would have falsified the
conjecture.
And if you find yourself in the state we are in now, would you
estimate that it is true, or leave it as an open problem?
Also, how
does this different from the current state of affairs?
>>The best you can do is say that a certain
>>section of mathematics is not a model of reality.
> You're not following my arguments. Maybe it's my fault.
Then please clarify.
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
>> I view the purpose of mathematics as extending the bounds
of logical
>>consequences. Some of these results have useful application
in modeling
>>aspects of the real world, but that is not necessarily the
intended
>>function of any given branch of mathematics.
> As I have pointed out, formal systems themselves are objects
that
> live in the world of computation. What's being questioned is
the
> interpretation of the formal sentences which are provable
within
> those formal systems. That is, mathematical statements can
have
> meaning, but not all statements in all formalisms have
meaning.
What is the difference between meaning and meaning?
> In particular, the assertion that there exists a
super-infinite
> world is being questioned. What does exists mean in that
formal
> implication of set theory?
Since you put exists in quotes, I don't know that you mean
what most of
us mean. I interpret it to mean a logical constract that may
or may not
have anything to do with reality. A concept exists.
>> To abandon the standard way of thinking about these sets
and adopt
>>your proposed approach appears to lead to disagreements
between the two
>>theories, yet you wish to claim they will be basically the
same in their
>>results.
> I am claiming that those who apply mathematics will see
little
> or no change.
Then why change?
>>I am unconvinced that your perspective will allow limits in
calculus to
>>be formally defined, given the problems above.
> To do calculus, we would first define the concepts we want to
> deal with, with the recognition of the notions of imprecision
> and uncertainty, and then abstract out those notions (i.e.
consider
> the limit as the uncertainty and imprecision go to zero)
I am not convinced that you can effectively define limits
without a
notion of the infinite. In particular, how will you guarantee
that
certain numbers exist, if you deal only with finite sets?
>>I don't know if these are issues you have thought about, but
until I can
>>see a resolution between standard mathematics and your
proposed new
>>foundation, it seems unlikely that I will be joining you in
advocating a
>>world of the finite.
> Well, advocating a world of the finite is a distortion of
> what I am doing.
> I don't want to push you. Maybe someday it will occur to you
> that there is a better way than what you are being taught.
There may be better ways than what I have studied, but the
ways I have
studied work as a whole. I am not convinced that your proposed
changes
will not cause all of mathematics to collapse or at least
require
enormous work to reconstruct.
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
> That is, mathematical statements can have
> meaning, but not all statements in all formalisms have
meaning.
> What is the difference between meaning and meaning?
By putting the word in quotes, I was suggesting to the reader
that he should give extra throught to the meaning of the word.
Mathematical statements can be meaningful in the sense that
they make predictions about the outcome of computational
experiments. But not all statements in all formalisms are
meaninful in that sense.
Subject: Re: Die Cantor Die
>>>That is, mathematical statements can have
>>>meaning, but not all statements in all formalisms have
meaning.
>>What is the difference between meaning and meaning?
> By putting the word in quotes, I was suggesting to the reader
> that he should give extra throught to the meaning of the
word.
I see. I interpreted it as meaning you wanted to add some
subtlety to
the standard definition without stating what it was.
> Mathematical statements can be meaningful in the sense that
> they make predictions about the outcome of computational
> experiments. But not all statements in all formalisms are
> meaninful in that sense.
Is that the only sense in which they are meaningful?
Is that the only sense in which they are meaningful to you?
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
>>>mathematics is more than just a game. There is an underlying
>>>reality to it, which is computation.
>>There is more than computation.
> I'm not sure you've thought this through.
> Try to find one single mathematical statement which would be
> of interest to those who apply mathematics, which could not
be
> interpreted as a statement about the world of computation.
There is a key concept here that you don't seem to understand:
there is
a difference between mathematics and applying mathematics. You
are
interested in only those areas of mathematics that can be
applied. Some
of us are not. Why should we restrict ourselves to your area
of interest?
be codified to be verified by a computer proof-checker, it
could also be
argued that all of math is *already* computation and that we
have
already achieved your vision. Perhaps your understanding of
computation
is more limited than ours?
Since you are arguing for a change, it appears you do not
accept the
second option.
>>Why should we not be interested in a
>>larger view?
> The world of fiction is not reasonably called a larger view
> than the world of reality.
Switching to literature: the world of fiction has driven many
of the
inventions that have become part of reality. The notion of
travel in
space was around long before flight was believed to be
possible for man.
The internet was in its infancy when virtual reality systems
were
being envisioned in stories. Linking computers to humans was
fantasy
for a long time, yet is becoming real now.
Fiction can give vision to what reality will become. In a
similar way
pure math can lead to breakthroughs in applied math. The issue
is:
we don't know in advance if a non-computational aspect of math
will lead
to an advance in computational math. Formal logic and set
theory were
developed to verify that other branches of mathematics were
valid. What
good is a computation if you don't know whether the results
are correct
or useful?
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
> There is a key concept here that you don't seem to
understand: there is
> a difference between mathematics and applying mathematics.
When trying to distinguish between productive lines of inquiry,
and mental masturbation, it's a good idea to consider the
possible
applications of your inquiry.
> we don't know in advance if a non-computational aspect of
math will lead
> to an advance in computational math.
There is some truth to this, although mostly you are arguing
with
your own misinterpretation of what I'm saying. As I have
pointed
out way too many times, the study of formal systems does fall
within
the scope of mathematics as the study of phenomena observable
in
the world of computation.
Subject: Re: Die Cantor Die
>>There is a key concept here that you don't seem to
understand: there is
>>a difference between mathematics and applying mathematics.
> When trying to distinguish between productive lines of
inquiry,
> and mental masturbation, it's a good idea to consider the
possible
> applications of your inquiry.
What applications have you generated so far?
>>we don't know in advance if a non-computational aspect of
math will lead
>>to an advance in computational math.
> There is some truth to this, although mostly you are arguing
with
> your own misinterpretation of what I'm saying. As I have
pointed
> out way too many times, the study of formal systems does
fall within
> the scope of mathematics as the study of phenomena
observable in
> the world of computation.
In which case, all current mathematics already falls under
your proposal
and nothing need change for you to be satisfied. What is it
you don't
like again?
Hint: the best you've done is state a few axioms you disagree
with, but
you haven't shown how you would formalize replacements (if
any), nor
have you shown how you would patch together the holes that
would be
left. As others have observed, your system will preserve
arithmatic,
but will lose the ability to prove certain other useful
statements
relating to computation. While I don't know which statements
those are,
it seems like it would be worthwhile for you to research that,
analyze
the value of what would be lost, and perhaps start formalizing
your
ideas. So far I've seen a great deal of rhetoric of a category
similar
to that produced by politicians. Mathematics, regardless of
how you do
it, must be a little bit clearer. Right now you have failed to
provide
a replacement for set theory. Until you can at least offer the
axioms
you wish to use, it is difficult to take you seriously,
especially if
this is the progress you have made in the past 15 years.
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
> My guess is that you object to the Axiom of Infinity, which
basicly says
> there is a set containing the natural numbers, and would
prefer it to be
> that the natural numbers form a proper class. If so, then go
ahead and
> develop a new set theory with an Axiom of Finitude or
similar. Or it
> may already exist.
Actually, what is wrong with doing this? Other than removing a
large
part of intresting mathematics. Does it actually damage our
ability to
deal with numbers in some way?
Subject: Re: Die Cantor Die
|> My guess is that you object to the Axiom of Infinity, which
basicly says
|> there is a set containing the natural numbers, and would
prefer it to be
|> that the natural numbers form a proper class. If so, then
go ahead and
|> develop a new set theory with an Axiom of Finitude or
similar. Or it
|> may already exist.
|
|Actually, what is wrong with doing this? Other than removing
a large
|part of intresting mathematics. Does it actually damage our
ability to
|deal with numbers in some way?
If in Zermelo-Frankel set theory (as well as various others, I
think) you
replace the axiom of infinity with an axiom saying that every
set is finite
the resulting system is basically equivalent to the axioms of
elementary
arithmetic. There's a sort of obvious model, and the elements
can be
encoded by integers.
By Goedel's theorem there are statements of arithmetic that
can't be
proven in this system. Some of them can be proven in ZF,
however.
Keith Ramsay
Subject: Re: Die Cantor Die
Discussion, linux)
> If in Zermelo-Frankel set theory (as well as various others,
I think) you
> replace the axiom of infinity with an axiom saying that
every set is
finite
> the resulting system is basically equivalent to the axioms
of elementary
> arithmetic. There's a sort of obvious model, and the
elements can be
> encoded by integers.
> By Goedel's theorem there are statements of arithmetic that
can't be
> proven in this system. Some of them can be proven in ZF,
however.
Any known interesting ones?
--
Jesse Hughes
Surround sound is going to be increasingly important in future
offices.
-- Microsoft marketing manager displays his keen insight
Subject: Re: Die Cantor Die
> Any known interesting ones?
This is a recurring question. It all depends on what one finds
interesting. There are no ordinary-style number-theoretic
theorems of
ZF (Dirichlet's theorem, Fermat, and so on) which are known to
be
unprovable in ZF without infinity (or, equivalently, in PA).
There are
however (apart from metamathematical statements) many
combinatorial
principles in finite mathematics which are known not to be
provable in
ZF without infinity, beginning with the Ramsey principle used
in the
Paris-Harrington theorem.
Subject: Re: Die Cantor Die
>> My guess is that you object to the Axiom of Infinity, which
basicly
>> says there is a set containing the natural numbers, and
would prefer
>> it to be that the natural numbers form a proper class. If
so, then go
>> ahead and develop a new set theory with an Axiom of
Finitude or
>> similar. Or it may already exist.
> Actually, what is wrong with doing this? Other than removing
a large
> part of intresting mathematics. Does it actually damage our
ability to
> deal with numbers in some way?
Nothing is wrong with doing this. It would be more useful than
Petry's
constant discussions about the problems of set theory as it is
usually
taught. What I propose, ultimately, is that he do some work
rather than
berate mathematicians for not accepting his ideas. He may find
that
what he thinks will happen doesn't. I don't particularly care
what set
of axioms a person chooses to investigate. I do care when they
try to
suggest, without doing the investigation, that one is somehow
superior
to another.
--
Will Twentyman
email: wtwentyman at copper dot net
Subject: Re: Die Cantor Die
> I'm not sure you've thought this through.
> Try to find one single mathematical statement which would be
> of interest to those who apply mathematics, which could not
be
> interpreted as a statement about the world of computation.
Elliptic geometry is consistent. It's of great intrest to
physicists.
Of course your use of the word interpreted gives you pretty
much free
reign, as any thing can be interpreted as anything else given
enough
effort.
Subject: Proof - erasure trapping technique
Can someone please help me to proof the following:
Theorem/Proposition:
When an erasure polynomial e(x) = a_ix^i + a_{h-1}x^{h-1} +
a_{h-2}x^{h-2} + ... + a_0 is added to a codeword c(x) =
b_{n-1}x^{n-1} + b_{n-2}x^{n-2} + ... + b_0 in C(x) (C(x) is
an (n,k)
MDS code generated by g(x)) where a_l = b_l or a_l = 0, e(x)
can be
determined by adding r(x) to s * c_1(x), where c_1(x) = x^i +
d_{h-1}x^{h-1} + d_{h-2}x^{h-2} + ... + d_0 in C(x) and s is a
scalar. Here, r(x) = f_{h-1}x^{h-1} + f_{h-1}x^{h-2} + ... +
f_0 is
the coset representative (or the remainder polynomial) of the
received
polynomial v(x) = c(x) + e(x), i.e. r(x) = v(x) mod g(x).
Furthermore, c_1(x) can be computed as x^i + x^i mod g(x). The
scalar
s can be found as d_j/f_j where the corresponding b_j is
known. We
assume that the number of erasures does not exceed d_{min} -
1. (h =
n-k = d_{min - 1})
(Thus, the erasure polynomial consists of a single burst
erasure of
length n-k, and one other single erasure. The received
polynomial
v(x) has zeros where the erasures occurred. To get the
polynomial
c1(x), we encode x^i, where x^i is the highest erased
coefficient.)
Proof:
One way to proof this that I thought of was to show that a
polynomial
p(x) = s*c1(x) + r(x) has the following properties:
1.) p(x) belongs to the same coset as r(x) => H*p^t = H*c1^t +
H*r^t
= 0 + r^t
Thus, p(x) belongs to the same coset as r(x). Here, H is a
parity-check matrix of C(x)
2.) p(x) has coefficients at the corresponding places where the
erasures occurred:
This is a bit tricky for me and I don't know how to accomplish
it. I
thought of using the parity check matrix again, where H =
[h_0,h_1,
..., h_{n-1}], h_i the i-th column of H.
Any help and/or suggestions will be greatly appreciated
Jaco Versfeld
Subject: Re: Hyperelliptic curves examples!!
> I would like to some examples of HyperElliptic Curves (HEC)
and
> supersingular curves too.. of different genus along wiht the
points
> for different fields.. so I can calculate the divisors and
test the
> arithmetic algorithm that I have written. Can some one give
me
> references as to where I can find them.
> I thought the term supersingular only applied to elliptic
curves.
> (so of genus 1).
> What does it mean to say that a general curve is
supersingular?
Supersingular applies to abelian varieties of any dimension, so
possibly the OP wants curves whose Jacobian varieties are
supersingular. But I suspect that the OP meant to say
superelliptic,
which is Serge Lang's term for curves of the form y^n=f(x). In
more
prosaic terms, these are cyclic covers of the projective line.
JS
Subject: Re: Cardano's formulas
>> What's the practical use of Cardano's formula?I don't see
any.
>> Then,why do we have to learn them?
>Just a week or two ago someone in this group was complaining
that he
>DIDN'T learn Cardano's formulas in school, but they should
have
>been included.
High school math teaching is really moving fast.
--
I'm not interested in mathematics that might have anything
to do with reality. -- Easterly, in sci.math
Subject: Bell curve distribution
I'd like to ask a favor of you guys. I need an algorithm that
will
analyze any set of data and put in into a normal bell-curve
distribution, by frequency groups. Or at the very least, I
need tips to
figure out such an algorithm (I'm not entirely mathematically
ignorant,
but I'm not exactly highly educated either).
In case what I'm looking for is not quite clear, let me
explain. I need
to be able to place a set of data into a bell curve and then
break the
bell curve up into a set number of frequency groups, so that I
have the
value range for each group. Here's a sample set:
Group 1 -- data points: 9; value range: 133-183
Group 2 -- data points: 23; value range: 184-388
Group 3 -- data points: 36; value range: 389-603
Group 4 -- data points: 45; value range: 604-822
Group 5 -- data points: 50; value range: 823-1,105
Group 6 -- data points: 45; value range: 1,106-1,590
Group 7 -- data points: 36; value range: 1,591-2,052
Group 8 -- data points: 23; value range: 2,053-2,818
Group 9 -- data points: 8; value range: 2,819-3,807
I don't know how good that sample set is -- that's just an
example of
what my boss is looking for (that's the sample set he gave
me). But, in
essence, what I'm looking for is an algorithm that will divide
any data
set into a distribution similar to that example, and it needs
to fall
into a normal bell curve as accurately as possible. Any help
you guys
can give me?
--
Mekkala, Atheist #2148
Atheism is ... the bed-rock of sanity in a world of madness.
--Emmett F. Fields
Subject: Re: mathematical trivia (was: How Did Euler Actually
Go About
Solving the Bridges of Konigsberg Problem?
> Where is this K.9anigsberg?
In a small piece of Russia on the Baltic Sea, surrounded by
Poland and
Lithuania.
> What is it called now?
Kalingrad.
> What country is it now in?
Modern-day Russia.
> What country was it in during Euler's time?
Prussia, I think.
Best punishment for bad student: Other students make bad
nickname for
> student and repeat name in front of class laughing. Good for
ego, too.
The best punishment for students is to be ignored by their own
teachers; as
far as I am concerned, when I am a teacher, I'll adopt the
following
behaviour :
1) The students want to work, I'll do my best to help
them.
2) The students don't want to work: I don't care.
As far as corporal punishments are concerned (in advanced
countries) i
guess that if a professor tries to hit a student with a stick,
then the
student will kick him to death, and i think that's a good
thing. In any
constitution of any civilized country, there's one thing
called the right
to
keep one's physical integrity intact. (our ancestrors fought
to win that
right)
Subject: Re: a little question of non-math
charset=iso-8859-7
Julien Santini
> Best punishment for bad student: Other students make bad
nickname for
> student and repeat name in front of class laughing. Good for
ego, too.
> The best punishment for students is to be ignored by their
own teachers;
as
> far as I am concerned, when I am a teacher, I'll adopt the
following
> behaviour :
> 1) The students want to work, I'll do my best to
help them.
> 2) The students don't want to work: I don't care.
I think you missed my point entirely. What was meant by make
bad
nickname,
means that society has its very own unique way to punish
aberrant school
behavior, by using pre-existing means. A student cannot be
respected more
than what he actually represents in grade school or high
school.
As school in general is a jungle which will host all sorts of
little
devils,
often those very devils will devise ways to crush other
students' egos. I
recall from my days in high school that we had a fat student
who we called
Swan Luncheon Meat. A poor student with 25 degrees of
nearsightedness
and
beer-bottle glasses who needed to stick his face against the
blackboard so
he could read. This student we called The Telescope. A student
who was
continuously dosing off, was called Hypnos (Sleep). I was
called The
Old
Man. Everyone got nicknamed according to his social
characteristics.
My point being that society and nature are relentless. If a
kid can survive
the kind of psychological abuse during school, it can survive
anything. Any
other punishment is pretty much unecessary, ideally.
> As far as corporal punishments are concerned (in advanced
countries)
i
> guess that if a professor tries to hit a student with a
stick, then the
> student will kick him to death, and i think that's a good
thing. In any
> constitution of any civilized country, there's one thing
called the right
to
> keep one's physical integrity intact. (our ancestrors fought
to win that
> right)
Baloney. When I was a grad student in Chicago, kids at the
local school
were
carrying guns. When students are noisy to the point of
disrupting the
ENTIRE
class and not let other students (who wish to) learn, I have a
God given
right to kick them out of class pronto.
There are students in every class who DESIRE to learn. When
these students'
desire (and right) is inhibited by assholes, I can acquire the
right to
expel them or use some other justifiable means to keep the
class going.
That's part of the teacher's job.
--
Ioannis Galidakis
http://users.forthnet.gr/ath/jgal/
------------------------------------------
Eventually, _everything_ is understandable
Subject: Re: New Knowledge
> How, when and why does *New Knowledge* come to humankind?
> The 'free-Mickey' lawyer (is his name Lessig?) said that all
creativity
is
> derivative. Darren.
> ......ahahaha......but derivate is retro active,
Standing on the shoulders of giants is retro active?
>hence does not point
> to anything that is really novel. No surprise that this
notion comes from
> a lawyer whose name is Essig = vinegar and who defends
copyrights
> to last an average of 95 years
he was trying to free Mickey, not keep him behind bars.
>So, much for an impetus to foster
> creativity, looking and searching for the
NEW..........ahahahaha....
> OTOH green s would love to have their permit charges and
> user fees last for 95 years. But it ain't gonna happen. The
creativity
> by others that the green turds feed off will be --- OUT
SOURCED ---.
> ** Outsourcing the only real legacy of environmentalism ***
> hanson
Subject: Re: Which math tutoring software?
> I'm a current college student in Trigonometry. I will be
taking
> Calculus I, II, III, Differential & Orthogonal Equations. I
am also a
> soon to be electrical engineering student which is going to
require I
> know how to solve many different types of problems. What math
> software does this group recommend that would help me on
problems I
> don't know how to solve, by showing me the complete steps to
solve the
> given problem?
If they had software like that, they wouldn't need electrical
engineers, would they? They'd just take the pimple farmer
out from behind the microphone at Jack In the Box, give
him the software, hand him a huge paycheck and whenever he had
a problem, he'd just type it in and read the answer back to
his boss.....or maybe the boss would just type in the question
himself.
Subject: Design Question
I need to extract and amplify only the components of two
signals that are
exactly in phase. Out of phase signals of any amplitude are
noise. I am
thinking this is sort of the opposite of common mode
rejection. Should I be
pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
there some way to utilize the phase part of an FFT?
Ron H.
Subject: Re: Design Question
Noise may exist or not exist. You also need an oscilloscope to
engineer this. Keep the signals separate and simply amplify.
If you
want to compare the signals, then the fact they are in phase
won't
produce cancellation. Common mode rejection occurs because the
signals
are identical, basically.
Subject: Re: Design Question
> I need to extract and amplify only the components of two
signals that are
> exactly in phase. Out of phase signals of any amplitude are
noise. I am
> thinking this is sort of the opposite of common mode
rejection. Should I
be
> pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
> there some way to utilize the phase part of an FFT?
Phase Locked Loop (PLL)?
> Ron H.
Subject: Re: Design Question
>I need to extract and amplify only the components of two
signals that are
>exactly in phase. Out of phase signals of any amplitude are
noise. I am
>thinking this is sort of the opposite of common mode
rejection. Should I
be
>pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
>there some way to utilize the phase part of an FFT?
>Ron H.
Isn't that what a lock-in amplifier does? It can pick out a
signal that's
swamped by orders of magnitude more noise.
--
We don't grow up hearing stories around the camp fire anymore
about
cultural figures. Instead we get them from books, TV or
movies, so the
characters that today provide us a common language are
corporate
creatures -- Rebecca Tushnet
Subject: Re: Design Question
> I need to extract and amplify only the components of two
signals
that are
> exactly in phase. Out of phase signals of any amplitude are
noise. I
am
> thinking this is sort of the opposite of common mode
rejection.
Should I be
> pursuing an electronic (hardware) solution or a math (DSP)
solution?
Is
> there some way to utilize the phase part of an FFT?
> Ron H.
Google on lock in amplifier (without the quotes).
Regards
Ian
Subject: Re: Design Question
>lock in amplifier (without the quotes)
>Ian Buckner
Owlett's right. Use quote marks for phrases.
http://www.google.com/help/refinesearch.html#phrase
Subject: Re: Design Question
>>lock in amplifier (without the quotes)
>>Ian Buckner
>Owlett's right. Use quote marks for phrases.
>http://www.google.com/help/refinesearch.html#phrase
Or used the Advanced Search Page and fill-in the blanks:
http://www.google.com/advanced_search?hl=en
...Jim Thompson
--
| James E.Thompson, P.E. | mens |
| Analog Innovations, Inc. | et |
| Analog/Mixed-Signal ASIC's and Discrete Systems | manus |
| Phoenix, Arizona Voice:(480)460-2350 | |
| E-mail Address at Website Fax:(480)460-2142 | Brass Rat |
| http://www.analog-innovations.com | 1962 |
Will you still need me, will you still feed me, when I'm
sixty-four?
Subject: Re: Design Question
>>I need to extract and amplify only the components of two
signals
> that are
>>exactly in phase. Out of phase signals of any amplitude are
noise. I
> am
>>thinking this is sort of the opposite of common mode
rejection.
> Should I be
>>pursuing an electronic (hardware) solution or a math (DSP)
solution?
> Is
>>there some way to utilize the phase part of an FFT?
>>Ron H.
> Google on lock in amplifier (without the quotes).
> Regards
> Ian
lock in amplifier or lock-in amplifier with quotes is even
better.
Jerry
--
Engineering is the art of making what you want from things you
can get.
[OS
lash]
[OSl
ash]
[OSl
ash]
[OSl
ash]
Subject: Re: Design Question
>>I need to extract and amplify only the components of two
signals
> that are
>>exactly in phase. Out of phase signals of any amplitude are
noise. I
> am
>>thinking this is sort of the opposite of common mode
rejection.
> Should I be
>>pursuing an electronic (hardware) solution or a math (DSP)
solution?
> Is
>>there some way to utilize the phase part of an FFT?
>>Ron H.
> Google on lock in amplifier (without the quotes).
> Regards
> Ian
Actually I think the result with quotes is better.
Three of the first four hits are just what you're looking for.
IMHO
Subject: Re: Design Question
> I need to extract and amplify only the components of two
signals that are
> exactly in phase. Out of phase signals of any amplitude are
noise. I am
> thinking this is sort of the opposite of common mode
rejection. Should I
be
> pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
> there some way to utilize the phase part of an FFT?
If E(total) = E(common mode) + E(non-common mode)
and a common mode circuit yields E(non-common mode)
then
E(common mode) = E(total) - E(non-common mode)
so
you can run the signal through a common mode rejection circuit
( Do a Google search on quadrature rejection ),
subtract the resultant signal from the original signal,
and then amplify the resultant.
Of course,
the best way to separate signal from noise,
is to cross-correlate the effect signal
with the cause signal.
A Google search on MTI RADAR might
show you how auto-correlation is done with analog circuits.
Correlation involves summing the product of two strings of
data.
If you multiply two random strings of data, and sum the
products,
you end up with zero in the case of pure noise.
The larger the sum of the products,
the greater the correlation between the two strings.
Note that the time delay (Phase in the case of periodic
signals)
has to be adjusted to make the data strings overlap,
as there is a time interval or phase difference between a
cause and an
effect.
--
Tom Potter http://tompotter.us
Subject: Re: Design Question
>I need to extract and amplify only the components of two
signals that are
>exactly in phase. Out of phase signals of any amplitude are
noise. I am
>thinking this is sort of the opposite of common mode
rejection. Should I
be
>pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
>there some way to utilize the phase part of an FFT?
Take a look at quadrature demodulation - in either hardware or
software. It is perfectly suited to extracting the in-phase
component
of a signal as well as the 90-degrees out-of-phase component.
-Robert Scott
Ypsilanti, Michigan
(Reply through this forum, not by direct e-mail to me, as
automatic reply
address is fake.)
Subject: Re: Design Question
> I need to extract and amplify only the components of two
signals that are
> exactly in phase. Out of phase signals of any amplitude are
noise. I am
> thinking this is sort of the opposite of common mode
rejection. Should I
be
> pursuing an electronic (hardware) solution or a math (DSP)
solution? Is
> there some way to utilize the phase part of an FFT?
Cross-Correlation
Subject: Re: This Week's Finds in Mathematical Physics (Week
202)
Here's a neat 18th century way of getting the Catalan numbers.
Consider the equation for Lisp S-expressions
S = A + S x S
meaning that an S-expression is either an atom or a pair of
S-expressions. Regard this as a quadratic equation for S and
solve it
by the quadratic formula. Expand the square root in power
series.
The coefficients of the powers of A are just the Catalan
numbers
giving the number of types of S-expression with a given number
of atoms. Of course, you get set isomorphisms as well.
I suppose this is a special case of John Baez's theory.
--
John McCarthy, Computer Science Department, Stanford, CA 94305
http://www-formal.stanford.edu/jmc/progress/
He who refuses to do arithmetic is doomed to talk nonsense.
Subject: Re: This Week's Finds in Mathematical Physics (Week
202)
X-MailScanner-SpamxCheck: not spam, SpamAssassin (score=-11.3,
required 5,
Lisp S-expressions
> S = A + S x S
Lisp has a very simple typing system so there isn't much
beyond S=A+SxS.
In a language with a type system such as Haskell, where types
are built
algebraically, then the method used to define types is
isomorphic to
writing
things like S=A+SxS apart from a bit of syntactic sugar. For
example a
simple tree type in Haskell is defined by
data Tree a = Leaf a | Branch (Tree a) (Tree a)
The type system in Haskell has been well studied from the
point of view of
category theory (Curry-Howard isomorphism connecting types with
intuitionistic logic etc.)
Haskell is also a great computer language for considering some
of the other
stuff JB has been talking about. For example you can think
about a monad
that allows you to 'lift' functions on classical systems to
functions on
quantum
systems and then directly implement a quantum computer
simulator in Haskell
using this monad. http://
www.cs.indiana.edu/~sabry/papers/quantum.pdf
(Monads form a fundamental part of Haskell BTW)
--
Torque
Subject: Re: Role of (Pressure/Volume) quotient in ideal Gas
Law
[..]
> However, I do not find this one variable U=P/V mentioned in
physics or
> thermodynamics in undergraduate classes.
[..]
> Can it be agreed that U= P / V is a valid physical quantity ?
It looks like you are rewriting some well known laws by
rearranging
the variables in new combinations.
Sometimes this can be useful, depending on what you want to do.
Thermodynamics has quite al lot of this, for example, you can
express a problem in terms of internal energy, enthalpy,
Helmholtz
free energy, or Gibbs free enthalpy, depending again on what
you want
to do. Recently quantities like Exergy have become popular.
But I don't see any use for the quantity U (which is usually
reserved for internal energy in thermodynamics). You need to
show how it can make solving certain problems more easy.
Gerard
Subject: Re: Differential equation
Dear Lee,
thank you for this nice posting!
> Then we consider the real subspace V of C^2 defined by
z_2=(z_1)*,
> and call *it* R^2, so that now (in some even more abusive
sense)
delta(z)delta(z*) means the point mass at the origin of V.
Why is this statement abusive? Such kind of hints don't help
me,
you have to explain to me what actually is wrong with this
abusive
physicist's notation! Otherwise you maybe satisfied, but I
didn't
learn anything new. And what is a point mass in the
mathematical meaning?
Sounds like a term from physics.
> Now we consider as our space of test functions something like
> the (complex valued) real-analytic functions on V (which of
course
> are very far from being like nice compactly supported test
functions
> which you would use to define distributions), any one of
which
> is of course the restriction to V of a genuine
complex-analytic
> function on some neighborhood of V in C^2 (and conversely).
Why does one need compactly supported test functions to define
distributions? In my undergraduate courses we defined the delta
distribution as a limit, of for example a properly normalized
Gaussian, or a rectangle with Area one.
> Even if any of this is right (or rightly interpreted), I too
don't
> see why you'd want to do it. But then, I don't pretend to
understand
> physic(ist)s.
Well, sometimes, doing some calculations in the coordinates z
= x+iy
and z* = x-iy is nicer to do than in other coordinates.
Ren.8e.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Re: Differential equation
>> Then we consider the real subspace V of C^2 defined by
z_2=(z_1)*,
>> and call *it* R^2, so that now (in some even more abusive
sense)
>delta(z)delta(z*) means the point mass at the origin of V.
>Why is this statement abusive?
The difficulties involved in multiplying distributions have
already been noted repeatedly. On top of that, here there's
the sleight-of-hand involved in calling V by the name R^2,
since R^2 is already present inside C^2 and not equal to V.
>Such kind of hints don't help me,
>you have to explain to me what actually is wrong with this
abusive
>physicist's notation! Otherwise you maybe satisfied, but I
didn't
>learn anything new. And what is a point mass in the
mathematical meaning?
>Sounds like a term from physics.
Well, what *I* meant by it was a measure supported on a
singleton,
such that the measure of the singleton is 1. Such a measure (or
any measure) can be interpreted as a functional on various
spaces
of functions (and, generally, not on other spaces of functions,
or things-like-functions; for instance, you'd have trouble
defining
delta on L^p for any p, since elements of L^p don't have
values at
particular points).
As soon as one starts slinging terms like distribution around,
one is, at least implicitly, making such an identification (it
seems to me); but a measure, pure and simple, is not such a
functional--at *best*, it's manifested as many different
functionals
on many different spaces.
>> Now we consider as our space of test functions something
like
>> the (complex valued) real-analytic functions on V (which of
course
>> are very far from being like nice compactly supported test
functions
>> which you would use to define distributions), any one of
which
>> is of course the restriction to V of a genuine
complex-analytic
>> function on some neighborhood of V in C^2 (and conversely).
>Why does one need compactly supported test functions to define
>distributions?
I dunno. Maybe you don't. It's a matter of terminology, of
course.
>In my undergraduate courses we defined the delta
>distribution as a limit, of for example a properly normalized
>Gaussian, or a rectangle with Area one.
That would be a limit in what topological (vector)space?
Given a measure m which is a point mass at the origin (as I
used the term above), and given an appropriate space F of
functions (on whatever space the origin is the origin *of*),
the functional on F which assigns to a function f in F the
the integral of f with respect to m is linear; when F is
equipped with a topology, this linear functional may (or
may not) be continuous; in that case (or, I suppose, even if
it isn't continuous, but is at least defined), you can ask
whether this linear functional is (or is not) the limit of
an appropriate sequence (or net...) of *other* linear
functionals,
perhaps defined by integration with respect to *other* measures
(but perhaps not).
>> Even if any of this is right (or rightly interpreted), I
too don't
>> see why you'd want to do it. But then, I don't pretend to
understand
>> physic(ist)s.
>Well, sometimes, doing some calculations in the coordinates z
= x+iy
>and z* = x-iy is nicer to do than in other coordinates.
Oh, sure, I know that's true in lots of cases, even cases I
have
a personal interest in. But I don't know why it's true (for
you)
in the case being discussed.
Lee Rudolph
Subject: Re: Differential equation
>>> Then we consider the real subspace V of C^2 defined by
z_2=(z_1)*,
>>> and call *it* R^2, so that now (in some even more abusive
sense)
>>delta(z)delta(z*) means the point mass at the origin of V.
>>Why is this statement abusive?
>The difficulties involved in multiplying distributions have
>already been noted repeatedly. On top of that, here there's
>the sleight-of-hand involved in calling V by the name R^2,
>since R^2 is already present inside C^2 and not equal to V.
>>Such kind of hints don't help me,
>>you have to explain to me what actually is wrong with this
abusive
>>physicist's notation! Otherwise you maybe satisfied, but I
didn't
>>learn anything new. And what is a point mass in the
mathematical meaning?
>>Sounds like a term from physics.
>Well, what *I* meant by it was a measure supported on a
singleton,
>such that the measure of the singleton is 1. Such a measure
(or
>any measure) can be interpreted as a functional on various
spaces
>of functions (and, generally, not on other spaces of
functions,
>or things-like-functions; for instance, you'd have trouble
defining
>delta on L^p for any p, since elements of L^p don't have
values at
>particular points).
>As soon as one starts slinging terms like distribution around,
>one is, at least implicitly, making such an identification (it
>seems to me); but a measure, pure and simple, is not such a
>functional--at *best*, it's manifested as many different
functionals
>on many different spaces.
Actually, modulo various technicalities, a measure is exactly
the same thing as a positive linear functional on the space
of continuous functions with compact support.
>>> Now we consider as our space of test functions something
like
>>> the (complex valued) real-analytic functions on V (which
of course
>>> are very far from being like nice compactly supported test
functions
>>> which you would use to define distributions), any one of
which
>>> is of course the restriction to V of a genuine
complex-analytic
>>> function on some neighborhood of V in C^2 (and conversely).
>>Why does one need compactly supported test functions to
define
>>distributions?
>I dunno. Maybe you don't. It's a matter of terminology, of
course.
There are various sorts of distributions. Roughly speaking,
distributions per se give the dual of the smooth functions
with compact support; since the functions have compact
support there's no restriction on the growth of the
distribution
at infinity (not that it makes sense to put it exactly that
way,
since the distribution is not a function.) There's a standard
class of rapidly decreasing or Schwarz functions, which
are required to die very fast at infinity; the dual of that
class is the tempered distributions, which roughly
speaking don't grow too fast at infinity. Then one can
consider smooth functions with no growth condition at
all; the dual of _that_ class is the distributions with
compact support.
>>In my undergraduate courses we defined the delta
>>distribution as a limit, of for example a properly normalized
>>Gaussian, or a rectangle with Area one.
>That would be a limit in what topological (vector)space?
>Given a measure m which is a point mass at the origin (as I
>used the term above), and given an appropriate space F of
>functions (on whatever space the origin is the origin *of*),
>the functional on F which assigns to a function f in F the
>the integral of f with respect to m is linear; when F is
>equipped with a topology, this linear functional may (or
>may not) be continuous; in that case (or, I suppose, even if
>it isn't continuous, but is at least defined), you can ask
>whether this linear functional is (or is not) the limit of
>an appropriate sequence (or net...) of *other* linear
functionals,
>perhaps defined by integration with respect to *other*
measures
>(but perhaps not).
>>> Even if any of this is right (or rightly interpreted), I
too don't
>>> see why you'd want to do it. But then, I don't pretend to
understand
>>> physic(ist)s.
>>Well, sometimes, doing some calculations in the coordinates
z = x+iy
>>and z* = x-iy is nicer to do than in other coordinates.
>Oh, sure, I know that's true in lots of cases, even cases I
have
>a personal interest in. But I don't know why it's true (for
you)
>in the case being discussed.
>Lee Rudolph
************************
Subject: Re: Differential equation
...
>>As soon as one starts slinging terms like distribution
around,
>>one is, at least implicitly, making such an identification
(it
>>seems to me); but a measure, pure and simple, is not such a
>>functional--at *best*, it's manifested as many different
functionals
>>on many different spaces.
>Actually, modulo various technicalities, a measure is exactly
>the same thing as a positive linear functional on the space
>of continuous functions with compact support.
As Herman Rubin often reminds us, no topology (and ipso facto,
no continuous functions or compact supports) is/are needed
to define a measure, and many interesting measures are defined
where topologists fear to tread. Now, if you put the right
adjective in front of the word measure (actually, I guess,
the right person's last name...but it's been a long time, and
I forget), I would agree with you without a quibble.
Lee Rudolph
Subject: Re: Differential equation
>...
>>>As soon as one starts slinging terms like distribution
around,
>>>one is, at least implicitly, making such an identification
(it
>>>seems to me); but a measure, pure and simple, is not such a
>>>functional--at *best*, it's manifested as many different
functionals
>>>on many different spaces.
>>Actually, modulo various technicalities, a measure is exactly
>>the same thing as a positive linear functional on the space
>>of continuous functions with compact support.
>As Herman Rubin often reminds us, no topology (and ipso facto,
>no continuous functions or compact supports) is/are needed
>to define a measure, and many interesting measures are defined
>where topologists fear to tread. Now, if you put the right
>adjective in front of the word measure (actually, I guess,
>the right person's last name...but it's been a long time, and
>I forget), I would agree with you without a quibble.
Well yes of course a measure does not require a topology;
that person's name (could be Mr regular Borel or Mr
Radon depending on who you talk to) was one of the
various technicalities I was referring to.
(In a context like the present, where we're talking about
analysis on R^2, people often just say measure in
place of regular Borel measure. _Also_ they often
say measure in place of complex Borel measure...)
>Lee Rudolph
************************
Subject: Re: Differential equation
>>>>You are right. Well, I think this is what I actually
forgot! The
>>>>manipulation
>>>>dd*ln(zz*) = d(1/z*) = d*(1/z)
>>> Typo there, of course.
>>Where is the typo?
> You actually meant dd*ln(zz*) = d(1/z*) = d*(1/z),
> not dd*ln(zz*) = d(1/z*) + d*(1/z)?
That's not a typo. Have a look at my sketch of the calculation
I did.
I now know, that ln (zz*) = lnz + lnz* is not true in general
as of
the problem with the ln of complex numbers. Is there any way
to make
this equation hold?
>>> Another possibility for the inconsistency is that dd* is
not exactly
>>> the Laplacian, it's the Laplacian times 4 or 1/4 or
something.
>>No, Polchinski definitely defines dd* as d/dz d/dzbar.
> What no? That's how I _assumed_ you were defining dd*.
> That's not the Laplacian.
You are right, I forgot to give you the definition of d and
d*. Its
d = d/dz = 1/2(d/dx -i d/dy)
and d* the complex conjugate of this, with all derivatives
being
partial ones. Thus dd* is 1/4 the laplacian.
Ren.8e.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Re: Differential equation
>>>>>You are right. Well, I think this is what I actually
forgot! The
>>>>>manipulation
>>>>>dd*ln(zz*) = d(1/z*) = d*(1/z)
>>>> Typo there, of course.
>>>Where is the typo?
>> You actually meant dd*ln(zz*) = d(1/z*) = d*(1/z),
>> not dd*ln(zz*) = d(1/z*) + d*(1/z)?
>That's not a typo.
Right - I realized later I wasn't reading carefully enough.
>Have a look at my sketch of the calculation I did.
>I now know, that ln (zz*) = lnz + lnz* is not true in general
as of
>the problem with the ln of complex numbers. Is there any way
to make
>this equation hold?
Like I said, I'm not sure. It's certainly ok away from the
origin, but
around the origin is the interesting part and that's exactyly
where it's not clear to me whether this makes sense or not.
>>>> Another possibility for the inconsistency is that dd* is
not exactly
>>>> the Laplacian, it's the Laplacian times 4 or 1/4 or
something.
>>>No, Polchinski definitely defines dd* as d/dz d/dzbar.
>> What no? That's how I _assumed_ you were defining dd*.
>> That's not the Laplacian.
>You are right, I forgot to give you the definition of d and
d*. Its
>d = d/dz = 1/2(d/dx -i d/dy)
>and d* the complex conjugate of this, with all derivatives
being
>partial ones. Thus dd* is 1/4 the laplacian.
>Ren.8e.
************************
Subject: Re: Differential equation
>>Have a look at my sketch of the calculation I did.
>>I now know, that ln (zz*) = lnz + lnz* is not true in
general as of
>>the problem with the ln of complex numbers. Is there any way
to make
>>this equation hold?
> Like I said, I'm not sure. It's certainly ok away from the
origin, but
> around the origin is the interesting part and that's exactyly
> where it's not clear to me whether this makes sense or not.
Actually, the application from where this problem arose
assumes r !=
0, and phi in [0,2pi] for z = r exp(iphi).
Ren.8e.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Re: Differential equation
>>>Have a look at my sketch of the calculation I did.
>>>I now know, that ln (zz*) = lnz + lnz* is not true in
general as of
>>>the problem with the ln of complex numbers. Is there any
way to make
>>>this equation hold?
>> Like I said, I'm not sure. It's certainly ok away from the
origin, but
>> around the origin is the interesting part and that's
exactyly
>> where it's not clear to me whether this makes sense or not.
>Actually, the application from where this problem arose
assumes r !=
>0, and phi in [0,2pi] for z = r exp(iphi).
Um, polar coordinates only make sense for r <> 0. If we're
wondering about how to show the laplacian of log(zz*) is
(whatever constant times) a point mass at the origin then
the problem is _at_ the origin. If we were only talking about
points away from the origin the problem wouldn't come
up in the first place; the laplacian of log(zz*) is obviously
0 away from the origin.
>Ren.8e.
************************
Subject: Re: Differential equation
>>> How to show this?
>>I looked up a text on distributions :-)
> One can give a cheap-hack sort of argument for this,
> first expressing the laplacian in polar coordinates
> (then recalling the _definition_ of the laplacian of
> a distribution) and integrating by parts.
And what is that definition? I really have not much knowledge
about
distribution theory, maybe some of you can enlighten me a
little bit
;-)
Ren.8e.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Re: Differential equation
>>>> How to show this?
>>>I looked up a text on distributions :-)
>> One can give a cheap-hack sort of argument for this,
>> first expressing the laplacian in polar coordinates
>> (then recalling the _definition_ of the laplacian of
>> a distribution) and integrating by parts.
>And what is that definition? I really have not much knowledge
about
>distribution theory, maybe some of you can enlighten me a
little bit
Say u is a distribution and L is the Laplacian. By definition
Lu is
the distribution such that
= is more proper
notation
for what people sometimes write as int(u phi). Another correct
alternative is u(phi).)
>;-)
>Ren.8e.
************************
Subject: Re: Differential equation
>>> It's well-known that -(log|z|)/2 pi
>>> is a fundamental solution of the Laplacian: i.e.
>>> Delta(log |z|) = -2 pi delta --- usual delta distribution!
>>> So d d*(log |z|^2) = -pi delta.
>> How to show this?
> I looked up a text on distributions :-)
Can you tell me some good ones? I never learned much about
distribution
theory except the stuff one learns as a physicist.
Ren.8e.
--
Ren.8e Meyer
Student of Physics & Mathematics
Zhejiang University, Hangzhou, China
Subject: Re: Differential equation
>>>> It's well-known that -(log|z|)/2 pi
>>>> is a fundamental solution of the Laplacian: i.e.
>>>> Delta(log |z|) = -2 pi delta --- usual delta distribution!
>>>> So d d*(log |z|^2) = -pi delta.
>>> How to show this?
>> I looked up a text on distributions :-)
>Can you tell me some good ones? I never learned much about
distribution
>theory except the stuff one learns as a physicist.
You can find a fairly gentle introduction in Folland Real
Analysis.
>Ren.8e.
************************
Subject: Re: Differential equation
>>>> It's well-known that -(log|z|)/2 pi
>>>> is a fundamental solution of the Laplacian: i.e.
>>>> Delta(log |z|) = -2 pi delta --- usual delta distribution!
>>>> So d d*(log |z|^2) = -pi delta.
>>> How to show this?
>> I looked up a text on distributions :-)
> Can you tell me some good ones?
The best is Hormander's volume 1. :-)
--
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: matlab matrix problem
I have a two dimensional matrix U which I'd like to power with
a. a has
different values depending on it's index a(k).
The result should be a three dimensional matrix (the two
dimensional matrix
powered by a) for each of the 100 different values of a(k).
The following program causes the error message: ??? In an
assignment
A(I)
= B, the number of elements in B and
I must be the same.
for k=1:100
H(k)= U.^a(k);
end
What is my mistake?
any help is appreciated
Nicolas
Subject: Re: matlab matrix problem
> I have a two dimensional matrix U which I'd like to power
with a. a has
> different values depending on it's index a(k).
> The result should be a three dimensional matrix (the two
dimensional
matrix
> powered by a) for each of the 100 different values of a(k).
> The following program causes the error message: ??? In an
assignment
A(I)
> = B, the number of elements in B and
> I must be the same.
It's a matrix-table allocation problem.
If H doesn't already exist,
the first assignment when k=1,
makes H a two-dimensional matix, by defauilt.
H=U.^a(1).
The usual correction is to define H
before you use it in a loop.
So use one of Matlabs matrix definition statements like:
H=zeros(size(U),100); to preallocate the matrix header table.
> for k=1:100
> H(k)= U.^a(k);
> end
> What is my mistake?
> any help is appreciated
> Nicolas
Subject: Re: matlab matrix problem
> I have a two dimensional matrix U which I'd like to power
with a. a has
> different values depending on it's index a(k).
> The result should be a three dimensional matrix (the two
dimensional
matrix
> powered by a) for each of the 100 different values of a(k).
> The following program causes the error message: ??? In an
assignment
A(I)
> = B, the number of elements in B and
> I must be the same.
> for k=1:100
> H(k)= U.^a(k);
> end
> What is my mistake?
> any help is appreciated
> Nicolas
If matrix H is already 3 dimensional, with first 2 dimensions
corresponding
to U:
for k=1:100
H(:,:,k)= U.^a(k);
end
Jeroen
Subject: Re: Candidate looking for Employment
> am looking for a job. Here's a copy of my resume
> Name : Suresh kumar Devanathan
> contact : mdsuresh ___@thing__ media __.thing__ mit
__.thing__
edu
Really want a job?
Lose the ethnicity. Change your name to John Smith, and be
sure not
to include a photo with your resume. Best to present them with
your
ethnicity at the same time you're dazzling them with your
brilliant
personality.
That's just a cynical suggestion, but make sure your resume is
picture
perfect because that's what will get you the interviews.
> Experience : MIT Media Lab
> Rutgers CAIP Lab
Good magic letters MIT, but doing what?
> Awards : AP Scholar With Distinction
> Other : Army Reserve
Do you still have an obligation? Could you be called back up
and sent
to Iraq next week? Employers won't want to hear this. In that
case
it's best not to highlight this.
> Worked for : Dr. Vishwani Agarwal( IEEE/ACM fellow) on VLSI
ATPG
> Dr. Edward Fredkin( CMU Distinguished professor) on
> Cellular Automata/ SARS-TA
Employers might want to know, What the hell is that!
> Public Projects:
> http://www.sourceforge.net/projects/atpg
> -----------------------------------------
> Built this project, as a PROOF of CONCEPT for
statistical
> techniques, i developed
> to test chips
> http://www.sourceforge.net/projects/blitz
> -----------------------------------------
> Patch work for blitz, a highly successful C++ math
library
> in use, by more than
> 15000 programmers worldwide
Go into a bit more detail.
> Job Skill : C++, C#, Unix, Cadence Design System, MATLAB,
.NET
How many years paid experience in each?
> Education : 3 years of college in Electrical Engineering
> Left college to jumpstart a partly successful
> entrepreneurship
Partly successful? Doesn't sound good. You may have to develop
a
strategy for peddling this.
They will want to know the real reason you dropped out of
school.
And be prepared to supply a transcript.
Three years of college usually just tells people you didn't
finish.
Of course McDonalds is always hiring. No questions asked!
Double-A
Subject: Question about associativity of cartesian product
As I understand it (according to
http://en.wikipedia.org/wiki/Cartesian_product),
1. the cartesian product of two sets gives a set of 2-tuples
(ordered
pairs), i.e. X*Y={(x,y):x in X and y in Y}
2. generally, the cartesian product of n sets gives a set of
n-tuples
3. cartesian product is associative.
But that third statement seems to contradict the first two: by
the
first two, (A*B)*(C*D) should give a set of pairs of sets of
pairs,
but by the third statement, that same expression should give a
set of
4-tuples.
Obviously I'm missing something obvious. Any enlightenment
would be
appreciated.
Subject: Re: Question about associativity of cartesian product
Adjunct Assistant Professor at the University of Montana.
>As I understand it (according to
>http://en.wikipedia.org/wiki/Cartesian_product),
>1. the cartesian product of two sets gives a set of 2-tuples
(ordered
>pairs), i.e. X*Y={(x,y):x in X and y in Y}
>2. generally, the cartesian product of n sets gives a set of
n-tuples
>3. cartesian product is associative.
>But that third statement seems to contradict the first two:
by the
>first two, (A*B)*(C*D) should give a set of pairs of sets of
pairs,
>but by the third statement, that same expression should give
a set of
>4-tuples.
>Obviously I'm missing something obvious. Any enlightenment
would be
>appreciated.
While (A*B)*(C*D) is not equal to A*B*C*D (the set of
4-tuples), there
is a natural identification that can be made by removing inner
parenthesis: identify the element ((a,b),(c,d)) with the
element
(a,b,c,d).
Using this identification, we think of things like A*(B*C) as
being
the same as (A*B)*C, etc.
--
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: Re: Question about associativity of cartesian product
>As I understand it (according to
>http://en.wikipedia.org/wiki/Cartesian_product),
>1. the cartesian product of two sets gives a set of 2-tuples
(ordered
>pairs), i.e. X*Y={(x,y):x in X and y in Y}
>2. generally, the cartesian product of n sets gives a set of
n-tuples
>3. cartesian product is associative.
>But that third statement seems to contradict the first two:
by the
>first two, (A*B)*(C*D) should give a set of pairs of sets of
pairs,
>but by the third statement, that same expression should give
a set of
>4-tuples.
Correct.
>Obviously I'm missing something obvious. Any enlightenment
would be
>appreciated.
They are not equal, they are *isomorphic*, that is there is a
*canonical, uniquely defined* isomorphism (that is, a set
bijection)
between those two cartesian products. Associativity is to be
understood in this sense, that is
(A*B)*C = A*(B*C)
where = means there exists a bijection (uniquely defined)
between the
LHS and the RHS.
With my best regards,
G. Rodrigues
Subject: Re: Cantor Paradox
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CDZcR10763;
>> As there are several proofs of The uncountability of the
set of reals,
>> what is the point of criticising just one proof?
>
>> The diagonal argument is the best known and it has been
>> used to prove a lot of things besides the uncountability of
>> the real numbers. It is important to know what it can prove
>> and what it can't prove.
>
>
>>
>> - 2 many 2 count
>
>
>What theorem, if any, has it been used to prove that has no
other proof?
>If, as I suspect, none, then what is the point of criticizing
it?
Virgil, you ask why if a conjecture doesn't actually prove a
theory but in
spirit implies the same result, why it is not just fine.
You question the criticism of such a result when another
method implies the
same result. The point of criticism of one is that, assuming
there are two,
that both must be criticized.
With the antidiagonal argument in considering mapping the
reals and
naturals,
the reals have dual representation, and the binary case is
sufficient and
required.
Applying the antidiagonal argument to an infinite binary
sequence for each
subset of the naturals in illustrating the set/powerset result
is a
different
result, where there is no dual representation of a subset as
an infinite
binary sequence.
In the consideration of the nested intervals and the
gaplessness of the
reals,
in an effort to show that inexistence of mapping reals to
naturals, there
are
certain piecewise functions, of a sort, that would not have
that result, as
was addressed and attributed to iota as a least positive real.
Criticism of the antidiagonal result is warranted because
people argue it
in-
correctly, acceptance of dual representation of reals show
that any list of
reals in the unit interval for simplicity could be reordered
to have an
anti-
diagonal that is actually a list element. The Sun rises each
day because
the
Earth orbits the Sun, not the other way around except
nominally as the Sun
has
mass much larger than the Earth as they spin in the cosmic
clockworks. The
Sun does rise and set each day, that's the definition of a day.
Applying that concept to the powerset as represented by the
infinite binary
sequence is actually quite a different approach than
application to the set
of
reals.
If a) the antidiagonal argument and b) the nested intervals
arguments are
each
found to be flawed, then, that doesn't necessarily imply a
flaw in an
argument
using still another method, but it does mean that they can not
be used to
sup-
port that theory.
The other argument of course is that a set can not map
bijectively to its
own
powerset and that any set that maps bijectively to its
powerset can not map
bijectively to it or else through composition of functions the
set would map
to its powerset.
In a model allowing the consideration of infinitesimal real
numbers directly
addressable as elements of the range or coimage of the
function with the do-
main of the naturals, and also one where reals have dual
representation as
binary sequences, there is not a result but Cantor-Bernstein
argument by
con-
tradiction.
In that case, various criticisms of the powerset result itself
are able to
focus on that result without false claims of other
non-results, or rather
ably
to easily dispatch those claims.
Some of the considerations of the set and its powerset go in
the direction
of
a set as ordinals, with multiple meanings of their structure
and inter-
pretation, and with as well considerations of the
recomposition of the sets
according to structural and ordering properties of their
elements.
The Cantor Paradox is that the set of all sets would be its
own powerset
and
thus map bijectively to itself by the identity function. If a
set is not
otherwise enjoined from mapping its own powerset then that
result of the
universal set being its own powerset is not a contradiction.
As well, when
within ZF the negation of Cantor's powerset theorem is shown,
or even
axiomatized, trivially, it doesn't necessarily cascade into
contradiction.
Personally I think a function on the naturals with a range
defined on [0,1],
a monotonically increasing function, is an excellent and even
fundamental
tool.
The problem with that is the expectation that the integration
of it would
have
a value of one half, and explaining why it is that instead of
the other
value,
determined by different methods, actually requires an
explanation of the
reals
as points and the reals as a continuous line.
The criticism of the antidiagonal argument as applied to reals
as for
example
through the properties of dual representation of reals and
sufficiency
and
necessity of the binary case in refuting the antidiagonal
argument in that
way is acceptible, or even acceptable, but mostly acceptible.
Combined with
description of other mappings from the integers to the reals
that are not
pro-
scribed by other results, an entire concept of mapping the
integers to
specific
sets of each element of intervals of the reals is opened for
mathematical
dis-
course free of pre-Galilean prejudices, and variously
browbeating,
iconoclasty,
useless restriction, moribund lassitude, and profound
ignorance.
Warm regards,
Ross F.
Subject: Re: Cantor Paradox
[already answered]
I owe Easterly an apology. I carelessly misread the ravings of
Ross A. Finlayson as being by .
Sorry. Should have known better.
Subject: Re: Cantor Paradox
>> As there are several proofs of The uncountability of the
set of
reals,
>> what is the point of criticising just one proof?
>
>> The diagonal argument is the best known and it has been
>> used to prove a lot of things besides the uncountability of
>> the real numbers. It is important to know what it can prove
>> and what it can't prove.
>
>
>>
>> - 2 many 2 count
>
>
>What theorem, if any, has it been used to prove that has no
other proof?
>If, as I suspect, none, then what is the point of criticizing
it?
> Virgil, you ask why if a conjecture doesn't actually prove a
theory but
in
> spirit implies the same result, why it is not just fine.
> You question the criticism of such a result when another
method implies
the
> same result. The point of criticism of one is that, assuming
there are
two,
> that both must be criticized.
> With the antidiagonal argument in considering mapping the
reals and
naturals,
> the reals have dual representation, and the binary case is
sufficient and
> required.
> Applying the antidiagonal argument to an infinite binary
sequence for each
> subset of the naturals in illustrating the set/powerset
result is a
different
> result, where there is no dual representation of a subset as
an infinite
> binary sequence.
The failure onf one proof for an alleged theorem does not
disprove the
theaorem, but the success of even one proof does prove it.
The issue is not whether there is a base in which Cantor's
diagonal
construction fails, but whether there is a base in which
Cantor's
diagonal construction succeeds, as it does for any integral
base larger
than three. The fact that it might fail with bases two and
three is
irrelevant.
> In the consideration of the nested intervals and the
gaplessness of the
> reals,
> in an effort to show that inexistence of mapping reals to
naturals, there
are
> certain piecewise functions, of a sort, that would not have
that result,
as
> was addressed and attributed to iota as a least positive
real.
It is mappings from naturals to reals, not the reverse, that
is at issue
here. Get your facts straight. And none of your blatherings on
the
subject invalidated the first proof anywhere but in your own
fuzzy mind.
> Criticism of the antidiagonal result is warranted because
people argue it
in-
> correctly,
But it only takes one to argue it correctly, as Cantor himself
did, to
establish the result. That others do it wrong is of no
consequence.
acceptance of dual representation of reals show that any list
of
> reals in the unit interval for simplicity could be reordered
to have an
anti-
> diagonal that is actually a list element. The Sun rises each
day because
the
> Earth orbits the Sun, not the other way around
The sun rises each day, NOT because it rotates about the sun,
but
because it rotates about itself. It is the seasons and years
that result
from earth's rotation about the sun. 's knowledge of astronomy
is on a par with his knowledge of mathematics.
except nominally as the Sun
> has
> mass much larger than the Earth as they spin in the cosmic
clockworks.
The
> Sun does rise and set each day, that's the definition of a
day.
> Applying that concept to the powerset as represented by the
infinite
> binary sequence is actually quite a different approach than
> application to the set of reals.
Which concept? Rotation about the sun?
> If a) the antidiagonal argument and b) the nested intervals
arguments
> are each found to be flawed, then, that doesn't necessarily
imply a
> flaw in an argument using still another method, but it does
mean that
> they can not be used to sup- port that theory.
But no one has shown either proof to have the slightest flaw,
except for
the difficulty that the stupid have in understanding them.
> The other argument of course is that a set can not map
bijectively to its
own
> powerset and that any set that maps bijectively to its
powerset can not
map
> bijectively to it or else through composition of functions
the set would
map
> to its powerset.
I think's short circuits are showing again.
> In a model allowing the consideration of infinitesimal real
numbers
directly
> addressable as elements of the range or coimage of the
function with the
do-
> main of the naturals, and also one where reals have dual
representation as
> binary sequences, there is not a result but Cantor-Bernstein
argument by
con-
> tradiction.
I think needs to reboot his internal operating system, and
check
for viruses.
> In that case, various criticisms of the powerset result
itself are
> able to focus on that result without false claims of other
> non-results, or rather ably to easily dispatch those claims.
Definitely needs a reboot or virus check.
[The rest deleted as incoherent]
Subject: Question on modular arithmetic
I have a question regarding modular arithmetic:
Is it possible to easily form a polynomial form of a modular
arithmetic series (much like arithmetic or geometric series)?
It'd
make the algorithm I'm working on much easier if it is.
Subject: Question about combining loop QG + m-theory
If you bump up m-theory to 12 dimensions it breaks down into a
quantum
mechanics m-theory of the microcosm in 12 dimensions and a
relativistic
m-theory of the macrocosm in 12 dimensions. In order to
combine loop
quantum
gravity with this 12 dimensional approach to m-theory I
speculate I would
have
to
cut the 12 dimensions up into 4 three dimensional slices. For
example:
3 dimensions of space (length ,width ,depth), 3 dimensions of
time(past,
present,
and future), 3 dimensions of mass (matter, antimatter,
supermatter), 3
dimensions
of solidity(atomic nucleus, the whole atom, and molecules).
The point being
you
could measure each 3 dimensional slice on an x,y,z coordinate
system.
my question is: Is there mathematics that lets you stack or
combine x,y,z
coordinate systems so you can represent 12 dimensions
mathematicly in order
to
combine loop Quantum gravity with m-theory
Subject: pi=2.83... ?
Was just wondering...
I have implemented a classical Bresenham midpoint circle
algorithm
(using 8-symetry). It works pretty well.
I have carefully counted (discarding possible overdraws) the
pixels
plotted by the function in order to draw a circle of radius r
(in
pixels).
This value is the perimeter of the discretized circle (in
pixels),
something close from 2*pi*r. By dividing this by 2*r... I may
obtain
an approximation of pi, but not.
Here is my problem, I find out a converging pi value of about
2.83...
I may be missing something *very big*
Any clues appreciated,
Thanks in advance.
Subject: Re: pi=2.83... ?
In-reply-to: cbergadler@yahoo.fr (Carlos Bergadler)
>I have implemented a classical Bresenham midpoint circle
algorithm
>(using 8-symetry). It works pretty well.
>I have carefully counted (discarding possible overdraws) the
pixels
>plotted by the function in order to draw a circle of radius r
(in
>pixels).
>This value is the perimeter of the discretized circle (in
pixels),
>something close from 2*pi*r. By dividing this by 2*r... I may
obtain
>an approximation of pi, but not.
>Here is my problem, I find out a converging pi value of about
2.83...
>I may be missing something *very big*
When computing arclength, you have to be careful that the
pieces you are
adding actually measure arclength. A pixel does not measure
arclength.
For example, the line from (0,0) to (100,100) contains
approximately 100
pixels. However, the distance between these points is between
141 and
142 pixel widths.
On parts of curves that are more horizontal, counting pixels
measures
the horizontal extent. On parts of curves that are more
vertical,
counting pixels measures the vertical extent. Between 0 and 45
degrees,
a circle is vertical; between 45 and 90 degrees, it is
horizontal.
Therefore, the number of pixels from 0 to 45 degrees is going
to be
radius/sqrt(2), and between 45 and 90 degrees, it is also
going to be
radius/sqrt(2). The circumference of a circle is
8*radius/sqrt(2) if
measured this way. Dividing this by a diameter of 2*radius
yields
4/sqrt(2) = 2.828427... for pi.
Rob Johnson Was just wondering...
> I have implemented a classical Bresenham midpoint circle
algorithm
> (using 8-symetry). It works pretty well.
> I have carefully counted (discarding possible overdraws) the
pixels
> plotted by the function in order to draw a circle of radius
r (in
> pixels).
> This value is the perimeter of the discretized circle (in
pixels),
> something close from 2*pi*r. By dividing this by 2*r... I
may obtain
> an approximation of pi, but not.
> Here is my problem, I find out a converging pi value of
about 2.83...
> I may be missing something *very big*
Are you in Indiana by any chance?
Bart
Subject: Re: pi=2.83... ?
>Was just wondering...
>I have implemented a classical Bresenham midpoint circle
algorithm
>(using 8-symetry). It works pretty well.
>I have carefully counted (discarding possible overdraws) the
pixels
>plotted by the function in order to draw a circle of radius r
(in
>pixels).
>This value is the perimeter of the discretized circle (in
pixels),
>something close from 2*pi*r. By dividing this by 2*r... I may
obtain
>an approximation of pi, but not.
>Here is my problem, I find out a converging pi value of about
2.83...
>I may be missing something *very big*
I'm not sure I understand. You're calculating the perimeter of
the
circle by counting the number of pixels that the algorithm
colors
when it draws the circle? If so there's really no reason that
should
give the perimeter - pixels are not round, the distance between
the centers of two adjacent pixels is either 1 or sqrt(2), etc.
If that's not what you're doing please explain.
>Any clues appreciated,
>Thanks in advance.
************************
Subject: Re: pi=2.83... ?
Originator: richard@cogsci.ed.ac.uk (Richard Tobin)
>Any clues appreciated,
Try using the same method with a 45-degree straight line to
get an
estimate of sqrt(2).
(For extra credit, determine why your approximation to pi is
roughly
2*sqrt(2).)
-- Richard
Subject: Division Rings Over Their Centers
Dear All,
Lets call an integer number n nice if there exists a division
ring
which is of dimension n (as a vector space) over its center.
Now the question is: Which integer numbers are nice!?
Thanks.
Subject: Re: Division Rings Over Their Centers
> Dear All,
> Lets call an integer number n nice if there exists a
division ring
> which is of dimension n (as a vector space) over its center.
> Now the question is: Which integer numbers are nice!?
The squares!
The buzzword is central simple algebras. See, for instance,
Peirce, _Associative Algebras_ or Patrick Morandi's notes at
http://sierra.nmsu.edu/morandi/notes/mathematicalnotes.html .
--
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: Is this travelling salesman problem version known?
> If you are required to return to the starting point,
> then this is the Hamiltonian cycle problem,
> which is knownto be NP-hard.
No. The path is allowed to be open (and in the problem I'm
working on,
it is always open except we have only one city)
I din't even see a proof of NP-hardness for the open path
version of
TSP. Here help is appreciated too.
--
Best regards,
Alex.
PS. To email me, remove loeschedies from the email address
given.
Subject: Re: matrix analysis: can I say matrix X=Y if I have
AXA=AYA?
>>1. the dimensions of A,X,Y should be compatible, in this
case I think
>>they should be all square matrices.
> Not at all.
Right you are!
--
Best regards,
Alex.
PS. To email me, remove loeschedies from the email address
given.
Subject: E and E{0} are not homeomorphic
Hi all,
Can anyone explain me how to prove that, given a normed space
E, E and
E{0} are not homeomorphic? Of course, I am not interested in
the
finite-dimensional case.
Best regards,
Jose Carlos Santos
Subject: Re: E and E{0} are not homeomorphic
>Can anyone explain me how to prove that, given a normed space
E, E and
>E{0} are not homeomorphic? Of course, I am not interested in
the
>finite-dimensional case.
It can't be proved, because it isn't always true. For instance,
Hilbert space H is homeomorphic (indeed, diffeomorphic) to
H{0},
and much, much more is true. Start from West, James E., The
diffeomorphic excision of closed local compacta from infinite-
dimensional Hilbert manifolds. Compositio Math. 21 1969
271--291
and work forwards and backwards through the literature. (If you
have broadband or plenty of time, download Kriegel and Michor's
free book at http://www.ams.org/online_bks/surv53/surv53.pdf --
it's 4.68 megabytes, loaded with relevant material.)
Lee Rudolph
Subject: Re: E and E{0} are not homeomorphic
>>Can anyone explain me how to prove that, given a normed
space E, E and
>>E{0} are not homeomorphic? Of course, I am not interested in
the
>>finite-dimensional case.
> It can't be proved, because it isn't always true. For
instance,
> Hilbert space H is homeomorphic (indeed, diffeomorphic) to
H{0},
> and much, much more is true. Start from West, James E., The
> diffeomorphic excision of closed local compacta from
infinite-
> dimensional Hilbert manifolds. Compositio Math. 21 1969
271--291
> and work forwards and backwards through the literature. (If
you
> have broadband or plenty of time, download Kriegel and
Michor's
> free book at http://www.ams.org/online_bks/surv53/surv53.pdf
--
> it's 4.68 megabytes, loaded with relevant material.)
Wow! Thanks a lot. I really thought that they were never
homeomorphic.
Best regards,
Jose Carlos Santos
Subject: Re: E and E{0} are not homeomorphic
>>>Can anyone explain me how to prove that, given a normed
space E, E and
>>>E{0} are not homeomorphic? Of course, I am not interested
in the
>>>finite-dimensional case.
>
>> It can't be proved, because it isn't always true. For
instance,
>> Hilbert space H is homeomorphic (indeed, diffeomorphic) to
H{0},
>> and much, much more is true.
...
>Wow! Thanks a lot. I really thought that they were never
homeomorphic.
Proving that they are diffeomorphic is (or used to be...)
rather
complicated. It's much less difficult to show that they are
homotopy equivalent, and even less difficult to show that they
are weakly homotopy equivalent (from which, if I remember
aright,
the strong homotopy equivalence follows easily with a bit more
work). First, in any normed space E, E{0} is homotopy
equivalent
to the unit sphere S of E (with respect to the norm). Now,
when we consider H to be the completion of its subvectorspace
R^{infty} (of sequences that are eventually 0), then we have
S as the completion of S', the intersection of S with
R^{infty};
and S' is the limit (in whatever the right technical sense is)
of its finite-dimensional subspheres S^k (where a point of S'
is in S^k if, as a sequence, it is 0 from the k+1st entry on).
But this limit is (weakly) contractible, because S^k contracts
to its basepoint within S' (the trace of its contraction can be
taken to be the upper hemisphere of S^{k+1}). Some handwaving
and you're done.
The proof that H is homeomorphic to H{0} is of intermediate
difficulty. I heard Bing lecture on it about 35 years ago.
The rough idea (which is all I got out of it at the time)
is to construct the homeomorphism from H{0} onto H as the
limit of a sequence of homeomorphisms f_i from H{0} onto
the open subset U_i = H{v_i}, where, in terms of the standard
basis e_1, e_2, ..., e_n, ... of H, v_i=e_1+....+e_i. Since
the sequence {v_i} doesn't converge in H, in the limit you
get a map f from H{0} onto H. Obviously there's something
tricky going on there (and you have to be careful to make
f exist, and to be a homeomorphism, and to be onto), but
when Bing waved his hands it was very convincing to the
younger me.
Lee Rudolph
Subject: (Nonisomorphic )Groups
charset=iso-8859-1
According to my book, there are 267 groups of order 64 (we
don't demand
that
they're isomorphic). Which are they?
I have found 11 only and i'd be delighted if somebody gave me
a hint on how
to find the rest.
I choose not to go insane and use Z3 x Z5 x Z11 denoted as [3
5 11]. I have
found following groups:
[2 2 2 2 2 2] [4 2 2 2 2] [8 2 2 2]
[16 2 2] [32 2] [64] [4 4 2 2]
[4 4 4] [8 4 2] [8 8] [16 4]
Even if i go wild and count [16 2 2] and [2 16 2] as different
groups i
still score 32, not as the book claims 267. I don't need
(don't want) a
full
list, of course but a hint on what groups i missed.
--
Kindly
Konrad
---------------------------------------------------
May all spammers die an agonizing death; have no burial places;
their souls be chased by demons in Gehenna from one room to
another for all eternity and more.
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sense to be lazy
---------------------------------------------------
Subject: Re: (Nonisomorphic )Groups
> According to my book, there are 267 groups of order 64 (we
don't demand
> that they're isomorphic). Which are they?
> I have found 11 only and i'd be delighted if somebody gave
me a hint on
> how to find the rest.
> I choose not to go insane and use Z3 x Z5 x Z11 denoted as
[3 5 11]. I
> have found following groups:
> [2 2 2 2 2 2] [4 2 2 2 2] [8 2 2 2]
> [16 2 2] [32 2] [64] [4 4 2 2]
> [4 4 4] [8 4 2] [8 8] [16 4]
> Even if i go wild and count [16 2 2] and [2 16 2] as
different groups i
> still score 32, not as the book claims 267. I don't need
(don't want) a
> full list, of course but a hint on what groups i missed.
You simply missed all non-abelian groups - your count for the
abelian ones
seems to be correct.
--
I'm on warm milk and laxatives
Cherry-flavored antacids
reverse my forename for mail! - saibot
Subject: Re: (Nonisomorphic )Groups
charset=iso-8859-1
> You simply missed all non-abelian groups - your count for
the abelian
ones
> seems to be correct.
I suspected it might be something like that. The problem is
that my book is
a little bit unclear on that subject (that, or i'm too limited
to get that
part).
For instance - Z(125) has following abelian groups (since 125
= 5*5*5):
Z(5^3), Z(5^2) x Z(5) and Z(5) x Z(5) x Z(5).
According to the printed thing there are two non-abelian.
Nice, i think to
my self, but which are they?!
I can only think of Z(5) x Z(5^2) but that ought to be
incorrect since
i'm still one group short...
--
Kindly
Konrad
---------------------------------------------------
May all spammers die an agonizing death; have no burial places;
their souls be chased by demons in Gehenna from one room to
another for all eternity and more.
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sense to be lazy
---------------------------------------------------
Subject: Re: (Nonisomorphic )Groups
>> You simply missed all non-abelian groups - your count for
the abelian
>> ones seems to be correct.
> I suspected it might be something like that. The problem is
that my book
> is a little bit unclear on that subject (that, or i'm too
limited to get
> that part).
> For instance - Z(125) has following abelian groups (since
125 = 5*5*5):
> Z(5^3), Z(5^2) x Z(5) and Z(5) x Z(5) x Z(5).
> According to the printed thing there are two non-abelian.
Nice, i think
to
> my self, but which are they?!
> I can only think of Z(5) x Z(5^2) but that ought to be
incorrect
since
> i'm still one group short...
All groups of the sort Z(n_1) x ... x Z(n_k) are abelian
because they are
products of abelian groups; to get one of the many non-abelian
groups of
order 64, take for example the dihedral group D_32.
(If you want, there exist arbitrary products in the category
of abelian
groups which are preserved under the canonical injection
functor
AbelianGroups ---> AllGroups. BTW, are products, coproducts
and limits
always conserved under functors? ;-)
--
I'm on warm milk and laxatives
Cherry-flavored antacids
reverse my forename for mail! - saibot
Subject: Re: (Nonisomorphic )Groups
charset=iso-8859-1
Thanks to both you and Arturo for the help. I see
more than clearly where i confused myself. Also,
my interest in going deeper into algebra has most
certainly increased. Thanks!
--
Kindly
Konrad
---------------------------------------------------
May all spammers die an agonizing death; have no burial places;
their souls be chased by demons in Gehenna from one room to
another for all eternity and more.
Sleep - thing used by ineffective people
as a substitute for coffee
Ambition - a poor excuse for not having
enough sense to be lazy
---------------------------------------------------
Subject: Re: (Nonisomorphic )Groups
Adjunct Assistant Professor at the University of Montana.
>> You simply missed all non-abelian groups - your count for
the abelian
ones
>> seems to be correct.
>I suspected it might be something like that. The problem is
that my book
is
>a little bit unclear on that subject (that, or i'm too
limited to get that
>part).
Describing all the groups of order 2^n (or p^n, p a prime) is
highly
nontrivial. See the review quoted at the end.
>For instance - Z(125) has following abelian groups (since 125
= 5*5*5):
>Z(5^3), Z(5^2) x Z(5) and Z(5) x Z(5) x Z(5).
>According to the printed thing there are two non-abelian.
Nice, i think to
>my self, but which are they?!
>I can only think of Z(5) x Z(5^2) but that ought to be
incorrect since
>i'm still one group short...
Z(5) x Z(5^2) is abelian. You already counted it: it is
isomorphic
(obviously so) to Z(5^2) x Z(5).
If p is a prime, then there are exactly two nonisomorphic
nonabelian
groups of order p^3.
If p = 2, you get the dihedral group of order 8
D_8 = If p is a prime, then there are exactly two nonisomorphic
nonabelian
> groups of order p^3.
> If p = 2, you get the dihedral group of order 8
> D_8 = and the quaternion group of order 8:
> Q_8 = <1,-1,i,-i,j,-j,k,-k : i^2 = j^2 = k^2 = -1, (ijk)^2 =
1
I've usually seen this with i^2 = j^2 = k^2 = ijk = -1.
Although I
may just be carrying over a bit too much from the non-abelian
extension
of the reals.
--
Daniel W. Johnson
panoptes@iquest.net
http://members.iquest.net/~panoptes/
039 53 36 N / 086 11 55 W
Subject: Re: Cantor Paradox
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CFItI24904;
>> Wow, sounds like Nathan has rediscovered Richard's Paradox
(1905). Not
>> bad for an 11-year-old, if he really did think of himself.
>He's not an 11-year-old. Whether he's ever read Richard's
Paradox or
>not, he clearly has the background to choose the issues
carefully
>here.
>(I hadn't seen the paradox before, so I appreciate having a
name
>attached to it.)
>--
>
>Contrariwise, continued Tweedledee, if it was so, it might
be, and
>if it were so, it would be; but as it isn't, it ain't. That's
logic!
> -- Lewis Carroll
I was going to dispute the fact that Nathan's paradox is
identical to
Richard's Paradox, but now I believe they are identical. I
also really
appreciate knowing the correct name for this fallacious but
very amusing
argument. Thanks, Fred Galvin. I used to read the news group
geometry_research a lot, and one of the delightful aspects of
that forum was
that often John Conway would supply the correct names for
things. Dr. Conway
is also highly adept at coining new names as the occasion
demands.
Danny Purvis
Subject: Re: Cantor Paradox
Danny Purvis says...
>I was going to dispute the fact that Nathan's paradox is
identical to
>Richard's Paradox, but now I believe they are identical.
On this web page, Nathan's paradox is explicitly connected with
Richard's paradox:
http://www.dpmms.cam.ac.uk/~wtg10/richardsparadox.html
--
Daryl McCullough
Ithaca, NY
Subject: Re: stochastic group S(2,4)
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CFIu424926;
>>> How to prove it is isomorphic to alternating group of
order 12?
>>Um, what is it?
>Never heard of it. S(2,4) is sometimes used for symplectic
group Sp(2,4),
>but that is the same as SL(2,4) and is isomorphic to
alternating group of
>order 60.
>It's not SO^+(2,4) or SO^-(2,4) either - they have orders 6
and 10.
>Derek Holt.
I've never heard of this group, but could this have something
to do
with stochastic matrices? These are matrices with all column
sums
equal to 1, or somehting like that. I don't know how one would
make a
group form them though.
Mark
Subject: Traces of matrices
I found the following exercise about matrices: show that there
are no
matrices A and B (over R or even over C) such that AB - BA =
I, I the
identity matrix. The solution I gave is very simple: For every
n x n
matrices A and B, we have Tr(AB - BA) = Tr(AB) - Tr(BA) =
Tr(AB) -
Tr(AB) = 0, where Tr means the trace of the matrix. Since
Tr(I) = n>0,
there can't be matrices A and B such that AB - BA = I. A
similar
argument shows that, in fact, if Tr(M)<>0, then there are no
matrices
A and B such that AB - BA = M.
Now, I'm trying to prove, or give a counter example, that if
Tr(M) =
0, then there are indeed matrices A and B such that AB - BA =
M. I got
stuck. I noticed that if D is a diagonal matrix, then, for
every
matrix A, DA and AD have the same main diagonal, so that the
main
diagonal of DA -AD is totally formed by 0s. Then I tried to
prove
that if Tr(M) = 0, then M is similar to a matrix with a null
diagonal,
but I'm not sure if this is true.
Any hints are welcome.
Thank yoy
Amanda
Subject: Re: Traces of matrices
> Now, I'm trying to prove, or give a counter example, that if
Tr(M) =
> 0, then there are indeed matrices A and B such that AB - BA
= M. I got
> stuck. I noticed that if D is a diagonal matrix, then, for
every
> matrix A, DA and AD have the same main diagonal, so that the
main
> diagonal of DA -AD is totally formed by 0s. Then I tried
to prove
> that if Tr(M) = 0, then M is similar to a matrix with a null
diagonal,
> but I'm not sure if this is true.
Hint: consider the matrices E_{ij} which have only 0's, except
for the
entry at column i and row j, which is equal to 1. Any M such
that Tr(M)
= 0 is a linear combination of some E_{ij} with i different
from j and
some E_{ii} - E_{jj}, again with i different from j.
Best regards,
Jose Carlos Santos
Subject: Re: AC <=> trichotomy
> Does anyone have a quick sketch of a proof that AC <=>
trichotomy?
The => implication follows since AC implies that every set is
well ordered
(use
transfinite induction or Zorn's Lemma) and hence it is easy to
find the
injection from one set to the other (or just use Zorn's Lemma
straight).
The <= implication follows, I recall, from something clever
called, maybe,
Hartogg's (spelling?) Lemma. This says that for any set there
exists an
ordinal
such that there is no injection from the ordinal to the set.
Trichotomy
then
implies that there exists an injection from the set to an
ordinal, hence it
is
well ordered. That every set is well ordered implies AC is
very easy.
My recollection is that the proof of Hartogg's Lemma is
something like this.
Given a set A, consider the collection of ordinals which map
injectively
into
the set A. Use the usual set axioms (e.g. replacement and
power set) to
show
that this collection is indeed a set. Hence this collection is
itself an
ordinal, and since an ordinal cannot be an element of itself,
this ordinal
satisfies the requirements.
Subject: Re: New Randomness Test
.
> This does not alter the fact that the sequence is not
> truly random, since it is entirely predictable given
> the knowledge of the appropriate formula.
Following this criterion do not exists finite random sequences,
because in the development of sqr(2) we can find any given
finite
sequence of digits.
As all we know the program for producing the digits of sqr(2)
is very
short.
L. Rodriguez
Subject: Re: New Randomness Test
> .
>
>> This does not alter the fact that the sequence is not truly
random,
>> since it is entirely predictable given the knowledge of the
appropriate
>> formula.
>
> Following this criterion do not exists finite random
sequences,
That's right. Is a sequence like 1 2 random? It seems to be
very
predictable but, do we know if it was randomly generated?
For finite sequences the best that one can do is to analyze
their
statistical properties - tapping a truly random source (e.g. a
decaying
radioactive sample) given enough time will produce any finite
sequence,
including a long chain of zero bits. The probability for this
to happen,
however, is minuscule, which is why statiscal tests will raise
a red flag:
if you happen to encounter such a sequence as produced by a
purported
source of randomness, chances are that it isn't.
Subject: Shafer-Dempster vs. Bayes
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CG4dw30832;
Dear Friends,
I am doing some research about treating uncertainties in
engineering
calculations. As one method I found the Shafer-Dempster Theory
of evidence.
I read that resulting from that theory, the Bayesian approach
of treating
uncertain data is a special case of the Shafer-Dempster Theory
and that is
the point where I got confused. Can anybody explain me that? I
also would
like to know what the advantages and disadvantages of both
approaches are.
Thanks,
Dan
Subject: Re: Documents about multiplicative order in general
and mersenne
numbers
>> Of course I am not just interested in exponents which are
composed
>> of exactly two exponents, but in all composed exponents in
>> general, because i know now that the divisors of mersenne
numbers
>> with prime exponents are Cunningham numbers, right?
>Sorry, never heard the expression, Cunningham number.
> By your address I guess you are a student?
Always have been, hope I always will be.
> Nobody really knows everything. I would be the last to
expect that,
> since I didn't even study math, but something else.
> If you are interested in Cunningham Numbers you will find
documents
> here:
> http://mathworld.wolfram.com/CunninghamNumber.html
So Mathworld defines a Cunningham number as any number of the
form
b^n + 1 or b^n - 1. But of course that makes every number a
Cunningham
number, since m = (m + 1)^1 - 1, so help me out a little here.
I know what the Cunningham project is, and I know what the
Cunningham
tables are, but I searched Math Reviews for Cunningham number
and
came up with nothing. It seems to be mathworld's invention,
rather
than a standard term.
> Mathworld is really a great site. I really can recomment it
to you.
It is a really great site, but I recommend great caution using
it.
I've seen too many posts to this newsgroup pointing out errors
in
mathworld. On the Cunningham number page, it says,
Primes of the form b^n - 1 are also very rare. The Mersenne
numbers
are known to be prime only for 37 values, the first few of
which are n
= 2, 3, 5, 7, 13, 17, 19, ... (Sloane's A000043). There are no
other
primes b^n - 1 for nontrivial b < 21 and 1 < n < 1001.
You know, I'll bet you can prove that there are no primes b^n
- 1
at all for b > 2 and n > 1. So this isn't exactly an error on
mathworld,
but it's certainly a silly thing for mathworld to say.
> In fact I wanted to reply using email, but it didn't work
isn't
> your correct address?
Yes, it is, and I don't know why it didn't work for you, but
you know,
there's a reason why I only put a disguised form of it in my
posts;
it has to do with keeping out spam. I'm not happy that you put
the
undisguised version in your post.
> I really got the impression that you felt offended by my
postings. If
> my feeling is right, I have to apologize. I never meant to
offend you
> or anybody else in this newsgroup and I surely won't waste
my time on
> flame wars.
My apologies if I lost my virtual temper.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Subject: Re: Point on the sphere equidistant from two others
Certainly it may be that there are no solutions to this
problem.
However, when there is, treat the sphere as a constraint and
use the
sum of the squares of the vectors as the function you're
trying to
minimize.
Use the method of Lagrange multipliers.
Example:
Let |Z| be a vector. THe sphere constraint is:
|Z| - C = 0
The sum of squares function is |A-Z|^2 + |B-Z|^2
Take the gradients of these two vector equations. i.e. take the
partial derivatives with respect to each coordinate, the
gradient is
.
Then find if the two gradients are parallel to each other.
That will
give you the critical points on the sphere. Since your sphere
has no
boundary in 3-space, any maximum or minimum will be inside the
sphere
(am I right on this?)
Enjoy,
-Greg
Subject: Re: Point on the sphere equidistant from two others
Axel,
You can derive an explicit formula for the solution
Doing this in cartesian coordinates seems easier to me (I
don't remember
the
spherical coordinates well enough, you might have to tinker
with the
formulae to accomodate for the usual denominations of N S E W
latitude and
longitude):
let B0 L0 and B1 L1 be your points expressed as latitude and
longitude, R
the radius of the earth (as R is a common factor in all
expressions we will
see afterwards, we could as well pose R=1)
their cartesian coordinates are
x_i = R cos Bi sin Li
y_i = R cos Bi cos Li
z_i = R sin Bi
the great circle of equidistant points belongs to the plane
which goes
through the center of the earth (origin of the coordinates
here) and has
the
vector going through both points as its normal.
Its equation is
(x1-x0) x + (y1 - y0) y + (z1 - z0) z =0
if B is the latitude of the parallel you are considering for
solutions, the
parallel lies on the plane
z = R sin B
substituting this in the first equations we have the
intersection of the
two
planes (supposing it exists, ie that we don't have x1-x0=0 and
y1-y2=0)
(x1-x0) x + (y1 - y0) y + (z1-z0) R sin B=0
we are left to find the intersections of this line with the
parallel circle
of latitude B
Let S be the longitude of a point on the parallel, we have
x= R cos B cos S
y= R cos B sin S
replacing this in the line equation, we get the formula for
the longitude
of
the solutions
(x1-x0) cos B cos S + (y1 - y0) cos B sin S + (z1-z0) sin B=0
(x1-x0) cos S + (y1 - y0) sin S + (z1-z0) tg B=0
which could be solved (for instance) by using the half tangent
equalities
posing t= tg (S/2)
(beware, there is a special case case when S=Pi)
we have cos S = (1- t^2) / (1+t^2)
sin S = 2t / (1+t^2)
and the equation becomes
(x1-x0) (1-t^2) + 2 (y1-y0)t + (z1-z0) tg B (1+t^2) =0
[(z1-z0) tg B - (x1-x0) ] t^2 + 2 (y1 - y0) t + [(z1-z0) tg B
+ (x1-x0) ]
=0
the discriminant is
D = (y1 - y0)^2 + (x1-x0)^2 - tg B ^2 (z1 - z0)^2
if it is positive the two solutions are (if all these ugly
calculations
were
correct)
S = 2 arctg( -(y1-y0) +/- sqrt(D) )
Finding the closest point is then very easy (it should be the
point with
the
smallest longitude difference, modulo 2pi, to any of the two
starting
points)
Hope this helps
Francois
85055d13.0403111233.29125b04@posting.google.com...
> I want a formula for the point on an arbitrary parallel of
latitude
> that is equidistant from two other points on the Earth
(assuming a
> spherical Earth). So far I used the great-circle-distance
expression
> to the two given points and equated them. After simplifying
there are
> two possible identities which can be solved recursively:
> Lx = arcsin (( s0 - s1 - (Dc cos Lx) )/Ds ) and
> Lx = arccos (( s0 - s1 - (Ds sin Lx) )/Dc ) where
> B0,L0 and B1,L1 are the latitude and longitude respectively
of the
> given points
> Bx is an arbitrary latitude
> Lx is the longitude of the point sought, Bx,Lx
> s0 = sin B0 sin Bx
> k0 = cos B0 cos Bx
> s1 = sin B1 sin Bx
> k1 = cos B1 cos Bx
> Dc = ( k1 cos L1 ) - k0 cos L0
> Ds = ( k1 sin L1 ) - k0 sin L0
> I start the recursions with Lx := ( L0 + L1 )/2.
> Sometimes one identity works while the other fails to
converge.
> Sometimes they both fail.
> Is there a procedure that always works?
> PS - Of course there are two points on the same parallel
that are
> equidistant to the two given places; I am looking for the
closer one.
Subject: Re: Point on the sphere equidistant from two others
> [ ... ]
>Simpler yet (once you're in Cartesian coordinates), add the
two
>unit vectors and normalize the result.
> Did you guys read beyond the original subject line? First
off, the set
of
> points equidistant from two others on a sphere is a great
circle. The OP
> wants to know where that great circle intersects a given
circle of
constant
> latitude. There are generally 2 such points, and he wants
the closer
one.
Phew! Thank you... Besides, Cartesian coordinates seem very
unnatural
to me.
Subject: Re: Point on the sphere equidistant from two others
<4050D590.23DA@mindspring.com>
>Do it in cartesian coordinates, and just find the lat and
long of the
>>midpoint of the straight line between the two points.
>Simpler yet (once you're in Cartesian coordinates), add the
two
>unit vectors and normalize the result.
> Did you guys read beyond the original subject line? First
off, the set
of
> points equidistant from two others on a sphere is a great
circle. The OP
> wants to know where that great circle intersects a given
circle of
constant
> latitude. There are generally 2 such points, and he wants
the closer
one.
> -- spud_demon -at- thundermaker.net
There are certainly cases where there is _no_ solution to the
OP's
problem.
Think of the two points lying on the same longitude
symmetrically to the
equator. Then the great circle that is the location of the
points
equidistant from both is the equator itsself, which obviously
never
intersects any other circle of constant latitude away from the
equator.
Similar for two points with same longitude: The great circle
passing
through
their midpoint will only cover longitudes nearer to the
equator, but
there
will be no solution in the region nearer to the poles than
(B0+B1)/2
(assuming the 2 points both on northern or both on southern
hemisphere).
Also the general case has regions, where there is no solution.
Hugo Pfoertner
Subject: Re: Point on the sphere equidistant from two others
> There are certainly cases where there is _no_ solution to
the OP's
> problem.
Thank you for your comments - however...
> Think of the two points lying on the same longitude
symmetrically to the
> equator. Then the great circle that is the location of the
points
> equidistant from both is the equator itsself, which
obviously never
> intersects any other circle of constant latitude away from
the equator.
The equator is indeed a circle of constant latitude: it just
happens
to be the only one that is a great circle. In this case the
desired
point would be Bx=0 and Lx=L0=L1 (in the notation of my first
post).
Of course you are right to say that the equator would not
intersect
the locus of points equidistant to the two given points, but
if you go
back to my OP you will see that I did not state the problem in
terms
of a locus-intersect. The idea was that given an arbitrary
latitude,
what is the longitude of the equidistant point?
> Similar for two points with same longitude: The great circle
passing
> through their midpoint will only cover longitudes nearer to
the equator,
> but there will be no solution in the region nearer to the
poles than
> (B0+B1)/2
> (assuming the 2 points both on northern or both on southern
hemisphere).
> Also the general case has regions, where there is no
solution.
Yes, I should have made the conditions of the problem
narrower. But
now, what if we specify that the latitude must be equal to (B0
+ B1)/2
instead of any arbitrary Bx? Are there still regions with no
solution?
Of course I am excluding the special case where the two given
points
are the North and South poles.
Subject: Re: Point on the sphere equidistant from two others
> There are certainly cases where there is _no_ solution to
the OP's
> problem.
> Thank you for your comments - however...
> Think of the two points lying on the same longitude
symmetrically to
the
> equator. Then the great circle that is the location of the
points
> equidistant from both is the equator itsself, which
obviously never
> intersects any other circle of constant latitude away from
the equator.
> The equator is indeed a circle of constant latitude: it just
happens
> to be the only one that is a great circle. In this case the
desired
> point would be Bx=0 and Lx=L0=L1 (in the notation of my
first post).
> Of course you are right to say that the equator would not
intersect
> the locus of points equidistant to the two given points, but
if you go
> back to my OP you will see that I did not state the problem
in terms
> of a locus-intersect. The idea was that given an arbitrary
latitude,
> what is the longitude of the equidistant point?
> Similar for two points with same longitude: The great circle
passing
> through their midpoint will only cover longitudes nearer to
the
equator,
> but there will be no solution in the region nearer to the
poles than
> (B0+B1)/2
> (assuming the 2 points both on northern or both on southern
hemisphere).
> Also the general case has regions, where there is no
solution.
> Yes, I should have made the conditions of the problem
narrower. But
> now, what if we specify that the latitude must be equal to
(B0 + B1)/2
> instead of any arbitrary Bx? Are there still regions with no
solution?
No, for this choice the solution can be found by adding the
cartesian
co-ordinates of the two points and scaling back to r=1 as
recommended
angles.
> Of course I am excluding the special case where the two
given points
> are the North and South poles.
You have to exclude all cases where the two points are exactly
opposite
to each other on the sphere. In this case the summation of the
two
vectors
gives 0. Any point on the great circle whose plane is
perpendicular to
both vectors is than a solution to your problem. P0 and P1 at
North and
South pole is just a special case with any point on the
equator being
a solution.
A more general formulation of the problem with the condition
of a given
latitude might ask for minimizing the difference of the
distances
abs(P0-Target) - abs(P1-Target) instead of making those
distances equal.
This problem will have a solution also in the cases mentioned
in my
previous posting.
Hugo
Subject: Trapezoidal Rule & Simpson's Rule - Follow Up Question
Thanks for the two examples. Over [-1,1] using n=2
subintervals, the
functions f(x)=x^(2/5) and f(x)=cos(pi x) are indeed examples
where the
Trapezoidal Rule better approximates the definite integral
than Simpson's
Rule. But for large values of n, Simpson's rule would be the
better of the
two.
Are there situations where the Trapezoidal Rule is better for
all n?
Thanks again -
L
Subject: Closure property question
Consider a collection of sets F. Define a set C_n of the form
oo oo
C_n = U / (B_njk), where B_njk belongs to F for all j and k.
j=1 k=1
{C_n} is the collection of all such sets. Question: Is {C_n}
closed
under countable unions? I believe it is since the resulting
iterated
unions can be expressed as one union, and the resulting set is
some
C_k. Is this correct, or is there some other reason {C_n} is
closed
under unions?
{C_n} would not necessarily be closed under countable
intersections?
Is this correct? All help appreciated.
Subject: Re: Closure property question
> Consider a collection of sets F. Define a set C_n of the form
> oo oo
> C_n = U / (B_njk), where B_njk belongs to F for all j and k.
> j=1 k=1
>
> {C_n} is the collection of all such sets. Question: Is {C_n}
closed
> under countable unions? I believe it is since the resulting
iterated
> unions can be expressed as one union, and the resulting set
is some
> C_k. Is this correct, or is there some other reason {C_n} is
closed
> under unions?
> {C_n} would not necessarily be closed under countable
intersections?
> Is this correct? All help appreciated.
Probably I do not understand your question correctly, because
this does not
make too much sense to me:
E_njk is indexed by the three parameters n, j and k. But what
does a
specific value of n have to do with all the other values of n?
Why should the resulting set be some C_k?
For example, set E_njk={1,...,n}. Then C_n={1,...,n}. But the
union of all
C_n is N. The same thing applies for intersections: set
E_njk=(N{1,...,n}). Then the intersection of all C_n is empty,
though
none
of the C_n is.
--
I'm on warm milk and laxatives
Cherry-flavored antacids
reverse my forename for mail! - saibot
Subject: Re: Strange Complex Variables Problem
> Oh my gosh, no I didn't say anything about the residue.
> I said something about the coefficient of 1/z in the Laurent
> expansion, but (i) that was when I was talking about
> functions in an _annulus_, not in D, (ii) that coefficient
> is _not_ a residue (except in the case when the annulus
> has zero inner radius.)
Yeah, that's true. The 2pi might not be the result of a
residue,
unless f is meromorphic in the region bounded by the outer
curve.
I didn't realize what you were saying. That's right, you can't
even
always get a laurent series expansion in my arbitrary domain
D. Well
you can but for small circles within D. I assumed you were
talking
about residues as f went around D's inner curve because it has
a pole
inside the inner curve.
No, basically the i*2PI is there to prop up the functions so
their
L2 norm could be bounded below by F'/F .
-Greg
Subject: Re: Strange Complex Variables Problem
I also wanted to mention again that in addition to what I
said, it's
useful to know that square-integrable analytic functions DO
form a
hilbert space on the domain, because these functions converge
in the
L2 norm iff they converge uniformly (i.e. to another analytic
function).
-Greg
Subject: Re: Strange Complex Variables Problem
>I also wanted to mention again that in addition to what I
said, it's
>useful to know that square-integrable analytic functions DO
form a
>hilbert space on the domain, because these functions converge
in the
>L2 norm iff they converge uniformly (i.e. to another analytic
>function).
That's nonsense. Uniform convergence does not imply convergence
in your Hilbert space, unless D has finite area. And
convergence
in your Hilbert space does not imply uniform convergence,
regardless of what D is.
The reason that that space is a Hilbert space, now that we've
finally got straight what space we're talking about, is that
convergence in that space implies uniform convergence
_on_ compact subsets of D (hence that the limit is analytic.)
>-Greg
************************
Subject: Re: Strange Complex Variables Problem
>Well, I'm talking about the L2 norm induced by the inner
product = INTEGRAL(over D) [ f(z) conj[g(z)] ] dz , which is simply
L2(f) =
>INTEGRAL(over D) [ |f(z)|^2 dz ].
> This is (in my experience) a strange use of notation.
I meant to indeed say INTEGRAL (over D) [ f(z) g(z)* ] dz dz*
[ I meant to say Green's theorem to integrate around the
boundary but
that's irrelevant. Let me try to start over because I don't
want to
confuse anyone. ]
I've got a hilbert space H of analytic functions defined on a
region
D, square-integrable on D. i.e. Integral [over D] ( |f|^2 dz
dz* ) or
alternatively Integral [over D] ( |f|^2 dx dy ) exists and is
finite.
Take the L2 norm of f as the square root of this integral, and
call it
L2(f).
The region D is doubly-connected. There is a function F
conformally
mapping the region D into an annulus centered around the
origin. The
radii of this annulus are not specified -- it's explained that
because
this is a conformal map, their ratio is always the same and
F'/F
sensitive only to their ratio. (e.g. G = 2F, G'/G = F'/F, etc.)
Now take the subset J of H consisting of functions with the
following
property: the function integrated along any curve going around
the
hole in the positive direction has a residue of 1. i.e. it has
the
value i*2pi .
F'/F is certainly square-integrable. How can I show that
L2(F'/F) <=
L2(j) for any j in J?
I hope that cleared things up. Sorry if I was misstating the
problem.
-Greg
PS: Yes, I am using standard lebesgue measure here to
integrate, I
believe you can even use the regular Riemann integral since we
are
always integrating analytic functions, which uses a measure
which
mathworld calls with the funny name of Jordan measure.
Subject: Re: Strange Complex Variables Problem
>>Well, I'm talking about the L2 norm induced by the inner
product >= INTEGRAL(over D) [ f(z) conj[g(z)] ] dz , which is simply
L2(f) =
>>INTEGRAL(over D) [ |f(z)|^2 dz ].
>
>> This is (in my experience) a strange use of notation.
>I meant to indeed say INTEGRAL (over D) [ f(z) g(z)* ] dz dz*
>[ I meant to say Green's theorem to integrate around the
boundary but
>that's irrelevant. Let me try to start over because I don't
want to
>confuse anyone. ]
>I've got a hilbert space H of analytic functions defined on a
region
>D, square-integrable on D. i.e. Integral [over D] ( |f|^2 dz
dz* ) or
>alternatively Integral [over D] ( |f|^2 dx dy ) exists and is
finite.
>Take the L2 norm of f as the square root of this integral,
and call it
>L2(f).
>The region D is doubly-connected. There is a function F
conformally
>mapping the region D into an annulus centered around the
origin. The
>radii of this annulus are not specified -- it's explained
that because
>this is a conformal map, their ratio is always the same and
F'/F
>sensitive only to their ratio. (e.g. G = 2F, G'/G = F'/F,
etc.)
>Now take the subset J of H consisting of functions with the
following
>property: the function integrated along any curve going
around the
>hole in the positive direction has a residue of 1. i.e. it
has the
>value i*2pi .
>F'/F is certainly square-integrable.
No, it's not true that F'/F is certainly square-integrable!
I explained this a few days ago. Suppose that D is the unit
disk
minus the origin. Then D is certainly doubly-connected, and
your annulus is D itself; F(z) = z, and then F'/F is
_not_ square-integrable over D.
Answer me a question. Is this a problem like in a book
somewhere? I know that you know the answer to _that_
question... if it is a problem in the book could you tell
us _exactly_ how the problem reads?
>How can I show that L2(F'/F) <=
>L2(j) for any j in J?
>I hope that cleared things up. Sorry if I was misstating the
problem.
>-Greg
>PS: Yes, I am using standard lebesgue measure here to
integrate, I
>believe you can even use the regular Riemann integral since
we are
>always integrating analytic functions, which uses a measure
which
>mathworld calls with the funny name of Jordan measure.
************************
Subject: Re: all the roots of a nonlinear equation
> I realize this is probably a dumb question but I've looked
at numerical
> analysis books and racked my brain but I must be missing
something
obvious.
> I have equations which are sums of sines and cosines an
example being:
> 2 Cos[x] - Cos[2 x] - Sin[x] + 3 Sin[2 x] - Sin[3 x] = 0
> I need to find all the roots of this equation in the range
-pi to pi.
> The algorithms I've looked at only give me one root and if
based on a
> bisection technique dont always give an answer in the range
I mentioned
> above.
> I tried using NSolve in mathematica for this above equation
and it
does'nt
> return any roots (there are 4 when looking at the plot).
> I am interested in an algorithm that I could implement in C
/ Fortran for
> later use that would give me all the roots of the above
types of
functions
> automatically.
> Is this possible? Or have I missed something very obvious?
> Thanks,
Would the following work? Write cos(nx) as (t^n + t^-n)/2,
write
sin(nx) as (t^n - t^-n)/2i, clear denominators, and find all
of the
complex roots of the resulting polynomial in t. Then you know
sin(x)
and cos(x), so easy to find x.
So your sample equation
2 Cos[x] - Cos[2 x] - Sin[x] + 3 Sin[2 x] - Sin[3 x] = 0
becomes
(t+t^-1) - (t^2+t^-2)/2 - (t-t^-1)/2i + 3(t^2-t^-2)/2i -
(t^3-t^-3)/2i
= 0.
Multiplying ty 2it^3 gives
1 - 3t + (-1+2i)t^4 + (1+2i)t^2 + (3 - i)t^5 - t^6 = 0.
The roots of this (according to PARI) are
-1.09973 + 0.261803550*I
-0.025318021 - 2.89325*I
0.233951633 - 0.649323431*I
-0.040888745 - 1.44359*I
0.938942371 - 0.047583210*I
-0.006953501 + 1.63036*I
and the corresponding [cos(x),sin(x)] are
[0.321614201, 0.086177678]
[-0.945089837, -0.239647076]
[1.00643, -0.764022744]
[0.121776434, -0.952180855]
[2.55438, -0.121637430]
[-0.059117022, 0.991309694]
Joe Silverman
Subject: Re: all the roots of a nonlinear equation
JHS escribi.97 en el mensaje
>> I realize this is probably a dumb question but I've looked
at
>> numerical
>> analysis books and racked my brain but I must be missing
something
>> obvious.
>> I have equations which are sums of sines and cosines an
example
>> being:
>> 2 Cos[x] - Cos[2 x] - Sin[x] + 3 Sin[2 x] - Sin[3 x] = 0
>> I need to find all the roots of this equation in the range
-pi to pi.
>> The algorithms I've looked at only give me one root and if
based on a
>> bisection technique dont always give an answer in the range
I
>> mentioned
>> above.
>> I tried using NSolve in mathematica for this above equation
and it
>> does'nt
>> return any roots (there are 4 when looking at the plot).
>> I am interested in an algorithm that I could implement in C
/
>> Fortran for
>> later use that would give me all the roots of the above
types of
>> functions
>> automatically.
>> Is this possible? Or have I missed something very obvious?
>> Thanks,
> Would the following work? Write cos(nx) as (t^n + t^-n)/2,
write
> sin(nx) as (t^n - t^-n)/2i, clear denominators, and find all
of the
> complex roots of the resulting polynomial in t. Then you
know sin(x)
> and cos(x), so easy to find x.
> So your sample equation
> 2 Cos[x] - Cos[2 x] - Sin[x] + 3 Sin[2 x] - Sin[3 x] = 0
> becomes
> (t+t^-1) - (t^2+t^-2)/2 - (t-t^-1)/2i + 3(t^2-t^-2)/2i -
(t^3-t^-3)/2i
> = 0.
> Multiplying ty 2it^3 gives
> 1 - 3t + (-1+2i)t^4 + (1+2i)t^2 + (3 - i)t^5 - t^6 = 0.
I get
- t^6 + t^5(3 - i) + t^4(2i - 1) + t^2(2i + 1) - t(i + 3) + 1
= 0
> The roots of this (according to PARI) are
> -1.09973 + 0.261803550*I
> -0.025318021 - 2.89325*I
> 0.233951633 - 0.649323431*I
> -0.040888745 - 1.44359*I
> 0.938942371 - 0.047583210*I
> -0.006953501 + 1.63036*I
> and the corresponding [cos(x),sin(x)] are
> [0.321614201, 0.086177678]
> [-0.945089837, -0.239647076]
> [1.00643, -0.764022744]
> [0.121776434, -0.952180855]
> [2.55438, -0.121637430]
> [-0.059117022, 0.991309694]
> Joe Silverman
But two of that solutions aren't in the (real) interval (-pi,
pi] ...
--
Best regards,
Ignacio Larrosa Ca.96estro
A Coru.96a (Espa.96a)
ilarrosaQUITARMAYUSCULAS@mundo-r.com
Subject: Re: Are you able?
> A pproblem in http://santinho-de-pau-a-fala.blogspot.com/
> Manuel Sousa, Portugal
The formula W/10000 = (1-1/n)^n gives 3678,77601766571 for
n=100000.
Subject: Re: Is this
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CGkI102949;
The open path problem, in the case where all cities must be
visited,
is the Hamiltonian path problem, which is also NP-hard.
>> If you are required to return to the starting point,
>> then this is the Hamiltonian cycle problem,
>> which is knownto be NP-hard.
>No. The path is allowed to be open (and in the problem I'm
working on,
>it is always open except we have only one city)
>I din't even see a proof of NP-hardness for the open path
version of
>TSP. Here help is appreciated too.
>--
>Best regards,
>Alex.
>PS. To email me, remove loeschedies from the email address
given.
Subject: Re: Cantor's Diagonal Argument
In sci.logic, |-|erc
:
>> Similarly :
>> Prob(exists j, for i = 1..n, b_i = c(j)_i)=1 -> Prob(exists
j,
>> for i = 1..n+1, b_i = c(j)_i)=1
>>
>> By induction on n :
>> for all n, exists j, b_1 = c(j)_1, b_2 = c(j)_2, ... b_n =
c(j)_n
>>
>> This suggests NotEqual(b, c(j)) will not halt.
>> It won't halt anyway; that's part of the problem.
>> Assume one is testing all b's. A UTM that enumerates
>> all b's can be created (the problem then becomes
>> slightly recursive!) and one can then run NotEqual(b, c(j))
>> on all j, while running simultaneously the b-generator UTM
>> and doing the comparisons with oodles of threads.
>> A pretty and complex problem -- but this problem will
>> not halt regardless, as there are an infinite number
>> of b's to test.
> That's not a concern. The initial formulation of Cantor's
proof is that
it
> finds ONE b! We take that number and however many digits we
compare
> its computable. Its effective to demonstrate a proof that
shows that
> the 1 formulated number is on the list.
b is only computable if the list of numbers is computable,
which
one has to assume if only because the mapping c(j) is
specified.
> Or as you originally put it :
>>Your problem is slightly different; you need to show that
for *all* b
> which is actually where my proof begins, covering the
extended problem
> of multiple b's.
> For a decimal matrix there are 8 to 9 possibilites for each
digit of b,
> the number of b's is just 9^digits to test. Parallel testing
is no
problem,
> but not necessary, if the Cantonian formulation is invalid
for one of the
> contenders the others will likely fail too.
Go for it.
>> It won't halt anyway; that's part of the problem.
> This is only a problem for multiple c(j), we can consider b
a constant.
> //Globals
> b = Cantor(C, random)
> cj = FindMatch(C, b, 1)
> DiagNotEqual () {
> DigitsEqual (b, cj, 1)
> Return TRUE
> }
> DigitsEqual (DE1, DE2, digit) {
> if (DE1_digit = DE2_digit) {
> DigitsEqual (DE1, FindMatch(C, b, digit+1), digit+1)
> }
> }
> Here FindMatch will tally through computable numbers until
it finds
> a number that matches the initial digits of b.
> j is allowed change to a different number during testing.
This algorithm
> will still determine whether a list of numbers does not
contain a new
> test number. The algorithm will not halt on the list of
computables
against
> a new number however it is constructed.
> Unfortunately the algorithm is too giving, it would never
determine that
pi
> was off the list of rationals, it would keep constructing
larger fractions
to
> continue the comparison.
> To me its enough that any number off the list is nonsensical,
> it only works when that number is infinitely long, which
voids
> it from any mathematical construction.
It doesn't work anyway as an algorithm. It only works as a
semialgorithm; the routine halts and announces failure, but
goes into an infinite loop upon success.
But that's not a problem, mathematically. All you have to do
is prove that DiagNotEqual(b, c) never halts for a
Cantorian-constructed b.
> Herc
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
Subject: Re: Cantor's Diagonal Argument
oo
____|mn
/ /_/ / _
/ K-9/ /_/ - www.YeOldeCoffeeShoppe.com -
/____/_____
--------------
>> Prob(exists j, for i = 1..n, b_i = c(j)_i)=1 -> Prob(exists
j,
>> for i = 1..n+1, b_i = c(j)_i)=1
>>
>> By induction on n :
>> for all n, exists j, b_1 = c(j)_1, b_2 = c(j)_2, ... b_n =
c(j)_n
>>
>> This suggests NotEqual(b, c(j)) will not halt.
>
>> It won't halt anyway; that's part of the problem.
>
>> Assume one is testing all b's. A UTM that enumerates
>> all b's can be created (the problem then becomes
>> slightly recursive!) and one can then run NotEqual(b, c(j))
>> on all j, while running simultaneously the b-generator UTM
>> and doing the comparisons with oodles of threads.
>
>> A pretty and complex problem -- but this problem will
>> not halt regardless, as there are an infinite number
>> of b's to test.
>
> That's not a concern. The initial formulation of Cantor's
proof is that
it
> finds ONE b! We take that number and however many digits we
compare
> its computable. Its effective to demonstrate a proof that
shows that
> the 1 formulated number is on the list.
> b is only computable if the list of numbers is computable,
which
> one has to assume if only because the mapping c(j) is
specified.
Ouch, I thought I defined the mapping given the usage of a
UTM. See
It was a long procedure I outlined, basically to overcome non
halting
programs in the list of all functions' outputs. Say all
programs halt,
then
UTM(1) = some number, UTM(2) = some number, UTM(3) = some
number.
The set of numbers output from UTM applied to N in turn,
contains all
computable
numbers. All I did was list the ones that halt 1st, testing
digit by digit,
to guarantee
that a digit is present on the diagonal.
> Or as you originally put it :
>>Your problem is slightly different; you need to show that
for *all* b
> which is actually where my proof begins, covering the
extended problem
> of multiple b's.
> For a decimal matrix there are 8 to 9 possibilites for each
digit of b,
> the number of b's is just 9^digits to test. Parallel testing
is no
problem,
> but not necessary, if the Cantonian formulation is invalid
for one of
the
> contenders the others will likely fail too.
> Go for it.
Ouch!! Doesn't this mean b = c(j)? and is it correctly derived?
for all n, exists j, b_1 = c(j)_1, b_2 = c(j)_2, ... b_n =
c(j)_n
This is equivalent to : for all n, b_n = c(j)_n
>> It won't halt anyway; that's part of the problem.
> This is only a problem for multiple c(j), we can consider b
a constant.
> //Globals
> b = Cantor(C, random)
> cj = FindMatch(C, b, 1)
> DiagNotEqual () {
> DigitsEqual (b, cj, 1)
> Return TRUE
> }
> DigitsEqual (DE1, DE2, digit) {
> if (DE1_digit = DE2_digit) {
> DigitsEqual (DE1, FindMatch(C, b, digit+1), digit+1)
> }
> }
> Here FindMatch will tally through computable numbers until
it finds
> a number that matches the initial digits of b.
> j is allowed change to a different number during testing.
This
algorithm
> will still determine whether a list of numbers does not
contain a new
> test number. The algorithm will not halt on the list of
computables
against
> a new number however it is constructed.
> Unfortunately the algorithm is too giving, it would never
determine that
pi
> was off the list of rationals, it would keep constructing
larger
fractions to
> continue the comparison.
> To me its enough that any number off the list is nonsensical,
> it only works when that number is infinitely long, which
voids
> it from any mathematical construction.
> It doesn't work anyway as an algorithm. It only works as a
semialgorithm; the routine halts and announces failure, but
> goes into an infinite loop upon success.
sprung, I was lazy with the construction. FindMatch goes to the
next number if there is no match, fails the next digit
comparison, falls out
of
the loop as not equal. I should write a general function that
works
with any list and a number.
> But that's not a problem, mathematically. All you have to do
> is prove that DiagNotEqual(b, c) never halts for a
> Cantorian-constructed b.
I think I came close, but if it proves diag is computable it
also proves
pi is rational. I'd had to overhaul my assumptions at this
point, work
'can be calculated' into my defn of computable.
Herc
Subject: algebra: additive,abelian
I am learning about additive, abelian category. The book quote
an
example:
Let C be a catgeory such that the object of C are pairs (A',A)
where A is
an
abelian group and A' is a subgroup of A. Morphism from (A',A)
to (B',B) is
a group homomorphism f: A to B where f(A') is in B'.
The book claim that it is an additive category having kernel
and cokernel
of
each morphism but it is not abelian.
Why is it true? Can anyone tell me how to see it?
Subject: Re: algebra: additive,abelian
Adjunct Assistant Professor at the University of Montana.
> I am learning about additive, abelian category. The book
quote an
>example:
>Let C be a catgeory such that the object of C are pairs
(A',A) where A is
an
>abelian group and A' is a subgroup of A. Morphism from (A',A)
to (B',B)
is
>a group homomorphism f: A to B where f(A') is in B'.
>The book claim that it is an additive category having kernel
and cokernel
of
>each morphism but it is not abelian.
>Why is it true? Can anyone tell me how to see it?
To see it is additive, you need the homomorphism sets to be an
additive abelian group in which composition is bilinear, which
will be
inherited from the category of Abelian groups; and a zero
object, and a
biproduct. The biproduct is just the direct sum on each
coordinate:
(A',A) oplus (B',B) = (A'oplus B', A oplus B)
which can be seen to have the necessary universal properties.
The zero
object is the pair (0,0).
The kernels should be the usual kernels: the kernel of
f:(A',A)->(B',B) will be (ker(f) intersect A', ker (f)).
The real difficulty is the cokernels. One might be tempted to
define
coker(f:(A',A)->(B',B)) = (B'/f(A'), B/f(A))
but that definition does not work because in general B'/f(A')
is not a
subgroup (or isomorphic to a subgroup) of B/f(A). Rather, you
need to
take [B'+f(A')]/f(A) = B'/[f(A) intersect B'].
So we define;
coker(f:(A',A) -> (B',B) ) = ([B'+f(A')]/f(A) , B/f(A))
To see it is a cokernel, we need to show that it is the
coequalizer of
f and the zero map from (A',A) to (B',B). That is, we need to
show
that coker(f)f = 0 [which it is], and that if h:(B',B) ->
(C',C) has
the property that hf = 0, then h = h'coker(f) for some unique
arrow
h'. If hf = 0, then f(A) is contained in the kernel of h, so
the map
B->C factors uniquely through B/f(A), which gives you the
property.
The usual universal properties of the kernel and cokernel in
Ab will
give you the necessary properties here.
Now, an abelian category is an Ab-category which has a zero
object,
binary biproducts, every arrow has a kernel and a cokernel,
and every
monic is a kernel and every epi a cokernel.
So in this case, what you want to show is that not every monic
is a
kernel, or not every epi is a cokernel (since the category
satisfies
all the others).
The key is that the cokernel is not (B'/f(A'), B/A). This is
what is
going to fail. So we can find an epi which is not a cokernel,
say by
finding a map which is an epi from A to B, but whose
restriction from
A' to B' is not.
So we look for a map which is surjective A->B, but where f(A)
intersect B' is larger than f(A'); say, by letting B'=B, but A'
smaller than A which does not map onto B' and is not contained
in the
kernel of f.
Consider A' = Z/2Zx{0}, A= (Z/2Z)x(Z/4Z), B' = B = Z/4Z, and
the map
f(a,b) = 2a+b.
I claim this is an epi. For if g,h:(B',B)->(C',C) satisfy gf =
hf,
then g(1) = g(f(1))=h(f(1))=h(1), so g=h, showing that f is
right
cancellable.
But the thing is that f is not a cokernel.
For assume that there is a map g:(C',C) -> (A',A) so that f is
the
coequalizer of g and the zero map 0:(C',C)->(A',A). Then that
means
that gf=0. So g(C) is contained in {(0,0), (1,2)}. In either
case,
g(C') = 0.
g cannot be the zero map, because the coequalizer of the zero
map with
itself is the identity. So g must map C onto {(0,0), (1,2)}
and C' to
0. Let N = {(0,0),(1,2)}.
If h:(A',A) -> (D',D) is any map such that hg = 0, then it
should
factor through f. But take the map h:(A',A)->([A'+N]/N,A/N})
given by the
cacnonical projection. Then hg=0, so the map should factor
through
f. That is, there exists h':(B',B) -> ([A'+N]/N, A/N) so that
h = h'f.
Since B'=B, the image of h' must be contained in
[A'+N]/N = {(a,b): a in Z/2Z, b=0 or 2}
But what is h'(1)? h'(1) = h'(f(0,1)) = h(0,1) = (0, 1+N),
which is
not in [A'+N]/N. So the map cannot factor through f, so h
cannot be a
cokernel.
--
==============================================================
========
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
==============================================================
========
Arturo Magidin
magidin@math.berkeley.edu
Subject: sine series
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CHxs611733;
The series
sin x = x - (x^3)/3! + (x^5)/5! - ...
doesnot works when x is large, eg. x = 80
Is there any series which is generalized for computing sine
for any x?
Subject: Re: sine series
> The series
> sin x = x - (x^3)/3! + (x^5)/5! - ...
> doesnot works when x is large, eg. x = 80
> Is there any series which is generalized for computing sine
for any x?
It does work, but it will not begin to converge before the
number of terms
is much larger than x (before, successive terms increase...).
If x is very
large, you need to calculate a lot of terms before you get
convergence, and
you might get an overflow of roundoff errors.
To be able to effectively use this series, substract an
appropriate
multiple
of 2 PI from x so that y = x - 2 k PI is between -PI and PI
and calculate sin(x) using y in the formula above. (you can do
a bit better
by reducing the argument to -PI/2 PI/2 and using the symetry
of the sine
function around PI/2)
Note also that if you want to calculate an approximation to
the sine
function using a polynomial of fixed degree, the above formula
is not the
best approximation.
Francois
provided you first reduce x to a smaller number, :
noting that sin (x+2kPI) = sin(x) for any x and integer k,
substract a
multiple of 2kPI from x, so that
Subject: Re: sine series
> The series
> sin x = x - (x^3)/3! + (x^5)/5! - ...
> Note also that if you want to calculate an approximation to
the sine
> function using a polynomial of fixed degree, the above
formula is not the
> best approximation.
Is a Hermite interpolation the best way of implementing sine
on a
computer? I know the name, but haven't figured it out yet.
--
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: sine series
> The series
> sin x = x - (x^3)/3! + (x^5)/5! - ...
> doesnot works when x is large, eg. x = 80
Depends on what you mean by work. The radius of convergence is
infinite, and therefore the series is mathematically correct
for all real
x. However, machine computation with limited precision may
fail when x
is large.
> Is there any series which is generalized for computing sine
for any x?
You mean, you want one series that works for all x without
significant
roundoff error? Not likely.
Instead of the MacLaurin series, you could try using the
Taylor series
expansion about an arbitrary x0. In the case of the sin
function there
is a particularly simple series when x0 = 2*pi*n:
sin(x-2*pi*n) = (x-2*pi*n) - (x-2*pi*n)^3/3! + (x-2*pi*n)^5/5!
-
...
This works for values of x that are reasonably close to
2*pi*n. Of
course, it's effectively the same as just performing a
trigonometric
reduction on x and then applying the MacLaurin series to the
reduced
argument.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
>Here's a new game: player 1 picks a number that may or may
not
>>be transcendental, player two wins if he can prove whether
the
>>number is transcendental or not.
>>I think player 1 will win almost every time, but if player 2
wins,
>>maybe he'll get a Nobel prize!
>Only if he can publish the proof in an economics journal.
Oops! Forgot about Mr. Nobel's grudge against mathematicians.
George
>Robert Israel israel@math.ubc.ca
>Department of Mathematics http://www.math.ubc.ca/~israel
>University of British Columbia
>Vancouver, BC, Canada V6T 1Z2
Subject: Matrix norms.
Is it possible to define an unbounded norm on matrices?
Subject: Re: Matrix norms.
>Is it possible to define an unbounded norm on matrices?
You must not be asking exactly the question you meant
to ask: _every_ matrix norm is unbounded!
(Proof: If ||.|| is a matrix norm and A <> 0 then ||A|| > 0.
But ||cA|| = c||A|| for c > 0, and c||A|| is unbounded.)
Trying to guess maybe what you meant: any norm is
_finite_, by definition.
************************
Subject: Re: Matrix norms.
> Is it possible to define an unbounded norm on matrices?
Perhaps that I am missing something here, but the only vector
that I know in which you can define a bounded norm is {0}.
Best regards,
Jose Carlos Santos
Subject: Unit pricing: Open-ended mutual funds
The value of individual units in open-ended mutual funds /
unit trusts
are given by the formula:
unit value = value of investment fund / number of units in
issue to
clients
An adjustment is made for spread to determine the sale and
repurchase price of the units. The value of the units
increases when
the value of the fund increases (and visa versa) The aim of the
mutual fund company is therefore to increase the value of the
investment fund (portfolio) in order to achieve an exceptable
return
for investors. The emphasis is therefore on increasing the
numerator
in the equation.
Gain Theory suggests that aditional gain can be achieved by
also
limiting the denominator in the equation, i.e. the number of
units
that are issued to clients. This can be easily achieved by
capitilising the spread into the value of the investment fund.
For more info refer to:
http://www.verismall.com/software/other/unittrustpricingmodel/
unittrustprici
ngmodel.htm
Francois Terblanche
Verismall Software
www.verismall.com
Subject: Conencted Small Graphs
Originator: fab@soda.csua.berkeley.edu (Fabio Rojas)
Not sure if this was posted the first time...
Is there a reference that tells me the % of connected graphs
with 30 nodes
or
less? How about equivalence classes of graphs? Formula would
be nice, if
it exists.
The table would look like:
Nodes Percent Connected
3 50%
.....
30 .000003%
Feel free to email me or post a citation.
Fabio
Subject: Accounting of downloads
The second volume of Buckminster Fuller's books Synergetics was
published in 1979, twenty five years ago. All of humanity had
to learn
things like the synergetics coordinate system to do more with
less
soon or it would be curtains for humanity according to Bucky,
but look
at these low numbers. And life still goes on.
Cliff Nelson
Begin forwarded message:
Dear Clifford Nelson,
At the moment, I only have access to the top 150 downloads.
You have
two
notebooks in the top 150. (I only have one, myself).
--Ed Pegg Jr.
Could you give me an accounting of the number of downloads of
my
notebooks from MathSource please?
http://library.wolfram.com/infocenter/search/?search_results=1
;search_person
_id=607
? Buckminster Fuller Notebooks [in /MathSource: Packages and
Programs/]
#129 in all time downloads. In the past year, notebook
downloaded
1118 times
? Bucky Number Mandelbrot [in /MathSource: Packages and
Programs/]
#86 in all time downloads. In the past year, notebook
downloaded 1335
? Four Triangle Fractals using Bucky Numbers and Synergetics
Coordinates [in /MathSource: Packages and Programs/]
Not in top 150
? Solving Matrix Problems Using Bucky Numbers [in /MathSource:
Packages and Programs/]
Not in top 150
Cliff Nelson
Subject: Re: Catholic or Protestant background ?
charset=iso-8859-1
> Hi Ha Ha Hanson, You said that you're not Jewish.
> Do you come from a Catholic or a Protestant background ?
> Do you believe in a Benevolent Creator ?
Your need to know is truly touching, Professor Relf.
Therefore, we are supremely honored to fulfill your splendid
inquisitorial nosiness, in great detail, in the CV that
follows.
We all do hope that the answers provided therin will ameliorate
your anxieties that originate from your vacuous curiosity.
First of all, hanson@quick.net is a cyber construct, a
composite from dozens of contributors who post under this
handle, to this & other such cyber parties aka News-groups.
Most of the posts do originate after board/court/R&D/prod
biz meetings to **vent**, or out of **boredom** from up
at some 35-45kft during the daily 17 hrs flight durations, or
from-to some godforsaken locations, at/in/to anywhere, to
**reassure** our dudes/dudettes, etc, etc, & of cousre, etc....
hanson@quick.net should be viewed from the perspective that it
has a reality equivalent/content like the following cyber
entity, which
you should study intensively, in particular its activities at
Ravencrag :
http://www.marvunapp.com/Appendix/cobrajal.htm#Hanson
http://www.marvunapp.com/Appendix/cobrajal.htm#Ravencrag
Now, to the details of your requests, Professor Relf:
> Hi Ha Ha Hanson, You said that you're not Jewish.
There is a general uncertainty about that, and as general Jew
propaganda likes to have it, I have no disprove that some
rabbi,
at one time, did not beat off into the family tree of my
European
ancestors. Stranger things have happened in the days of yore,
whore, yondern that makes you wondern. --- Hallelujah, praise
the Load! --- To boot, I am still looking for a Jewish Ersatz
Grandpa. One like that would come in handy, if you wish to
make points with fanatical Jews. I scoured Yiddishe olde kacker
homes for an Ersatz Granpa, for weeks, but they all wanted
a big time fee for it, upfront, even before signing on, with no
guarantee for standing up for me when it would count, cyber or
otherwise. So, go figure, as they say, ...and oy vey......
> Do you come from a Catholic or a Protestant background ?
They used to send me to some ing Sunday Shul, were
the old, holy looking broad was distributing pretty postage
stamps with Jesus+halo on it so that we kiddies would keep
quiet. I have no idea what this crock was all about.
Then, during my 1st high intensity testosterone season, the
religo/spiritual establishment enslaved me again into another
gig of theirs, attempting boring protesta/barmitz/commun type
brainwashing, which was so full of transparent lies and crock
that was read out of an old fat book and a roller that looked
like a 17th century toilet paper roll. .....And then these
wailing,
ing songs!........Jesus-H-ing Kriste ...........depressing.
If it would have been the Beatles or Elvis....well........
So anyway, there was one good thing about all this, ....the
wife
of the preacher was some good looking, pretty, beautiful
ass- and boob wiggling bitch. Every time she looked at me I got
an instant hard on. And she knew it and moaned while passing
by me. ...But the day of rapture did come! The Load was smiling
upon me then. Passion overcame beautiful Mrs. Preacher and
she grabbed & pushed me onto the altar and gave me one of
the great and memorable rides......Just as she hollered and
her juices ran down over my crotch the preacherman came in
and asked what we where doing.....---ex-communication, etc..
Went to a bunch of schools, the military, behaved, (well,
maybe)
till, still drunk, right after the doctoration ceremonies, my
ex-high
school cronies at the bar told me that I would never catch up
with
their earnings until I would be 55, if lucky....... Having
just gone
thru 4 years of penniless academic slavery, the realization of
this
negative $$-aspect-future made me so furiously mad that marched
right away over to my now ex-professor/advisor's office where I
placed my cum laude diploma carefully into my square tassel
hat, took
a onto it and poured a half a quart of good vodka over it.
Threw
a match at it and watched the beautiful faint blue flame evolve
into a yellow sodium fire that began to stink more & more.
Just as I was celebrating my achievement professor schmuckface,
as I called him, came waltzing in and asked me what I was
doing.....
deja vue........---ex-communication, etc....... left town in a
jiffy.
Never looked back .......went to see the
world.......ahahahahaha.......
So, as you can see, professor Relf, they did drag you to
Brazil to
proselytize FOR somebody.....I, OTOH, dragged my own ass also
to
far away stray corners of the globe, BUT I proselytized for
myself
and for my own cause......biz........
Moral of the story, both of us had a very banal youth, didn't
we.
Such then, this question of yours did not furnish you anything
new.
> Do you believe in a Benevolent Creator ?
Why, Professor Relf? Can't you, you of all people, can't you
see that I am one of them. -- May I administer to you, my son?
What are your spiritual needs, today, son Jeff? However, you
must know that there are my personal permit charges and user
fees attached to such services. Payable up front. Like the
govt's.
envrionmental tax levies.....the creation is green, after
all....
However, on a deeper, more useless level I can answer your
question for free, but with equal benevolence:
Everything, all actions or any person that bestows benevolence
upon you is a Benevolent Creator. You know, Professor Relf,
there
are some very nice people and many good opportunities around
that are benevolent, even if they are not Jewish.
Or even more general, a case can be made that everything which
can kick your ass, take away your control, determine your
behavior
etc. is a benevolent creatoration. With benevolent being a term
that is higlhy relative when seen from and in different
frames.......
However, if the Benevolent Creator must take on some mystical
dimension for your existential salvation validification then I
do
recommend for you to say:
Hallelujah, Praise the Load!, Professor Relf
ahahahahaha.........AHAHahahhahaha...... ahahahahanson
Subject: Harvard.EDU .
Hi Ha Ha Hanson, You called me Professor Relf .
Actually, I'm the president of Harvard University ...
http://www.president.Harvard.EDU/
. . . but I'll let that slide.
You said,
With ' benevolent ' being a term that is highly relative
when seen from and in different frames . . . .
Ah, so you are a theoretical physicist then.
Stevie Hawking perhaps ?
Subject: Re: Three positive integer triples with equal sums
and products
> I am looking at directions for solving the following problem,
> Given a set of three strictly positive integers {a,b,c}
(a>0,b>0,c>0)
> Under which conditions can I find two other sets of three
positive
integers
> {d,e,f}and {g,h,i}, with the same sum and product
> ie
> a+b+c=d+e+f=g+h+i
> abc=def=ghi
> with sets {a,b,c} {d,e,f} and {g,h,i} all different (not
permutations of
the
> same three numbers)
> (this could be generalised to 4, 5 ... triples, ad nauseam)
> I have been working on this for a while, and could prove a
few basic
> theorems on these sets (eg we cannot have a=d, neither can
we have a=b=1,
> nor a=b=c, etc...), but could not find a constructive way of
building
either
> all or some of the solutions to this problem.
Do you allow duplicates within a triplet as long as the
triplets
themselves are unique? The following three triplets have a sum
of
49 and product of 3600
12 25 12
10 15 24
9 20 20
Or must all 9 numbers be unique, as in the following sum of 49,
product of 3024?
9 12 28
8 14 27
7 18 24
> Does anyone have an idea on how to deal with this?
I generated mine using an Access database, but that's probably
not
what you want.
> In case someone is interested, it came as a math puzzle,
about three
> children, receiving three presents in three different boxes,
all of which
> had ineger dimensions, and noticing that the three boxes had
the same
> volume, and that the pieces of string packing them had the
same
lengths...)
> Thanks in advance
> Francois
Subject: Re: Three positive integer triples with equal sums
and products
> Do you allow duplicates within a triplet as long as the
triplets
> themselves are unique? The following three triplets have a
sum of
> 49 and product of 3600
> 12 25 12
> 10 15 24
> 9 20 20
yes, so far the three triplets are different, it is ok. It is
possible to
prove, though
- that there cannot be a solution with the same integer in two
different
triplets (else, the conditions would become a+b=d+e and ab=de,
which mean
the two triplets are equal)
- that in no solution can a triple be a cube (a a a)
- that although you can have one or two triplets with two
duplicate
integers, as in your example, you cannot have three triplets
with two
duplicate integers in one solution
Francois
Subject: Re: Three positive integer triples with equal sums
and products
> I am looking at directions for solving the following problem,
> Given a set of three strictly positive integers {a,b,c}
(a>0,b>0,c>0)
> Under which conditions can I find two other sets of three
positive
integers
> {d,e,f}and {g,h,i}, with the same sum and product
> ie
> a+b+c=d+e+f=g+h+i
> abc=def=ghi
> with sets {a,b,c} {d,e,f} and {g,h,i} all different (not
permutations of
the
> same three numbers)
> (this could be generalised to 4, 5 ... triples, ad nauseam)
> I have been working on this for a while, and could prove a
few basic
> theorems on these sets (eg we cannot have a=d, neither can
we have a=b=1,
> nor a=b=c, etc...), but could not find a constructive way of
building
either
> all or some of the solutions to this problem.
> Does anyone have an idea on how to deal with this?
This is discussed in problem D16 in Guy, Unsolved Problems in
Number
Theory. The 3rd edition is in press. The 2nd edition says
Schinzel has
proved that you can have as many triples as you like of
positive
integers with the same sum and the same product, but it
doesn't give
a citation. It does give some other information that might
interest you,
and references to Problem E2872 of the American Math Monthly,
89 (1982)
499-500 and 88 (1981) 148.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
Subject: Re: Sound levels
charset=iso-8859-1
> It seems to me that if there are x machines each with a
sound level
> of L(1) then the level of x machines should be
> L(x)=x{10log[I(1)/I(0)]}
> rather than what was given to me as above.
> I don't even know if I'm missing the concept or the maths!
The concept. A level *is* the logarithm of a ratio of
whatever. If
multiply something with 10, you could as well call it a 10 dB
increase,
if you multiply something with 100, you call it a 20 dB
increase,
multiplying with 1000 gives 30 dB and so on.
And when something is twice as much than something else, you
could say
it's got 3 dB more, because
10 lg 2 = 3 (well, more or less)
> Please explain as if to a 5 year old *smiles* ... this one
has really
> got me confused!
I like this. :-)
You can now inmprss your friends in the kindergarten by
telling them you
have 3 dB more marbles than they have.
Best regards
Steffen
Subject: Re: Prime factors of number near googolplexplex
> Phil,
> near googolplex ( http://www.alpertron.com.ar/GOOGOL.HTM ).
This
> thread is about the new page about factors of number near
> googolplexplex ( http://www.alpertron.com.ar/GOOGOLP.HTM ).
Of course,
> prime factors of googolplexplex-1, -10 or +10 are very easy
to find
> using the methods you mention above.
Aha, my window was too narrow!EItherthat ore I thought there
was an echoecho.
Just out of curiosity, I started generating factors of
10^10^10^100-1,
and there aren't as many as I'd thought.
The biggest so far is:
2900240811398724360969936242327094078063964843751
I'm generating them at about 5 seconds per prime, but that will
slow down as they get larger. Shall I just mail you the whole
log
in a day, or do you want me to use the form on the web-page?
Oooh:
27985968577282083202817375422455370426177978515625000001
To avoid anyone else duplicating my search space, it's:
(10k+{1,3,7,9}) * 2^a * 5^b + 1 k=0..99; a+b<1000)
10^10^10^100-100 looks like a more interesting beast so I'll
look at that next. However, 10^10^10^100-10 looks like it's a
complete orgy of factors, and you may as well generate those
locally, or your inbox will fill quicker than a quickly-filling
thing. 10^10^10^100+100 looks interesting too.
10^10^10^100+1 looks, ahem, either interesting or incredibly
boring. I won't be looking there.
Ooooh:
410207670714926138814098853947048839430067346256691962480545043
94531251
Phil
393168275162152283817205516936940656869126087258337065577507019
04296875001
--
1st bug in MS win2k source code found after 20 minutes:
scanline.cpp
2nd and 3rd bug found after 10 more minutes: gethost.c
Both non-exploitable. (The 2nd/3rd ones might be, depending on
the CRTL)
Subject: Re: ZZCrap
In sci.math, ZZBunker
:
>> In sci.math, ZZBunker
>> > :
>>> In sci.math, ZZBunker
>>> >> :
>>>
>>
>>>>to isn't a verb.
>>>>
>>>> The last time I heard that to wasn't an intransitive
>>>> verb was the last time that math wasn't spelled
>>>> in the moronic fashion maths.
>>>
>>>> What's math?
>>>
>>> Nobody knows, Since like we've been telling morons
>>> for several thousand years, now:
>>> Maths is philosophy not science.
>>> See Plato for philosophy and festive pillar work.
>>
>>> Math is an abstract model, nothing more. Occasionally it's
>>> useful; we've even harnessed the Queen of Uselessness,
>>> number theory, into modern cryptography. :-) Who says
>>> primes aren't useful? :-)
>>
>>> However, '1' cannot be captured in a butterfly net. 'pi',
>>> 'e', and i=sqrt(-1) are even more elusive. Good luck
>>> finding a Taylor or MacLaurin series running around in the
wild.
>>
>> Nobody ever said that you can find *anything* related
>> to moronic *Calculus* running around in the wild.
>> We have merely stated that you can find *real*
>> *Turing machines* running around in the wild.
>
>> Models only, and imperfect ones at that. The ideal Turing
>> machine has an infinite length tape (most of which is
blank).
>> All modern machines have finite tapes -- if they have tapes
>> at all.
> The ideal Turing machine never needed an infinite
> tape to begin with. Since neither Turing nor any
> other mathematician before him proved anything
> conclusive about the word finite, other than
> their token prayers to Euclid.
True; running a Turing machine with an infinite empty tape
would be slightly pointless, even were the machine the
simple one:
S=0: S <= 0, '0', Right
S=0: S <= 1, '0', Right
S=1: S <= 0, '1', Right
However, from a theoretical standpoint one has to specify
something along the lines of has a tape longer than needed.
As for finite: everything's finite. Even the symbol '+oo'
is a token of the concept we think of as infinity; it is
not, however, infinite in itself.
(The only infinite thing is God, if such exists at all, and
even that's highly debatable -- but probably not here. :-) )
>>
>>
>>> There are examples of Nature taking advantage of Fibonacchi
>>> numbers (e.g., sunflower seed spirals, plant stalk
branchings) but
>>> that doesn't mean Fibonacchi numbers are out there.
>>
>>> The main requirement for math is that it be
self-consistent.
>>
>> The *only* requirement for math is that it
>> exist in a non-existent vaccuum.
>> Other than that it's the most trivial
>> thing in the universe.
>
>> Math does not exist except as a series of thought patterns
>> and as ink on paper and bits on storage devices and
traveling
>> through cables and air -- and those are also imperfect
models.
> Since models of math are also math,
> we have never claimed that cables
> are even approximately anything
> other than exactly what they are,
> which is cables. But when you
> get somebody other than a
> retarded mathematican using cables, you also
> get a free subscription to
> GPS TV, with the cable. So
> again whatever math dorks
> consider perfect is up to
> math dorks. The rest of us are
> still going to Mars. No
> matter what Feynmann & heads Inc. say.
Cables can be (at least) the following:
[1] Information conduits.
[2] Power conduits.
[3] Garrotes.
[4] Structural supports (e.g., suspension bridges)
[5] Directional indicators (i.e., paths)
Of course we are going to Mars. The question is when.
(I'm not familiar with GPS TV or math dorks. Are these
related to the talk.origins howler monkeys? :-) )
>> In short, there's no there there. It is merely a way of
>> coloring one's perceptions of the Universe; without math,
>> we'd look at the Universe differently -- perhaps using
>> mysticism? -- and draw conclusions in a different fashion.
> Well you've obviously never heard of Descartes.
> He said the same thing and nobody believed him.
> Since his use of the principle of extension,
> was second only to Zeno's in being the
> most overrated piece of philosophy ever written.
That it is. I prefer, as Leonard Nimoy once said,
the concrete, the graspable, the provable.
I think, but amness (to coin a word) may not depend thereon,
although on Usenet, one has to think to write, although the
amount of thinking can be miniscule; I could try to be highly
intelligent and string words into elegant sentence structure,
or just exercise the minimum amount of thought necessary to
hold down the 's' key and generate a string of
sssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss.
..
until I decide to release the muscles holding down that key.
(Most computers/computer keyboards have auto-repeat.)
Infinity is not provable unless carefully specified. At best,
it's mathematical sophistry for the statement
for any integer N, there's always a bigger integer N+1.
(which is a variant of weak induction).
At worst, it's lazy thinking.
There are admittedly some interesting questions, though; can
one create/specify a Turing machine that, given an input tape
and an alphabet {0,1,2,3,4,5,6,7,8,9,.,+,-}, with a valid
number (encoded in the usual way), generate output with
{0,1,2,3,4,5,6,7,8,9}, and for every unique input the
output is valid (either '0' or begins with a nonzero) and
unique?
And is there another turing machine that reverses the process,
and for every unique input the output is valid
(begins with an optional sign, contains exactly one '.',
and there's either a single '0', nothing, or a string
of digits starting with nonzero before the '.') and unique?
The existence of such machines would prove that the map
between N and R would (a) exist, and (b) be computable.
Most mathematicians would probably be of the opinion that such
machines do not exist (see, for example,
http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_
proof )
and might characterize the problem as the infinities are
different.
--
#191, ewill3@earthlink.net
It's still legal to go .sigless.
Subject: Re: SCHOENFELDS RANDOM THEOREM
every possible finite sequence of digits occurs in pi (and
most other
> irrational numbers), many times,
Does say 0320495307103004298010 qualify as a sequence of 8
'zeroes' or
must the 00000000's be consecutive?
regards bill j.
Subject: Re: SCHOENFELDS RANDOM THEOREM
<404fb4fd$0$8358$afc38c87@news.optusnet.com.au
<7DO3c.71$2f4.61910@news.uswest.net
>> Any finite sequence of digits of pi is truly random with a
uniform
>>> distribution.
>>> Proof? URL? Anything?
>>http://mathworld.wolfram.com/PiDigits.html
>> Please quote the relevant part of that page.
> Is your web browser broken?
Is the copy-n-paste features of your browser/newsclient/window
manager/whatever broken?
You said that page proves your claim. Richard doesn't see how
anything on that page does so[1]. Is it so terribly hard to
point out
which part of that text proves that Any finite sequence of
digits of
pi is truly random with a uniform distribution?
Do you think you've proved the point?
Footnotes:
[1] Beats me if Richard is right or wrong.
--
It's your choice though, if you do not believe in mathematics,
in the
importance of its healthiness and correctness, then you can
just walk
away now. -- James S Harris, on the Pythagorean Oath
Subject: Re: SCHOENFELDS RANDOM THEOREM
>>>> Any finite sequence of digits of pi is truly random with
a uniform
>>>> distribution.
>>>> Proof? URL? Anything?
>>>http://mathworld.wolfram.com/PiDigits.html
>>> Please quote the relevant part of that page.
>> Is your web browser broken?
> Is the copy-n-paste features of your
browser/newsclient/window
> manager/whatever broken?
> You said that page proves your claim. Richard doesn't see how
> anything on that page does so[1]. Is it so terribly hard to
point out
> which part of that text proves that Any finite sequence of
digits of
> pi is truly random with a uniform distribution?
> Do you think you've proved the point?
What I see on the page is:
It is not known if pi is normal (Wagon 1985, Bailey and
Crandall
2001),
Paraphrasing the question by asking if any finite sequence of
digits of
pi [is] truly random with a uniform distribution misses the
point.
Any finite sequence of digits of pi has a degenerate
distribution with
exactly one possibility.
The first 5 digits are 31415 with probability 1.
The 6th through 10th digits are 92653 with probability 1.
The 11th through 15th are 58979 with probability 1.
And so on.
No distributions are involved unless you count these degenerate
distributions.
Using the term random seems to miss the point as well. It isn't
relevant whether pi is random. What we're talking about is
the limiting distribution of values of m digit contiguous
sequences
selected uniformly from the n-m possible starting points
within the
first n digits of pi as n increases without bound.
This limiting distribution (if it exists) will be a discrete
distribution. It may be uniform. Or it may be non-uniform. But
it will not be random.
John Briggs
Subject: Re: SCHOENFELDS RANDOM THEOREM
>Note that this is not a cop out as there should exist a way
>>to prove this question (Unlike proving or disproving god).
> Why should there exist a way to prove it? It might be
undecidable.
Because if it's undecidable, then this world is not the best
of all
possible worlds, which contradicts that God is omnipotent and
all-good.
Duh.
--
Jesse Hughes
Depression hits more people than thought.
--headline in Lexington, KY newspaper, as reported on
NPR's Morning Edition
Subject: Re: SCHOENFELDS RANDOM THEOREM
Originator: richard@cogsci.ed.ac.uk (Richard Tobin)
>> Why should there exist a way to prove it? It might be
undecidable.
>Because if it's undecidable, then this world is not the best
of all
>possible worlds, which contradicts that God is omnipotent and
>all-good.
I'll apply for research funds from the Vatican immediately.
-- Richard
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: fNa4c-6550-45-16359@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: 8edfdc81 6d0270bf 6c788614 5ed82e5e 547bc3da
>I'm posting because I'd hate to see a misunderstanding
between two
>decent people escalate into something uglier than it need be
--
>
>[big snip]
>> There is nothing wrong with believing in what you're trying
to prove
>> as long as you don't confuse your belief for a proof. It is
a fine
>> line here, since in science
>[snip comment by GR_Learner, as I want to focus on what Mati
said]
>> 1) Saying I believe this is true is a standard operating
procedure.
>> 2) Saying It is true since I believe in it marks you as a
cracpot.
>
>> You obviously did not interpret what I said correctly.
>And I think you misread Mati. It's quite clear he was not
trying to
>attack *you* personally, at least not here; his use of you
here is
>idiomatic English, meaning one. The context is very general;
he's
>trying to characterize *science*, not you.
I though it is quite clear that this is the case.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: fNa4c-6550-45-16359@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: 0a19bb5a bcd07a8f 73384e51 fcf40a5b d1dc2a7d
>>
>>>
>GR_Learner@GR.grv
>>>>
>>>>>
>>>>>
>>>>> >> Any finite sequence of digits of pi is truly random
with
>a
>>>>uniform
>>>>> >> distribution.
>>>>>
>>>>> >> Proof? URL? Anything?
>>>>>
>>>>> >http://mathworld.wolfram.com/PiDigits.html
>>>>>
>>>>> > Please quote the relevant part of that page.
>>>>>
>>>>>
>>>>>
>>>>> Quoting from the page GR_Learner@GR.grv referred you to
>>>>> http://mathworld.wolfram.com/PiDigits.html
>>>>>
>>>>> The following distribution of decimal digits d is found
for
>the
>>>first
>>>significant
>>>>> departure from a uniform distribution.
>>>>
>>>>> 10^n is not a statement about the entire series.
>>>>
>>>>What part of finite sequence of digits of pi don't you
>understand?
>>>>The first 10n is a finite sequence.
>>>>
>>>> There is *no* proof present in the above for statement
made. A
>>>> specific sequence is *not* any specific sequence. End of
story.
>>>
>>>So... do you think that the first 10^n decimals are special
in that
>they
>>>show no significant departure from a uniform distribution?
>>
>>> The answer is, you don't know.
>>
>>That is true of all things. However, Occam's razor will
allow me to
>place a
>>bet with confidence. If Kip can bet a bottle of wine on
Black Holes, I
>>could do the same for this.
>> You can place bets with whatever confidence you wish but a
bet is not
>> a proof either.
>Prove to me it is not.
:-)))
>>> The sequence a_n = exp(kn)/n decreases for n < (1/k), then
it starts
>>> increasing and tends to infinity. picking k of, say,
10^(-1000),
>>> you'll be going a very long way seeing it decreasing,
before it
starts
>>> increasing. After the first 10 billion or so terms, you'll
sure be
>>> ready to state that, this one tends to zero. But, it ain't.
>>
>>> A finite number of cases does not make for a proof, in
math. It may
>>> inspire a quest for a proof, but that's all.
>>
>>This is a statment of the statistical measruments of the
randomness of
>Pi.
>>http://www.super-computing.org/pi-decimal_current.html
>>
>> What part of the word proof you don't understand?
>Let me ask you somthing... do you apply the same standard or
definition of
>proof in physics as you do in math?
No. You don't prove things in physics, you only confirm the
validity
of models within a given range of paramters and to given
tolerances.
> Does either have more merit in describing reality?
Math, per se, does not describe reality. it is a set of logical
models. You can apply mappings between some of these models
and some
observable aspects of reality but then the correctness of the
mappings
is a separate issue from the correctness of the math theorems.
>Note that in physics, a proof of the validity of a model as
it applies to
>reality can never be found.
Indeed.
>Even if there is a absolute proof in certain aspects of
mathematics (and
>this is not the case for all aspects as you know) that does
not mean it
has
>any validity for reality.
Just what I said.
>As a physicist, there is a certain point where with enough
data I would
say
>that the theory is proved
You were doing well so far, now you blew it again. No, there
is no
point where a physical theory is proved, nor can there be such
point.
Mind you, Newtonian mechanics passed with flying colors all
possible
tests for upward of 200 years. Until, the point has been
reached that
it didn't.
>... make use of the results... and move on (Until
>there is evidence that the theory does not apply).
Meaning, it wasn't proved. I'm afraid you just don't
understand what
proved means.
> The data is beyond convincing in the case of pi.
Two points:
1) There are known cases of mathematical properties which only
become
true for truly astronomic numbers. So your convincing is
meaningless.
2) More important, we're not debating here , one opinion
against
another. There are acknowledged standards of what is and what
isn't a
mathematical proof. You're not being asked for your opinion of
what
they should be. You're just being told what they're. Now, you
made
a specific statement and provided as a proof something which is
anything but. I suggest you acknowledge this and move on. I've
neither the time, nor the inclination for Dionysian debates.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: fNa4c-6550-45-16359@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: f5eca798 dfc04500 7856c7db 9298e031 6ebca380
>> The answer is, you don't know.
>
>> The sequence a_n = exp(kn)/n decreases for n < (1/k), then
it starts
>> increasing and tends to infinity. picking k of, say,
10^(-1000),
>> you'll be going a very long way seeing it decreasing,
before it starts
>> increasing. After the first 10 billion or so terms, you'll
sure be
>> ready to state that, this one tends to zero. But, it ain't.
>
>> A finite number of cases does not make for a proof, in
math. It may
>> inspire a quest for a proof, but that's all.
>All of which brings to mind an old joke:
>Physicist's induction: 3 is prime, 5 is prime, 7 is prime;
therefore all
>odd numbers are prime
>Engineer's induction: 3 is prime; therefore all odd numbers
are prime
>Chemist's induction: 3 is prime, 5 is prime, 7 is prime, 9 is
prime, 11 is
>prime; therefore all odd numbers are prime
:-))) And oldone but still a good one. I recall that there was
a
variation including a computer scientist, as well, but I
forgot the
details.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: fNa4c-6550-45-16359@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: bee46d54 1b29faca a729adfd 9585abe7 75bfbbbf
>[...]
>> It may strongly suggest so, but this still is not a proof.
>>You might say show me the proof and I can with equal
validity say
show me
>>the disproof.
>He made the affirmative statement, ________ is... Burden of
proof is
his.
>Your only affirmative statement was that he had not proven
his. You met
your
>burden of proof by demonstrating his data was not logical
evidence for his
>unqualified claim.
Yes, for sure. I think that he may simply not be aware that
math uses
a precise language in which .. it is ... means much more than
just
I think it is... or I believe it is....
>> Sorry, that's not how math works.
>Not how the logic of verbal statements works, either.
>Sorry about horning in on your thread, but the poster seems
to be avoiding
>my replies. I wonder why. :-)
I wonder too:-) Still, I've seen many cases in which the order
in
which posts appear on one server differ (quite significantly,
at
times) from they time ordering on another. So, give it some
more
time.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: Nso4c-8668-45-22794@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: f198cb1e e31ccbe6 90d0a26b 0ace8a76 00b4d845
>>
>>[...]
>>
>>> It may strongly suggest so, but this still is not a proof.
>>
>>>You might say show me the proof and I can with equal
validity
say
>show me
>>>the disproof.
>>
>>He made the affirmative statement, ________ is... Burden of
proof
is
>his.
>>
>>Your only affirmative statement was that he had not proven
his. You met
>your
>>burden of proof by demonstrating his data was not logical
evidence for
>his
>>unqualified claim.
>>
>> Yes, for sure. I think that he may simply not be aware that
math uses
>> a precise language in which .. it is ... means much more
than just
>I think it is... or I believe it is....
>>
>>> Sorry, that's not how math works.
>>
>>Not how the logic of verbal statements works, either.
>>
>>Sorry about horning in on your thread, but the poster seems
to be
>avoiding
>>my replies. I wonder why. :-)
>>
>> I wonder too:-) Still, I've seen many cases in which the
order in
>> which posts appear on one server differ (quite
significantly, at
>> times) from they time ordering on another. So, give it some
more
>> time.
>I killfiled Servo if that is what you are wondering.
There is no accounting for taste.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
[...]
>>Sorry about horning in on your thread, but the poster seems
to be
>>avoiding my replies. I wonder why. :-)
>>
>> I wonder too:-) Still, I've seen many cases in which the
order in
>> which posts appear on one server differ (quite
significantly, at
>> times) from they time ordering on another. So, give it some
more
>> time.
>I killfiled Servo if that is what you are wondering.
> There is no accounting for taste.
True. GR_Learner is the one who responded to Evil Al's 100
lines
ROTFLMAO!
Imagine the fun he'll have when /Mein Kampf/ is published in
comic
book form.
Being killfiled by someone who finds the diseased sewage of
Carbuncle
Al funny is an honor. It saddens me only insofar as there is a
slight
chance of redemption for GR_Learner given sufficient
education, against
which he innoculates himself by excommunicating the opposition.
(That last sentence reminds me of a problem I have with usenet
and
gender-specific pronouns. GR_Learner could be a female, for
all I know.
And, so could Mati. Or Servo. If so, my apologies to anyone
who may
be sensitive to being mislabeled while still preferring gender
anonymity.
English can be most annoying in its deficiencies. It is not
widely
regarded as a polite substitute.
I deal with this regularly in my writing, but will be damned
before
I rethink every sentence of usenet postings. Too much like
work.)
Servo
Subject: Re: SCHOENFELDS RANDOM THEOREM
>[...]
>> It may strongly suggest so, but this still is not a proof.
>
>>You might say show me the proof and I can with equal
validity say
show me
>>the disproof.
>He made the affirmative statement, ________ is... Burden of
proof is
his.
>Your only affirmative statement was that he had not proven
his. You met
your
>burden of proof by demonstrating his data was not logical
evidence for
his
>unqualified claim.
> Yes, for sure. I think that he may simply not be aware that
math uses
> a precise language in which .. it is ... means much more
than just
I think it is... or I believe it is....
>> Sorry, that's not how math works.
>Not how the logic of verbal statements works, either.
>Sorry about horning in on your thread, but the poster seems
to be
avoiding
>my replies. I wonder why. :-)
> I wonder too:-) Still, I've seen many cases in which the
order in
> which posts appear on one server differ (quite
significantly, at
> times) from they time ordering on another. So, give it some
more
> time.
I killfiled Servo if that is what you are wondering.
Subject: Re: SCHOENFELDS RANDOM THEOREM
[...]
> I wonder too:-) Still, I've seen many cases in which the
order in
> which posts appear on one server differ (quite
significantly, at
> times) from they time ordering on another. So, give it some
more
> time.
> I killfiled Servo if that is what you are wondering.
Oh, THAT'S right, this was the little fellow who was lecturing
me
on just *PLONK*ing Evil Al if I don't like the bile he spews.
His loss.
Servo
Subject: Re: SCHOENFELDS RANDOM THEOREM
[...]
> You can place bets with whatever confidence you wish but a
bet is not
> a proof either.
> Prove to me it is not.
You're becoming incoherent. Do you wish Mati to prove that a
bet is not a
proof, of that some series of pi is not truly random?
[...]
>This is a statment of the statistical measruments of the
randomness of
>Pi.
>http://www.super-computing.org/pi-decimal_current.html
>
> What part of the word proof you don't understand?
> Let me ask you somthing... do you apply the same standard or
definition
of
> proof in physics as you do in math?
No.
> Does either have more merit in describing reality?
Mathematics is not concerned with physical realities.
Mathematics is a
broad-
ranging field of study in which the properties and
interactions of
idealized
objects are examined.
http://mathworld.wolfram.com/Mathematics.html
> Even if there is a absolute proof in certain aspects of
mathematics (and
> this is not the case for all aspects as you know) that does
not mean it
has
> any validity for reality.
It does for mathematical reality.
[Snip rehash of empiricsm...]
> The data is beyond convincing in the case of pi.
Different folks and applications have different requirements
for what might
be convincing. In US criminal law it's reasonable doubt, while
in
civil
law it's preponderance.
You may be convinced by the preponderance of evidence, but you
are still
offering an UNPROVEN assertion if you say Any finite sequence
of digits
of pi is truly random with a uniform distribution.
You might decide from a preponderance of evidence in civil
court that eight
accountants are guilty of embezzlement, but would be
committing the fallacy
of unwarranted generalization if you said that 'Any accountant
is a crook.'
Physics is based upon a preponderance of evidence.
Mathematics draws conclusions which are usually beyond a
reasonable doubt.
> Now a bit of a digression to ask you a question... feel free
to bring
back
> on topic if you wish.
> I am not certain that there is any proof of randomness for
any finite
> sequence of numbers. Is this true?
> Also, I am not certain if there is any test that can be done
to determin
if
> a finite sequence of number generated by any mathematical
procedure or
> algorythem is random.
We've been beating this to death in other threads. Definition
matters.
> In fact, I have read that true randomness is can only be
properly defined
> and described by the actiions of natural systems, such as
radioactive
decay,
> and that there is not mathematical procedure to generate a
truley random
> sequence.
Google on chaos theory, BBP algorith and the pi series. That
nature thing
is
no foregone conclusion, either.
> Now, if this is true, then perhaps the definition of proof
for the
> randomness of pi's digits should be of the emperical type,
not the
> mathematical?
No problem, if you say Empirical evidence suggests that 'any
finite sequence of digits of pi is truly random with a uniform
distribution,' with the clear understanding that any
counterexample
would falsify that belief. That's empirical science.
Still does not make the unqualified Any finite sequence of
digits
of pi is truly random with a uniform distribution a legitimate
assertion, however.
> You obviously did not interpret what I said correctly. The
fault for
> that is my fault.
No, he interpreted the word any with complete accuracy. True,
though,
saying any was your fault.
[Snip personal silliness.]
> Fermat's last theory. Wiles would not have bothered if he
did not
believe
> that it was true. In a colloquium I had the privilege to sit
in on he
> stated that since he was a kid he believed that FLT was
true... that he
knew in his heart of hearts that the theory could not possibly
been
> incorrect, and that this belief propelled him into a career
in
mathematics
> and the ultimate solution of the problem.
All of which says nothing about the truth of FLT or the
randomness of the
pi series. The history of mathematics is replete with the
corpses of those
who were wrong about their hunches. Even astrologers and
stopped clocks are
right once in a while.
> You mean that to disprove a theory is not valid in
mathematics?
> Mati... please prove to me that any finite sequence of the
decimals of pi
is
> NOT truly random.
Fallacy of shifting the burden of proof. He NEVER said it
wasn't--only that
your assertion of any was unproven.
> In fact, I will be satisfied if you could outline how such a
proof would
be
> attempted.
Hell, so would Mati. A Fields Medal nomination would surely be
in order.
Following your line of thinking, I could assert that there are
an
infinite
number of twin primes. After all, they're everywhere we look
so far. I
could demand, when challenged, that my critic produce a proof
that there
AREN'T. I would be:
1) making a logically unproven statement, and
2) diverting attention from my error using the fallacy of
shifting the
burden of proof.
> So... do you think I am a fool?
One sure way to prove you're not would be to admit that your
unqualified
use of any was an error or misstatement. It's not hard--I do
it all the
time.
Two things will each lower our IQ thirty points over a
lifetime--avoiding
admission of errors, and raising children.
Servo
Subject: Re: SCHOENFELDS RANDOM THEOREM
> Trivially disproven by example in Hofstadter's Godel, Escher,
Bach.
> It's only 777 pages long. If you look at each page for one
second
you
> can find the table within 12 minutes.
> Uncle Al is wrong, and transistively, so are you. Try
independent
thought.
> No, Uncle Al is right, as he very often (but not always) is,
and you
> are wrong. All you can prove is that the probability of your
statement
> being correct is 1, and conversely, the probability of it
being false
> is 0.
> Now, if x is a number chosen at random out of the set of all
> intergers/rational numbers/real numbers, the probability
that x=0 is
> zero. Have I proved that 0 does not exist?
I said that R the random sequence is an INTEGER sequence BOUND
by n
and m.
Let R = infinite sequence of random integers bound by n and m
Let x = an integer such that n <= x <= m
The probability of drawing x from R at any point is 1/(m-n + 1)
The probability of NOT drawing x from r at any point is 1 -
1/(m-n +
1)
Let x be the last element of some arbitrary sequence of
integers S
such that for all y in S, n <= y <= m
The probability of drawing S from R starting at any point is
1/(m-n +
1)^|S|
The probability of NOT drawing S from R starting at any point
is
1 - 1/(m-n + 1)^|S|
Now, the probability of NEVER drawing S from R is
lim j->|R| (1 - 1/(m-n + 1)^|S|)^j
Now let T = 1/(m-n + 1)^|S|
We see that 0 < T < 1
We also see that 0 < 1 - T < 1
Since |R| = +inf,
Then,
lim j->+inf (1 - T)^j = 0
Therefore the probability that S does not occur in R is 0,
transistively, the probability that S DOES occur in R is 1 - 0
= 1.
PROOF
> I'll drink to transistivity (hic!), especially of the
bipolar junction
> kind,
> Cheers,
> Zigoteau.
JS
Subject: Re: SCHOENFELDS RANDOM THEOREM
> Therefore the probability that S does not occur in R is 0,
> transistively, the probability that S DOES occur in R is 1 -
0 = 1.
> PROOF
With the greatest of respect, that is exactly what I thought
your
proof would end up demonstrating, and in fact by your use of
the word
'random', it was the only sort of thing it _could_ end up
demonstrating.
However if you choose an element at random from an infinite
set, the
probability of choosing any particular element is zero. Hence
proving
that an event has probability zero does not prove that it is
impossible.
Looks like the Nobel Prize for Mathematics has to be postponed
for yet
another year. However, transistively speaking, you might get
the Nobel
Prize for Electronics.
Cheers,
Zigoteau.
Subject: Re: SCHOENFELDS RANDOM THEOREM
I'm posting because I'd hate to see a misunderstanding between
two
decent people escalate into something uglier than it need be --
[big snip]
> There is nothing wrong with believing in what you're trying
to prove
> as long as you don't confuse your belief for a proof. It is
a fine
> line here, since in science
[snip comment by GR_Learner, as I want to focus on what Mati
said]
> 1) Saying I believe this is true is a standard operating
procedure.
> 2) Saying It is true since I believe in it marks you as a
cracpot.
> You obviously did not interpret what I said correctly.
And I think you misread Mati. It's quite clear he was not
trying to
attack *you* personally, at least not here; his use of you
here is
idiomatic English, meaning one. The context is very general;
he's
trying to characterize *science*, not you.
> It is much worse than a cop out. Mathematiclly speaking,
that's ...
> well, I don't want to get offensive, yet.
> You already have become ofensive Mati.
No, that was your misreading.
I will look past that. Also...
> please indicate where you have snipped text so it will not
be taken out
of
> context.
> Mati Meron | When you argue with a fool,
> meron@cars.uchicago.edu | chances are he is doing just the
same
> So... do you think I am a fool?
It's his *.sig*, for goshsakes. It appears in all of Mati's
postings.
Don't take everything so personally.
(OTOH the things you are saying here about mathematical proof
*are* foolish. Asking Mati to Prove... that a bet isn't a
proof?
That's just silly. Smart people say foolish things sometimes,
and
I would say that for you this is one of those times. IMO.)
Subject: Re: SCHOENFELDS RANDOM THEOREM
X-SessionID: fNa4c-6550-45-16359@news.uchicago.edu
X-Hash-Info: post-filter,v:1.4
X-Hash: 0746b2e5 33b1ebb2 34205f41 26a4b484 6a4cb538
>> I'm posting because I'd hate to see a misunderstanding
between two
>> decent people escalate into something uglier than it need
be --
>Thanks. I stand corrected with regard to saying that the
digits of pi are
>randomly distributed. Matis is right that no matter how many
digits are
>sampled that still does not constitute a mathematical proof.
>My apologies if I insulted Mati.
You didn't. I was trying to set the terminology straight since
this
issue, of proof versus confirmation versus indication etc. is
one of the eternal sources of confusion on this ng. So, I'm
glad we
got it cleared up.
Mati Meron | When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same
Subject: Re: SCHOENFELDS RANDOM THEOREM
> I'm posting because I'd hate to see a misunderstanding
between two
> decent people escalate into something uglier than it need be
--
Thanks. I stand corrected with regard to saying that the
digits of pi are
randomly distributed. Matis is right that no matter how many
digits are
sampled that still does not constitute a mathematical proof.
My apologies if I insulted Mati.
Subject: Re: SCHOENFELDS RANDOM THEOREM
> I'm posting because I'd hate to see a misunderstanding
between two
> decent people escalate into something uglier than it need be
--
> Thanks. I stand corrected with regard to saying that the
digits of pi
are
> randomly distributed. Matis is right that no matter how many
digits are
> sampled that still does not constitute a mathematical proof.
> My apologies if I insulted Mati.
Well done. More should follow this example.
Servo
Subject: expectation of the product of two dependent random
variables
Hi sorry to bother you guys here.
Let random variable X = a_1r_1+a_2r_2+...+a_nr_n, and
random variable Y = b_1r_1+b_2r_2+...+b_nr_n, where a_i and b_i
(i=1...n) are
constants, and r_i is i.i.d. random variable chosen from
N(0,1). Note
that r_i in both X and Y are the same.
So how to compute the expectation of exp(p times X times Y),
that
is
E[exp(p times X times Y)] = ?
Thanks a lot
roy
Subject: re:expectation of the product of two dependent random
variables
> exp(p times X times Y)
Please clarify what this expression means.
Posted Via Usenet.com Premium Usenet Newsgroup Services
----------------------------------------------------------
** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **
----------------------------------------------------------
http://www.usenet.com
Subject: Re: (-2/3)^(-2/3) = (3/2)^(2/3)?
> Note that GrafEq plots no points for y = (-2)^(x^x) where -1
< x < 0.
> Is there a mathematical difference between the systems:
> |y = (-2)^u |y = (-2)^(x^x)
> |u = x^x and |-1 < x < 0
> |-1 < x < 0
> According to GrafEq the graph of the first system consists
of four
> curves and infinite vertical line (probably x = 0). The
graph of the
> second consists of point (probably (-1;-0.5)) and infinite
vertical
> line (probably x = 0).
> So everything is OK?
> As far as I can tell, everything is OK with GrafEq. But your
statements
> about what the graphs consist of according to GrafEq are
subtly
incorrect.
I agree with the last sentance. But there are cases in which
GrafEq
doesn't plot correctly (it doesn't plot at all) the graph of
mathematical equation or system.
For example from the following system:
|u * u = 2
|x = ((-3)^u)^u
|x belongs to R
=> x = -9. Right?
On that basis the graph of y = ((-2)^sqrt(2))^sqrt(x) should be
visually identical to y = -(2^sqrt(2))^sqrt(x). Right?
Calvin
Subject: re:cantor's theorem
I dont agree yet to the old hat:
As far as I know uses different types of sets, all belonging
to different levels.
In my set theory, there is only one kind or type of set. But I
use
types or levels for the element relation.
In my theory, a set can be element of itself - ins theory that
is not allowed.
But my meta set theory is not completely ready yet, as for
example I
could construct the natural numbers as sets, but not the set
of all
natural numbers...
Posted Via Usenet.com Premium Usenet Newsgroup Services
----------------------------------------------------------
** SPEED ** RETENTION ** COMPLETION ** ANONYMITY **
----------------------------------------------------------
http://www.usenet.com
Subject: Re: puzzle: GCDs of Infinite Set of Integer Pairs
>>)>If you pick two numbers randomly from [0..1], is it
possible that those
>>)>two numbers are the same ?
>>) Yes. But it is not possible that either of those numbers
is 1.5.
>>So even though they won't be the same no matter how many
times you try,
>>you say that it's still possible ?
Hey I went away for 3 weeks to NZ, and you guys have managed
to turn this
into
one of the longest threads in the history of usenet, but
AFAICS you still
haven't resolved the basic point for me, even though the
discussion has got
back
to pretty well exactly the same place.
My contention is:
You can say 'select a real randomly from the range [0,1]' but
you can't
actually
do it. It's like saying 'Imagine you can travel faster than
the speed of
light'.
A fine thought experiment, and you can draw all sorts of
conclusions about
a
universe that allows it, but this is not that universe.
I contend that:
1. There is no physical way of doing it (spin a pencil etc
fails due to the
quantum nature of reality)
2. There is no way to describe a finite process which results
in selecting
such
a number randomly
3. There is no way to accurately describe the number you
selected. If you
describe the number precisely, I'm entitled to call you a
liar, and assert
with
certainty that you didn't choose randomly.
On this basis, I assert that it's overly optimistic at best to
say choose
a
real number at random in the interval [0,1]
Oh, and I can't agree with this obviously:
>Who says they won't be the same no matter how many times I
try? That's
>the way to bet, but it's not a certainty.
If ever there was a certainty, this is it. You can't even
choose a number
*once*
let alone twice. Even if I am eventually persuaded that you
can in fact
choose a
number at all, you definitely can't pick it again.
'Probability zero' has
to
count for something in our little mathematical universe, or
it's a useless
little universe.
--
Patrick Hamlyn posting from Perth, Western Australia
Windsurfing capital of the Southern Hemisphere
Moderator: polyforms group (polyforms-subscribe@egroups.com)
Subject: Re: General algorithm for vector cross-product
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2C1JVE04033;
The cross product is a convenient way to find a vector that is
normal to the plane spanned by two non-collinear vectors, in 3
dimensions. Just as Don mentioned, IF you have n-1, n
dimensional
vectors, then the vector that is perpendicular to these is
found by
computing the determinant of the n x n matrix consisting of
the n-1
vectors and the e_1, e_2, ..., e_n basis. However, say I have
two
'k' dimensional vectors and I want to find the vector that is
perpendicular to them. In general, there is no _one_ unique
vector.
In fact, in general, this problem usually has a solution that
is a
vector space. In particular, if I have m non-collinear 'n'
domensional vectors (with mHey all,
>Alas, there doesn't seem to be a good newsgroup for
mathematical
algorithms,
>so I'm posting here.
>I am using the cross-product to find a normal that us
perpendicular
to two
>vectors. All of the examples and code snippets that I have
ween deal
with
>the 3D case. Correct me if I am wrong, but we can find the
cross-product of
>other dimensions (either lower or higher), can we not?
>To handle higher dimensions, we could recursively find the
determinent of
>the vectors, but if we have n-dimensional vectors, do we need
n-1
vectors to
>fill the n x n matrix?
>Also, how would we lower dimensions? I imagine that the cross
of
>one-dimensional vectors (i.e. numbers) is just their
arithmetic
product.
>What about R^2?
>So is there some general algorithm for handling all cases?
>Dave
Subject: Intergrals
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2C1JVr04029;
int [sqrt(36-x^2)] dx
help! i dont know wat to substitute x as!
Subject: Re: Intergrals
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CH8lQ06002;
>int [sqrt(36-x^2)] dx
>help! i dont know wat to substitute x as!
You mean without actually looking in a text book? :)
Try the substitution 6x= sin(theta). The point of that is then
sqrt(36-x^2)= sqrt(36(1-sin^2(theta))= 6cos(theta).
Also 6 dx = cos(theta)dtheta or dx= (1/6)cos(theta)dtheta so
the
integral becomes int cos^2(theta)dtheta. Use the trig identity
cos^2(theta)= (1/2)(1+ cos(2theta)) to do that integral.
Any calculus text book should have a section on trigonometric
substitutions in the chapter of methods of integration.
Subject: Re: Intergrals
>int [sqrt(36-x^2)] dx
>help! i dont know wat to substitute x as!
> You mean without actually looking in a text book? :)
> Try the substitution 6x= sin(theta).
That would be counterproductive. Try x = 6*sin(theta).
> The point of that is then
> sqrt(36-x^2)= sqrt(36(1-sin^2(theta))= 6cos(theta).
> Also 6 dx = cos(theta)dtheta or dx= (1/6)cos(theta)dtheta so
the
> integral becomes int cos^2(theta)dtheta.
or indeed 6 times this.
> Use the trig identity
> cos^2(theta)= (1/2)(1+ cos(2theta)) to do that integral.
> Any calculus text book should have a section on trigonometric
> substitutions in the chapter of methods of integration.
--
P.A.C. Smith
The vast majority of Iraqis want to live in a peaceful, free
world.
And we will find these people and we will bring them to
justice.
Subject: Re: Intergrals
> int [sqrt(36-x^2)] dx
> help! i dont know wat to substitute x as!
Try this: let 6 (sqrt(36)) be the hypotenuse and x the
opposite side.
It's
not guaranteed to work (well, not all the time, with all
possible
combinations of stuff under the square root sign), but you can
keep messing
around with it and eventually something will work.
Jon Miller
Subject: Re: Intergrals
Content-transfer-encoding: 8bit
> int [sqrt(36-x^2)] dx
> help! i dont know wat to substitute x as!
x = 6sin(v)
--
Paul Sperry
Columbia, SC (USA)
Subject: Re: sequence of rationals
> Let {x_n} be a sequence of rational numbers that converges
to an
> irrational number, x. i.e. x_n -> x. Then the denominators
of each
> of the x_n must necessarily grow as n goes to infinity.
> This fact is obviously true, but I am having a hard time
proving it.
> Thanks for any help.
Assume that M is an integer upperbound on the denominators.
Then M!*x_n, where M! is factorial M, must be an integer for
each n, so
we have the sequence of integers {M!*x_n} converging to the
still
irrational value M!*x.
I don't think so.
Subject: need help too (probability)
Let X1, Z be independent where X1 is N(0; 1) and P(Z = 1) =
P(Z =
?1) = 1/2. Let X2 = X1Z. Show that X2 is N(0; 1), and that X1
and X2 are
uncorrelated (that is, have zero covariance) but not
independent. Why does
this not contradict the result given in class, that
uncorrelated
multivariate
normal random variables are independent?
Is there any clue to show that X2 is also N(0,1)
How come X1 and X2 are uncorrelated but not independent? By
the definition
if the covariance is zero, correlation should also be zero.
And they are
independent. But why it is not?
Can anyone help me ~~????
Thanks!
Subject: Re: Non-AP Calculus?
> Should high schools teach Regular Calculus courses for
students who
> are not at the level to handle the Advanced Placement
Calculus course?
> My instinct is to say that those students would benefit more
from
> additional precalculus preparation. Is that the way college
> professors would prefer things, or would they rather have
students who
> have seen some calculus in a watered-down class.
> I ask this question because I am the mathematics department
chair at
> my school, and we are looking at what courses we teach and
what
> courses we should teach.
> Thank you for any insights you can provide!
> Michael Brown
I think it is a good idea. As a matter of fact, I did it for
about 25
years. The non-AP course that we taught covered the AB
curriculum plus
many
of the BC topics. We covered through advanced methods of
integration,
applications of definite integrals including arc length,
surface area,
parametric equations. Basically what was skipped were infinite
series,
power series, and ODE's that used to be a bigger part of the
BC topics
list.
The difference was that we did not have to be done by the time
that the AP
exam was offered. The additional time allowed for better
development and
reinforcement. The students who took this course did one of
three things
when entering college: a) Skipped one semester, b) Entered an
honors or
more rapidly paced program, or c) Did the whole course over.
Students in
all of the above groups reported back positively about the
experience. One
student who had struggled to get a C- took his calc at a
prestige school as
a pass/fail because he was worried about his HS course. He got
an A+ and
could only call it a pass. The confidence developed by most
seems to have
made a big difference and of course seeing things for a second
time helped.
Also, the additional contact hours available in a HS setting
allows time
for
firming up ideas from the pre-calc topics. This done in
context of a calc
environment seemed to work better for us than simply redoing
pre-calc
topics
alone.
It worked extremely well for me/us and the students.
Ken
Subject: Re: Non-AP Calculus?
> Should high schools teach Regular Calculus courses for
students who
> are not at the level to handle the Advanced Placement
Calculus course?
> My instinct is to say that those students would benefit more
from
> additional precalculus preparation. Is that the way college
> professors would prefer things, or would they rather have
students who
> have seen some calculus in a watered-down class.
Why?
Seriously, the AB course covers the first semester of college
calculus in a
year. That means that a student who is prepared can go to
class and do an
average of about half an hour of homework a day (more during
the push for
the exam). If they aren't willing to work that hard, will they
be prepared
for the class? So will they get anything out of the
watered-down
version?
I think the answer to both those questions is no.
If they aren't ready for calculus, they should either do more
to get ready
for calculus or study something that they find more
interesting.
Jon Miller
Subject: Re: Non-AP Calculus?
Content-transfer-encoding: 8bit
> Should high schools teach Regular Calculus courses for
students who
> are not at the level to handle the Advanced Placement
Calculus course?
> My instinct is to say that those students would benefit more
from
> additional precalculus preparation. Is that the way college
> professors would prefer things, or would they rather have
students who
> have seen some calculus in a watered-down class.
> I ask this question because I am the mathematics department
chair at
> my school, and we are looking at what courses we teach and
what
> courses we should teach.
> Thank you for any insights you can provide!
> Michael Brown
I can't claim to speak for all College/University Calculus
instructors,
but I think that many would prefer _no_ High School Calculus -
AP or
otherwise.
While there are certainly exceptions, it is the a little
knowledge is
a dangerous thing syndrome. Students coming out of AP Calculus
tend to
think they know more than they actually do. The AP students,
having
coasted through AP Calculus, think that they will continue
coasting in
College/University Calculus and are sometimes slow to wake up.
I think a solid trig course with _proofs_, a solid geometry
course with
_proofs_ and a very solid pre-calc algebra course would serve
your
students much better.
You might want to ask your question on the sci.math newsgroup.
--
Paul Sperry
Columbia, SC (USA)
Subject: Re: Non-AP Calculus?
> Should high schools teach Regular Calculus courses for
students who
> are not at the level to handle the Advanced Placement
Calculus course?
> My instinct is to say that those students would benefit more
from
> additional precalculus preparation. Is that the way college
> professors would prefer things, or would they rather have
students who
> have seen some calculus in a watered-down class.
What would you consider to be appropriate precalculus
preparation?
Subject: Re: generalization of stokes' and divergence theorems?
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CH8kd05998;
>Stokes theorem says SS[ curl F dS ] over S = S[ F dr ] over
C, and
the
>divergence theorem says SSS[ div F dV] over E = SS[ F dS ]
over S.
>Is there a theorem that says, SSSS[ something(F) dH ] over H
= SSS[ F
dV]
>over V. Wouldn't V have to have an orientation? It seems that
in
R^3 we
>couldn't have that, but in R^4 I'd think you could define
something
like
>that.
>Is there a general something where I can write:
>SS...n[ something(F) dA ] over A = SS...(n-1)[ F dB] over B
>or something similar?
>Thanks,
>Jeremy
The most general something is Poincare's formulation:
Let M be a n-dimensional manifold with boundary dM (so dM is an
(n-1)-dimensional manifold. Let w be an (n-1)-order
differential
form on dM with derivative (co-boundary) dw (an n-order
differential form). If M is simply connected then the integral
of dw
over M is equal to the integral of w over M.
If M is not simply connected, then the difference between the
two
integrals is related to the homotopy groups.
Subject: Re: tensors
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CHagW09393;
>I'm understanding tensors are things who obey certain
transformation
>properties and algebraic properties and everything, but I
don't
understand
>what they physically/geometrically mean. What is the physical
significance
>of a covariant, contravariant, or mixed tensor of a specific
order?
I.e.,
>what's the physical difference between A^ij, A_ij, and A^i_j?
I'm
also
>confused as to why we write a dot product of vectors in tensor
notation as
>a_i b^i, why not a_i b_i, or a^i b^i? Is there a difference
between
a^i b_i
>and a_i b^i? I guess I'm not really understanding the
difference
between
>covariant, contravariant, and mixed, other than that the
differences
in how
>they transform.
>I'd really appreciate it if someone could help me here!
>Thanks,
>Jeremy
That's asking a lot! The problem is that you are used to
working
with vectors in Euclidean space with a cartesian coordinate
system.
Tensors are regularly applied to spaces or surfaces in which
there is
NO cartesian coordinate system (for example, the surface of a
sphere).
If you were to work only in Euclidean space and always used
cartesian
coordinate systems, there would be no difference at all
between the
contravariant and covariant coordinates of a tensor.
Notice I said contravariant and covariant coordinates,
NOT contravariant and covariant tensors. Strictly speaking any
tensor has both covariant and contravariant coordinates.
Here's a
simple example. Take flat, two dimensional, Euclidean space
and take
the x-axis as the u coordinate axis but take the line y= x as
the
v coordinate axis. Now there are two different ways of
labelling
any point (u,v): (1), drop a perpendicular from the point to
each axis
and take the distance to the v-axis as the u coordinate and
take the
distance to the u-axis as the v coordinate. (2) draw lines
through
the point parallel to the coordinate axes. Measure the
distance along
the line parallel to the u-axis to the v-axis and make that
the u
coordinate and similarly for the v coordinate. In cartesian
coordinates, of course, those would give exactly the same
thing. In
skew coordinates they are not the same and are the
contravariant and
covariant coordinates respectively.
As for u_i v^i as dot product, do you know what the dual space
is?
For any vector space, the dual space is the set of all linear
functions from the vector space to the real numbers. That is,
if v is
a vector and f a dual vector, then f(v) is a number. One can
show
that, in Euclidean space (i.e. flat with cartesian
coordinates), the
dual space is isomorphic to the vector space. That is, there
is a
natural (given the coordinate system) way to associate each
vector
with such a function. When we right u.v we really mean f(v)
where f
is the function associated that way with u. In more general
spaces,
there is no such association, but we CAN associate the
contravariant
components with functions on the covariant components and
vice-versa.
Subject: toplogy
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CJ6Y119302;
Hello all
Just like to know something about toplogy:-
1) Where i can find information about an Old magic which
magician use
to do in which they wear e.g. two shirts .... one is black (
or of any
colour) and over that they wear a green shirt ( or of any
color this
is just an example) and by a topological trick they were able
to
takeoff the inner shirt in our case the black one... how this
magic
trick is related to toplogy?
as iam using internet cafe ..not mailing from my own pc so i
request u
all if u have the link or info plz mail me...
thankyou
waqar
Subject: Computer Science student seeks mathematic aid
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CJ6Yu19292;
Since I dont have a background in math, I figure that I can
more
easily describe my question with a little practical
background. I'm
creating a program to find curves; three dimensional curves
with
specific endpoints and slope at those endpoints. The flow of
the
program will be this:
Input.
1. Endpoints
2. Three dimensional unit vector at each endpoint describing
the
motion of the curve at that endpoint.
3. Length of the curve.
Output.
1. Curve fuction (My biggest problem. I dont know if it is
possible to
create one function or colleciton of functions from the
limited input.
I've mildly studied polynomial interpolation and splines)
Requirements.
1. Goes through both endpoints
2. Path of the curve must be along the unit vector at the
endpoint
3. Must have stated curve length.
The optimal curve will :
1. Be smooth
2. Minimize concavity
3. Minimize rate of change of concavity
(I'm just saying the less curve the better. But the parts that
match
up to the unit vector at the endpoints need to be smooth. More
broad
curves are better than fewer sharper curves, too.)
Can anyone point me in any direction? I'm currently in design
time
with the program and I need to find out my options. I dont
mind doing
guess-n-check for these values either. So if someone can give
me a
template for a curve which fulfills the requirements, I can
use trial
and error to find the most optimal of these curves.
I didn't know where to look for help other than here. I hope
you all
find enjoyment on helping me with this project. I've talked to
people
at my university but those were student tutors, not professors.
Thanks all in advance!
Derek Adams
Student, Computer Science
Cal Poly Pomona
Subject: Re: Computer Science student seeks mathematic aid
> Since I dont have a background in math, I figure that I can
more
> easily describe my question with a little practical
background. I'm
> creating a program to find curves; three dimensional curves
with
> specific endpoints and slope at those endpoints. The flow of
the
> program will be this:
> Input.
> 1. Endpoints
> 2. Three dimensional unit vector at each endpoint describing
the
> motion of the curve at that endpoint.
> 3. Length of the curve.
> Output.
> 1. Curve fuction (My biggest problem. I dont know if it is
possible to
> create one function or colleciton of functions from the
limited input.
> I've mildly studied polynomial interpolation and splines)
If the allowable arc length is greater than the distance
between
starting point and end point, the curve is possible.
If the allowable arc length is equalto the distance between
starting
point and end point, and the direction at each end is exactly
towards
the other end, the curve is possible, and will be a line
segment.
Otherwise the curve is impossible.
> Requirements.
> 1. Goes through both endpoints
> 2. Path of the curve must be along the unit vector at the
endpoint
> 3. Must have stated curve length.
> The optimal curve will :
> 1. Be smooth
> 2. Minimize concavity
I think by concavity you mean what mathematicians would call
the
curvature
> 3. Minimize rate of change of concavity
I'm not sure whether you mean to minimize torsion or rate of
change of
curvature wrt arc length. They are quite different. You might
possibly
want to minimize both.
Both an arc of a circle and a section of a helix have constant
positive
curvature, but every plane curve has zero torsion, and a helix
has
constant positive torsion.
> (I'm just saying the less curve the better. But the parts
that match
> up to the unit vector at the endpoints need to be smooth.
More broad
> curves are better than fewer sharper curves, too.)
The Frenet formulas of differential geometry are useful in
analysing
parametrically represented space curves, including curvature
and torsion
of such curves.
> Can anyone point me in any direction? I'm currently in
design time
> with the program and I need to find out my options. I dont
mind doing
> guess-n-check for these values either. So if someone can
give me a
> template for a curve which fulfills the requirements, I can
use trial
> and error to find the most optimal of these curves.
> I didn't know where to look for help other than here. I hope
you all
> find enjoyment on helping me with this project. I've talked
to people
> at my university but those were student tutors, not
professors.
> Thanks all in advance!
> Derek Adams
> Student, Computer Science
> Cal Poly Pomona
Subject: Re: Computer Science student seeks mathematic aid
You REALLY need the Mathematics background. The machine can
not help you
if
you lack the Mathematics. Your project absolutely requires you
to
understand
the mathematics.
G C
>Since I dont have a background in math, I figure that I can
more
>easily describe my question with a little practical
background.
>I didn't know where to look for help other than here. I hope
you all
>find enjoyment on helping me with this project. I've talked
to people
>at my university but those were student tutors, not
professors.
> Derek Adams
> Student, Computer Science
> Cal Poly Pomona
Subject: New source of the math!!!
I have come across a source providing good support to those
experiencing difficulties with doing homework in maths
including
arithmetic(http://www.bymath.com/studyguide/ari/form1.htm),
geometry
(http://www.bymath.com/studyguide/geo/pro/pro1/pro1.htm),
algebra
(http://www.bymath.com/studyguide/alg/pro/pro1/pro1.htm),
functions
and graphics
(http://www.bymath.com/studyguide/fun/pro/pro.htm),
principles of
analysis(http://www.bymath.com/studyguide/ana/pro/pro1/pro1.
htm),
all this is also supported by a good deal of examples and
illustrations.
Subject: Re: Mupad syntax
<1Ko3c.1895$Or1.84@news.chello.at
> sobald man statt a eine Zahl einsetzt z.B. 2 das Ergeniss
nicht
richtig
> formatiert wird.
> also
> simplify(2^v/2)
Bug: a^v/a is actually a^v*a^(-1), but 2^v/2 is actually
2^v*1/2
(where 1/2 is a rational number of type DOM_RAT) and combine
and
simplify don't realize this can be simplified. I suggest
posting a
bug report at www.mupad.de/bugs.html.
--
+--+
+--+|
|+-|+ Christopher Creutzig (ccr@mupad.de)
+--+ Tel.: 05251-60-5525
Subject: Re: Sum evaluation with Wilf-Zeilberger method
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i2CDZe210830;
Thank you very much,
this was exactly what i needed to prove the Lemma.
The function G can be simplified to
G =
(-1)^(n-k+1)*k*m*binomial(m-k,k)*binomial(m-2*k,n-k+1)/((n+1)*
(m-n-1))
Frank Haefner
frankdothaefner@web.de
Subject: Re: homeomorphisms S^3 --> S^3
Originator: israel@math.ubc.ca (Robert Israel)
> it would be great if someone could help me with the
following question:
> How do I find all isotopy-classes of homeomorphisms S^3 -->
S^3?
> Is it true, that all homeomorphism, which preserve
orientation, are
> isotopic to the identity?
Yes. Have a look at the archived sci.math.research discussion
at
http://www.math.niu.edu/~rusin/known-math/99/aut_S3 .
Near the bottom of the page, you'll find the following answer
by Linus
Kramer:
> Let Diff(S^3) denote the diffeomorphism group of S^3 (with
the
> C^infty topolgy), Top(S^3) its homeomorphism group
(c-o-topology)
> and O(4) the rotation group in R^4. There are natural maps
> O(4) --> Diff(S^3) --> Top(S^3).
> Hatcher proved [Ann. Math. 117] that both maps are homotopy
> equivalences. This settled the so-called Smale conjecture,
> and it's a very difficult theorem! Also, the result is not
> true any more for spheres of higher dimensions.
-- Marc Nardmann
(To reply, remove every occurrence of a certain letter from my
e-mail
address.)
Subject: Re: homeomorphisms S^3 --> S^3
Originator: israel@math.ubc.ca (Robert Israel)
> it would be great if someone could help me with the
following question:
> How do I find all isotopy-classes of homeomorphisms S^3 -->
S^3?
> Is it true, that all homeomorphism, which preserve
orientation, are
> isotopic to the identity?
There have been several replies to this question, mentioning
Hatcher's
proof of the Smale conjecture. However, an affirmative answer
to the
original question is older (and is part of background for
Hatcher's
work). First, that homeomorphisms are isotopic to PL
homeomorphisms (and
hence to diffeomorphisms) goes back to Moise's work in the
late 1950s.
(Much of this is reworked in his book, Geometric Topology in
Dimension 2
and 3; I think that Shalen's paper: A piecewise-linear method
for
triangulating $3$-manifolds. Adv. in Math. 52 (1984), no. 1,
34--80,
would give a proof more accessible to modern readers.) The
proof that
an orientation preserving diffeomorphism of the 3-sphere is
isotopic to
the identity is due to Jean Cerf, from the early 60's: Sur les
diffeomorphismes de la sphere de dimension trois $(Gamma
sb{4}=0)$.
(French) Lecture Notes in Mathematics, No. 53.
Daniel Ruberman
Subject: Re: Fixed Point Arithmetic Resources ?
Originator: israel@math.ubc.ca (Robert Israel)
> hi there,
> from past few days i have been doing research on using Fixed
Point
> arithmetic using Integers, i was able to find out some good
> resources(google) but they were related to simple
> Addition-Subtraction-Multiplication, i m looking for some
good resources
on
> doing Trignometric functions(sin,cos..) through Fixed Point
Atithmetic
Knuth's METAFONT program implements all the standard
trigonometric
functions using 32-bit integer arithmetic. It's fairly painful
but it
works.
Subject: Shafer-Dempster vs. Bayes
Epigone-thread: zeuwhingri
Originator: israel@math.ubc.ca (Robert Israel)
Dear Folks,
I am currently trying to collect information about the
different ways
to consider uncertainties in engineering calculations and
finally met
the Shafer-Dempster Theory of evidence. At some point of the
theory
there is the conclusion the the Bayes approach is a special
case of
the Shafer-Dempster theory. I think that I don't understand
that. What
are the differences of Bayes and Shafer-Dempster (how is Bayes
a
special case?) and what are the advantages and disadvantages in
treating uncertainties?
Thanks,
Daniel
Subject: Re: Pointwise Ergodic Theorem
Originator: israel@math.ubc.ca (Robert Israel)
Sorry, I was not precise enough:
As to the first objection noted I mean that v is the ONLY
ergodic
measure and as far as the moderators note I intend the map to
be
nonsingular so that the interval [0,1) cannot map to the point
1.
Thanks,
E
>Suppose you have a map T: X -> X for which the probability
measure v
>is invariant and ergodic. Also suppose v << m where m is
lebesgue
>measure. Is it possible to show convergence in the pointwise
ergodic
>theorem for m almost every x in X. Obviously the ergodic
theorem
>itself only talks about v almost every convergence.
> I don't know anything about this ergodic stuff, but surely
the
> answer is clearly not?
> Maybe X = [0,2], T maps [0,1) to [0,1) and maps [1,2] to
[1,2],
> and v is concentrated on [1,2]. Then the hypotheses say
> nothing whatever about how T behaves on [0,1), so they
> certainly don't imply convergence a.e. on [0,1).
> [ moderator's note:
> Perhaps the OP meant to assume that v is the only ergodic
probability
> measure. But still there would be easy counterexamples, e.g.
> with X = [0,2], where T maps [0,1) to 1, v must be
concentrated on [1,2],
> and the behaviour of the iterates of points in [0,1) depends
on the
> iterates of the single point 1, which may be atypical.
> - ri
> ]
>Thanks,
>E
> ************************
>
Subject: Re: Law of large number in a finite set with distance
measure
Originator: israel@math.ubc.ca (Robert Israel)
>>Consider a discrete set X in which a distance d(x,y) that
satisfies the
>>triangle inequality is defined.
>>Suppose there is a probability measure P of X such that the
>>mean of P is m. The mean can be defined by, say,
>>m = argmin_{x in X} sum_{y in X} d(y, x) P(y)
> This is more like the median than the mean. It may not
> be unique. Whether d satisfies the triangle inequality
> or is even symmetric is not of great importance, although
> it might be in looking at the asymptotics if X is
> infinite; the triangle inequality would enable a
> somewhat easier proof, but not solve all the problems.
>>Now, based on P, we get a sample of size D from X,
>>D = {x[1], x[2], ..., x[n]}.
>>We then find the mean of D. This can be defined by
>>hat{m} = argmin_{ x in X } sum_i d( x[i], x )
>>When n tends to infinity, I expect hat{m} tends to m.
> Certainly if X is finite, hat{m} will tend to the
> set of medians almost surely.
Yeh... I also notice about that shortly after I posted the
question.
>>Is this true? If yes, Can you give me some references that
>>contain the proof? If no, is it possible to impose
>>additional assumptions on P to make hat{m} go to m?
>>Also, can anything be said about
>>the rate of convergence, if it can be defined after all?
> The probability that hat{m} is not in the set of medians
> tends to 0 exponentially in the finite case. If there are
> two medians, they are roughly equally likely for large n,
> but if there are more, I do not know the asymptotics.
This is an interesting property. May I ask where can I
find the proofs of these facts? Thank you.
Subject: Re: Are analytic Divisors locally algebraic?
Originator: israel@math.ubc.ca (Robert Israel)
> Let V be an analytic divisor in a polydysc D in C^n (i.e.
V={f=0} with
> f analytic)
> that pass through 0 (i.e. f(0)=0)
> Is there (up to shrinking D) a biholomorphism G:D->D, that
fixes 0 and
> such that the image of V is algebraic (i.e. G(V)={g=0} with g
> polynomial) ?
> Of course the question is trivial if V is irreducible and
smooth at 0.
> Thanks in advance for any hint.
Your notations are somewhat unclear for me. Do you mean it
as analytic germs (with natural structure through f) or
as analytic sets (with the reduced structure)?
A common formulation would be 'hypersurface singularity'
and your Q means to look at it as analytic algebras,
asking whether they are 'analytifications' of polynomial
ones.
For the natural structure i think there are counter examples.
Sorry, i do not have one, but suggest to look at
( exp(x) - 1 )^2 - y*exp(x*y)^3 or x^2-y*exp(y*x)^3
(a modification of Neil's parabola x^2-y^3).
Playing with Maple it should be singular in 0 (evaluate
the Hessian) but to decompose exp(x*y) (in polynomials)
should fail.
The other suggestion is to search for 'affine GAGA' or
similar. May be papers of Greuel or Loijenga are also a
good starting point. Or literature on local analytic
algebra or geometry (elder ones will be in French).
---
remove the no for mail
Subject: Re: Are analytic Divisors locally algebraic?
Originator: israel@math.ubc.ca (Robert Israel)
>Let V be an analytic divisor in a polydysc D in C^n (i.e.
V={f=0} with
>f analytic)
>that pass through 0 (i.e. f(0)=0)
>Is there (up to shrinking D) a biholomorphism G:D->D, that
fixes 0 and
>such that the image of V is algebraic (i.e. G(V)={g=0} with g
>polynomial) ?
(Probably you mean e.g., not i.e.)
>Of course the question is trivial if V is irreducible and
smooth at 0.
If you really want the biholomorphism to be onto D, then it
may not be
as trivial as you assert.
You should look at the Weierstrass Preparation Theorem; for
references
see Gunning or Griffiths-Harris. (This assumes that you're
willing to
accept a biholomorphism from D to an open neighborhood of 0.)
Subject: Re: Are analytic Divisors locally algebraic?
Originator: israel@math.ubc.ca (Robert Israel)
Thanks for your answer.
>Let V be an analytic divisor in a polydysc D in C^n (i.e.
V={f=0} with
>f analytic)
>that pass through 0 (i.e. f(0)=0)
>Is there (up to shrinking D) a biholomorphism G:D->D, that
fixes 0 and
>such that the image of V is algebraic (i.e. G(V)={g=0} with g
>polynomial) ?
> (Probably you mean e.g., not i.e.)
I don't see any other definition of algebraic divisor. But
anyway it
would be good for me if the image of V can be written as zero
of a
polynomial.
>Of course the question is trivial if V is irreducible and
smooth at 0.
> If you really want the biholomorphism to be onto D, then it
may not be
> as trivial as you assert.
Sorry about that. As you said, it does not need to be onto.
On the other hand I don't see how Weierstrass thorem helps. I
want g
(as above) to be algebraic with respect to all the variables,
and
induction doesn't seem to work so naturally.