mm-1001
===
Subject: Difference between the set of all the finite subsets
of an infinite
series and all subsets
Would someone mind explaining to me the difference between
the set of
all finite subsets of an infinite series and the 
set of all
subsets of
an infinite series.
Obviously one set includes the infinite subsets while the 
other
doesnÕt. I think my problem is that I canÕt 
picture a place
where the
finite subsets end and the infinite subsets begin 
(so to speak).
Hopefully this is intelligible (if not, well, at least I made
someone
as confused myself ;-).
The context of my question is in showing that the set of all
finite
subsets of a countable series is countable itself, while all
the
subsets (including those infinite I guess) is not in fact
countable
(ie: the power set 2^A where A is a countable set is not in
fact
countable)
===
Subject: Re: Difference between the set of all the finite
subsets of an
infinite series and all subsets
> Would someone mind explaining to me the difference between
the set of
> all finite subsets of an infinite series and the 
set of all
subsets of
> an infinite series.
It may be less confusing to think in terms of infinite sets
and not
mention series here, since that brings in some unnecessary
baggage. In
fact, itÕs not entirely clear what you even mean by a subset
of a series.
> Obviously one set includes the infinite subsets while the
other
> doesnÕt. I think my problem is that I canÕt 
picture a place
where the
> finite subsets end and the infinite subsets 
begin (so to
speak).
ThatÕs the advantage of dealing with sets here rather than
series. They
donÕt have an order.
> The context of my question is in showing that the set of
all finite
> subsets of a countable series is countable itself, while
all the
> subsets (including those infinite I guess) is not in fact
countable
> (ie: the power set 2^A where A is a countable set is not in
fact
> countable)
Looks like your question is really about sets and not about
series as far
as I can tell. The power set of the integers includes lots of
sets that
are infinite, such as the set of primes, the set of squares,
the set of
odd numbers, and lots of others that donÕt happen to have
convenient
names.
For each finite n, the set of all n-element subsets of N is
countable.
There is a standard proof that a countable union of countable
sets is
countable.
--
Dave Seaman
Judge YohnÕs mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&
bookid=228>
===
Subject: help for the cross-correlation using FFT in Image
analysis purpose
hi all,
if anybody knows how to use FFT to realize cross-correlation
function,
which is for image processing and analysis purpose: exactly as
normcorr2function in matlab image processing toolbox, find the
correlation position of image B, on another image A(image
size B is
much smaller than image size A). however, i have to do it in
C or C++.
unfortunatelly size of A and B are both not power of 2. could
it be
possible be done in FFTW? i do download the FFTW2.1.5, but i
do not
know how to install it in windows(athere are only source
code....) and
how to include it in my project which is built in MS c++.net.
i read the message from spasmous, howver i could not make it
work....
it is quite important and urgent for me, coz this is a bridge
part of
my project work!!without it i could not continue my work. any
help and
suggestion is greatly appreciated.
thank a lot!
Bobbie
===
Subject: Re: help for the cross-correlation using FFT in
Image analysis
purpose
> <snip unfortunatelly size of A and B are both not power of
2. could it be
> possible be done in FFTW? i do download the FFTW2.1.5, but
i do not
> know how to install it in windows(athere are only source
code....) and
> how to include it in my project which is built in MS
c++.net.
> i read the message from spasmous, howver i could not make it
> work....
Please state what the error with fftw compilation is. Or try
the other
FFT code referred to on the other thread.
===
Subject: Re: help for the cross-correlation using FFT in
Image analysis
purpose
> unfortunatelly size of A and B are both not power of 2.
could it be
> possible be done in FFTW?
Yes.
> i do download the FFTW2.1.5, but i do not
> know how to install it in windows(athere are only source
code....) and
> how to include it in my project which is built in MS
c++.net.
See fftw.org/windows.html for precompiled Windows binaries of
FFTW 2.x
and FFTW 3.x, as well as VC++ project files to compile them
yourself.
===
Subject: Re: Nonlinear curve fit for saturation data
> Have I misunderstood what you mean by fitting
> to lowest sum of squared absolute error?
Doubtful, since you understand English. Maybe
my SSQ fitting algorithm needs work, eh? I
could tighten up the iteration stop parameters.
James
===
Subject: Which columns are inside?
Given an mxn matrix A of rank m, let C(A) denote the convex
hull of
the columns of A . I would like to determine which columns
(if any)
are in the interior of C(A). Answers, appropriate references
would be
appreciated. Please post (a copy of) your response to me. I
may not
see it on sci.math.num-analysis.
===
Subject: Eigenvalues of reduced kronecker product
here is a problem I gave some thought to, but didnÕt come up
with a
solution
for the general case. The background is the implementation of
the
Carleman approximation for a nonlinear system.
I have a square matrix A_2x2 and calculate itÕs eigenvalues
la1 and la2.
Now I build a sum of kronecker products of A with an identity
matrix I of
the same dimension as A.
B = A x I + I x A (x denoting the kronecker product)
Matrix B now has the dimension (4 x 4) and its eigenvalues
can be
calculated as linear combinations of eigenvalues of A
(lb1 = 2*la1, lb2 = lb3 = la1+la2, lb4 = 2*la2).
For the application in mind one row and one column (i.e. the
2nd or the
3rd row and column) is redundant. So we are processing the
reduced matrix
Br_3x3, consisting of 3 row and columns of B.
So I am wondering if there is any simple relation between the
eigenvalues
of Br and the eigenvalues of B. Or stated differently: Can we
apply an
operation (maybe just scaling) on Br to give the same
eigenvalues as B
but with no repetition (as for B), i.e. 2la1,2la2,la1+la2
I observed that for the discussed 2x2 example rescaling the
offdiagonal
elements of Br with sqrt(2) gives the desired effect, but I
did not find
something for the case where A is a nxn matrix.
Also of interest is the eigenvalue relation for reducing the
matrix
B2 = A x I x I + I x A x I + I x I x A
and its generalisation (n summands each consisting of n-1
kronecker
products).
Hope thats not to trivial for this group and I thank in
advance for any
ideas or references,
Heinz
===
Subject: vectors satisfying vT*A*v=0 for matrix A
Is there any lore (including algorithms) concerning the set of
vectors v associated with a given (square) matrix A which
satisfy vT * A
* v = 0? Note that this set includes, but is not necessarily
limited to,
the null space of A. For instance, do these vectors have a
name? A is
real symmetric in my application...
Eric Henry
eric@helix.nih.gov
===
Subject: Re: vectors satisfying vT*A*v=0 for matrix A
Distribution: inet
> Is there any lore (including algorithms) concerning the set
of vectors
v
> associated with a given (square) matrix A which satisfy vT
* A * v = 0?
> Note that this set includes, but is not necessarily limited
to, the null
> space of A. For instance, do these vectors have a name? A
is real
symmetric
> in my application...
> Eric Henry
In connection with Lorentz manifolds one calls such vectors
lightlike
Nicolas.
===
Subject: Re: vectors satisfying vT*A*v=0 for matrix A
>> Is there any lore (including algorithms) concerning the
set of vectors
v
>>associated with a given (square) matrix A which satisfy vT
* A * v = 0?
>>Note that this set includes, but is not necessarily limited
to, the null
>>space of A. For instance, do these vectors have a name? A
is real
symmetric
>>in my application...
The set of all such points has a name. It is called a
(projective)
conic section.
> In connection with Lorentz manifolds one calls such vectors
lightlike
... if A is symmetric and nonsingular with exactly one
negative eigenvalue, and v is nonzero.
Arnold Neumaier
===
Subject: Re: vectors satisfying vT*A*v=0 for matrix A
> Is there any lore (including algorithms) concerning the set
of
>vectors v associated with a given (square) matrix A which
satisfy vT * A
>* v = 0? Note that this set includes, but is not necessarily
limited to,
>the null space of A. For instance, do these vectors have a
name? A is
>real symmetric in my application...
>Eric Henry
>eric@helix.nih.gov
if A is not only symmetric but also positive semidefinite,
then {v} is
exactly
the nullspace of A. But in the indefinite case this is not
true and the
structure of this set : iti s simply a level set for an
indefinite
quadratic
form ....
hth
peter
===
Subject: Five Parameter Logistics
IÕm looking for a formula to fit a set of data 
to a so called
five
parameter logistic curve. IÕve already done this using a 
four
parameter logistic using
the following formula:
A - D
y = ( ------------- ) + D
1 + (x/C)^B
...And now IÕm urgently looking for the 5 param. version.
Joerg
===
Subject: Re: Five Parameter Logistics
> IÕm looking for a formula to fit a set of data 
to a so
called five
> parameter logistic curve. IÕve already done this using a
four
> parameter logistic using
> the following formula:
> A - D
> y = ( ------------- ) + D
> 1 + (x/C)^B
> ...And now IÕm urgently looking for the 5 param. version.
> Joerg
If you expect your data to follow the functional form
exactly, it
seems you have a set of nonlinear equations to solve. If your
data has
noise, you have a nonlinear least squares problem. There
exists
software for both types of problems.
===
Subject: Re: Five Parameter Logistics
>IÕm looking for a formula to fit a set of data 
to a so called
five
>parameter logistic curve. IÕve already done this using a 
four
>parameter logistic using
>the following formula:
> A - D
> y = ( ------------- ) + D
> 1 + (x/C)^B
>...And now IÕm urgently looking for the 5 param. version.
>Joerg
so what is the fundamental difference?
you will use a nonlinear least squares estimator anyway.
see
http://plato.la.asu.edu/topics/problems/nlolsq.html
hth
peter
===
Subject: Order of the complex numbers and its consequences
Order of the complex numbers and its consequences
In this work we define the complex number order and the power
of complex
number base and real number index by means of the limit of a
monotonic
trigonometric function;and from these we define the pi number.
In: http://www.telecable.es/personales/carloman/
===
Subject: Partitioning by hyperplanes
sorry if this looks out of place here, but it is related to
linear
optimization, which after all is kind of related to numerical
analysis...
The question itself also looks trivial, but I canÕt 
find this
explicitly stated. I will be grateful for the answer or a
pointer to a
source:
Given m hyperplanes in R^n (m>n), what is the maximal number
of
non-empty disjoint convex segments that may be created?
A recursive answer I was able to come up with looks very
messy, even
for n=2, so I am looking for something more elegant. I am
sure there
must be a simple and nice formula for this somewhere.
Radek Zakrzewski
===
Subject: Re: HELP!!! a set theory problem.
>IÕve got a problem:
>Given R is the set of real numbers.
>define: (x1, y1) < (x2, y2) if x1 < x2 OR ( x1 = x2 AND y1 <
y2)
>define: f is monotone increasing if f((x1, y1)) < f((x2, y2))
for every
>(x1, y1) < (x2, y2)
>Question: Is there such a monotone increasing function f
which is also a
>bijection from R x R to R ??? Prove it.
There is no monotone increasing function, even without
bijection. For all x, f(x, 1) must be strictly greater
than f(x, 0), so this gives uncountably many disjoint
open intervals.
--
This address is for information only. I do not claim that
these views
are those of the Statistics Department or of Purdue
University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
===
Subject: Re: small eigenvalues
> Suppose 0<=l1 <= l2 <= ... <=ln are the eigenvalues of a
real
> symmetric matrix. Let lr be the first non-zero eigenvalue,
i.e.
> 0=l1=...=l(r-1)<lr<=...<=ln.
> Obviously, numerically, the 0 eigenvalues are not exactly 0.
> There must be a mathematical argument or a heuristic for
deciding
> what r is. Forgive my ignorance, IÕm not a numerical
analysis
> person, but trying to learn these tricks as I run into them.
For a nonnegative real symmetric n x n matrix I use the zero
treshold tolerance
macheps*10*sqrt(n)*|maxeigenvalue|
where macheps=10^{-16} in typical double precision work.
Probably as good as any other empirical formula for
typical n (10^2 to 10^7). Since maxeigenvalue is not usually
computed for large n, it has to be estimated from a matrix
norm.
===
Subject: Re: small eigenvalues
> this is the decision concerning the rank of your matrix, so
you need to
know
> how much larger the eigenvalue lr will be above the
roundoff level.
> you can assume that the inßueneces of rounding errors on the
determination of
> eigenvalues of a symmetric mtrix is like the inßuence of a
symmetric
> eps-perturbation in the original matrix. and there is a
theorem wich says
> if A and B are symmetric and the eigenvalues of A and B are
ordered in
magnitude
> then the difference
> abs(eigenvalue_i(A)-eigenvalue_i(B)) <= norm(A-B)
> hence in the order of some units in the last digit, that
means
> n*eps*norm(A)
> for example in this application.
> hth
> peter
I donÕt really know what an eps-perturbation is, so 
IÕll just
try to apply all you said as a black box. IÕm assuming in 
the
formula you came up with, n is the dimension of A, eps is the
matlab eps, and norm(A) is the L2 norm of A. Is that correct?
WhatÕs a good place to read about these things?
Silviu Minut
===
Subject: wigner-ville distribution
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i14McRG10159;
sir,
can you help me find a web site where i can find 
the
applications,
algorithms and theories of wigner-ville distribution.your
help is
greatly appreciated.
===
Subject: Re: wigner-ville distribution
> sir,
> can you help me find a web site where i can find 
the
applications,
> algorithms and theories of wigner-ville distribution.your
help is
> greatly appreciated.
Try searching on short term Fourier transform and
time-frequency
analysis.
Good luck,
OUP
===
Subject: numerical boundary conditions
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i150HqJ18894;
i am trying to solve a hyperbolic PDE, using the Lax-Wendroff
method.
I have an initial condition, and a boundary condition at one
edge.
However, with the stencil used, i need to know the solution
at the
other edge as well. How would i best calculate this?
My teacher suggests a linear extrapolation, like this:
u(L) = 2u(L-1) - u(L-2)
But isnÕt this only first degree accuracy? As i 
see it, this
error
will creep in and eventually destroy the whole solution?
(lax-wendroff
is 2nd order) I would prefer something like second order
extrapolation:
u(L) = 3u(L-1) - 3u(L-2) + u(L-3)
Please can someone tell me, which is more correct. If i
wasnÕt clear
enough, i can explain in more detail. The PDE is modeling a
heat
Paramo
===
Subject: Re: C routines for Special Functions
<SNIPError Handling
>IÕm not a fan of the Ôint status = 
gsl_function(...); if
(status)
>{...Õ method of error handling for functions that return
doubles.
>I am especially not a fan of having default error handlers
call
>abort(), like the assert macro in C. Of course IÕve had the
head
>of IT tear me a new asshole when a programmer of mine left
an assert
>in production code that got called 15 minutes before market
open,
>so I canÕt claim to be unbiased.
If that is about automated buying of shares....
Would you have preferred the program to buy shares with a
limit
of $4294967295 ($-1) ?
>GSL does have Ônatural formÕ calling 
conventions available
(more
>code on their side to maintain) but they do not permit error
>checking. I prefer returning NaNÕs that have embedded error
>information. If you donÕt check the return codes, at least
your
>program is more likely to return gibberish than to crash.
Which is a bad thing, of course. (I mean returning gibberish.)
The proper thing to do in a real time show must go on scenario
(like radar tracking of a missile ms before impact) is to
catch the error and substitute a reasonable value, and hope
for
the best. Not to send your defense missile astray.
In the context of mathematical functions I just want to know
that the programmer goofed. If you as a project leader deleted
the programmers assertÕs you would have a very bad time with
me.
If your head of IT agrees with you, I would take my business
elsewhere.
--