| Topic: |
Science > Physics |
| User: |
"Cheng Cosine" |
| Date: |
10 Mar 2005 05:32:39 PM |
| Object: |
? Measuring eigenvalues |
Hi:
Students are taught in school many theories about PDE and eigenvalues
and eigenfunctions. We also know that one can use appropriate PDE/ODE
to approximate the nature. But then do eigenvalues and eigenfunction have
practical physical meaning can can be measured by experiment?
Thanks,
by Cheng Cosine
Mar/10/2k5 Ut
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| User: "tadchem" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:23:19 PM |
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Cheng Cosine wrote:
Hi:
Students are taught in school many theories about PDE and
eigenvalues
and eigenfunctions. We also know that one can use appropriate PDE/ODE
to approximate the nature. But then do eigenvalues and eigenfunction
have
practical physical meaning can can be measured by experiment?
PDEs and ODEs are the basis of almost all modern physics. When Newton
defined force it was as the time derivative of momentum.
ODEs describe one-dimensional problems. PDEs describe problems in two,
three, or more dimensions.
Eigenfunctions are functions that satisfy linear systems of
differential equations, and eigenvalues are the associated scalar
factors that make eigenfunctions work in those systems of equations.
<http://mathworld.wolfram.com/Eigenvalue.html>
<http://mathworld.wolfram.com/Eigenfunction.html>
The art of the mathematical physicist is to construct the equations in
such a way that they are linear.
This approach has been immensely successful in solving complex problems
regarding things such as electronic energy levels in atoms and
molecules, molecular vibrations, and mechanical vibrations in strings,
membranes, and 3-dimensional structures such as office buildings,
bridges, automobiles, and airplanes.
You see that it is also useful to chemists, musicians, and engineers.
To a chemist, every line in a spectrum is an eigenvalue. To a
musician, every note is an eigenvalue. To an engineer, every vibration
al frequency is an eigenvalue.
Finite element analysis (digital solutions to eigenvalue and
eigenfunction problems) is also used to *approximate* the behavior of
bodies composed of infinitesimal elements (anything with curved
surfaces).
HTH
Tom Davidson
Richmond, VA
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| User: "Gregory L. Hansen" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:42:38 PM |
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In article <d0qlen$o9e$1@vegh.ks.cc.utah.edu>,
Cheng Cosine <acosine@ms13.url.com.tw> wrote:
Hi:
Students are taught in school many theories about PDE and eigenvalues
and eigenfunctions. We also know that one can use appropriate PDE/ODE
to approximate the nature. But then do eigenvalues and eigenfunction have
practical physical meaning can can be measured by experiment?
Thanks,
by Cheng Cosine
Mar/10/2k5 Ut
James had something useful buried in his reply. When we measure line
spectra of a gas, we're directly measuring the energy eigenvalues of
atomic and molecular orbitals. The wavefunctions are the eigenfunctions
corresponding to those energies. We don't NEED to express orbitals in
that way; we could use plane waves (which are momentum eigenfunctions),
Hermite polynomials (which are energy eigenfunctions of the harmonic
oscillator), or other functions.
Standing waves on a string are a simple example of eigenfunctions, as are
the analogous normal modes of a drum head or other systems that can
vibrate. But it's hard to come up with a similar interpretation for
Fourier transforming a heat flow problem, except that it just makes the
math easier.
We can choose to solve for energy eigenvalues or angular momentum
eigenvalues or others that directly represent observable things. If we're
trying to find e.g. energy levels, energy eigenfunctions are a natural
choice. But sometimes they're "just math" and help us to solve a
difficult problem.
--
"I'm giving you the chance to look fate in those pretty eyes of hers
and say, 'Step off, *****. This is my party and you're not invited.'"
-- Chris Shugart, _Testosterone Magazine_
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 02:22:02 AM |
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Let's go back to something more physical.
Suppose we have some irregular geometry and we know there is
diffusion process going on. However, all we know about the
diffusion coefficient is that it is not a constant. Then how can we
determine the e-pairs based upon some measurement?
Moreover, since we don't have the precise PDE, that is, the
system might be governed by diff(u)/pdiff(t) = div( k(t,x)*grad(u) )
or diffusion-reaction eqn diff(u)/pdiff(t) = div( k(t,x)*grad(u) )+g(t,x)*u.
Then how do we determine enough e-pairs to approximately reconstruct
the system responses?
by Cheng Cosine
Mar/11/2k5 UT
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 02:27:22 AM |
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"Cheng Cosine" <acosine@ms13.url.com.tw> wrote in message
news:d0rkfb$6ss$1@vegh.ks.cc.utah.edu...
Let's go back to something more physical.
Suppose we have some irregular geometry and we know there is
diffusion process going on. However, all we know about the
diffusion coefficient is that it is not a constant. Then how can we
determine the e-pairs based upon some measurement?
Moreover, since we don't have the precise PDE, that is, the
system might be governed by diff(u)/pdiff(t) = div( k(t,x)*grad(u) )
or diffusion-reaction eqn diff(u)/pdiff(t) = div(
k(t,x)*grad(u) )+g(t,x)*u.
Then how do we determine enough e-pairs to approximately reconstruct
the system responses?
Let's be more precise, suppose we were Fourier who is going to develop
the conduction law. That is, all we have are experimental data, how do we
first
determine what kind of governing equation that describes the system and then
how do we extract the e-pairs to approximte the system response so that we
need not to do the experiment again and again?
by Cheng Cosine
Mar/11/2k5 UT
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 03:26:39 AM |
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In article <d0rkpb$71h$1@vegh.ks.cc.utah.edu>, "Cheng Cosine" <acosine@ms13.url.com.tw> writes:
"Cheng Cosine" <acosine@ms13.url.com.tw> wrote in message
news:d0rkfb$6ss$1@vegh.ks.cc.utah.edu...
Let's go back to something more physical.
Suppose we have some irregular geometry and we know there is
diffusion process going on. However, all we know about the
diffusion coefficient is that it is not a constant. Then how can we
determine the e-pairs based upon some measurement?
Moreover, since we don't have the precise PDE, that is, the
system might be governed by diff(u)/pdiff(t) = div( k(t,x)*grad(u) )
or diffusion-reaction eqn diff(u)/pdiff(t) = div(
k(t,x)*grad(u) )+g(t,x)*u.
Then how do we determine enough e-pairs to approximately reconstruct
the system responses?
Let's be more precise, suppose we were Fourier who is going to develop
the conduction law. That is, all we have are experimental data, how do we
first
determine what kind of governing equation that describes the system and then
how do we extract the e-pairs to approximte the system response so that we
need not to do the experiment again and again?
The governing equation comes, essentially, from conservation of energy
(for heat) and an assumption of linearity.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 03:24:31 AM |
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In article <d0rkfb$6ss$1@vegh.ks.cc.utah.edu>, "Cheng Cosine" <acosine@ms13.url.com.tw> writes:
Let's go back to something more physical.
Suppose we have some irregular geometry and we know there is
diffusion process going on. However, all we know about the
diffusion coefficient is that it is not a constant. Then how can we
determine the e-pairs based upon some measurement?
Moreover, since we don't have the precise PDE, that is, the
system might be governed by diff(u)/pdiff(t) = div( k(t,x)*grad(u) )
or diffusion-reaction eqn diff(u)/pdiff(t) = div( k(t,x)*grad(u) )+g(t,x)*u.
Then how do we determine enough e-pairs to approximately reconstruct
the system responses?
If the diffusion coefficient is constant then the equation is not
linear and, in general, the concept of eigenvalues and functions does
not apply. In some cases (when k is not an explicit finction of x,t,
only a function of u, the equation can be still brought to a linear
form. If it cannot, forget e-values and solve numerically.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 09:39:04 AM |
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<mmeron@cars3.uchicago.edu> wrote in message
news:jtdYd.25$45.3921@news.uchicago.edu...
...
If the diffusion coefficient is constant then the equation is not
linear and, in general, the concept of eigenvalues and functions does
not apply. In some cases (when k is not an explicit finction of x,t,
only a function of u, the equation can be still brought to a linear
form. If it cannot, forget e-values and solve numerically.
Guess you are trying to say: If the diffusion coefficient is not constant
then the equation is not linear.
But how can that be try? When the diffusion coef is not const, then
the PDE can be parabolic or or types depending on the coef at different
locations. But don't see why it becomes nonlinar. There are litereatures
giving the Green's function for some simple nonconst diffusion coef,
e.g. diffusion in layered media. Since Green's function exist, it is linear.
Furthermore, one can use series expansion to express the Green's function,
then those functions used can be e-functions (at most after some change of
bases).
by Cheng Cosine
Mar/11/2k5 UT
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 10:01:23 PM |
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In article <d0se2o$f5r$1@vegh.ks.cc.utah.edu>, "Cheng Cosine" <acosine@ms13.url.com.tw> writes:
<mmeron@cars3.uchicago.edu> wrote in message
news:jtdYd.25$45.3921@news.uchicago.edu...
...
If the diffusion coefficient is constant then the equation is not
linear and, in general, the concept of eigenvalues and functions does
not apply. In some cases (when k is not an explicit finction of x,t,
only a function of u, the equation can be still brought to a linear
form. If it cannot, forget e-values and solve numerically.
Guess you are trying to say: If the diffusion coefficient is not constant
then the equation is not linear.
Oh, yes, sorry. lost the "not".
But how can that be try? When the diffusion coef is not const, then
the PDE can be parabolic or or types depending on the coef at different
locations. But don't see why it becomes nonlinar.
I should've been more specific, it depends what the diffusion coef
depends on. A common practical situation I had in mind is this of
heat propagation when the conductivity coefficient is a function of
the temperature. In this case, obviously, the equation is not linear.
Mind you, it can be made linear.
There are litereatures
giving the Green's function for some simple nonconst diffusion coef,
e.g. diffusion in layered media. Since Green's function exist, it is linear.
Indeed.
Furthermore, one can use series expansion to express the Green's function,
then those functions used can be e-functions (at most after some change of
bases).
That doesn't necessarily follow. It is quite common to express the
Green's function as a series of eigenfunctions of some other equation.
In this case you've a series expansion but the functions used are not
the e-functions of your equation.
Now, if by "some change of bases" you mean that your e-functions, if
not directly those used in the Green's function, can be expressed as
the linear combinations of those, then sure, this is true, but it is
true for nearly every function.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 10:17:53 AM |
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Yes, that's right: Even if the diffusion coefficient changes with
position, the equation is still linear. Green functions still exist,
as do eigenvalues/-functions. They are, however, much harder to find!
Linearity is the key requirement, meaning that if y1 and y2 are each
solutions of the equation, then a*y1+b*y2 is also a solution.
As for determining what kind of equation controls a system---that's
where insight and intuition trump any amount of canned logic any day.
I wish I could just plug my results into a computer and get what kind
of equation it is out, but instead you almost always start with some
mental picture of what might be going on, and work out the results of
that model in detail. You then test to see if your conclusions match
what is going on in the experiment. Very slow, but it's the only way
we've managed to get to work. The view in physics textbooks of
following a predefined path to a set goal works only after the problem
has been around for at least fifty years.
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| User: "David Cross" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 05:58:36 PM |
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On Thu, 10 Mar 2005 16:32:39 -0700, "Cheng Cosine" <acosine@ms13.url.com.tw>
wrote:
Hi:
Students are taught in school many theories about PDE and eigenvalues
and eigenfunctions. We also know that one can use appropriate PDE/ODE
to approximate the nature. But then do eigenvalues and eigenfunction have
practical physical meaning can can be measured by experiment?
If you mean the solutions to systems of differential equations, the
eigenvalues give the degree of dependence on the exponential term in the
solutions. Does this help?
---
David Cross
dcross1 AT shaw DOT ca
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 06:00:42 PM |
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You have absolutely no idea. If there is any mathematical idea you
will learn that can be applied to physics and will be useful in the
future, it will be eigenvalues, eigenvectors, and eigenfunctions.
One of the main reasons for this is because higher-dimensional systems
can be modelled using multiple partial differential equations. For
linear systems, or for nonlinear systems near a a stationary point,
many PDEs can be turned into linear algebra problems, which are best
solved by finding the eigenvectors of matricies.
In mechanical systems, for instance, the laws of motion can often be
turned into eigenvalue problems, where the resulting eigenvalues are
the fundamental frequencies of oscillation of the system, and the
eigenvectors show how the system osscilating at that frequency would
move.
In electromagnetic systems, Poissons equation, with appropriate
boundary conditions, can often be used to solve for the voltage in an
empty region of space. Poissons equation can be expressed as an
eigenvalue problem. The eigenfunction is the electric potential in
that region of space. If the system is time-dependent, then the
eigenvalue equations will often return frequencies of osscilation much
like in mechanical systems.
Quantum mechanics is based largely on the concept of operators. These
operators can often be expressed as matricies, but may also be
differential operators. The eigenvalues of these operators are things
that can actually be measured, such as energy or momentum of a system,
and the eigenfunctions are the quantum-mechanical wave-functions that
can be directly used to predict probabilities and cross-sections, and
so on. For instance, the energy levels of the hydrogen atom are
eigenvalues of the so-called Hamiltonian operator (better understood as
the operator which returns the energy of a system.).
These are just some of the most common uses for eigenvalues and
eigenvectors in modern physics. Knowledge of them is absolutely
essential nowadays for anything beyond sophomore level physics in
College.
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 11:22:44 PM |
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But is there any general procedure or some common features in
measuring the e-value/-vector/-function? For example, what do
the e-values of a diffusion process mean? What about the e-values
of Navier-Stokes equation?
Thanks,
by Cheng Cosine
Mar/10/2k5 UT
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| User: "David Cross" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:09:29 AM |
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"Cheng Cosine" <acosine@ms13.url.com.tw> wrote in message
news:d0r9v6$3ai$1@vegh.ks.cc.utah.edu...
But is there any general procedure or some common features in
measuring the e-value/-vector/-function? For example, what do
the e-values of a diffusion process mean? What about the e-values
of Navier-Stokes equation?
Well, in matrix algebra, finding the eigenvalues is a process that can be done
without error as long as you know what a determinant is and can factor a
polynomial.
--
David Cross
dcross1 AT shaw DOT ca
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:18:41 AM |
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In article <tCaYd.639852$6l.521553@pd7tw2no>, "David Cross" <nospam@spammenot.com> writes:
"Cheng Cosine" <acosine@ms13.url.com.tw> wrote in message
news:d0r9v6$3ai$1@vegh.ks.cc.utah.edu...
But is there any general procedure or some common features in
measuring the e-value/-vector/-function? For example, what do
the e-values of a diffusion process mean? What about the e-values
of Navier-Stokes equation?
Well, in matrix algebra, finding the eigenvalues is a process that can be done
without error as long as you know what a determinant is and can factor a
polynomial.
Even that much isn't needed and, in fact, for matrices larger than
3x3 or 4x4 (in special cases) finding the eigenvalues this way is
tedious. Algorithms exist where one diagonalzes a matix using nothing
more than matrix multiplications.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:55:56 AM |
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<mmeron@cars3.uchicago.edu> wrote in message
news:5LaYd.22$45.3942@news.uchicago.edu...
In article <tCaYd.639852$6l.521553@pd7tw2no>, "David Cross"
<nospam@spammenot.com> writes:
Well, in matrix algebra, finding the eigenvalues is a process that can be
done
without error as long as you know what a determinant is and can factor a
polynomial.
Even that much isn't needed and, in fact, for matrices larger than
3x3 or 4x4 (in special cases) finding the eigenvalues this way is
tedious. Algorithms exist where one diagonalzes a matix using nothing
more than matrix multiplications.
Well, simply because it has been proven that poly of order > 5 have no
radical solution. Regarding to diagonalizing the matrix, that is equivalent
in some sense to solve a finite-dim linear system. But a PDE is inf-dim
system.
So there is a question of how many e-functions should one take to ensure
most characteristics of the system are properly modeled.
Lastly, back to my original question, given a diffusion process, how to
experimentally determine the e-values/e-function since we know the analytic
expression of the solution in terms of e-functions is only available for
problem with simple geometry and bcs. While they are interesting as
classroom
discussions, how to determine them for more practical situations, e.g.
non-constant
diffusion coef, irregular geometry, etc.
by Cheng Cosine
Mar/10/2k5 Ut
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 01:22:56 AM |
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In article <d0rfdt$5kn$1@vegh.ks.cc.utah.edu>, "Cheng Cosine" <acosine@ms13.url.com.tw> writes:
<mmeron@cars3.uchicago.edu> wrote in message
news:5LaYd.22$45.3942@news.uchicago.edu...
In article <tCaYd.639852$6l.521553@pd7tw2no>, "David Cross"
<nospam@spammenot.com> writes:
Well, in matrix algebra, finding the eigenvalues is a process that can be
done
without error as long as you know what a determinant is and can factor a
polynomial.
Even that much isn't needed and, in fact, for matrices larger than
3x3 or 4x4 (in special cases) finding the eigenvalues this way is
tedious. Algorithms exist where one diagonalzes a matix using nothing
more than matrix multiplications.
Well, simply because it has been proven that poly of order > 5 have no
radical solution.
That's not the problem. In practice, it is exceedingly rare that you
solve even a 3rd order polynomial equation by radicals, numerical
solutions are faster. And the numerical algorithms work for an
arbitrary order of polynomials, no cutoff at 4. However, for large
systems diagonalization of a matrix is both faster and more
computationally stable than factoring a polynomial.
Regarding to diagonalizing the matrix, that is equivalent
in some sense to solve a finite-dim linear system. But a PDE is inf-dim
system.
Neverhteless, quite often you can get all the eigenvaluesof a PDE,
much faster than finding those of a matrix, if you make an appropriate
use of symmetries. That's cheating abit, of course (as it is akin to
picking the base in which the matrix is already diagonalized). But,
for nice symmetrical situations it works well. And when you can't do
it, well, the tradition of approaching an infinitely dimensional
system by a finite dimensional one and taking a limit goes all the way
back to Newton.
So there is a question of how many e-functions should one take to ensure
most characteristics of the system are properly modeled.
That can only be decided on a cse by case basis.
Lastly, back to my original question, given a diffusion process, how to
experimentally determine the e-values/e-function since we know the analytic
expression of the solution in terms of e-functions is only available for
problem with simple geometry and bcs.
Indeed.
While they are interesting as
classroom
discussions, how to determine them for more practical situations, e.g.
non-constant
diffusion coef, irregular geometry, etc.
Again, case by case basis. You can often get quite far by replacing
the irregular geometry by a sufficently close symmetrical one ("assume
a spherical cow":-)) And when all fails you forget about e-functions
and do finite element analysis.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "Randy Poe" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:52:35 PM |
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Cheng Cosine wrote:
But is there any general procedure or some common features in
measuring the e-value/-vector/-function? For example, what do
the e-values of a diffusion process mean? What about the e-values
of Navier-Stokes equation?
I don't know about general statements.
If you have a covariance matrix, for instance on the
errors in measuring a position (lat,lon), then I believe
the eigenvectors give you the major and minor axes
of the ellipse of uncertainty, and the eigenvalues
are related to the widths of those axes.
In radar, there is huge interest in what is called the
covariance matrix E[XX'] where X is a vector of
observations (indexed by, for instance, time, distance,
angle, and array element). The eigenvalues and
eigenvectors characterize the background clutter,
and knowledge of them is used to pull signals out
of that clutter.
- Randy
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| User: "Gregory L. Hansen" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 01:50:44 PM |
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In article <1110567155.730927.318860@l41g2000cwc.googlegroups.com>,
Randy Poe <poespam-trap@yahoo.com> wrote:
Cheng Cosine wrote:
But is there any general procedure or some common features in
measuring the e-value/-vector/-function? For example, what do
the e-values of a diffusion process mean? What about the e-values
of Navier-Stokes equation?
I don't know about general statements.
If you have a covariance matrix, for instance on the
errors in measuring a position (lat,lon), then I believe
the eigenvectors give you the major and minor axes
of the ellipse of uncertainty, and the eigenvalues
are related to the widths of those axes.
I'm going to expand on that because I think it was kind of vague. At
least as it concerns error propagation, since that's the only place where
I know how to use a covariance matrix.
The typical physics student will (or should) come away from lab classes
knowing that if he has a set of measured values x_i with uncertainties
sigma_i, the derived uncertainty for a function f(x1,x2...) is calculated
by
sigma_f^2 = (@f/@x_i)^2 * sigma_i^2
where @f/@x is a partial derivative.
That procedure rolls in a few assumptions, including the assumption that
the measured parameters are independent. Independence is expressed by a
diagonal covariance matrix. The expression above is a simplification of
the more general expression
sigma_f^2 = (@f/@x_i)(@f/@x_j) * sigma_ij^2
where the sigma_ij are the elements of the covariance matrix. The values
for i != j give an indication of the strength of the dependency between
x_i and x_j.
--
"The polhode rolls without slipping on the herpolhode lying in the
invariable plane." -- Goldstein, Classical Mechanics 2nd. ed., p207.
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 11:46:11 PM |
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There *are* limits to how useful the eigenvalues, etc. are. Not all
problems are easily converted to a linear algebra problem. One of the
prime requirements, in fact, is that the operator is linear, or almost
linear. Another one is that the equation(s) is(are) homogenous, e.g.
can be expressed as A x = a x, not A x = a x + b.
I don't think I have ever seen the diffusion equation expressed in
terms of e-values, etc., and I'm not sure if it's possible (it's almost
certainly not useful). The Navier-Stokes equation is non-linear, and
so unless you make some pretty serious approximations, this technique
is utterly useless for Navier-Stokes. The "eigenfunction" of this
equation is completely undefined. What you would want are the
solutions, which are a whole other story.
So I guess that the answer to your question is no. There is no general
way to turn physics problems into eigenvalue problems. But when you
can do it, the tools you have available are *much* more powerful than
otherwise.
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| User: "Cheng Cosine" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 11:58:42 PM |
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<jamesahart79@gmail.com> wrote in message
news:1110519971.012885.82690@g14g2000cwa.googlegroups.com...
There *are* limits to how useful the eigenvalues, etc. are. Not all
problems are easily converted to a linear algebra problem. One of the
prime requirements, in fact, is that the operator is linear, or almost
linear. Another one is that the equation(s) is(are) homogenous, e.g.
can be expressed as A x = a x, not A x = a x + b.
....
This is odd. For whatever linear system, the solution, if exists and
satisfies somoe convergent conditions, can always be expressed in terms of
eigenfunctions. This is the idea of generalized Fourier expansion.
As a specific example, the diffusion equation can be solved with separation
of
variables and then expressed in terms of eigenfunctions. The problem now is
how to setup a practical experimental proceudre to determine the
eigenvalues.
by Cheng Cosine
Mar/10/2k5 Ut
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
10 Mar 2005 11:59:20 PM |
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In article <1110519971.012885.82690@g14g2000cwa.googlegroups.com>, writes:
There *are* limits to how useful the eigenvalues, etc. are. Not all
problems are easily converted to a linear algebra problem. One of the
prime requirements, in fact, is that the operator is linear, or almost
linear. Another one is that the equation(s) is(are) homogenous, e.g.
can be expressed as A x = a x, not A x = a x + b.
Not quite true, as knowledge of the solutions of the homogenous allows
you to construct the Green's function you use to solve the
inhomogenous.
I don't think I have ever seen the diffusion equation expressed in
terms of e-values, etc., and I'm not sure if it's possible (it's almost
certainly not useful). The Navier-Stokes equation is non-linear, and
so unless you make some pretty serious approximations, this technique
is utterly useless for Navier-Stokes. The "eigenfunction" of this
equation is completely undefined. What you would want are the
solutions, which are a whole other story.
So I guess that the answer to your question is no. There is no general
way to turn physics problems into eigenvalue problems. But when you
can do it, the tools you have available are *much* more powerful than
otherwise.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:20:58 AM |
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Mati Meron:
Touche'! Yep, that's what you do. This is my first time posting, and
I'm gonna have to learn to slow down a bit.
Cheng Cosine:
What you say about the diffusion equation is fair enough. One thing to
remember is that the eigenfunctions depend on the boundary conditions.
However, it seems to me that the physical meaning is actually pretty
straight-forward: The eigenvalue of the diffusion equation is the
inverse of the amount of time it will take for the magnitude of the
eigenfunction to decay by a factor e. In order for the equation to be
self-consistent, it must also be the square of the spatial wave-length,
e.g. the time component will be exp(-Ct) and the space component will
be sin(sqrt(C) t ) or cos( sqrt(C) t ). Thus an experiment to measure
the eigenvalue would measure the decay rate of any disturbances in
whatever medium is diffusing.
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| User: "" |
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| Title: Re: ? Measuring eigenvalues |
11 Mar 2005 12:54:49 AM |
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In article <1110522058.052202.210750@z14g2000cwz.googlegroups.com>, writes:
Mati Meron:
Touche'! Yep, that's what you do. This is my first time posting, and
I'm gonna have to learn to slow down a bit.
Oh, you're doing fine. No prob.
Cheng Cosine:
What you say about the diffusion equation is fair enough. One thing to
remember is that the eigenfunctions depend on the boundary conditions.
However, it seems to me that the physical meaning is actually pretty
straight-forward: The eigenvalue of the diffusion equation is the
inverse of the amount of time it will take for the magnitude of the
eigenfunction to decay by a factor e. In order for the equation to be
self-consistent, it must also be the square of the spatial wave-length,
e.g. the time component will be exp(-Ct) and the space component will
be sin(sqrt(C) t ) or cos( sqrt(C) t ). Thus an experiment to measure
the eigenvalue would measure the decay rate of any disturbances in
whatever medium is diffusing.
That should do, indeed. What confuses matters is that you've many
eigenvalues. So, what you'll measure this way will be the dominant
one for the specific situation.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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