Type - valuation of the object being auctioned. Like on page 28 here: http://books.google.ru/books?id=fIPSYiNYvK4C Simplest setup of the Maximum principle gives you an ODE so there might be a lot of examples.
yup, eigenmodes of the Schrodinger equation for instance (which may be represented as eigenvectors of a Hamiltonian matrix), but also normal modes of acoustic/photonic cavities, or even the linear heat/diffusion equation. these all do have in common, though, that they are 'steady-state' solutions of a temporal problem, so they aren't totally unrelated to time.
ah, since they describe steady-states, I guess I should say "the principal eigenvector of the Hamiltonian". (Or it is eigenfunction?)
Is the Hamiltonian matrix a dynamics matrix, which tells you how to iterate the system (i.e. like the transition matrix for a Markov Chain)? If so, that would imply that the system can be described as a finite-dimensional vector.
Hmm, not quite a transition matrix. The equation for quantum wave evolution is the schrodinger equation. (See http://en.wikipedia.org/wiki/Schrodinger_equation under "General Quantum System", the first equation).
To convert the Hamiltonian (H) into a time-iteration matrix (call that M) for a time step dt, you'd have M = exp(-i*dt*H/hbar) (http://en.wikipedia.org/wiki/Matrix_exponentiation#Linear_differential_equations) which is approximately 1-i*dt*H/hbar for short timesteps dt. All the eigenvalues of M are complex numbers with magnitude 1 (M is a unitary matrix)... so they are all principal eigenvalues.
Usually, Hamiltonians are inifinite-dimensional since they model (for example) a particle which can occupy an infinite number of positions. Physicists seem to get away with using infinite dimensional matrices due to some math details I never learned (Hilbert spaces and the spectral theorem and such).
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
krja![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
Re: differential equations
Date: 2010-03-01 11:43 am (UTC)Re: differential equations
Date: 2010-03-01 06:28 pm (UTC)Re: differential equations
Date: 2010-03-06 01:04 pm (UTC)Simplest setup of the Maximum principle gives you an ODE so there might be a lot of examples.
(no subject)
Date: 2010-03-01 06:20 pm (UTC)(no subject)
Date: 2010-03-01 06:23 pm (UTC)(no subject)
Date: 2010-03-01 06:54 pm (UTC)tangentially
Date: 2010-03-01 07:14 pm (UTC)Is the Hamiltonian matrix a dynamics matrix, which tells you how to iterate the system (i.e. like the transition matrix for a Markov Chain)? If so, that would imply that the system can be described as a finite-dimensional vector.
Re: tangentially
Date: 2010-03-02 06:21 am (UTC)To convert the Hamiltonian (H) into a time-iteration matrix (call that M) for a time step dt, you'd have M = exp(-i*dt*H/hbar) (http://en.wikipedia.org/wiki/Matrix_exponentiation#Linear_differential_equations) which is approximately 1-i*dt*H/hbar for short timesteps dt. All the eigenvalues of M are complex numbers with magnitude 1 (M is a unitary matrix)... so they are all principal eigenvalues.
Usually, Hamiltonians are inifinite-dimensional since they model (for example) a particle which can occupy an infinite number of positions. Physicists seem to get away with using infinite dimensional matrices due to some math details I never learned (Hilbert spaces and the spectral theorem and such).
Re: tangentially
Date: 2010-03-02 06:38 am (UTC)After some confusion, I now see how you can have many principal eigenvectors if the eigenvalues aren't restricted to the reals.