702 research outputs found
Solving the Dirichlet problem constructively
The Dirichlet problem is of central importance in both applied and
abstract potential theory. We prove the (perhaps surprising) result that the existence
of solutions in the general case is an essentially nonconstructive proposition: there
is no algorithm which will actually compute solutions for arbitrary domains and
boundary conditions. A corollary of our results is the nonexistence
of constructive
solutions to the NavierStokes
equations of fluid flow. But not all the news is
bad: we provide reasonable conditions, omitted in the classical theory but easily
satisfied, which ensure the computability of solutions
An Algorithm for Finding the Periodic Potential of the Three-dimensional Schrodinger Operator from the Spectral Invariants
In this paper, we investigate the three-dimensional Schrodinger operator with
a periodic, relative to a lattice {\Omega} of R3, potential q. A special class
V of the periodic potentials is constructed, which is easily and constructively
determined from the spectral invariants. First, we give an algorithm for the
unique determination of the potential q in V of the three-dimensional
Schrodinger operator from the spectral invariants that were determined
constructively from the given Bloch eigenvalues. Then we consider the stability
of the algorithm with respect to the spectral invariants and Bloch eigenvalues.
Finally, we prove that there are no other periodic potentials in the set of
large class of functions whose Bloch eigenvalues coincides with the Bloch
eigenvalues of q in V
Different Approaches on Stochastic Reachability as an Optimal Stopping Problem
Reachability analysis is the core of model checking of time systems. For
stochastic hybrid systems, this safety verification method is very little supported mainly
because of complexity and difficulty of the associated mathematical problems. In this
paper, we develop two main directions of studying stochastic reachability as an optimal
stopping problem. The first approach studies the hypotheses for the dynamic programming
corresponding with the optimal stopping problem for stochastic hybrid systems.
In the second approach, we investigate the reachability problem considering approximations
of stochastic hybrid systems. The main difficulty arises when we have to prove the
convergence of the value functions of the approximating processes to the value function
of the initial process. An original proof is provided
An exact solution method for 1D polynomial Schr\"odinger equations
Stationary 1D Schr\"odinger equations with polynomial potentials are reduced
to explicit countable closed systems of exact quantization conditions, which
are selfconsistent constraints upon the zeros of zeta-regularized spectral
determinants, complementing the usual asymptotic (Bohr--Sommerfeld)
constraints. (This reduction is currently completed under a certain vanishing
condition.) In particular, the symmetric quartic oscillators are admissible
systems, and the formalism is tested upon them. Enforcing the exact and
asymptotic constraints by suitable iterative schemes, we numerically observe
geometric convergence to the correct eigenvalues/functions in some test cases,
suggesting that the output of the reduction should define a contractive
fixed-point problem (at least in some vicinity of the pure case).Comment: flatex text.tex, 4 file
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