15,201 research outputs found
IFSM representation of Brownian motion with applications to simulation
Several methods are currently available to simulate paths of the Brownian
motion. In particular, paths of the BM can be simulated using the properties of
the increments of the process like in the Euler scheme, or as the limit of a
random walk or via L2 decomposition like the Kac-Siegert/Karnounen-Loeve
series.
In this paper we first propose a IFSM (Iterated Function Systems with Maps)
operator whose fixed point is the trajectory of the BM. We then use this
representation of the process to simulate its trajectories. The resulting
simulated trajectories are self-affine, continuous and fractal by construction.
This fact produces more realistic trajectories than other schemes in the sense
that their geometry is closer to the one of the true BM's trajectories. The
IFSM trajectory of the BM can then be used to generate more realistic solutions
of stochastic differential equations
Two-Dimensional Supersymmetric Quantum Mechanics: Two Fixed Centers of Force
The problem of building supersymmetry in the quantum mechanics of two
Coulombian centers of force is analyzed. It is shown that there are essentially
two ways of proceeding. The spectral problems of the SUSY (scalar) Hamiltonians
are quite similar and become tantamount to solving entangled families of Razavy
and Whittaker-Hill equations in the first approach. When the two centers have
the same strength, the Whittaker-Hill equations reduce to Mathieu equations. In
the second approach, the spectral problems are much more difficult to solve but
one can still find the zero-energy ground states.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
Solitary Waves in Massive Nonlinear S^N-Sigma Models
The solitary waves of massive (1+1)-dimensional nonlinear S^N-sigma models
are unveiled. It is shown that the solitary waves in these systems are in
one-to-one correspondence with the separatrix trajectories in the repulsive
N-dimensional Neumann mechanical problem. There are topological (heteroclinic
trajectories) and non-topological (homoclinic trajectories) kinks. The
stability of some embedded sine-Gordon kinks is discussed by means of the
direct estimation of the spectra of the second-order fluctuation operators
around them, whereas the instability of other topological and non-topological
kinks is established applying the Morse index theorem
State determination: an iterative algorithm
An iterative algorithm for state determination is presented that uses as
physical input the probability distributions for the eigenvalues of two or more
observables in an unknown state . Starting form an arbitrary state
, a succession of states is obtained that converges to
or to a Pauli partner. This algorithm for state reconstruction is
efficient and robust as is seen in the numerical tests presented and is a
useful tool not only for state determination but also for the study of Pauli
partners. Its main ingredient is the Physical Imposition Operator that changes
any state to have the same physical properties, with respect to an observable,
of another state.Comment: 11 pages 3 figure
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