4,779 research outputs found
Open Boundaries for the Nonlinear Schrodinger Equation
We present a new algorithm, the Time Dependent Phase Space Filter (TDPSF)
which is used to solve time dependent Nonlinear Schrodinger Equations (NLS).
The algorithm consists of solving the NLS on a box with periodic boundary
conditions (by any algorithm). Periodically in time we decompose the solution
into a family of coherent states. Coherent states which are outgoing are
deleted, while those which are not are kept, reducing the problem of reflected
(wrapped) waves. Numerical results are given, and rigorous error estimates are
described.
The TDPSF is compatible with spectral methods for solving the interior
problem. The TDPSF also fails gracefully, in the sense that the algorithm
notifies the user when the result is incorrect. We are aware of no other method
with this capability.Comment: 21 pages, 4 figure
Pairs of Frequency-based Nonhomogeneous Dual Wavelet Frames in the Distribution Space
In this paper, we study nonhomogeneous wavelet systems which have close
relations to the fast wavelet transform and homogeneous wavelet systems. We
introduce and characterize a pair of frequency-based nonhomogeneous dual
wavelet frames in the distribution space; the proposed notion enables us to
completely separate the perfect reconstruction property of a wavelet system
from its stability property in function spaces. The results in this paper lead
to a natural explanation for the oblique extension principle, which has been
widely used to construct dual wavelet frames from refinable functions, without
any a priori condition on the generating wavelet functions and refinable
functions. A nonhomogeneous wavelet system, which is not necessarily derived
from refinable functions via a multiresolution analysis, not only has a natural
multiresolution-like structure that is closely linked to the fast wavelet
transform, but also plays a basic role in understanding many aspects of wavelet
theory. To illustrate the flexibility and generality of the approach in this
paper, we further extend our results to nonstationary wavelets with real
dilation factors and to nonstationary wavelet filter banks having the perfect
reconstruction property
Wavelets as a variational basis of the XY model
We use Daubechies' orthonormal compact wavelets as a variational basis for
the model in two and three dimensions. Assuming that the fluctuations of
the wavelet coefficients are Gaussian and uncorrelated, minimization of the
free energy yields the fluctuation strength of wavelet coefficients at
different scales, from which observables can be computed. This model is able to
describe the low-temperature phase and makes a prediction about the phase
transition temperature.Comment: 3 pages, postscript. Contribution to the Lattice 93 workshop (Dallas,
Texas, October 1993
Surface Comparison with Mass Transportation
We use mass-transportation as a tool to compare surfaces (2-manifolds). In
particular, we determine the "similarity" of two given surfaces by solving a
mass-transportation problem between their conformal densities. This mass
transportation problem differs from the standard case in that we require the
solution to be invariant under global M\"obius transformations. Our approach
provides a constructive way of defining a metric in the abstract space of
simply-connected smooth surfaces with boundary (i.e. surfaces of disk-type);
this metric can also be used to define meaningful intrinsic distances between
pairs of "patches" in the two surfaces, which allows automatic alignment of the
surfaces. We provide numerical experiments on "real-life" surfaces to
demonstrate possible applications in natural sciences
A Deterministic Analysis of Decimation for Sigma-Delta Quantization of Bandlimited Functions
We study Sigma-Delta () quantization of oversampled bandlimited
functions. We prove that digitally integrating blocks of bits and then
down-sampling, a process known as decimation, can efficiently encode the
associated bit-stream. It allows a large reduction in the
bit-rate while still permitting good approximation of the underlying
bandlimited function via an appropriate reconstruction kernel. Specifically, in
the case of stable th order schemes we show that the
reconstruction error decays exponentially in the bit-rate. For example, this
result applies to the 1-bit, greedy, first-order scheme
Wavelet-induced renormalization group for the Landau-Ginzburg model
The scale hierarchy of wavelets provides a natural frame for renormalization.
Expanding the order parameter of the Landau-Ginzburg/ model in a basis
of compact orthonormal wavelets explicitly exhibits the coupling between scales
that leads to non-trivial behavior. The locality properties of Daubechies'
wavelets enable us to derive the qualitative renormalization flow of the
Landau-Ginzburg model from Gaussian fluctuations in wavelet space.Comment: LATTICE99(Renormalization), LaTeX, 3 page
Weak Coherent State Path Integrals
Weak coherent states share many properties of the usual coherent states, but
do not admit a resolution of unity expressed in terms of a local integral. They
arise e.g. in the case that a group acts on an inadmissible fiducial vector.
Motivated by the recent Affine Quantum Gravity Program, the present article
studies the path integral representation of the affine weak coherent state
matrix elements of the unitary time-evolution operator. Since weak coherent
states do not admit a resolution of unity, it is clear that the standard way of
constructing a path integral, by time slicing, is predestined to fail. Instead
a well-defined path integral with Wiener measure, based on a continuous-time
regularization, is used to approach this problem. The dynamics is rigorously
established for linear Hamiltonians, and the difficulties presented by more
general Hamiltonians are addressed.Comment: 21 pages, no figures, accepted by J. Math. Phy
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