888 research outputs found
Scattering and delay time for 1D asymmetric potentials: the step-linear and the step-exponential cases
We analyze the quantum-mechanical behavior of a system described by a
one-dimensional asymmetric potential constituted by a step plus (i) a linear
barrier or (ii) an exponential barrier. We solve the energy eigenvalue equation
by means of the integral representation method, classifying the independent
solutions as equivalence classes of homotopic paths in the complex plane.
We discuss the structure of the bound states as function of the height U_0 of
the step and we study the propagation of a sharp-peaked wave packet reflected
by the barrier. For both the linear and the exponential barrier we provide an
explicit formula for the delay time \tau(E) as a function of the peak energy E.
We display the resonant behavior of \tau(E) at energies close to U_0. By
analyzing the asymptotic behavior for large energies of the eigenfunctions of
the continuous spectrum we also show that, as expected, \tau(E) approaches the
classical value for E -> \infty, thus diverging for the step-linear case and
vanishing for the step-exponential one.Comment: 14 pages, 10 figure
Semi-regular continued fractions and an exact formula for the moments of the Minkowski question mark function
This paper continues investigations on the integral transforms of the
Minkowski question mark function. In this work we finally establish the
long-sought formula for the moments, which does not explicitly involve regular
continued fractions, though it has a hidden nice interpretation in terms of
semi-regular continued fractions. The proof is self-contained and does not rely
on previous results by the author.Comment: 8 page
Modelling the habitat of the endangered Carpentarian Grasswren (Amytornis dorotheae): The importance of spatio-temporal habitat availability in a fire prone landscape
Species distribution modelling (SDM), a tool increasingly adopted to quantify geographic range size, often predicts species’ distributions as static. However, habitat availability may exhibit spatial and temporal variation when dynamic processes, such as fire, determine suitability. Static SDM approaches may not satisfactorily represent this dynamic process. We investigated the potential use of SDM to quantify dynamic habitat availability by applying the MaxEnt SDM technique to model the habitat of the Carpentarian Grasswren (Amytornis dorotheae), an endangered Australian passerine dependent on long unburnt vegetation in a fire prone system. By adjusting a typical SDM approach to incorporate the dynamic nature of fire, we modelled the spatio-temporal variation of suitable habitat over 12 years and compared it to a static modelling approach. Incorporating fire as a dynamic process increased the importance of the fire variable to models (from 35% permutation importance) and improved model performance, as evaluated by the AUC using cross-validation. Our dynamic model revealed sizeable temporal variation in the area and spatial arrangement of suitable habitat that was not apparent in the static model. This result may partly solve the mystery of why the species occurs as widely separated populations despite the presence of seemingly suitable intervening habitat. In areas where the species is no longer found, habitat availability was less consistent due to frequent fire, and fire refugia was more limited and isolated, when compared to sites with recent records. These results demonstrate that, when compared to a static approach, a dynamic SDM approach can lead to improved understanding of dynamic ecological processes, and their impact on a species
Special functions associated to a certain fourth order differential equation
We develop a theory of "special functions" associated to a certain fourth
order differential operator on depending
on two parameters . For integers with
this operator extends to a self-adjoint operator on
with discrete spectrum. We find a closed
formula for the generating functions of the eigenfunctions, from which we
derive basic properties of the eigenfunctions such as orthogonality,
completeness, -norms, integral representations and various recurrence
relations.
This fourth order differential operator arises as the
radial part of the Casimir action in the Schr\"odinger model of the minimal
representation of the group , and our "special functions" give
-finite vectors
Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
We prove sharp stability estimates for the variation of the eigenvalues of
non-negative self-adjoint elliptic operators of arbitrary even order upon
variation of the open sets on which they are defined. These estimates are
expressed in terms of the Lebesgue measure of the symmetric difference of the
open sets. Both Dirichlet and Neumann boundary conditions are considered
Resonant Magnetic Vortices
By using the complex angular momentum method, we provide a semiclassical
analysis of electron scattering by a magnetic vortex of Aharonov-Bohm-type.
Regge poles of the -matrix are associated with surface waves orbiting around
the vortex and supported by a magnetic field discontinuity. Rapid variations of
sharp characteristic shapes can be observed on scattering cross sections. They
correspond to quasibound states which are Breit-Wigner-type resonances
associated with surface waves and which can be considered as quantum analogues
of acoustic whispering-gallery modes. Such a resonant magnetic vortex could
provide a new kind of artificial atom while the semiclassical approach
developed here could be profitably extended in various areas of the physics of
vortices.Comment: 6 pages, 7 figure
Transfinite thin plate spline interpolation
Duchon's method of thin plate splines defines a polyharmonic interpolant to
scattered data values as the minimizer of a certain integral functional. For
transfinite interpolation, i.e. interpolation of continuous data prescribed on
curves or hypersurfaces, Kounchev has developed the method of polysplines,
which are piecewise polyharmonic functions of fixed smoothness across the given
hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has
introduced boundary conditions of Beppo Levi type to construct a semi-cardinal
model for polyspline interpolation to data on an infinite set of parallel
hyperplanes. The present paper proves that, for periodic data on a finite set
of parallel hyperplanes, the polyspline interpolant satisfying Beppo Levi
boundary conditions is in fact a thin plate spline, i.e. it minimizes a Duchon
type functional
Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function
A new method is presented for Fourier decomposition of the Helmholtz Green
Function in cylindrical coordinates, which is equivalent to obtaining the
solution of the Helmholtz equation for a general ring source. The Fourier
coefficients of the Helmholtz Green function are split into their half
advanced+half retarded and half advanced-half retarded components. Closed form
solutions are given for these components in terms of a Horn function and a
Kampe de Feriet function, respectively. The systems of partial differential
equations associated with these two-dimensional hypergeometric functions are
used to construct a fourth-order ordinary differential equation which both
components satisfy. A second fourth-order ordinary differential equation for
the general Fourier coefficent is derived from an integral representation of
the coefficient, and both differential equations are shown to be equivalent.
Series solutions for the various Fourier coefficients are also given, mostly in
terms of Legendre functions and Bessel/Hankel functions. These are derived from
the closed form hypergeometric solutions or an integral representation, or
both. Numerical calculations comparing different methods of calculating the
Fourier coefficients are presented
An optimal series expansion of the multiparameter fractional Brownian motion
We derive a series expansion for the multiparameter fractional Brownian
motion. The derived expansion is proven to be rate optimal.Comment: 21 pages, no figures, final version, to appear in Journal of
Theoretical Probabilit
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