160 research outputs found
On the new translational shape invariant potentials
Recently, several authors have found new translational shape invariant
potentials not present in classic classifications like that of Infeld and Hull.
For example, Quesne on the one hand and Bougie, Gangopadhyaya and Mallow on the
other have provided examples of them, consisting on deformations of the
classical ones. We analyze the basic properties of the new examples and observe
a compatibility equation which has to be satisfied by them. We study particular
cases of such equation and give more examples of new translational shape
invariant potentials.Comment: 9 pages, uses iopart10.clo, version
Convergence, Hemiplasy, and Correlated Evolution Impact Morphological Diversity Related to a Web-Less Lifestyle in the Two-Clawed Spiders
Traits that independently evolve many times are important for testing hypotheses about correlated evolution and understanding the forces shaping biodiversity. However, population genetics processes can cause hemiplasies (traits determined by genes whose topologies do not match the species tree), leading to a false impression of convergence (homoplasy) and potentially misleading inferences of correlated evolution. Discerning between homoplasies and hemiplasies can be important in cases of rapid radiations and clades with many gene tree incongruences. Here, focusing on two-clawed spiders (Dionycha) and close relatives, we evaluate if the observed distribution of characters related to a web-less lifestyle could be better explained as synapomorphies, homoplasies, or hemiplasies. We find that, although there are several convergences, hemiplasies are also sometimes probable. We discuss how these hemiplasies could affect inferences about correlation and causal relationship of traits. Understanding when and where in the tree of life hemiplasy could have happened is important, preventing false inference of convergent evolution. Furthermore, this understanding can provide alternative hypotheses that can be tested with independent data. Using traits related to the climbing ability of spiders we show that, when hemiplasy is unlikely, adequate model testing can be used to better understand correlated evolution, and propose hypotheses to be tested using controlled behavioral and mechanical experiments.Fil: Azevedo, Guilherme H. F.. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Museo Argentino de Ciencias Naturales "Bernardino Rivadavia"; ArgentinaFil: Bougie, Tierney. San Diego State University; Estados UnidosFil: Carboni, Martín Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Museo Argentino de Ciencias Naturales "Bernardino Rivadavia"; ArgentinaFil: Hedin, Marshal. San Diego State University; Estados UnidosFil: Ramirez, Martin Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Parque Centenario. Museo Argentino de Ciencias Naturales "Bernardino Rivadavia"; Argentin
Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation
In supersymmetric quantum mechanics, shape invariance is a sufficient
condition for solvability. We show that all conventional additive shape
invariant superpotentials that are independent of obey two partial
differential equations. One of these is equivalent to the one-dimensional Euler
equation expressing momentum conservation for inviscid fluid flow, and it is
closed by the other. We solve these equations, generate the set of all
conventional shape invariant superpotentials, and show that there are no others
in this category. We then develop an algorithm for generating all additive
shape invariant superpotentials including those that depend on
explicitly.Comment: 4 page
Time resolved particle dynamics in granular convection
We present an experimental study of the movement of individual particles in a
layer of vertically shaken granular material. High-speed imaging allows us to
investigate the motion of beads within one vibration period. This motion
consists mainly of vertical jumps, and a global ordered drift. The analysis of
the system movement as a whole reveals that the observed bifurcation in the
flight time is not adequately described by the Inelastic Bouncing Ball Model.
Near the bifurcation point, friction plays and important role, and the branches
of the bifurcation do not diverge as the control parameter is increased. We
quantify the friction of the beads against the walls, showing that this
interaction is the underlying mechanism responsible for the dynamics of the
flow observed near the lateral wall
Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
We study a class of quantum two-dimensional models with complex potentials of
specific form. They can be considered as the generalization of a recently
studied model with quadratic interaction not amenable to conventional
separation of variables. In the present case, the property of shape invariance
provides the equidistant form of the spectrum and the algorithm to construct
eigenfunctions analytically. It is shown that the Hamiltonian is
non-diagonalizable, and the resolution of identity must include also the
corresponding associated functions. In the specific case of anharmonic
second-plus-fourth order interaction, expressions for the wave functions and
associated functions are constructed explicitly for the lowest levels, and the
recursive algorithm to produce higher level wave functions is given.Comment: 17 p.
Method for Generating Additive Shape Invariant Potentials from an Euler Equation
In the supersymmetric quantum mechanics formalism, the shape invariance
condition provides a sufficient constraint to make a quantum mechanical problem
solvable; i.e., we can determine its eigenvalues and eigenfunctions
algebraically. Since shape invariance relates superpotentials and their
derivatives at two different values of the parameter , it is a non-local
condition in the coordinate-parameter space. We transform the shape
invariance condition for additive shape invariant superpotentials into two
local partial differential equations. One of these equations is equivalent to
the one-dimensional Euler equation expressing momentum conservation for
inviscid fluid flow. The second equation provides the constraint that helps us
determine unique solutions. We solve these equations to generate the set of all
known -independent shape invariant superpotentials and show that there
are no others. We then develop an algorithm for generating additive shape
invariant superpotentials including those that depend on explicitly,
and derive a new -dependent superpotential by expanding a Scarf
superpotential.Comment: 1 figure, 4 tables, 18 page
Supersymmetrical Separation of Variables for Scarf II Model: Partial Solvability
Recently, a new quantum model - two-dimensional generalization of the Scarf
II - was completely solved analytically by SUSY method for the integer values
of parameter. Now, the same integrable model, but with arbitrary values of
parameter, will be studied by means of supersymmetrical intertwining relations.
The Hamiltonian does not allow the conventional separation of variables, but
the supercharge operator does allow, leading to the partial solvability of the
model. This approach, which can be called as the first variant of
SUSY-separation, together with shape invariance of the model, provides
analytical calculation of the part of spectrum and corresponding wave functions
(quasi-exact-solvability). The model is shown to obey two different variants of
shape invariance which can be combined effectively in construction of energy
levels and wave functions.Comment: 6 p.p., accepted for publication in EP
NMR Experiments on a Three-Dimensional Vibrofluidized Granular Medium
A three-dimensional granular system fluidized by vertical container
vibrations was studied using pulsed field gradient (PFG) NMR coupled with
one-dimensional magnetic resonance imaging (MRI). The system consisted of
mustard seeds vibrated vertically at 50 Hz, and the number of layers N_ell <= 4
was sufficiently low to achieve a nearly time-independent granular fluid. Using
NMR, the vertical profiles of density and granular temperature were directly
measured, along with the distributions of vertical and horizontal grain
velocities. The velocity distributions showed modest deviations from
Maxwell-Boltzmann statistics, except for the vertical velocity distribution
near the sample bottom which was highly skewed and non-Gaussian. Data taken for
three values of N_ell and two dimensionless accelerations Gamma=15,18 were fit
to a hydrodynamic theory, which successfully models the density and temperature
profiles including a temperature inversion near the free upper surface.Comment: 14 pages, 15 figure
Role of friction in pattern formation in oscillated granular layers
Particles in granular flows are often modeled as frictionless (smooth)
inelastic spheres; however, there exist no frictionless grains, just as there
are no elastic grains. Our molecular dynamics simulations reveal that friction
is essential for realistic modeling of vertically oscillated granular layers:
simulations of frictionless particles yield patterns with an onset at a
container acceleration about 30% smaller than that observed in experiments and
simulations with friction. More importantly, even though square and hexagonal
patterns form for a wide range of the oscillation parameters in experiments and
in our simulations of frictional inelastic particles, only stripe patterns form
in the simulations without friction, even if the inelasticity is increased to
obtain as much dissipation as in frictional particles. We also consider the
effect of particle friction on the shock wave that forms each time the granular
layer strikes the container. While a shock wave still forms for frictionless
particles, the height and time dependence of the hydrodynamic fields differ for
the cases with and without friction.Comment: final version appeared in Phys. Rev.
Exceptional orthogonal polynomials and new exactly solvable potentials in quantum mechanics
In recent years, one of the most interesting developments in quantum
mechanics has been the construction of new exactly solvable potentials
connected with the appearance of families of exceptional orthogonal polynomials
(EOP) in mathematical physics. In contrast with families of (Jacobi, Laguerre
and Hermite) classical orthogonal polynomials, which start with a constant, the
EOP families begin with some polynomial of degree greater than or equal to one,
but still form complete, orthogonal sets with respect to some positive-definite
measure. We show how they may appear in the bound-state wavefunctions of some
rational extensions of well-known exactly solvable quantum potentials. Such
rational extensions are most easily constructed in the framework of
supersymmetric quantum mechanics (SUSYQM), where they give rise to a new class
of translationally shape invariant potentials. We review the most recent
results in this field, which use higher-order SUSYQM. We also comment on some
recent re-examinations of the shape invariance condition, which are independent
of the EOP construction problem.Comment: 21 pages, no figure; communication at the Symposium Symmetries in
Science XV, July 31-August 5, 2011, Bregenz, Austri
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