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

Continuum simulations of shocks and patterns in vertically oscillated granular layers

We study interactions between shocks and standing-wave patterns in vertically oscillated layers of granular media using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order. We simulate a layer of grains atop a plate that oscillates sinusoidally in the direction of gravity. Standing waves form stripe patterns when the accelerational amplitude of the plate's oscillation exceeds a critical value. Shocks also form with each collision between the layer and the plate; we show that pressure gradients formed by these shocks cause the flow to reverse direction within the layer. This reversal leads to an oscillatory state of the pattern that is subharmonic with respect to the plate's oscillation. Finally, we study the relationship between shocks and patterns in layers oscillated at various frequencies and show that the pattern wavelength increases monotonically as the shock strength increases.Comment: 12 pages, 9 figure

Supersymmetry, shape invariance and the Legendre equations

In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard solutions. This is a very general and important result and appears in many problems in physics (for example, the multipole expansion etc). We study these equations from an operator point of view, much like the harmonic oscillator, and show that there is an underlying shape invariance symmetry in these systems responsible for their solubility. We bring out various interesting features resulting from this analysis from the shape invariance point of view.Comment: 4 pages, 1 figure; to appear in PL

Onset of Patterns in an Ocillated Granular Layer: Continuum and Molecular Dynamics Simulations

We study the onset of patterns in vertically oscillated layers of frictionless dissipative particles. Using both numerical solutions of continuum equations to Navier-Stokes order and molecular dynamics (MD) simulations, we find that standing waves form stripe patterns above a critical acceleration of the cell. Changing the frequency of oscillation of the cell changes the wavelength of the resulting pattern; MD and continuum simulations both yield wavelengths in accord with previous experimental results. The value of the critical acceleration for ordered standing waves is approximately 10% higher in molecular dynamics simulations than in the continuum simulations, and the amplitude of the waves differs significantly between the models. The delay in the onset of order in molecular dynamics simulations and the amplitude of noise below this onset are consistent with the presence of fluctuations which are absent in the continuum theory. The strength of the noise obtained by fit to Swift-Hohenberg theory is orders of magnitude larger than the thermal noise in fluid convection experiments, and is comparable to the noise found in experiments with oscillated granular layers and in recent fluid experiments on fluids near the critical point. Good agreement is found between the mean field value of onset from the Swift-Hohenberg fit and the onset in continuum simulations. Patterns are compared in cells oscillated at two different frequencies in MD; the layer with larger wavelength patterns has less noise than the layer with smaller wavelength patterns.Comment: Published in Physical Review

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

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 $a$, it is a non-local condition in the coordinate-parameter $(x, a)$ 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 $\hbar$-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 $\hbar$ explicitly, and derive a new $\hbar$-dependent superpotential by expanding a Scarf superpotential.Comment: 1 figure, 4 tables, 18 page

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.

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

Canvass: a crowd-sourced, natural-product screening library for exploring biological space

NCATS thanks Dingyin Tao for assistance with compound characterization. This research was supported by the Intramural Research Program of the National Center for Advancing Translational Sciences, National Institutes of Health (NIH). R.B.A. acknowledges support from NSF (CHE-1665145) and NIH (GM126221). M.K.B. acknowledges support from NIH (5R01GM110131). N.Z.B. thanks support from NIGMS, NIH (R01GM114061). J.K.C. acknowledges support from NSF (CHE-1665331). J.C. acknowledges support from the Fogarty International Center, NIH (TW009872). P.A.C. acknowledges support from the National Cancer Institute (NCI), NIH (R01 CA158275), and the NIH/National Institute of Aging (P01 AG012411). N.K.G. acknowledges support from NSF (CHE-1464898). B.C.G. thanks the support of NSF (RUI: 213569), the Camille and Henry Dreyfus Foundation, and the Arnold and Mabel Beckman Foundation. C.C.H. thanks the start-up funds from the Scripps Institution of Oceanography for support. J.N.J. acknowledges support from NIH (GM 063557, GM 084333). A.D.K. thanks the support from NCI, NIH (P01CA125066). D.G.I.K. acknowledges support from the National Center for Complementary and Integrative Health (1 R01 AT008088) and the Fogarty International Center, NIH (U01 TW00313), and gratefully acknowledges courtesies extended by the Government of Madagascar (Ministere des Eaux et Forets). O.K. thanks NIH (R01GM071779) for financial support. T.J.M. acknowledges support from NIH (GM116952). S.M. acknowledges support from NIH (DA045884-01, DA046487-01, AA026949-01), the Office of the Assistant Secretary of Defense for Health Affairs through the Peer Reviewed Medical Research Program (W81XWH-17-1-0256), and NCI, NIH, through a Cancer Center Support Grant (P30 CA008748). K.N.M. thanks the California Department of Food and Agriculture Pierce's Disease and Glassy Winged Sharpshooter Board for support. B.T.M. thanks Michael Mullowney for his contribution in the isolation, elucidation, and submission of the compounds in this work. P.N. acknowledges support from NIH (R01 GM111476). L.E.O. acknowledges support from NIH (R01-HL25854, R01-GM30859, R0-1-NS-12389). L.E.B., J.K.S., and J.A.P. thank the NIH (R35 GM-118173, R24 GM-111625) for research support. F.R. thanks the American Lebanese Syrian Associated Charities (ALSAC) for financial support. I.S. thanks the University of Oklahoma Startup funds for support. J.T.S. acknowledges support from ACS PRF (53767-ND1) and NSF (CHE-1414298), and thanks Drs. Kellan N. Lamb and Michael J. Di Maso for their synthetic contribution. B.S. acknowledges support from NIH (CA78747, CA106150, GM114353, GM115575). W.S. acknowledges support from NIGMS, NIH (R15GM116032, P30 GM103450), and thanks the University of Arkansas for startup funds and the Arkansas Biosciences Institute (ABI) for seed money. C.R.J.S. acknowledges support from NIH (R01GM121656). D.S.T. thanks the support of NIH (T32 CA062948-Gudas) and PhRMA Foundation to A.L.V., NIH (P41 GM076267) to D.S.T., and CCSG NIH (P30 CA008748) to C.B. Thompson. R.E.T. acknowledges support from NIGMS, NIH (GM129465). R.J.T. thanks the American Cancer Society (RSG-12-253-01-CDD) and NSF (CHE1361173) for support. D.A.V. thanks the Camille and Henry Dreyfus Foundation, the National Science Foundation (CHE-0353662, CHE-1005253, and CHE-1725142), the Beckman Foundation, the Sherman Fairchild Foundation, the John Stauffer Charitable Trust, and the Christian Scholars Foundation for support. J.W. acknowledges support from the American Cancer Society through the Research Scholar Grant (RSG-13-011-01-CDD). W.M.W.acknowledges support from NIGMS, NIH (GM119426), and NSF (CHE1755698). A.Z. acknowledges support from NSF (CHE-1463819). (Intramural Research Program of the National Center for Advancing Translational Sciences, National Institutes of Health (NIH); CHE-1665145 - NSF; CHE-1665331 - NSF; CHE-1464898 - NSF; RUI: 213569 - NSF; CHE-1414298 - NSF; CHE1361173 - NSF; CHE1755698 - NSF; CHE-1463819 - NSF; GM126221 - NIH; 5R01GM110131 - NIH; GM 063557 - NIH; GM 084333 - NIH; R01GM071779 - NIH; GM116952 - NIH; DA045884-01 - NIH; DA046487-01 - NIH; AA026949-01 - NIH; R01 GM111476 - NIH; R01-HL25854 - NIH; R01-GM30859 - NIH; R0-1-NS-12389 - NIH; R35 GM-118173 - NIH; R24 GM-111625 - NIH; CA78747 - NIH; CA106150 - NIH; GM114353 - NIH; GM115575 - NIH; R01GM121656 - NIH; T32 CA062948-Gudas - NIH; P41 GM076267 - NIH; R01GM114061 - NIGMS, NIH; R15GM116032 - NIGMS, NIH; P30 GM103450 - NIGMS, NIH; GM129465 - NIGMS, NIH; GM119426 - NIGMS, NIH; TW009872 - Fogarty International Center, NIH; U01 TW00313 - Fogarty International Center, NIH; R01 CA158275 - National Cancer Institute (NCI), NIH; P01 AG012411 - NIH/National Institute of Aging; Camille and Henry Dreyfus Foundation; Arnold and Mabel Beckman Foundation; Scripps Institution of Oceanography; P01CA125066 - NCI, NIH; 1 R01 AT008088 - National Center for Complementary and Integrative Health; W81XWH-17-1-0256 - Office of the Assistant Secretary of Defense for Health Affairs through the Peer Reviewed Medical Research Program; P30 CA008748 - NCI, NIH, through a Cancer Center Support Grant; California Department of Food and Agriculture Pierce's Disease and Glassy Winged Sharpshooter Board; American Lebanese Syrian Associated Charities (ALSAC); University of Oklahoma Startup funds; 53767-ND1 - ACS PRF; PhRMA Foundation; P30 CA008748 - CCSG NIH; RSG-12-253-01-CDD - American Cancer Society; RSG-13-011-01-CDD - American Cancer Society; CHE-0353662 - National Science Foundation; CHE-1005253 - National Science Foundation; CHE-1725142 - National Science Foundation; Beckman Foundation; Sherman Fairchild Foundation; John Stauffer Charitable Trust; Christian Scholars Foundation)Published versionSupporting documentatio

Effects of Thermal Noise on Pattern Onset in Continuum Simulations of Shaken Granular Layers

The author investigates the onset of patterns in vertically oscillated layers of dissipative particles using numerical solutions of continuum equations to Navier-Stokes order. Above a critical accelerational amplitude of the cell, standing waves form stripe patterns which oscillate subharmonically with respect to the cell. Continuum simulations neglecting interparticle friction yield pattern wavelengths consistent with experiments using frictional particles. However, the critical acceleration for standing wave formation is approximately 10% lower in continuum simulations without added noise than in molecular dynamics simulations. This report incorporates fluctuating hydrodynamics theory into continuum simulations by adding noise terms with no fit parameters; this modification yields a critical acceleration in agreement with molecular dynamics simulations.Comment: 5 pages, 4 figure