Group scheduling with general position-dependent effect

In order to investigate more realistic group scheduling problems with position-dependent effects,the model of general position-dependent group scheduling is proposed,where the actual group setup times and actual processing times are described by general functions of the normal group setup time and position in the sequence.These general functions are not assumed to have specific function structures,and are not restricted to be monotone.By mathematical analysis and proof,each considered problem is decomposed into a group scheduling process and a job scheduling process,and each scheduling process is transferred into the classic assignment problem or the classic single-machine sequence problem,and then the computational complexity to solve the considered problem is analyzed.Analysis results show that,even with general position-dependent job processing times,both the single machine makespan minimization group scheduling problems and the parallel-machine total load minimization group scheduling problems remain polynomially solvable.

Non-hypergeometric Type of Polynomials and Solutions of Schrodinger Equation with Position-Dependent Mass

Using the coordinate transformation method, we study the polynomial solutions of the Schr6dinger equation with position-dependent mass （PDM）. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The issues related to normalization of the wavefunetions and Hermiticity of the Hamiltonian are also analyzed.

Generation of Exactly Solvable Potentials of Position-Dependent Mass Schrödinger Equation from Hulthen Potential

Exactly Solvable Potentials (ESPs) of Position-Dependent Mass (PDM) Schrodinger equation are generated from Hulthen Potential (parent system) by using Extended Transformation (ET) method. The method includes a Co-ordinate Transformation (CT) followed by Functional Transformation (FT) of wave function. Mass function of parent system gets transformed to that of generated system. Two new ESPs are generated. The explicit expressions of mass functions, energy eigenvalues and corresponding wave functions for newly generated potentials (systems) are derived. System specific regrouping method is also discussed.

Coordinate Transformation and Exact Solutions of Schrodinger Equation with Position-Dependent Effective Mass

Using the coordinate transformation method, we solve the one-dimensional Schrodinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term Vm, which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of Vm for several mass functions are discussed.

On superintegrable systems with a position-dependent mass in polar-like coordinates

For a superintegrable system defined in plane polar-like coordinates introduced by Szumiński et al. and studied by Fordy, we show that the system with a position-dependent mass is separable in three distinct coordinate systems. The corresponding separation equations and additional integrals of motion are derived explicitly. The closure algebra of integrals is deduced. We also make a generalization of this system by employing the classical Jacobi method. Lastly a sufficient condition which ensures flatness of the underlying space is derived via explicit calculation.

Localization of s-Wave and Quantum Effective Potential of a Quasi-free Particle with Position-Dependent Mass

The properties of the 8-wave for a quasl-free partide with position-dependent mass （PDM） have been discussed in details. Differed from the system with constant mass in which the localization of the s-wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the 8-waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.

Bound states of the Dirac equation with position-dependent mass for the Eckart potential

Studying with the asymptotic iteration method, we present approximate solutions of the Dirac equation for the Eckart potential in the case of position-dependent mass. The centrifugal term is approximated by an exponential form, and the relativistic energy spectrum and the normalized eigenfunctions are obtained explicitly.

Shannon information entropies for position-dependent mass Schrdinger problem with a hyperbolic well

The Shannon information entropy for the Schrodinger equation with a nonuniform solitonic mass is evaluated for a hyperbolic-type potential. The number of nodes of the wave functions in the transformed space z are broken when recovered to original space x. The position Sx and momentum S p information entropies for six low-lying states are calculated. We notice that the Sx decreases with the increasing mass barrier width a and becomes negative beyond a particular width a,while the Sp first increases with a and then decreases with it. The negative Sx exists for the probability densities that are highly localized. We find that the probability density ρ（x） for n = 1, 3, 5 are greater than 1 at position x = 0. Some interesting features of the information entropy densities ρs（x） and ρs（p） are demonstrated. The Bialynicki-Birula-Mycielski（BBM）inequality is also tested for these states and found to hold.

Generalized Harmonic Oscillator and the Schrdinger Equation with Position-Dependent Mass

We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties corresponding effective potentials for several mass functions, for the system with PDM are also discussed. We give the the systems with such potentials are isospectral to the usual harmonic oscillator.

Semiclassical Method to Schroedinger Equation with Position-Dependent Effective Mass

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schroedinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schroedinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.

Approximate energies and thermal properties of a position-dependent mass charged particle under external magnetic fields

We solve the Schrodinger equation with a position-dependent mass（PDM） charged particle interacted via the superposition of the Morse-plus-Coulomb potentials and is under the influence of external magnetic and Aharonov–Bohm（AB） flux fields. The nonrelativistic bound state energies together with their wave functions are calculated for two spatially-dependent mass distribution functions. We also study the thermal quantities of such a system. Further, the canonical formalism is used to compute various thermodynamic variables for second choosing mass by using the Gibbs formalism. We give plots for energy states as a function of various physical parameters. The behavior of the internal energy, specific heat, and entropy as functions of temperature and mass density parameter in the inverse-square mass case for different values of magnetic field are shown.

Energy states of the Hulthen plus Coulomb-like potential with position-dependent mass function in external magnetic fields

We need to solve a suitable exponential form of the position-dependent mass （PDM） Schr6dinger equation with a charged particle placed in the Hulthen plus Coulomb-like potential field and under the actions of the external magnetic and Aharonov-Bohm （AB） flux fields. The bound state energies and their corresponding wave functions are calculated for the spatially-dependent mass distribution function of interest in physics. A few plots of some numerical results with respect to the energy are shown.

Genome-wide association study in historical and contemporary U.S.winter wheats identifies heightreducing loci

Plant height has been a major target for selection of high-yielding varieties in wheat.Two height-reducing loci(Rht-B1 and Rht-D1)have been widely used since the Green Revolution.However,these genes also negatively affect other agronomic traits such as kernel weight.Identifying alternative height-reducing loci could benefit wheat improvement.This study focused on the genetics of plant height in 260 historical and contemporary winter wheat accessions via genome-wide association studies using 38,693 single nucleotide polymorphism(SNP)markers generated through genotyping by sequencing,two Kompetitive Allele Specific Polymorphismmarkers,and phenotypic data recorded in two seasons(2016 and 2018).The 260 accessions showed wide variation in plant height.Most accessions developed after 1960 were shorter than earlier accessions.The broad-sense heritability for plant height was high(H2=0.82),whichwas also supported by a high correlation(r=0.82,P<0.0001)between heights from the two years.We detected a total of 16 marker–trait associations(MTAs)for plant height at–lg(P)≥4.0 on chromosomes 1A,2B,2D,3B,4D,5A,5D,6A,6B,7A,and 7D.We detected three of the MTAs(QPLH-2D,QPLH-4B.2,and QPLH-4D)consistently in individual-year and combined-year analyses.These MTAs individually explained 10%–16%of phenotypic variation.The heightreducing alleles at these threeMTAs appeared after 1960 and increased in frequency thereafter.Among the genes near these loci were gibberellic acid insensitive(GAI)and GRAS transcription factor(GIBBERELLIC-ACID INSENSITIVE(GAI),REPRESSOR of GAI(RGA),and SCARECROW(SCR)).The evidence from this study and previous reports suggests that QPLH-2D is Rht8.A gene encoding a GRAS transcription factor is located near QPLH-2D.Validation of the QPLH-2D locus and associated candidate genes awaits further study.

The bound state solution for the Morse potential with a localized mass profile

We investigate an analytical solution for the Schr o¨dinger equation with a position-dependent mass distribution, with the Morse potential via Laplace transformations. We considered a mass function localized around the equilibrium position.The mass distribution depends on the energy spectrum of the state and the intrinsic parameters of the Morse potential. An exact bound state solution is obtained in the presence of this mass distribution.

Vibrational resonance in a multistable system with position-dependent mass

The occurrence of vibrational resonance(VR)in a dual-frequency-driven multistable system with a spatially varying mass modelling particle with position-dependent mass(PDM)and evolving in a one-dimensional symmetric periodic potential has been investigated and reported in this paper.We numerically compute and analyze the response amplitude,the effects of the PDM parameters(m0,a)on the potential structure,the occurrence of VR and the bifurcation of the equilibrium points.It is shown that the PDM parameters,besides controlling VR,can induce unconventional resonance patterns through the variation of the potential well depth.The resonant states can be influenced through the cooperation of the PDM parameters and the external forcing leading the system from multiresonance state into single and double resonance states.The contributions of the fixed rest mass m0 on the VR and the PDM-induced resonance are determined by threshold conditions imposed by the magnitude of the mass nonlinear strength a.

Quantum carpets:efficiently probing fractional revivals in position-dependent mass systems

Position-dependent-mass systems are of great importance in many physical situations,such as the transport of charge carriers in semiconductors with non-uniform composition and in the theory of many-body interactions in condensed matter.Here we investigate,numerically and analytically,the phenomenon of fractional revivals in such systems,which is a generic characteristic manifested by the wave-packet evolution in bounded Hamiltonian systems.Identifying the fractional revivals using specific probes is an important task in the theory of quantum measurement and sensing.We numerically simulate the temporal evolution of probability density and information entropy density,which manifest self-similarly recurring interference patterns,namely,quantum carpets.Our numerical results show that the quantum carpets not only serve as an effective probe for recognizing the fractional revivals of various order but they efficiently describe the effect of spatially-varying mass on the structure of fractional revivals,which is manifested as a symmetry breaking in their designs.

The fractional features of a harmonic oscillator with position-dependent mass

In this study,a harmonic oscillator with position-dependent mass is investigated.Firstly,as an introduction,we give a full description of the system by constructing its classical Lagrangian;thereupon,we derive the related classical equations of motion such as the classical Euler–Lagrange equations.Secondly,we fractionalize the classical Lagrangian of the system,and then we obtain the corresponding fractional Euler–Lagrange equations(FELEs).As a final step,we give the numerical simulations corresponding to the FELEs within different fractional operators.Numerical results based on the Caputo and the Atangana-Baleanu-Caputo(ABC)fractional derivatives are given to verify the theoretical analysis.

The spontaneous emission of an excited atom embedded in photonic crystals with two atomic position-dependent bands

The spontaneous emission of an excited atom embedded in photonic crystals with two atomic position-dependent bands is investigated.The distribution of the density of states between two bands depends on the atomic position in a unit cell of the photonic crystal and is described with an atomic position-dependent parameter.The result shows that the emitted field and the time evolution of the upper-level population are affected by the atomic position and the gap width.The spontaneous emission spectrum in free space can be shifted and narrowed with the photonic reservoir and the gap width.

Coherent States of Nonlinear Oscillators with Position-Dependent Mass:Temporal Stability and Fractional Revivals

We develop generalized coherent states for a class of nonlinear oscillators with position-dependent effective mass in the context of the Gazeau–Klauder formalism and discuss some of their properties. In order to investigate the temporal evolution we first explore the statistical properties by means of weighting distribution and the Mandel parameter. It is found that the temporal evolution of the coherent states may exhibit the phenomena of quantum revivals and fractional revivals for a particular choice of position-dependent mass oscillator.

Barut–Girardello Coherent States for Nonlinear Oscillator with Position-Dependent Mass

Using ladder operators for the non-linear oscillator with position-dependent effective mass, realization of the dynamic group SU(1,1) is presented. Keeping in view the algebraic structure of the non-linear oscillator, coherent states are constructed using Barut–Girardello formalism and their basic properties are discussed. Furthermore, the statistical properties of these states are investigated by means of Mandel parameter and second order correlation function. Moreover,it is shown that in the harmonic limit, all the results obtained for the non-linear oscillator with spatially varying mass reduce to corresponding results of the linear oscillator with constant mass.