Pseudo-Hermiticity and some consequences of a generalized quantum condition

We exploit the hidden symmetry structure of a recently proposed non-Hermitian Hamiltonian and of its Hermitian equivalent one. This sheds new light on the pseudo-Hermitian character of the former and allows access to a generalized quantum condition. Special cases lead to hyperbolic and Morse-like potentials in the framework of a coordinate-dependent mass model.Comment: 10 pages, no figur

Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT symmetry

We develop a systematic approach to construct novel completely solvable rational potentials. Second-order supersymmetric quantum mechanics dictates the latter to be isospectral to some well-studied quantum systems. $\cal PT$ symmetry may facilitate reconciling our approach to the requirement that the rationally-extended potentials be singularity free. Some examples are shown.Comment: 13 pages, no figure, some additions to introduction and conclusion, 4 more references; to be published in Special issue of Pramana - J. Phy

Graded extension of SO(2,1) Lie algebra and the search for exact solutions of Dirac equation by point canonical transformations

SO(2,1) is the symmetry algebra for a class of three-parameter problems that includes the oscillator, Coulomb and Morse potentials as well as other problems at zero energy. All of the potentials in this class can be mapped into the oscillator potential by point canonical transformations. We call this class the "oscillator class". A nontrivial graded extension of SO(2,1) is defined and its realization by two-dimensional matrices of differential operators acting in spinor space is given. It turns out that this graded algebra is the supersymmetry algebra for a class of relativistic potentials that includes the Dirac-Oscillator, Dirac-Coulomb and Dirac-Morse potentials. This class is, in fact, the relativistic extension of the oscillator class. A new point canonical transformation, which is compatible with the relativistic problem, is formulated. It maps all of these relativistic potentials into the Dirac-Oscillator potential.Comment: Replaced with a more potrable PDF versio

A general scheme for the effective-mass Schrodinger equation and the generation of the associated potentials

A systematic procedure to study one-dimensional Schr\"odinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schr\"odinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.Comment: 16 pages + 1 figure; minor changes + new "free-particle" problem; version published in Mod. Phys. Lett.

Inference of population splits and mixtures from genome-wide allele frequency data

Many aspects of the historical relationships between populations in a species are reflected in genetic data. Inferring these relationships from genetic data, however, remains a challenging task. In this paper, we present a statistical model for inferring the patterns of population splits and mixtures in multiple populations. In this model, the sampled populations in a species are related to their common ancestor through a graph of ancestral populations. Using genome-wide allele frequency data and a Gaussian approximation to genetic drift, we infer the structure of this graph. We applied this method to a set of 55 human populations and a set of 82 dog breeds and wild canids. In both species, we show that a simple bifurcating tree does not fully describe the data; in contrast, we infer many migration events. While some of the migration events that we find have been detected previously, many have not. For example, in the human data we infer that Cambodians trace approximately 16% of their ancestry to a population ancestral to other extant East Asian populations. In the dog data, we infer that both the boxer and basenji trace a considerable fraction of their ancestry (9% and 25%, respectively) to wolves subsequent to domestication, and that East Asian toy breeds (the Shih Tzu and the Pekingese) result from admixture between modern toy breeds and "ancient" Asian breeds. Software implementing the model described here, called TreeMix, is available at http://treemix.googlecode.comComment: 28 pages, 6 figures in main text. Attached supplement is 22 pages, 15 figures. This is an updated version of the preprint available at http://precedings.nature.com/documents/6956/version/

L2 series solutions of the Dirac equation for power-law potentials at rest mass energy

We obtain solutions of the three dimensional Dirac equation for radial power-law potentials at rest mass energy as an infinite series of square integrable functions. These are written in terms of the confluent hypergeometric function and chosen such that the matrix representation of the Dirac operator is tridiagonal. The "wave equation" results in a three-term recursion relation for the expansion coefficients of the spinor wavefunction which is solved in terms of orthogonal polynomials. These are modified versions of the Meixner-Pollaczek polynomials and of the continuous dual Hahn polynomials. The choice depends on the values of the angular momentum and the power of the potential.Comment: 13 pages, 1 Tabl

A basis for variational calculations in d dimensions

In this paper we derive expressions for matrix elements (\phi_i,H\phi_j) for the Hamiltonian H=-\Delta+\sum_q a(q)r^q in d > 1 dimensions. The basis functions in each angular momentum subspace are of the form phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2}, i >= 0, p > 0, t > 0. The matrix elements are given in terms of the Gamma function for all d. The significance of the parameters t and p and scale s are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form b/r^a, a,b > 0, perturbed Coulomb potentials -D/r + B r + Ar^2, and potentials with weak repulsive terms, such as -g r^2 + r^4, g > 0.Comment: 22 page

Reconstruction of major maternal and paternal lineages of the Cape Muslim population

The earliest Cape Muslims were brought to the Cape (Cape Town - South Africa) from Africa and Asia from 1652 to 1834. They were part of an involuntary migration of slaves, political prisoners and convicts, and they contributed to the ethnic diversity of the present Cape Muslim population of South Africa. The history of the Cape Muslims has been well documented and researched however no in-depth genetic studies have been undertaken. The aim of the present study was to determine the respective African, Asian and European contributions to the mtDNA (maternal) and Y-chromosomal (paternal) gene pool of the Cape Muslim population, by analyzing DNA samples of 100 unrelated Muslim males born in the Cape Metropolitan area. A panel of six mtDNA and eight Y-chromosome SNP markers were screened using polymerase chain reaction-restriction fragment length polymorphisms (PCR-RFLP). Overall admixture estimates for the maternal line indicated Asian (0.4168) and African mtDNA (0.4005) as the main contributors. The admixture estimates for the paternal line, however, showed a predominance of the Asian contribution (0.7852). The findings are in accordance with historical data on the origins of the early Cape Muslims.Web of Scienc

PFTAIRE Kinase L63 Interactor 1A (Pif1A Protein) Is Required for Actin Cone Movement during Spermatid Individualization in Drosophila melanogaster

A useful model for determining the mechanisms by which actin and actin binding proteins control cellular architecture is the Drosophila melanogaster process of spermatogenesis. During the final step of spermatogenesis, 64 syncytial spermatids individualized as stable actin cones move synchronously down the axonemes and remodel the membranes. To identify new genes involved in spermatid individualization, we screened a collection of Drosophila male-sterile mutants and found that, in the line Z3-5009, actin cones formed near to the spermatid nuclei but failed to move, resulting in failed spermatid individualization. However, we show by phalloidin actin staining, electron microscopy and immunocytochemical localization of several actin binding proteins that the early cones had normal structure. We sequenced the genome of the Z3-5009 line and identified mutations in the PFTAIRE kinase L63 interactor 1A (Pif1A) gene. Quantitative real-time PCR showed that Pif1A transcript abundance was decreased in the mutant, and a transgene expressing Pif1A fused to green fluorescent protein (GFP) was able to fully rescue spermatid individualization and male fertility. Pif1A-GFP localized to the front of actin cones before initiation of movement. We propose that Pif1A plays a pivotal role in directing actin cone movement