A Numerical Methodology for the Painlevé Equations

The six Painlevé transcendents PI – PVI have both applications and analytic properties that make them stand out from most other classes of special functions. Although they have been the subject of extensive theoretical investigations for about a century, they still have a reputation for being numerically challenging. In particular, their extensive pole fields in the complex plane have often been perceived as ‘numerical mine fields’. In the present work, we note that the Painlevé property in fact provides the opportunity for very fast and accurate numerical solutions throughout such fields. When combining a Taylor/Padé-based ODE initial value solver for the pole fields with a boundary value solver for smooth regions, numerical solutions become available across the full complex plane. We focus here on the numerical methodology, and illustrate it for the PI equation. In later studies, we will concentrate on mathematical aspects of both the PI and the higher Painlevé transcendents

Towards understanding Regge trajectories in holographic QCD

We reassess a work done by Migdal on the spectrum of low-energy vector mesons in QCD in the light of the AdS-QCD correspondence. Recently, a tantalizing parallelism was suggested between Migdal's work and a family of holographic duals of QCD. Despite the intriguing similarities, both approaches face a major drawback: the spectrum is in conflict with well-tested Regge scaling. However, it has recently been shown that holographic duals can be modified to accomodate Regge behavior. Therefore, it is interesting to understand whether Regge behavior can also be achieved in Migdal's approach. In this paper we investigate this issue. We find that Migdal's approach, which is based on a modified Pade approximant, is closely related to the issue of quark-hadron duality breakdown in QCD.Comment: 17 pages, 1 figure. Typos fixed, references added, improved discussion. Minor changes to match the journal versio

Chemical Abundances of Planetary Nebulae in the Sagittarius Dwarf Elliptical Galaxy

Spectrophotometry and imaging of the two planetary nebulae He2-436 and Wray16-423, recently discovered to be in the Sagittarius dwarf elliptical galaxy, are presented. Wray16-423 is a high excitation planetary nebula (PN) with a hot central star. In contrast He2-436 is a high density PN with a cooler central star and evidence of local dust, the extinction exceeding that for Wray16-423 by E(B-V)=0.28. The extinction to Wray16-423, (E(B-V)=0.14), is consistent with the extinction to the Sagittarius (Sgr) Dwarf. Both PN show Wolf-Rayet features in their spectra, although the lines are weak in Wray16-423. Images in [O III] and H-alpha+[N II], although affected by poor seeing, yield a diameter of 1.2'' for Wray16-423 after deconvolution; He~2-436 was unresolved. He2-436 has a luminosity about twice that of Wray16-423 and its size and high density suggest a younger PN. In order to reconcile the differing luminosity and nebular properties of the two PN with similar age progenitor stars, it is suggested that they are on He burning tracks The abundance pattern is very similar in both nebulae and shows an oxygen depletion of -0.4 dex with respect to the mean O abundance of Galactic PN and [O/H] = -0.6. The Sgr PN progenitor stars are representative of the higher metallicity tail of the Sgr population. The pattern of abundance depletion is similar to that in the only other PN in a dwarf galaxy companion of the Milky Way, that in Fornax, for which new spectra are presented. However the abundances are larger than for Galactic halo PN suggesting a later formation age. The O abundance of the Sgr galaxy deduced from its PN, shows similarities with that of dwarf ellipticals around M31, suggesting that this galaxy was a dwarf elliptical before its interaction with the Milky Way.Comment: 24 pages, Latex (aas2pp4.sty) including 5 postscript figures. To appear in Ap

Non-Hermitian Hamiltonians with real eigenvalues coupled to electric fields: from the time-independent to the time dependent quantum mechanical formulation

We provide a reviewlike introduction into the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PT-symmetry and pseudo-Hermiticity. We discuss the time-evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, ii) the spiked harmonic oscillator, which exhibits explicit supersymmetry and iii) the -x^4-potential, which serves as a toy model for the quantum field theoretical phi^4-theory.Comment: 14 pages, 3 figures, to appear in Laser Physics, minor typos correcte

Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement

Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time

Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme

In this paper, we consider the numerical simulations of an extended nonlinear form of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4) schemes proposed by Cox and Matthew (J Comput Phys 176:430-455, 2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233, 2005), for the time integration of spatially discretized partial differential equations. We demonstrate the supremacy of ETDRK4 over the existing exponential time differencing integrators that are of standard approaches and provide timings and error comparison. Numerical results obtained in this paper have granted further insight to the question "What is the minimal size of the spatial domain so that the population persists?" posed by Kierstead and Slobodkin (J Mar Res 12:141-147, 1953 ), with a conclusive remark that the popula- tion size increases with the size of the domain. In attempt to examine the biological wave phenomena of the solutions, we present the numerical results in both one- and two-dimensional space, which have interesting ecological implications. Initial data and parameter values were chosen to mimic some existing patternsScopus 201

Accuracy of Risk Estimates from the iPrevent Breast Cancer Risk Assessment and Management Tool.

BACKGROUND: iPrevent is an online breast cancer (BC) risk management decision support tool. It uses an internal switching algorithm, based on a woman's risk factor data, to estimate her absolute BC risk using either the International Breast Cancer Intervention Study (IBIS) version 7.02, or Breast and Ovarian Analysis of Disease Incidence and Carrier Estimation Algorithm version 3 models, and then provides tailored risk management information. This study assessed the accuracy of the 10-year risk estimates using prospective data. METHODS: iPrevent-assigned 10-year invasive BC risk was calculated for 15 732 women aged 20-70 years and without BC at recruitment to the Prospective Family Study Cohort. Calibration, the ratio of the expected (E) number of BCs to the observed (O) number and discriminatory accuracy were assessed. RESULTS: During the 10 years of follow-up, 619 women (3.9%) developed BC compared with 702 expected (E/O = 1.13; 95% confidence interval [CI] =1.05 to 1.23). For women younger than 50 years, 50 years and older, and BRCA1/2-mutation carriers and noncarriers, E/O was 1.04 (95% CI = 0.93 to 1.16), 1.24 (95% CI = 1.11 to 1.39), 1.13 (95% CI = 0.96 to 1.34), and 1.13 (95% CI = 1.04 to 1.24), respectively. The C-statistic was 0.70 (95% CI = 0.68 to 0.73) overall and 0.74 (95% CI = 0.71 to 0.77), 0.63 (95% CI = 0.59 to 0.66), 0.59 (95% CI = 0.53 to 0.64), and 0.65 (95% CI = 0.63 to 0.68), respectively, for the subgroups above. Applying the newer IBIS version 8.0b in the iPrevent switching algorithm improved calibration overall (E/O = 1.06, 95% CI = 0.98 to 1.15) and in all subgroups, without changing discriminatory accuracy. CONCLUSIONS: For 10-year BC risk, iPrevent had good discriminatory accuracy overall and was well calibrated for women aged younger than 50 years. Calibration may be improved in the future by incorporating IBIS version 8.0b