Solving the radial Dirac equations: a numerical odyssey

We discuss, in a pedagogical way, how to solve for relativistic wave functions from the radial Dirac equations. After an brief introduction, in Section II we solve the equations for a linear Lorentz scalar potential, V_s(r), that provides for confinement of a quark. The case of massless u and d quarks is treated first, as these are necessarily quite relativistic. We use an iterative procedure to find the eigenenergies and the upper and lower component wave functions for the ground state and then, later, some excited states. Solutions for the massive quarks (s, c, and b) are also presented. In Section III we solve for the case of a Coulomb potential, which is a time-like component of a Lorentz vector potential, V_v(r). We re-derive, numerically, the (analytically well-known) relativistic hydrogen atom eigenenergies and wave functions, and later extend that to the cases of heavier one-electron atoms and muonic atoms. Finally, Section IV finds solutions for a combination of the V_s and V_v potentials. We treat two cases. The first is one in which V_s is the linear potential used in Sec. II and V_v is Coulombic, as in Sec. III. The other is when both V_s and V_v are linearly confining, and we establish when these potentials give a vanishing spin-orbit interaction (as has been shown to be the case in quark models of the hadronic spectrum).Comment: 39 pages (total), 23 figures, 2 table

Sustainability of water resources management in the Indus Basin under changing climatic and socio economic conditions

Pakistan is highly dependent on water resources originating in the mountain sources of the upper Indus for irrigated agriculture which is the mainstay of its economy. Hence any change in available resources through climate change or socio-economic factors could have a serious impact on food security and the environment. In terms of both ratio of withdrawals to runoff and per-capita water availability, Pakistan's water resources are already highly stressed and will become increasingly so with projected population changes. Potential changes to supply through declining reservoir storage, the impact of waterlogging and salinity or over-abstraction of groundwater, or reallocations for environmental remediation of the Indus Delta or to meet domestic demands, will reduce water availability for irrigation. <br><br> The impact of climate change on resources in the Upper Indus is considered in terms of three hydrological regimes – a nival regime dependent on melting of winter snow, a glacial regime, and a rainfall regime dependent on concurrent rainfall. On the basis of historic trends in climate, most notably the decline in summer temperatures, there is no strong evidence in favour of marked reductions in water resources from any of the three regimes. Evidence for changes in trans-Himalayan glacier mass balance is mixed. Sustainability of water resources appears more threatened by socio-economic changes than by climatic trends. Nevertheless, analysis and the understanding of the linkage of climate, glaciology and runoff is still far from complete; recent past climate experience may not provide a reliable guide to the future

Equity of access to renal transplant waiting list and renal transplantation in Scotland: cohort study

Objective To examine the access to the renal transplant waiting list and renal transplantation in Scotland. Design Cohort study. Setting Renal and transplant units in Scotland. Participants 4523 adults starting renal replacement therapy in Scotland between 1 January 1989 and 31 December 1999. Main outcome measures Impact of age, sex, social deprivation, primary renal disease, renal or transplant unit, and geography on access to the waiting list and renal transplantation. Results 1736 of 4523 (38.4%) patients were placed on the waiting list for renal transplantation and 1095 (24.2%) underwent transplantation up to 31 December 2000, the end of the study period. Patients were less likely to be placed on the list if they were female, older, had diabetes, were in a high deprivation category, and were treated in a renal unit in a hospital with no transplant unit. Patients living furthest away from the transplant centre were listed more quickly. The only factors governing access to transplantation once on the list were age, primary renal disease, and year of listing. A significant centre effect was found in access to the waiting list and renal transplantation. Conclusions A major disparity exists in access to the renal transplant waiting list and renal transplantation in Scotland. Comorbidity may be an important factor

Concussions in the National Basketball Association: Analysis of Incidence, Return to Play, and Performance From 1999 to 2018.

Background: The effect of concussions on professional athletes has been investigated in many sports. However, few studies have evaluated concussions in National Basketball Association (NBA) players. Hypothesis: We hypothesized that concussion incidence has increased, yet the return-to-play (RTP) rate will remain high following the institution of the NBA concussion policy (NBACP). We also hypothesized that the incidence of repeat concussions will be similar to first occurrences and that player performance and game availability will not be significantly affected by sustaining a concussion. Study Design: Descriptive epidemiology study. Methods: Publicly available records were searched to identify all concussions from NBA seasons 1999-2000 to 2017-2018. Player demographics and information regarding career history were tabulated. Incidence of concussion and RTP timing were evaluated before and after institution of the NBACP (2011). Minutes per game and game score per minute were evaluated pre- versus postconcussion. Player availability and performance were also compared with an age-, body mass index-, position-, and experience-matched control group of players who did not sustain a concussion. Results: A total of 189 concussions were reported in the NBA from 1999 to 2018, with a mean +/- SD incidence of 9.7 +/- 7.3 concussions per season. Following implementation of the NBACP, incidence significantly increased from 5.7 +/- 2.8 to 16.7 +/- 7.5 concussions per season (P = .007). All players returned to play following first-time concussion after missing 7.7 +/- 8.6 days and 3.5 +/- 4.1 games. RTP time was not significantly different after implementation of the NBACP (games missed, P = .24; days missed, P = .27), and there was no difference in concussion-free time interval (P = .29). Game score per minute and minutes per game were not significantly affected by sustaining a concussion (both P \u3e .05). Conclusion: Concussion incidence in NBA players is approximately 17 instances per season since the 2011 institution of a league-wide concussion policy. The number of reported concussions significantly increased following the policy, in line with trends seen in other professional sports leagues. Players have retained a high rate of RTP after 3 to 4 missed games. Player performance and availability are not affected by sustaining a concussion following successful RTP

Algorithm for normal random numbers

We propose a simple algorithm for generating normally distributed pseudo random numbers. The algorithm simulates N molecules that exchange energy among themselves following a simple stochastic rule. We prove that the system is ergodic, and that a Maxwell like distribution that may be used as a source of normally distributed random deviates follows when N tends to infinity. The algorithm passes various performance tests, including Monte Carlo simulation of a finite 2D Ising model using Wolff's algorithm. It only requires four simple lines of computer code, and is approximately ten times faster than the Box-Muller algorithm.Comment: 5 pages, 3 encapsulated Postscript Figures. Submitted to Phys.Rev.Letters. For related work, see http://pipe.unizar.es/~jf

A Detailed Cloud Fraction Climatology of the Upper Indus Basin and Its Implications for Near-Surface Air Temperature*

implications for near surface air temperature. Journal of Climate 2015, 28(9)

Phase diffusion as a model for coherent suppression of tunneling in the presence of noise

We study the stabilization of coherent suppression of tunneling in a driven double-well system subject to random periodic $\delta-$function ``kicks''. We model dissipation due to this stochastic process as a phase diffusion process for an effective two-level system and derive a corresponding set of Bloch equations with phase damping terms that agree with the periodically kicked system at discrete times. We demonstrate that the ability of noise to localize the system on either side of the double-well potenital arises from overdamping of the phase of oscillation and not from any cooperative effect between the noise and the driving field. The model is investigated with a square wave drive, which has qualitatively similar features to the widely studied cosinusoidal drive, but has the additional advantage of allowing one to derive exact analytic expressions.Comment: 17 pages, 4 figures, submitted to Phys. Rev.

Multidisciplinary Applications of Detached-Eddy Simulation to Separated Flows at High Reynolds Numbers

We focus on multidisciplinary applications of detached-eddy simulation (DES), principally flight mechanics and aeroelasticity. Specifically, the lateral instability (known as abrupt wing stall) of the preproduction F/A-18E is reproduced using DES, including the unsteady shock motion. The presence of low frequency pressure oscillations due to shock motion in the current simulations and the experiments motivated a full aircraft calculation, which showed low frequency high-magnitude rolling moments that could be a significant contributor to the abrupt wing stall phenomenon. DES is also applied to the F-18 high angle of attack research vehicle (HARV) at a moderate angle of attack to reproduce the vortex breakdown leading to vertical stabilizer buffet. Unsteady tail loads are compared to flight test data. This work lays the foundation for future deforming grid calculations to reproduce the aero-elastic tail buffet seen in flight test. Solution based grid adaption is used on unstructured grids in both cases to improve the resolution in the separated region. Previous DoD Challenge work has demonstrated the unique ability of the DES turbulence treatment to accurately and efficiently predict flows with massive separation at flight Reynolds numbers. DES calculations have been performed using the Cobalt code and on unstructured grids, an approach that can deal with complete configurations with very few compromises. A broad range of flows has been examined in previous Challenge work, including aircraft forebodies, airfoil sections, a missile afterbody, vortex breakdown on a delta wing, and the F-16 and F-15E at high angles-of-attack. All DES predictions exhibited a moderate to significant improvement over results obtained using traditional Reynolds-averaged models and often excellent agreement with experimental/flight-test data. DES combines the efficiency of a Reynolds-averaged turbulence model near the wall with the fidelity of Large-Eddy Simulation (LES) in separated regions. Since it uses Large-Eddy Simulation in the separated regions, it is capable of predicting the unsteady motions associated with separated flows. The development and demonstration of improved methods for the prediction of flight mechanics and aeroelasticity in this Challenge is expected to reduce the acquisition cost of future military aircraft