Harmonically trapped jellium

We discuss the model of a $D$-dimensional confined electron gas in which the particles are trapped by a harmonic potential. In particular, we study the non-interacting kinetic and exchange energies of finite-size inhomogeneous systems, and compare the resulting Thomas-Fermi and Dirac coefficients with various uniform electron gas paradigms. We show that, in the thermodynamic limit, the properties of this model are identical to those of the $D$-dimensional Fermi gas.Comment: 6 pages, 2 figures, 1 table, invited paper for Peter Taylor's 60th anniversary, submitted to Molecular Physic

An extension of Laplace's method

Asymptotic expansions are obtained for contour integrals of the form $$\int_a^b \exp \left( - zp(t) + z^{\nu /\mu } r(t) \right)q(t)dt,$$ in which $z$ is a large real or complex parameter, $p(t)$, $q(t)$ and $r(t)$ are analytic functions of $t$, and the positive constants $\mu$ and $\nu$ are related to the local behaviour of the functions $p(t)$ and $r(t)$ near the endpoint $a$. Our main theorem includes as special cases several important asymptotic methods for integrals such as those of Laplace, Watson, Erd\'elyi and Olver. Asymptotic expansions similar to ours were derived earlier by Dingle using formal, non-rigorous methods. The results of the paper also serve to place Dingle's investigations on a rigorous mathematical foundation. The new results have potential applications in the asymptotic theory of special functions in transition regions, and we illustrate this by two examples.Comment: 19 pages, 2 figures, revised version, accepted for publication in Constructive Approximatio

Hyperasymptotic solutions for certain partial differential equations

We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel summability, which allows us to find integral representations of solutions for such PDEs.Comment: 17 page

An EWMA control chart for the multivariate coefficient of variation

This is the peer reviewed version of the following article: Giner-Bosch, V, Tran, KP, Castagliola, P, Khoo, MBC. An EWMA control chart for the multivariate coefficient of variation. Qual Reliab Engng Int. 2019; 35: 1515-1541, which has been published in final form at https://doi.org/10.1002/qre.2459. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] Monitoring the multivariate coefficient of variation over time is a natural choice when the focus is on stabilising the relative variability of a multivariate process, as is the case in a significant number of real situations in engineering, health sciences, and finance, to name but a few areas. However, not many tools are available to practitioners with this aim. This paper introduces a new control chart to monitor the multivariate coefficient of variation through an exponentially weighted moving average (EWMA) scheme. Concrete methodologies to calculate the limits and evaluate the performance of the chart proposed and determine the optimal values of the chart's parameters are derived based on a theoretical study of the statistic being monitored. Computational experiments reveal that our proposal clearly outperforms existing alternatives, in terms of the average run length to detect an out-of-control state. A numerical example is included to show the efficiency of our chart when operating in practice.Generalitat Valenciana, Grant/Award Number: BEST/2017/033 and GV/2016/004; Ministerio de Economia y Competitividad, Grant/Award Number: MTM2013-45381-PGiner-Bosch, V.; Tran, KP.; Castagliola, P.; Khoo, MBC. (2019). An EWMA control chart for the multivariate coefficient of variation. Quality and Reliability Engineering International. 35(6):1515-1541. https://doi.org/10.1002/qre.2459S15151541356Kang, C. W., Lee, M. S., Seong, Y. J., & Hawkins, D. M. (2007). A Control Chart for the Coefficient of Variation. Journal of Quality Technology, 39(2), 151-158. doi:10.1080/00224065.2007.11917682Amdouni, A., Castagliola, P., Taleb, H., & Celano, G. (2015). Monitoring the coefficient of variation using a variable sample size control chart in short production runs. The International Journal of Advanced Manufacturing Technology, 81(1-4), 1-14. doi:10.1007/s00170-015-7084-4Amdouni, A., Castagliola, P., Taleb, H., & Celano, G. (2017). A variable sampling interval Shewhart control chart for monitoring the coefficient of variation in short production runs. International Journal of Production Research, 55(19), 5521-5536. doi:10.1080/00207543.2017.1285076Yeong, W. C., Khoo, M. B. C., Tham, L. K., Teoh, W. L., & Rahim, M. A. (2017). Monitoring the Coefficient of Variation Using a Variable Sampling Interval EWMA Chart. Journal of Quality Technology, 49(4), 380-401. doi:10.1080/00224065.2017.11918004Teoh, W. L., Khoo, M. B. C., Castagliola, P., Yeong, W. C., & Teh, S. Y. (2017). Run-sum control charts for monitoring the coefficient of variation. European Journal of Operational Research, 257(1), 144-158. doi:10.1016/j.ejor.2016.08.067Sharpe, W. F. (1994). The Sharpe Ratio. The Journal of Portfolio Management, 21(1), 49-58. doi:10.3905/jpm.1994.409501Van Valen, L. (1974). Multivariate structural statistics in natural history. Journal of Theoretical Biology, 45(1), 235-247. doi:10.1016/0022-5193(74)90053-8Albert, A., & Zhang, L. (2010). A novel definition of the multivariate coefficient of variation. Biometrical Journal, 52(5), 667-675. doi:10.1002/bimj.201000030Aerts, S., Haesbroeck, G., & Ruwet, C. (2015). Multivariate coefficients of variation: Comparison and influence functions. Journal of Multivariate Analysis, 142, 183-198. doi:10.1016/j.jmva.2015.08.006Bennett, B. M. (1977). On multivariate coefficients of variation. Statistische Hefte, 18(2), 123-128. doi:10.1007/bf02932744Underhill, L. G. (1990). The coefficient of variation biplot. Journal of Classification, 7(2), 241-256. doi:10.1007/bf01908718Boik, R. J., & Shirvani, A. (2009). Principal components on coefficient of variation matrices. Statistical Methodology, 6(1), 21-46. doi:10.1016/j.stamet.2008.02.006MacGregor, J. F., & Kourti, T. (1995). Statistical process control of multivariate processes. Control Engineering Practice, 3(3), 403-414. doi:10.1016/0967-0661(95)00014-lBersimis, S., Psarakis, S., & Panaretos, J. (2007). Multivariate statistical process control charts: an overview. Quality and Reliability Engineering International, 23(5), 517-543. doi:10.1002/qre.829Yeong, W. C., Khoo, M. B. C., Teoh, W. L., & Castagliola, P. (2015). A Control Chart for the Multivariate Coefficient of Variation. Quality and Reliability Engineering International, 32(3), 1213-1225. doi:10.1002/qre.1828Lim, A. J. X., Khoo, M. B. C., Teoh, W. L., & Haq, A. (2017). Run sum chart for monitoring multivariate coefficient of variation. Computers & Industrial Engineering, 109, 84-95. doi:10.1016/j.cie.2017.04.023Roberts, S. W. (1966). A Comparison of Some Control Chart Procedures. Technometrics, 8(3), 411-430. doi:10.1080/00401706.1966.10490374Roberts, S. W. (1959). Control Chart Tests Based on Geometric Moving Averages. Technometrics, 1(3), 239-250. doi:10.1080/00401706.1959.10489860Lucas, J. M., & Saccucci, M. S. (1990). Exponentially Weighted Moving Average Control Schemes: Properties and Enhancements. Technometrics, 32(1), 1-12. doi:10.1080/00401706.1990.10484583Wijsman, R. A. (1957). Random Orthogonal Transformations and their use in Some Classical Distribution Problems in Multivariate Analysis. The Annals of Mathematical Statistics, 28(2), 415-423. doi:10.1214/aoms/1177706969The general sampling distribution of the multiple correlation coefficient. (1928). Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 121(788), 654-673. doi:10.1098/rspa.1928.0224Paolella, M. S. (2007). Intermediate Probability. doi:10.1002/9780470035061WalckC.Handbook on statistical distributions for experimentalists. Tech. Rep. SUFPFY/96‐01 Stockholm   Particle Physics Group Fysikum University of Stockholm;2007. http://inspirehep.net/record/1389910BROOK, D., & EVANS, D. A. (1972). An approach to the probability distribution of cusum run length. Biometrika, 59(3), 539-549. doi:10.1093/biomet/59.3.539Castagliola, P., Celano, G., & Psarakis, S. (2011). Monitoring the Coefficient of Variation Using EWMA Charts. Journal of Quality Technology, 43(3), 249-265. doi:10.1080/00224065.2011.11917861Vining, G. (2009). Technical Advice: Phase I and Phase II Control Charts. 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Remarks on the method of comparison equations (generalized WKB method) and the generalized Ermakov-Pinney equation

The connection between the method of comparison equations (generalized WKB method) and the Ermakov-Pinney equation is established. A perturbative scheme of solution of the generalized Ermakov-Pinney equation is developed and is applied to the construction of perturbative series for second-order differential equations with and without turning points.Comment: The collective of the authors is enlarged and the calculations in Sec. 3 are correcte

Causality and the AdS Dirichlet problem

The (planar) AdS Dirichlet problem has previously been shown to exhibit superluminal hydrodynamic sound modes. This problem is defined by bulk gravitational dynamics with Dirichlet boundary conditions imposed on a rigid timelike cut-off surface. We undertake a careful examination of this set-up and argue that, in most cases, the propagation of information between points on the Dirichlet hypersurface is nevertheless causal with respect to the induced light cones. In particular, the high-frequency dynamics is causal in this sense. There are however two exceptions and both involve boundary gravitons whose propagation is not constrained by the Einstein equations. These occur in i) AdS$_3$, where the boundary gravitons generally do not respect the induced light cones on the boundary, and ii) Rindler space, where they are related to the infinite speed of sound in incompressible fluids. We discuss implications for the fluid/gravity correspondence with rigid Dirichlet boundaries and for the black hole membrane paradigm.Comment: 29 pages, 5 figures. v2: added refs. v3: minor clarification

Jacobi Polynomials as su(2, 2) Unitary Irreducible Representation

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