Lambert W Function for Applications in Physics

The Lambert W(x) function and its possible applications in physics are presented. The actual numerical implementation in C++ consists of Halley's and Fritsch's iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued-logarithm recursion.Comment: 9 pages, 12 figures. Extended version of arXiv:1003.1628, updated link to source

Yukawa Potential Orbital Energy: Its Relation to Orbital Mean Motion as well to the Graviton Mediating the Interaction in Celestial Bodies

Research on gravitational theories involves several contemporary modified models that predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. In this paper we consider a Yukawa potential and we calculate the time rate of change of the orbital energy as a function of the orbital mean motion for circular and elliptical orbits. In both cases we find that there is a logarithmic dependence of the orbital energy on the mean motion. Using that, we derive an expression for the mean motion as a function of the Yukawa orbital energy, as well as specific Yukawa potential parameters. Furthermore, various special cases are examined. Lastly, expressions for the Yukawa range and coupling constant are also derived. Finally, an expression for the mass of the graviton mediating the interaction is calculated using the expression its Compton wavelength (i.e., the potential range ).Numerical estimates for the mass of the graviton mediating the interaction are finally obtained at various eccentricity values and in particular at the perihelion and aphelion points of Mercury’s orbit around the sun

Computing the Lambert W function in arbitrary-precision complex interval arithmetic

We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in interval arithmetic, which has been implemented in the Arb library. The classic 1996 paper on the Lambert W function by Corless et al. provides a thorough but partly heuristic numerical analysis which needs to be complemented with some explicit inequalities and practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure

A framework for second-order parton showers

A framework is presented for including second-order perturbative corrections to the radiation patterns of parton showers. The formalism allows to combine O(alphaS^2)-corrected iterated 2->3 kernels for "ordered" gluon emissions with tree-level 2->4 kernels for "unordered" ones. The combined Sudakov evolution kernel is thus accurate to O(alphaS^2). As a first step towards a full-fledged implementation of these ideas, we develop an explicit implementation of 2->4 shower branchings in this letter.Comment: 11 pages, 3 figure

Electromagnetic surface wave propagation in a metallic wire and the Lambert WW function

We revisit the solution due to Sommerfeld of a problem in classical electrodynamics, namely, that of the propagation of an electromagnetic axially symmetric surface wave (a low-attenuation single TM$_{01}$ mode) in a cylindrical metallic wire, and his iterative method to solve the transcendental equation that appears in the determination of the propagation wave number from the boundary conditions. We present an elementary analysis of the convergence of Sommerfeld's iterative solution of the approximate problem and compare it with both the numerical solution of the exact transcendental equation and the solution of the approximate problem by means of the Lambert $W$ function.Comment: REVTeX double column, 9 pages, 3 figures, minor differences between v3 and published version; "Editor's Pick" for June 2019 edition of AJ

On an Estimation in quantum hypothesis testing via Powers-St{\o}rmer's Inequality

Regarding finding possible upper bounds for the probability of error for discriminating between two quantum states, it is known that $$\mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big)$$ holds for every positive matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive matrices $A$ and $B$. We show that the class of functions satisfying this inequality contains additional elements and we provide a better estimation concerning the quantum Chernoff bound. We also investigate the case of matrix monotone decreasing functions and present a matrix inequality, when the matrix perspective function is considered

An invertible dependence of the speed and time of the induction machine during no-load direct start-up

Novel expression for time–speed curve of IM during no-load direct start-up An invertible dependence for speed–time curve of IM during no-load direct start-up Simulation results Experimental results Conclusion Disclosure statement References Full Article Figures & data References Citations Metrics Licensing Reprints & Permissions PDF AbstractFormulae display:MathJax Logo? In this paper, an invertible dependence of the speed and time of the induction machine during no-load direct start-up is presented. Namely, based on the parameters of the induction machine equivalent circuit as well as on the basic, well-known, equation for machine torque, the analytical expression for the induction machine time-speed dependence during direct start-up is derived. On the other hand, in order to obtain inverse i.e. speed-time dependence, the derived time-speed expression is rearranged in one nonlinear equation. As the derived nonlinear equation does not have an analytical solution, a novel iterative procedure, based on the usage of Lambert W function, is proposed for its solving. The results obtained by using the developed expressions for speed-time or time-speed curves are compared with the corresponding results obtained by using expressions known in the literature as well as with the results obtained by using a numerical time-domain computation method. Moreover, the results obtained by using the developed expressions have been compared with the corresponding experimental results to demonstrate the accuracy of the derived expressions. The Matlab code developed for solving the presented iterative procedure, as well as the Matlab code for induction machine speed-time curve determination, is also provided

Non-iterative simulation methods for virtual analog modelling

The simulation of nonlinear components is central to virtual analog simulation. In audio effects, circuits often include devices such as diodes and transistors, mostly operating in a strongly nonlinear regime. Mathematical models are in the form of systems of nonlinear ordinary differential equations (ODEs), and traditional integrators, such as the trapezoid and midpoint methods, can be employed as solvers. These methods are fully implicit and require the solution of a nonlinear algebraic system at each time step, introducing further complications regarding the existence and uniqueness of the solution, as well as the choice of halting conditions for the iterative root finder. On the other hand, fast explicit methods such as Forward Euler, are prone to unstable behaviour at standard audio sample rates. For these reasons, in this work, a family of linearly-implicit schemes is presented. These schemes take the form of a perturbation expansion, making the construction of higher-order schemes possible. Compared with classic implicit designs, the proposed methods have the advantage of efficiency, since the update is computed in a single iteration. Furthermore, the existence and uniqueness of the update are proven by simple inspection of the update matrix. Compared to classic explicit designs, the proposed schemes display stable behaviour at standard audio sample rates. In the case of a single scalar ODE, sufficient conditions for numerical stability can be derived, imposing constraints on the choice of the sampling rate. Several theoretical results are provided, as well as numerical examples for typical stiff equations used in virtual analog modelling