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

Numerical Algorithms - We describe an algorithm to evaluate all the complex branches of the Lambert W function with rigorous error bounds in arbitrary-precision interval arithmetic or ball...     

Solution of the self-adjoint radiative transfer equation on hybrid computer systems

A new technique for simulating three-dimensional radiative energy transfer for the use in the software designed for the predictive simulation of plasma with high energy density on parallel computers is proposed. A highly scalable algorithm that takes into account the angular dependence of the radiation intensity and is free of the ray effect is developed based on the solution of a second-order equation with a self-adjoint operator. A distinctive feature of this algorithm is a preliminary transformation of rotation to eliminate mixed derivatives with respect to the spatial variables, simplify the structure of the difference operator, and accelerate the convergence of the iterative solution of the equation. It is shown that the proposed method correctly reproduces the limiting cases—isotropic radiation and the directed radiation with a δ-shaped angular distribution.     

Rank–Crank-type PDEs and generalized Lambert series identities

We show how Rank–Crank-type PDEs for higher order Appell functions due to Zwegers may be obtained from a generalized Lambert series identity due to the first author. Special cases are the Rank–Crank PDE due to Atkin and the third author and a PDE for a level 5 Appell function also found by the third author. These two special PDEs are related to generalized Lambert series identities due to Watson, and Jackson, respectively. The first author’s Lambert series identity is a common generalization. We also show how Atkin and Swinnerton-Dyer’s proof using elliptic functions can be extended to prove these generalized Lambert series identities.     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.     

A new characteristic of Möbius transformations by use of polygons having type A

In the hyperbolic plane Möbius transformations can be characterized by Lambert quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals must be Möbius. In this paper we generalize this result to the case of polygons with n sides having type A, that is, having exactly two non-right interior angle.     

Inferences based on a bivariate distribution with von Mises marginals

There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.     

Optimization of Approximation Models for Solving Kinetic Equations

We consider the problem of cancer treatment using radiation therapy. A deterministic Boltzmann equation is considered as the particle transport model. The P_N system of moment equations is derived using angular discretization of the transport equation. This P_N system is solved using StaRMAP [3]. An optimization problem to get the desired particle distribution is formulated and solved using adjoint equations. The algorithm is validated with several simulations. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Factorization theorems for generalized Lambert series and applications

The Ramanujan Journal - We prove new variants of the Lambert series factorization theorems studied by the authors which correspond to a more general class of Lambert series expansions of the form...     

Optimality of the fully discrete filtered backprojection algorithm for tomographic inversion

Although the filtered backprojection algorithm (FBA) has been the standard reconstruction algorithm in 2D computerized tomography for more than 30 years, its convergence behavior is not completely settled so far. Relying on convergence results by Rieder and Faridani for the semi-discrete FBA [SIAM J. Numer. Anal., 41(3), 869–892, 2003], we show optimality of the fully discrete version for reconstructing sufficiently smooth density distributions. Further, we introduce MFBA, a modified version of FBA, and prove its optimality under weaker smoothness requirements. Remarkably MFBA may have a larger convergence order in the angular than in the lateral variable, thus allowing optimal convergence in case of angular under- sampling. Moreover, MFBA can be seen as a limit of the phantom view method introduced to increase angular resolution.     

LambertW函数对一阶复时滞动力系统的稳定性分析

Lambert W函数具有的一些性质以及现今成熟的数学软件Maple等使得它能很好地应用于时滞微分方程的稳定性判别中.通过应用Lambert W函数对一阶复系数时滞微分方程渐近稳定性的判别命题,分析了一类参数反馈控制复系数时滞微分方程的稳定性,得到了更加精细的结果.相比已往的方法,新方法更简单、计算更方便并能快速有效的给出判定结果.     

Lambert's W, infinite divisibility and Poisson mixtures

The Lambert W function is shown to be the Laplace exponent of a positive infinitely divisible law (i.e. W is a Bernstein function) called the standard Lambert law. This law is a generalized gamma convolution. At least three Poisson mixture families are defined in terms of W. One of these is the generalized Poisson laws which are shown to be generalized negative-binomial convolutions. Mixing with positive stable laws yields further generalizations.     

Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

A note on Chrystal’s equation

The exact, explicit form of the transcendental solution of Chrystal’s equation, a first order nonlinear ordinary differential equation (ODE) of degree two, is derived in terms of the Lambert W-function. It is shown that this case of the general solution is dual-valued over a finite interval and that, for a special case of the coefficients, its zeros involve the Golden ratio. Additionally, a number of applications involving special cases of this ODE are noted and the main properties of the Lambert W-function are briefly reviewed.     

基于CT图像重建的多重集合分裂可行性问题应用分析

为了较好地应用CQ算法解决稀疏角度CT 图像重建的问题,提出了一种新的实时的分块逐次混合算法．首先将稀疏角度CT 图像重建的重建问题转化成分裂可行性问题．其次,通过分析非空闭凸集C和Q的不同的定义,在N维实空间中分别针对不同的CQ算法给出了7种不同的实现方案．通过试验,分别对不同算法及其方案的重建精度和收敛速度进行了对比分析,并对多重集合分裂可行性问题算法中约束权因子的选取及其对输出的影响进行了研究,从而给出了CQ算法在稀疏角度CT图像重建问题中应用的最佳凸集定义方案．以此为基础,给出了所提出算法的最佳实现方案．试验结果表明,该算法收敛速度快,重建精度高,为多重集合分裂可行性问题及其改进算法在该重建问题上的应用提供了参考．     

On three identities of Ramanujan

The Ramanujan Journal - In this paper, we represent the generating function of the rank function as a summation of four parts—a constant, two Lambert series and a product. Applying it to...     

Deriving Karmarkar’s LP algorithm using angular projection matrix

Understanding Karmarkar’s algorithm is both desirable and necessary for its efficient implementation, for further improvement and for carrying out complexity analysis. In this report an algorithm based on the concept of angular projection matrix, to solve linear programming problems is derived. Surprisingly, this algorithm coincides with the affine version of Karmarkar’s algorithm.     

Selected experimental methods for determining droplet parameters from the intensity distribution of scattered light: A review

The present paper summarizes some experimental methods, which have been developed using the results of calculations based on the theory of Gustav Mie. As examples two regions of the scattered light are considered in more detail. The intensity distribution in the forward hemisphere for scattering angles 30 ≤ θ ≤ 60 shows maxima, which can be identified as regular fringes on a screen. The spacing between the fringes is a measure for the droplet size. In the intensity distribution of the backward hemisphere the region of the first rainbow can be found. In the rainbow region a main maximum, subsidiary maxima and a ripple structure can be identified. The angular position of the main maximum is a measure for the refractive index of the droplet. The angular distance between the subsidiary maxima and the angular distance between the ripples both are a function of droplet size. A comparison with other size measurement methods gives information about the sphericity of the droplet. An evaporating droplet shows for instance oscillations of the rainbow position due to morphology dependent resonances. The oscillation frequency is proportional to the evaporation rate.     

Some new transformations for Bailey pairs and WP-Bailey pairs

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.     

Development of the parametric tolerance modeling and optimization schemes and cost-effective solutions

Most of previous research on tolerance optimization seeks the optimal tolerance allocation with process parameters such as fixed process mean and variance. This research, however, differs from the previous studies in two ways. First, an integrated optimization scheme is proposed to determine both the optimal settings of those process parameters and the optimal tolerance simultaneously which is called a parametric tolerance optimization problem in this paper. Second, most tolerance optimization models require rigorous optimization processes using numerical methods, since closed-form solutions are rarely found. This paper shows how the Lambert W function, which is often used in physics, can be applied efficiently to this parametric tolerance optimization problem. By using the Lambert W function, one can express the optimal solutions to the parametric tolerance optimization problem in a closed-form without resorting to numerical methods. For verification purposes, numerical examples for three cases are conducted and sensitivity analyses are performed.