Numerical propagation of partially coherent laser beams through optical systems

The propagation of partially coherent beams through optical systems is computed numerically in one transverse dimension. The optical system is divided into different elementary segments, through which the propagation of light can be calculated by appropriate operators, working on the coherence function or the Wigner distribution function respectively. For the necessary changes between these two functions describing the partially coherent beams, the use of the remarkable z-transform seems to be an advantage. With this algorithm the grid and the resolution in the spatial frequency domain can be arbitrarily chosen in contrast to the usual Fourier transform, the influence of phase aberrations on the focusability of Gauss-Schell model beams is discussed as an application example of the numerical model. With the help of this tool, practical beam guiding systems can be simulated for use with multimode laser radiation.     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

Application of a Near-Optimal Reinforcement Learning Controller to a Robotics Problem in Manufacturing: A Hybrid Approach

Optimization theory provides a framework for determining the best decisions or actions with respect to some mathematical model of a process. This paper focuses on learning to act in a near-optimal manner through reinforcement learning for problems that either have no model or the model is too complex. One approach to solving this class of problems is via approximate dynamic programming. The application of these methods are established primarily for the case of discrete state and action spaces. In this paper we develop efficient methods of learning which act in complex systems with continuous state and action spaces. Monte-Carlo approaches are employed to estimate function values in an iterative, incremental procedure. Derivative-free line search methods are used to obtain a near-optimal action in the continuous action space for a discrete subset of the state space. This near-optimal control policy is then extended to the entire continuous state space via a fuzzy additive model. To compensate for approximation errors, a modified procedure for perturbing the generated control policy is developed. Convergence results under moderate assumptions and stopping criteria are established.     

斜孔气膜冷却数值模拟分析

本文通过数值模拟分析圆柱孔和扩散孔的单斜孔气膜冷却特性,考察复合角、孔型和吹风比对流场和气膜冷却效果的影响。结果表明,复合角的引入使气膜侧向分布更宽,但冷却效果沿主流方向衰减更快。适中吹风比得到的气膜能更有效的保护壁面。在相同吹风比和复合角条件下,扩散孔的气膜冷却效率比圆柱孔更好,且冷却更为均匀持久。     

Bases,spanning sets,and the axiom of choice

Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F, every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ?₂ has a basis implies that every well‐ordered collection of two‐element sets has a choice function. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Computation of Figure-eight Solution for 3-body Problems

The main goal of this paper is to compute the Figure-eight solutions for the planar Newtonian 3-body problem with equal masses by finding the critical points of the functional associated with the motion equations of 3-body in plane R2. The algorithm adopted here is the steepest descent method, which is simple but very valid for our problem.     

含腔电大尺寸导体目标电磁散射的一体化数值模拟

应用等效原理，通过引入口面上等效磁流将含腔导电目标电磁散射简化为腔内、外两个等效 问题. 腔内问题分段求解并应用级联法获得口面等效导纳矩阵；腔内外的耦合关系应用近似 边界元方法描述并由此获得口面等效磁流；最后，这一具有混合源的腔体内外一体化散射问 题则应用所提出的广义混合场积分方程方法建立电磁模型，并用多层快速多极子方法实现高 效数值求解. 实例计算结果与测试结果具有很好的一致性. 关键词： 含腔目标 电磁散射 混合场积分方程 数值分析     

Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

ℙmax variations related to slaloms

We prove the iteration lemmata, which are the key lemmata to show that extensions by Pmax variations satisfy absoluteness for Π₂-statements in the structure 〈H (ω ₂), ∈, NS_{ω 1}, R 〉 for some set R of reals in L (ℝ), for the following statements: (1) The cofinality of the null ideal is ℵ₁. (2) There exists a good basis of the strong measure zero ideal. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)