Geometrical Optics and Models of Computer Memory Fragmentation

We analyze a two-dimensional difference equation that arose in a stochastic model of memory fragmentation in computers. We use the ray method of geometrical optics, and other singular perturbation methods, to solve this equation asymptotically. The asymptotic limit corresponds to one of heavy system usages. We interpret the results from the ray method probabilistically.     

Buffer Overflow Probabilities for a Markov-Modulated Fluid Model

Markov modulated fluid models are widely used in modelling communications and computer systems. In the AMS (Annick, Mitra, Sohndi) model, heterogeneous, bursty sources modeled by multidimensional Markov processes are superimposed or multiplexed together to drive a fluid buffer. The performance of the system is measured by the steady state probability that the buffer exceeds a high level. The exact solution to this problem derived by AMS requires too much computation to be used on-line. Here we derive an upper bound for the above probability which is fast to compute and accurate enough for practical use.     

Pricing Options under Two-Factor Markov-Modulated Stochastic Volatility Models

In this paper, we investigate the valuation of European-style call options under an extended two-factor Markov-modulated stochastic volatility model, where the first stochastic volatility component is driven by a mean-reversion square-root process and the second stochastic volatility component is modulated by a continuous-time, finite-state Markov chain. The inverse Fourier transform is adopted to obtain analytical pricing formulae. Numerical examples are given to illustrate the discretization of the pricing formulae and the implementation of our model.     

Integrable Equations in Nonlinear Geometrical Optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole–Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov–Novikov equation for the refractive index. It is demonstrated that the Veselov–Novikov hierarchy is amenable to the quasiclassical     -dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev–Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.     

Reflections on Refraction (Geometrical Optics)

Consider an optical system made of an unknown number N of layers of homogeneous transparent plates with different unknown refraction indices. Observing beams of monochromatic light through the system, find the number N of plates together with their respective indices and their thicknesses. The mathematical analysis of the problem involves the so-called Hadamard quotient of two power series. We shall also discuss fractal optical systems consisting of infinitely many infinitely thin plates. If the index of refraction varies in an erratic way there may be multiple refraction. These systems could be called “refractals". We conclude the paper with independent considerations on a general system consisting of one plate with continuous varying index n(x, y) ≥ 1. To determine the function seems to be a difficult problem. Our contribution to solving it is thus very modest. Submitted: December 20, 2007. Accepted: February 20, 2008. à Jacques Peyrière     

Blossoming: A Geometrical Approach

A geometrical approach of a notion of blossom for piecewise smooth Chebyshev functions is developed by considering convenient intersections of osculating flats. A subblossoming principle allows us to obtain all the expected properties and leads to the notion of blossom for splines based on a given piecewise smooth Chebyshev function. January 7, 1997. Date revised: October 1, 1997. Date accepted: December 22, 1997.     

Stochastic Mean-Field Approach to Fluid Dynamics

It is shown that the incompressible Navier–Stokes equation can be derived from an infinite-dimensional mean-field stochastic differential equation.     

Variational Approach to the Inviscid Compressible Fluid

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.      

Modulation Equations of Weakly Nonlinear Geometrical Optics in Media Exhibiting Mixed Nonlinearity

The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propagating in materials exhibiting mixed nonlinearity is derived with the resonant wave interactions inherent in the system. Our analysis differs from the results of Hunter et al., in that we have employed a different scaling, keeping in view the delayed effects of nonlinearity in certain thermodynamic systems exhibiting mixed nonlinearity. The result is to modify the transport equations obtained by Hunter et al. by the addition of certain cubic nonlinear terms. Through the method of averaging, the secular terms are eliminated. However, the averaging process is carried out in two steps; first, along manifolds of codimension two giving an advection equation, the solution of which is then averaged in a direction transverse to the above-mentioned manifold.     

A Minimax Approach to Errors-in-Variables Linear Models

The paper considers a simple Errors-in-Variables (EiV) model Y_i = a + bX_i + εξ_i; Z_i= X_i + σζ_i, where ξ_i, ζ_i are i.i.d. standard Gaussian random variables, X_i ∈ ? are unknown non-random regressors, and ε, σ are known noise levels. The goal is to estimates unknown parameters a, b ∈ ? based on the observations {Y_i, Z_i, i = 1, …, n}. It is well known [3] that the maximum likelihood estimates of these parameters have unbounded moments. In order to construct estimates with good statistical properties, we study EiV model in the large noise regime assuming that n → ∞, but \({\epsilon ^2} = \sqrt n \epsilon _ \circ ^2,{\sigma ^2} = \sqrt n \sigma _ \circ ^2\) with some \(\epsilon_\circ^2, \sigma_\circ^2>0\). Under these assumptions, a minimax approach to estimating a, b is developed. It is shown that minimax estimates are solutions to a convex optimization problem and a fast algorithm for solving it is proposed.     

基于几何光学法和惠更斯原理的微波暗室性能分析

微波暗室在导弹导引仿真中已广泛使用,针对导弹导引仿真试验中用到的微波暗室的静区性能评估问题,分别建立了基于几何光学法和惠更斯原理的评价分析模型.此模型考虑了暗室空间内各个壁面电磁辐射的相互影响,将壁面看作次级余弦辐射体,建立了描述壁面任一点垂直辐射强度的积分微分方程.通过离散方法将问题转化为求解线性方程组的数值问题.计算得到暗室壁面辐射强度分布以及性能指标γ.结果表明,垂直反射率为0.5时不满足要求,而0.05时满足要求.最后考察了模型的收敛性,随着离散度的增高,计算结果趋于稳定.     

Geometric Optics Approach to First-Passage Distributions: Caustic Boundaries and Exponentially Small Eigenvalues

We develop singular perturbation methods for computing the first passage time distribution for one-dimensional diffusion processes. Detailed results are given for an Ornstein–Uhlenbeck process, and the method is sketched for more general problems. For some parameter values, we find the presence of caustic boundaries; whereas, for other parameter values, there are exponentially small eigenvalues. We use the ray method of geometric optics and asymptotic matching.     

Passage Times in Fluid Models with Application to Risk Processes

An efficient quadratically convergent algorithm has been derived earlier by Ahn and Ramaswami for computing the busy period distribution of the canonical fluid flow model. In this paper, we derive formulae for a variety of passage time distributions in the canonical fluid flow model in terms of its busy period distribution and that of its reflection about the time axis. These include several passage time distributions with taboo not only of the fluid level 0 but also of a set [a, ∞) of levels. These are fundamental to the analysis of a large set of complex applied probability models, and their use is illustrated in the context of a general insurance risk model with Markovian arrival of claims and phase type distributed claim sizes, a context in which we have also introduced some new ideas that make the analysis very transparent.      

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Summary. In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables. Received November 8, 2000; accepted January 6, 2001 Online publication March 23, 2001     

Fluid Models of the Solar Wind

The paper analyses the fluid dynamical models that have beenconstructed to describe the solar wind since Parker's (1958)first model. A brief historical development of the subject issketched, and an account is given of the gross properties ofthe solar wind medium. Based on certain simplifying postulates,the fluid equations of continuity, momentum, and energy aregiven for the idealized solar wind. Parker's polytropic modelis presented, followed by an analysis of the various modelsincorporating thermal conduction. Viscous and two-fluid solarwind models are examined. A discussion of the applicabilityof the hydrodynamic regime to the solar wind is finally given.     

An Interactive,Graphical Approach to Linear Financial Models

Early attempts to utilize linear algebraic (L.P.) techniques in financial planning met with little success. This was probably due to a misunderstanding of the process by which financial and corporate plans were formulated. The planning process is now better understood and the use of computer-based models is well accepted. However current modelling systems perform only simple arithmetic evaluations. There is a prima facie case for re-examining linear algebraic methods to exploit the considerable amount of information which is available within a plan. This paper describes the work done in the South of Scotland Electricity Board to develop an interactive financial modelling system.     

An Algebraic Approach to Knowledge Base Models Informational Equivalence

In this paper we study the notion of knowledge from the positions of universal algebra and algebraic logic. We consider first order knowledge which is based on first order logic. We define categories of knowledge and knowledge base models. These notions are defined for the fixed subject of knowledge. The key notion of informational equivalence of two knowledge base models is introduced. We use the idea of equivalence of categories in this definition. We prove that for finite models there is a clear way to determine whether the knowledge base models are informationally equivalent.