On the Distribution of Overlaps in the Sherrington–Kirkpatrick Spin Glass Model

This paper describes some of the analytic tools developed recently by Ghirlanda and Guerra in the investigation of the distribution of overlaps in the Sherrington–Kirkpatrick spin glass model and of Parisi's ultrametricity. In particular, we introduce to this task a simplified (but also generalized) model on which the Gaussian analysis is made easier. Moments of the Hamiltonian and derivatives of the free energy are expressed as polynomials of the overlaps. Under the essential tool of self-averaging, we describe with full rigour, various overlap identities and replica independence that actually hold in a rather large generality. The results are presented in a language accessible to probabilists and analysts.     

Optimal resolution with extreme depth of focus

When the focal depth is required to be much larger than the wavelength, λ, the effective NA of the beam with optimal resolution is much less than unity. An aperture that is much larger than this beam's footprint is then of no consequence. Such beams that maximize the mean encircled energy fraction within a cylindrical focal region are shown to depend on only a single parameter, ω, that is proportional to the ratio of the square of the cylinder's radius to the product of its length and λ. A linear combination of Hermite– or Laguerre–Gaussian modes is used to represent these fields in two and three dimensions, respectively. For small ω, the results are compared both to asymptotic expansions and to optimal Gaussian and Bessel–Gauss beams.     

The molecular electrostatic potential of some simple molecules

The calculation of the molecular electrostatic potential from simplified models of the electron density is considered. Results are shown for water, hydrogen fluoride and ammonia. Little loss of accuracy is evident when the density is represented by a linear sum of well-chosen Gaussians. When these are further simplified into sets of point charges the inner parts of the molecule are poorly represented. More elaborate point moments make the representation worse. On the other hand a mixed representation with point charges and one diffuse Gaussian gives all the essential features of the potential of these molecules.     

Spectral distributions of harmonic generation from electron oscillation driven by intense femtosecond laser pulses

The characteristics of harmonic radiation due to electron oscillation driven by an intense femtosecond laser pulse are analyzed considering a single electron model. An interesting modulated structure of the spectrum is observed and analyzed for different polarization. Higher order harmonic radiations are possible for a sufficiently intense driving laser pulse. We have shown that for a realistic pulsed photon beam, the spectrum of the radiation is red shifted as well as broadened because of changes in the longitudinal velocity of the electrons during the laser pulse. These effects are more pronounced at higher laser intensities giving rise to higher order harmonics that eventually leads to a continuous spectrum. Numerical simulations have further shown that by increasing the laser pulse width broadening of the high harmonic radiations can be limited.     

高斯光束在任意分布的径向梯度折射率介质中传输的计算方法

根据弱波导近似,本文建立了处理高斯光束在任意分布的径向梯度折射率介质中传输的计算方法,并具体计算和讨论了高斯光束腰斑在沿径向呈抛物型分布的介质中随传输距离的变化情况,其数值结果与解析近似符合。     

基于衍射模型的Z扫描理论

以菲涅耳－基尔霍夫衍射理论为基础，建立了非线性光学介质对高斯光束的衍射模型，从一种新的角度解释了Z扫描现象，这种理论不仅在小非线性相移的情况下与传播Z扫描理论的结果完全一致，而且还适用于大非线性相移的情况。数值计算表明，随着非线性相移的增大，原来近似对称的Z扫描曲线发生畸变，谷的透过率被饱和抑制而峰的透过率被增强。通过开孔的光功率随着入射激光功率的增加会出现振荡衰减现象。     

Disintegration of Gaussian measures and average-case optimal algorithms

It is shown that a Gaussian measure in a given infinite-dimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator. This covers a similar statement made earlier in [Lee and Wasilkowski, Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2(1) (1986) 12–43.] for the case of finite-rank operators.     

Rounding Error Analysis in Solving M-Matrix Linear Systems of Block Hessenberg Form

The problem of solving large M-matrix linear systems with sparse coefficient matrix in block Hessenberg form is here addressed. In previous work of the authors a divide-and-conquer strategy was proposed and a backward error analysis of the resulting algorithm was presented showing its effectiveness for the solution of computational problems of queueing theory and Markov chains. In particular, it was shown that for block Hessenberg M-matrices the algorithm is weakly backward stable in the sense that the computed solution is the exact solution of a nearby linear system, where the norm of the perturbation is proportional to the condition number of the coefficient matrix. In this note a better error estimate is given by showing that for block Hessenberg M-matrices the algorithm is even backward stable.     

An equivalent polynomial approximation technique in nonlinear structural dynamics

Summary A new technique is proposed to obtain an approximate probability density for the response of a general nonlinear system under Gaussian white noise excitations. In this new technique, the original nonlinear system is replaced by another equivalent nonlinear system, structured by the polynomial formula, for which the exact solution of stationary probability density function is obtainable. Since the equivalent nonlinear system structured in this paper originates directly from certain classes of real nonlinear mechanical systems, the technique is applied to some very challenging nonlinear systems in order to show its power and efficiency. The calculated results show that applying the technique presented here can yield exact stationary solutions for the nonlinear oscillators. This is obtained by using an energy-dependent system, and for a nonlinearity of a more complex type. A more accurate approximate solution is then available, and is compared with the approximation. Application of the technique is illustrated by examples.