Approximate solution for some stochastic differential equations involving both Gaussian and Poissonian white noises

By combining the Kramers-Moyal expansion with fractional Brownian motion of order n, in a formal symbolic calculus, one can obtain an approximation for the solution of some stochastic differential equations involving both Gaussian and Poissonian white noises, in terms of rotating Gaussian white noises on the grid defined by the complex roots of the unity. Illustrative examples are outlined.     

1H NMR spectra of alkane‐1,3‐diols in benzene: GIAO/DFT shift calculations

The proton nuclear magnetic resonance (NMR) spectra of propane‐1,3‐diol, 2‐methylpropane‐1,3‐diol, 2,2‐dimethylpropane‐1,3‐diol, butane‐1,3‐diol, 3‐methylbutane‐1,3‐diol, pentane‐2,4‐diols (dl and meso), 2‐methylpentane‐2,4‐diol and cyclohexane‐1,3‐diols (cis and trans) in benzene have been analysed. The conformer distribution and the NMR shifts of these diols have been computed on the basis of density functional theory, the solvent being included by means of the integral equation formalism phase continuum model (IEFPCM) implemented in Gaussian 09. Relative Gibbs energies of all conformers are calculated at the Perdew, Burke and Ernzerhof (PBE)0/6‐311 + G(d,p) level, and NMR shifts by the gauge‐including atomic orbital method with the PBE0/6‐311 + G(d,p) geometry and the cc‐pVTZ basis set. Vicinal coupling constants for 1,2‐ and 1,3‐diols are rationalised in terms of relative conformer populations and geometries. The NMR shifts of hydrogen‐bonded protons in individual conformers of alkane‐1,n‐diols show a very rough correlation with the OH?OH distances. The computed overall NMR shifts for CH protons in 1,2‐ and 1,3‐diols are systematically high but correlate very well with the experimental values, with a gradient of 1.07 ± 0.01. Some values for nonequivalent methylene protons in 1,3‐diols are reversed, calculation giving enhanced values for the proton anti to the C? OH bonds. Errors in the NMR shifts computed for the OH protons of nonsymmetrical diols appear to be related to relative populations of conformers where one or other of the OH groups is the donor. Some results based on the second‐order Møller–Plesset approach, the Becke three‐parameter Lee‐Yang‐Parr method and on the IEFPCM solvation model implemented in Gaussian 03 are included. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of a Gaussian beam profile passing through a Kerr medium

The circular Gaussian beam shape has been simulated by numerical evaluation on beam shape based on z-scan method through beam radius. This method is useful in many studies such as z-scan experiments. The validity of the relation between peak and valley separation in close aperture data and the diffraction length of the beam has been investigated by this method for some real cases. The results are in good agreement with the experimental studies.     

Uniqueness of Gaussian quadrature formula for computed tomography

This note answers positively a question raised by B. Bojanov and G. Petrova. Namely, the Gaussian quadrature formula for computed tomography among the given type is unique.     

Density estimates for a random noise propagating through a chain of differential equations

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the n−1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on … , so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.     

A Note on the Difference of Gaussian Measure of Two Balls in Hilbert Spaces

§１．　Ｉｎｔｒｏｄｕｃｔｉｏｎ　　ＬｅｔＨｂｅａｓｅｐａｒａｂｌｅＨｉｌｂｅｒｔｓｐａｃｅ，μｂｅａｓｙｍｍｅｔｒｉｃＧａｕｓｓｉａｎｍｅａｓｕｒｅｏｎＨａｎｄλ１≥λ２≥…０ｂｅｔｈｅｅｉｇｅｎｖａｌｕｅｓｏｆｔｈｅｃｏｖａｒｉａｎｃｅμＴ．ＺａｋｇｉｖｅｓｔｈｅｅｓｔｉｍａｔｅｏｆｔｈｅｄｉｆｆｅｒｅｎｃｅｏｆＧａｕｓｓｉａｎｍｅａｓｕｒｅｏｆｔｗｏｂａｌｌ．μ｛ｘ∈Ｈ：ｙｘｙ≤ｔ｝－μ｛ｘ∈Ｈ：ｙｘ＋ｒｙ≤ｔ｝ｉｎ［１，２］．Ｉｎｔｈｉｓｎｏｔｅｗｅｏｂｔａｉｎａｆｉｎｅｒｅｓｔｉｍａｔ…     

On the efficiency of a testimator for the mean of an inverse Gaussian distribution

Though the sample mean is a natural estimator for the mean μ of an Inverse Gaussian (IG) distribution having another parameter λ, when a guess μ₀ for μ seems plausible, an alternative adaptive estimator which shrinks towards μ₀ when a preliminary test for H₀:μ = μ₀ is tenable or else towards , is considered a suitable competitor. Certain numerical illustrations are presented showing higher efficiency of such a testimator over in several situations when the sample size is small and λ is either known or unknown.     

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.