Stability of Zernike coefficients solved by the covariance matrix method in the analysis of the wave front aberration

The deduction of Zernike coefficients is usually influenced by the finite number of sampling dots on interferogram and their inherited measurement errors. In this paper, a simplified Gram-Schmidt method for solving the Zernike polynomial with the higher fitting precision is presented and used to analyze the wave front aberrations for the circle interference fringe of the fine polished aluminum disk surface captured by a Twyman-Green interferometer system. We find the stability of the Zernike coefficients changes with changing the Zernike term, which has lead to the wrong expression for the wave front aberration. By analyzing the condition number of the coefficients matrix and the fitting precision of the method, it is indicated that the instability can be avoided when the Zernike term is lower than 14. Such an analysis will be valuable in solving the Zernike polynomial for the wave front aberration analysis in optical testing.     

ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OF ESTIMATES OF THE COVARIANCE MATRIX

Let X1,…XN(where N>m)be independent Nm(μ,∑）random vectors,and put X^-=1/N ∑i=1^N Xi and T‘T=A=∑i=1^N(Xi-X^-)(Xi-X^-)‘,where T is upper-triangular with positive diagonal elements.The author considers the problem of estimating ∑,and restricts his attention to the class of estimates D={T‘△^*T+Nb^*X^-X^‘·△^* is any diagonal matrix and b^* is any nonnegative constant}because it has the following attractive features:(a)Its elements are all quadratic forms of the sufficient and complete statistics(X^-,T).(b)It contains all estimates of the form αA+NbX^-X^-‘(α≥0 and b≥0),which construct a complete subclass of the class of nonnegative quadratic estimates D^8={X‘BX:B≥0}(where X=(X1,…,XN)‘)for any strict convex loss function.(c)It contains all invariant estimates under the transformation group of upper-triangular matrices.The author obtains the characteristics for an estimate of the form.T‘△T+NbX^-X^-‘(△=diag{δ1,…，δm}≥0 and b≥0)of ∑ to be admissible in D when the loss function is chosen as tr(∑^-1∑-I)^2,and shows,by an example,that αA+NX^-X^-‘(α≥0 and b≥0)is admissible in D^* can not imply its admissibility in D.     

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.     

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.     

Pre-Envelope Covariance Differential Equations For White and Nonwhite Input Processes

The pre-envelope process is a complex process whose statistics are strictly related to the statistics of the envelope of a given process. The paper deals with the evaluation of the covariances of the pre-envelope output process of classically and nonclassically damped linear systems subjected to stationary and nonstationary white and nonwhite pre-envelope input process. More precisely, the pre-envelope covariances for nonwhite complex input processes are evaluated as solution of a set of first order differential equations. Furthermore, in the paper the pre-envelope of the white input process is defined, and for such input the pre-envelope covariance differential equations are determined by means of an extension to the complex field of the stochastic differential calculus. Sommario.Il processo 'pre-inviluppo' é un processo complesso i cui parametri statistici sono strettamente legati a quelli del processo 'inviluppo'. Il lavoro riguarda la valutazione delle covarianze del processo pre–inviluppo della risposta di sistemi dinamici classicamente e non classicamente smorzati soggetti a processi pre–inviluppo bianchi e filtrati, stazionari e non stazionari. Piprecisamente, nel caso di un processo filtrato pre–inviluppo le covarianze della risposta sono valutate come soluzione di un sistema di equazioni differenziali del primo ordine. Inoltre, nel lavoro definito il processo pre–inviluppo di un processo bianco e per tale caso sono presentate le equazioni differenziali delle covarianze della risposta ottenute attraverso l'estensione al campo complesso del classico calcolo differenziale stocastico.     

A well-conditioned estimator for large-dimensional covariance matrices

Many applied problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator—the sample covariance matrix—is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte Carlo confirm that the asymptotic results tend to hold well in finite sample.     

On general and restricted covariance in general relativity

We reconsider the principle of general covariance and give a rigorous formulation of a principle ofrestricted covariance. We give a number of examples of preferred coordinate systems, considered in the literature, and in each case demonstrate the applicability of the notion of restricted covariance proposed.     

共形不变旋量方程及其应用

按照H.Weyl对旋量粒子方程的理解,本文第一次推求出明显SU(2,2)不变旋量方程组并且证明了它们相当于通常的共形Dirac方程。还讨论了共形旋量的物理意义,最后给出了关于gamma矩阵的Pauli基本公式。