Quality oriented multimode processes monitoring based on a novel hierarchical common and specific structure with different order information

Due to higher demands on product diversity,flexible shift between productions of different products in one equipment becomes a popular solution,resulting in existence of multiple operation modes in a single process.In order to handle such multi-mode process,a novel double-layer structure is proposed and the original data are decomposed into common and specific characteristics according to the relationship between variables among each mode.In addition,both low and high order information are considered in each layer.The common and specific information within each mode can be captured and separated into several subspaces according to the different order information.The performance of the proposed method is further validated through a numerical example and the Tennessee Eastman(TE)benchmark.Compared with previous methods,superiority of the proposed method is validated by the better monitoring results.

Higher order asymptotic behaviour of partial maxima of random sample from generalized Maxwell distribution under power normalization

In this article, the higher order asymptotic expansions of cumulative distribution function and probability density function of extremes for generalized Maxwell distribution are established under nonlinear normalization. As corollaries, the convergence rates of the distribu- tion and density of maximum are obtained under nonlinear normalization.

A theorem for quantum operator correspondence to the solution of the Helmholtz equation

We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where V （X1 ,X2,X3） is the solution to the Helmholtz equation △2V ＋λ2V = 0, the symbol ： ： denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac＇s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.

EXISTENCE AND MULTIPLICITY OF POSITIVE SOLUTIONS TO THIRD-ORDER PERIODIC BOUNDARY VALUE PROBLEM

The existence and multiplicity of positive solutions to a periodic boundary value problem for nonlinear third-order ordinary differential equation are established, based on the zero point theorem concerning cone expansion and compression of order type. Our main approach is different from the previous papers on the existence of multiple positive solutions to the similar problem.