The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize “approximation level” to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis.
Defines two-valued logic p-measure in probability space whose power is 2 and randomized truth degree of propositions in classical logic system which is popularized from truth degree of propositions (see Wang Guojun in Science in China (series A) 45(9):1106-1116, 2002). This paper also testifies that the set of randomized truth degree of all formulas in the range of [0, 1] is dense when p=1/3, and gives the general expression of formula randomized truth degree. It defines similarity degree between formulas using randomized truth degree, and educes a kind of pseudo-distance in the set of all formulas, so provides a possible structure of proximity reasoning theory.
In order to overcome the well-known multicollinearity problem, we propose a new Stochastic Restricted Liu Estimator in logistic regression model. In the mean square error matrix sense, the new estimation is compared with the Maximum Likelihood Estimation, Liu Estimator Stochastic Restricted Maximum Likelihood Estimator etc. Finally, a numerical example and a Monte Carlo simulation are given to explain some of the theoretical results.
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