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Analytical Evaluation of Non-Elementary Integrals Involving Some Exponential, Hyperbolic and Trigonometric Elementary Functions and Derivation of New probability measures Generalizing the Gamma-Type and Gaussian-Type Distributions

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Advances in Pure Mathematics 2020年第7期10卷 371-392页

作者： Victor Nijimbere School of Mathematics and Statistics Carleton University Ottawa ON Canada

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, and where α,η and β are real or complex constants are evaluated in terms of the confluent hypergeometric function 1F1 and the hypergeometric function 1F2. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions 1F1 and 1F2. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and Gaussian distributions are also obtained. The obtained generalized probability distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square (x2) statistical tests and other statistical tests constructed based on the central limit theorem (CLT)), while avoiding the use of computational approximations (or methods) which are in general expensive and associated with numerical errors.

关键词： Non-Elementary Integrals Hypergeometric Function Confluent Hypergeometric Function probability measure Generalized Gamma-Type Distributions Generalized Gaussian-Type Distributions

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Optimal searching for a Helix target motion

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Science China Mathematics 2015年第4期58卷 749-762页

作者： MOHAMED Abd Allah El-Hadidy Department of Mathematics Faculty of Science Tanta University

This paper discusses a search problem for a Helix target motion in which any information of the target position is not available to the searchers. There exist three searchers start searching for the target from the origin. The purpose of this paper is to formulate a search model and finds the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite. Also, the existence of the optimal search plan that minimizes the expected value of the first meeting time is shown. Furthermore,this optimal search plan is found. The effectiveness of this method is illustrated by using an example with numerical results.

关键词： Helix motion optimal search plan first meeting time probability measure multiobjective nonlinear programming problem

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DIMENSIONS OF measure ON GENERAL SIERPINSKI CARPET

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Acta Mathematica Scientia 1999年第1期19卷 81-85页

作者：李文侠肖冬梅 Department of Mathematics Central China Normal University Wuhan 430079 China

Let S = Pi(i=1)(infinity){0, 1, ..., r - 1} and (R) over bar the general Sierpinski carpet, Let mu be the induced probability measure on (R) over bar of on S by phi, where phi is the natural surjection from S onto (R) over bar and is the infinite product probability measure corresponding to probability vector (b(0), ..., b(r-1)) with b(i) = a(i)(logn) (m-1)/m(alpha). Authors show that dim(H) mu = (C) under bar(L)(mu) = (C) over bar(L)(mu) = (C) under bar(mu) = (C) over bar C(mu) = alpha.

关键词： general Sierpinski carpet dimension of measure probability measure

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On the Abundance of Chaotic Behavior for Generic One-Parameter Families of Maps

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Acta Mathematica Sinica,English Series 1996年第4期12卷 398-412页

作者： Zheng Zhiming Department of Mathematics Peking University Beijing, 100871 China Department of Mathematics Peking University Beijing China

In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that if fa:I→I, a∈P is a regular family satisfying some conditions described in the next section, then there exists a Borel set Ω(?)P of positive Lebesgue measure such that for every a∈Ω, fa admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in [1] and [7], follows a similar line.

关键词： Chaos Generic regular family probability measure

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Simulating Random Fuzzy Variable

Simulating Random Fuzzy Variable

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The Eighth International Conference on Information and Management Sciences

作者： Danni Guo~1 and Renkuan Guo~2 1 Kirstenbosch Research Center South African National Biodiversity Institute Private Bag X7 Claremont 7735 Cape Town South Africa 2 Department of Statistical Sciences University of Cape Town Private Bag Rondebosch 7701 Cape Town South Africa

＜正＞In this paper,we discuss a statistical simulation approach to obtain a sequence of realizations of a random fuzzy ***,we extend the simulation scheme to random fuzzy Brownian process and the spatial random fuzzy... 详细信息

关键词： Credibility measure probability measure Fuzzy variable Random fuzzy variable Statistical simulation

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A Sufficient and Necessary Condition for Chance measure

A Sufficient and Necessary Condition for Chance Measure

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第一届中国智能计算大会

作者： Xiang Li Baoding Liu Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China

＜正＞Chance theory is a mathematical tool to deal with fuzzi- ness and randomness *** important con- cept in chance theory is chance measure,which is a hybrid of credibility measure and probability measure. This paper proves a sufficient and necessary condition for chance *** addition,the’chance extension theo- rem is proved,which provides a new method of defining a chance measure.

关键词： probability measure Credibility measure Chance measure Chance extension theorem.

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