This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's t...This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.展开更多
We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a c...We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].展开更多
One of the most important challenges in the design of the foundation of the Earth layer below the surface is the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desir...One of the most important challenges in the design of the foundation of the Earth layer below the surface is the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desired user. Based on this analysis and design criteria of two successive ruptures (load bearing) and settlement, due to the nature of non-homogeneous soil and its parameters uncertainty, relying on one number as the amount of foundation settlement doesn’t seem logical. This is while in the methods of the probability distribution function by taking the probability for each of the input parameters, or the characteristics of each parameter, the parameter values are likely to have the chance of occurrence. In this research, effort is made using the method of probabilistic Monte Carlo simulation, the effect of the uncertainty of parameters influencing the mechanical behavior following the successive layers of earth and examined. In the event that non-deterministic model input variables for describing, not non-deterministic model output as well. So the output of each method to analysis of the concept of the probability distribution function for the input variables is a function of the probability distribution for the target function. In this study, the reliability of the settlement for the three modes of settlement center, corner of rigid foundation is fitted with two types of normal probability distribution and the log-normal distributions. For this purpose, the parameters of the effect of the transition on the analysis of soil modulus of elasticity of foundation, such as settlement and the coefficient of Poisson ratio distribution in probability using probabilistic log-normal and normal have been considered. Analysis indicated that the settlement in the center of the wake is flexible critical than the other two and has a higher probability of occurrence of the settlement in this part of the foundation. In the case of the normal distribution and the normal distribution graph of the log was used, the probab展开更多
One of the most important challenges in the design of the foundation of the Earth layer below the surface, the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desired...One of the most important challenges in the design of the foundation of the Earth layer below the surface, the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desired user. Based on this analysis and design criteria of two successive ruptures (load bearing) and settlement. The design of foundation is usually the amount of designated critical foundation than the amount of force that sought the ability to transfer to the soil below. Informal mode using the average values of the parameter, transition effects on mechanical behavior of soil, a number of settlement any part of the amounts are determined by the foundation. Due to the nature of non-homogeneous soil and its parameters uncertainty, relying on one number as the amount of foundation settlement doesn’t seem logical. This is while in the methods of the probability distribution function by taking the probability for each of the input parameters, or the characteristics of each parameter, the parameter values are likely to have the chance of occurrence. In this research, effort is made using the method of probabilistic Monte Carlo simulation, the effect of the uncertainty of parameters influencing the mechanical behaviour following the successive layers of earth and examined. This method is a kind of simulation is that the uncertainty in the different aspects of the issue to be obvious and a bit of the show. Monte Carlo simulation method for the determination of model uncertainty, a little bit for each of the input random variables is a function of the probability distribution which is considered. In the event that non-deterministic model input variables for describing, not non-deterministic model output as well. So the output of each method to analysis of the concept of the probability distribution function for the input variables is a function of the probability distribution for the target function. In this study, the reliability of the settlement for the three modes of settlement Center, corner of rigid foun展开更多
The uncertainty distribution can more effectively express the uncertainty of decision makers' judgments during a pairwise comparison of any alternatives. This paper investigates the priority models of group intuit...The uncertainty distribution can more effectively express the uncertainty of decision makers' judgments during a pairwise comparison of any alternatives. This paper investigates the priority models of group intuitionistic fuzzy preference relations with normal uncertainty distribution. The mathematical equivalence between the membership, non-membership degree interval fuzzy preference relation and the intuitionistic fuzzy preference relation is constructed, showing that there exists an inverse relationship between the priority of alternatives using these two types of interval preference relations. The new optimal models regarded the event that the deviation between the ideal judgement meeting the multiplicative consistency and the actual judgement obeying normal uncertainty distribution shall not exceed a threshold value under the given belief degree as a constraint, and regarded the minimum sum of all the threshold values as the objective function. The chance constraint was introduced to measure the degree to which multiplicative consistency can be realized under different belief degrees. The priority model provides a new method for simulating uncertainty and fuzziness in the real-world decision making environment.展开更多
The primary results have been obtained by the analyses on the deviation of macroscopic epicenters from relevant microscopic epicenters determined rapidly after the 139 earthquake events occunrred during 1989-2004. Fur...The primary results have been obtained by the analyses on the deviation of macroscopic epicenters from relevant microscopic epicenters determined rapidly after the 139 earthquake events occunrred during 1989-2004. Further study has proved the normal distributions exist in both the east and west parts of Chinese mainland partitioned along meridian 105°E. The empirical two-dimensional normal distribution functions for the deviations are presented in the paper. The results are of significance in reducing uncertainty of losses assessment immediately after a destroyed earthquake.展开更多
基金supported by National Basic Research Program of China (Grant No.2007CB814900)(Financial Risk)
文摘This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It's type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.
基金support from The National Basic Research Program of China(973 Program)grant No.2007CB814900(Financial Risk)
文摘We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].
文摘One of the most important challenges in the design of the foundation of the Earth layer below the surface is the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desired user. Based on this analysis and design criteria of two successive ruptures (load bearing) and settlement, due to the nature of non-homogeneous soil and its parameters uncertainty, relying on one number as the amount of foundation settlement doesn’t seem logical. This is while in the methods of the probability distribution function by taking the probability for each of the input parameters, or the characteristics of each parameter, the parameter values are likely to have the chance of occurrence. In this research, effort is made using the method of probabilistic Monte Carlo simulation, the effect of the uncertainty of parameters influencing the mechanical behavior following the successive layers of earth and examined. In the event that non-deterministic model input variables for describing, not non-deterministic model output as well. So the output of each method to analysis of the concept of the probability distribution function for the input variables is a function of the probability distribution for the target function. In this study, the reliability of the settlement for the three modes of settlement center, corner of rigid foundation is fitted with two types of normal probability distribution and the log-normal distributions. For this purpose, the parameters of the effect of the transition on the analysis of soil modulus of elasticity of foundation, such as settlement and the coefficient of Poisson ratio distribution in probability using probabilistic log-normal and normal have been considered. Analysis indicated that the settlement in the center of the wake is flexible critical than the other two and has a higher probability of occurrence of the settlement in this part of the foundation. In the case of the normal distribution and the normal distribution graph of the log was used, the probab
文摘One of the most important challenges in the design of the foundation of the Earth layer below the surface, the Summit Foundation, which can be a very large impact on the sustainability and the structure of the desired user. Based on this analysis and design criteria of two successive ruptures (load bearing) and settlement. The design of foundation is usually the amount of designated critical foundation than the amount of force that sought the ability to transfer to the soil below. Informal mode using the average values of the parameter, transition effects on mechanical behavior of soil, a number of settlement any part of the amounts are determined by the foundation. Due to the nature of non-homogeneous soil and its parameters uncertainty, relying on one number as the amount of foundation settlement doesn’t seem logical. This is while in the methods of the probability distribution function by taking the probability for each of the input parameters, or the characteristics of each parameter, the parameter values are likely to have the chance of occurrence. In this research, effort is made using the method of probabilistic Monte Carlo simulation, the effect of the uncertainty of parameters influencing the mechanical behaviour following the successive layers of earth and examined. This method is a kind of simulation is that the uncertainty in the different aspects of the issue to be obvious and a bit of the show. Monte Carlo simulation method for the determination of model uncertainty, a little bit for each of the input random variables is a function of the probability distribution which is considered. In the event that non-deterministic model input variables for describing, not non-deterministic model output as well. So the output of each method to analysis of the concept of the probability distribution function for the input variables is a function of the probability distribution for the target function. In this study, the reliability of the settlement for the three modes of settlement Center, corner of rigid foun
基金This work was supported by the National Natural Science Foundation of China under Grant Nos. 71571104 and 71171115a project funded by the Priority Academic Program Developmerit of Jiangsu Higher Education Institutions and the Natural Science Foundation of Jiangsu, China under Grant No. BK20141481the fifth issue of the "333 Project" funded research project under Grant No. BRA2017456.
文摘The uncertainty distribution can more effectively express the uncertainty of decision makers' judgments during a pairwise comparison of any alternatives. This paper investigates the priority models of group intuitionistic fuzzy preference relations with normal uncertainty distribution. The mathematical equivalence between the membership, non-membership degree interval fuzzy preference relation and the intuitionistic fuzzy preference relation is constructed, showing that there exists an inverse relationship between the priority of alternatives using these two types of interval preference relations. The new optimal models regarded the event that the deviation between the ideal judgement meeting the multiplicative consistency and the actual judgement obeying normal uncertainty distribution shall not exceed a threshold value under the given belief degree as a constraint, and regarded the minimum sum of all the threshold values as the objective function. The chance constraint was introduced to measure the degree to which multiplicative consistency can be realized under different belief degrees. The priority model provides a new method for simulating uncertainty and fuzziness in the real-world decision making environment.
基金National Natural Science Foundation of China (40474023).
文摘The primary results have been obtained by the analyses on the deviation of macroscopic epicenters from relevant microscopic epicenters determined rapidly after the 139 earthquake events occunrred during 1989-2004. Further study has proved the normal distributions exist in both the east and west parts of Chinese mainland partitioned along meridian 105°E. The empirical two-dimensional normal distribution functions for the deviations are presented in the paper. The results are of significance in reducing uncertainty of losses assessment immediately after a destroyed earthquake.
