In order to reveal the complex network feature and its evolution principle of aviation network of China,probability distribution and its evolution trace of average length of node path of aviation network of China were...In order to reveal the complex network feature and its evolution principle of aviation network of China,probability distribution and its evolution trace of average length of node path of aviation network of China were researched according to statistics data in years 1988,1994,2001,2008 and 2015 of civil aviation of China.Floyd algorithm to calculate the path length between any two nodes of network was applied and average length of node path of aviation network was obtained according to this algorithm.It was discovered that average length of node path to other nodes had normal distribution function in each year.At meantime,the location parameter and scale parameter of normal distribution function had linear evolution trace.Airline rate was an index to describe the density of airline.It was found that average length of node path of aviation network of China evolved synchronously with airline rate and they had linear relationship.展开更多
In order to reveal the complex network characteristics and evolution principle of China aviation network,the probability distribution and evolution trace of node nearest neighbor average degree of China aviation netwo...In order to reveal the complex network characteristics and evolution principle of China aviation network,the probability distribution and evolution trace of node nearest neighbor average degree of China aviation network were studied based on the statistics data of China civil aviation network in 1988,1994,2001,2008 and 2015.According to the theory and method of complex network,the network system was constructed with the city where the airport was located as the network node and the route between cities as the edge of the network.Based on the statistical data,the nearest neighbor average degrees of nodes in China aviation network in 1988,1994,2001,2008 and 2015 were calculated.Using the probability statistical analysis method,it was found that the nearest neighbor average degree had the probability distribution of normal function and the position parameters and scale parameters of the probability distribution had linear evolution trace.展开更多
Statistical properties of winds near the Taichung Harbour are investigated. The 26 years'incomplete data of wind speeds, measured on an hourly basis, are used as reference. The possibility of imputation using simu...Statistical properties of winds near the Taichung Harbour are investigated. The 26 years'incomplete data of wind speeds, measured on an hourly basis, are used as reference. The possibility of imputation using simulated results of the Auto-Regressive (AR), Moving-Average (MA), and/ or Auto-Regressive and Moving-Average (ARMA) models is studied. Predictions of the 25-year extreme wind speeds based upon the augmented data are compared with the original series. Based upon the results, predictions of the 50- and 100-year extreme wind speeds are then made.展开更多
In this paper, by using the wave data from a few oceanographic observation stations in the coastal zone of the Yellow Sea, the East China Sea and the South China Sea, the long-term joint distribution of the one-tenth ...In this paper, by using the wave data from a few oceanographic observation stations in the coastal zone of the Yellow Sea, the East China Sea and the South China Sea, the long-term joint distribution of the one-tenth large (or significant) wave height with average period is studied. The statistical data demonstrate that the long- term distribution of the one- tenth wave height or average period fits the log-normal distribution, thus the joint distribution also fits the two-dimensional log-normal distribution. Then the conditional probability distribution of the average period is derived, and the range as well as the mode of the average wave period corresponding to a certain return period of wave height can be calculated easily.展开更多
By analyzing the structures of circuits,a novel approach for signal probability estimation of very large-scale integration(VLSI)based on the improved weighted averaging algorithm(IWAA)is proposed.Considering the failu...By analyzing the structures of circuits,a novel approach for signal probability estimation of very large-scale integration(VLSI)based on the improved weighted averaging algorithm(IWAA)is proposed.Considering the failure probability of the gate,first,the first reconvergent fan-ins corresponding to the reconvergent fan-outs were identified to locate the important signal correlation nodes based on the principle of homologous signal convergence.Secondly,the reconvergent fan-in nodes of the multiple reconverging structure in the circuit were identified by the sensitization path to determine the interference sources to the signal probability calculation.Then,the weighted signal probability was calculated by combining the weighted average approach to correct the signal probability.Finally,the reconvergent fan-out was quantified by the mixed-calculation strategy of signal probability to reduce the impact of multiple reconvergent fan-outs on the accuracy.Simulation results on ISCAS85 benchmarks circuits show that the proposed method has approximate linear time-space consumption with the increase in the number of the gate,and its accuracy is 4.2%higher than that of the IWAA.展开更多
Peak to Average Power Ratio (PAPR) is defined as the instantaneous power (maximum value) to the average power ratio. PAPR is considered to be a major problem in OFDM systems. This problem can cause radical unexpected ...Peak to Average Power Ratio (PAPR) is defined as the instantaneous power (maximum value) to the average power ratio. PAPR is considered to be a major problem in OFDM systems. This problem can cause radical unexpected behavior of the signal fluctuation. This fluctuation is constituted by a large number of power states. The enormous number of these states leads to an additional complexity of ADCs and DACs. This research addresses the previous problem in OFDM systems utilizing Turbo Codes. μLaCP technique is employed for the purpose of decreasing PAPR. Moreover, our OFDM system was simulated in the presence of an AWGN channel with four types of codes (without the presence of ADCs and DACs). These were constituted of PCCC (typical and new), SCCC, and Convolutional Codes. Our Turbo Coded OFDM exhibited unchanged BER performance before and after the use of μLaCP technique. This was accomplished by modifying our previous PAPR reduction technique without sacrificing greatly its attributes.展开更多
Recently,machine learning has become a powerful tool for predicting nuclear charge radius RC,providing novel insights into complex physical phenomena.This study employs a continuous Bayesian probability(CBP)estimator ...Recently,machine learning has become a powerful tool for predicting nuclear charge radius RC,providing novel insights into complex physical phenomena.This study employs a continuous Bayesian probability(CBP)estimator and Bayesian model averaging(BMA)to optimize the predictions of RCfrom sophisticated theoretical models.The CBP estimator treats the residual between the theoretical and experimental values of RCas a continuous variable and derives its posterior probability density function(PDF)from Bayesian theory.The BMA method assigns weights to models based on their predictive performance for benchmark nuclei,thereby accounting for the unique strengths of each model.In global optimization,the CBP estimator improved the predictive accuracy of the three theoretical models by approximately 60%.The extrapolation analyses consistently achieved an improvement rate of approximately 45%,demonstrating the robustness of the CBP estimator.Furthermore,the combination of the CBP and BMA methods reduces the standard deviation to below 0.02 fm,effectively reproducing the pronounced shell effects on RCof the Ca and Sr isotope chains.The studies in this paper propose an efficient method to accurately describe RCof unknown nuclei,with potential applications in research on other nuclear properties.展开更多
The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed....The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed.Random sea surface is treated as a narrow-band stochastic process,and the stochastic parametric excitation is studied based on the effective wave theory.The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function.By using the stochastic averaging method,the differential equation of motion is transformed into Ito’s stochastic differential equation.The steady-state probability density function of roll motion is obtained,and the results are validated with the numerical simulation and model test.展开更多
Shape Memory Alloy(SMA)is a typical material with memory effect,and it is widely used in many engineering fields.Based on the elastic theory and Galerkin method,a vibration system of SMA beam with rigid constraints is...Shape Memory Alloy(SMA)is a typical material with memory effect,and it is widely used in many engineering fields.Based on the elastic theory and Galerkin method,a vibration system of SMA beam with rigid constraints is proposed.The non⁃smooth transformation was employed to deal with the discontinuous position,and the original system was turned into an approximate equivalent system associated with the Dirac function.Then,using the stochastic averaging method,the drift and diffusion coefficients of the corresponding Fokker Planck Kolmogorov equation were described.Lastly,the approximate probability response of the system was formulated analytically.Meanwhile,numerical simulation was carried out to verify the effectiveness of analytical results.Furthermore,stochastic bifurcation was discussed.Results show that the stationary probability response of the system was affected by the increase of noise amplitude and restitution force,and a certain restitution value and damping could induce P⁃bifurcation.展开更多
The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied....The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokl^er-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.展开更多
In this paper,average bit error probability(ABEP)bound of optimal maximum likelihood(ML)detector is first derived for ultra massive(UM)multiple-input-multiple-output(MIMO)system with generalized amplitude phase modula...In this paper,average bit error probability(ABEP)bound of optimal maximum likelihood(ML)detector is first derived for ultra massive(UM)multiple-input-multiple-output(MIMO)system with generalized amplitude phase modulation(APM),which is confirmed by simulation results.Furthermore,a minimum residual criterion(MRC)based lowcomplexity near-optimal ML detector is proposed for UM-MIMO system.Specifically,we first obtain an initial estimated signal by a conventional detector,i.e.,matched filter(MF),or minimum mean square error(MMSE)and so on.Furthermore,MRC based error correction mechanism(ECM)is proposed to correct the erroneous symbol encountered in the initial result.Simulation results are shown that the performance of the proposed MRC-ECM based detector is capable of approaching theoretical ABEP of ML,despite only imposing a slightly higher complexity than that of the initial detector.展开更多
We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We al...We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"展开更多
Wireless communications have to face to several different security issues in practice due to the nature of broadcast.The information theory is well known to provide efficient approaches to address security issues in w...Wireless communications have to face to several different security issues in practice due to the nature of broadcast.The information theory is well known to provide efficient approaches to address security issues in wireless communications,which attracts much attention in both industry and academia in recent years.In this paper,inspired by information theory,we study the outage probability of the opportunistic relay selection based on cognitive decode-and-forward relaying with the secrecy consideration.Specifically,the closed-form expression of the outage probability is proposed.Moreover,the asymptotic performance evaluation on the basis of the analytical results is investigated.The simulation results show that the relay selection can reduce the outage probability in accordance with our theoretical analysis.展开更多
近年来,机器学习方法被广泛应用于对原子核质量的预测中。基于连续型贝叶斯概率(Continuous Bayesian Probability,CBP)估计器,结合贝叶斯模型平均(Bayesian Model Averaging,BMA)改进了理论模型对核质量的描述。在CBP方法中,核质量理...近年来,机器学习方法被广泛应用于对原子核质量的预测中。基于连续型贝叶斯概率(Continuous Bayesian Probability,CBP)估计器,结合贝叶斯模型平均(Bayesian Model Averaging,BMA)改进了理论模型对核质量的描述。在CBP方法中,核质量理论值与实验值的差异被视为连续变量,通过核密度估计(Kernel Density Estimation,KDE)生成其先验和似然概率密度函数,并以贝叶斯定理确定后验概率密度函数。在全局优化和外推分析中,CBP方法显著提升了理论模型预测的精确度。此外,BMA方法基于各模型对基准核的预测表现为每个模型分配权重,平衡了各模型在不同区域的优势。通过预测Ca同位素双中子滴线位置,评估了BMA方法修正结果的可靠性。CBP方法结合BMA方法为预测未知区域的核质量提供了一种有效途径,并可应用于对其他核性质的研究。展开更多
The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest f...The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.展开更多
文摘In order to reveal the complex network feature and its evolution principle of aviation network of China,probability distribution and its evolution trace of average length of node path of aviation network of China were researched according to statistics data in years 1988,1994,2001,2008 and 2015 of civil aviation of China.Floyd algorithm to calculate the path length between any two nodes of network was applied and average length of node path of aviation network was obtained according to this algorithm.It was discovered that average length of node path to other nodes had normal distribution function in each year.At meantime,the location parameter and scale parameter of normal distribution function had linear evolution trace.Airline rate was an index to describe the density of airline.It was found that average length of node path of aviation network of China evolved synchronously with airline rate and they had linear relationship.
文摘In order to reveal the complex network characteristics and evolution principle of China aviation network,the probability distribution and evolution trace of node nearest neighbor average degree of China aviation network were studied based on the statistics data of China civil aviation network in 1988,1994,2001,2008 and 2015.According to the theory and method of complex network,the network system was constructed with the city where the airport was located as the network node and the route between cities as the edge of the network.Based on the statistical data,the nearest neighbor average degrees of nodes in China aviation network in 1988,1994,2001,2008 and 2015 were calculated.Using the probability statistical analysis method,it was found that the nearest neighbor average degree had the probability distribution of normal function and the position parameters and scale parameters of the probability distribution had linear evolution trace.
基金The project is partly supported by the National Science Council, Contract Nos. NSC-89-261 l-E-019-024 (JZY), and NSC-89-2611-E-019-027 (CRC).
文摘Statistical properties of winds near the Taichung Harbour are investigated. The 26 years'incomplete data of wind speeds, measured on an hourly basis, are used as reference. The possibility of imputation using simulated results of the Auto-Regressive (AR), Moving-Average (MA), and/ or Auto-Regressive and Moving-Average (ARMA) models is studied. Predictions of the 25-year extreme wind speeds based upon the augmented data are compared with the original series. Based upon the results, predictions of the 50- and 100-year extreme wind speeds are then made.
文摘In this paper, by using the wave data from a few oceanographic observation stations in the coastal zone of the Yellow Sea, the East China Sea and the South China Sea, the long-term joint distribution of the one-tenth large (or significant) wave height with average period is studied. The statistical data demonstrate that the long- term distribution of the one- tenth wave height or average period fits the log-normal distribution, thus the joint distribution also fits the two-dimensional log-normal distribution. Then the conditional probability distribution of the average period is derived, and the range as well as the mode of the average wave period corresponding to a certain return period of wave height can be calculated easily.
基金The National Natural Science Foundation of China(No.61502422)the Natural Science Foundation of Zhejiang Province(No.LY18F020028,LQ15F020006)the Natural Science Foundation of Zhejiang University of Technology(No.2014XY007)
文摘By analyzing the structures of circuits,a novel approach for signal probability estimation of very large-scale integration(VLSI)based on the improved weighted averaging algorithm(IWAA)is proposed.Considering the failure probability of the gate,first,the first reconvergent fan-ins corresponding to the reconvergent fan-outs were identified to locate the important signal correlation nodes based on the principle of homologous signal convergence.Secondly,the reconvergent fan-in nodes of the multiple reconverging structure in the circuit were identified by the sensitization path to determine the interference sources to the signal probability calculation.Then,the weighted signal probability was calculated by combining the weighted average approach to correct the signal probability.Finally,the reconvergent fan-out was quantified by the mixed-calculation strategy of signal probability to reduce the impact of multiple reconvergent fan-outs on the accuracy.Simulation results on ISCAS85 benchmarks circuits show that the proposed method has approximate linear time-space consumption with the increase in the number of the gate,and its accuracy is 4.2%higher than that of the IWAA.
文摘Peak to Average Power Ratio (PAPR) is defined as the instantaneous power (maximum value) to the average power ratio. PAPR is considered to be a major problem in OFDM systems. This problem can cause radical unexpected behavior of the signal fluctuation. This fluctuation is constituted by a large number of power states. The enormous number of these states leads to an additional complexity of ADCs and DACs. This research addresses the previous problem in OFDM systems utilizing Turbo Codes. μLaCP technique is employed for the purpose of decreasing PAPR. Moreover, our OFDM system was simulated in the presence of an AWGN channel with four types of codes (without the presence of ADCs and DACs). These were constituted of PCCC (typical and new), SCCC, and Convolutional Codes. Our Turbo Coded OFDM exhibited unchanged BER performance before and after the use of μLaCP technique. This was accomplished by modifying our previous PAPR reduction technique without sacrificing greatly its attributes.
基金supported by the National Natural Science Foundation of China(Nos.12475135,12035011,and 12475119)the Shandong Provincial Natural Science Foundation,China(No.ZR2020MA096)the Fundamental Research Funds for the Central Universities(No.22CX03017A)。
文摘Recently,machine learning has become a powerful tool for predicting nuclear charge radius RC,providing novel insights into complex physical phenomena.This study employs a continuous Bayesian probability(CBP)estimator and Bayesian model averaging(BMA)to optimize the predictions of RCfrom sophisticated theoretical models.The CBP estimator treats the residual between the theoretical and experimental values of RCas a continuous variable and derives its posterior probability density function(PDF)from Bayesian theory.The BMA method assigns weights to models based on their predictive performance for benchmark nuclei,thereby accounting for the unique strengths of each model.In global optimization,the CBP estimator improved the predictive accuracy of the three theoretical models by approximately 60%.The extrapolation analyses consistently achieved an improvement rate of approximately 45%,demonstrating the robustness of the CBP estimator.Furthermore,the combination of the CBP and BMA methods reduces the standard deviation to below 0.02 fm,effectively reproducing the pronounced shell effects on RCof the Ca and Sr isotope chains.The studies in this paper propose an efficient method to accurately describe RCof unknown nuclei,with potential applications in research on other nuclear properties.
基金the State Administration of Science,Technology and Industry for National Defense of China(Grant No.B2420132001).
文摘The paper studies the parametric stochastic roll motion in the random waves.The differential equation of the ship parametric roll under random wave is established with considering the nonlinear damping and ship speed.Random sea surface is treated as a narrow-band stochastic process,and the stochastic parametric excitation is studied based on the effective wave theory.The nonlinear restored arm function obtained from the numerical simulation is expressed as the approximate analytic function.By using the stochastic averaging method,the differential equation of motion is transformed into Ito’s stochastic differential equation.The steady-state probability density function of roll motion is obtained,and the results are validated with the numerical simulation and model test.
基金Sponsored by the National Natural Science Foundation of China(Grant No.11302158)the Natural Science Foundation of Shaanxi Province,China(Grant No.2018JM1044)
文摘Shape Memory Alloy(SMA)is a typical material with memory effect,and it is widely used in many engineering fields.Based on the elastic theory and Galerkin method,a vibration system of SMA beam with rigid constraints is proposed.The non⁃smooth transformation was employed to deal with the discontinuous position,and the original system was turned into an approximate equivalent system associated with the Dirac function.Then,using the stochastic averaging method,the drift and diffusion coefficients of the corresponding Fokker Planck Kolmogorov equation were described.Lastly,the approximate probability response of the system was formulated analytically.Meanwhile,numerical simulation was carried out to verify the effectiveness of analytical results.Furthermore,stochastic bifurcation was discussed.Results show that the stationary probability response of the system was affected by the increase of noise amplitude and restitution force,and a certain restitution value and damping could induce P⁃bifurcation.
基金Project supported by the National Natural Science Foundation of China(No.11025211)the Zhejiang Provincial Natural Science Foundation of China(No.Z6090125)the Special Fund for National Excellent Ph.D.Dissertation and Research Grant Council of Hong Kong City(No.U115807)
文摘The nonstationary probability densities of system response of a single-degree- of-freedom system with lightly nonlinear damping and strongly nonlinear stiffness subject to modulated white noise excitation are studied. Using the stochastic averaging method based on the generalized harmonic functions, the averaged Fokl^er-Planck-Kolmogorov equation governing the nonstationary probability density of the amplitude is derived. The solution of the equation is approximated by the series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. According to the Galerkin method, the time-dependent coefficients can be solved from a set of first-order linear differential equations. Then, the semi-analytical formulae of the nonstationary probability density of the amplitude response as well as the nonstationary probability density of the state response and the statistic moments of the amplitude response can be obtained. A van der Pol-Duffing oscillator subject to modulated white noise is given as an example to illustrate the proposed procedures. The effects of the system parameters, such as the linear damping coefficient and the nonlinear stiffness coefficient, on the system response are discussed.
基金supported in part by the National Key Research and Development Program of China under Grant 2019YFB1803400in part by the National Science Foundation of China under Grant 62001179in part by the Fundamental Research Funds for the Central Universities under Grant 2020kfyXJJS111.
文摘In this paper,average bit error probability(ABEP)bound of optimal maximum likelihood(ML)detector is first derived for ultra massive(UM)multiple-input-multiple-output(MIMO)system with generalized amplitude phase modulation(APM),which is confirmed by simulation results.Furthermore,a minimum residual criterion(MRC)based lowcomplexity near-optimal ML detector is proposed for UM-MIMO system.Specifically,we first obtain an initial estimated signal by a conventional detector,i.e.,matched filter(MF),or minimum mean square error(MMSE)and so on.Furthermore,MRC based error correction mechanism(ECM)is proposed to correct the erroneous symbol encountered in the initial result.Simulation results are shown that the performance of the proposed MRC-ECM based detector is capable of approaching theoretical ABEP of ML,despite only imposing a slightly higher complexity than that of the initial detector.
文摘We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"
基金This work is partially supported by National Major Project of China(No.2010ZX03006-006)National Natural Science Foundation of China(No.61571241)+4 种基金the Ministry of Education-China Mobile Research Foundation,China(No.MCM20170205)the Communication Soft Science Research Project of Ministry of Industry and Information Technology,China(No.2017-R-34)the Scientific Research Foundation of the Higher Education Institutions of Jiangsu Province,China(No.15KJA510002 and 17KJB510043)the Research Foundation for Advanced Talents,Nanjing University of Posts and Telecommunications(No.NY217146)Open Foundation of the Remote Measuring and Control Key Lab of Jiangsu Province,China(No.2242015k30005).
文摘Wireless communications have to face to several different security issues in practice due to the nature of broadcast.The information theory is well known to provide efficient approaches to address security issues in wireless communications,which attracts much attention in both industry and academia in recent years.In this paper,inspired by information theory,we study the outage probability of the opportunistic relay selection based on cognitive decode-and-forward relaying with the secrecy consideration.Specifically,the closed-form expression of the outage probability is proposed.Moreover,the asymptotic performance evaluation on the basis of the analytical results is investigated.The simulation results show that the relay selection can reduce the outage probability in accordance with our theoretical analysis.
文摘近年来,机器学习方法被广泛应用于对原子核质量的预测中。基于连续型贝叶斯概率(Continuous Bayesian Probability,CBP)估计器,结合贝叶斯模型平均(Bayesian Model Averaging,BMA)改进了理论模型对核质量的描述。在CBP方法中,核质量理论值与实验值的差异被视为连续变量,通过核密度估计(Kernel Density Estimation,KDE)生成其先验和似然概率密度函数,并以贝叶斯定理确定后验概率密度函数。在全局优化和外推分析中,CBP方法显著提升了理论模型预测的精确度。此外,BMA方法基于各模型对基准核的预测表现为每个模型分配权重,平衡了各模型在不同区域的优势。通过预测Ca同位素双中子滴线位置,评估了BMA方法修正结果的可靠性。CBP方法结合BMA方法为预测未知区域的核质量提供了一种有效途径,并可应用于对其他核性质的研究。
基金a grant from National Natural Science Foundation of China,Grant No.10671072Doctoral Program Foundation of the Ministry of Education of China,Grant No.20060269016+1 种基金"Shu Guang"project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation,Grant No.04SG27the National Basic Research Program of China (973 Program),Grant No.2007CB814904
文摘The authors consider two discrete-time insurance risk models. Two moving average risk models are introduced to model the surplus process, and the probabilities of ruin are examined in models with a constant interest force. Exponential bounds for ruin probabilities of an infinite time horizon are derived by the martingale method.
