This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which adm...This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.展开更多
In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)...In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.展开更多
By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of t...By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.展开更多
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the proce...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.展开更多
The general mutual information （GMI） and general conditional mutual information （GCMI） are considered to measure lag dependences in nonlinear time series. Both of the measures have the property of invariance with ...The general mutual information （GMI） and general conditional mutual information （GCMI） are considered to measure lag dependences in nonlinear time series. Both of the measures have the property of invariance with transform. The statistics based on GMI and GCMI are estimated using the correlation integral. Under the hypothesis of independent series, the estimators have Gaussian asymptotic distributions. Simulations applied to generated nonlinear series demonstrate that the methods appear to find frequently the correct lags.展开更多
We study the continuous dependence on coefficients of solutions of the nonlinear nonhomogeneous Fourier boundary value problems involving the p(x)-Laplace operator.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10371098, 10447007 and 10475055）, the Natural Science Foundation of Shaanxi Province of China （Grant No 2005A13）.
文摘This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt = A（u, ux）uxx＋B（u, ux, ut） which admits the derivative- dependent functional separable solutions （DDFSSs）. We also extend the concept of the DDFSS to cover other variable separation approaches.
文摘In this paper we consider a sequence of Markov dependent bivariate trials whose each component results in an outcome success (0) and failure (1) i.e. we have a sequence {(Xn/Yn), n>=0} of S={(0/0),(0/1),(1/0),(1/1)}-valued Markov dependent bivariate trials. By using the method of conditional probability generating functions (pgfs), we derive the pgf of joint distribution of (X0n,k10,X1n,k11;Y0n,k20,Y1n,k21) where for i=0,1,Xin,k1i denotes the number of occurrences of i-runs of length k1i in the first component and Yin,k2i denotes the number of occurrences of i-runs of length k2i in the second component of Markov dependent bivariate trials. Further we consider two patterns Λ1 and Λ2 of lengths k1 and k2 respectively and obtain the pgf of joint distribution of (Xn,Λ 1,Yn,Λ2 ) using method of conditional probability generating functions where Xn,Λ1(Yn,Λ2) denotes the number of occurrences of pattern Λ1(Λ2 ) of length k1 (k2) in the first (second) n components of bivariate trials. An algorithm is developed to evaluate the exact probability distributions of the vector random variables from their derived probability generating functions. Further some waiting time distributions are studied using the joint distribution of runs.
基金Project supported by the National Natural Science Foundation of China(Grant No.10671156)the Natural Science Foundation of Shaanxi Province of China(Grant No.SJ08A05)
文摘By using the approximate derivative-dependent functional variable separation approach, we study the quasi-linear diffusion equations with a weak source ut = （A（u）Ux）x ＋ eB（u, Ux）. A complete classification of these perturbed equations which admit approximate derivative-dependent functional separable solutions is listed. As a consequence, some approxi- mate solutions to the resulting perturbed equations are constructed via examples.
基金the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012)Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
文摘We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
基金Supported by the National Natural Science Foundation of China (Grant Nos.60375003 60972150)the Science and Technology Innovation Foundation of Northwestern Polytechnical University (Grant No.2007KJ01033)
文摘The general mutual information （GMI） and general conditional mutual information （GCMI） are considered to measure lag dependences in nonlinear time series. Both of the measures have the property of invariance with transform. The statistics based on GMI and GCMI are estimated using the correlation integral. Under the hypothesis of independent series, the estimators have Gaussian asymptotic distributions. Simulations applied to generated nonlinear series demonstrate that the methods appear to find frequently the correct lags.
文摘We study the continuous dependence on coefficients of solutions of the nonlinear nonhomogeneous Fourier boundary value problems involving the p(x)-Laplace operator.
