At each time n∈N,let Y^(n)(ξ)=(y^(n)_(1)(ξ),y^(n)_(2)(ξ),…)be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=(ξ_(n))n∈N.The existence and uniqueness of the non-nega...At each time n∈N,let Y^(n)(ξ)=(y^(n)_(1)(ξ),y^(n)_(2)(ξ),…)be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=(ξ_(n))n∈N.The existence and uniqueness of the non-negative fixed points of the associated smoothing transformation in random environment are considered.These fixed points are solutions to the distributional equation for a.e.ξ,Z(ξ)=dΣ_(i∈N_(+))y^(0)_(i)(ξ)Z^(1)_(i)(ξ),where{Z^(1)_(i):i∈N_(+)}are random variables in random environment which satisfy that for any environmentξ,under P_(ξ),{Z^(1)_(i)(ξ):i∈N_(+)}are independent of each other and Y^(0)(ξ),and have the same conditional distribution P_(ξ)(Z^(1)_(i)(ξ)∈·)=P_(Tξ)(Z(Tξ)∈·),where T is the shift operator.This extends the classical results of J.D.Biggins[J.Appl.Probab.,1977,14:25-37]to the random environment case.As an application,the martingale convergence of the branching random walk in random environment is given as well.展开更多
Discusses a technique for the computation of hypersonic flow of air with chemical reactions over concave corners. Details of smooth transformation of domain; Use of finite difference method; Numerical results.
The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques ...The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or of the proposed algorithms is studied given. uniform grids, the convergence behavior and a collection of numerical results is give.展开更多
基金the National Key Research and Development Program of China(No.2020YFA0712900)the National Natural Science Foundation of China(Grant No.11971062)the Scientific Research Foundation for Young Teachers in Capital University of Economics and Business(NO.XRZ2021035).
文摘At each time n∈N,let Y^(n)(ξ)=(y^(n)_(1)(ξ),y^(n)_(2)(ξ),…)be a random sequence of non-negative numbers that are ultimately zero in a random environmentξ=(ξ_(n))n∈N.The existence and uniqueness of the non-negative fixed points of the associated smoothing transformation in random environment are considered.These fixed points are solutions to the distributional equation for a.e.ξ,Z(ξ)=dΣ_(i∈N_(+))y^(0)_(i)(ξ)Z^(1)_(i)(ξ),where{Z^(1)_(i):i∈N_(+)}are random variables in random environment which satisfy that for any environmentξ,under P_(ξ),{Z^(1)_(i)(ξ):i∈N_(+)}are independent of each other and Y^(0)(ξ),and have the same conditional distribution P_(ξ)(Z^(1)_(i)(ξ)∈·)=P_(Tξ)(Z(Tξ)∈·),where T is the shift operator.This extends the classical results of J.D.Biggins[J.Appl.Probab.,1977,14:25-37]to the random environment case.As an application,the martingale convergence of the branching random walk in random environment is given as well.
基金This work was partially supported by the North Carolina Supercomputing Center.
文摘Discusses a technique for the computation of hypersonic flow of air with chemical reactions over concave corners. Details of smooth transformation of domain; Use of finite difference method; Numerical results.
基金Acknowledgements The authors are grateful to the referees for many helpful remarks and suggestions. This work was supported by the Estonian Science Foundation （Grant No. 9104）.
文摘The numerical solution of linear Volterra integral equations of the second kind is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using suitable smoothing techniques and polynomial splines on mildly graded or of the proposed algorithms is studied given. uniform grids, the convergence behavior and a collection of numerical results is give.
