— Transmission of packets over computer networks is subject to packet-level errors, which appear as " bursts " of bit-level errors and are not well modeled by memoryless binary channels. Using a standard scrambling technique  transmission of packets can be modeled by the q-ary symmetric channel (q-SC) with alphabet size q and error probability p.… (More)
We consider a path-following problem in which the goal is to ensure that the error between the system output and the geometric path be asymptotically less than a prespecified constant, while guaranteeing a forward motion along the path and boundedness of all states. A solution to this problem was given in  for a class of nonlinear systems and for paths… (More)
In this paper, we prove that on every Finsler n-sphere (S n , F) for n ≥ 6 with reversibility λ and flag curvature K satisfying λ λ+1 2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exist [ n 2 ] − 2 closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n − 3 closed… (More)
As a model for multiscale systems under random influences on physical boundary, a stochastic partial differential equation under a fast random dynamical boundary condition is investigated. An effective equation is derived and justified by reducing the random dynamical boundary condition to a random static boundary condition. The effective system is still a… (More)
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described… (More)
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey's Theorem for colorings of pairs could help in simplifying the… (More)
Complex projection bodies were introduced by Abardia and Bernig, recently. In this paper some geometric inequalities for mixed complex projection bodies which are analogs of inequalities for mixed real projection bodies are established.
In this paper, let Σ ⊂ R 6 be a compact convex hypersurface. We prove that if Σ carries only finitely many geometrically distinct closed characteristics, then at least two of them must possess irrational mean indices. Moreover, if Σ carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.
Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences. Invariant manifolds often provide geometric structure for understanding stochastic dynamics. In this paper, a random invariant manifold reduction principle is proved for a class of stochastic partial differential equations. The… (More)