In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types.

Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure.

Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented.

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC³ boundary, under the assumption that |a|_L²(Θ) + |f|_L¹(0,∞;L²(Θ)) or |∇a|_L²(Θ) + |f|_L²(0,∞;L²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed.

In this paper, we obtain the following main theorem for a free quadratic bialgebraJ: (a)

Forp≠0,J is a pointed cosemisimple coalgebra. Forp=0,J is a hyperalgebra.

Forp≠0 andq≠0,J has antipodeS iffp·q+2=0 andS(x)=x. Forp=0 orq=0,J has antipode andS(x)=×.

All leftJ^*-modules are rational.

Also, we give some applications in homological theory and algebraicK-theory.

Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a nonattracting closed orbit. Smith^[3] considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine^[4] is given. The stability problem of closed orbits raised in [5] and [6] is resolved.

In this paper, we consider certain mappings, $${\mathcal {M}}$$ , sufficiently close to an integrable one, which is weakly reversible with respct to the mappings $${\mathcal {G}}$$ sufficiently close to an involution of tye (m, n), where m, n∈Z₊ are arbitrary. Under some weak non-degeneracy condition, we construct a uniform KAM iteration for providing the existence of a Cantor family of m-tori invariant under the reversible mappings $${\mathcal {M}}$$ and the reversing mapping $${\mathcal {G}}$$ .

In this paper we introduce the concept of quasinormal and subnormal operators on a Krein space and prove that every quasinormal operator is subnormal. And some conditions for an operator on a Hilbert space to be a subnormal operator in the Krein space sense are obtained.

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations: $$\left\{ \begin{gathered} \psi _t = - (\sigma - \alpha )\psi - \sigma \theta _x + \alpha \psi _{xx} , \hfill \\ \theta _t = - (1 - \beta )\theta + v\psi _x + (\psi \theta )_x + \beta \theta _{xx} . \hfill \\ \end{gathered} \right.$$