Let X be a compact polyhedron, H a normal subgroup of the fundamental group $\pi_1(X)$ of X and $f : X \longrightarrow X$ a selfmap such that $f_piH \subset H$, where f_\pi : \pi_1(X) \longrightarrow \pi_1(X)$ is the induced homomorphism by f.

We consider the reaction-diffusion system of two-component model in one-dimensional space described by $$ (1) u_s = d_1 u_{xx} + f(u, \upsilon) \upsilon_t = d_2\upsilon_{xx} + \gammag(u, \upsilon) $$ where $d_1$ and $d_2$ are the diffusion rates of u and $\upsilon$, and $\gamma$ is the ration of reaction rates. It is interesting the case of that there are differences in the diffusion and reaction rates of u and $\upsilon$.

This paper is concerned with the problem given below $$ (1.1) i\frac{dx}{du_1(x,\lambda)} + q1(x)u_2(x,\lambda) = \lambdau_1(x,\lambda) 0 \leq x < \infty - i\frac{dx}{du_2(x,\lambda)} + q2(x)u_1(x,\lambda) = \lambdau_2(x,\lambda), $$ $$ (2) u_2(0,\lambda) - hu_1(0,\lambda) = 0 $$ where $\lambda$ is a complex parameter and h is a non-zero complex number.

Let S be a set and B be a $\sigma$-field on S. We consider $(\Omega = S^Z, T = B^z, P)$ as the basic probability space. We denote by T the left shift on $\Omega$. We assume that P is invariant under T, i.e., $PT^{-1} = P$, and that T is ergodic. We denote by $X = \cdots, X_-1, X_0, X_1, \cdots$ the coordinate maps on $\Omega$. From our assumptions it follows that ${X_i}_{i \in Z}$ is a stationary and ergodic process.

A fundamental problem is to construct linear systems with given transfer functions. This problem has a well known solution for unitary linear systems whose state spaces and coefficient spaces are Hilbert spaces. The solution is due independently to B. Sz.-Nagy and C. Foias [15] and to L. de Branges and J. Ball and N. Cohen [4]. Such a linear system is essentially uniquely determined by its transfer function. The de Branges-Rovnyak construction makes use of the theory of square summable power series with coefficients in a Hilbert space. The construction also applies when the coefficient space is a Krein space [7].

In this paper a capacitance matrix method is developed for the Poisson equation on a rectangle $$ (1-1) Lu \equiv -(u_{xx} + u_{yy} = f, (x, y) \in \Omega \equiv (0, 1) \times (0, 1) $$ with the homogeneous Dirichlet boundary condition $$ (1-2) u = 0, (x, y) \in \partial\Omega $$ where $\partial\Omega$ is the boundary of the region $\Omega$.

A complex n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called a comples space form, which is denoted by $M_n(c)$. A complete and simply connected complex space form consists of a complex projective space $P_nC$, a complex Euclidean space $C^n$ or a complex hyperbolic space $H_nC$, according as c > 0, c = 0 or c < 0. The induced almost contact metric structure of a real hypersurface M of $M_n(c)$ is denoted by $(\phi, \zeta, \eta, g)$.

In this paper we discuss the existence and uniqueness of singular solutions for equations of the form $$ (F) u_t = u{xx} - $\mid$u$\mid$^{q-1} u_x - $\mid$u$\mid$^{p-1}u, p,q > 1, $$ in the domain $Q = {(x,t) : x \in R, t > 0}$. This equation represents a model of diffusion-convection with absorption.

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

Let $M^n$ be a hypersurface in the Euclidean space $R^{n+1}$ with position vector field x and unit normal vector field G.

Let M be a compact smooth manifold of dimension n and let E be a flat (complet) vector bundle over M of rank r.

There is much literature on the study of matrices over a finite Boolean algebra. But many results in Boolean matrix theory are stated only for binary Boolean matrices. This is due in part to a semiring isomorphism between the matrices over the Boolean algebra of subsets of a k element set and the k tuples of binary Boolean matrices. This isomorphism allows many questions concerning matrices over an arbitrary finite Boolean algebra to be answered using the binary Boolean case. However there are interesting results about the general (i.e. nonbinary) Boolean matrices that have not been mentioned and they differ somwhat from the binary case.

The nonlinear Klein Gordon equation $$ (1) \frac{\partial t^2}{\partial^2 u} - \Delta u + V_u(u) = f $$ where $\Delta$ is the Laplacian operator in $R^d (d = 1, 2, 3), V_u(u)$ is the derivative of the "potential function" V, and f is a source term independent of the solution u, in various areas of mathematical physics.l physics.

In a Recent paper [3], D. Chinea, M. Delon and J. C. Marrero proved that a cosymplectic manifold is formal and constructed an example of compact cosymplectic manifold which is not a global product of a Kaehler manifold with the circle. In this paper we study the compact cosymplectic manifolds with critical Riemannian metrics.

The recent work of Hopkins, Kuhn and Ravenel [H-K-R] indicates the Morava K-theory, $K(n)^*(-)$, occupy an important and fundamental place in homology theory. In particular $K(n)^*(BG)$ for classifying spaces of finite groups are studied by many authors [H-K-R], [R], [T-Y 1, 2] and [Hu].

J.Milnor[Mi2] has introduced the notion of directional entropy in his study of Cellular Automata. Cellular Automaton map can be considered as a continuous map from a space $K^Z^n$ to itself which commute with the translation of the lattice $Z^n$. Since the space $K^Z^n$ is compact, map S is uniformly continuous. Hence S is a block map(a finite code)[He]. (S is said to have a finite memory.) In the case of n = 1, we have a shift map, T on $K^Z$, and a block map S and they together generate a $Z^2$ action.

Unless otherwise stated, we always assume that X is a Banach space, and $1 < p, q < \infty with \frac{p}{1}+\frac{q}{1} = 1$. We use S(X) and B(X) to denote the unit sphere and the unit ball in X respectively.

The study of self-adjoint operator algebras on Hilbert space is well established, with a long history including some of the strongest mathematicians of the twentieth century. By contrast, non-self-adjoint CSL-algebras, particularly reflexive algebras, are only begins to be studied by W. B. Wrveson [1] 1974.

Let (M, g) be a complete Riemannian manifold and G be a closed (connected) subgroup of the group of isometries of M. Then the union ${\MM}$ of all principal orbits is an open dense subset of M and the quotient map ${\MM} \longrightarrow {\BB} := {\MM}/G$ becomes a Riemannian submersion for the restriction of g to ${\MM}$ which gives the quotient metric on ${\BB}$. Namely, B is a singular (complete) Riemannian space such that $\partialB$ consists of non-principal orbits.

Consider a function m defined in $R^n$ and the associated convolution operator T given by the Fourier transform $\hat{T} = m\hat{f}$.

Classical Fatous' theorem asserts that the non-tangential limit of the Poisson integral of an integrals function on $[\pi, \pi]$ exists a.e.

Let $H$ denote a separable, infinite dimensional complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$. A dual algebra is a subalgebra of $L(H)$ that contains the identity operator $1_H$ and is closed in the $weak^*$ operator topology on $L(H)$. For $T \in L(H)$, let $A_T$ denote the smallest subalgebra of $L(H)$ that contains T and $1_H$ and is closed in the $weak^*$ operator topology.