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M. Junger, G. Reinelt, and W. R. Pulleyblank asked the following questions ([2]). (1) Is it true that every simple planar 2-edge connected bipartite graph has a 3-partition in which each component consists of the edge set of a simple path? (2) Does every simple planar 2-edge connected graph have a 3-partition in which every component consists of the edge set of simple paths and triangles? The purpose of this paper is to provide a positive answer to the second question for simple outerplanar 2-vertex connected graphs and a positive answer to the first question for simple planar 2-edge connected bipartite graphs one set of whose bipartition has at most 4 vertices.

In this paper, we first find a necessary and sufficient condition for the existence of multiple orthogonal polynomials by the moments of a pair of measures and then give representations for multiple orthogonal polynomials. We also prove four term recurrence relations for multiple orthogonal polynomials of type II and several interesting relations for multiple orthogonal polynomials are given. A generalized recurrence relation for multiple orthogonal polynomials of type I is found and then four term recurrence relations are obtained as a special case.

In this paper we establish limsup and liminf theorems for the increments of partial sum processes of a dependent station- ary Gaussian sequence.

Let (2;2;2;2) be ramiflcation indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass P function. Lattµes (7) gives a rational function R(z) = z4+ 1 g2z 2 + 1 16 g 2 2 4z3ig2z which is chaotic on C and is induced by the Weierstrass P function and the linear map L(z) = 2z on complex plane C. It is also known that there exist regular branched covering maps from T 2 onto " C if and only if the ramiflcation indices are (2;2;2;2), (2;4;4), (2;3;6) and (3;3;3), by the Riemann-Hurwitz formula. In this paper we will construct regular branched covering maps corresponding to the ramiflcation indices (2;4;4), (2;3;6) and (3;3; 3), as well as chaotic maps induced by these regular branched cov- ering maps.

In this paper we obtained the following : Let H be a Hilbert space and L be a subspace lattice on H. Let X and Y be operators acting on H. If the range of X is dense in H, then the following are equivalent: (1) there exists an operator A in AlgL such that AX = Y , (2) sup ‰ kE ? Y fk kE ? Xfk : f 2 H; E 2 L ae = K < 1: Moreover, if condition (2) holds, we may choose the operator A such that kAk = K.