We find a sufficient condition on weights $u(.)$ and $v(.)$ for which the fractional maximal operator $M_{\alpha }$, , is bounded from $L_{dec}^p(v(x)dx)$ to $L^q(u(x)dx)$. Here , and $L_{dec}^p(v(x)dx)$denotes the set of all radial and nonincreasing functions which belong to the weighted Lebesgue space $L^p(v(x)dx)$. Actually a characterization of this boundedness is obtained whenever the weight $v(\cdot )$ satisfies some reverse doubling condition.

In this paper we study the Cauchy problem for the cubic derivative nonlinear Schrödinger equation involving at least one derivative in the nonlinear term. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.

In this paper we introduce a bottom-up pushdown tree transducer (b-PDTT) which results from generalizing a bottom-up tree transducer by adding a pushdown storage (where the pushdown storage have the form of trees, i.e., a tree-pushdown storage) and may be considered as a dual concept of the top-down pushdown tree transducer (t-PDTT). After proving some fundamental properties of b-PDTT, such that any b-PDTT can be realized by a linear stack with single state and converted into G-type normal form which corresponds to Greibach normal form in a context-free grammar, we compare the translational capability of a b-PDTT with that of a t-PDTT.

In this paper we treat vector bundles of rank 2 on a smooth cubic surface whose restrictions to the 27 lines are rigid. We investigate $\mu$-stability of these bundles and the relation between these bundles and other bundles on surfaces which are given by contractions of mutually skew lines on a cubic surface.

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. Let $K_4^{-}$ stand for the graph obtained from $K_4$ by removing one edge. It is proved by the third author of this paper that if $k$ is odd, then a $k$-connected graph which does not contain $K_4^{-}$ has a $k$-contractible edge. The same conclusion does not hold when $k$ is even. In this paper, we prove that if a $k$-connected graph with no $K_4^{-}$ does not have a $k$-contractible edge, then $k$ is even and each vertex in $G$ is contained in at least two triangles.