In this paper, the weak $L\log L$ estimates for the commutators of a class of Littlewood-Paley operators with real parameter are established by using a technique of the sharp function.

Using methods and results of decomposition theory we construct minimal actions of groups of homeomorphisms of some classical fractals (the Sierpi\'nski carpet and its generalizations, and the Menger curve) with positive entropy. Suspending these group actions we get minimal foliated spaces which have positive geometric entropy and are modelled on these fractals.

In this paper we prove the global existence of mild solutions for the semilinear parabolic equation $u_t=\Delta u+a|\nabla u|^q+b|u|^{p-1}u, t>0, x \in \mathbb R^n, n \geq 1,$ $a \in \mathbb R, b \in \mathbb R, p>1+(2/n),$ $(n+2)/(n+1)<q<2$ and $q\leq p(n+2)/(n+p),$ with small initial data with respect to a norm related to the equation. We also prove that some of these global solutions are asymptotic to self--similar solutions of the equations $u_t=\Delta u+\nu a|\nabla u|^q+\mu b|u|^{p-1}u,$ with $\nu, \mu=0$ or $1.$ The values of $\nu$ and $\mu$ depend on the decaying of the initial data and on the position of $q$ with respect to $2p/(p+1).$ Our results apply for the viscous Hamilton--Jacobi equation: $u_t=\Delta u+a|\nabla u|^q$ and hold without sign restriction neither on $a$ nor on the initial data. We prove that if $(n+2)/(n+1)<q<2$ and the initial data behaves, near infinity, like $c|x|^{-\alpha}, (2-q)/(q-1) \leq \alpha<n, c$ is a small constant, then the resulting solution is global. Moreover, if $(2-q)/(q-1)<\alpha<n,$ the solution is asymptotic to a self--similar solution of the linear heat equation. Whereas, if $(2-q)/(q-1)=\alpha<n,$ the solution is asymptotic to a self--similar solution of the viscous Hamilton--Jacobi equation. The asymptotics are given, in particular, in $W^{1,\infty}(\mathbb R^n).$

We shall construct involutions, in the symmetric group $S_{24}$, which generate $M_{22}$, $M_{23}$ and $M_{24}$.

For Hopf hypersurfaces in a nonflat complex space form $M^n(c; \Bbb{C})$, integral curves of their characteristic vector fields are ''nice'' curves in the sense that their extrinsic shapes in $M^n(c; \Bbb{C})$ are K\"ahler circles. In this paper we mainly study geodesic spheres in a nonflat complex space form $M^n(c; \Bbb{C})$. On these geodesic spheres we classify smooth curves whose extrinsic shapes are K\"ahler circles in $M^n(c; \Bbb{C}),c\not=0$. We also give a characterization of complex space forms among K\"ahler manifolds by extrinsic shapes of integral curves of characteristic vector fields on their geodesic spheres.

Let $G$ be an abelian group. A subgroup $A$ of $G$ is said to be {\it purifiable} in $G$ if, among the pure subgroups of $G$ containing $A$, there exists a minimal one. Suppose that $A$ is purifiable in $G$ and $H$ is a pure subgroup of $G$ containing $A$. Then $A$ need not be purifiable in $H$. In this note, we ask for conditions that guarantee that $A$ is purifiable in the intermediate group $H$. First, we prove that if $A$ is a torsion--free purifiable subgroup of a group $G$ and $H$ is a direct summand of $G$ containing $A$, then $A$ is purifiable in $H$. Next, we characterize the pure subgroups $K$ of a group $G$ with the property that a torsion--free finite rank subgroup $A$ of $K$ is purifiable in $K$ if $A$ is purifiable in $G$.

This article is concerned with local solvability of the Cauchy problem for a quasilinear cubic wave equation in dimension $d=3$. Here, we improve the index of regularity of the initial data compared to the one given by classical energy methods.

The new class of weights called $A_p^{\dy ,m}$ weights is introduced. We prove that a characterization and an unconditional basis of the weighted $L^p$ space $L^p(\mathbb R^n , w(x)dx)$ with $w \in A_p^{\dy ,m}$ $(1<p<\infty)$ are given by the Haar wavelets and the Haar scaling function. As an application of these results, we establish a greedy basis by using the Haar wavelets and the Haar scaling function again.

Adaptive networks are emerging a lot of fields of science, recently. In this paper, we consider about a mathematical model for adaptive network made by the plasmodium. The organism contains a tube network by means of which nutrients and signals circulate through the body. The tube network changes its shape to connect two exits through the shortest path when the organism is put in a maze and food is placed at two exits. Recently, a mathematical model for this adaptation process of the plasmodium has been proposed. Here we analyze it mathematically rigorously. In ring-shaped network and Wheatstone bridge-shaped network, we mainly show that the globally asymptotically stable equilibrium point of the model corresponds to the shortest path connecting two special points on the network in the case where the shortest path is determined uniquely. From the viewpoint of mathematical technique, especially in the case of Wheatstone bridge-shaped network, we show that there is a simple but novel device used here by which we prove the global asymptotic stability, even when Lyapunov function cannot be constructed.