By using the Ruscheweyh operator $D^n f(z)$, $z\in U$, we obtain sharp superordinations results related to some normalized holomorphic functions in the unit disk $U$.

Let $G$ be a finite group. We define a Lefschets module $L(G, \Pi)$ which consists of equivalent classes of all $\Pi$-maps and prove that it is isomorphic to the Burnside module $\Omega(G, \Pi)$.

Let $G$ is a group. In the case where $G$ is finite, Oliver-Petrie defined a Burnside module $\Omega(G, {\cal F})$ consisting of all equivalent classes of $\cal F$-complex. The purpose of this paper is to define the universal Burnside module $U(G, {\cal F})$. If $G$ is finite, we have $U(G, {\cal F}) \cong \Omega(G, {\cal F})$.

We prove a uniqueness theorem for meromorphic functions sharing three values with small weights which improves some known results. We also exhibit some applications of the main result.

We represent the $105$ automorphic forms on the $5$-dimensional complex ball $\mathbb B^5$ constructed by Matsumoto-Terasoma as the products of four linear combinations of the pull backs of theta constants under an embedding of $\mathbb B^5$ into the Siegel upper half space of degree $6$. They were used to describe the inverse of the period map for the family of the $4$-fold coverings of the complex projective line branching at eight points.

Let $M_b$ be the operator of pointwise multiplication by $b$, that is $M_b f=bf$. Set $[A,B]=AB-BA$. The Reisz potentials are the operators $R_\alpha f(x)=\int f(x-y)\frac{dy}{\abs y ^{\alpha} }, 0<\alpha<1.$ They map $L^p\mapsto L^q$, for $1-\alpha+\frac{1}{q}=\frac{1}{p}$, a fact we shall take for granted in this paper. A Theorem of Chanillo [6] states that one has the equivalence $||[M_b,R_\alpha]||_{p\to q}\simeq ||b||_{BMO}$ with the later norm being that of the space of functions of bounded mean oscillation. We discuss a proof of this result in a discrete setting, and extend part of the equivalence above to the higher parameter setting.

In this note, we show that some families of complex exponentials are either Riesz sequences or not basic sequences in $L^2[-\pi,\pi]$. Besides, we show that every incomplete complex exponential system satisfying some condition can be complemented up to a complete and minimal system of complex exponentials in $L^2[-\pi,\pi]$.

We study pulse dynamics in one-dimensional heterogeneous media. In particular we focus on the case where the pulse is close to the singularity of codim 2 type consisting of drift and saddle-node instabilities in a parameter space. We assume that the heterogeneity is of jump type, namely one of the coefficients of the system undergoes an abrupt change at one point in the space. Depending on the height of this jump, the responses of pulse behavior are penetration, splitting, and rebound. Taking advantage of the fact that pulse is close to the singularity, the PDE dynamics can be reduced to a finite-dimensional system, which displays the three behaviors. Moreover it takes a universal form independent of model systems, and is valid for much more general heterogeneities such as bump, periodic, and random cases.

We consider generalized wave equations for the $p$--Laplacian and prove the local in time existence of solutions to the Cauchy problem. We give an estimate of the life-span of the solution, and show by a generic　counter-example that global in time solutions can not be expected.

This is an erratum to the paper 'On the group-homological description of the second Johnson homomorphism' (Hokkaido Math. J. 30 (2001), 605--613).

A uniform local energy decay property is discussed to a linear hyperbolic equation with spatial variable coefficients. We shall deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we assume algebraic order weight restrictions as $\vert x\vert \to +\infty$ on the initial data in order to derive the uniform local energy decay, and its proof is quite simple.

Examination of automorphism groups of forms is undertaken by many authors. Sometimes the description of such groups is a difficult task. It turns out that a representation of a form as a sum of powers of linear forms may be very helpful, especially when this representation is unique. We show this in the case of $\Sigma_{n+1}-$invariant symmetric trilinear form $\Theta_n$ considered by Egawa and Suzuki.

In this paper, we discuss the rigidity of $Sp(n)$ as a Riemannian submanifold of $M(n,n;\mathbb{H})$. We prove that the inclusion map $\pmb{f}_0$, which is called the canonical isometric imbedding of $Sp(n)$, is rigid in the following strongest sense: Any isometric immersion $\pmb{f}_1$ of a connected open set $U (\subset Sp(n))$ into $\pmb{R}^{4n^2}\,(\cong M(n,n;\mathbb{H}))$ coincides with $\pmb{f}_0$ up to a euclidean transformation of $\pmb{R}^{4n^2}$, i.e., there is a euclidean transformation $a$ of $\pmb{R}^{4n^2}$ satisfying $\pmb{f}_1=a\pmb{f}_0$ on $U$.

Let $S$ be a Sierpinski-like fractal with the compression ratio $\frac{1}{3}$, $N$ be the set of all the basic triangles to generate $S$. In this paper, by the mass distribution principle, the exact value of the Hausdorff measure of $S$, $H(S)=1$, is obtained, and the fact that the Hausdorff measure of $S$ can be determined by the net measure $H_N(S)$ is shown, and the best coverings of $S$ that are nontrivial are also obtained.