The purpose of this paper is to introduce the spaces of sequences that are strongly almost $(w, \lambda)$-summable with respect to an Orlicz function. We give some relations related to these sequence spaces. It is also shown if a sequence is strongly $(w, \lambda)$-summable with respect to an Orlicz function, then it is $\bar{S}_{\lambda}$-statistically convergent.

Consider the following discrete model of nonautonomous Lotka-Volterra type: \[\left\{ \begin{array}{ll} N_i(p+1) = N_i(p) \exp \{ r_i(p)(c_i(p)-b_i(p)N_i(p)- \dis \sum_{j=1}^n \dis \sum_{l=0}^m b_{ij}^l(p) N_j(p-l)) \}, \\ \hspace{8cm} p=0,1,2,\cdots, \ 1 \leq i \leq n, \\ N_i(0) = N_{i0} > 0 \ and \ N_i(-l)=N_{i(-l)} \geq 0, \ 1 \leq i \leq n, \ 1 \leq l \leq m, \\ \end{array} \right. \] where \[ \left\{ \begin{array}{ll} r_i(p)>0, \ for \ p \geq 0, \ \dis \liminf_{p \to \infty}r_i(p)>0, \\ b_i(p)>0, \ b_{ii}^0(p) = 0, \ 1 \leq i \leq n, \ b_{ij}^l(p) \geq 0, \ 1 \leq i \leq j \leq n, \ 0 \leq l \leq m. \\ \end{array} \right. \] In this paper, to the above discrete system, we establish two type sufficient conditions that ensure the "contractivity" of solutions which are sufficient conditions of the global asymptotic stability of system. This is an extension of the former work (2002, ${\it J. Math. Anal. Appl.} {\bf 270}, 602-635$) for $n=1$ to $n \geq 2$. In particular, for autonomous case of the above system, the sufficient condition for the global asymptotic stability of the positive equilibrium offered by Wang et al. (2002, ${\it J. Math. Anal. Appl.} {264}, 147-169$), is proved to satisfy this contractivity of the positive equilibrium.

Some boundedness results are established for some rough singular integral operators and fractional singular integral operators on the Homogeneous Morrey-Herz Spaces. And also we get some boundedness results on the weak homogeneous Morrey-Herz Spaces.

In the context of the axiomatic potential theory, we introduce the notions of polyharmonic functions and polypotentials on a Brelot harmonic space $\Omega$. For these functions,we prove some results analogous to the Riesz decomposition, balayage, domination principle, etc., which are usually associated with harmonic and superharmonic functions on $\Omega$. We also consider the polyharmonic classifications of the harmonic spaces.

In [7], we have proved that $P^2({\mathbf Cay})$ cannot be isometrically immersed into ${\mathbf R}^{25}$ even locally. In this paper, we investigate isometric immersions of $P^2({\mathbf Cay})$ into ${\mathbf R}^{26}$ and prove that the canonical isometric imbedding ${\mathbf f}_0$ of $P^2({\mathbf Cay})$ into ${\mathbf R}^{26}$, which is defined in Kobayashi~\cite{kobayashi}, is rigid in the following strongest sense: Any isometric immersion ${\mathbf f}_1$ of a connected open set $U (\subset P^2({\mathbf Cay}))$ into ${\mathbf R}^{26}$ coincides with ${\mathbf f}_0$ up to a euclidean transformation of ${\mathbf R}^{26}$, i.e., there is a euclidean transformation $a$ of ${\mathbf R}^{26}$ satisfying ${\mathbf f}_1=a{\mathbf f}_0$ on $U$.

In this paper, we classify all affine semiparallel surfaces in ${\mathbb R}^3$ with constant Pick invariant. As its application, we can characterize affine semiparallel surfaces of constant Pick invariant but of which shape operators are not parallel.

In this paper the geometries of surfaces in the 3-dimensional Heisenberg group with a left invariant metric are classified from a stand point of the Grassmann geometry, and for each of them the existence or nonexistence of surfaces with constant mean curvature is clarified.

We give a formula for the one-parameter strongly continuous semigroup $e^{-tL},\,t>0$, generated by the Hermite operator $L$ on the Heisenberg group $\H1$ in terms of Weyl transforms, and use it to obtain an $L^2$ estimate for the solution of the initial value problem for the heat equation governed by $L$ in terms of the $L^p$ norm of the initial data for $1\leq p\leq \infty.$

We study the scattering theory for the coupled Klein-Gordon-Schrödinger equation with the Yukawa type interaction in two space dimensions. The scattering problem for this equation belongs to the borderline between the short range case and the long range one. We show the existence of the wave operators to this equation without any size restriction on the Klein-Gordon component of the final state and any restriction on the support of the Fourier transform of the final state.

In this paper we consider the modified maximal operator on the separable metric space. Define $M_kf(x)= \sup_{r>0} \frac{1}{\mu(B(x,kr))}\int_{B(x,r)}|f(y)|d\mu(y)$ and $ M_{k,uc}f(x) = \sup_{x \in B(y,r)} \frac{1}{\mu(B(y,kr))}\int_{B(y,r)}|f(z)|d\mu(z)$ respectively. We investigate in what parameter $k$ the weak $(1,1)$-inequality holds for $M_k$ and $M_{k,uc}$ in general metric space and Euclidean space. The proofs are sharper than the method of Vitali's covering lemma. This attempt is partially done by Yutaka Terasawa [9] before. When we investigate ${\mathbf R}^d$, we prove a new covering lemma of ${\mathbf R}^d$. We also show that our condition on parameter $k$ is sharp. In connection with this we consider the dual inequality of Stein type and its applications.

We define a pluriharmonic map from a complex manifold with a complex affine connection to a manifold with an affine connection and obtain some fundamental results which generalize those for a pluriharmonic map from a K\"{a}hler manifold to a Riemannian manifold. Especially, by using an associated family, we find a sufficient condition for the product of two $(1,1)$-geodesic affine immersions to an affine space to be a complex affine immersion from the manifold to the product of affine spaces with a certain complex structure.

In this paper we study the twisted de Rham cohomology for an arrangement of hyperplanes. We prove formulas for basic constructions of arrangements: the twisted Künneth formula for products or decompositions, the long exact sequence for triple constructions, and the relation for coning and deconing constructions. Moreover, we obtain the vanishing, dimension, and basis of the twisted de Rham cohomology for an arrangement in general position and a generic arrangement with arbitrary weight.