In this article, we deal with the uniqueness problems of meromorphic functions concerning differential polynomials and prove the following result: Let $f$ and $g$ be two nonconstant meromorphic functions and let $n(\geq 14)$ be an integer such that $n+1$ is not divisible by $3$. If $f^{n}(f^{3}-1)f'$ and $g^{n}(g^{3}-1)g'$ share $(1,2)$ or $``(1,2)"$, then $f\equiv g$. If $\overline{E}_{4)}(1,f^{n}(f^{3}-1)f')=\overline{E}_{4)}(1,g^{n}(g^{3}-1)g')$ and $E_{2)}(1,f^{n}(f^{3}-1)f')=E_{2)}(1,g^{n}(g^{3}-1)g')$, then $f\equiv g$.

We consider a class of Ruelle type operators which play an important role in the study of singular perturbation of symbolic dynamics via thermodynamic formalism. We study the eigenvalues of those operators with maximal modulus and obtain necessary and suffcient conditions for them to be semisimple.

We show that the Lebesgue space with a variable exponent $L_{p(\cdot )}$ is a rearrangement--invariant space if and only if $p$ is constant. In addition, we give a necessary and sufficient condition on a variable exponent for a martingale inequality to hold.

We prove two general formulas for a two-parameter family of hypergeometric $\3F2(z)$ functions over a finite field $\F_q$, where $q$ is a power of an odd prime. Each formula evaluates a $\3F2$ in terms of a $\2F1$ over $\F_{q^2}$. As applications, we evaluate infinite one-parameter families of $\3F2(\frac{1}{4})$ and $\3F2(-1)$, thereby extending results of J. Greene--D. Stanton and K. Ono, who gave evaluations in special cases.

We develop a group-theoretic method to generalize the Laplace-Beltrami operators on the classical domains. In K. Okamoto, "Harmonic analysis on homogeneous vector bundles," Lecture Notes in Mathematics, Springer-Verlag, 266 (1971), 255–271, inspired by Helgason's paper, "A duality for symmetric spaces with applications to group representations," Advan. Math. 5 (1970), 1–154, we defined the "Poisson transforms" for homogeneous vector bundles over symmetric spaces. In K. Okamoto, M. Tsukamoto and K. Yokota, "Generalized Poisson and Cauchy kernel functions on classical domains," Japan. J. Math. 26 No. 1 (2000), 51–103., we defined the generalized Poisson-Cauchy transforms for homogeneous holomorphic line bundles over hermitian symmetric spaces and computed explicitly the kernel functions for each type of the classical domains. In E. Imamura, K. Okamoto, M. Tsukamoto and A. Yamamori, "Generalized Laplacians for Generalized Poisson-Cauchy transforms on classical domains," Proc. Japan Acad., 82, Ser. A (2006), 167–172., making use of the Casimir operator, we defined the "generalized Laplacians" on homogeneous holomorphic line bundles over hermitian symmetric spaces and showed that the generalized Poisson-Cauchy transforms give rise to eigenfunctions of the "generalized Laplacians". In this paper, using the canonical coordinates for each type of the classical domains, we carry out the direct computation to obtain the explicit formulas of (line bundle valued) invariant differential operators which we call the generalized Laplacians and compute their eigenvalues evaluated at the generalized Poisson-Cauchy kernel functions

We define a space of holomorphic functions $O_{1}(U,E\mid F)$ on a domain of holomorphy $U$ of ${\Bbb C}^{n}$, taking their values in quotient bornological spaces $E\mid F$ as the kernel of a sheaf-morphism. We show that if $E$ is a Schwartz $b$-space and $F$ is a Fréchet-Schwartz $b$-space, then $O_{1}(U,E\mid F)$ and $O\left( U,E\right) \mid O\left( U,F\right)$ are naturally isomorphic.

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized to $\alpha$-resolvability and affine $\alpha$-resolvability in 1963. These arguments are mostly done for a class of balanced incomplete block designs. The present paper will make the combinatorial investigation on affine $\alpha$-resolvable partially balanced incomplete block designs with two associate classes. The characterization of parameters in a closed form will be given and then existence problems with construction methods will be discussed. Comprehensive and useful results on combinatorics are obtained. Several methods of construction are newly presented with some illustrations.