Let $b$ be an odd prime, $m, r \in \mathbf{N}$ with $2 \mid m$ and $2 \nmid r$, $r > 1$, and define the integers $U_r$, $V_r$ by $(m + \sqrt{-1} )^r = V_r + U_r\sqrt{-1}$. In this paper, we prove that if $a = |V_r|$, $b = |U_r|$, $c = m^2 + 1$, and $b > 8 \cdot 10^6$, $b \equiv 3 \pmod{4}$, then the Diophantine equation $x^2 + b^y = c^z$ has only the positive integer solution $(x, y, z) = (a, 2, r)$.

The existence theorems for (1) a differential inclusion in a Banach space and (2) a variational problem geverned by it are presented. In order to solve this problem, some implications of the weak convergence in the space of vector-valued absolutely continuous functions are also explored.

We show an elliptic curve over $\mathbf{Q}(t)$ of rank at least 9 with a non-trivial 2-torsion point.

Let $\nu$ be a positive measure on a space $Y$ with $\nu(Y) \neq 0$ and let $f_j$ ($j = 1, 2, \dots, n$) be positive $\nu$-integrable functions on $Y$. For some positive real numbers $\alpha_j$ ($j = 1, 2, \dots, n$), $\beta_j$ ($j= 1, 2, \dots, k < n$) and a measurable subset $Y_1$ of $Y$, we have some inequalities. From these results, we refine Hölder's inequality.

In this paper we calculate 2-primary components of homotopy groups of the homogeneous spaces $F_4/G_2$, $F_4/\mathrm{Spin}(9)$ and $E_6/F_4$.

We prove that infinite convolution products of complex probability measures with bounded total variation converge to a hyperfunction under a weak assumption on supports.

We consider boundedness of a function defined by an infinite product which is used to study a uniqueness theorem on a plane domain and the point separation problem of a two-sheeted covering Riemann surface. We show that there is such an infinite product that it converges but the function defined by it is not bounded on arbitrary Zalcman domain.