Tahara, Vanhecke and Watanabe constructed a family of almost Her- mitian structure $(J_1;G_1)$ and two almost complex structures $J_2$, $J_3$ on the tangent bundle $TM$ over an almost Hermitian manifold $M$. In this paper, we define Riemannian metrics $G_2$ and $G_3$ on $TM$ which make $TM$ an almost Hermitian manifold and determine the conditions for $(J_i, G_i) (i = 1; 2; 3)$ so that $(TM; J_i, G_i)$ belongs to each of the sixteen classes established by Gray and Hervella.

We present a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there exist infinitely many ways of constructing the complete ordered field of real numbers. As an application of our results, we prove the irrationality of certain numbers.

We give a simultaneous extension of Selberg and Buzano inequalities: If $y1, y2$ and nonzero vectors $\{ z_i; i = 1, 2, \dots , n \}$ in a Hilbert space $\mathscr{H}$ satisfy the orthogonality condition $\langle y_k; z_i \rangle = 0$ for $i = 1, 2, \dots , n$ and $k = 1, 2,$ then \[ | \langle x, y_1 \rangle \langle x, y_2 \rangle | + \mathit{B} (y_1, y_2) \sum_i \frac{| \langle x, z_i \rangle |^2}{\sum_j | \langle z_i, Z-j \rangle |} \leq \mathit{B} (y_1, y_2) \|x\|^2 \] holds for all $x \in \mathscr{H}$, where $\mathit{B} (y1; y2) = \frac{1}{2} (\|y_1\| \|y_2\| + | \langle y_1, y_2 \rangle |)$. As an application, we discuss some refinements of the Heinz-Kato-Furuta inequality and the Bernstein inequality.

We prove a stronger version of Egoroff's theorem in the non-commutative setting

We describe the automorphism groups of curves appearing in a classification list of smooth plane curves with at least two Galois points. One of them is an ordinary curve whose automorphism group exceeds the Hurwitz bound.