A geometric algorithm is introduced for finding a symplectic basis of the first integral homology group of a compact Riemann surface, which is a $p$-cyclic covering of $\mathbf{C}P_1$ branched over 3 points. The algorithm yields a previously unknown symplectic basis of the hyperelliptic curve defined by the affine equation $w^2=z^{2g+1}-1$ for genus $g \ge 2$. We then explicitly obtain the period matrix of this curve, its entries being elements of the $(2g+1)$-st cyclotomic field. In the proof, the details of our algorithm play no significant role.

We study the Prohorov distance between the distance matrix distributions of two metric measure spaces. We prove that it is not smaller than 1-box distance between two metric measure spaces and also prove that it is not larger than 0-box distance between two metric measure spaces.

Nikisin [6] proved the linear independence of special values of polylogarithm functions over the rational number field. (see Theorem 0.1). In this paper, we give a generalization of the above result of Nikisin to the case of the Lerch function.

For an imaginary abelian field $K$, Hasse [3, Satz 45] obtained a criterion for the relative class number to be odd in terms of the narrow class number of the maximal real subfield $K^+$ and the prime numbers which ramify in $K$, by using the analytic class number formula. In [4], we gave a refined version (= "$\Delta$-decomposed version") of Satz 45 by an algebraic method. In this paper, we give one more algebraic proof of the refined version.

We show that three numerical semigroups $\langle 5, 6, 7, 8\rangle, \langle3, 7, 8\rangle$ and $\langle 3, 5\rangle$ are of double covering type, i.e., the Weierstrass semigroups of ramification points on double covers of curves. Combining the result with [7] and [4] we can determine the Weierstrass semigroups of the ramification points on double covers of genus two curves.

Ki and Nakagawa [Tôhoku Math. J., 39, 27-40 (1987)] characterized the minimal isoparametric hypersurfaces in a sphere that have three distinct constant principal curvatures from the stand point of the Ricci tensor. In this paper we generalize their results and characterize the isoparametric hypersurfaces, again in a sphere and with three distinct constant principal curvatures, that are either minimal or parallel to a minimal hypersurface.

We give a method for finding a minimal point adjacent to 1 of the reduced lattice in cubic number fields using an isotropic vector of the quadratic form and two-dimensional lattice.

In [3], a classification is given of the exceptional $\mathbf{Z}_2 \times \mathbf{Z}_2$-symmetric spaces $G/K$, where $G$ is an exceptional compact Lie group or $Spin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms (involutions) $\tilde{\sigma}$, $\tilde{\tau}$ of $G$ concretely and determine the structure of group $G^\sigma \cap G^\tau$ corresponding to Lie algebra $\mathfrak{g}^\sigma\cap \mathfrak{g}^\tau$, where $G$ is an exceptional compact Lie group. Thereby, we realize exceptional $\mathbf{Z}_2 \times \mathbf{Z}_2$-symmetric spaces, globally.