We obtain new trigonometric identities. We show that the coefficients of Laurent expansions of the identities give rise to the relation between special values of Hurwitz zeta function and Bernoulli numbers. Then we look into in detail the parameterized cotangent sums appearing in the identities.

We are studying the action of Galois groups on the pro-$l$ completion of the fundamental group of $\mathbf{P}^1_{\overline{\mathbf{Q}(\mu_n)}} \backslash \{0,\mu_n,\infty\}$. If $n=2p$, where $p$ is an odd prime number then the Lie algebra of derivations associated to the image of $\text{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}(\mu_{2p\cdot l^\infty}))$ has $\frac{p-1}{2}$ generators in each even degree and $\frac{p-1}{2}$ generators in each odd degree greater than $1.$ We shall show that generators in even degrees generate a free Lie algebra.

The hyper-Kloosterman code was first defined as a trace code by Chinen-Hiramatsu [1]. In this article, two basic parameters of it, the minimum distance and the dimension are estimated. Analysis of the dimension shows that it is one of few examples of trace codes, of which the dimensions do not reduce when taking the trace, and are determined explicitly. It is also shown that the hyper-Kloosterman code can be realized as a quasi-cyclic code. It implies a method of explicit construction of quasi-cyclic codes of a new type.

In this article, we improve and generalize the result of [O]; the one is the necessary and sufficient condition of the minimality of Lagrangian submanifolds in adjoint orbits, which are Hermitian symmetric spaces, and the other is the necessary and sufficient condition of the Hamiltonian stability of minimal Lagrangian submanifolds in adjoint orbits, which are not necessarily Hermitian symmetric spaces.

In this paper, an upper bound of the average curvature of a convex curve in a simply connected surface is obtained.

Analytic singularities of local solutions to the nonlinear ordinary differential equation $(|u_x|^{p-2}u_x)_x + |u|^{q-2}u = 0$ are obtained through Briot-Bouquet type nonlinear analytic differential equations with regular singularity.

We classify the conformally flat, semi-Riemannian manifolds satisfying $R(X,Y) \cdot Q = 0$, where $R$ and $Q$ are the curvature tensor and the Ricci operator, respectively. As the cases which do not occur in the Riemannian manifolds, the Ricci operator $Q$ has pure imaginary eigenvalues or it satisfies $Q^2 = 0$.