In this paper we announce fundamental results of the Ruelle zeta function for odd dimensional hyperbolic manifolds with cusps; the meromorphic extension over $\mathbf{C}$, its functional equation and the singularity at $s=0$.

For a bounded linear operator $A$ on a Hilbert space $\mathcal{H}$, let $\| A \|$ denote the operator norm and $w(A)$ the numerical radius. It is well-known that \begin{equation*} \frac{1}{2} \| A \| ≤q w(A) ≤q \| A \|. \end{equation*} For equalities, we consider linear operators $A$ with $A^{2} = 0$ and normaloid matrices.

Let $g \geq 2$ and $n \geq 1$ be integers. In this paper, we shall show that there are infinitely many imaginary quadratic fields whose class number is divisible by $2g$ and whose discriminant has only two prime divisors. As a corollary, we shall show that there are infinitely many imaginary quadratic fields whose 2-class group is a cyclic group of order divisible by $2^{n}$.

We are concerned with the solvablity of certain nonlinear partial differential equation (PDE), which is derived from the optimal investment problem under the random risk process. The equation describes the evolution of the Arrow-Pratt coefficient of absolute risk aversion with respect to the optimal value function. Employing the fixed point approach combined with the convergence argument we show the existence of solutions.

The arcsine law for random walks on the line is well known and it was extended greatly by Lamperti for a class of discrete-time stochastic processes. In the present paper we treat its extreme case where the excursion intervals have very heavy tail probabilities. The result is a refinement of Lamperti’s theorem. A functional limit theorem is also discussed.