We consider a positively definite self-adjoint operator $P$ on a separable Hilbert space $H$ which has a compact resolvent. Then a specific example of the Ikehara Tauberian theorem is extended to the case where the zeta function of $P$ only has simple poles. In such circumstances, we can obtain the asymptotic behavior of the counting function of eigenvalues with remainder terms. And we have their applications to some partial differential operators.

Let $X_1$ and $X_2$ be two independent Hunt processes which take values in a metric space and have the same transition density functions with respect to a reference measure. We describe explicit conditions on the transition density functions so that $X_1$ and $X_2$ have collisions with positive probability or with probability one or do not have any collision. The applications to Lévy processes, diffusions driven by s.d.e.'s and Brownian motions on fractals are exhibited.

Certain exact results concerning the Bernstein-Nikolsky inequality are established in this paper.

Let $\rho_1,\rho_2,\ldots,\rho_{2g+1}$ be rotation vectors for periodic points of a homeomorphism on an orientable surface of genus $g>1$. Assume that the convex hull of the set $\{\rho_1,\rho_2,\ldots,\rho_{2g+1}\}$, Conv$(\rho_1,\rho_2,\ldots,\rho_{2g+1})$, has nonempty interior. We will give a sufficient condition for the existence of a dense subset of Conv$(\rho_1,\rho_2,\ldots,\rho_{2g+1})$ that is realized by periodic points.