For prime $l$ we study the structure of the $2$-part of the ideal class group Cl of $\,\mathbb{Q}(\zeta_l)$. We prove that Cl$\, \otimes \,{\mathbb {Z}}_2$ is a cyclic Galois module for all $l < 10000$ with one exception and compute the explicit structure in several cases.

It is shown that iteration of analytic set-valued functions can be used to generate composite Julia sets in ${\bf C}^N$. Then it is shown that the composite Julia sets generated by a finite family of regular polynomial mappings of degree at least 2 in ${\bf C}^N$, depend analytically on the generating polynomials, in the sense of the theory of analytic set-valued functions. It is also proved that every pluriregular set can be approximated by composite Julia sets. Finally, iteration of infinitely many polynomial mappings is used to give examples of pluriregular sets which are not composite Julia sets and on which Markov's inequality fails.

Associated with a Hamiltonian holomorphic vector field on a compact Kähler manifold, a nice functional on a space of Kähler metrics will be constructed as an integration of the bilinear pairing in [FM] contracted with the Hamiltonian holomorphic vector field. As applications, we have functionals $\hat{\mu}$, $\hat{\nu}$ whose critical points are extremal Kähler metrics or "Kähler-Einstein metrics" in the sense of [M4], respectively. Finally, the same method as used by [G1] allows us to obtain, from the convexity of $\hat{\nu}$, the uniqueness of "Kähler-Einstein metrics" on nonsingular toric Fano varieties possibly with nonvanishing Futaki character.

In Part I of this work, we have shown that the stochastic process $Z^{(\mu)}$ defined by (8.1) below is a diffusion process, which may be considered as an extension of Pitman's $2M-X$ theorem. In this Part II, we deduce from an identity in law partly due to Dufresne that $Z^{(\mu)}$ is intertwined with Brownian motion with drift $\mu$ and that the intertwining kernel may be expressed in terms of Generalized Inverse Gaussian laws.

We study the behavior of Hilbert-Kunz multiplicity for powers of an ideal, especially the case of stable ideals and ideals in local rings of dimension $2$. We can characterize regular local rings by certain equality between Hilbert-Kunz multiplicity and usual multiplicity.

We show that rings with "minimal" Hilbert-Kunz multiplicity relative to usual multiplicity are "Veronese subrings" in dimension $2$.

It is completely known the characterization of all Hilbert functions and all graded Betti numbers for $3$-codimensional arithmetically Gorenstein subschemes of $\mathbb{P}^r$ (works of Stanley [St] and Diesel [Di]. In this paper we want to study how geometrical information on the hypersurfaces of minimal degree containing such schemes affect both their Hilbert functions and graded Betti numbers. We concentrate mainly on the case of general hypersurfaces and of irreducible hypersurfaces, for which we find strong restrictions for the Hilbert functions and graded Betti numbers of their subschemes.

This paper investigates the existence of positive solutions of nonresonant singular boundary value problem of second order differential equations. A necessary and sufficient condition for the existence of $C[0,1]$ positive solutions as well as $C^1[0,1]$ positive solutions is given by means of the method of lower and upper solutions with the fixed point theorems.

In this paper, we prove that if a complete Riemannian manifold $M$ has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded $\cal A$-harmonic functions on $M$ is one to one corresponding to $\mathbf{R}^l$, where $\cal A$ is a nonlinear elliptic operator of type $p$ on $M$ and $l$ is the number of $p$-nonparabolic ends of $M$. We also prove that if a complete Riemannian manifold $M$ is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded $\cal A$-harmonic functions on $M$ is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor'yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

Fractional moments of the passage-times are considered for positively recurrent Markov chains with countable state spaces. A criterion of the finiteness of the fractional moments is obtained in terms of the convergence rate of the transition probability to the stationary distribution. As an application it is proved that the passage time of a direct product process of Markov chains has the same order of the fractional moments as that of the single Markov chain.

We consider a signal detection problem for the continuous-time stationary diffusion processes. The optimal decision region is given by Neyman-Pearson's lemma. We establish certain large deviation estimates, and with the help of it we show that the error probability of the second kind of the signal detection tends to zero or one exponentially fast, depending on the fixed exponent of the decay of the error probability of the first kind, as the observation time goes to infinity.