In this paper we mainly investigate the radial distribution of Julia sets of Newton’s method of entire solutions of some complex linear differential equations. Under certain conditions, we find the lower bound of it and also obtain some related results.

An explicit construction is provided for embedding $n$ positive eigenvalues in the spectrum of a Schrödinger operator on the half-line with a Dirichlet boundary condition at the origin. The resulting potential is of von Neumann-Wigner type, but can be real-valued as well as complex-valued.

We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

For a prime number $p$, we say that a number field $F$ satisfies the Hilbert-Speiser condition $(H_{p})$ if each tame cyclic extension $N/F$ of degree $p$ has a normal integral basis. In this note, we determine the real abelian number fields satisfying $(H_{p})$ for odd prime numbers $p$ with $h(\mathbf{Q}(\sqrt{-p}))=1$.

We shall deal with infinite towers of cyclic fields of genus number 1 in which a prime number $l\geq 5$ is totally ramified. Our main result states that, if $m$ is a positive divisor of $l-1$ less than $(l-1)/2$, then for any positive integer $n$, the cyclic field of degree $ml^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean. In particular, it follows that, for any positive integer $n$, the (real) cyclic field of degree $l^{n}$ with conductor $l^{n+1}$ is not norm-Euclidean and that the (imaginary) cyclic field of degree 14 with conductor 49, whose class number is known to equal 1, is not norm-Euclidean.