It is known that the set of possible cofinalities pcf($A$) has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that pcf($A$) has no good properties in the presence of a measurable cardinal, or in generic extensions by Prikry-type forcing.

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a totally geodesic submanifold. In this paper, we classify congruent classes of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms, and consider their moduli spaces. As a consequence, we show that some moduli spaces of isometric immersions between space forms which one of the same constant curvature are non-Hausdorff.

We introduce the notion of a strong minuscule element, which is a dominant minuscule element $w$ in the Weyl group for which there exists a unique (dominant) integral weight $\Lambda$ such that $w$ is $\Lambda$-minuscule. We prove that the dominant integral weight associated to a strong minuscule element is the fundamental weight corresponding to a short simple root (in this paper, all simple roots in the simply-laced cases are treated as short roots). In addition, we enumerate the strong minuscule elements explicitly. As an application of this enumeration, we determine the dimension of certain Demazure module in the finite-dimensional irreducible module whose highest weight is a minuscule weight.

Let $\beta > 1$. In this paper, we define the odometer associated with a $\beta$-numeration system, and study its properties. As a result, we characterize the finiteness or positive finiteness property of this numeration system by using this odometer. We also study the computability of this odometer via natural transducer.

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

Linear diffusions similar to Bessel processes are studied. The asymptotic behavior of the transition probability in the long term is discussed under the condition that the left boundary is regular-killing or exit-not-entrance.

In this paper, we study perfectness of Higson type compactifications and give a necessary and sufficient condition that the subpower and the sublinear Higson compactification of a locally connected proper metric space are perfect. Consequently, we show that the sublinear Higson corona of the half open interval with the usual metric is an indecomposable continuum which is an affirmative answer to Iwamoto's question [9].