In this paper, we discuss inequalities for the numerical radius of the product of two Hilbert space operators. The significance of the obtained results is in the way they interpolate and improve several known inequalities in the literature. The equality cases are also discussed.

We construct compact parametrix implementations of covariant derivations on the quantum annulus.

Source algebras of block ideals are one of the main subjects in the modular representation theory of finite groups. Especially their bimodule structures attract much interest. In this paper, we shall treat block ideals with defect groups isomorphic to extraspecial $p$-groups of order $p^3$ and exponent $p$. We shall first analyze bimodule structures of these block ideals; we shall give a direct sum decomposition. We shall then prove that the images of the transfer maps on the cohomology rings of defect groups defined by the source algebras coincide with the cohomology rings of the block ideals in concern.

In this paper, we discuss a role of the potential term on the $L^2$ estimate of the solution itself for wave equations with/without a damping term. In the case of free waves, it is known ([10]) that the $L^2$-norm of the solution itself generally increases to infinity (as $t \to \infty$) in $1$ and $2$ dimensions. However, in this paper, we report that such a grow-up property can be controlled by adding a potential term with a generous condition. This idea can also be applied to damped wave equations with potential terms, especially in one dimension, where faster energy decay rates are observed than in the case of ordinary damped wave equaions. Applications to heat and plate equations with a potential can also be treated. In this paper, the low dimensional case is a main target.

We pose a normal form of transition functions along some Levi-flat hypersurfaces obtained by suspension. By focusing on methods in circle dynamics and linearization theorems, we give a sufficient condition to obtain a normal form as a geometrical analogue of Arnol'd's linearization theorem.

Let $M$ be a complete Kähler Manifold, whose universal covering is biholomorphic to a ball in $\mathbb{C}^m$. We prove that if two linearly nondegenerate meromorphic mappings $f$ and $g$ from $M$ into $\mathbb{P}^n(\mathbb{C})$ share two different families of hyperplanes $\{H_j\}_{j=1}^q$ and $\{L_j\}_{j=1}^q$ without multiplicity then there is a linear projective transformation $\mathcal L$ of $\mathbb{P}^n(\mathbb{C})$ into itself such that $\mathcal L(g)\equiv f$ and $\mathcal L(L_j)=H_j$ $(1\le j\le q)$ for $q$ large enough.

Under the C-admissible condition, Hoshi and Mochizuki proved a geometric version of the Grothendieck conjecture for the geometric outer monodromy representation associated to the configuration space of a hyperbolic curve. In the present paper, under conditions concerning the number of cusps of the hyperbolic curve and the dimension of the configuration space, we remove the C-admissible condition from the result of Hoshi and Mochizuki.