We treat the extended rings according to Beurling. Especially, Theorems VII and X in [1] are extended to the case of $n$-dimensional euclidean space $\mathbf{R}^n$.

In [7], we gave a classification of the configurations of singularities of irreducible sextic of torus type. In this paper, we give a classification of the configurations of singularities on reducible sextics of torus type. We determine the component types and the geometry of the components for each configuration.

In this paper, we state two results on the coarse index for simple models of the coarse geometry. The first one is a coarse index theorem on trivial bundles over even dimensional closed Riemannian manifolds. It comes from the Poincar\'e duals for these base manifold in the trivial bundles. We next show some correspondence between the generator of the coarse cohomology of $\mathbf{R}^n$ and the Riesz transformation on $\mathbf{R}^n$.

Applying Jaco's Handle Addition Lemma, we give a condition for a $3$-manifold to have an incompressible boundary. As an application, we show that the boundary of the exterior of a minimally knotted planar graph is incompressible.

We concern ourselves firstly with the abstract evolution equation \begin{equation*} \left\{ \begin{array}{@{\hspace*{2pt}}l} \mathbf{x}^\prime(t)= (A + B) \mathbf{x}(t)\,,\quad t>0\,;\quad \mathbf{x}(0)=\mathbf{x}_0\in D\,, \\ \rho ( \mathbf{x}(t))^\prime \leq g( \rho( \mathbf{x}(t) ))\,,\quad t>0 \end{array} \right. \end{equation*} in a Banach space $X$, where $A$ is m-quasidissipative, but $B$ satisfies no global conditions of quasidissipativity. We assume that a secondary topology (and a related notion of limits) can be introduced, through a lower semicontinuous functional $\rho$, such that $B$ is locally Lipschitz continuous with respect to this topology. Under appropriate assumptions, the first product formula \begin{equation*} \mathbf{x}(t) = (D,\varphi)\text{-}\lim_{h \downarrow 0}\{( I - h A)^{-1}( I + hB)\}^{[ t/h]} \mathbf{x}_0 \,,\quad t\geq 0 \end{equation*} can be defined, and we show that this provides unique solutions in a generalized sense to our original equation. Here $(D,\varphi)\text{-}\lim$ refers to our new notion of limit. For approximations $A_h$ of $A$, we also show convergence for the second product formula, \begin{equation*} \mathbf{x}(t) = (D,\varphi)\text{-}\lim_{h \downarrow 0}\{( I - h A_h)^{-1}( I + hB)\}^{[ t/h]} \mathbf{x}_0\,, \end{equation*} and use this to generate solutions to a class of advection reaction diffusion systems. As a concrete example, a mathematical model for HIV infection is studied.

We develop the fundamental properties of multiplicative functional $\mathcal{M}$ defined on a semidynamical system $(X,\mathcal{B},\Phi,w)$. We give a characterization of semigroups which are subordinate to the deterministic semigroup $\mathbb{H}$ and we show that they are generated by a multiplicative functional. We study the case when a multiplicative semigroup defined on a measurable space $(X,\mathcal{B})$ is deterministic.

A counter example to a conjecture of Y. Ito concerning direct summands of the integers is presented.

Convolution semigroups of probability measures with parameter in a cone in a Euclidean space generalize usual convolution semigroups with parameter in $[0,\infty)$. A characterization of such semigroups is given and examples are studied. Subordination of cone-parameter convolution semigroups by cone-valued cone-parameter convolution semigroups is introduced. Its general description is given and inheritance properties are shown. In the study the distinction between cones with and without strong bases is important.

In this paper, we shall generalize the Gauss-Bonnet and Chern-Lashof theorems to compact submanifolds in a simply connected symmetric space of non-positive curvature. Those proofs are performed by applying the Morse theory to squared distance functions because height functions are not defined.

We describe the Poincar\'e formula for two complex submanifolds in irreducible Hermitian symmetric spaces. That can be expressed as a constant times the product of the volumes of two submanifolds under some conditions.