The weights of a two-dimensional weighted homogeneous polynomial $f$ of degree $h$ corresponding to an isolated singularity are arithmetically characterized by Prof. K. Saito and are called a regular system of weights. Let $m_0$ be the dimension of the vector space of the elements of degree $h$ of the Jacobi ring of $f$. It is shown that $m_0$ is determined by weights and is estimated from below by using the genus of the central curve and the number of branches of a minimal good resolution of the corresponding singularity.

We define and construct holomorphic $C$-existence families and holomorphic integrated semigroups and discuss their relationships with holomorphic $C$-semigroups and each other. We prove simple Hille-Yosida type theorems, involving the rate of growth of $\lVert(w-A)^{-1}\rVert$ or $\lVert(w-A)^{-1}C\rVert$ in a sector, for an operator to generate any of these families of operators.

It is known that if $X$ is a Lebesgue probability space, $T:X\to X$ an ergodic measure preserving automorphism, and $n$ a fixed nonzero integer, then a coboundary for the automorphism $T^n$ is also a coboundary for $T$. In this paper, the result is extended to include the case where the exponent $n=m(x)$ is an arbitrary integrable integer valued function on $X$.

A nonstationary problem of convection in the exterior domain to a heated sphere is studied. In the Boussinesq approximation, convection phenomena are governed by the system of the Navier-Stokes and heat equations. We find sufficient conditions on boundary and initial data to ensure the global existence of $L^p$-solutions for this system.

The structure of the space of multipliers of the range of a composition operator $C_\phi$ on the Hardy space is studied. We provide necessary and/or sufficient conditions in terms of appropriate measurements of the distance of $|\phi|$ to 1 for the containment or the inclusion of the space of multipliers in standard spaces.

We study a smooth structure on a non-commutative 3-sphere $S_\Theta^3$ defined as a deformed $C^*$-algebra of $C(S^3)$ by a continuous function $\Theta$. We then consider the subalgebra $(S_\Theta^3)^\infty$ of all smooth elements of $S_\Theta^3$. It is a non-commutative version of $S^{3}$ as a smooth manifold. We also construct a smooth linear map from $(S_\Theta^3)^\infty$ to the algebra $C^\infty(S^3)$ of all smooth functions on $S^3$ so that the Lie algebra $\mathfrak{su}(2)$ acts on $(S_\Theta^3)^\infty$ with a twisted Leibniz's rule. Finally we find a Haar measure on $S_\Theta^3$ and show its uniqueness.