Let $B$ be an $F$-space and $B^\ast _1$ the unit ball of the dual space. A sequence $(\phi _n)$ in $B^\ast _1$ is called $\ell ^1$-interpolating if for every sequence $(w_n)$ in $\ell ^1$ there exists an element $f$ in $B$ such that $\phi _n(f)=w_n$ for all $n$. In order to study an interpolation problem for $\ell ^1$, we introduce two quantities $\rho _n$ and ${\prod_{k\ne n}}\sigma (\phi _n,\phi _k)$. For arbitrary Banach space, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n}\rho _n>0$. Moreover, when a Banach space has a predual, we show that if ${\inf_n\prod_{k\ne n}}\sigma(\phi_n, \phi_k)>0$ then $(\phi_n)$ is an $\ell^1$-interpolating sequence. When $(\phi _n)$ is embeded in the open unit disc in the complex plane, we show that $(\phi _n)$ is an $\ell ^1$-interpolating sequence if and only if ${\inf_n \prod_{k\ne n}}\sigma (\phi _n,\phi _k)>0$, for a Hardy space $H^p(D)(1\leq p\leq \infty )$ and the Smirnov class $N_+(D)$.

We study the combinatorial and structural properties of the circle map sequences. We introduce an embedding procedure which gives a map $\Phi: \Omega \to W:= \{ R, L \}^{\bf N}$ from the hull(closure of the set of translates) to the sequence of embedding operations through which we study the structure of $\Omega$. We also study the set of admissible words and classify them in terms of their appearance.

We prove that the Steiner ratios for complete locally compact length spaces having $n$ ends are less than or equal to $n/2(n-1)$. In particular, the Steiner ratio of a complete simply connected surface with a pole satisfying the Visibility axiom is $1/2$.

We study the structure of the Penrose tiling (PT, in short) constructed by the matching rule, and deduce directly a substitution rule from that, which gives us (i)local configuration of the tiles,(ii) elementary proofs of the aperiodicity, the locally isomorphic property, and the uncountability,(iii) alternative proof of the fact that all PT's obtained by the matching rule can be constructed via the up-down generation.

Let $\cal M$ be a von Neumann algebra. We will show that for two normal semifinite faithful weights $\phi$, $\psi$ on $\cal M$, the corresponding non-commutative $L^p$-spaces $L^p({\cal M},\phi)$ and $L^p({\cal M},\psi)$ are isometrically isomorphic.