We show that the classical Hölder inequality between means of order $\alpha$, $0<\alpha\leqq 1$, can be improved on the assumption that the terms are not too often of comparable size. As an application, we derive a general, optimal bound for the entropy of a probability distribution.

In 1983, J. H. Conway and C. McA. Gordon showed in [1] that every embedding of the complete graph $K_7$ in the three-dimensional Euclidean space $\mathbf{R}^3$ contains a knotted cycle. In this paper we generalize their method and show that every embedding of the complete bipartite graph $K_{5,5}$ in $\mathbf{R}^3$ contains a knotted cycle.

Let $A_\theta$ be an irrational rotation algebra. In the present paper we will show that automorphisms of $A_\theta$ with some properties can be extended to inner automorphisms of an AF-algebra. In other words, there are a monomorphism $\rho$ of $A_\theta$ into an AF-algebra $B$ and a unitary element $w\in B$ such that $\rho(\alpha(x))=w\rho(x)w^*$ for any $x\in A_\theta$.