When $K$ is an algebraically closed field of characteristic $p=3$, we shall investigate the existence of a regular factorial subring $R^{\prime }$ of $R=K[x,y]$ containing $R^p=K[x^p,y^p]$ and the existence of a $p$-basis of $R$ over $R^{\prime }$.

We find the fundamental units of a family of cubic fields introduced by Ishida. Using the family, we also construct a family of biquadratic fields whose 3-class field tower has length greater than 1.

For any $x$,$y\in ℬ$, a unital Banach algebra and $n\ge 1$ we show that

$$\Vert y^n-x^n\Vert \le n\Vert y-x\Vert \underset{0}{\overset{1}{{\displaystyle \int }}}\Vert (1-t)x+ty{\Vert }^{n-1}dt.$$

Upper bounds for quantities such as

$$\Vert f(x)-f(y)\Vert ,\Vert f(xy)-f(yx)\Vert ,$$

and

$$\Vert \frac{f(x)+f(y)}{2}-f(\frac{x+y}{2})\Vert$$

that can naturally be associated with the analytic function $f(\lambda ):=\sum _{j=0}^{\infty }{\alpha }_j{\lambda }^j$ defined on the open disk $D(0,R)$ and the elements $x$ and $y$ of the unital Banach algebra $ℬ$ are given. Some applications for functions of interest such as the exponential map on $ℬ$ and the resolvent function are provided as well.

The main focus of therapeutic confirmatory clinical trials, generally, is to demonstrate the efficacy of a new or experimental treatment in terms of mean(s) of the primary endpoint(s). In some cases, however, variance(s) of the endpoints also may be the focus of interest. For example, given two treatments with equal efficacy in terms of the means, a treatment with smaller variability, and thus more predictable efficacy, may be preferable. In this paper we consider single and multiple comparison procedures for partial covariance matrices of endpoints in clinical trials that can be used to demonstrate the superiority of a new treatment to an active comparator in terms of the variability. First, we review and discuss the existing methods for comparing two covariance matrices based on the union-intersection test procedure. Second, we propose a single comparison procedure and a multiple comparison procedure for partial covariance matrices that limits the number of comparisons and compare its performance with the tests discussed in the first section, using Monte Carlo simulations. The simulation results suggest that power of the proposed procedure is generally higher than those of the previous methods, while keeping the type I error rates nearly within the nominal level.