Of concern are some operators inequalities arising in quantum chemistry. Let $A$ be a positive operator on a Hilbert space $\mathcal{H}$. We consider the minimization of $||U-A||_{p}$ as $U$ ranges over the unitary operators in $\mathcal{H}$ and prove that in most cases the minimum is attained when $U$ is the identity operator. The norms considered are the Schatten $p$-norms. The methods used are of independent interest; application is made of noncommutative differential calculus.

The fractional parts of the Bernoulli numbers are dense in the interval $(0,1)$. For every positive integer $k$, the set of all $m$ for which $B_{2m}$ has the same fractional part as $B_{2k}$ has positive asymptotic density.

An example is constructed showing that best uniform approximation (local or global) from $R_{n}^{m}(\mathbf{c})$ can not be characterized by linearization techniques or by alternation properties of the error function. A class of local best approximations are characterized, and used to demonstrate approximation properties of $R_{n}^{m}(\mathbf{c})$.

The problem considered is to characterize those integers $m$ such that $m = \mathrm{det}(C)$, $C$ an integral $n \times n$ circulant. It is shown that if $(m,n) = 1$ then such circulants always exist, and if $(m,n)> 1$ and $p$ is a prime dividing $(m,n)$ such that $p^{t}||n$, then $p^{t+1}|m$. This implies for example, that $n$ never occurs as the determinant of an integral $n \times n$ circulant, if $n > 1$.

This paper deals with minimal topologies on Riesz spaces. A minimal topology is a Hausdorff locally solid topology that is coarser than any other Hausdorff locally solid topology on the space. It is shown that every minimal topology satisfies the Lebesgue property, that an Archimedean Riesz space can admit a locally convex-solid topology that is minimal if and only if the space is discrete, that $C[0,1]$ and $L([0,1])$ do not admit a minimal topology, and that the topology of convergence in measure on $Lp([0,1])(0 < p < \infty)$ is a minimal topology. A similar result is shown for certain Orlicz spaces.