Let $\Omega_n$ be the set of all permutations of the set $N_n = \{1, 2, \dots, n\}$ and let us suppose that each permutation $\omega = (a_1, \dots, a_n) \in \Omega_n$ has probability $1/n!$. For $\omega = (a_1, \dots, a_n)$ let $X_{nj} = |a_j - a_{j+1}|$, $j \in N_n$, $a_{n+1} = a_1$, $M_n = \max\{X_{n1}, \dots, X_{nn}\}$. We prove herein that the random variable $M_n$ has asymptotically the Weibull distribution, and give some remarks on the domains of attraction of the Fréchet and Weibull extreme value distributions.

We define invariants of two dimensional projectively Anosov diffeomorphisms. More precisely, we show the space of circles tangent to the invariant subbundle has a kind of Morse decomposition and the homotopy type of its one point compactification is preserved under any homotopy of projectively Anosov diffeomorphisms. We also calculate the invariants for some examples.

Let $(R, \boldsymbol{m})$ be a Noetherian complete local ring with unequal characteristic, and let $(P, pP)$ be a discrete valuation ring contained in $R$. Then, under some assumptions of separability on the residue fields, the following conditions are equivalent: (1) $R$ is a regular local ring and $p \notin \boldsymbol{m}^2$. (2) The $\boldsymbol{m}$-adic higher differential algebra $\widehat{D}_t(R/P, \boldsymbol{m})$ is a polynomial ring over $R$ for some $t~(1 \leq t)$.

We slightly extend a uniqueness theorem for generalized Radon transforms $f \mapsto Rf$ proved in [B2], by giving sharp decay conditions on the function $f$ at infinity.