Every transitive family of subspaces of a vector space of finite dimension $n\ge 2$ over a field $\mathbb {F}$ contains a subfamily which is transitive but has no proper transitive subfamily. Such a subfamily is called minimally transitive. Each has at most $n^2-n+1$ elements. On ${{\mathbb {C}}}^n, n\ge 3$ , a minimally transitive family of subspaces has at least four elements and a minimally transitive family of one-dimensional subspaces has $\tau $ elements where $n+1\le \tau \le 2n-2$ . We show how a minimally transitive family of one-dimensional subspaces arises when it consists of the subspaces spanned by the standard basis vectors together with those spanned by $0$ – $1$ vectors. On a space of dimension four, the set of nontrivial elements of a medial subspace lattice has five elements if it is minimally transitive. On spaces of dimension $12$ or more, the set of nontrivial elements of a medial subspace lattice can have six or more elements and be minimally transitive.

A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$ . G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$ .

Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ${\omega }$ -homogeneous but not ${\omega }$ -transitive permutation group on a cardinal ${\lambda }$ provided

1. (i) ${\lambda }<{\omega }_{\omega }$ , or

2. (ii) $2^{\omega }<{\lambda }$ , and ${\mu }^{\omega }={\mu }^+$ and $\Box _{\mu }$ hold for each ${\mu }\le {\lambda }$ with ${\omega }=\operatorname {cf}({\mu })<{{\mu }}$ , or

3. (iii) our model was obtained by adding $(2^{\omega })^+$ many Cohen generic reals to some ground model.

For ${\kappa }>{\omega }$ we give a method to construct large ${\kappa }$ -homogeneous, but not ${\kappa }$ -transitive permutation groups. Using this method we show that there exist ${\kappa }^+$ -homogeneous, but not ${\kappa }^+$ -transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$ .

We study the reflexivity and transitivity of a double triangle lattice of subspaces in a Hilbert space. We show that the double triangle lattice is neither reflexive nor transitive when some invertibility condition is satisfied (by the restriction of a projection under another). In this case, we show that the reflexive lattice determined by the double triangle lattice contains infinitely many projections, which partially answers a problem of Halmos on small lattices of subspaces in Hilbert spaces.

Patients with apraxia are more impaired when performing transitive pantomimes than intransitive gestures. This dissociation might be related to the differences in movement complexity. Alternatively, the programs for intransitive gestures might be better defined, more widely distributed, or easier to activate than are those for transitive pantomimes. The purpose of this study was to test the complexity versus representational hypotheses. Twenty right-handed normal subjects both performed and discriminated correct from incorrect transitive pantomimes and intransitive gestures. The discrimination was performed by having subjects point at illustrations of hand postures. The subjects performed better when discriminating postures than when performing gestures or pantomimes. On both the production and discrimination tests, subjects performed better with intransitive gestures than transitive pantomimes. Although the finding that even normal subjects had more difficulty performing transitive pantomimes than intransitive gestures might appear to support the complexity hypothesis, that subjects also had more difficulty discriminating transitive than intransitive postures supports the representational activation hypothesis. (JINS, 2002, 8, 958–963.)

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms.

As a consequence we prove that, for $3$-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal.

We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.

AMS 2000 Mathematics subject classification: Primary 37D25; 37C70; 37C20; 37C29

The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions.

It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions.

We believe that this approach, and the combinatorial primitives that have been developed for the “cut and join” analysis, will also assist with the general case.