In this paper, we obtain a new result for overtwisted contact $(+1/n)$-surgery. We also give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery.

Pathogen spread within pig host populations can vary depending on within-herd interactions among pigs also called the contact structure. The recommended batch farrowing management, allowing for a fixed-interval mating for groups of sows of equal size, called batches, leads to an all-in/all-out management of pigs in which animals in different batches have no contact. To maintain a profitable pig delivery, producers have to deliver groups of pigs at a given weight, what needs sometimes herd management adaptations. However, producers’ adaptations that avoid delivering pigs below slaughtering weight (out-of-range pigs), result in increasing the contact between animals from different batches. To study the influence of herd management on contact structure and on pig delivery, a stochastic mathematical model representing population dynamics within a farrow-to-finish herd was elaborated. Sixteen management systems were represented combining or not the all-in/all-out management system with producers’ decisions: batch mixing, use of an extra room, suppression of the drying period and sale of post-weaning batches. Two types of contact were considered: via the animals themselves, when batch mixing occurred; and via the room, when decontamination was not complete. The impact of producers’ decisions on contact structure and on pig delivery, differed radically when pig growth was normal and when it was slow (i.e. mean age at slaughtering weight increased by 20%). When pig growth was normal, the all-in/all-out management prevented both contact via the animals and via the room but resulted in 9% of pigs delivered out of range. The use of an extra room or batch mixing decreased this percentage, the latter resulting in very frequent contact between batches via the animals. When pig growth was slow, the all-in/all-out management led to a very high percentage of pigs delivered out of range (almost 80%). The suppression of the drying period at the end of the finishing period and the sale of post-weaning batches induced a significant decrease in this percentage (down to 2% to 20%), the latter allowing to reduce the percentage of batches that made contact via the room (40% instead of 80%). This pig herd model helped to understand the compromise for producers between implementing internal biosecurity or maintaining a profitable pig delivery. Our results show that there was no unique optimal system and that efficient producers’ decisions (for biosecurity and delivery) may differ, depending on pig growth.

A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.

The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606).

The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

The set N of all null geodesics of a globally hyperbolic $(d + 1)$-dimensional spacetime $(M, g)$ is naturally a smooth $(2d - 1)$-dimensional contact manifold. The sky of an event x in M is the subset X of N consisting of all null geodesics through x, and is an embedded Legendrian submanifold of N diffeomorphic to $S^{(d - 1)}$. It was conjectured by Low that for $d = 2$ two events x and y are causally related if and only if X and Y are linked (in an appropriate sense). We use the contact structure and knot polynomial calculations to prove this conjecture in certain particular cases, and suggest that for $d = 3$ smooth linking should be replaced with Legendrian linking.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and ${{H}^{*}}$-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

We introduce a certain population contact structure and derive, in three different ways, the final size equation for a quite general superimposed epidemic process. The contact structure is characterized by the following two properties: (i) each individual contacts exactly k other individuals; (ii) these k acquaintances are a random sample of the (infinite) population.