Volume 7, Issue 3

We relate the theory of the middle convolution functor MC λ to the study of algebraic solutions of the sixth Painlevé equation PVI. We interpret some recently found algebraic solutions by Boalch in terms of the middle convolution. Then we show how to obtain algebraic solutions of PVI by starting with triples in GL 2 and applying MC λ . The effect on the underlying Fuchsian systems can be described in a very simple manner.

In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by [U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18 (1985), 651–670. MR839689 (87j:53058) Zbl 0595.53043], [A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4) 21 (1988), 593–622. MR982335 (90d:53049) Zbl 0662.53032] and [S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669–682. MR1277451 (95c:53049) Zbl 0953.53027] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature.

Indicator sets arising from symplectic line-spreads of a 3-dimensional projective space are characterized. Moreover, locally Hermitian ovoids of a Hermitian surface (3, q 2 ) are studied in terms of “non-equivalent” associated indicator sets. Translation locally Hermitian ovoids of (3, q 2 ) arising from regular spreads of PG(3, q ), Kantor semifield spreads of PG(3, q ) and semifield spreads of PG(3, q ) with center are also investigated.

We solve a question of William M. Kantor by proving that dual good translation generalized quadrangles in odd characteristic always are elation generalized quadrangles.

Let . A p-set of equi-isoclinic planes with parameter λ in is a set of p planes spanning each pair of which has the same non-zero angle arccos . We prove that the maximum number of equi-isoclinic planes in is nine and that if λ > ¼, the maximum number of equi-isoclinic planes in with parameter λ is equal to that one of equiangular lines in with angle arccos .

To any point on an Alexandrov surface homeomorphic to the sphere one can associate a minimal subtree of the cut locus containing all farthest points. It is called the antipodal tree. Two points of a compact orientable Alexandrov surface are called mutually critical if each of them is critical with respect to the other. All points which are mutually critical with a given point form a set. In this paper we show that this set, as well as the set of endpoints of any antipodal tree, are finite.

We investigate the family of intersections of balls in a finite-dimensional vector space with a polyhedral norm. The spaces for which is closed under Minkowski addition are completely determined. We characterize also the polyhedral norms for which is closed under adding a ball. A subset of consists of the Mazur sets K , defined by the property that for any hyperplane H not meeting K there is a ball containing K and not meeting H . We characterize the Mazur sets in terms of their normal cones and also as summands of closed balls. As a consequence, we characterize the polyhedral spaces with only trivial Mazur sets as those whose unit ball is indecomposable.

Here we construct smooth curves X of genus g defined over and with prescribed geometrical properties: a suitable plane model or a degree k morphism defined over and with a fiber formed by k elements of .

We build a new class of embedded hypersurfaces in which have nonzero constant mean curvature and are O( m )×O( n )-invariant. For this we consider a 1-parameter family of planar vector fields associated to the differential equation of mean curvature.

We describe how to obtain the projective coordinates of the points of the Ree–Tits generalized octagon over a perfect field K in the 25-dimensional embedding of [K. Coolsaet, Algebraic structure of the perfect Ree-Tits generalized octagons. Innov. Incidence Geom. 1 (2005), 67–131. MR2213954 (2007a:51004) Zbl 05031723] from their Van Maldeghem coordinates (as defined in [H. Van Maldeghem, Generalized polygons. Birkhäuser 1998. MR1725957 (2000k:51004) Zbl 0914.51005, Section 3.6]). As an important bonus the explicit formulas which result from this may now serve as the definition of a 25-dimensional embedding also in the non-perfect case. We also give explicit formulas for the operations Ψ i and Φ j of the octagonal octanary ring associated with the octagon and its dual. (The latter are new.) As an application we derive the dimensions of subspaces of this embedding generated by various special subsets of points of the octagon: the sets of points at a fixed distance from a given point or a given line, the Suzuki suboctagons, the traces and the hypertraces. In some of the cases these dimensions turn out to depend on whether K is the field of 2 elements, or not.

We consider Courant and Courant–Jacobi brackets on the stable tangent bundle TM × ℝ h of a differentiable manifold and corresponding Dirac, Dirac–Jacobi and generalized complex structures. We prove that Dirac and Dirac–Jacobi structures on TM × ℝ h can be prolonged to TM × ℝ k , k > h , by means of commuting infinitesimal automorphisms. Some of the stable generalized complex structures are a natural generalization of the normal almost contact structures; they are expressible by a system of tensors ( P , θ, F , Z a , ξ a ) ( a = 1, …, h ), where P is a Poisson bivector field, θ is a 2-form, F is a (1, 1)-tensor field, Z a are vector fields and ξ a are 1-forms, which satisfy conditions that generalize the conditions satisfied by a normal almost contact structure ( F , Z , ξ). We prove that such a generalized structure projects to a generalized complex structure of a space of leaves and we characterize the structure by means of the projected structure and of a normal bundle of the foliation. Like in the Boothby–Wang theorem about contact manifolds, principal torus bundles with a connection over a generalized complex manifold provide examples of this kind of generalized normal almost contact structures.