Let M be a closed, irreducible, genus two 3–manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M . Then each component manifold Mi of M − F has handle number one, i.e. admits a Heegaard splitting obtained by attaching a single 1–handle to one or two components of ∂Mi. This result also holds for a decomposition of M along a maximal ...

The category of admissible (in an appropriately modified sense of representation theory of totally disconnected groups) semi-linear representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of the field of algebraic complex numbers is described. Let k be a field of characteristic zero containing all ℓ-primary roots of unity for a prime ℓ, F ...

Let R be a regular ring essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–struct...

The category of admissible (in the appropriately modified sense of representation theory of totally disconnected groups) semi-linear representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of the field of algebraic complex numbers is described. Let k be a field of characteristic zero containing all ℓ-primary roots of unity for a prime ℓ, F...

The category of admissible (in the appropriately modified sense of representation theory of totally disconnected groups) semi-linear representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of the field of algebraic complex numbers is described. Let k be a field of characteristic zero containing all ℓ-primary roots of unity for a prime ℓ, F...

It is known that the treewidth of a planar graph with a dominating set of size d is O( √ d) and this fact is used as the basis for several fixed parameter algorithms on planar graphs. An interesting question motivating our study is if similar bounds can be obtained for larger minor closed graph families. We say that a graph family F has the domination-treewidth property if there is some functio...

Let R be a regular ring, essentially of finite type over a perfect field k. An R–module M is called a unit R[F ]–module if it comes equipped with an isomorphism F M −→ M, where F denotes the Frobenius map on SpecR, and F e∗ is the associated pullback functor. It is well known that M then carries a natural DR–module structure. In this paper we investigate the relation between the unit R[F ]–stru...

1.1 Definition. Let X be a nonempty set. A set c ⊂ P(X) is club in P(X) if there is a function f : X → X for which c is the set of A ⊂ X closed under f . Given a cardinal κ ≤ |X|, c is club in [X] (or in [X]) if c is the set of A ∈ [X] (or [X]) closed under f . A set a ⊂ P(X) is stationary in P(X) if it intersects every club subset of P(X), and stationary in [X] (or [X]) if it intersects every ...

We generalize the notions of F-regular and F-pure rings to those for a pair (R, a) of a ring R and an ideal a ⊂ R with real exponent t > 0, and investigate those properties. Those “F-singularities of pairs” correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that fo...

Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V whic...