Global smooth fibrations of R3 by oriented lines

A smooth fibration of ³ by oriented lines is given by a smoothunit vector field V on ³ all of whose integral curves are straightlines. Such a fibration is said to be nondegenerate if dV vanishesonly in the direction of V. Let be the space of oriented linesof ³ endowed with its canonical pseudo-Riemannian neutral metric.We characterize the nondegenerate smooth fibrations of ³ byoriented lines as the closed (in the relative topology) definiteconnected surfaces in . In particular, local conditions on imply the existence of a global fibration. Besides, for anysuch fibration the base space is diffeomorphic to the open discand the directions of the fibers form an open convex set ofthe two-sphere. We characterize as well, in a similar way, thesmooth (possibly degenerate) fibrations.     

Asymptotic behavior of maximum likelihood estimators in a branching diffusion model

In this paper the consistency and asymptotic normality of maximum-likelihood estimations for a super-critical branching diffusion model are obtained under certain conditions on its drift, variance and reproduction law. We proceeded by first studying the limit behavior of the Fisher information measure and related processes, and then verifying conditions established in Barndorff-Nielsen and Sørensen (Int stat Rev 62:133–165, 1994). This in turn uses the Martingale Law of Large Numbers as well as the Martingale Central Limit Theorem.     

Skew category, Galois covering and smash product of a -category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.

     

An implicit preconditioning strategy for large-scale generalized Sylvester equations

Large-scale generalized Sylvester equations appear in several important applications. Although the involved operator is linear, solving them requires specialized techniques. Different numerical methods have been designed to solve them, including direct factorization methods suitable for small size problems, and Krylov-type iterative methods for large-scale problems. For these iterative schemes, preconditioning is always a difficult task that deserves to be addressed. We present and analyze an implicit preconditioning strategy specially designed for solving generalized Sylvester equations that uses a preconditioned residual direction at every iteration. The advantage is that the preconditioned direction is built implicitly, avoiding the explicit knowledge of the given matrices. Only the effect of the matrix-vector product with the given matrices is required. We present encouraging numerical experiments for a set of different problems coming from several applications.     

Residual algorithm for large-scale positive definite generalized eigenvalue problems

In the positive definite case, the extreme generalized eigenvalues can be obtained by solving a suitable nonlinear system of equations. In this work, we adapt and study the use of recently developed low-cost derivative-free residual schemes for nonlinear systems, to solve large-scale generalized eigenvalue problems. We demonstrate the effectiveness of our approach on some standard test problems, and also on a problem associated with the vibration analysis of large structures. In our numerical results we use preconditioning strategies based on incomplete factorizations, and we compare with and without preconditioning with a well-known available package.     

Rings Defined by Extensions

In this article, we study subrings of the Ext-algebra of a graded module over a graded ring R. We show these subrings can be defined by equivalence relations on exact sequences over the ring. In particular, the shriek ring, R ^!, and the even part of an Ext-algebra of a d-Koszul algebra can be defined by equivalence relations on exact sequences.     

On Moduli Spaces for Abelian Categories

We show that if 𝒜 is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.