Bott's vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example:

--- differential groupoids,

--- principal bundles,

--- vector bundles,

--- actions of Lie groups on manifolds,

--- transversally complete foliations,

--- nonclosed Lie subgroups,

--- Poisson manifolds,

--- some complete closed pseudogroups.

We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

     

On the resolution of relative Thue equations

An efficient algorithm is given for the resolution of relative Thue equations. The essential improvement is the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.

Recently relative Thue equations have gained an important application, e.g., in computing power integral bases in algebraic number fields. The presented methods can surely be used to speed up those algorithms.

The method is illustrated by numerical examples.

     

Abelian categories, almost split sequences, and comodules

The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length.

(a) Each indecomposable injective has a simple subobject.

(b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented.

(c) The category has left almost split sequences.

     

Inequalities for products of spectral radii

It is shown that submultiplicative inequalities for spectral radii often imply supermultiplicative inequalities, and vice versa.

     

Error analysis for the computation of zeros of regular Coulomb wave function and its first derivative

In 1975 one of the coauthors, Ikebe, showed that the problem of computing the zeros of the regular Coulomb wave functions and their derivatives may be reformulated as the eigenvalue problem for infinite matrices. Approximation by truncation is justified but no error estimates are given there.

The class of eigenvalue problems studied there turns out to be subsumed in a more general problem studied by Ikebe et al. in 1993, where an extremely accurate asymptotic error estimate is shown.

In this paper, we apply this error formula to the former case to obtain error formulas in a closed, explicit form.

     

On transience of card shuffling

We present simple proofs of transience/recurrence for certain card shuffling models, that is, random walks on the infinite symmetric group.

     

Essentiality for Mönch type maps

A new fixed point theorem for Mönch maps on locally convex spaces is given. In addition, a continuation theorem for Mönch maps is presented.

     

A sampling theorem on homogeneous manifolds

We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points.

The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.

     

Fifteen consecutive integers with exactly four prime factors

We describe a successful search for a sequence of fifteen consecutive integers, each the product of exactly four prime factors. Fifteen is best possible.

     

A similarity principle for complex vector fields and applications

This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.

     

The curve of ``Prym canonical' Gauss divisors on a Prym theta divisor

The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties:

1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve;

2) those divisors in the Gauss system parametrized by the canonical curve are reducible.

In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties:

Theorem. For all smooth doubly covered nonhyperelliptic curves of genus , the general Prym canonical Gauss divisor is normal and irreducible.

Corollary. For all smooth doubly covered nonhyperelliptic curves of genus , the Prym canonical curve is not dual to the branch divisor of the Gauss map.

     

Standard realizations of crystal lattices via harmonic maps

An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.

     

On the Grushin operator and hyperbolic symmetry

Complexity of geometric symmetry for differential operators with mixed homogeniety is examined here. Sharp Sobolev estimates are calculated for the Grushin operator in low dimensions using hyperbolic symmetry and conformal geometry.

     

Cohomology of projective space seen by residual complex

A subcomplex of a residual complex on projective space is constructed for computing the cohomology modules of locally free sheaves. A constructive new proof of the Bott formula is given by explicitly exhibiting bases for the cohomology modules.

     

Galois embeddings for linear groups

A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear group covering.

     

A generalization of Bendixson's criterion

Bendixson's condition on the nonexistence of periodic solutions for planar ordinary differential equations is extended to higher dimensional ordinary differential equations with first integrals to preclude the existence of certain invariant Lipschitz compact submanifolds for those equations.

     

An example on the Mann iteration method for Lipschitz pseudocontractions

An example of a Lipschitz pseudocontractive map with a unique fixed point is constructed for which the Mann iteration sequence fails to converge. This resolves a long standing open problem.

     

A characterization of bilateral operator weighted shifts being Cowen-Douglas operators

It is characterized when a bilateral operator weighted shift is a Cowen-Douglas operator.

     

A useful semistability criterion

We provide a short proof for a semistability criterion which is crucial to the construction of master spaces which has drawn interest in recent research in Geometric Invariant Theory.

     

Explicit upper bounds for exponential sums over primes

We give explicit upper bounds for linear trigonometric sums over primes.