More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.     

Flat laguerre planes admitting 4-dimensional groups of automorphisms that fix at least two parallel classes

We take a further step toward the classification of all flat (2-dimensional) Laguerre planes of group dimension 4 by determining, up to isomorphism, all such Laguerre planes that admit 4-dimensional groups of automorphisms that fix at least two parallel classes. It is shown that these planes occur among those flat Laguerre planes of generalised shear type that admit a 3-dimesional kernel.     

Eight and seven point degenerations of Miquel's Theorem in Laguerre planes

For M?bius planes Schaeffer [9] has proved that all seven point degenerations of Miquel's Theorem characterize miquelian M?bius planes. For Laguerre planes we have several degenerations of Miquel's Theorem with eight and seven points. We prove that all except one of these degenerations characterize miquelian Laguerre planes. The remaining degeneration characterizes elation Laguerre planes. Received 14 September 2001; revised 1 September 2002.     

Locally projective spaces which satisfy the Bundle Theorem

We give a complete and short proof of KAHN's Theorem that every locally projective space (M,M) with dim M3 satisfying the Bundle Theorem is embeddable in a projective space. The central tool of KAHN's proof is the fact that (M,M) is locally projective, while we use mainly the Bundle Theorem.Dedicated to Professor Dr. H. Mäurer on the occasion of his sixtieth birthday     

A remark on two famous theorems concerning polynomials

A relation between Gauss-Lucas Theorem and Laguerre Theorem concerning the zeros of a polynomial in complex domain is discussed.     

Weighted permutation problems and Laguerre polynomials

The connection between integrals of products of Laguerre polynomials, power series coefficients of certain rational functions of several variables, and certain numbers of weighted permutation problems is investigated. A combinatorial proof of our main result would be very desirable since this could lead the way to more general result, q-analogs, and perhaps even a q-analog of MacMahon's Master Theorem.     

A family of flat Laguerre planes of Kleinewillingh?fer type IV.A

Just like Lenz–Barlotti classes reflect transitivity properties of certain groups of central collineations in projective planes, Kleinewillingh?fer types reflect transitivity properties of certain groups of central automorphisms in Laguerre planes. In the case of flat Laguerre planes, Polster and Steinke have shown that some of the conceivable types cannot exist, and they gave models for most of the other types. Only few types are still in doubt. Two of them are types IV.A.1 and IV.A.2, whose existence we prove here. In order to construct our models, we make systematic use of the restrictions imposed by the group generated by all central automorphisms guaranteed in type IV. With these models all simple Kleinewillingh?fer types with respect to Laguerre homologies and also with respect to Laguerre homotheties are now accounted for, and the number of open cases of Kleinewillingh?fer types (with respect to Laguerre homologies, Laguerre translations and Laguerre homotheties combined) is reduced to two.     

Sisterhoods of flat Laguerre planes

Every flat Laguerre plane of shear type over a pair of skew parabolae is related to a flat Laguerre plane of translation type over a pair of skew parabolae and vice versa. The relationship is defined using the connection between flat Laguerre planes and three-dimensional generalized quadrangles.Dedicated to Prof. H. R. Salzmann on his 65th birthday     

On the Kleinewillinghöfer types of flat Laguerre planes

Kleinewillinghöfer classified in [7] Laguerre planes with respect to central automorphisms and obtained a multitude of types. For finite Laguerre planes many of these types are known to be empty. In this paper we investigate the Kleinewillinghöfer types of flat Laguerre planes with respect to the full automorphism groups of these planes and completely determine all possible types of flat Laguerre planes with respect to Laguerre translations.     

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.     

Flat Circle Planes as Integrals of Flat Affine Planes

In a previous article (Arch. Math. {64} (1995), 75–85) we showed that flat Laguerre planes can be constructed by'integrating' flat affine planes. It turns out that'most' of the known flat Laguerre planes arise in this manner. In this paper we show that similar constructions are also possible in the case of the other two kinds of flat circle planes, that is, the flat Möbius planes and the flat Minkowski planes. In particular, we show that many of the known flat Möbius planes can be constructed by integrating a closed strip taken from a flat affine plane. We also show how flat Minkowski planes arise as integrals of two flat affine planes. All known flat Minkowski planes can be constructed in this manner.     

The apollonius problem in flat laguerre planes

The number of circles of a flat Laguerre plane touching three given circles or points depends only on the given geometric configuration but not on the Laguerre plane.Dedicated to Prof. J. Joussen on his 60^th birthday     

Semiclassical Topological Flat Laguerre Planes Obtained by Pasting Along A Circle

A new rather large family of locally compact 2-dimensional topological Laguerre planes is introduced here. This family consists exactly of those Laguerre planes which can be obtained by pasting together two halves of the classical real Laguerre plane along a circle suitably. Isomorphism classes and automorphism groups of these planes are determined. Together with [9] this gives a complete classification of all semicalssical topological flat Lguerre planes.     

The Two-Dimensional,Linear Orthotropic Plate

We First define a continuous extension of the Laquerre polynomials and give some properties of this continuous extension. Then we define a continuous Laguerre transform for square integrable functions, give some properties of this transform and give a sampling theorem that is similar to the well known Shannon-Whittaker Sampling Theorem for Fourier transform. The inverse of this transform is also given.     

A localization theorem for Laguerre expansions

Regularity properties of Laguerre means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localization theorem for Laguerre expansions.     

A proof of the Bundle Theorem for certain semimodular locally projective lattices of rank 4

The Bundle Theorem is proved for geometric locally projective lattices of rank 4 which for every given line (rank 2 element) do not contain too many lines that are on a common plane (rank 3 element) with this line, but on no common point (rank 1 element). By a result of J. Kahn (Math. Z.175 (1980), 219–247), this implies that these lattices are projectively embeddable.     

Three-dimensional quadrangles and flat Laguerre planes

Every three-dimensional generalized quadrangle can be constructed from flat Laguerre planes.Dedicated to Prof. H. Salzmann on his 60th birthday     

On weak-injective modules over integral domains

We show that a weak-injective module over an integral domain need not be pure-injective (Theorem 2.3). Equivalently, a torsion-free Enochs-cotorsion module over an integral domain is not necessarily pure-injective (Corollary 2.4). This solves a well-known open problem in the negative.In addition, we establish a close relation between flat covers and weak-injective envelopes of a module (Theorem 3.1). This yields a method of constructing weak-injective envelopes from flat covers (and vice versa). Similar relation exists between the Enochs-cotorsion envelopes and the weak dimension ?1 covers of modules (Theorem 3.2).     

Meromorphic quadratic differentials with prescribed singularities

We study the singular flat structure associated to any meromorphic quadratic differential on a compact Riemann surface to prove an existence theorem as follows. There exists a meromorphic quadratic differential with given orders of the poles and zeros and orientability or non orientability of the horizontal foliation, iff these prescribed topological data are admissible according to the Gauss-Bonnet Theorem, the Residue Theorem and certain conditions arising from local orientability or non orientablity considerations. Some few exceptional cases remain excluded. Thus, we generalize two previous results. One due to Masur & Smillie, which assumes that poles are at most simple; and a second one due to Muciño-Raymundo, which assumes that the horizontal foliation is orientable.Partially supported by DGAPA-UNAM and CONACYT 28492-E.     

Fourth Moment Theorems for Markov diffusion generators

Inspired by the insightful article [4], we revisit the Nualart–Peccati criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards Gamma and Beta distributions under moment conditions is also discussed.