A non-miquelian Laguerre plane satisfying a theorem of miquelian type

This note shows that a theorem of miquelian type known as (M2) holds in a certain non miquelian Laguerre plane of shear type as defined by Löwen and Pfüller[1].Dedicated to Professor H. Karzel on the occasion of his 70th birthday     

Four-point characterizations of real inner product spaces

Generalized euclidean spaces have been characterized among metric spaces by the requirement that each member of certain classes of quadruples of points of the metric space be congruent to a quadruple of points of a euclidean space. The present paper strengthens earlier characterizations which only require the embedding of certain classes of quadruples which contain a linear triple and in which some three of the six distances between pairs of points are equal. These results generalize some similar characterizations of euclidean spaces among normed linear spaces. Received 4 January 1999; revised 12 August 2002.     

A Local Limit Theorem on Certain p-AdicGroups and Buildings

 We present a local limit theorem for a measure on the special linear group with entries in a local field. (Received 20 March 2000; in revised form 8 September 2000)     

The Gergonne and Nagel centers of a tetrahedron

The Gergonne center of a triangle is the intersection of the cevians through the points where the incircle touches the sides. This does not admit a direct generalization to tetrahedra since the cevians of a tetrahedron through the points where the insphere touches the faces are not necessarily concurrent. This article introduces an alternative definition of the Gergonne center that coincides with the previous definition for the triangle and that admits a generalization to tetrahedra. The same is done for the Nagel center. Received 4 February 2000; revised 1 May 2000.     

4-Transitivity and 6-figures in some Moufang-Klingenberg planes

In this paper, the Moufang-Klingenberg plane over a local alternative ring R of dual numbers is studied. It is shown that its collineation group is transitive on quadrangles. It is therefore shown that the coordinatization of these Moufang-Klingenberg planes is independent of the choice of the coordinatization quadrangle. Also, the concept of 6-figures is extended to these Moufang-Klingenberg planes and it is shown that any 6-figure corresponds to only one inversible m ∈ R.     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

Jordan Homomorphisms and Harmonic Mappings

 We show that each Jordan homomorphism R → R′ of rings gives rise to a harmonic mapping of one connected component of the projective line over R into the projective line over R′. If there is more than one connected component then this mapping can be extended in various ways to a harmonic mapping which is defined on the entire projective line over R. Received December 7, 2001; in revised form April 28, 2002 Published online January 7, 2003     

Remarks on my paper: packing of incongruent circles on the sphere

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.     

Projective resolutions associated to projections

In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal. Received: 22 February 1999     

Solid varieties of arbitrary type

A solid variety is an equational class in which every identity holds as a hyperidentity as well, meaning that it is satisfied not just by the fundamental operations but also by all terms of the appropriate arity. For type (2), an infinite number of solid varieties (of semigroups) are known, but for other types very few examples of solid varieties are known. In this paper we present several constructions which produce infinite chains of solid varieties. One construction generalizes the normalization of a variety, and gives a method to produce a chain of solid varieties from any given solid variety of type (n). The second construction generalizes the rectangular nilpotent varieties of type (2) to type (n). Finally, we use identities which are consequences of idempotency to construct an infinite chain of solid varieties of any fixed type. Received March 23, 2001; accepted in final form July 11, 2002. RID="h1" ID="h1"Research of the third author was supported by NSERC of Canada.     

Locally finite Lie algebras¶with unitary highest weight representations

In this paper we essentially classify all locally finite Lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. It turns out that these Lie algebras have many interesting relations to geometric structures such as infinite-dimensional bounded symmetric domains and coadjoint orbits of Banach–Lie groups which are strong K?hler manifolds. In the present paper we concentrate on the algebraic structure of these Lie algebras, such as the Levi decomposition, the structure of the almost reductive and locally nilpotent part, and the structure of the representation of the almost reductive algebra on the locally nilpotent ideal. Received: 2 August 2000 / Revised version: 10 January 2001     

Explaining the Gentzen–Takeuti reduction steps: a second-order system

Using the concept of notations for infinitary derivations we give an explanation of Takeuti's reduction steps on finite derivations (used in his consistency proof for Π¹ ₁-CA) in terms of the more perspicious infinitary approach from [BS88]. Received: 27 April 1999 / Published online: 21 March 2001     

Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces

Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers. Received: 28 November 2000 / Published online: 23 May 2002     

Symplectic structures¶on Heisenberg-type nilmanifolds

We use cohomological methods to study the existence of symplectic structures on nilmanifolds associated to two-step nilpotent Lie groups. We construct a new family of symplectic nilmanifolds with building blocks the quaternionic analogue of the Heisenberg group, determining the dimension of the space of all left invariant symplectic structures. Such structures can not be K?hlerian. Also, we prove that the nilmanifolds associated to H type groups are not symplectic unless they correspond to the classical Heisenberg groups. Received: 26 May 1999 / Revised version: 10 April 2000     

Triangle inequalities associated with the vertex angles and dihedral angles of a simplex and their applications

In this paper, a class of triangle inequalities associated with the vertex angles and dihedral angles of a simplex is given. As an application, we improve an inequality on the volume of the pedal simplex. Received 3 July 2000.     

The Number of Linearly Independent Binary Vectors with Applications to the Construction of Hypercubes and Orthogonal Arrays,Pseudo (<Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">s</Emphasis>)-Nets and Linear Codes

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1kn. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.This revised version was published online in October 2004 with a corrected Received date.     

Small Transitive Families of Dense Operator Ranges

In complex, separable, infinite-dimensional Hilbert space there exist 5 proper dense operator ranges with the property that every operator leaving each of them invariant is a scalar multiple of the identity. The algebra of operators leaving a pair of proper dense operator ranges invariant can have an infinite nest of invariant subspaces. A slight extension of Foiaş' Theorem shows that it can also have a non-trivial reducing subspace. Submitted: July 13, 2001? Revised: December 6, 2001.     

On the Napoleon-Torricelli configuration in Affine Cayley-Klein planes

There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ω_k-and anti-Ω_k-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived. The author is partially supported by grant VU-MI-204/2006.