Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Algebraic K-theory of non-linear projective spaces

In the spirit of “The Fundamental Theorem for the algebraic K-theory of spaces: I” (J. Pure Appl. Algebra 160 (2001) 21–52) we introduce a category of sheaves of topological spaces on n-dimensional projective space and present a calculation of its K-theory, a “non-linear” analogue of Quillen's isomorphism K_i(P_Rⁿ)₀ⁿK_i(R).     

Subregular Spreads of PG (5, 2e)

The theory of subregular spreads of PG(3,q) was developed by R.H. Bruck (1969, in “Combinatorial Mathematics and Its Applications,” Chap. 27, pp. 426–514. Univ. of North Carolina Press, Chapel Hill). An extension of these results was provided to the higher-dimensional case by the author (1998, Finite Fields Appl.4, 362–380); however, examples of such spreads were only constructed in PG(5, q) for q odd. In this paper, we give a construction of subregular spreads for PG(5, q), where q is even.     

Mixed partitions of PG(3,q)

A mixed partition of PG(2n−1,q²) is a partition of the points of PG(2n−1,q²) into (n−1)-spaces and Baer subspaces of dimension 2n−1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n−1,q²) can be used to construct a (2n−1)-spread of PG(4n−1,q) and hence a translation plane of order q²ⁿ. In this paper, we provide several new examples of such mixed partitions in the case when n=2.     

Switching Sets Regolari E Cubiche Rigate Di P G (5, q ) Piani Di Tralsazione Ad Essi Associati G (5, q ) Piani Di Tralsazione Ad Essi Associati

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ. The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ⊂ Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ − Φ) ∪ Φ′. The planes Π, of orderq ³, are a generalization of the finite Hall planes.     

An n-dimensional Wiener-Plancherel formula

The n-dimensional analogues of Wiener's “s-function” and difference operator associated to rectangular limit processes are defined. The Wiener-Plancherel formula is proved in this context for functions having bounded quadratic means.     

Linear representation of a class of projective planes in a four dimensional projective space

Summary The theory of linear representations of projective planes developed by Bruck and one of the authors (Bose) in two earlier papers [J. Algebra1 (1964), pp. 85–102 and4 (1966), pp. 117–172] can be further extended by generalizing the concept of incidence adopted there. A linear representation is obtained for a class of non-Desarguesian projective planes illustrating this concept of generalized incidence. It is shown that in the finite case, the planes represented by the new construction are derived planes in the sense defined by Ostrom [Trans. Amer. Math Soc.111 (1964), pp. 1–18] and Albert [Boletin Soc. Mat. Mex,11 (1966), pp, 1–13] of the dual of translation planes which can be represented in a 4-space by the Bose-Bruck construction. An analogous interpretation is possible for the infinite case. This research was sponsored by the National Science Foundation under Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-68-1406. This research was conducted while the author was visiting professor at the University of North Carolina at Chapel Hill. His research was also partially supported by C.N.R. Entrata in Redazione il 28 maggio 1970.     

Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

Piecewise regular spreads

A spread $\cal S$ of the real projective 3-space PG(3,?) is called piecewise regular, if, roughly speaking, the Klein image of $\cal S$ is composed of two elliptic caps and z elliptic zones (z ∈ {0,1,2,…}); we say that $\cal S$ is of segment number z + 2. We use piecewise regular spreads in order to give explicit examples of rigid and hyperrigid spreads. A spread $\cal T$ of a projective 3-space II is called rigid, if the only collineation of II leaving $\cal T$ invariant is the identity. A rigid spread ? is said to be hyperrigid, if there is no duality of II leaving ? invariant. We exhibit a 3-parameter family S″ of rigid piecewise regular spreads of segment number 4 and show that S″ contains spreads which represent non-isomorphic rigid 4-dimensional translation planes. Finally, we construct a 7-parameter family H of explicit examples of hyperrigid piecewise regular spreads of segment number 5. In H there are at least four spreads which represent mutually non-isomorphic rigid 4-dimensional translation planes.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified. Received 10 February 1997; in final form 19 December 1997     

Projective Planes, Coverings and a Network Problem

A problem concerning packet switched networks leads to a natural combinatorial design problem, whose solution involves (k, n)-arcs in projective planes, 3-dimensional linear codes, the theory of fractional matchings and designs which approximate projective planes.     

Note on Weakly n -Dimensional Spaces

 Weakly n-dimensional spaces were first distinguished by Karl Menger. In this note we shall discuss three topics concerning this class of spaces: universal spaces, products, and the sum theorem. We prove that there is a universal space for the class of all weakly n-dimensional spaces, present a simpler proof of Tomaszewski’s result about the dimension of a product of weakly n-dimensional spaces, and show that there is an n-dimensional space which admits a pairwise disjoint countable closed cover by weakly n-dimensional subspaces but is not weakly n-dimensional itself.     

The invariant polynomials characterizing U(n) tensor operators p, q,…, q, 0,…, 0 having maximal null space

We make use of the “path sum” function to prove that the family of stretched operator functions characterized by the operator irrep labels p,q,…,q, 0,…, 0 satisfy a pair of general difference equations. This family of functions is a generalization of Milne's p,q,…,q, 0, functions for U(n) and Biedenharn and Louck's p,q, 0 functions for U(3). The fact that this family of stretched operator functions are polynomials follows from a detailed study of their symmetries and zeros. As a further application of our general difference equations and symmetry properties we give an explicit formula for the polynomials characterized by the operator irrep labels p, 1, 0,…, 0.     

Ovoids and translation planes revisited

Kantor has previously described the translation planes which may be obtained by projecting sections of ovoids in ⁺(8, q)-spaces to ovoids in corresponding ⁺(6, q)-spaces. Since the Klein correspondence associates spreads in 4-dimensional vector spaces with ovoids in ⁺(6, q)-spaces, there are corresponding translation planes of order q ² and kernel containing GF(q). In this article, we revisit some of these translation planes and give some presentations of the spreads. Motivated by various properties of the planes, we study, in general, translation planes which admit certain homology groups and/or elation groups. In particular, we develop new constructions of projective planes of Lenz-Barlotti class II-1.Finally, we show how certain projective planes of order q ² of Lenz-Barlotti class II-1 may be considered equivalent to flocks of quadratic cones in PG(3, q).This work was partially supported by NSF grant DMS-8800843.     

Regulus-free Spreads of PG(3,q)

An old conjecture of Bruck and Bose is that every spread of =PG(3,q) could be obtained by starting with a regular spread and reversing reguli. Although it was quickly realized that this conjecture is false, at least forq even, there still remains a gap in the spaces for which it is known that there are spreads which are regulus-free. In several papers Denniston, Bruen, and Bruen and Hirschfeld constructed spreads which were regulus-free, but none of these dealt with the case whenq is a prime congruent to one modulo three. This paper closes that gap by showing that for any odd prime powerq, spreads ofPG(3,q) yielding nondesarguesian flag-transitive planes are regulus-free. The arguments are interesting in that they are based on elementary linear algebra and the arithmetic of finite fields.Dedicated to Hanfried Lenz on the occasion of his 80th birthdayThis work was partially supported by NSA grant MDA 904-95-H-1013.This work was partially supported by NSA grant MDA 904-94-H-2033.     

Connections in fiberings associated with the Grassman manifold and the space of centered planes

In the present paper we study connections in the fiberings associated with the Grassmann manifold and the space of the centered planes. The work is related to the studies in differential geometry. In the paper, we use the method of continuations and scopes of G. F. Laptev which generalizes the moving frame method and the exterior forms method of Cartan; the method depends on calculation of exterior differential forms. In the paper, we develop a new method of research in Grassman manifolds and some generalization of the method which includes the theory of the induced connections of the spaces of planes and centered planes in the n-dimensional projective space.     

On the Spectra of Certain Graphs Arising from Finite Fields

Cayley graphs on a subgroup ofGL(3,p),p>3 a prime, are defined and their properties, particularly their spectra, studied. It is shown that these graphs are connected, vertex-transitive, nonbipartite, and regular, and their degrees are computed. The eigenvalues of the corresponding adjacency matrices depend on the representations of the group of vertices. The “1-dimensional” eigenvalues can be completely described, while a portion of the “higher dimensional” eigenfunctions are discrete analogs of Bessel functions. A particular subset of these graphs is conjectured to be Ramanujan and this is verified for over 2000 graphs. These graphs follow a construction used by Terras on a subgroup ofGL(2,p). This method can be extended further to construct graphs using a subgroup ofGL(n, p) forn≥4. The 1-dimensional eigenvalues in this case can be expressed in terms of the 1-dimensional eigenvalues of graphs fromGL(2,p) andGL(3,p); this part of the spectra alone is sufficient to show that forn≥4, the graphs fromGL(n, p) are not in general Ramanujan.     

Linking (n − 2)-dimensional panels inn-space II: (n − 2, 2)-frameworks and body and hinge structures

An (n – 1, 2)-framework inn-space is a structure consisting of a finite set of (n – 2)-dimensional panels and a set of rigid bars each joining a pair of panels using ball joints. A body and hinge (or (n + 1,n – 1)-) framework inn-space consists of a finite set ofn-dimensional bodies articulated by a set of (n – 2)-dimensional hinges, i.e., joints in 2-space, line hinges in 3-space, plane-hinges in 4-space, etc. In this paper we characterize the graphs of all rigid (n – 1, 2)- and (n + 1,n – 1)-frameworks inn-space. Rigidity here is statical rigidity or equivalently infinitesimal rigidity.     

G-Loops and Permutation Groups

A G-loop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about G-loops. In particular, we study G-loops of order pq, where p < q are primes and p  (q − 1). In the case p = 3, the only G-loop of order 3q is the group of order 3q. The notion “G-loop” splits naturally into “left G-loop” plus “right G-loop.” There exist non-group right G-loops and left G-loops of order n iff n is composite and n > 5.