Determination of generalized quadrangles with distinct elation points

In this paper, we classify the finite generalized quadrangles of order (s,t), s,t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L, M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42     

A generalized quadrangle of order (s, t) with center of transitivity is an elation quadrangle if s ≤ t

We show that a generalized quadrangle of order (s, t) with a center of transitivity is an elation generalized quadrangle if s ≤ t. In order to obtain this result, we generalize Frohardt’s result on Kantor’s conjecture from elation quadrangles to the more general case of quadrangles with a center of transitivity.      

Blocking sets of Hermitian generalized quadrangles

Some infinite families of minimal blocking sets on Hermitian generalized quadrangles are constructed.     

The automorphism group of Payne derived generalized quadrangles

We solve a long-standing open problem by proving that the automorphism group of any thick Payne derived generalized quadrangle with ambient quadrangle S a thick generalized quadrangle of order s, s?5 and odd, with a center of symmetry, is induced by the automorphism group of S.     

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q²) and H(4,q²) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q²). We also construct in H(3,q²), q=2^2h+1, h≥ 1, maximal partial spreads of size smaller than the size q²+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q²) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008     

Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), and . For , and , we are able to describe small weight codewords geometrically. For , q odd, and for , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]     

Covers of generalized quadrangles, 2. Kantor-Knuth covers and embedded ovoids

In this paper, which is a sequel to [12], we proceed with our study of covers and decomposition laws for geometries related to generalized quadrangles. In particular, we obtain a higher decomposition law for all Kantor-Knuth generalized quadrangles which generalizes one of the main results in [12]. In a second part of the paper, we study the set of all Kantor-Knuth ovoids (with given parameter) in a fixed finite parabolic quadrangle, and relate this set to embeddings of parabolic quadrangles into Kantor-Knuth quadrangles. This point of view gives rise to an answer of a question posed in [11].     

On the nonexistence of certain Hughes generalized quadrangles

In this paper, we prove that the Hermitian quadrangle is the unique generalized quadrangle Γ of order (q ², q ³) containing some subquadrangle of order (q ², q) isomorphic to such that every central elation of the subquadrangle is induced by a collineation of Γ. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

Combinatorial generalizations of generalized quadrangles of order (2, 2)

We study substructures of a projective space PG(n, 2) represented in terms of elementary combinatorics of finite sets, which generalize the Sylvester’s representation of the generalized quadrangle of order (2, 2). Their synthetic properties are established and automorphisms are characterized.      

Generalized quadrangles admitting a sharply transitive Heisenberg group

All known finite generalized quadrangles that admit an automorphism group acting sharply transitively on their point set arise by Payne derivation from thick elation generalized quadrangles of order s with a regular point. In these examples only two groups occur: elementary abelian groups of even order and odd order Heisenberg groups of dimension 3. In [2] the authors determined all generalized quadrangles admitting an abelian group with a sharply transitive point action. Here, we classify thick finite generalized quadrangles admitting an odd order Heisenberg group of dimension 3 acting sharply transitively on the points. In fact our more general result comes close to a complete solution of classifying odd order Singer p-groups.      

Affine ovoids and extensions of generalized quadrangles

A set Δ of vertices of a generalized quadrangle of order (s, t) is said to be a hyperoval if any line intersects Δ in either 0, or 2 points. A hyperoval Δ is called an affine ovoid if |Δ|=2st. It is well known that μ-subgraphs in triangular extensions of generalized quadrangles are hyperovals. In the present paper we prove that ifS is a triangular extension forGQ(s, t) with totally regular point graph Γ such that μ=2st, thens is even, Γ is an τ-antipodal graph of diameter 3 with τ=1+s/2, and eithers=2, ort=s+2. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 266–271, August, 2000.     

Ovoids of generalized quadrangles of order and Delsarte cocliques in related strongly regular graphs

We investigate strongly regular graphs for which Hoffman's ratio bound and Cvetcović's inertia bound are equal. This means that , where v is the number of vertices, k is the regularity, is the smallest eigenvalue, and is the multiplicity of . We show that Delsarte cocliques do not exist for all Taylor's 2‐graphs and for point graphs of generalized quadrangles of order for infinitely many q. For cases where equality may hold, we show that for nearly all parameter sets, there are at most two Delsarte cocliques.     

Slender partial quadrangles and their automorphisms

The partial quadrangle PQ(s, t, μ) is an incidence system consisting of points and lines in which every line contains s + 1 points, every point sits on t + 1 lines (two lines meet in at most one point), and the meet of the neighborhoods of any two non-adjacent points in the collinearity graph is a μ-coclique. We provide a classification for partial quadrangles PQ(s, t, μ) with t ⩽ 6, and study into their automorphisms. Supported by RFBR grant No. 05-01-00046 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 603–619, September–October, 2006.     

Intriguing sets in partial quadrangles

The point‐line geometry known as a partial quadrangle (introduced by Cameron in 1975) has the property that for every point/line non‐incident pair (P, ?), there is at most one line through P concurrent with ?. So in particular, the well‐studied objects known as generalized quadrangles are each partial quadrangles. An intriguing set of a generalized quadrangle is a set of points which induces an equitable partition of size two of the underlying strongly regular graph. We extend the theory of intriguing sets of generalized quadrangles by Bamberg, Law and Penttila to partial quadrangles, which gives insight into the structure of hemisystems and other intriguing sets of generalized quadrangles. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:217‐245, 2011     

Generalized quadrangles admitting PSL(2,q) × PSL(2,q)

In this paper, we classify the generalized quadrangles of order (s,t), s ≠ 1 ≠ t, which admit the natural action of PSL(2,s) × PSL(2,s) on a subGQ of order (s,1). This generalizes a recent result of J. De Kaey and H. Van Maldeghem 3 , by whom the classification was obtained for the case s = t. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

NOTES ON GENERALIZED EXPONENTIAL DICHOTOMIES

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Generalized monotonicity and generalized convexity

Generalized monotonocity of bifunctions or multifunctions is a rather new concept in optimization and nonsmooth analysis. It is shown in the present paper how quasiconvexity, pseudoconvexity, and strict pseudoconvexity of lower semicontinuous functions can be characterized via the quasimonotonicity, pseudomonotonicity, and strict pseudomonotonicity of different types of generalized derivatives, including the Dini, Dini-Hadamard, Clarke, and Rockafellar derivatives as well.This research was supported by the National Science Foundation of Hungary, Grant No. OTKA 1313/1991.