Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Compact Generalised Quadrangles with Parameter 1 and Large Group of Automorphisms

If the group of automorphisms of a compact generalised quadrangle with parameter 1 has dimension at least 6, the quadrangle is the real symplectic quadrangle or its dual. There are nonclassical compact generalised quadrangles with parameter 1 whose group of automorphism has dimension 5.     

On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elationgroups on finite generalized quadrangles and contain a fullgroup of symmetries about some line through the base point.Such groups are related to the translation groups of translationtransversal designs with parameters depending on those of thequadrangles. Using results on the structure of p-groups which act as translationgroups on transversal designs and results on the index of theHughes subgroups of finite p-groups, we can show how restrictedthe structure of elation groups of finite generalized quadrangleswith symmetries is. One of our main results is that G is necessarily an elementaryabelian 2-group, provided that G has even cardinality. In particular,the elation generalized quadrangle coordinatized by G is a translationgeneralized quadrangle with G as translation group, that is,G contains full groups of symmetries about every line throughthe base point.     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

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.      

Linearly small elation quadrangles

We show that each elation generalized quadrangle with parameters (p, p), where p is a prime, is isomorphic to the symplectic quadrangle W(p) or its dual Q(4, p). Our results cover the more general case of linearly small elation generalized quadrangles. In particular, we obtain a characterization of the symplectic quadrangle over the field of complex numbers among compact connected quadrangles. We prove that every root elation quadrangle (Q, c, H _F ) is a skew translation quadrangle.     

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

The Fixity of Groups of Prime-Power Order

It is well known that the only groups of prime-power order whichcan act fixed point freely on a complex linear space are thecyclic or generalised quaternion groups. Given a positive integerf and a prime p exceeding f, we determine the p-groups whichhave a faithful complex representation such that the dimensionsof the spaces of fixed points of non-trivial elements are atmost f.     

Central aspects of skew translation quadrangles, 1

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles \(\mathcal {H}(4,q^2)\), is an elation generalized quadrangle for which the elation point is a center of symmetry—that is, is a “skew translation generalized quadrangle” (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are “central” (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite “generic STGQs,” a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs.     

Weighted intriguing sets of finite generalised quadrangles

We construct and analyse interesting integer valued functions on the points of a generalised quadrangle which lie in the orthogonal complement of a principal eigenspace of the collinearity relation. These functions generalise the intriguing sets introduced by Bamberg et al. (Combinatorica 29(1):1?C17, 2009), and they provide the extra machinery to give new proofs of old results and to establish new insight into the existence of certain configurations of generalised quadrangles. In particular, we give a geometric characterisation of Payne??s tight sets, we give a new proof of Thas?? result that an m-ovoid of a generalised quadrangle of order (s,s ²) is a hemisystem, and we give a bound on the values of m for which it is possible for an m-ovoid of the four dimensional Hermitian variety to exist.     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

Norm statistics and the complexity of clustering problems

In this paper we introduce a new class of clustering problems. These are similar to certain classical problems but involve a novel combination of ?_p-statistics and ?_q norms. We discuss a real world application in which the case p=2 and q=1 arises in a natural way. We show that, even for one dimension, such problems are NP-hard, which is surprising because the same 1-dimensional problems for the ‘pure’ ?₂-statistic and ?₂ norm are known to satisfy a ‘string property’ and can be solved in polynomial time. We generalize the string property for the case p=q. The string property need not hold when q≤p−1 and we show that instances may be constructed, for which the best solution satisfying the string property does arbitrarily poorly. We state some open problems and conjectures.     

On a conjecture of Brouwer involving the connectivity of strongly regular graphs

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components.We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called Δ-spaces are counterexamples to Brouwer?s Conjecture. Using J.I. Hall?s characterization of finite reduced copolar spaces, we find that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field Fq (for any q prime power), and the strongly regular graphs constructed from the hyperbolic quadrics O⁺(2r,2) and from the elliptic quadrics O⁻(2r,2) over the field F₂, respectively, are counterexamples to Brouwer?s Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall?s characterization theorem for Δ-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of Δ-spaces and thus, yield other counterexamples to Brouwer?s Conjecture.We prove that Brouwer?s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles GQ(q,q) graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue −2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components.     

Symmetries of BLT-Sets

We do the tentative beginnings of a study of BLT-sets of generalised quadrangles via their symmetries. In particular, the study of whorls about a line leads us to hyperbolic reflections preserving a BLT-set of Q(4, q).     

A note on isoparametric generalized quadrangles

We generalize a result of Kramer, see [7, 10.7 and 10.10], on generalized quadrangles associated with isoparametric hypersurfaces of Clifford type to Tits buildings of type C₂ derived from arbitrary isoparametric hypersurfaces with four distinct principal curvatures in spheres: two distinct points p and q of a generalized quadrangle associated with an isoparametric hypersurface in the unit sphere of a Euclidean vector space can be joined by a line K if and only if (p − q)/||p − q|| is a line. This line is orthogonal to K. Dually, two distinct lines L and K intersect if and only if (L − K)/||L − K|| is point. Received: 14 October 2005     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

Ordering groups and validity in lattice-ordered groups

An inductive characterization is given of the subsets of a group that extend to the positive cone of a right order on the group. This characterization is used to relate validity of equations in lattice-ordered groups (?-groups) to subsets of free groups that extend to the positive cone of a right order. As a consequence, new proofs are obtained of the decidability of the word problem for free ?-groups and generation of the variety of ?-groups by the ?-group of automorphisms of the real line. An inductive characterization is also given of the subsets of a group that extend to the positive cone of an order on the group. In this case, the characterization is used to relate validity of equations in varieties of representable ?-groups to subsets of relatively free groups that extend to the positive cone of an order.     

On codewords in the dual code of classical generalised quadrangles and classical polar spaces

In [J.L. Kim, K. Mellinger, L. Storme, Small weight codewords in LDPC codes defined by (dual) classical generalised quadrangles, Des. Codes Cryptogr. 42 (1) (2007) 73-92], the codewords of small weight in the dual code of the code of points and lines of Q(4,q) are characterised. Inspired by this result, using geometrical arguments, we characterise the codewords of small weight in the dual code of the code of points and generators of Q⁺(5,q) and H(5,q²), and we present lower bounds on the weight of the codewords in the dual of the code of points and k-spaces of the classical polar spaces. Furthermore, we investigate the codewords with the largest weights in these codes, where for q even and k sufficiently small, we determine the maximum weight and characterise the codewords of maximum weight. Moreover, we show that there exists an interval such that for every even number w in this interval, there is a codeword in the dual code of Q⁺(5,q), q even, with weight w and we show that there is an empty interval in the weight distribution of the dual of the code of Q(4,q), q even. To prove this, we show that a blocking set of Q(4,q), q even, of size q²+1+r, where 0<r<(q+4)/6, contains an ovoid of Q(4,q), improving on [J. Eisfeld, L. Storme, T. Sz?nyi, P. Sziklai, Covers and blocking sets of classical generalised quadrangles, Discrete Math. 238 (2001) 35-51, Theorem 9].     

On constructing normal and non-normal Cayley graphs

Bamberg and Giudici (2011) showed that the point graphs of certain generalised quadrangles of order $(q?1,q+1)$, where $q=p^k$ is a prime power with $p\ge 5$, are both normal and non-normal Cayley graphs for two isomorphic groups. We call these graphs BG-graphs. In this paper, we show that the Cayley graphs obtained from a finite number of BG-graphs by Cartesian product, direct product, and strong product also possess the property of being normal and non-normal Cayley graphs for two isomorphic groups.