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).     

Constructions of intriguing sets of polar spaces from field reduction and derivation

The concepts of a tight set of points and an m-ovoid of a generalised quadrangle were unified recently by Bamberg, Law and Penttila under the title of intriguing sets. This unification was subsequently extended to polar spaces of arbitrary rank. The first part of this paper deals with a method of constructing intriguing sets of one polar space from those of another via field reduction. In the second part of this paper, we generalise an ovoid derivation of Payne and Thas to a derivation of intriguing sets.      

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.     

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order (q−1,q+1) obtained by Payne derivation from the classical symplectic quadrangle W(3,q). For q=pf with f?2 we obtain at least two nonisomorphic groups when p?5 and at least three nonisomorphic groups when p=2 or 3. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial p-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.     

Tight sets,weighted <Emphasis Type="Italic">m</Emphasis>-covers,weighted <Emphasis Type="Italic">m</Emphasis>-ovoids,and minihypers

Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q⁻(5,3), to 56-caps contained in this elliptic quadric Q⁻(5,3).      

Generalised dual arcs and Veronesean surfaces, with applications to cryptography

We start by defining generalised dual arcs, the motivation for defining them comes from cryptography, since they can serve as a tool to construct authentication codes and secret sharing schemes. We extend the characterisation of the tangent planes of the Veronesean surface in PG(5,q), q odd, described in [J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, New York, 1991], as a set of q²+q+1 planes in PG(5,q), such that every two intersect in a point and every three are skew. We show that a set of q²+q planes generating PG(5,q), q odd, and satisfying the above properties can be extended to a set of q²+q+1 planes still satisfying all conditions. This result is a natural generalisation of the fact that a q-arc in PG(2,q), q odd, can always be extended to a (q+1)-arc. This extension result is then used to study a regular generalised dual arc with parameters (9,5,2,0) in PG(9,q), q odd, where we obtain an algebraic characterisation of such an object as being the image of a cubic Veronesean.     

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.     

A functional equation arising from multiplication of quantum integers

For the quantum integer [n]_q=1+q+q²+?+qⁿ⁻¹ there is a natural polynomial multiplication such that [m]_q⊗_q[n]_q=[mn]_q. This multiplication leads to the functional equation f_m(q)f_n(q^m)=f_mn(q), defined on a given sequence of polynomials. This paper contains various results concerning the construction and classification of polynomial sequences that satisfy the functional equation, as well open problems that arise from the functional equation.     

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].     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Relative hemisystems on the Hermitian surface

Let S be a generalized quadrangle of order (q ²,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of S?S′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle $\mathcal{H}(3,q^{2})$ admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.     

Strong convergence of an iterative method for nonexpansive and accretive operators

Let X be a Banach space and A an m-accretive operator with a zero. Consider the iterative method that generates the sequence {xn} by the algorithm xn+1=αnu+(1−αn)Jrnxn, where {αn} and {rn} are two sequences satisfying certain conditions, and Jr denotes the resolvent ⁻¹(I+rA) for r>0. Strong convergence of the algorithm {xn} is proved assuming X either has a weakly continuous duality map or is uniformly smooth.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

On global attractor for nonlinear parabolic equations of m-Laplacian type

Existence and some regularity results of global attractor in Lq, q?1, for m-Laplacian type quasilinear parabolic equation with a perturbation like a(x)(α|u|u−β|u|u)+f(x) with α>β?0, a(x)?0 are proved. For the proofs Moser's technique is used extensively.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

Punctured plane partitions and the q-deformed Knizhnik-Zamolodchikov and Hirota equations

We consider partial sum rules for the homogeneous limit of the solution of the q-deformed Knizhnik-Zamolodchikov equation with reflecting boundaries in the Dyck path representation of the Temperley-Lieb algebra. We show that these partial sums arise in a solution of the discrete Hirota equation, and prove that they are the generating functions of τ²-weighted punctured cyclically symmetric transpose complement plane partitions where τ=−(q+q−1). In the cases of no or minimal punctures, we prove that these generating functions coincide with τ²-enumerations of vertically symmetric alternating sign matrices and modifications thereof.     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2-(mn²+1,mn,(n+1)(mn−1)) design starting from a Steiner system S(2,m+1,mn²+1) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.