q -Bilattices

In the present paper, we consider the concepts of q-semilattices, q-lattices, and q-bilattices with unary operation and prove the existence of an epimorphism between q-bilattices of some varieties and the superproduct of two lattices.     

A characterization of the Grassmann embedding of H(q), with q even

In this note, we characterize the Grassmann embedding of H(q), q even, as the unique full embedding of H(q) in PG(12, q) for which each ideal line of H(q) is contained in a plane. In particular, we show that no such embedding exists for H(q), with q odd. As a corollary, we can classify all full polarized embeddings of H(q) in PG(12, q) with the property that the lines through any point are contained in a solid; they necessarily are Grassmann embeddings of H(q), with q even.     

On 1-Systems of Q(6, q), q Even

In this paper, a method is developed to study locally hermitian 1-systems of Q(6, q), q even, by associating a kind of flock in PG(4, q) to them. This method is applied to a known locally hermitian 1-system of Q(6, 2^2e ), which was discovered by Offer as a spread of the hexagon H(2^2e ). The results concerning this spread appear to be suitable for generalization and enable us to find new classes of 1-systems of Q(6, q), q even. We also prove that a locally hermitian 1-system of Q(6, q), q even, which is not contained in a 5-dimensional subspace, is semi-classical if and only if it belongs to the new classes we describe. Finally, from the new classes of 1-systems arise new classes of semipartial geometries.     

Caps in PG(n,q),q even,n⩾3

Letm′₂(3,q) be the largest value ofk(k<q ²+1) for which there exists a completek-cap in PG(3,q),q even. In this paper, the known upper bound onm′₂(3,q) is improved. We also describe a number of intervals, fork, for which there does not exist a completek-cap in PG(3,q),q even. These results are then used to improve the known upper bounds on the number of points of a cap in PG(n, q),q even,n?4.     

On (q 2 + q + 1)-sets of class [1, m,n]2 in PG(3, q)

A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q ² + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.     

Linear (q+1)-fold Blocking Sets in PG(2, q4)

A (q+1)-fold blocking set of size (q+1)(q⁴+q²+1) in PG(2, q⁴) which is not the union of q+1 disjoint Baer subplanes, is constructed     

On the Action of the Groups GL(n+1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) on PG(n, q t)

Let q be a prime power and let n ≥ 0, t ≥ 1 be integers. We determine the sizes of the point orbits of each of the groups GL(n + 1, q), PGL(n + 1, q), SL(n + 1, q) and PSL(n + 1, q) acting on PG(n, q ^t) and for each of these sizes (and groups) we determine the exact number of point orbits of this size.     

(∈, ∈∨q)-fuzzy subnear-rings and (∈, ∈∨q)-fuzzy ideals of near-rings

In this paper, we introduce the notions of (∈, ∈∨q)-fuzzy subnear-ring, (∈, ∈∨q)-fuzzy ideal and (∈, ∈∨q)-fuzzy quasi-ideal of near-rings and find more generalized concepts than those introduced by others. The characterization of such (∈, ∈ ∨q)-fuzzy ideals are also obtained.     

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.     

A spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd

This article presents a spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd. We prove that for every integer k in an interval of, roughly, size [q ²/4, 3q ²/4], there exists such a minimal blocking set of size k in PG(3, q), q odd. A similar result on the spectrum of minimal blocking sets with respect to the planes of PG(3, q), q even, was presented in Rößing and Storme (Eur J Combin 31:349–361, 2010). Since minimal blocking sets with respect to the planes in PG(3, q) are tangency sets, they define maximal partial 1-systems on the Klein quadric Q ⁺(5, q), so we get the same spectrum result for maximal partial 1-systems of lines on the Klein quadric Q ⁺(5, q), q odd.     

q—Type Sampling Theorems

This paper introduces q—versions of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon as well as Kramer’s analytic theorem. q—type band-limited signals are defined in terms of Jackson’s q—integral and their sampling representations are given. The sampling points are the zeros of a q—sine type function, derived by Bustoz and Cardoso (2001).     

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B ₃(q), C ₃(q), and D ₄(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.     

L q harmonic functions on graphs

We prove an analogue of Yau’s Caccioppoli-type inequality for nonnegative subharmonic functions on graphs. We then obtain a Liouville theorem for harmonic or nonnegative subharmonic functions of class L ^q , 1 ≤ q < ∞, on any graph, and a quantitative version for q > 1. Also, we provide counterexamples for Liouville theorems for 0 < q < 1.     

The ranks of primitive parabolic permutation representations of the simple groups B l (q), C l (q), and D l (q)

We determine the ranks of the permutation representations of the simple groups B _l (q), C _l (q), and D _l (q) on the cosets of the parabolic maximal subgroups.     

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 sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

On the q -Convolution on the Line

The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients.     

A characterisation of tangent subplanes of PG(2, q 3)

In “Barwick and Jackson (Finite Fields Appl. 18:93–107 2012)”, the authors determine the representation of Order-q-subplanes s and order-q-sublines of PG(2, q ³) in the Bruck–Bose representation in PG(6, q). In particular, they showed that an Order-q-subplanes of PG(2, q ³) corresponds to a certain ruled surface in PG(6, q). In this article we show that the converse holds, namely that any ruled surface satisfying the required properties corresponds to a tangent Order-q-subplanes of PG(2, q ³).