A Bose-Burton Theorem for Elliptic Polar Spaces

In bose&burton, Bose and Burton determined the smallest point sets of PG(d, q) that meet every subspace of PG(d, q) of a given dimension c. In this paper an equivalent result for quadrics of elliptic type is obtained. It states the folloing. For 0 c n - 1 the smallest point set of the elliptic quadric Q ^-(2n + 1, q) that meets every singular subspace of dimension c of Q ^-(2n + 1, q) has cardinality (q ⁿ⁺¹ + q ^c )(q ^n-c - 1)/(q - 1). Furthermore, the point sets of the smallest cardinality are classified.     

Lower bounds for the cardinality of minimal blocking sets in projective spaces

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r₂(q) be the number such that q+r₂(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most q^m+q^m−1+…+q+1+r₂(q)·(∑_i=2m−n′^m−1qⁱ) points for an integer n′ satisfying mn′2m, then the dimension of B is at most n′. If the dimension of B is n′, then the following holds. The cardinality of B equals q^m+q^m−1+…+q+1+r₂(q)(∑_i=2m−n′^m−1qⁱ). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q16 the set B is a Baer cone. If q is odd and |B|<q^m+q^m−1+…+q+1+r₂(q)(q^m−1+q^m−2), it follows from this result that the subspace generated by B has dimension at most m+1. Furthermore we prove that in this case, if , then B is an m-dimensional subspace or a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r₂(q)+1 in a plane skew to the vertex. For q=p^3h, p7 and q not a square we show this assertion for |B|q^m+q^m−1+…+q+1+q^2/3·(q^m−1+…+1).     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

Partial Ovoids in Classical Finite Polar Spaces

Ovoids in finite polar spaces are related to many other objects in finite geometries. In this article, we prove some new upper bounds for the size of partial ovoids in Q ^–(2n+1,q) and W(2n+ 1,q). Further, we give a combinatorial proof for the non-existence of ovoids of H(2n +1,q ²) for n>q ³.     

On 1-Blocking Sets in PG(n,q), n ≥ 3

In this article we study minimal1-blocking sets in finite projective spaces _PG(n,q),n 3. We prove that in PG(n,q ²),q = p ^h , p prime, p > 3,h 1, the second smallest minimal 1-blockingsets are the second smallest minimal blocking sets, w.r.t.lines, in a plane of PG(n,q ²). We also study minimal1-blocking sets in PG(n,q ³), n 3, q = p ^h, p prime, p > 3,q 5, and prove that the minimal 1-blockingsets of cardinality at most q 3 + q ² + q + ¹ are eithera minimal blocking set in a plane or a subgeometry PG(3,q).     

On the Diameter of Wenger Graphs

Let q be a prime power, the field of q elements, and n≥1 a positive integer. The Wenger graph W _n (q) is defined as follows: the vertex set of W _n (q) is the union of two copies P and L of (n+1)-dimensional vector spaces over , with two vertices (p ₁,p ₂,…,p _n+1)∈P and [l ₁,l ₂,…,l _n+1]∈L being adjacent if and only if l _i +p _i =p ₁ l _i−1 for 2≤i≤n+1. Graphs W _n (q) have several interesting properties. In particular, it is known that when connected, their diameter is at most 2n+2. In this note we prove that the diameter of connected Wenger graphs is 2n+2 under the assumption that 1≤n≤q−1.     

On constants in some inequalities for intermediate derivatives on a finite interval

Let L_q (1q<∞) be the space of functions f measurable on I=[−1,1] and integrable to the power q, with normL_∞ is the space of functions measurable on I with normWe denote by AC the set of all functions absolutely continuous on I. For nN, q[1,∞] we setW_n,q={f:f⁽ⁿ⁻¹⁾AC, f⁽ⁿ⁾L_q}.In this paper, we consider the problem of accuracy of constants A, B in the inequalities (1)|| f^(m)||_qA|| f||_p+B|| f^(m+k+1)||_r, mN, kW; p,q,r[1,∞], fW_m+k+1,r.     

Exponential Sums for O+(2n,q) and Their Applications

For a nontrivial additive character of the finite field with q elements and each positive integer r, the exponential sums ( ( trw )^r ) over w SO ⁺(2n,q) and over w O ⁺(2n,q) are considered. We show that both of them can be expressed as polynomials in q involving certain new exponential sums. Estimates on those new exponential sums are given. Also, we derive from these expressions the formulas for the number of elements w in SO ⁺(2n,q) and O ⁺(2n,q) with (trw) ^r = , for each in the finite field with q elements.     

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

Theq-gamma function forx<0

F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!) _q =1(1+q)(1+q+q ²)...(1+q+q ²+...+q ^n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x –1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of _q asq changes and showed that asq1^–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of _q (x) forx<0 and allq>0. In addition to some basic properties of _q , we will study the behavior of the sequence {x _n (q)} of critical points asn orq changes.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

On the flocks of Q+(3,q)

A complete characterization of the flocks of Q⁺(3, q) is given. As an application, it follows that if q is odd, q11, 23, 59, there exist no maximal exterior sets of Q⁺ (2n–1, q) for n>2.Paper written with partial financial support of M.P.I.; the authors are members of G.N.S.A.G.A. of C.N.R.     

On an Isomorphism Problem of Some Dual Hyperovals in PG(2d + 1, q) with q even

Let q = 2^l with l≥ 1 and d ≥ 2. We prove that any automorphism of the d-dimensional dual hyperoval over GF(q), constructed in [3] for any (d + 1)-dimensional GF(q)-vector subspace V in GF(qⁿ) with n≥ d + 1 and for any generator σ of the Galois group of GF(qⁿ) over GF(q), always fixes the special member X(∞). Moreover, we prove that, in case V = GF(q^d+1), two dual hyperovals and in PG(2d + 1,q), where σ and τ are generators of the Galois group of GF(q^d+1) over GF(q), are isomorphic if and only if (1) σ = τ or (2) σ τ = id. Therefore, we have proved that, even in the case q > 2, there exist non isomorphic d-dimensional dual hyperovals in PG(2d + 1,q) for d ≥ 3.     

On blocking sets of quadrics

We determine the three smallest blocking sets with respect to lines of the quadric Q(2n, q) withn 3 and the two smallest blocking sets with respect to lines of the quadric Q⁺(2n+1,q) withn 2. These results will be used in a forthcoming paper for determining the smallest blocking sets with respect to higher dimensional subspaces in the quadrics Q(2n, q) and Q⁺(2n+ 1, q).     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

Visibility queries in a polygonal region

In this paper, we consider the problem of computing the region visible to a query point located in a given polygonal domain. The polygonal domain is specified by a simple polygon with m holes and a total of n vertices. We provide two bounds on the complexity of this problem. One approach constructs a data structure with space complexity O(n²) in time O(n²lgn) and yields a query time of O((1+min(m,|V(q)|))lg²n+m+|V(q)|). Here, V(q) represents the set of vertices of the visibility polygon of a query point q, and |E| denotes the number of edges in the visibility graph. The other approach provides a data structure with space complexity O(min(|E|,mn)+n) in time O(T+|E|+nlgn) with the query time of O(|V(q)|lgn+m). Here, T is the time to triangulate the given polygonal region (which is O(n+mlg¹⁺m) for a small positive constant >0). In both of these approaches, O(m) additive factor in the query time is eliminated with an additional O((min(|E|,mn))²) space and an additional O(m(min(|E|,mn))²) preprocessing time.     

On the Spectrum of the Sizes of Maximal Partial Line Spreads in <Emphasis Type="Italic">PG</Emphasis>(2<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>), <Emphasis Type="Italic">n</Emphasis> ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51–63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75–91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97–106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135–150]. In [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N 5, are given. In PG(2n,q), n 3, the largest possible size for a partial line spread is q^2n-1+q^2n-3+...+q³+1. The largest size for the maximal partial line spreads constructed in [A. Gács and T. Sznyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123–129] is (q²ⁿ⁺¹–q)/(q²–1)–q³+q²–2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q 9.J. Eisfeld: Supported by the FWO Research Network WO.011.96NP. Sziklai: The research of this author was partially supported by OTKA D32817, F030737, F043772, FKFP 0063/2001 and Magyary Zoltan grants. The third author is grateful for the hospitality of Ghent University.     

Sublines of Prime Order Contained in the Set of Internal Points of a Conic

In [2] it was shown that if q ≥ 4n²−8n+2 then there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qⁿ), q odd, n≥ 3. In this article we improve this bound in the case where q is prime to , and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in PG(3,qⁿ), ovoids of Q(4,qⁿ), rank two commutative semifields, and eggs in PG(4n−1,q). AMS Classification:11T06, 05B25, 05E12, 51E15