On the number of lines in planar spaces

SupposeS is a planar space withv>4 points and letq be the positive real number such thatv=q ³+q²+q+1. Assuming a weak non-degeneracy condition, we shall show thatS has at least (q²+1)(q²+q+1) lines with equality iffq is a prime power andS=PG(3,q).     

A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd

The incidence structure NQ⁺(3, q) has points the points not on a non-degenerate hyperbolic quadric Q⁺(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q⁺(3, q). It is easy to show that NQ⁺(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ⁺(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ⁺(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ⁺(3, q).     

On a Decomposition of Complete Graphs

It is shown, among other results, that for any prime power q, the complete graph on 1+q+q ²+q ³ vertices can be decomposed into a union of 1+q Siamese Strongly Regular Graphs S R G(1+q+q ²+q ³,q+q ²,q–1,q+1) sharing 1+q ² cliques of size 1+q. Acknowledgments.The authors are indebted to a referee for a very extensive report and for many suggestions which improved the presentation of the paper tremendously.AMS Subject Numbers: 05B05, 05B20, 05E30This work was completed while the first author was on sabbatical leave visiting Institute for studies in theoretical Physics and Mathematics, (IPM), in Tehran, Iran. Support and hospitality is appreciated. Supported by an NSERC operating grant.     

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

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     

Complete and Stable O(p+1)×O(q+1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space ℝ p+q+2

We classify the zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in the euclidean space ℝ ^p+q+2, p,q > 1, analyzing whether they are embedded and stable. The Morse index of the complete hypersurfaces show the existence of embedded, complete and globally stable zero scalar curvature O(p+1)×O(q+1)-invariant hypersurfaces in ℝ ^p+q+2, p+q≥ 7, which are not homeomorphic to ℝ ^p+q+1. Such stable examples provide counter-examples to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature. Mathematics Subject Classifications (2000): 53A10, 53C42,49005.     

Characterisations of A p+2 and L 2(2 q )

Let G be a transitive group of degree p+2 with p|G| where p ≧ 5 is a prime number, then (i) G is isomorphic to S _p+2 or A _p+2, if G has an element of order 4, (ii) G is isomorphic to L ₂(2 ^q ) or P Γ L ₂ (2 ^q ), if 2 ^q − 1=p is a Mersenne prime and G has no element of order 4.     

Optimal pseudo-cyclic codes and caps in PG (3,q)

Existence and uniqueness of pseudo-cyclic [q ²+1,q ²–3, 4]-codes over GF(q) are proved. Elliptic quadrics are characterized as those (q ²+1)-caps in PG(3,q) whose corresponding [q ²+1,q ²–3, 4]-codes are pseudo-cyclic.     

Varietà di Segre e ovoidi dello spazio polare Q+(7, q)

Summary In this paper, for q even, we construct an ovoid O ₃ and a spread S of the finite classical polar space Q⁺(7, q) determinated by a hyperbolic quadric Q⁺ of PG(7, q) such that there is a subgroup of PGO ₊ ⁸ (q) isomorphic to PGL₂(q ³), which maps O ₃ in itself and S in S and is 3-transitive on O ₃ and on S; for q>2, S is not a Desarguesian spread of Q⁺(7, q) and O ₃ is a Desarguesian ovoid.

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Varietà di Segre e ovoidi dello spazio polare Q⁺(7, q)

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Al Prof. Adriano Barlotti in occasione del suo 60^o compleanno     

Commuting polarities and maximal partial ovoids of H(4,q2)

In PG(4,q²), q odd, let Q(4,q²) be a non‐singular quadric commuting with a non‐singular Hermitian variety H(4,q²). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q₀ in a Baer subgeometry Σ₀ of PG(4,q²). It is proved that any maximal partial ovoid of H(4,q²) intersecting Q₀ in an ovoid has size at least 2(q²+1). Further, given an ovoid O of Q₀, we construct maximal partial ovoids of H(4,q²) of size q³+1 whose set of points lies on the hyperbolic lines 〈P,X〉 where P is a fixed point of O and X varies in O\{P}. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009     

Local geometry of L 1 ν L ∞ and L 1+L ∞

Extreme points of the unit sphere S (L ¹+L ^∞ ) of LL ¹+L ^∞ under the classical norm used in the interpolation theory were characterized in [8] and [11], while extreme points of S(L ¹ ∩ L ^∞) under the classical norm were characterized in [7]. In this paper extreme points of the unit sphere of L ¹+L ^∞ and L ¹ ∩ L ^∞ under the “dual” norms are characterized. Moreover, all the extreme points in L ¹ ∩ L ^∞ and L ¹+L ^∞ (under both kinds of norms) are examined if they are exposed, H-points, or strongly exposed. Smooth points in both these spaces for both the norms are also characterized. Finally, it is proved that in general the spaces L ^p +L ^q and L ^p ∩ L ^q are not isometric to Orlicz spaces, provided that 1<p<q<+∞. The paper was written while the first named author was visiting The University of Memphis The third named author is supported by KBN-Grant 2 PO3A 050 09.     

An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377–386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into R ^n+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L ^n/2 Ricci curvature norm in R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.     

On the pseudoachromatic index of the complete graph

Let q = 2^β be, for some β∈?, and let n = q² + q+ 1. By exhibiting a complete coloring of the edges of K_n, we show that the pseudoachromatic number ψ(G_n) of the complete line graph G_n = L(K_n)—or the pseudoachromatic index of K_n, if you will—is at least q³ + q. This bound improves the implicit bound of Jamison [Discrete Math 74 (1989), 99–115] which is given in terms of the achromatic number: ψ(G_n)≥α(G_n)≥q³ + 1. We also calculate, precisely, the pseudoachromatic number when q+ 1 extra points are added: Copyright © 2010 John Wiley Periodicals, Inc. J Graph Theory 66:89‐97, 2011     

On maximal curves in characteristic two

The genus g of an q-maximal curve satisfies g=g ₁≔q(q−1)/2 or . Previously, q-maximal curves with g=g ₁ or g=g ₂, q odd, have been characterized up to q-isomorphism. Here it is shown that an q-maximal curve with genus g ₂, q even, is q-isomorphic to the non-singular model of the plane curve ∑ _{i =1}} ^t y ^{q /2 i} =x ^{q +1}, q=2 ^t , provided that q/2 is a Weierstrass non-gap at some point of the curve. Received: 3 December 1998     

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).     

Study of Some Fundamental Projective Quadrics Associated with a Standard Pseudo-Hermitian Space H p,q

This self-contained short note deals with the study of the properties of some real projective compact quadrics associated with a a standard pseudo-hermitian space H _p,q , namely [(Q(p, q))\tilde], [(Q_2p+1,1)\tilde], [(Q_1,2q+1)\tilde], [(H_p,q)\tilde].  [(Q(p, q))\tilde]{\widetilde{Q(p, q)}, \widetilde{Q_{2p+1,1}}, \widetilde{Q_{1,2q+1}}, \widetilde{H_{p,q}}. \, \widetilde{Q(p, q)}} is the (2n – 2) real projective quadric diffeomorphic to (S ^2p–1 × S ^2q–1)/Z ₂. inside the real projective space P(E ₁), where E ₁ is the real 2n-dimensional space subordinate to H _p,q . The properties of [(Q(p, q))\tilde]{\widetilde{Q(p, q)}} are investigated. [(H_p,q)\tilde]{\widetilde{H_p,q}} is the real (2n – 3)-dimensional compact manifold-(projective quadric)- associated with H _p,q , inside the complex projective space P(H _p,q ), diffeomorphic to (S ^2p–1 × S ^2q–1)/S ¹. The properties of [(H_p,q)\tilde]{\widetilde{H_{p,q}}} are studied. [(Q_2p+1,1)\tilde]{\widetilde{Q_{2p+1,1}}} is a 2p-dimensional standard real projective quadric, and [(Q_1,2q+1)\tilde]{\widetilde{Q_{1,2q+1}}} is another standard 2q-dimensional projective quadric. [(Q_2p+1,1)\tilde] è[(Q_1,2q+1)\tilde]{\widetilde{Q_{2p+1,1}} \cup \widetilde{Q_{1,2q+1}}}, union of two compact quadrics plays a part in the understanding of the "special pseudo-unitary conformal compactification" of H _p,q . It is shown how a distribution y → D _y , where y ? H\{0},H{y \in H\backslash\{0\},H} being the isotropic cone of H _p,q allows to [(H_p+1,q+1)\tilde]{\widetilde{H_{p+1,q+1}}} to be considered as a "special pseudo-unitary conformal compactified" of H _p,q × R. The following results precise the presentation given in [1,c].     

Factorization of the Cyclotomic Polynomialx+ 1 over Finite Fields

The aim of this note is to show that the (well-known) factorization of the 2ⁿ⁺¹th cyclotomic polynomialx2ⁿ+ 1 over GF(q) withq≡ 1 (mod 4) can be used to prove the (more complicated) factorization of this polynomial over GF(q) withq≡ 3 (mod 4).     

Average CaseL∞-Approximation in the Presence of Gaussian Noise

We consider the average caseL_∞-approximation of functions fromC^r([0, 1]) with respect to ther-fold Wiener measure. An approximation is based onnfunction evaluations in the presence of Gaussian noise with varianceσ²>0. We show that the n th minimal average error is of ordern^{−(2r+1)/(4r+4)} ln^1/2 n, and that it can be attained either by the piecewise polynomial approximation using repetitive observations, or by the smoothing spline approximation using non-repetitive observations. This completes the already known results forL_q-approximation withq<∞ andσ0, and forL_∞-approximation withσ=0.     

On Boundary Regularity of the Navier–Stokes Equations

Abstract

We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L ^{p, q} with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.     

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).