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1.
This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s
2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3
e
),e3, is constructed. Then, by the duality betweenQ(4, 3
e
) and the classical generalized quadrangleW (3
e
), we get line spreads of PG(3, 3
e
) and hence translation planes of order 32e
. These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q
2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q
2),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q). 相似文献
2.
Paolo Terenzi 《Israel Journal of Mathematics》1998,104(1):51-124
lcub;x n rcub; with lcub;x n ,x* n rcub; biorthogonal is a “uniformly minimal basis with quasifixed brackets and permutations” of a Banach spaceX if lcub;x n rcub; andx* n rcub; are both bounded. Moreover, there is an increasing sequence lcub;q m rcub; of positive integers such that, for eachx′ ofX, settingq′(0)=0, $$x' = \sum\limits_{m = 0}^\infty { \sum\limits_{n = q'(m) + 1}^{q'(m + 1)} {x_{\pi '(n)}^ * (x')x_{\pi '(n)} ,} } $$ , where, for eachm≥1,q(m)+1≤q′(m)≤q(m+1) while $$\left\{ {\pi '(n)} \right\}_{n = q(m) + 1}^{q(m + 1)} is a permutation of \left\{ n \right\}_{n = q(m) + 1}^{q(m + 1)} .$$ . Then, for each subspaceY of a separable Banach spaceX, there exists a uniformly minimal basis with quasi-fixed brackets and permutations ofY, which can be extended to a uniformly minimal basis with quasi-fixed brackets and permutations ofX. 相似文献
3.
The maximum number m
2(n, q) of points in PG(n, q), n2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q
n–1
–1/2q
n–2 is obtained for q even when n4 and q>2. A necessary preliminary is an improved upper bound for m2(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that
and that m2(3, 4)=14. 相似文献
4.
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 相似文献
5.
The structure of
n– 1-sets inPG(n, q) with more thanq – 1 nuclei is investigated. It is shown that classification of these sets with the maximal numberq
n– 1-q
n– 2 of nuclei is equivalent to the classification of (q + l)-sets inPG(2,q) havingq –1 nuclei.Dedicated to Professer Walter Benz for his 60th birthday 相似文献
6.
Starting from a linear collineation of PG(2n–1,q) suitably constructed from a Singer cycle of GL(n,q), we prove the existence of a partition of PG(2n–1,q) consisting of two (n–1)-subspaces and caps, all having size (qn–1)/(q–1) or (qn–1)/(q+1) according as n is odd or even respectively. Similar partitions of quadrics or hermitian varieties into two maximal totally isotropic subspaces and caps of equal size are also obtained. We finally consider the possibility of partitioning the Segre variety
of PG(8,q) into caps of size q2+q+1 which are Veronese surfaces. 相似文献
7.
Klaus Metsch 《Journal of Geometry》2000,67(1-2):188-207
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). 相似文献
8.
Kenneth L. Clarkson Herbert Edelsbrunner Leonidas J. Guibas Micha Sharir Emo Welzl 《Discrete and Computational Geometry》1990,5(1):99-160
We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges boundingm cells in an arrangement ofn lines is (m
2/3
n
2/3 +n), and that it isO(m
2/3
n
2/3
(n) +n) forn unit-circles, where(n) (and later(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up toO(m
3/5
n
4/5
(n) +n). The same bounds (without the(n)-terms) hold for the maximum sum of degrees ofm vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees ofm vertices in an arrangement ofn spheres in three dimensions isO(m
4/7
n
9/7
(m, n) +n
2), in general, andO(m
3/4
n
3/4
(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances amongm points in three dimensions isO(m
3/2
(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.The research of the second author was supported by the National Science Foundation under Grant CCR-8714565. Work by the fourth author has been supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a research grant from the NCRD, the Israeli National Council for Research and Development. A preliminary version of this paper has appeared in theProceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988. 相似文献
9.
Daniel S. Moak 《Aequationes Mathematicae》1980,21(1):179-191
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. 相似文献
10.
J. Wu 《Monatshefte für Mathematik》1994,117(3-4):303-322
Let (n) be the number of all prime divisors ofn and (n) the number of distinct prime divisors ofn. We definev
q
(x)=|{nx(n)–(n)=q}|. In this paper, we give an asymptotic development ofv
q
(x); this improves on previous results. 相似文献
11.
The differential operatorly=y+q(x)y with periodic (antiperiodic) boundary conditions that are not strongly regular is studied. It is assumed thatq(x) is a complex-valued function of classC
(4)[0, 1] andq(0)q(1). We prove that the system of root functions of this operator forms a Riesz basis in the spaceL
2(0, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 558–563, October, 1998. 相似文献
12.
We show that the number of critical positions of a convex polygonal objectB moving amidst polygonal barriers in two-dimensional space, at which it makes three simultaneous contacts with the obstacles but does not penetrate into any obstacle isO(kn
s
(kn)) for somes6, wherek is the number of boundary segments ofB,n is the number of wall segments, and
s
(q) is an almost linear function ofq yielding the maximal number of breakpoints along the lower envelope (i.e., pointwise minimum) of a set ofq continuous functions each pair of which intersect in at mosts points (here a breakpoint is a point at which two of the functions simultaneously attain the minimum). We also present an example where the number of such critical contacts is (k
2
n
2), showing that in the worst case our upper bound is almost optimal.Work on this paper by the second author has been supported by Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. 相似文献
13.
In this paper we introduce and study a family An(q)\mathcal{A}_{n}(q) of abelian subgroups of GLn(q){\rm GL}_{n}(q) covering every element of GLn(q){\rm GL}_{n}(q). We show that An(q)\mathcal{A}_{n}(q) contains all the centralizers of cyclic matrices and equality holds if q>n. For q>2, we obtain an infinite product expression for a probabilistic generating function for |An(q)||\mathcal{A}_{n}(q)|. This leads to upper and lower bounds which show in particular that
c1q-n £ \frac|An(q)||GLn(q)| £ c2q-nc_1q^{-n}\leq \frac{|\mathcal{A}_n(q)|}{|\mathrm{GL}_n(q)|}\leq c_2q^{-n} 相似文献
14.
Paola De Vito 《Ricerche di matematica》2011,60(1):39-43
We prove that if q = p
h
, p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q
k
and 1 < k < n, determining N directions where
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