On finite generalized quadrangles of even order

In this paper, we establish the following two results: (1) a skew translation generalized quadrangle of even order is a translation generalized quadrangle, (2) a generalized quadrangle of even order does not admit a point regular automorphism group. The first result confirms a conjecture of Payne (1975) based on earlier work of Ott (2021), and the second result confirms a conjecture of Ghinelli (1992).Comment: revised version, incorporated referees' comments, 20 page

Central aspects of skew translation quadrangles, I

Except for the Hermitian buildings $\mathcal{H}(4,q^2)$, up to a combination of duality, translation duality or Payne integration, every known finite building of type $\mathbb{B}_2$ satisfies a set of general synthetic properties, usually put together in the term "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify finite skew translation generalized quadrangles. In the first installment of the series, as corollaries of the machinery we develop in the present paper, (a) we obtain the surprising result that any skew translation quadrangle of odd order $(s,s)$ is a symplectic quadrangle; (b) we determine all skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (c) we develop a structure theory for root-elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root-elations for each member, and hence all members are "central" (the main property needed to control STGQs, as which will be shown throughout); (d) we solve the Main Parameter Conjecture for a class of STGQs containing the class of the previous item, and which conjecturally coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013

Ovoids and spreads of finite classical generalized hexagons and applications

One intuitively describes a generalized hexagon as a point-line geometry full of ordinary hexagons, but containing no ordinary n-gons for n<6. A generalized hexagon has order (s,t) if every point is on t+1 lines and every line contains s+1 points. The main result of my PhD Thesis is the construction of three new examples of distance-2 ovoids (a set of non-collinear points that is uniquely intersected by any chosen line) in H(3) and H(4), where H(q) belongs to a special class of order (q,q) generalized hexagons. One of these examples has lead to the construction of a new infinite class of two-character sets. These in turn give rise to new strongly regular graphs and new two-weight codes, which is why I dedicate a whole chapter on codes arising from small generalized hexagons. By considering the (0,1)-vector space of characteristic functions within H(q), one obtains a one-to-one correspondence between such a code and some substructure of the hexagon. A regular substructure can be viewed as the eigenvector of a certain (0,1)-matrix and the fact that eigenvectors of distinct eigenvalues have to be orthogonal often yields exact values for the intersection number of the according substructures. In my thesis I reveal some unexpected results to this particular technique. Furthermore I classify all distance-2 and -3 ovoids (a maximal set of points mutually at maximal distance) within H(3). As such we obtain a geometrical interpretation of all maximal subgroups of G2(3), a geometric construction of a GAB, the first sporadic examples of ovoid-spread pairings and a transitive 1-system of Q(6,3). Research on derivations of this 1-system was followed by an investigation of common point reguli of different hexagons on the same Q(6,q), with nice applications as a result. Of these, the most important is the alternative construction of the Hölz design and a subdesign. Furthermore we theoretically prove that the Hölz design on 28 points only contains Hermitian and Ree unitals (previously shown by Tonchev by computer). As these Hölz designs are one-point extensions of generalized quadrangles, we dedicate a final chapter to the characterization of the affine extension of H(2) using a combinatorial property

Central aspects of skew translation quadrangles, 1

Modulo a combination of duality, translation duality or Payne integration, every known finite generalized quadrangle except for the Hermitian quadrangles H(4, q2), is an elation generalized quadrangle for which the elation point is a center of symmetry-that is, is a "skew translation generalized quadrangle" (STGQ). In this series of papers, we classify and characterize STGQs. In the first installment of the series, (1) we obtain the rather surprising result that any skew translation quadrangle of finite odd order (s, s) is a symplectic quadrangle; (2) we determine all finite skew translation quadrangles with distinct elation groups (a problem posed by Payne in a less general setting); (3) we develop a structure theory for root elations of skew translation quadrangles which will also be used in further parts, and which essentially tells us that a very general class of skew translation quadrangles admits the theoretical maximal number of root elations for each member, and hence, all members are "central" (the main property needed to control STGQs, as which will be shown throughout); and (4) we show that finite "generic STGQs," a class of STGQs which generalizes the class of the previous item (but does not contain it by definition), have the expected parameters. We conjecture that the classes of (3) and (4) contain all STGQs

Characterization of some 4-gonal configurations of AHRENS-SZEKERES type

Motivated by the Ahrens-Szekeres-Quadrangles I present a variation of the 4-gonal families method of construction introduced by Kantor in 1980. Since a long time it has been known the relation between generalized quadrangles of order (s,s) and of order (s-1,s+1). A geometrical description of this interrelation was given by Payne in 1971 and rests on the notion of regular points or rather of regular lines. In this paper I develop these connections algebraically in the hope of getting more insight into them from the group theoretical point of view. In this way I am able to characterize two classes of known 4-gonal configurations

Characterization of some 4-gonal configurations of Ahrens–Szekeres type

Motivated by the Ahrens–Szekeres Quadrangles, we shall present a variation of the 4-gonal family method of construction introduced by Kantor in 1980. The relation between generalized quadrangles of order (s, s) and of order (s − 1, s + 1) has been known for a long time. A geometrical description of this interrelation was given by Payne in 1971 and rests on the notion of regular points or of regular lines. In this paper we wish to develop these connections algebraically in the hope of getting more insight into them from the group-theoretical point of view. In this way we are able to characterize two classes of known 4-gonal configurations

Risk Reduction Strategies for Assisted Conception in Women with Polycystic Ovary Syndrome

Superovulation in assisted conception can create Ovarian Hyperstimulation Syndrome (OHSS). Morbidity and even mortality that can occur with OHSS should be avoided by using the lowest risk and safest treatment strategy. Women with Polycystic Ovary Syndrome (PCOS) are at high risk of over response due to the ample number of antral follicles capable of responding to stimulation. Diagnosis of PCOS is based on a collection of subjective symptoms, signs and laboratory investigations. Anti-Mullerian Hormone (AMH), produced by the granulosa cells of the antral follicles, is elevated in women with PCOS. In a consecutive series of women presenting to an infertility clinic, the finding of increased AMH in those with PCOS was confirmed. Furthermore, AMH was shown to correlate with anovulation and hyperandrogenism. A single AMH value is interchangeable with any of the Rotterdam diagnostic criteria. Proposed values are 29pmol/L for polycystic ovarian morphology and 45pmol/L for either anovulation or hyperandrogenism using the generation II assay. Metformin has been shown to reduce the risk of OHSS in an agonist IVF cycle. The antagonist cycle is recommended for those at high risk of over-response. In a randomised double-blinded placebo controlled trial on 153 recruited patients; metformin was shown to have no effect on the incidence of OHSS in an antagonist cycle. There was no improvement in clinical pregnancy or live birth rate. The trial highlighted the discrepancy in clinical outcome between a White Caucasian and South Asian population. Avoidance of superovulation is an attractive option offered by in vitro maturation (IVM). A pilot study of 30 IVM cycles proved that immature oocytes can mature and fertilise in vitro at similar published rates. Unfortunately, no clinical pregnancies were created despite adequate transferred embryo quality. Although no incidence of OHSS, IVM appears to have been superseded by alternative approaches with replicable higher pregnancy rates