Collineation Research Articles

Point-primitive, line-transitive generalised quadrangles of holomorph type

Abstract Let G be a group of collineations of a finite thick generalised quadrangle Γ. Suppose that G acts primitively on the point set 𝒫 ${\mathcal{P}}$ of Γ, and transitively on the lines of Γ. We show that the primitive action of G on 𝒫 ${\mathcal{P}}$ cannot be of holomorph simple or holomorph compound type. In joint work with Glasby, we have previously classified the examples Γ for which the action of G on 𝒫 ${\mathcal{P}}$ is of affine type. The problem of classifying generalised quadrangles with a point-primitive, line-transitive collineation group is therefore reduced to the case where there is a unique minimal normal subgroup M and M is non-Abelian.

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.

Generalised Polygons Admitting a Point-Primitive Almost Simple Group of Suzuki or Ree Type

Let $G$ be a collineation group of a thick finite generalised hexagon or generalised octagon $\Gamma$. If $G$ acts primitively on the points of $\Gamma$, then a recent result of Bamberg et al. shows that $G$ must be an almost simple group of Lie type. We show that, furthermore, the minimal normal subgroup $S$ of $G$ cannot be a Suzuki group or a Ree group of type $^2\mathrm{G}_2$, and that if $S$ is a Ree group of type $^2\mathrm{F}_4$, then $\Gamma$ is (up to point-line duality) the classical Ree-Tits generalised octagon.

On the rigidity of the Figueroa replacement in PG(2, q 3)

Let ź0 be a subplane of order q of PG(2,q3) and let G be the copy of PGL(3,q) preserving ź0. The Figueroa plane Fig(q3) is constructed by replacing some parts of the lines of PG(2,q3) external to ź0 by suitable q-subgeometries of PG(2,q3). Moreover, Fig(q3) inherits G from PG(2,q3). We show that this is the unique replacement for the external lines to ź0 yielding a projective plane of order q3 admitting G as a collineation group.

Desargues configurations: minors and ambient automorphisms

We determine all possible minors of the Desargues configuration, their embeddings in projective spaces, and their ambient automorphism groups (i.e., the group of all projective collineations that leave the embedded configuration invariant) in Pappian projective spaces. Examples show that the latter problem becomes hard if the extra condition (Pappian) is dropped.

Regular $3$-dimensional parallelisms of $\mbox{{\rm PG}}(3,\mathbb R)$

In [8] the collineation groups of some known 5-, 4- and 3-dimensional topological regular parallelisms of $PG(3,\mathbb R)$ were determined. In the present article we concentrate on 3-dimensional regular parallelisms and prove: the 3-dimensional regular parallelisms are exactly those which can be constructed from generalized line stars, see [3]. We determine the collineation groups of 3-dimensional regular parallelisms and show that only group dimension 1 or 2 is possible. If the collineation group is 2-dimensional, then the parallelism is rotational which means that there is a rotation group $SO_2(\mathbb R)$ about some axis leaving the parallelism invariant. We give a construction method for the generalized line stars which induce these parallelisms.

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, qs) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, qs) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, qs) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).

Superbounce and loop quantum cosmology ekpyrosis from modified gravity

As is known, in modified cosmological theories of gravity many of the cosmologies which could not be generated by standard Einstein gravity, can be consistently described by $F(R)$ theories. Using known reconstruction techniques, we investigate which $F(R)$ theories can lead to a Hubble parameter describing two types of cosmological bounces, the superbounce model, related to supergravity and non-supersymmetric models of contracting ekpyrosis and also the Loop Quantum Cosmology modified ekpyrotic model. Since our method is an approximate method, we investigate the problem at large and small curvatures. As we evince, both models yield power law reconstructed $F(R)$ gravities, with the most interesting new feature being that both lead to accelerating cosmologies, in the large curvature approximation. The mathematical properties of the some Friedmann-Robertson-Walker spacetimes $M$, that describe superbounce-like cosmologies are also pointed out, with regards to the group of curvature collineations $CC(M)$.

Reduced three-dimensional nasal airway volume in treacher collins syndrome and its association with craniofacial morphology.

Airway insufficiency decreases quality of life and may be life threatening in patients with Treacher Collins syndrome. The authors calculated the three-dimensional nasal airway volume in patients with Treacher Collins syndrome to identify correlations between nasal airway volume and craniofacial morphology and provide guidance for surgical planning. Thirty nonoperated patients with Treacher Collins syndrome were compared with 35 unaffected age- and gender-matched controls. Anatomic variables of the cranial base, the maxilla complex, and internal diameters of nasal airway were compared between patients and control subjects using three-dimensional craniometric analyses. In the Treacher Collins group, the relation of craniofacial morphology to nasal airway volume was assessed separately. Statistical analyses were performed using independent sample t tests and Pearson correlation coefficient analyses. Nasal airway volume was decreased 38.6 percent in patients with Treacher Collins syndrome relative to controls (p = 0.001). A positive correlation of maxillary position and nasal airway volume was shown in Treacher Collins patients (r = +0.463, p = 0.013). Maxillary, nasal bone, and orbitale width were also positively correlated with nasal airway volume (r = +0.582, p = 0.001; r = +0.408, p = 0.035; and r = +0.677, p < 0.001, respectively). Shortened internal diameters of the nasal airway all positively correlated with nasal airway volume. Nasal airway volume is reduced in patients with Treacher Collins syndrome. Reduced projection of the maxilla and transverse maxillary deficiency are correlated with reduced nasal airway volume and are primarily responsible for obstruction of the nasal airway. Risk, II.

Parabolic unitals in a family of commutative semifield planes

In the commutative semifield planes constructed in Zhou and Pott (2013), we obtain a family of parabolic transitive unitals. For any unital in this family, we prove that there is a collineation group fixing it and acts sharply transitively on its affine points. We also consider its dual unital, the collinearity of its feet and show that as a design it is always resolvable. In particular, we give a necessary and sufficient condition under which a unital in our family is equivalent to a Ganley unital derived from a unitary polarity.

Symmetries of Type N Pure Radiation Fields

The geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector l are considered. These symmetries have been translated into the language of Newman-Penrose formalism for pure radiation (PR) type N fields. It is seen that for such fields, conformal, special conformal and homothetic motions degenerate to motion. The concept of free curvature, matter curvature and matter affine collineations have been discussed and the conditions under which PR type N fields admit such collineations have been obtained. Moreover, it is shown that the projective collineation degenerate to matter affine, special projective, conformal, special conformal, null geodesic and special null geodesic collineations. It is also seen that type N pure radiation fields admit Maxwell collineation along the propagation vector l.

Linear representations of subgeometries

The linear representation \(T_n^*(\mathcal {K})\) of a point set \(\mathcal {K}\) in a hyperplane of \(\mathrm {PG}(n+1,q)\) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations \(T_n^*(\mathcal {K})\) and \(T_n^*(\mathcal {K}')\), under a few conditions on \(\mathcal {K}\) and \(\mathcal {K}'\). First, we prove that an isomorphism between \(T_n^*(\mathcal {K})\) and \(T_n^*(\mathcal {K}')\) is induced by an isomorphism between the two linear representations \(T_n^*(\overline{\mathcal {K}})\) and \(T_n^*(\overline{\mathcal {K}'})\) of their closures \(\overline{\mathcal {K}}\) and \(\overline{\mathcal {K}'}\). This allows us to focus on the automorphism group of a linear representation \(T_n^*(\mathcal {S})\) of a subgeometry \(\mathcal {S}\cong \mathrm {PG}(n,q)\) embedded in a hyperplane of the projective space \(\mathrm {PG}(n+1,q^t)\). To this end we introduce a geometry \(X(n,t,q)\) and determine its automorphism group. The geometry \(X(n,t,q)\) is a straightforward generalization of \(H_{q}^{n+2}\) which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of \(X(n,t,q)\) as a coset geometry we extend this result and prove that \(X(n,t,q)\) and \(T_n^*(\mathcal {S})\) are isomorphic. Finally, we compare the full automorphism group of \(T^*_n(\mathcal {S})\) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space.

Collineation groups of topological parallelisms

Abstract Let P be a topological parallelism of the real projective 3-space PG3ℝ. We investigate the group AutP of all collineations of PG3ℝ leaving P invariant. We prove that AutP is a Lie group whose dimension is at most 6. The main result of the article says: if the dimension equals 6, then P is a Clifford parallelism. Furthermore, we show that there are no topological parallelisms in PG3ℝ with a 5-dimensional group.

Groups of projective planes with differing numbers of point and line orbits

We give examples of infinite projective planes with collineation groups having differing numbers of orbits on points and on lines, solving a problem posed by Cameron [9] and attributed to Kantor. Some of our examples are Desarguesian.

On Small Complete Arcs and Transitive A5-Invariant Arcs in the Projective Plane PG(2,q)

Let q be an odd prime power such that q is a power of 5 or (mod 10). In this case, the projective plane admits a collineation group G isomorphic to the alternating group A5. Transitive G-invariant 30-arcs are shown to exist for every . The completeness is also investigated, and complete 30-arcs are found for . Surprisingly, they are the smallest known complete arcs in the planes , and . Moreover, computational results are presented for the cases and . New upper bounds on the size of the smallest complete arc are obtained for .

On unitals in $$PG(2,q^2)$$ P G ( 2 , q 2 ) stabilized by a homology group

The linear collineation group of a classical unital of $$\mathrm{PG}(2,q^2)$$ PG ( 2 , q 2 ) contains a group of homologies of order $$q+1$$ q + 1 . In this paper we prove that if $$\mathcal{U }$$ U is a unital of PG $$(2,q^2)$$ ( 2 , q 2 ) stabilized by a homology group of order $$q+1$$ q + 1 and $$q$$ q is a prime number, then $$\mathcal{U }$$ U is classical.

On unitals in stabilized by a homology group

Abstract The linear collineation group of a classical unital of PG ( 2 , q 2 ) contains a subgroup of homologies of order q + 1 . If U is a unital of PG ( 2 , q 2 ) stabilized by a homology group of order q + 1 and q is a prime number, then U is classical.

On the genus of a cyclic curve over a finite field

Abstract Cyclic curves, i.e. curves fixed by a cyclic collineation group, play a central role in the investigation of cyclic arcs in Desarguesian projective planes. In this paper, the genus of a cyclic curve arising from a cyclic k -arc of Singer type is computed.

Hyperovals on [formula omitted] left invariant by a group of order [formula omitted

Three new infinite families of hyperovals on the generalized quadrangle H(3,q2) (q=ph, p a prime) of sizes 6(q+1), 12(q+1)2 (if q>7) and 6(q+1)2 (if p>3) are constructed. Furthermore they turn out to be invariant under the action of a linear collineation group of order 6(q+1)3 that fixes no point or line in a secant plane of H(3,q2). In particular the hyperovals of size 6(q+1)2 are transitive.

Antibiotic Killing Method In Dispute

Just when it seemed that scientists had a handle on how antibiotics kill bacteria and how their power could be amplified, new evidence has emerged that casts doubt on that approach. In 2007, James J. Collins and coworkers at Boston University presented evidence that all bacteria-killing antibiotics do their jobs by inducing the formation of reactive oxygen species (ROS) ( CE Cell, DOI: 10.1016/j.cell.2007.06.049). The reactive molecules damage bacterial cells. The three antibiotics the Collins group used—norfloxacin (a quinolone), ampicillin (a β-lactam), and kanamycin (an aminoglycoside)—target different bacterial functions. The scientists hoped to show that a common mechanism could provide a way to make bacteria more sensitive to antibiotics. But now it seems the research results may have been too good to be true. Two studies from other research groups report that the same antibiotics used in Collins’ study don’t kill bacteria with ROS after ...