Diagonal groups and arcs over groups

Partially supported by Simons Foundation Collaboration Grant 359872 and by Fundacaopara a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) grant PTDC/MAT-PUR/31174/2017. Australian Research Council Discovery Grant DP160102323.In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for m≥2, a set of m+1 partitions of a set Ω, any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if m=2), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have m+r partitions with r≥2, any m of which are minimal elements of a Cartesian lattice. If m=2, this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For m>2, things are more restricted. Any m+1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality m+r in the (m−1)-dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.Publisher PDFPeer reviewe

The geometry of diagonal groups

Part of the work was done while the authors were visiting the South China University of Science and Technology (SUSTech), Shenzhen, in 2018, and we are grateful (in particular to Professor Cai Heng Li) for the hospitality that we received.The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Groups, representations and applications: new perspectives (supported by EPSRC grant no.EP/R014604/1), where further work on this paper was undertaken. In particular we acknowledge a Simons Fellowship (Cameron) and a Kirk Distinguished Visiting Fellowship (Praeger) during this programme. Schneider thanks the Centre for the Mathematics of Symmetry and Computation of The University of Western Australia and Australian Research Council Discovery Grant DP160102323 for hosting his visit in 2017 and acknowledges the support of the CNPq projects Produtividade em Pesquisa (project no.: 308212/2019-3) and Universal (project no.:421624/2018-3).Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied in general. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over arbitrary groups, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our combinatorial axioms. However, for m≥3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in great profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set Ω, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions turn out to be equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group as the diagonal semilattice except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes, and is potentially of interest. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to the synchronization property of permutation groups.PostprintPeer reviewe

Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also explore geometric properties such as connectivity, firmness and transitivity conditions to determine when they are preserved under the quotienting operation. We show that the class of coset pregeometries, which contains all flag-transitive geometries, is closed under an appropriate quotienting operation.Comment: 26 pages, 5 figure

Bounds on the diameter of Cayley graphs of the symmetric group

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.Comment: 17 pages, 6 table

Abnormal expression of p27kip1 protein in levator ani muscle of aging women with pelvic floor disorders – a relationship to the cellular differentiation and degeneration

BACKGROUND: Pelvic floor disorders affect almost 50% of aging women. An important role in the pelvic floor support belongs to the levator ani muscle. The p27/kip1 (p27) protein, multifunctional cyclin-dependent kinase inhibitor, shows changing expression in differentiating skeletal muscle cells during development, and relatively high levels of p27 RNA were detected in the normal human skeletal muscles. METHODS: Biopsy samples of levator ani muscle were obtained from 22 symptomatic patients with stress urinary incontinence, pelvic organ prolapse, and overlaps (age range 38–74), and nine asymptomatic women (age 31–49). Cryostat sections were investigated for p27 protein expression and type I (slow twitch) and type II (fast twitch) fibers. RESULTS: All fibers exhibited strong plasma membrane (and nuclear) p27 protein expression. cytoplasmic p27 expression was virtually absent in asymptomatic women. In perimenopausal symptomatic patients (ages 38–55), muscle fibers showed hypertrophy and moderate cytoplasmic p27 staining accompanied by diminution of type II fibers. Older symptomatic patients (ages 57–74) showed cytoplasmic p27 overexpression accompanied by shrinking, cytoplasmic vacuolization and fragmentation of muscle cells. The plasma membrane and cytoplasmic p27 expression was not unique to the muscle cells. Under certain circumstances, it was also detected in other cell types (epithelium of ectocervix and luteal cells). CONCLUSIONS: This is the first report on the unusual (plasma membrane and cytoplasmic) expression of p27 protein in normal and abnormal human striated muscle cells in vivo. Our data indicate that pelvic floor disorders are in perimenopausal patients associated with an appearance of moderate cytoplasmic p27 expression, accompanying hypertrophy and transition of type II into type I fibers. The patients in advanced postmenopause show shrinking and fragmentation of muscle fibers associated with strong cytoplasmic p27 expression

Schreier type theorems for bicrossed products

We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} (G,*)$. Moreover, if we fix the group $H$ and the automorphism \sigma \in \Aut(H) then any $\sigma$-invariant isomorphism $H {}_{\alpha} \bowtie_{\beta} G \cong H {}_{\alpha'} \bowtie_{\beta'} G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given.Comment: 21 pages, final version to appear in Central European J. Mat

Expertise in surgical neuro-oncology. Results of a survey by the EANS neuro-oncology section

Introduction: Technical advances and the increasing role of interdisciplinary decision-making may warrant formal definitions of expertise in surgical neuro-oncology. Research question: The EANS Neuro-oncology Section felt that a survey detailing the European neurosurgical perspective on the concept of expertise in surgical neuro-oncology might be helpful. Material and methods: The EANS Neuro-oncology Section panel developed an online survey asking questions regarding criteria for expertise in neuro-oncological surgery and sent it to all individual EANS members. Results: Our questionnaire was completed by 251 respondents (consultants: 80.1%) from 42 countries. 67.7% would accept a lifetime caseload of &gt;200 cases and 86.7% an annual caseload of &gt;50 as evidence of neuro-oncological surgical expertise. A majority felt that surgeons who do not treat children (56.2%), do not have experience with spinal fusion (78.1%) or peripheral nerve tumors (71.7%) may still be considered experts. Majorities believed that expertise requires the use of skull-base approaches (85.8%), intraoperative monitoring (83.4%), awake craniotomies (77.3%), and neuro-endoscopy (75.5%) as well as continuing education of at least 1/year (100.0%), a research background (80.0%) and teaching activities (78.7%), and formal interdisciplinary collaborations (e.g., tumor board: 93.0%). Academic vs. non-academic affiliation, career position, years of neurosurgical experience, country of practice, and primary clinical interest had a minor influence on the respondents’ opinions. Discussion and conclusion: Opinions among neurosurgeons regarding the characteristics and features of expertise in neuro-oncology vary surprisingly little. Large majorities favoring certain thresholds and qualitative criteria suggest a consensus definition might be possible

Ireland: Submerged Prehistoric Sites and Landscapes

Evidence of Ireland's drowned landscapes and settlements presently comprises 50 sites spread across the entire island. These comprise mainly intertidal find spots or small collections of flint artefacts. A handful of fully subtidal sites are known, generally from nearshore regions and consisting, with one exception, of isolated single finds. Evidence of organic remains is also sparse, with the exception of Mesolithic and Neolithic wooden fish traps buried in estuarine sediments under Dublin. The relatively small number of sites is probably due to lack of research as much as taphonomic issues, and thus the current evidence hints at the potential archaeological record which may be found underwater. Such evidence could contribute to knowledge of the coastal adaptations and seafaring abilities of Ireland's earliest inhabitants. Nonetheless, taphonomic considerations, specifically relating to Ireland's history of glaciation, sea-level change and also modern oceanographic conditions likely limit the preservation of submerged landscapes and their associated archaeology. Realistically, the Irish shelf is likely characterised by pockets of preservation, which makes detection and study of submerged landscapes difficult but not impossible. A range of potential routes of investigation are identifiable, including site-scale archaeological survey, landscape-scale seabed mapping, archival research and community engagement