Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order $(5^2,5)$ and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order $(s^2,s)$ for $s \le 11$. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle $H(3,q^2)$.Comment: slight revisions made following referee's reports, and included raw dat

Family carers' experience of caring for an older parent with severe and persistent mental illness

While the burden of caring for older people with chronic medical illness and dementia has been well documented, considerably less is known about how carers develop the strength and resilience to sustain this important role with older family members with mental illness. The aim of the study was to understand the lived experience of primary caregivers of older people with severe and persistent mental illness, and to explore what, if anything, helps to sustain them in their caring role. An interpretative phenomenological analysis approach was adopted, and qualitative interviews were used with 30 primary caregivers. Two overarching themes, and related subthemes, were abstracted from the data. First, caring is a difficult and demanding responsibility. It affects carers adversely, emotionally, physically, socially, and financially, and their lifestyle in general. This is reflected in three subthemes: (i) physically and emotionally draining; (ii) grieving about the loss; (iii) and adverse effects on lifestyle and social relationships. Second, carers develop resilience in caring, which helps sustain them in their role, as illustrated in three subthemes: (i) caring as purposeful and satisfying; (ii) harnessing social support from others; and (iii) purposefully maintaining their own well-being. Community mental health nurses have a key role in assessing carers’ needs and supporting them in their caring role

Simple groups, product actions, and generalized quadrangles

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group$G$preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on$G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that$G$cannot haveholomorph compoundO’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.</jats:p

Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201