11 research outputs found
On commutative association schemes and associated (directed) graphs
Let denote the Bose--Mesner algebra of a commutative -class
association scheme (not necessarily symmetric), and
denote a (strongly) connected (directed) graph with adjacency matrix . Under
the assumption that belongs to , in this paper, we describe the
combinatorial structure of .
Among else, we show that, if is a commutative -class
association scheme that is not an amorphic symmetric scheme, then we can always
find a (directed) graph such that the adjacency matrix of
generates the Bose--Mesner algebra of
Classification of flocks of the quadratic cone in PG(3,64)
Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q ≤ 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2, 64)
Pseudo-ovals of elliptic quadrics as Delsarte designs of association schemes
A pseudo-oval of a finite projective space over a finite field of odd order q is a configuration of equidimensional subspaces that is essentially equivalent to a translation generalised quadrangle of order (q^n,q^n) and a Laguerre plane of order (for some n). In setting out a programme to construct new generalised quadrangles, Shult and Thas asked whether there are pseudo-ovals consisting only of lines of an elliptic quadric Q^-(5,q) , non-equivalent to the classical example, a so-called pseudo-conic. To date, every known pseudo-oval of lines of Q^-(5,q) is projectively equivalent to a pseudo-conic. Thas characterised pseudo-conics as pseudo-ovals satisfying the perspective property, and this paper is on characterisations of pseudo-conics from an algebraic combinatorial point of view. In particular, we show that pseudo-ovals in Q^-(5,q) and pseudo-conics can be characterised as certain Delsarte designs of an interesting five-class association scheme. These association schemes are introduced and explored, and we provide a complete theory of how pseudo-ovals of lines of Q^-(5,q) can be analysed from this viewpoint
Distance-regular graphs with classical parameters that support a uniform structure: case
Let denote a finite, simple, connected, and
undirected non-bipartite graph with vertex set and edge set .
Fix a vertex , and define , where denotes the path-length
distance in . Observe that the graph is
bipartite. We say that supports a uniform structure with respect to
whenever has a uniform structure with respect to in the
sense of Miklavi\v{c} and Terwilliger \cite{MikTer}.
Assume that is a distance-regular graph with classical parameters
and diameter . Recall that is an integer such
that . The purpose of this paper is to study when
supports a uniform structure with respect to . We studied the case
in \cite{FMMM}, and so in this paper we assume . Let denote
the Terwilliger algebra of with respect to . Under an additional
assumption that every irreducible -module with endpoint is thin, we show
that if supports a uniform structure with respect to , then either
or , , and .Comment: arXiv admin note: substantial text overlap with arXiv:2305.0893
Neuroticism and Conscientiousness Moderate the Effect of Oral Medication Beliefs on Adherence of People with Mental Illness during the Pandemic
Background. After the declaration of the pandemic status in several countries, the continuity of face-to-face visits in psychiatric facilities has been delayed or even interrupted to reduce viral spread. Little is known about the personality factors associated with medication beliefs and adherence amongst individuals with mental illness during the COVID-19 pandemic. This brief report describes a preliminary naturalistic longitudinal study that explored whether the Big Five personality traits prospectively moderate the effects of medication beliefs on changes in adherence during the pandemic for a group of outpatients with psychosis or bipolar disorder. Methods. Thirteen outpatients undergoing routine face-to-face follow-up assessments during the pandemic were included (41 observations overall) and completed the Revised Italian Version of the Ten-Item Personality Inventory, the Beliefs about Medicines Questionnaire, the Morisky Medication Adherence Scale-8-item and the Beck Depression Inventory-II. Results. Participants had stronger concerns about their psychiatric medications rather than beliefs about their necessity, and adherence to medications was generally low. Participants who had more necessity beliefs than concerns had better adherence to medications. People scoring higher in Conscientiousness and Neuroticism traits and more concerned about the medication side effects had poorer adherence. Conclusions. These preliminary data suggest the importance of a careful assessment of the adherence to medications amongst people with psychosis/bipolar disorder during the pandemic. Interventions aimed to improve adherence might focus on patients' medication beliefs and their Conscientiousness and Neuroticism personality traits
Reconstructing a generalized quadrangle with a hemisystem from a class association scheme
In 2013, van Dam, Martin and Muzychuk constructed a cometric antipodal
class association scheme from a GQ of order , odd, which have
a hemisystem. In this paper we characterize this scheme by its Krein array. The
techniques which are used involve the triple intersection numbers introduced by
Coolsaet and Juri\v{s}i\'c
Reconstructing a generalized quadrangle from the Penttila-Williford class association scheme
Penttila and Williford constructed a class association scheme from a
generalized quadrangle with a doubly subtended subquadrangle. We show that an
association scheme with appropriate parameters and satisfying some assumption
about maximal cliques must be the Penttila-Williford scheme
Classification of flocks of the quadratic cone in PG(3,64)
Flocks are an important topic in the field of finite geometry, with many
relations with other objects of interest. This paper is a contribution to the
difficult problem of classifying flocks up to projective equivalence. We
complete the classification of flocks of the quadratic cone in PG(3,q) for q <=
71, by showing by computer that there are exactly three flocks of the quadratic
cone in PG(3,64), up to equivalence. The three flocks had previously been
discovered, and they are the linear flock, the Subiaco flock and the Adelaide
flock. The classification proceeds via the connection between flocks and herds
of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in
PG(2,64)
Classification of spreads of Tits quadrangles of order 64
Brown et al. provide a representation of a spread of the Tits quadrangle
, an oval of , even, in
terms of a certain family of ovals of . By combining this
representation with the Vandendriessche classification of hyperovals in
and the classification of flocks of the quadratic cone in
, recently given by the authors, in this paper, we classify
all the spreads of , an oval of , up to equivalence. These complete the classification of spreads of
for