Tridiagonal realization of the anti-symmetric Gaussian β\beta-ensemble

The Householder reduction of a member of the anti-symmetric Gaussian unitary ensemble gives an anti-symmetric tridiagonal matrix with all independent elements. The random variables permit the introduction of a positive parameter $\beta$, and the eigenvalue probability density function of the corresponding random matrices can be computed explicitly, as can the distribution of $\{q_i\}$, the first components of the eigenvectors. Three proofs are given. One involves an inductive construction based on bordering of a family of random matrices which are shown to have the same distributions as the anti-symmetric tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg integral theory. A second proof involves the explicit computation of the Jacobian for the change of variables between real anti-symmetric tridiagonal matrices, its eigenvalues and $\{q_i\}$. The third proof maps matrices from the anti-symmetric Gaussian $\beta$-ensemble to those realizing particular examples of the Laguerre $\beta$-ensemble. In addition to these proofs, we note some simple properties of the shooting eigenvector and associated Pr\"ufer phases of the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal transformation proof for both cases (Method III

Stigma in youth with Tourette's syndrome: a systematic review and synthesis

Tourette's syndrome (TS) is a childhood onset neurodevelopmental disorder, characterised by tics. To our knowledge, no systematic reviews exist which focus on examining the body of literature on stigma in association with children and adolescents with TS. The aim of the article is to provide a review of the existing research on (1) social stigma in relation to children and adolescents with TS, (2) self-stigma and (3) courtesy stigma in family members of youth with TS. Three electronic databases were searched: PsycINFO, PubMed and Web of Science. Seventeen empirical studies met the inclusion criteria. In relation to social stigma in rating their own beliefs and behavioural intentions, youth who did not have TS showed an unfavourable attitude towards individuals with TS in comparison to typically developing peers. Meanwhile, in their own narratives about their lives, young people with TS themselves described some form of devaluation from others as a response to their disorder. Self-degrading comments were denoted in a number of studies in which the children pointed out stereotypical views that they had adopted about themselves. Finally, as regards courtesy stigma, parents expressed guilt in relation to their children's condition and social alienation as a result of the disorder. Surprisingly, however, there is not one study that focuses primarily on stigma in relation to TS and further studies that examine the subject from the perspective of both the 'stigmatiser' and the recipient of stigma are warranted

Hypergeometric solutions to the q-Painlev\'e equation of type A4(1)A_4^{(1)}

We consider the q-Painlev\'e equation of type $A_4^{(1)}$ (a version of q-Painlev\'e V equation) and construct a family of solutions expressible in terms of certain basic hypergeometric series. We also present the determinant formula for the solutions.Comment: 16 pages, IOP styl