The Hasse Norm Principle For Biquadratic Extensions

We give an asymptotic formula for the number of biquadratic extensions of the rationals of bounded discriminant that fail the Hasse norm principle.Comment: 19 pages. Proof of Theorem 1 improved/simplified. Accepted by Journal de Th\'eorie des Nombres de Bordeau

A Positive Proportion of Hasse Principle Failures in a Family of Ch\^atelet Surfaces

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop an asymptotic formula for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.Comment: 13 pages, comments welcome. To appear in International Journal of Number Theor

Average Bateman--Horn for Kummer polynomials

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.Comment: V2: Minor correction

Asymptotics of the kk-free diffraction measure via discretisation

We determine the diffraction intensity of the $k$-free integers near the origin

Intervening against mental illness stigma and its internalisation: an organising framework

Reviews of interventions targeting of interpersonal stigma and internalised stigma have each identified several methods. Education about mental illness, contact between people with and without experience of mental illness, and protest against stigma have been identified as three means of reducing interpersonal stigma. While there is evidence that education and contact can be effective both separately and in combination, protest has been discouraged because of evidence suggesting that it can be counterproductive. Further there is little research directly addressing the question of whether education and contact are effective for structural level discrimination. On the other hand, the effectiveness of some types of protests against stigmatising organisational decisions suggests researchers should give further consideration to protest. Reviews of interventions targeting internalised stigma identified the following methods as the most used ones in effective interventions: cognitive; narrative; behavioural decision making, and psychoeducational. Since these reviews, recent work has begun to identify contact as effective for reducing internalised stigma. This article aims to synthesise these fields with the following objectives: (i) to highlight the similarities between interventions targeted to interpersonal and internalised stigma and the implications of these similarities; (ii) to draw attention to the need to evaluate structural level interventions; (iii) to create a comprehensive model for intervening against stigma using the ‘cycle of oppression' model which is widely applied in diversity and inclusion training. This model is proposed to be useful both to inform decisions about designing and targeting interventions, but can also be used as content for an intervention to reduce internalised stigma and help people with mental illness and their ‘allies' to intervene against stigma themselves