Let Γ be a finite graph and let $A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph $\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that $\Gamma_{\mathcal B}$ always contains Γ as a subgraph. This provides an effective way to reconstruct the defining graph Γ from the cohomology of $A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of $A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.

How closely related are modern anti-democratic beliefs among white Americans, and to what extent are these beliefs shaped by exclusionary racial attitudes? Using data from the Political Unrest Study, the Collaborative Multiracial Post-Election Survey (CMPS), Democracy Fund + UCLA Nationscape, and the Survey of the Performance of American Elections (SPAE), we find that support for voting restrictions, opposition to voting expansions, belief in widespread voter fraud, and support for overturning democratic election results load onto a single underlying dimension. While the prevalence of anti-democratic beliefs among white Americans has remained stable over the past decade, these beliefs have become increasingly interconnected. Furthermore, racial attitudes towards out-groups—including racial resentment, anti-immigrant sentiment, and white racial grievance—strongly correlate with anti-democratic beliefs, whereas in-group racial attitudes do not. Analysis of multiple waves of the American National Election Studies (ANES) reveals that racial resentment and white grievance now explain twice as much variation in anti-democratic beliefs as they did in 2012. Experimental evidence also demonstrates that white Americans react negatively to voting expansions when the racial implications of these reforms are made explicit. These findings underscore the growing alignment between anti-democratic beliefs and racial attitudes in contemporary U.S. politics.

We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.

We consider interacting urns on a finite directed network, where both sampling and reinforcement processes depend on the nodes of the network. This extends previous research by incorporating node-dependent sampling and reinforcement. We classify the sampling and reinforcement schemes, as well as the networks on which the proportion of balls of either colour in each urn converges almost surely to a deterministic limit. We also investigate conditions for achieving synchronisation of the colour proportions across the urns and analyse fluctuations under specific conditions on the reinforcement scheme and network structure.

Dark patterns that manipulate consumer behaviour are now a pervasive feature of digital markets. Depending on the choice architecture utilised, they can affect the perception, behaviour and purchasing patterns of online consumers. Using a novel empirical design, we find strong evidence that individuals across all groups are susceptible to dark patterns, and only weak evidence that user susceptibility is materially affected by commonly used general proxies for consumer vulnerability (such as income, educational attainment or age). Our conclusions provide empirical support for broad restrictions on the use of dark patterns, such as those contained in the EU’s Digital Services Act, that protect all consumer groups. Our study also finds that added friction, in the form of required payment action following successful deployment of dark patterns, reduces their effectiveness. This insight highlights the instances in which dark patterns would be most effective – when no further action is required by the user. Consumer vulnerability is therefore more pronounced when dealing with online providers who store users’ payment details and can rely on a ‘single click’ to complete the purchase.

Veterinarians are custodians of animal welfare, ensuring practices remain current and effective in the face of the ever-changing demands of the profession. Continuing Professional Development (CPD) is essential for protecting animal welfare, however access to quality CPD is a challenge in many countries. India has one of the fastest growing veterinary industries globally, emphasising the importance of accessible CPD opportunities that are relevant to this setting. This study used focus groups to explore how Indian veterinarians identify relevant CPD, barriers they encounter, and their experiences with CPD. We describe three themes: (1) ‘career vs calling’, where veterinarians’ extrinsic and intrinsic motivational factors were identified, such as their desire to protect animal welfare; (2) being ‘willing to learn but can’t’, with context-specific barriers, such as accessing reliable CPD information; and (3) ‘real interactions matter’, where participants described how pedagogical design influenced their choices, e.g. being able to observe animal welfare improvements through practical teaching. We have three recommendations: firstly, to improve CPD learning opportunities informed by evidence-based methods, to meet knowledge and skills gaps such as the high demand for practically focused training; secondly, the development of a unified accreditation and quality assurance framework to assess content, relevance and delivery standards of available CPD options to veterinarians; and, lastly, improved support from employers to address current barriers and facilitate attendance. These findings contribute to the current knowledge gap of factors that influence Indian veterinarians’ experiences of attaining relevant, accessible CPD and makes suggestions to improve standards of veterinary care and, ultimately, patient welfare.

This essay is a write-up of my Professorial Inaugural Lecture, delivered at the London School of Economics on 9 December 2024. Herein, I describe how I became involved and have helped develop the field of behavioural public policy (BPP). I detail how the intellectual architecture of BPP – its journal, Annual International Conference and Association – came into existence, and allude to my hopes for how the field might develop as we go forward.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.