Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$. Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.

We study a growth model with two types of agents who are heterogeneous in their degree of family altruism. We prove that every equilibrium path of consumption, bequests, and capital converges to a unique steady state and study the effect of altruism on the properties of steady-state equilibrium. We show that aggregate income is positively related to both level of altruism and altruism heterogeneity. When altruism heterogeneity is low or moderate, income inequality follows an inverse U-shaped pattern relative to the level of altruism. These observations are consistent with the cross-country Kuznets curve linking different steady-state levels of income to steady-state levels of inequality. When altruism heterogeneity is high, income inequality decreases with the level of altruism. Our results suggest that heterogeneous altruism is a possible mechanism linking economic growth and income inequality.

A novel microstrip filtering antenna with slot coupling feed is presented in this work. An asymmetric interdigital coupling structure is used for the feed to excite the patch antenna through gap-fed coupling. Introducing a U-shaped slot on the patch surface modifies the current path to attain different resonant modes. The asymmetric coupled fingers in the low-frequency band generate a radiation null of −64 dBi, while additional resonances introduced in the high band broaden the bandwidth from 4.9 to 5.3 GHz. A horizontally shorted microstrip branch produces another null point of −28 dBi between the bands, enabling steeper roll-off and further improvement in frequency selectivity. The proposed filtering antenna provides a flexible filter response without requiring extra filtering circuits, with appreciable peak gains (6.1 dBi and 7.1 dBi) and stable radiation characteristics. This makes it suitable for WLAN (Wireless Local Area Network) and ISM (Industrial Scientific Medical) systems applications.

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.