International Centre for Mathematical Sciences

Completing earlier work on three dimensional (3D) N=1 supergravity with curvature-squared terms, we construct the general supergravity extension of cosmological massive gravity theories. We expand about supersymmetric anti-de Sitter vacua, finding the conditions for bulk unitarity and the critical points in parameter space at which the spectrum changes. We discuss implications for the dual conformal field theory.

Twisted bilayers of two-dimensional materials contain a moir\'e superatomic lattice domain that can be bounded by a domain boundary network. The authors show here that the nodes of this network can be considered topological defects. While the nodes in previously studied twisted moir\'e materials are vortices, the authors discover antivortices formed in strained moir\'e systems using transmission electron micrographs. This work provides insight into the underlying rules governing moir\'e structures, inspiring the construction of new types of moir\'e interfaces.

We construct, at the linearized level, the three-dimensional (3D) supersymmetric 'general massive supergravity' and the maximally supersymmetric 'new massive supergravity'. We also construct the maximally supersymmetric linearized topologically massive supergravity, although we expect to be maximal at the nonlinear level.

Abstract The symmetry of the whole experimental setups, including specific sample environments and measurables, can be compared with that of specimens for observable physical phenomena. We, first, focus on one-dimensional (1D) experimental setups, independent from any spatial rotation around one direction, and show that eight kinds of 1D objects (four; vector-like, the other four; director-like), defined in terms of symmetry, and their dot and cross products are an effective way for the symmetry consideration. The dot products form a Z2 × Z2 × Z2 group with Abelian additive operation, and the cross products form a Z2 × Z2 group with Abelian additive operation or Q8, a non-Abelian group of order eight, depending on their signs. Those 1D objects are associated with characteristic physical phenomena. When a 3D specimen has symmetry operational similarity (SOS) with (identical or lower, but not higher, symmetries than) an 1D object with a particular phenomenon, the 3D specimen can exhibit the phenomenon. This SOS approach can be a transformative and unconventional avenue for symmetry-guided materials designs and discoveries.

Statistical models that can predict graft and patient survival outcomes following kidney transplantation could be of great clinical utility. We sought to appraise existing clinical prediction models for kidney transplant survival outcomes that could guide kidney donor acceptance decision-making. We searched for clinical prediction models for survival outcomes in adult recipients with single kidney-only transplants. Models that require information anticipated to become available only after the time of transplantation were excluded as, by that time, the kidney donor acceptance decision would have already been made. The outcomes of interest were all-cause and death-censored graft failure, and death. We summarised the methodological characteristics of the prediction models, predictive performance and risk of bias. We retrieved 4,026 citations from which 23 articles describing 74 models met the inclusion criteria. Discrimination was moderate for all-cause graft failure (C-statistic: 0.570–0.652; Harrell’s C: 0.580–0.660; AUC: 0.530–0.742), death-censored graft failure (C-statistic: 0.540–0.660; Harrell’s C: 0.590–0.700; AUC: 0.450–0.810) and death (C-statistic: 0.637–0.770; Harrell’s C: 0.570–0.735). Calibration was seldom reported. Risk of bias was high in 49 of the 74 models, primarily due to methods for handling missing data. The currently available prediction models using pre-transplantation information show moderate discrimination and varied calibration. Further model development is needed to improve predictions for the purpose of clinical decision-making. Systematic Review Registration: https://osf.io/c3ehp/l .

This Note contains the recommendation for a mathematical basis for the apportionment of the seats in the European Parliament between the Member States of the European Union. This is the unanimous recommendation of the Participants in the Cambridge Apportionment Meeting, held at the instigation of the Committee on Constitutional Affairs at the Centre for Mathematical Sciences, University of Cambridge, on 28-29 January 2011.

1 Abstract Objective to determine the statistical relationship between reported deaths and infections in the UK coronavirus outbreak Design Publicly available UK government data is used to determine a relationship between reported cases and deaths, taking into account various UK regions, age profiles and prevalence of the variant of concern (VOC) B.1.1.7. Main Outcome Measures Establishing a simple statistical relationship between detected cases and subsequent mortality. Results Throughout October and November 2020, deaths in England are well described as 1/55 th of detected cases from 12 days previously. After that, the relationship no longer holds and deaths are significantly higher. This is especially true in regions affected by the VOC B.1.1.7 Conclusions In early December, some new factor emerged to increase the case-fatality rate in the UK. Summary Box What is already known on this topic The infection-mortality ratio enables one to predict future deaths based on current infections. Incomplete monitoring of infection may be sufficient to predict future trends. What the study adds For the specific case of the second wave of coronavirus infection in the UK, we show a clear mathematical relationship between detected infections (positive tests) and subsequent deaths. This relationship begins to fail in December, with unexpectedly high death rates. This may be correlated in time and region with the emergence of the Variant of Concern B 1.1.7.

BACKGROUND: The United Nations' Sustainable Development Goal 4, and particularly target 4.2, which seeks to ensure that, by 2030, all children have access to quality early childhood development, care, and preprimary education so that they are ready for primary education, is far from being achieved. The COVID-19 pandemic compromised progress by disrupting education, reducing access to well-being resources, and increasing family violence. Evidence from low- and middle-income countries suggests that in-person parenting interventions are effective at improving child learning and preventing family violence. However, scaling up these programs is challenging because of resource constraints. Integrating digital and human-delivered intervention components is a potential solution to these challenges. There is a need to understand the feasibility and effectiveness of such interventions in low-resource settings. OBJECTIVE: This study aims to determine the feasibility and effectiveness of a digital parenting program (called Naungan Kasih in Bahasa Melayu [Protection through Love]) delivered in Malaysia, with varying combinations of 2 components included to encourage engagement. The study is framed around the following objectives: (1) to determine the recruitment, retention, and engagement rates in each intervention condition; (2) to document implementation fidelity; (3) to explore program acceptability among key stakeholders; (4) to estimate intervention costs; and (5) to provide indications of the effectiveness of the 2 components. METHODS: This 10-week factorial cluster randomized trial compares ParentText, a chatbot that delivers parenting and family violence prevention content to caregivers of preschool-aged children in combination with 2 engagement components: (1) a WhatsApp support group and (2) either 1 or 2 in-person sessions. The trial aims to recruit 160 primary and 160 secondary caregivers of children aged 4-6 years from 8 schools split equally across 2 locations: Kuala Lumpur and Negeri Sembilan. The primary outcomes concern the feasibility and acceptability of the intervention and its components, including recruitment, retention, and engagement. The effectiveness outcomes include caregiver parenting practices, mental health and relationship quality, and child development. The evaluation involves mixed methods: quantitative caregiver surveys, digitally tracked engagement data of caregivers' use of the digital intervention components, direct assessments of children, and focus group discussions with caregivers and key stakeholders. RESULTS: Overall, 208 parents were recruited at baseline December 2023: 151 (72.6%) primary caregivers and 57 (27.4%) secondary caregivers. In January 2024, of these 208 parents, 168 (80.8%) enrolled in the program, which was completed in February. Postintervention data collection was completed in March 2024. Findings will be reported in the second half of 2024. CONCLUSIONS: This is the first factorial cluster randomized trial to assess the feasibility of a hybrid human-digital playful parenting program in Southeast Asia. The results will inform a large-scale optimization trial to establish the most effective, cost-effective, and scalable version of the intervention. TRIAL REGISTRATION: OSF Registries; https://osf.io/f32ky. INTERNATIONAL REGISTERED REPORT IDENTIFIER (IRRID): DERR1-10.2196/55491.

BACKGROUND: Violence against children (VAC) is a serious public health concern with long-lasting adverse effects. Evidence-based parenting programs are one effective means to prevent VAC; however, these interventions are not scalable in their typical in-person group format, especially in low- and middle-income countries where the need is greatest. While digital delivery, including via chatbots, offers a scalable and cost-effective means to scale up parenting programs within these settings, it is crucial to understand the key pillars of user engagement to ensure their effective implementation. OBJECTIVE: This study aims to investigate the most effective and cost-effective combination of external components to optimize user engagement with ParentText, an open-source chatbot-led parenting intervention to prevent VAC in Mpumalanga, South Africa. METHODS: This study will use a mixed methods design incorporating a 2 × 2 factorial cluster-randomized controlled trial and qualitative interviews. Parents of adolescent girls (32 clusters, 120 participants [60 parents and 60 girls aged 10 to 17 years] per cluster; N=3840 total participants) will be recruited from the Ehlanzeni and Nkangala districts of Mpumalanga. Clusters will be randomly assigned to receive 1 of the 4 engagement packages that include ParentText alone or combined with in-person sessions and a facilitated WhatsApp support group. Quantitative data collected will include pretest-posttest parent- and adolescent-reported surveys, facilitator-reported implementation data, and digitally tracked engagement data. Qualitative data will be collected from parents and facilitators through in-person or over-the-phone individual semistructured interviews and used to expand the interpretation and understanding of the quantitative findings. RESULTS: Recruitment and data collection started in August 2023 and were finalized in November 2023. The total number of participants enrolled in the study is 1009, with 744 caregivers having completed onboarding to the chatbot-led intervention. Female participants represent 92.96% (938/1009) of the sample population, whereas male participants represent 7.03% (71/1009). The average participant age is 43 (SD 9) years. CONCLUSIONS: The ParentText Optimisation Trial is the first study to rigorously test engagement with a chatbot-led parenting intervention in a low- or middle-income country. The results of this study will inform the final selection of external delivery components to support engagement with ParentText in preparation for further evaluation in a randomized controlled trial in 2024. TRIAL REGISTRATION: Open Science Framework (OSF); https://doi.org/10.17605/OSF.IO/WFXNE. INTERNATIONAL REGISTERED REPORT IDENTIFIER (IRRID): DERR1-10.2196/52145.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

BACKGROUND: Intimate partner violence (IPV) and violence against children are global issues with severe consequences. Intersections shared by the 2 forms of violence have led to calls for joint programming efforts to prevent both IPV and violence against children. Parenting programs have been identified as a key entry point for addressing multiple forms of family violence. Building on the IPV prevention material that has been integrated into the parenting program ParentText, a digital parenting chatbot, this pilot study seeks to explore parents' engagement with the IPV prevention content in ParentText and explore preliminary changes in IPV. OBJECTIVE: This study aimed to assess parents' and caregivers' level of engagement with the IPV prevention material in the ParentText chatbot and explore preliminary changes in experiences and perpetration of IPV, attitudes toward IPV, and gender-equitable behaviors following the intervention. METHODS: Caregivers of children aged between 0 and 18 years were recruited through convenience sampling by research assistants in Cape Town, South Africa, and by UNICEF (United Nations Children's Fund) Jamaica staff in 3 parishes of Jamaica. Quantitative data from women in Jamaica (n=28) and South Africa (n=19) and men in South Africa (n=21) were collected electronically via weblinks sent to caregivers' phones using Open Data Kit. The primary outcome was IPV experience (women) and perpetration (men), with secondary outcomes including gender-equitable behaviors and attitudes toward IPV. Descriptive statistics were used to report sociodemographic characteristics and engagement outcomes. Chi-square tests and 2-tailed paired dependent-sample t tests were used to investigate potential changes in IPV outcomes between pretest and posttest. RESULTS: The average daily interaction rate with the program was 0.57 and 0.59 interactions per day for women and men in South Africa, and 0.21 for women in Jamaica. The rate of completion of at least 1 IPV prevention topic was 25% (5/20) for women and 5% (1/20) for men in South Africa, and 21% (6/28) for women in Jamaica. Exploratory analyses indicated significant pre-post reductions in overall IPV experience among women in South Africa (P=.01) and Jamaica (P=.01) and in men's overall harmful IPV attitudes (P=.01) and increases in men's overall gender-equitable behaviors (P=.02) in South Africa. CONCLUSIONS: To the best of our knowledge, this is the first pilot study to investigate user engagement with and indicative outcomes of a digital parenting intervention with integrated IPV prevention content. Study findings provide valuable insights into user interactions with the chatbot and shed light on challenges related to low levels of chatbot engagement. Indicative results suggest promising yet modest reductions in IPV and improvements in attitudes after the program. Further research using a randomized controlled trial is warranted to establish causality.

Half-supersymmetric geometries of N = 2 five-dimensional gauged supergravity have recently been fully classified using spinorial geometry techniques. We use this classification to determine all possible regular half-supersymmetric near-horizon geometries, assuming that all of the gauge-invariant spinor bilinears are regular at the horizon. Four geometries are found, two of which have been found previously in Kunduri and Lucietti (2007 J. High Energy Phys. JHEP02(2007)026), and Kunduri and Lucietti (2007 J. High Energy Phys. JHEP12(2007)015).

In a traditional livestock farming, the livelihood of animals depends highly on existing plant biomass, which is affected by the level and intensity of temperature, rainfall, humidity and other meteorological variables. Understanding the interaction of such meteorological factors and agricultural production in general is an important aspect in planning at the macro and micro levels. Particularly livestock agriculture is heavily affected by the changing climate, and hence the variation in major meteorological variables. However, there is still limited research regarding the impacts of meteorological variables on livestock production in each particular region. Soil moisture is one of the main factors in agricultural production and hydrological cycles with better memory of previous weather conditions. It also involves complex structural characteristics and meteorological factors. In this study, a soil moisture dependent mathematical model for the interaction of plants and herbivores is developed and analysed. The Soil Moisture Active Passive level 4 satellite soil moisture data is used in the model to simulate the possible spatial distribution of plants and the corresponding potential livestock production level for Botswana. A global dynamic sensitivity analysis is employed to study the sensitivity of the solution of the model with a variation in the involved parameter values. The results of the simulations of the model show that estimated livestock harvest in wet regions is more than triple as compared to what is estimated for dry regions. If some important parameters are properly estimated and the soil moisture data is available for each region, it is possible to estimate the livestock production level for each spatial region with better accuracy using the proposed model.

Abstract We present a method for rapid calculation of coronavirus growth rates and R-numbers tailored to the publicly available data in the UK. The R-number is derived from time-series of case data, using bespoke data processing to remove systematic and errors and stochastic fluctuations. In principle, growth rate can be obtained by differentiating the reported case numbers, but in fact daily stochastic fluctuations disqualify this method. We therefore assume that the case data comprises a smooth, underlying trend which is differentiable and a noise term. The approach produces, up-to-date estimates of the R-number throughout the period of data availability. Our method is validated against published consensus R-numbers from the UK government, and shown to produce comparable results. A significant advantage of our method is that it is stable up to the most recent data, this enables us to make R-number estimates available over two weeks ahead of the published consensus. The short-lived peaks observed in the R-number and case data cannot be explained by a well-mixed model and are suggestive of spread on a localised network. Such a localised spread model tends to give an Rt number close to 1, regardless of how large R0 is. The case-driven approach is combined with Weight-Shift-Scale (WSS) methods to monitor trends in the epidemic and for medium term predictions. Using case-fatality ratios, we create a narrative for trends in the UK epidemic increased infectiousness of the alpha and delta variants, and the effectiveness of vaccination in reducing severity of infection.

Abstract We prove an arithmetic ‘path integral’ formula for the inverse ‐adic absolute values of Kubota–Leopoldt ‐adic ‐functions at roots of unity.

ABSTRACT We prove an analog of Kida's formula for the Iwasawa invariants of the Mazur–Tate elements attached to elliptic curves over . Let be an odd prime and let be a Galois extension of abelian number fields with ‐power Galois group. For an elliptic curve , we study the Mazur–Tate elements over the finite layers of the cyclotomic ‐extensions of and . We show that the vanishing of the ‐invariant is preserved in the extension: if the level‐ Mazur–Tate element over has , then the corresponding element over also has . Moreover, the associated ‐invariants satisfy an explicit transition formula. This parallels the work of Hachimori–Matsuno on Selmer groups and of Matsuno on ‐adic ‐functions. As an application, we obtain an analog of Kida's formula for the analytic Iwasawa invariants associated with Pollack's signed ‐adic ‐functions. Since our results apply to elliptic curves with any reduction type at under mild hypotheses, including those with additive reduction, we also obtain a Kida‐type formula for the ‐adic ‐functions constructed by Delbourgo for elliptic curves with unstable additive reduction. In particular, because Mazur–Tate elements approximate ‐adic ‐functions in the limit, our results simultaneously unify all previously known cases of Kida's formula for analytic Iwasawa invariants.

On February 6, 2023, a magnitude 7.8 earthquake struck southern and central Türkiye and parts of Syria, followed by a 7.5 event.The destruction was geographically vast: tens of thousands of lives were lost, universities were shut down, and students were thrown into uncertainty.The question that haunted me was how could I help those who were affected.As a pure mathematician, my first response was a form of helplessness.I was not a doctor, nurse, cook, or professional relief worker.I could not clear rubble or repair cell towers.Yet it is a mistake to equate "immediate material necessity" with "the whole space of necessity."Human beings do not survive solely by calories and shelter; they also survive by reasons to keep a day structured, by the option of thinking beyond the next hour, and by experiencing being addressed as a human rather than merely as a victim.My only reliable resource was my mathematical expertise and a network of people who shared it.The project that eventually emerged -Matematik Dayanışma Platformu (the Math Solidarity Platform) -began from that narrow but genuine premise: if we could restore even a small piece of continuity for affected students, we might offer something that is not food or medicine, but is still a lifeline.A precursor of this platform was a program that already existed.Before the earthquake, I had been participating in the Directed Reading Program (DRP) Türkiye: an independent, online initiative run by volunteer graduate students and early-career researchers.DRP Türkiye pairs undergraduates in Türkiye with mentors around the world to read a piece of mathematical research together during summer months, typically on topics not covered in standard undergraduate courses.The program's implicit philosophy is that mathematical growth is not only additive (more techniques) but also exploratory (new objects, new questions, new styles of thinking), and that mentorship can make such exploration possible.After the disaster, two of DRP Türkiye's founding members, Şefika Kuzgun and Feride Ceren Köse, asked a practical question that was almost a moral one: could this network, built for intellectual exploration, be repurposed -quickly and responsibly -into a form of disaster support?They reached out to me, and together with Tekin Karadağ we met to think concretely about what could be done that was neither symbolic nor naïve.That conversation

Sparse autoencoders (SAEs) have proven effective for extracting monosemantic features from large language models (LLMs), yet these features are typically identified in isolation. However, broad evidence suggests that LLMs capture the intrinsic structure of natural language, where the phenomenon of "feature splitting" in particular indicates that such structure is hierarchical. To capture this, we propose the Hierarchical Sparse Autoencoder (HSAE), which jointly learns a series of SAEs and the parent-child relationships between their features. HSAE strengthens the alignment between parent and child features through two novel mechanisms: a structural constraint loss and a random feature perturbation mechanism. Extensive experiments across various LLMs and layers demonstrate that HSAE consistently recovers semantically meaningful hierarchies, supported by both qualitative case studies and rigorous quantitative metrics. At the same time, HSAE preserves the reconstruction fidelity and interpretability of standard SAEs across different dictionary sizes. Our work provides a powerful, scalable tool for discovering and analyzing the multi-scale conceptual structures embedded in LLM representations.

Free-response observer performance studies are of great importance for accuracy evaluation and comparison in tasks related to the detection and localization of multiple targets or signals. The free-response receiver operating characteristic (FROC) curve and many similar curves based on the free-response observer performance assessment data are important tools to display the accuracy of detection under different thresholds. The true positive rate at a fixed false positive rate and summary indices such as the area under the FROC curve are also commonly used as the figures of merit in the statistical evaluation of these studies. Motivated by a free-response observer performance assessment research of a Software as a Medical Device (SaMD), we propose a unified method based on the initial-detection-and-candidate model to simultaneously estimate a smooth curve and derive confidence intervals for summary indices and the true positive rate at a fixed false positive rate. A maximum likelihood estimator is proposed and its asymptotic normality property is derived. Confidence intervals are constructed based on the asymptotic normality of our maximum likelihood estimator. Simulation studies are conducted to evaluate the finite sample performance of the proposed method. We apply the proposed method to evaluate the diagnostic performance of the SaMD for detecting pulmonary lesions.

Grothendieck talked about “algebraic geometry in its most fascinating aspect for me—the “arithmetic” aspect, apprehended by intuition, concepts, and techniques all of “geometric” nature.” (ReS, p. 372) This has become a very prominent development in recent mathematics, and the new scenario for the interplay of key ideas (and tensions) within pure math, like the relations geometry/number, continuous/discrete. It seems to us that our grasp of mathematical ideas (even basic ideas like number, space) can be transformed when a very innovative practice is established. Otherwise said, they can be transformed by immersion in, or interplay with, new structures. The traditional antithesis between discrete and continuous can be bridged, smoothed out, and integrated by means of rich hybrid structures such as schemes. But such developments raise basic questions, in particular – why is arithmetic geometry geometry? The evolution of ideas of space and geometry is an extremely complicated subject, covering as it does all of human history. In this talk, we will reflect on the meandering evolution of these ideas, which accelerated with 20th century mathematical abstractions, and attempt some answer to the key question.