Baillie’s identity is (see [1] or [2], p. 240). Its integral analogue, , is not difficult to prove (see Lemma 1, below). In this Article, we prove a generalisation of the latter result (see Theorem 1). Theorems 2 and 3 are extensions, involving in addition, powers of cosines in the integrand. Theorem 4 answers a question raised after the proof of Theorem 1, and Theorem 5 collects together the preceding results in the form of three identities between trigonometric integrals. Theorem 6 gives a further generalisation.

Flows enabled by phoretic mechanisms are of significant interest in several biological and biomedical processes, such as bacterial motion and targeted drug delivery. Here, we develop a homogenisation-based macroscopic boundary condition that describes the effective flow across a diffusio-phoretic microstructured membrane, where the interaction between the membrane walls and the solute particles is modelled via a potential approach. We consider two cases where potential variations occur (i) at the pore scale and (ii) only in the close vicinity of the boundary, allowing for a simplified version of the macroscopic flow description, in the latter case. Chemical interactions at the microscale are rigorously upscaled to macroscopic phoretic solvent velocity and solute flux contributions, and added to the classical permeability and diffusivity properties of the membrane. These properties stem from the solution of Stokes advection–diffusion problems at the microscale, some of them forced by an interaction potential term. Eventually, we show an application of the macroscopic model to develop minimal phoretic pumps, showcasing its suitability for efficient design and optimisation procedures.

Early pregnancy diagnosis in buffaloes enhances reproductive efficiency and dairy production. This study aimed to develop and validate a duplex TaqMan-based RT-qPCR assay using two interferon-stimulated genes (ISGs), ISG15 and LGALS3BP, for early pregnancy diagnosis in buffaloes. Whole blood samples were collected from artificially inseminated buffaloes before (day 0) and after (days 20, 25 and 40) artificial insemination (AI). On 40 days, post-insemination ultrasonography was performed to categorize animals as pregnant or non-pregnant. RNA from peripheral blood mononuclear cells (PBMCs) was isolated and converted into cDNA. A duplex TaqMan-based RT-qPCR assay was developed to predict pregnancy based on the expression of ISG15 and LGALS3BP in PBMCs. Gene expression was evaluated in a test dataset at various time points. The assay's performance was validated using two additional datasets, under which blood samples were collected randomly at 20 ± 2 days to evaluate its utility for predicting pregnancy. Colour Doppler ultrasonography was utilized to confirm pregnancy at 40 days post-insemination for all the animals. Significantly different abundance of transcripts of ISG15 and LGALS3BP, in pregnant buffaloes, was observed on day 20 post-insemination. A statistically significant fold change (p < 0.05) of ISG15 and LGALS3BP transcripts was observed between pregnant and non-pregnant buffaloes. The receiver operating characteristic curve of validation datasets demonstrated the AUC = 0.95 and AUC = 0.90. The negative predictive value and positive predictive value range from 90% to 95% and 75% to 85%. In conclusion, the developed duplex RT-qPCR-based assay demonstrates high sensitivity and specificity in detecting early pregnancy in buffaloes.

We study constant Q-curvature metrics conformal to the the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov–Hausdorff topology, has the local structure of a real algebraic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighbourhood in the moduli spaces is actually a smooth, real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighbourhood of the moduli space embeds as a Lagrangian submanifold in the parameter space. We remark that our construction of the symplectic structure is quite different from the one in the scalar curvature setting, due to the fact that the associated partial differential equation is fourth-order rather than second-order.

Individual-level solutions, such as lifestyle changes, are often regarded as cost-effective complements to systemic policies like government regulations in addressing climate change. However, some argue that exposure to these solutions may shift focus away from systemic interventions, potentially crowding-out support for large-scale policies. Across two experiments in Iran (N = 303) and Australia (N = 308 and 625), we tested whether exposure to different numbers of individual and systemic policies would affect individuals’ perception of systemic climate solutions. Results showed that, regardless of country or exposure type, whether through daily life or experimental manipulation, participants consistently supported systemic policies and changes, attributed greater responsibility to the government, perceived minimal conflict between systemic and individual approaches, rated systemic policies as more effective than individual ones, and viewed individual actions as only moderately substitutable for systemic solutions. These findings challenge the crowding-out hypothesis, suggesting that exposure to individual solutions does not diminish support for systemic policies or structural reforms.

Maximising creativity requires an enriched imagination that uses all five senses. This study explored the effects of a reduced visual–auditory multisensory stimuli environment on creativity. Nineteen participants took the Alternative Uses Test (AUT) under the nine decreased visual–auditory multisensory stimuli conditions. Fluency and originality were evaluated as a part of the creativity assessment. The number of ideas from the AUT determined the fluency level, and the three judges’ evaluations determined originality. A study on associative conceptual network analysis explored the word associations of selected nouns from the AUT under nine reduced visual–auditory multisensory stimuli experimental conditions, revealing outdegree centrality scores to evaluate creative potential. The results suggest that the decreasing visual stimuli inhibit fluency whereas auditory stimuli do not, and that originality is enhanced when stimuli are reduced, whether visual or auditory, unless there is a notable divergence between the visual and auditory conditions. These results highlight the importance of perceptual focus and cognitive load regulation in fostering creative potential.

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan – Leith – Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12, 439–450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

My Mathematical Association Presidential Address focused on maths education in the context of the current environment in England. However, many of the issues are relevant to other UK nations and maths education more widely.

This research examines in detail the complex nonlinear forces generated when steep waves interact with vertical cylindrical structures, such as those typically used as offshore wind turbine foundations. These interactions, particularly the nonlinear wave forces associated with the secondary load cycle, present unanswered questions about how they are triggered. Our experimental campaigns underscore the occurrence of the secondary load cycle. We also investigate how the vertical distributions of the scattering force, pressure field and wave field affect the nonlinear wave forces associated with the secondary load cycle phenomena. A phase-based harmonic separation method isolates harmonic components of the scattering force’s vertical distribution, pressure field and wave field. This approach facilitates the clear separation of individual harmonics by controlling the phase of incident waves, which offers new insights into the mechanisms of the secondary load cycle. Our findings highlight the importance of complex nonlinear wave–structure interactions in this context. In certain wave regimes, nonlinear forces are locally larger than the linear forces, highlighting the need to consider the secondary load cycle in structural design. In addition, a novel discovery emerges from our comparative analysis, whereby very high-frequency (over the fifth in harmonic and order) oscillations, strongly correlated to wave steepness, have the potential to play a role in structural fatigue. This new in-depth analysis provides a unique insight regarding the complex interplay between severe waves and typical cylindrical offshore structures, adding to our understanding of the secondary load cycle for applications related to offshore wind turbine foundations.

Colostrum (CM) and transition milk (TM) are crucial for neonatal development but knowledge about the changes in their composition in dairy goats is still limited. The aim of this study was to characterise the immunological, lipid and microbial changes from CM to TM in Guanzhong dairy goats, an indigenous, highly productive Chinese breed. Milk samples were collected from eight primiparous goats at day 0 (CM) and day 7 (TM) postpartum. Immunoglobulin concentrations (IgG, IgA, IgM) were measured by ELISA, long-chain saturated fatty acids (FA) were analysed by gas chromatography, and bacterial communities were profiled by 16S rRNA gene amplicon sequencing. Microbial functions were inferred using PICRUSt2 (Phylogenetic Investigation of Communities by Reconstruction of Unobserved States) software based on KEGG (Kyoto Encyclopedia of Genes and Genomes) orthologs. Immunoglobulin concentrations declined significantly from CM to TM, reflecting a transition from immunoprotection to food supply. The TM had increased concentrations of long-chain saturated FA, particularly palmitic acid (C16:0) and stearic acid (C18:0) and higher concentrations of branched-chain FA, suggesting an adaptation to the growing energy demands of the newborn. Although overall microbial diversity was stable, taxonomic shifts were observed. CM was enriched in Emticicia, which may be involved in oligosaccharide degradation, while TM had higher abundance of Faecalibacterium, Ruminococcus and Salinicoccus, genera associated with SCFA production and intestinal development. Functional predictions indicated enrichment of cold shock and haem biosynthetic pathways in CM and enhanced carbohydrate degradation pathways in TM. Correlation analyses revealed relationships between specific FA and bacterial genera, suggesting interactions between host, microbes and nutrients that may influence milk functionality. This integrative analysis of milk components during early lactation provides new insights into the compositional and functional dynamics of goat milk.

This study examined the effects of fat content (0.1%, 1.5%, 3%, and 3.5%), heat treatment (pasteurization and UHT), and vitamin D supplementation on the physicochemical, microbiological, and sensory characteristics of yoghurt over 21 days. Yoghurts with higher fat content generally received higher sensory acceptability scores (7–8), exhibited less syneresis (14.75–22.48%) and higher whiteness index; 0.1%-fat had the highest syneresis (51.28%) and lowest whiteness (80.77) (p < 0.05). Although pasteurised yoghurt exhibited slightly better firmness (1.77 N) than UHT (1.61 N), no significant effect of heat treatment was found (p > 0.05). Sensory preference was for 3% and 3.5%-fat and pasteurised variety, especially for consistency. All yoghurts maintained viable lactic acid bacteria counts >10⁷ cfu/g throughout the storage period. The vitamin D addition resulted in a 6% average higher syneresis rate (p > 0.05) and a nearly 4% higher whiteness index (p < 0.05). This study shows that it is possible to obtain yoghurts with enhanced properties by optimizing the fat content and heat treatment conditions applied, and it also directs future studies on the effect of vitamin D addition on yoghurt properties.

Michael Sata’s presidency in Zambia (2011–14) marked a notable attempt to revive statist development ideas rooted in the country’s postindependence era. While the preceding MMD government had begun reintroducing limited state intervention, its commitment remained constrained. Sata, by contrast, articulated a more assertive vision of state-led development, echoing the UNIP-era model under Kenneth Kaunda. Drawing on policy documents, speeches, and survey data, this article situates Sata’s politics and policies within broader public dissatisfaction with neoliberal reforms and highlights enduring tensions in Africa’s poststructural adjustment era between market-oriented policies and demands for greater state involvement.

This communication addresses the hypothesis that silver nanoparticles obtained by green synthesis (G-AgNPs), using an aqueous extract of chicory, have antibacterial activity against bovine mastitis-associated strains and may inhibit their adherence capacity. Bovine mastitis is among the diseases with the greatest economic impact worldwide. Multifactorial in nature, it is usually associated with infection caused by bacterial species such as Staphylococcus aureus, coagulase-negative staphylococci, Streptococcus uberis, and Escherichia coli. Antibiotics are currently the main line of treatment, but in order to offset the emergence of resistance, alternative compounds and nanomaterials are being researched for prevention and control. In this study, bacteria that cause bovine mastitis were molecularly identified by groEL gene PCR-RFLP. Then, G-AgNPs were produced using an aqueous extract of chicory as a reducing and stabilizing agent. The nanoparticles showed powerful antibacterial activity, but they did not seem to reduce the adherence of pathogens to MAC-T cells that had been preincubated with a sub-MIC concentration of G-AgNPs. Nevertheless, very low concentrations (21-30 pM) were needed for the antibacterial effect to take place, which makes G-AgNPs promising candidates for the control of bovine mastitis.

Part of a series of longitudinal studies of churchgoing in British towns and cities, this article traces the statistical history of church attendance in London over the past two hundred years. Among other sources, it utilises national religious censuses in 1851 and 1979–2005 and metropolitan ones in 1886–7, 1902–3, 1933–4 and 2012, presenting results as an index of attendance (IA), expressing attendances as percentage of population. Throughout the late nineteenth and twentieth centuries, London’s IAs fell continuously and were mostly below those in other conurbations. Following an uptick around the millennium, net decline resumed after 2012.

An arbitrary Lagrangian–Eulerian finite element method and numerical implementation for curved and deforming lipid membranes is presented here. The membrane surface is endowed with a mesh whose in-plane motion need not depend on the in-plane flow of lipids. Instead, in-plane mesh dynamics can be specified arbitrarily. A new class of mesh motions is introduced, where the mesh velocity satisfies the dynamical equations of a user-specified two-dimensional material. A Lagrange multiplier constrains the out-of-plane membrane and mesh velocities to be equal, such that the mesh and material always overlap. An associated numerical inf–sup instability ensues, and is removed by adapting established techniques in the finite element analysis of fluids. In our implementation, the aforementioned Lagrange multiplier is projected onto a discontinuous space of piecewise linear functions. The new mesh motion is compared to established Lagrangian and Eulerian formulations by investigating a pre-eminent numerical benchmark of biological significance: the pulling of a membrane tether from a flat patch and its subsequent lateral translation.

The dignity owed to every person should not cease with death.1 The processes by which individuals and societies across cultures and religions care for, honour and mourn their dead provide the necessary closure to families and their communities. When this is disrupted through improper protection and/or disrespectful treatment of the dead, it harms individuals and societies and, in the case of unlawful deaths, it undermines or impedes the victims’ rights to truth, justice and reparation. With the increasing complexity of mass fatality incidents, especially as a result of conflict, migration, pandemics and natural disasters (including those caused by climate change), the need to respectfully protect the dead is of growing importance.2

The specific means by which protection of the dead occurs in practice are usually contextually adapted to the beliefs and customs of each community and State. To be universally effective, however, it is recommended that they always be guided by principles of respect, dignity and decency toward the dead and their families. This will ensure the fulfilment of applicable international human rights, humanitarian law and criminal law obligations, and help to ease the pain that families, communities and societies face with the loss of their loved ones.

The authors of this article believe that the time is ripe for the development of a set of guiding principles, framed under international human rights law (IHRL), for the dignified management and protection of the dead, and they propose seven key areas to build such principles upon. In addition to filling a conspicuous gap in human rights protection, these guiding principles will help fulfil States’ duties to respect and protect the rights of families of the deceased under IHRL. While not exhaustive, the authors believe that these guiding principles, which should be read as lex ferenda, develop a framework for stronger IHRL protections of deceased persons.

This article builds on a recent thematic report on the protection of the dead presented to the United Nations Human Rights Council in June 2024 by one of the co-authors.3