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.

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.

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.

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.

In 1408, as the Great Western Schism entered its fourth decade, the Parisian community of Celestine monks asked Pierre d’Ailly to write a life of their founder, Pietro Morrone, who served as Pope Celestine V in 1294. D’Ailly found in him an example that suited his times: a pope could resign without damage to the Church at large. With cardinals from both obediences ready to withdraw support from Avignon’s Benedict XIII and Rome’s Gregory XII, both of whom resisted resignation, d’Ailly made his case for papal resignation via his life of Celestine V.