A formula for the determinant of a partitioned matrix, possibly with singular submatrices, is derived and applied to some psychometric and numerical problems.

Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi $ be the unique quadratic multiplicative character of the finite field $\mathbb {F}_q$. If the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$, then we prove that

$$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$

This confirms a conjecture of Zhi-Wei Sun.

Motivated by the work initiated by Chapman [‘Determinants of Legendre symbol matrices’, Acta Arith. 115 (2004), 231–244], we investigate some arithmetical properties of generalised Legendre matrices over finite fields. For example, letting $a_1,\ldots ,a_{(q-1)/2}$ be all the nonzero squares in the finite field $\mathbb {F}_q$ containing q elements with $2\nmid q$, we give the explicit value of the determinant $D_{(q-1)/2}=\det [(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then

$$ \begin{align*}\bigg(\frac{\det D_{(p-1)/2}}{p}\bigg)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4,\\ (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}\end{align*} $$

where $(\frac {\cdot }{p})$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb {Q}(\sqrt {-p})$.

Globally, the prevalence of those living with obesity (≥30 kg/m2) is rising, with this trend expected to continue if firm and decisive policy interventions are not introduced. Across Europe, despite many consecutive policies aiming to reverse rising trends in weight status over recent decades, no country is currently on track to halt and reverse current trends in the coming years. This is evident in Ireland too, whereby the reporting of nationally representative weight status data show that targets have not been achieved since reporting began. The aim of this review is to critically appraise recent evidence relating to the key determinants of obesity including weight status, diet quality and physical activity with an emphasis on socioeconomic inequalities. And to consider these in the context of respective policy measures and propose future-focused recommendations. Furthermore, as with the complex nature of obesity, multifaceted approaches that shift the focus from the individual and place responsibility at a societal level will be reviewed.

There is a dearth of data on Se status in very old adults. The aims of this study were to assess Se status and its determinants in 85-year-olds living in the Northeast of England by measuring serum Se and selenoprotein P (SELENOP) concentrations and glutathione peroxidase 3 (GPx3) activity. A secondary aim was to examine the interrelationships between each of the biomarkers. In total, 757 participants (463 women, 293 men) from the Newcastle 85+ Study were included. Biomarker concentrations were compared with selected cut-offs (serum Se: suboptimal 70 µg/l and deficient 45 µg/l; SELENOP: suboptimal 4·5 mg/l and deficient 2·6 mg/l). Determinants were assessed using linear regressions, and interrelationships were assessed using restricted cubic splines. Median (inter-quartile range) concentrations of serum Se, SELENOP and of GPx3 activity were 53·6 (23·6) µg/l, 2·9 (1·9) mg/l and 142·1 (50·7) U/l, respectively. Eighty-two percentage and 83 % of participants had suboptimal serum Se (< 70 µg/l) and SELENOP (< 4·5 mg/l), and 31 % and 40 % of participants had deficient serum Se (< 45 µg/l) and SELENOP (< 2·6 mg/l), respectively. Protein intake was a significant determinant of Se status. Additional determinants of serum Se were sex, waist:hip ratio, self-rated health and disease, while sex, BMI and physical activity were determinants of GPx3 activity. There was a linear association between serum Se and SELENOP, and nonlinear associations between serum Se and GPx3 activity and between SELENOP and GPx3 activity. These findings indicate that most participants had suboptimal Se status to saturate circulating SELENOP.

Representative school data on SARS-CoV-2 past-infection are scarce, and differences between pupils and staff remain ambiguous. We performed a nation-wide prospective seroprevalence study among pupils and staff over time and in relation to determinants of infection using Poisson regression and generalised estimating equations. A cluster random sample was selected with allocation by region and sociodemographic (SES) background. Surveys and saliva samples were collected in December 2020, March, and June 2021, and also in October and December 2021 for primary pupils. We recruited 885 primary and 569 secondary pupils and 799 staff in 84 schools. Cumulative seroprevalence (95% CI) among primary pupils increased from 11.0% (7.6; 15.9) at baseline to 60.4% (53.4; 68.3) in December 2021. Group estimates were similar at baseline; however, in June they were significantly higher among primary staff (38.9% (32.5; 46.4)) compared to pupils and secondary staff (24.2% (20.3; 28.8)). Infections were asymptomatic in 48–56% of pupils and 28% of staff. Seropositivity was associated with individual SES in pupils, and with school level, school SES and language network in staff in June. Associations with behavioural characteristics were inconsistent. Seroconversion rates increased two- to four-fold after self-reported high-risk contacts, especially with adults. Seroprevalence studies using non-invasive sampling can inform public health management.

Vitamin D deficiency is common in Irish adults, though there is limited research on its determinants, knowledge of vitamin D or indications for testing. We aimed to explore the determinants of vitamin D status in adults and examine knowledge and reasons for testing. The study population comprised adults who had serum 25-hydroxyvitamin D tested by general practitioners request at a Dublin Hospital in 2020. Questionnaires detailing dietary intake, sun exposure, ethnicity, biophysical factors and vitamin D knowledge were sent to a sample stratified by age, sex and vitamin D status. In total, there were 383 participants, mean age 56·0 (sd 16·6) years. Wintertime deficiency disproportionally affected non-white v. white (60 % v. 24 %, P < 0·001). The greatest predictors of deficiency were low vitamin D intake (< 10 μg/d) (P < 0·001) and non-white ethnicity (P = 0·006), followed by sun avoidance (P = 0·022). It was also more prevalent in those with lower body exposure when outdoors. The majority (86 %) identified vitamin D as important for bone health. However, 40 % were tested for non-clinical indications and half were not aware of the recommended daily allowance (RDA). Low vitamin D intake was the most important determinant of deficiency, but ethnicity and sun exposure habits were also significant predictors. The majority had no clear indication for testing and were not aware of the RDA. Public health policies to improve knowledge and vitamin D intake, especially for those of non-white ethnicity and with reduced sun exposure, should be considered.

Dietary diversity in children may be influenced not only by individual circumstances but also by the features of the community in which they live. Our study aimed to assess community and individual-level determinants of minimum dietary diversity among children aged 6–23 months in Ethiopia. We included 2960 children aged 6–23 months from the recent Ethiopia Demographic and Health Survey. A minimum dietary diversity was defined as the consumption of at least five food groups out of the eight reference food groups within 24 h by children aged 6–23 months. Multilevel logistic regression was used to investigate the drivers of minimum dietary diversity in Ethiopian children aged 6–23 months. About 12⋅5 % of children met the bare minimum of dietary diversification. Age of the child (9–11 months AOR, 3⋅3 (95 % CI 1⋅8, 5⋅6), 12–17 months AOR, 4⋅0 (95 % CI 2⋅4, 6⋅7), 18–23 months AOR, 3⋅5 (95 % CI 2⋅0, 5⋅8)), caregiver listening radio at least once a week AOR, 1⋅6 (95 % CI 1⋅1, 2⋅4) and wealth quantiles (Second AOR, 1⋅8 (95 % CI 1⋅1, 3⋅1), Fourth AOR, 2⋅9 (95 % CI 1⋅6, 5⋅2) and Highest AOR, 2⋅2 (95 % CI 1⋅1, 4⋅2)) were individual characteristics associated with dietary diversity. Place of residence was the only community-level characteristic associated with children's dietary diversity (Rural AOR, 0⋅4 (95 % CI 0⋅2, 0⋅6)). The minimum dietary diversity among Ethiopian children is suboptimal. Nutrition programmes aimed at enhancing dietary diversity should be strengthened in this population, particularly for those from poor families and residing in rural areas.

Environmental features such as the ‘foodscape’ defined as the physical, sociocultural and economic space in which people encounter meals and foods, might be associated with dietary intake and health outcomes. This review focuses mainly on the spatial approach of the foodscape, i.e. all the local shops, markets, restaurants and sales outlets that provide food supplies in a given area. This review aims to explore the evidence on relationships of urban foodscape with diet and health outcomes and to highlight the limitations in studying these relationships as well as suggestions for future studies. Many systematic reviews on characteristics of the foodscape in relation to weight status outcomes emerged over the last decade and results are equivocal. There is not a direct association between the foodscape and weight status of the individual, rather any association is a distant one. Therefore, it is more appropriate to focus on associations between foodscape and intermediate, more proximal outcomes, such as dietary behaviours. Research on the role of the foodscape in promoting or hindering healthy dietary behaviours are also numerous, and results are again mixed. The diversity of methodologies might partly explain the heterogeneity of these results. Focusing on overall diet quality rather than fruit and vegetable consumption, taking into account multiple characteristics of the foodscape, as well as socioeconomic and contextual differences, might be part of the solution for more consistent results. Consequently, results of such studies could help shape foodscapes, which present a great opportunity for promoting healthier and eventually more sustainable diets.

Food insecurity is an important public health concern; however, research into this phenomenon within the Netherlands is limited. Food insecurity is not solely related to individual factors, but can also be influenced by various factors in the social and physical environment. Therefore, this study aimed to identify determinants of food insecurity within the personal, social and physical environment, based on the social ecological model (SEM), and to identify their relative importance for experiencing food insecurity. The study population consisted of 307 participants living in disadvantaged neighbourhoods of the Dutch city The Hague, of which approximately one-quarter were food insecure. Participant characteristics showing bivariate associations P < 0⋅20 were placed in a predetermined level of the SEM, after which a multivariate logistic regression was performed for each level and the Nagelkerke pseudo R2 was presented. Determinants of food insecurity were BMI, gross monthly income, highest educational attainment, smoking status, diet quality, employment status, marital status and religion (P < 0⋅05). The results showed that 29⋅7 % of the total variance in food insecurity status was explained by all included determinants together. The personal, social and physical environment explained 20⋅6, 14⋅0 and 2⋅4 % of the total variance, respectively. Our findings suggest that determinants within the personal environment are most important for explaining differences in experienced food insecurity. The present study contributes to furthering the knowledge about the relative importance of the personal, social and physical environment, indicating that determinants within the personal environment may be most promising for developing targeted interventions to reduce food insecurity.

Anthropometric studies have given much attention to the impact of industrialization and urbanization on the biological standards of living of urban populations. Instead, we know less about the evolution of height and the disparities within the rural world and how they have changed during the modern economic growth process. This article analyzes the evolution and the determining factors that would explain the inequality of the biological welfare of a group of rural populations in Mediterranean Spain. Using a database of the heights of military conscripts (N = 146,041) of the study area, a comparison is made of the biological well-being of the cohorts born between 1840 and 1965 in different rural environments (irrigated vs. dry farming). The results show that the recruits residing in irrigated areas were taller than those in dry farming areas and that the nutritional differences were greater among the latter. The advantage of the heights in irrigated areas widened with the development of commercial agriculture at the end of the nineteenth century and, although it began to reduce from the early decades of the twentieth century, the anthropometric gap persisted throughout the period analyzed. The data also suggest that the distribution of income was also more unequal in the dry farming areas, where the diet was less varied and rich than in the irrigated areas. This situation could be largely explained by the existence of low productivity agriculture in these dry farming areas, among other possible factors.

The prevalence of anaemia and its continuous growth, especially among women, is a global health concern. The present study aims to examine the prevalence of anaemia and its determinants in the North Eastern region of India. The study used secondary data from the National Family Health Survey 2015-2016 on women of reproductive age group in India. The data were adjusted for sampling weight, stratification and cluster sampling design for analysis. Binary logistic regression and multivariable regression analysis were performed to determine factors associated with anaemia. Of the 65941 participants, 25993 (40%) had anaemia. High prevalence of mild or moderate anaemia was found among women with following socio-demographic characteristics: residing in the rural area (41.20%), having no education (43.07%), belonging to the low-income family (43.39%), having a well as source of drinking water (46.29%), using the traditional method of contraception (44.55%), underweight (42.18%) and those who had first delivery before 20 years of age (40.66%). Logistic regression (adjusted odds ratio) showed that women in the following categories were more likely to develop anaemia: in the age-group of 35-49 years, with no education, with poor wealth, having low nutrition levels; using traditional contraception and women who ate pulses or fish once a week. Alarming rates of anaemia (two in every five), found in the study, need effective strategies for fortification of iron supplement among women. Generation of mass awareness in this regard by utilizing the ongoing adolescent, maternal, child health and nutrition programmes will help in reducing the incidence of anaemia among women.

The present study explored whether motivational constructs for diet and physical activity (PA) cluster and how these motivational constructs relate to dietary and PA behaviour. Data of 1142 participants were used from a randomised controlled trial examining the effects of a web-based diet and PA promotion intervention based on self-determination theory and motivational interviewing. Motivation was assessed using the Treatment Self-Regulation Questionnaire and Behavioural Regulation in Exercise Questionnaire. The dietary outcomes were measured using an adapted Food Frequency Questionnaire. PA was assessed using the Short QUestionnaire to ASsess Health. Spearman rank-order correlations showed large correlation coefficients (rs ≥ 0⋅63) between similar motivational constructs between the two lifestyle domains, except for intrinsic motivation where a medium correlation coefficient was found (rs = 0⋅41). Furthermore, the exploratory factor analysis illustrated that more self-determined forms of motivation seem to be more domain-specific. In contrast, non-self-determined forms of motivation seem to be domain-independent. Last, regression analyses demonstrated that intrinsic motivation towards PA was the only motivational construct significantly positively associated with all PA sub-behaviours (standardised regression coefficients ranging from 0⋅17 to 0⋅28, all P < 0⋅0125). Intrinsic motivation to eat healthily was significantly positively associated with fruits, vegetables and fish intake (standardised regression coefficients ranging from 0⋅11 to 0⋅16, all P < 0⋅0125), but not with unhealthy snacks. Insight of this exploratory study is useful for understanding the interrelationships of motivational induced behaviours, the development of interventions targeting multiple behaviours, and the construction of questionnaires.