Convex drawings of the complete graph: topology meets geometry

In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characterized by excluding two of the five drawings of $K_5$. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted)

Sources of Salt-Water Pollution in Western Tom Green County

Tom Green County is situated within the discharge zone of the Permian Basin regional flow system in West Texas. Analysis of hydrochemical facies and ionic ratios of major chemical constituents suggests that a significant portion of the saline groundwater in the area is a combination of subsurface brine flowing eastward from the Permian Basin and locally recharged, shallowly circulating meteoric water. Aquifers that typically contain relatively fresh water in outcropping Paleozoic rocks exhibit the presence of brine and hydrocarbons at shallow depths ranging from 200 to 900 feet (60 to 270 meters) just a few tens of miles to the west. The chemical composition of groundwater is closely linked to the outcrop of Paleozoic formations from which brine is discharged.Bureau of Economic Geolog

Graphs with at most one crossing

The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one

Drawings of complete graphs in the projective plane

Hill's Conjecture states that the crossing number $\text{cr}(K_n)$ of the complete graph $K_n$ in the plane (equivalently, the sphere) is $\frac{1}{4}\lfloor\frac{n}{2}\rfloor\lfloor\frac{n-1}{2}\rfloor\lfloor\frac{n-2}{2}\rfloor\lfloor\frac{n-3}{2}\rfloor=n^4/64 + O(n^3)$. Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely $n^4/64+O(n^3)$, thus matching asymptotically the conjectured value of $\text{cr}(K_n)$. Let $\text{cr}_P(G)$ denote the crossing number of a graph $G$ in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of $K_n$ is $(n^4/8\pi^2)+O(n^3)$. In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if $\lim_{n\to\infty} \text{cr}_P(K_n)/n^4=1/8\pi^2$. We construct drawings of $K_n$ in the projective plane that disprove this

Exploring the Synergies of Single‐Molecule Fluorescence and 2D Materials Coupled by DNA

The world of 2D materials is steadily growing, with numerous researchers attempting to discover, elucidate, and exploit their properties. Approaches relying on the detection of single fluorescent molecules offer a set of advantages, for instance, high sensitivity and specificity, that allow the drawing of conclusions with unprecedented precision. Herein, it is argued how the study of 2D materials benefits from fluorescence-based single-molecule modalities, and vice versa. A special focus is placed on DNA, serving as a versatile adaptor when anchoring single dye molecules to 2D materials. The existing literature on the fruitful combination of the two fields is reviewed, and an outlook on the additional synergies that can be created between them provided

Global development assistance for early childhood care and education in 134 low- and middle-income countries, 2007-2021

Providing quality early childhood care and education (ECCE) is widely recognized as a fundamental strategy for lifelong individual and societal benefits. However, the expansion of ECCE in low- and middle-income countries (LMICs) is impeded by limited financial resources. Little is known about the levels and trends of global development assistance (DA) allocated to ECCE in LMICs between 2007 and 2021. We extracted data from the Creditor Reporting System (CRS) on aid projects in 134 LMICs from 2007 to 2021. Using keyword-searching and funding-allocation methods, we generated two estimates of ECCE aid: one for DA primarily focused on ECCE, and another for DA both primarily and partially focused on ECCE as well as DA not explicitly targeting ECCE but benefiting ECCE. We analyzed the patterns and time trends of ECCE aid by donors, recipients, CRS sectors, implementing agencies, and aid-flow types. We calculated ECCE aid as a percentage of educational aid at both annual and aggregative levels, comparing it to UNICEF’s recommended allocation of at least 10% of educational aid to the ECCE sector in LMICs. Additionally, we explored the alignment of aid with addressing children’s learning losses during global crises by examining ECCE aid to projects involving COVID-19 prevention and mitigation, along with ECCE aid to conflict-affected countries. From 2007 to 2021, primary ECCE-focused DA amounted to US dollars 3,646 million, accounting for 1.7% of the total US dollars 213,279 million allocated to education. The World Bank led all donors with US dollars 1,944 million in ECCE aid (53.3%). Low-income countries consistently received less ECCE aid per child before 2016, then started to catch up, but subsequently experienced a decrease from US dollars 0.8 (2020) to US dollars 0.6 (2021). In contrast, lower-middle-income and upper-middle-income countries saw an increase from US dollars 0.4 (2020) to US dollars 0.6 (2021) and from US dollars 1.0 (2020) to US dollars 1.3 (2021), respectively. ECCE aid to projects with COVID-19 activities declined from US dollars 50 million in 2020 to US dollars 37 million in 2021, representing 11.4% and 6.6% of annual ECCE aid. Over 15 years, conflict-affected countries received an average of US dollars 0.3 per child, only a quarter of the aid received by non-conflict-affected countries (US dollars 1.2 per child). Although ECCE aid increased significantly between 2007 and 2021, its share of total educational aid remained small and fell far short of the recommended minimum of 10%. Our recommendations include increasing the share of ECCE aid in total educational aid, raising aid to low-income and conflict-affected countries, and investing more in preparing ECCE programs for future global crises

Postnatal depressive symptoms and child psychological development at 10 years: a prospective study of longitudinal data from the South African Birth to Twenty cohort.

BACKGROUND: In high-income countries, maternal postnatal depression is associated with adverse outcomes in the child. However, few studies have investigated this relation in countries of low and middle income. Furthermore, to our knowledge, no studies have followed up cohorts into later childhood. We aimed to investigate whether maternal depression 6 months after birth is associated with psychological difficulties in a socioeconomically disadvantaged South African cohort of children at age 10 years. METHODS: Birth to Twenty is a prospective, longitudinal, birth-cohort study based in the Soweto area of Johannesburg, South Africa. Mothers and children in this cohort have been followed up at timepoints ranging from before birth to age 10 years. Maternal mood was measured at 6 months with the Pitt depression inventory and at 10 years with the Centre for Epidemiologic Studies depression scale (CES-D). Child psychological functioning was assessed at 10 years with the South African child assessment schedule (SACAS). Our primary outcome was psychological development of children at age 10 years, measured by total score on the SACAS. Secondary outcomes were scores on externalising and internalising subscales of the SACAS. We used t tests to compare psychological outcomes between children whose mother had postnatal depression at 6 months and those whose mother did not have postnatal depression. We examined associations between maternal postnatal depression and child psychological outcomes by multivariate linear-regression analysis, adjusting for socioeconomic status and maternal depression at 10 years, and we used logistic regression to provide odds ratios for associations identified by linear regression. FINDINGS: 1866 mothers completed the Pitt depression inventory 6 months after the birth of their child; of these, 453 (24%) had symptoms of postnatal depression. At the 10-year assessment, 1012 mothers completed the CES-D questionnaire, of whom 747 (74%) were judged to have depression. Sociodemographic characteristics did not differ between mothers with and without depression at both 6 months and 10 years. After adjusting for socioeconomic status and maternal depression at 10 years, children whose mothers had postnatal depression at 6 months were more than twice as likely to have significant psychological difficulties 10 years later compared with children whose mothers did not have postnatal depression at 6 months (adjusted odds ratio 2·26, 95% CI 1·23-4·16). INTERPRETATION: Maternal postnatal depression is associated with adverse psychological outcomes in children up to 10 years later in countries of low and middle income. In view of the increased prevalence of postnatal depression in these settings, this finding has important implications for policy and interventions for children and their mothers. FUNDING: Wellcome Trust (UK), Medical Research Council of South Africa, Human Science Research Council (South Africa), University of the Witwatersrand