A residual-set framework is introduced for analyzing additive prime conjectures, with particular emphasis on the Strong Goldbach Conjecture (SGC). For each even integer $E_n\ge 4$, the residual set $ℛ\left(E_n\right)=\{E_n-p\mid$ $p<E_n,p\in \mathbb{P}\}$ is defined, and the universal residual set ${ℛ}_{\mathbb{E}}={{{\displaystyle \cup }}^{\text{}}}_{E_n}ℛ\left(E_n\right)$ is constructed. It is shown that ${ℛ}_{\mathbb{E}}$ contains infinitely many primes. A nontrivial constructive lower bound is derived, establishing that the number of Goldbach partitions satisfies $G\left(E\right)\ge 2$ for all $E\ge 8$, and that the cumulative partition count satisfies ${\displaystyle \sum }_{E\le N}G\left(E\right)\gg \frac{N^2}{{\text{log}}^{4}N}$. An optimized deterministic algorithm is implemented to verify the SGC for even integers up to 16,000 digits. Each computed partition $E_n=p+q$ is validated using elliptic curve primality testing, and no exceptions are observed. Runtime variability observed in the empirical tests corresponds with known fluctuations in prime density and modular residue distribution. A recursive construction is formulated for generating Goldbach partitions, using residual descent and leveraging properties of the residual sets. The method extends naturally to Lemoine's Conjecture, asserting that every odd integer $n\ge 7$ can be expressed as $n=p+2q$, where $p,q\in \mathbb{P}$. A corresponding residual formulation is developed, and it is proven that at least two valid partitions exist for all $n\ge 9$. Comparative analysis with the Hardy-Littlewood and Chen estimates is provided to contextualize the cumulative growth rate. The residual-set methodology offers a deterministic, scalable, and structurally grounded approach to additive problems in prime number theory, supported by both theoretical results and large-scale computational evidence.

The Mambaka watershed is extends between latitudes 1 3°45'E and 14°15'E and longitudes 7°16'N and 6°45'N. The geology, various tectonic and structural events that have affected the Adamawa Plateau in Cameroon make it rich in multi-substance mining. The objective of this study is to map rare earth (REE) geochemical anomalies in the sediments of the watershed streams west of Mambaka, and to trace their origins and geochemical processes. Predictive maps from inverse distance interpolations (IDW), factor analysis (F1) or principal component analysis (PCA) and hierarchical bottom-up classification maps provided a better understanding of the central tendency, distribution and dispersion of REE in the samples and in the study area, based on standard deviation and variance values that generated two factors F1 (Ho-Tm-Er-Yb-Lu-Dy-Tb-Gd-Eu-Sm) and F2 (Pr-Nd-Ce-La-Sm) representing 92.44% of the total cumulative variance. The ratios Ce/Ce* > 0.78 and Eu/Eu* > 1 demonstrate positive anomalies in Ce and Eu, and clear differentiation. The normalized concentrations used to calculate fractionation ratios show that the values for LaN/YbN (0.58 to 1.34), LaN/SmN (0.61 to 0.88) and LaN/LuN (0.62 to 1.43) suggest higher fractionation in SS09 and lower fractionation in SS01. Similarly, the ratios La/Lu (61.71 to 143.46), La/Yb (9.00 to 20.72), La/Sm (4.02 to 5.83) and La/ Lu (61.71 to 143.46) confirm these higher ratios in SS09 and lower in SS01. The REE in the study area comes from hydrothermal processes based on high lineament densities at sampling points in igneous rocks with a mean ∑REE value of between 174-219 ppm.

Opuntia ficus-indica (L.) Mill., commonly known as prickly pear, is a versatile plant widely used for food, medicinal, and cosmetic purposes in various regions of the Mediterranean and North Africa. This study provides a comparative ethnobotanical review of prickly pear uses in Algeria, Morocco, Tunisia, and Italy. In total, 74 articles were selected from ethnobotanical uses in Italy, Algeria, Morocco and Tunisia. In the Maghreb, the plant is used for fruit production and processed products, such as jams, oils, and flours, as well as for traditional medicinal purposes. In Algeria and Morocco, the fruits and cladodes are used to treat digestive disorders, diabetes, and skin diseases. In Tunisia, products derived from O. ficus-indica are also applied in the cosmetic industry and for erosion control. In Italy, particularly in Sicily and Calabria, the prickly pear is a vital resource, used for both food consumption and for medicinal purposes. The cladodes, rich in mucilage, are applied as topical remedies for skin problems, while the fruits are a key ingredient in the preparation of traditional desserts. Furthermore, O. ficus-indica has historically been used as forage and to produce natural dyes. Results indicate that the versatility of this species, combined with its ability to adapt to extreme climates, makes it a valuable resource for the development of new nutraceutical and cosmetic products. However, further scientific research is necessary to explore the bio-functional potential of this plant and to promote its broader and more sustainable use, especially in arid and semi-arid regions.

Background: Financial well-being is a key domain of overall well-being, encompassing an individual's ability to meet financial obligations, secure their financial future, and maintain a sense of financial freedom. Education is often viewed as a critical pathway to enhancing financial well-being. However, the returns of education on financial well-being are not uniform across racial, ethnic, and nativity groups. The theory of Minorities’ Diminished Returns (MDRs) suggests that the positive effects of education on outcomes such as income and financial security are weaker for marginalized groups, including Black individuals, Latinos, and immigrants. Objective: This study examines the diminished returns of education on financial well-being among Black, Latino, and immigrant populations in the United States. We aim to investigate how structural inequalities contribute to weaker financial returns on education for these groups compared to their White and native-born counterparts. Methods: We utilized data from the Understanding America Study (UAS 2014) to conduct a cross-sectional analysis of adult respondents. The study assessed financial well-being outcomes (e.g., income, savings, and financial security) and their association with educational attainment across racial, ethnic, and nativity groups. Regression models were employed to test for interaction effects between education and race/ethnicity/nativity, adjusting for sociodemographic factors such as age, gender, employment, and family structure. Results: Our analysis included 8,121 individuals. The mean age of the respondents was 48 years (SD = 16). High education was associated with higher financial well-being (B = 1.284, 95% CI: 1.157, 1.410). The interaction terms between education and immigrant status (B = -0.507, 95% CI: -0.930, -0.084), race (Black) (B = -0.770, 95% CI: -1.208, -0.331), and ethnicity (Hispanic) (B = -0.589, 95% CI: -0.969, -0.210) were all significant, suggesting that immigrant, Black, and Hispanic individuals experience diminished returns on education in terms of financial well-being, relative to US-born non-Hispanic White individuals. The significant negative interactions between education and minority statuses (Black, Hispanic, and immigrant) indicate that while education generally improves financial well-being, the magnitude of this improvement is substantially smaller for these marginalized groups. Conclusion: Understanding how education translates to financial well-being across different racial, ethnic, and nativity groups is critical for addressing persistent financial disparities.