Necessary and sufficient conditions for the existence of the solutions of a class of scalar and mainly for operator-valued moment problems are reviewed. This was the first motivation for proving our constrained extension results for linear operators. Polynomial approximations on bounded and on unbounded closed subsets are very useful in proving the uniqueness of the solution. We also reviewed earlier results on the extension of positive linear functional and operators. Such results are applied to ensure the extension of our linear solution from the subspace of polynomials to a larger function space. In most of the cases from below, this is made using polynomial approximation in one and several variables. Besides positivity, our solution is bounded from above by a dominating linear, sublinear or only convex continuous operator, on the entire domain space or only on its positive cone. This allows estimating the norm of the linear solution.

Let \(E(G)\) and \(d_x\) denote the edge set and degree of a vertex \(x\) in \(G\), respectively. Recently, the elliptic Sombor index has been defined as \[ESO(G) = \sum_{xy \in E(G)} (d_x + d_y) \sqrt{d_x^2 + d_y^2}\,.\] A molecular tree is a tree in which the maximum degree does not exceed \(4\). In this paper, we establish sharp upper and lower bounds for the \(ESO\) index in the class of molecular trees with order \(n\) and exactly \(k\) vertices of maximum degree \(\Delta \geq 2\). Moreover, we completely characterize the extremal trees attaining these bounds. Our findings contribute to the structural analysis of molecular trees and further the understanding of the elliptic Sombor index in chemical graph theory.

Background:Teledermatology consists of a trained clinical photographer taking a series of digital images of a patient’s skin lesion. The images are remotely reviewed by a dermatologist to decide if additional in-office care is required. Objective: To analyze the scope and patterns of cutaneous cases received through social media websites. Patients and Methods: The research involved male and female patients of all ages—infants, children, and adults—who presented clinical photographs of dermatological lesions through WhatsApp. Results: The clinical information and photographs showed a varied range of dermatological disorders. Conclusion: Social media has revolutionized medical communication significantly, lowering distance-related barriers. All consultations, diagnoses, treatments, and follow-ups were provided for free, as a service to society. Alhamdulillah.

Zaslavsky (1991) characterized all single-element coextensions of graphic matroids in terms of a graphical structure called a biased graph. In this paper we characterize all orientations of a single-element coextension of a graphic matroid in terms of graphically defined orientations of its associated biased graph.

A series of novel 2-methyl-5-[2-(substituted)phenyl]-1,3,4-oxadiazole derivatives (6a–6i) was synthesized via a multi-step protocol starting from commercially available salicylic acid (1). The initial esterification of salicylic acid using thionyl chloride and ethanol at 80 °C for 12 h yielded ethyl 2-hydroxybenzoate (2), which was subsequently converted to 2-hydroxybenzohydrazide (3) upon treatment with hydrazine monohydrate in ethanol at 80 °C for 2 h. Acetylation of intermediate 3 with acetic anhydride afforded N’-acetyl-2-hydroxybenzoate (4), which was reacted with various halo compounds (4a–4i) to produce a series of N’-acetyl-2-(substituted)oxybenzohydrazides (5a–5i). These key intermediates were cyclized using triphenylphosphine, triethylamine, carbon tetrachloride, and acetonitrile at 100 °C for 1 h to furnish the final oxadiazole derivatives (6a–6i). The compounds were purified using appropriate chromatographic techniques and fully characterized by ¹H NMR, ¹³C NMR, FTIR, and mass spectrometry. Biological screening of the synthesized compounds revealed that several derivatives, particularly 6c, 6d, and 6g, exhibited promising antimicrobial and antioxidant activities. Notably, compound 6a demonstrated significant cytotoxicity against HeLa cancer cells. Molecular docking studies further supported the biological potential of the compounds, with 6e displaying a high docking score of –5.66 kcal/mol.

Shea butter is a natural moisturizer used to improve skin health. This study investigates its effects on skin barrier function, hy-dration, and lipid profile, using analytical chemistry methodologies. Trans-Epithelial Water Loss (TEWL), corneometry, im-pedance spectroscopy, and gas chromatography-mass spectrometry (GC-MS) were used to assess skin barrier function, hydra-tion, and lipid profile after shea butter application. Results show that TEWL decreased by 37.8% after 24 hours (p < 0.01), Skin hydration increased by 58% after 24 hours (p< 0.001). Impedance spectroscopy showed a 33% increase in skin imped-ance, GC-MS analysis revealed a balanced fatty acid composition in shea butter, ceramide profiling showed six subclasses, with Ceramide 1 and 2 being the most abundant. The results demonstrate shea butter's efficacy in improving skin barrier func-tion, hydration, and lipid profile. The rapid decrease in TEWL and increase in skin hydration suggest immediate effects on skin lipids. Shea butter's fatty acid composition contributes to its moisturizing and barrier-enhancing properties. This study provides evidence for the benefits of shea butter in maintaining healthy skin. Its natural moisturizing properties make it a valuable in-gredient in skincare products.