Several fixed point results of mappings on bipolar metric space have been discussed in the literature, and this has become an interesting area to many researchers because of its theoretical tool to allow diverse uses in various disciplines, such as biology, game theory, engineering and so on,. In this paper, we make further extension of some existing mappings in the literature to bipolar metric space and also introduce Hardy and Roger mappings on bipolar metric spaces with applications to integral equations. The results obtain generalized and complements some existing works in literature.

We introduce the Laguerre-type 2nd kind hypergeometric Bernoulli polynomials and numbers. After showing their recursive computation, we exploit the Laguerre-type Blissard problem to derive a representaion formula of the relevant numbers in terms of Bell’s polynomials.

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.