It is well known that positive Green’s operators are not necessarily positivity preserving. This result is important, because many physical problems require positivity in their solutions in order to make sense. In this paper we investigate the matter of just how far from being positivity preserving a positive Green’s operator can be. In particular, we will see that there exists positive Green’s operators that takes some positive functions to functions with negative mean values. We will also identify a broad class of Green’s operators that are not necessarily positivity preserving but have properties related to positivity preservation that one expects from positivity preserving Green’s operators. Finally, we will compare the results contained in this paper with those that already exist in the literature on the subject.

Faces in graphs play a crucial role in understanding the structural properties of planar graphs. They represent the regions or bounded areas formed by the edges of the graph when it is embedded in the plane. The concept of faces provides insights into the connectivity and layout of systems, helping analyze the geometry and topology of networks, communication systems, and various real-world applications. In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances. In graph theory, the concept of resolvability plays a significant role in identifying distinct elements within a graph based on distances. Let \( G \) be a connected planar graph with vertex \( V(G) \), edge set \( E(G) \), and face set \( F(G) \). The distance between a face \( f \) and a vertex \( v \) is defined as the minimum distance from \( v \) to any vertex incidence to \( f \). In this work, we introduce a new resolvability parameter for connected planar graphs, referred to as the face metric dimension. A face-resolving set \( R \subseteq V(G) \) is a set of vertices such that for every pair of distinct faces \( f_1, f_2 \in F(G) \), there exists at least one vertex \( r \in R \) for which the distances \( d(f_1, r) \) and \( d(f_2, r) \) are distinct. The face metric dimension of \( G \), denoted \( \ fmd(G) \), is the minimum cardinality of a face-resolving set. This new metric provides insight into the structure of planar graphs and offers a novel perspective on the analysis of graph resolvability.

This research investigates the influence of the Cattaneo-Christov double diffusive flow of ferromagnetic hybrid nanofluids, taking into account heterogeneous-homogeneous chemical reactions, heat radiative flux, and the Soret-Dufour effect. The mathematical modeling of the system of equations results in the formulation of partial differential equations (PDEs). These PDEs were subsequently transformed into total differential equations (ODEs) via the application of similarity transformation. The resultant modified ODEs were addressed utilizing an innovative approach known as the spectral relaxation method (SRM). This methodology was employed to solve the system of ODEs in an iterative manner, following the Gauss-Seidel procedure. The findings of this investigation indicate that the heterogeneous-homogeneous chemical reaction significantly influences the fluid concentration, leading to a reduction in the concentration profile. An elevated level of thermal radiation was found to enhance both the fluid temperature and the velocity contour. Conversely, an increase in the magnetic field strength was noted to diminish the velocity contour. The current analysis was compared with previous studies and was found to exhibit a strong correlation.

In this paper, by using a new identity we establish some trapezoidal type inequalities for functions whose modulus of the first derivatives are \( \left( s,m\right)\)-preinvex via Caputo fractional derivatives.

This review provides a comprehensive overview of the synthesis process of nanoscale materials and highlights key characterization methods used for nanomaterials and biomaterials. It emphasizes the importance of effective techniques for investigating materials at the nanoscale, as these are too small for the human eye to detect. The review also explores various approaches to producing nanoscale materials and offers insights into the application, development, advantages, and limitations of different experimental methods for nanoparticle characterization. A particular focus is placed on advanced characterization techniques and their role in data interpretation, aiming to guide novice researchers in the field of nanoscience and nanotechnology.