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After the appearance of relation-theoretic contraction principle proved in a metric space equipped with an amorphous binary relation (often termed as relational metric space), various core fixed point results have been proved in the setting of different relational distance spaces by varying underlying contraction conditions. In proving such results, the notions of completeness of ambient space, continuity of involved mapping and d-self-closedness of underlying binary relation are of paramount importance. The aim of this paper is to further refine the relation-theoretic contraction principle by relaxing the conditions of completeness and continuity by replacing their respective relation-theoretic analogues. Moreover, we observe that the notion of d-self-closedness utilized in relation-theoretic contraction principle is more general than the concepts of regularity and strong regularity utilized by earlier authors.

Microscopy is mostly the method of choice to analyse biofilms. Due to the high local heterogeneity of biofilms, single and punctual analyses only give an incomplete insight into the local distribution of biofilms. In order to retrieve statistically significant results a quantitative method for biofilm thickness measurements was developed based on confocal laser scanning microscopy and the programming language R. The R‐script allows the analysis of large image volumes with little hands‐on work and outputs statistical information on homogeneity of surface coverage and overall biofilm thickness. The applicability of the script was shown in microbial fuel cell experiments. It was found that Geobacter sulfurreducens responds differently to poised anodes of different material so that the optimum potential for MFC on poised ITO anodes had to be identified with respect to maximum current density, biofilm thickness and MFC start‐up time. Thereby, a positive correlation between current density and biofilm thickness was found, but with no direct link to the applied potential. The optimum potential turned out to be +0.1 V versus SHE. The script proved to be a valuable stand‐alone tool to quantify biofilm thickness in a statistically valid manner, which is required in many studies.

In the present research note, we discuss the energy–momentum squared gravity model F ( R , T 2) coupled with perfect fluid. We obtain the equation of state for the perfect fluid in the F ( R , T 2) -gravity model. Furthermore, we deal with the energy–momentum squared gravity model F ( R , T 2) coupled with perfect fluid, which admits the hyperbolic Ricci solitons with a conformal vector field. We provide a clue in this series to determine the density and pressure in the radiation and phantom barrier periods, respectively. Also, we investigate the rate of change in hyperbolic Ricci solitons within the same vector field. In addition, we determine the different energy conditions, black holes and singularity conditions for perfect fluid attached to F ( R , T 2) -gravity in terms of hyperbolic Ricci solitons. Lastly, we deduce the Schrödinger equation for the potential U n with hyperbolic Ricci solitons in the F ( R , T 2) -gravity model coupled with perfect fluid and a phantom barrier.

In the present research note, we explore the nature of the conformal Ricci solitons on the energy–momentum squared gravity model F(R,T2) that is a modification of general relativity. Furthermore, we deal with a subcase of the F(R,T2)=R+λT2-gravity model coupled with a perfect fluid, which admits conformal Ricci solitons with a time-like concircular vector field. Using the steady conformal Ricci soliton, we derive the equation of state for the perfect fluid in the F(R,T2)-gravity model. In this series, we convey an indication of the pressure and density in the phantom barrier period and the stiff matter era, respectively. Finally, using a conformal Ricci soliton with a concircular vector field, we study the various energy constraints, black holes, and singularity circumstances for a perfect fluid coupled to F(R,T2)-gravity. Lastly, employing conformal Ricci solitons, we formulate the first law of thermodynamics, enthalpy, and the particle production rate in F(R,T2)-gravity and orthodox gravity.

In this paper, we introduce the concept of R-nonexpansive self-mappings defined on a suitable subset K of a Banach space, wherein R stands for a transitive binary relation on K, and utilize the same to prove a relation-theoretic variant of classical Browder–Göhde fixed point theorem. As consequences of our newly proved results, we are able to derive several core fixed-point theorems existing in the literature.

This article explores wormhole solutions within the framework of Finsler geometry and the modified gravity theory. Modifications in gravitational theories, such as f( R , T ) gravity, propose alternatives that potentially avoid the exotic requirements. We derive the field equations from examining the conditions for Finslerian wormhole existence and investigate geometrical and material characteristics of static wormholes using a polynomial shape function in Finslerian space–time. Furthermore, we address energy condition violations for different Finsler parameters graphically. We conclude that the proposed models, which assume a constant redshift function, satisfy the necessary geometric constraints and energy condition violations indicating the presence of exotic matter at the wormhole throat. We also discuss the anisotropy factors of the wormhole models. The results are validated through analytical solutions and 3-D visualizations, contributing to the broader understanding of wormholes in Finsler-modified gravity contexts.

In this research paper, we investigate a Finsler-Randers spacetime in the context of a Bianchi type-V model of universe within the framework of Lyra geometry, employing a modified f( R , T ) gravity theory that incorporates a cosmological constant Λ . We have derived the corresponding anisotropic Friedmann equations for the Finsler–Randers Bianchi type-V model of universe with modified f( R , T ) gravity in Lyra geometry, including the contributions of the cosmological constant and Randers anisotropic terms b0(t) and obtained analytical solutions. Further, we have examined the behavior of various dynamical parameters, commonly used in cosmological analysis, both geometrical and graphical interpretations have been provided. Furthermore, we have derived the Raychaudhuri equation in terms of the cosmological constant as a function of the cosmic time t. Our analysis reveals that the shear scalar σ2 and the scalar expansion θ decrease with cosmic time and tend to zero at late times, indicating the isotropization of the universe in the presence of the cosmological constant; however, the Hubble parameter approaches a constant value rather than vanishing, while the energy density ρ , pressure P , and the Lyra gauge function β remain finite and non-zero even at large cosmic times. Ultimately, we conclude that the universe described by this framework exhibits continuous acceleration, as indicated by the negative value of the deceleration parameter q.

The importance of 16S rRNA gene amplicon profiles for understanding the influence of microbes in a variety of environments coupled with the steep reduction in sequencing costs led to a surge of microbial sequencing projects. The expanding crowd of scientists and clinicians wanting to make use of sequencing datasets can choose among a range of multipurpose software platforms, the use of which can be intimidating for non-expert users. Among available pipeline options for high-throughput 16S rRNA gene analysis, the R programming language and software environment for statistical computing stands out for its power and increased flexibility, and the possibility to adhere to most recent best practices and to adjust to individual project needs. Here we present the Rhea pipeline, a set of R scripts that encode a series of well-documented choices for the downstream analysis of Operational Taxonomic Units (OTUs) tables, including normalization steps, alpha - and beta -diversity analysis, taxonomic composition, statistical comparisons, and calculation of correlations. Rhea is primarily a straightforward starting point for beginners, but can also be a framework for advanced users who can modify and expand the tool. As the community standards evolve, Rhea will adapt to always represent the current state-of-the-art in microbial profiles analysis in the clear and comprehensive way allowed by the R language. Rhea scripts and documentation are freely available at https://lagkouvardos.github.io/Rhea .

f( R ,T) -gravity is a generalization of Einstein’s field equations (EFEs) and f( R ) -gravity. In this research article, we demonstrate the virtues of the f( R ,T) -gravity model with Einstein solitons (ES) and gradient Einstein solitons (GES) . We acquire the equation of state of f( R ,T) -gravity, provided the matter of f( R ,T) -gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a f( R ,T) -gravity filled with perfect fluid admits an Einstein soliton (g,ρ,λ) and the Einstein soliton vector field ρ of (g,ρ,λ) is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the f( R ,T) -gravity model. Next, we prove that if a f( R ,T) -gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the f( R ,T) -gravity model together with gradient Einstein soliton.

Camera trapping has become increasingly common in ecological studies, but is hindered by analyzing large datasets. Recently, artificial intelligence (deep learning models in particular) has emerged as a promising solution. However, applying deep learning for images processing is complex and often requires programming skills in Python, reducing its accessibility. Some authors addressed this issue with user‐friendly software, and a further progress was the transposition of deep learning to R, a statistical language frequently used by ecologists, enhancing flexibility and customization of deep learning models without advanced computer expertise. We aimed to develop a user‐friendly workflow based on R scripts to streamline the entire process, from selecting to classifying camera trap images. Our workflow integrates the MegaDetector object detector for labelling images and custom training of the state‐of‐the‐art YOLOv8 model, together with potential for offline image augmentation to manage imbalanced datasets. Inference results are stored in a database compatible with Timelapse for quality checking of model predictions. We tested our workflow on images collected within a project targeting medium and large mammals of Central Italy, and obtained an overall precision of 0.962, a recall of 0.945, and a mean average precision of 0.913 for a training set of only 1000 pictures per species. Furthermore, the custom model achieved 91.8% of correct species‐level classifications on a set of unclassified images, reaching 97.1% for those classified with > 90% confidence. YOLO, a fast and light deep learning architecture, enables application of the workflow even on resource‐limited machines, and integration with image augmentation makes it useful even during early stages of data collection. All R scripts and pretrained models are available to enable adaptation of the workflow to other contexts, plus further development. Our manuscript presents a R workflow to streamline camera trap images classification at the species level, combining image augmentation and deep learning models in a user‐friendly environment. Tested on a dataset from Central Italy, the procedure achieved over 90% accuracy in classifying 17 target mammals, humans, and vehicles, starting from just 1000 images per category, significantly reducing classification time compared to manual methods.