Cone Metric Research Articles

Metabolic, Microvascular, and Structural Predictors of Long-Term Functional Changes Evaluated by Multifocal Electroretinogram in Type 1 Diabetes

Background: This study aimed to analyze the potential pathogenic connection between metabolic factors, photoreceptor cell rearrangements, retinal microvascular perfusion, and functional parameters through multifocal electroretinography (mfERG) in type 1 diabetes mellitus (DM1). Methods: This prospective observational cohort study enrolled DM1 patients (40.5 ± 9.1 years) with mild nonproliferative diabetic retinopathy followed for 4 years. Patients were subjected to multimodal imaging, which included color fundus photography, optical coherence tomography (OCT), OCT angiography, adaptive optics (AO), and mfERG. OCTA slabs were analyzed using ImageJ software (software version 2.3.0/1.53f) to calculate perfusion density (PD) at both superficial (SCP) and deep (DCP) capillary plexuses, as well as flow deficit percentage (FD%) at the choriocapillaris (CC). To calculate cone metrics on AO at the parafovea, including cone density (CD), linear dispersion index (LDi), and heterogeneity packing index (Hpi%) in the parafovea, the images were post-processed using a MATLAB algorithm. The mfERG P1 implicit time (IT) and N1-P1 response amplitude density (RAD) from R1 (foveal area), R2 (parafoveal area), and the unified rings R1 + R2 were evaluated. Results: A total of 22 patients (22 eyes) were enrolled. No significant differences were noted in central mfERG amplitude and implicit time-averaged values (p > 0.05, all). The main factor influencing R1 IT was HbA1c, while R1 RAD was affected by Hpi and CC FD%. R1 + R2 IT was influenced by Hpi, LDi (p > 0.001, all), and modifications in the perfusion density in the SCP (p < 0.001) and DCP (p = 0.03) at the parafovea. In contrast, R1 + R2 RAD were associated with HbA1c (p = 0.02) and Hpi (p < 0.001). Conclusions: Microvascular changes and glucometabolic factors are key elements influencing the long-term morphofunctional alterations at the photoreceptor level in DM1.

Common Fixed Point Theorems in Dislocated Cone Metric Spaces

Common Fixed Point Theorems in Dislocated Cone Metric Spaces

Projectively induced Kähler cones over regular Sasakian manifolds

Motivated by a conjecture in Loi et al. (Math Zeit 290:599–613, 2018) we prove that the Kähler cone over a regular complete Sasakian manifold is Ricci-flat and projectively induced if and only if it is flat. We also obtain that, up to Da\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D_a$$\\end{document}—homothetic transformations, Kähler cones over homogeneous compact Sasakian manifolds are projectively induced. As main tool we provide a relation between the Kähler potentials of the transverse Kähler metric and of the cone metric.

Trudinger–Moser Inequalities on a Closed Riemann Surface with a Symmetric Conical Metric

Trudinger–Moser Inequalities on a Closed Riemann Surface with a Symmetric Conical Metric

The Liouville current of holomorphic quadratic differential metrics

In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J. Leininger proved in Bankovic and Leininger (Trans Am Math Soc 370:1867–1884, 2018) that for a fixed closed surface, there is an injection map from the space of flat cone metrics to the space of geodesic currents. We manage to show that metrics coming from holomorphic quadratic differentials can be distinguished from other flat metrics by just looking at the geodesic currents. The key idea is to analyze the support of Liouville current, which is a topological invariant independent of the metric, and get information about cone angles and holonomy. The holonomy part involves some subtlety of relationship between singular foliation and geodesic lamination. We also obtain a new proof of a classical result that almost all simple geodesics of a quadratic differential metric will be dense in the surface. Furthermore, for other flat cone metrics, there is no simple dense geodesic.

Schauder estimates for equations with cone metrics, II

Schauder estimates for equations with cone metrics, II

Analytical Analysis of Common Fixed Point Results in Fuzzy Cone Metric Spaces

The concept of fuzzy sets was introduced by Zadeh (1965) which marked the beginning of the evolution of fuzzy mathematics. The introduction of uncertainty in the theory of sets in a non-probabilistic manner opened up new possibilities for research in this field. Since then, many authors have explored the theory of fuzzy sets and its applications, leading to successful advancements in various fields such as mathematical programming, model theory, engineering sciences, image processing, and control theory. In this paper, we aim to improve and generalize some common fixed point theorems in fuzzy cone metric spaces, an extension of the well-known results given by Saif Ur Rahman and Hong Xu-Li (2017).

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings With Application

In this article, we prove some new fixed point theorems for contractive type mappings in the setting of complete cone metric spaces and provide examples to illustrate the concepts and results developed in the article. We consider some consequences of our results to establish fixed point theorems in the context of cone metric spaces. As an application of our results, periodic point results for the contractive type mappings are proved.

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings

The purpose of this article, is to establish some fixed point results for contractive type mappings in cone metric spaces. Examples are provided to support results and concepts presented herein. As an application of our results, we deduce other established fixed point theorems in cone metric spaces.

Metrization of soft metric spaces and its application to fixed point theory

<abstract><p>Soft set theory has attracted many researchers from several different branches. Sound theoretical improvements are accompanied with successful applications to practical solutions of daily life problems. However, some of the attempts of generalizing crisp concepts into soft settings end up with completely equivalent structures. This paper deals with such a case. The paper mainly presents the metrizability of the soft topology induced by a soft metric. The soft topology induced by a soft metric is known to be homeomorphic to a classical topology. In this work, it is shown that this classical topology is metrizable. Moreover, the explicit construction of an ordinary metric that induces the classical topology is given. On the other hand, it is also shown that soft metrics are actually cone metrics. Cone metrics are already proven to be an unsuccessful attempt of generalizing metrics. These results clarify that most, if not all, properties of soft metric spaces could be directly imported from the related classical theory. The paper concludes with an application of the findings, i.e., a new soft fixed point theorem is stated and proven with the help of the obtained homemorphism.</p></abstract>

Some Fixed Point Results of (α,β,φ,δ) Contractions in Extended Cone b-Metric Spaces

This paper investigates the existence and uniqueness of fixed points for a class of generalized contractive mappings defined on extended cone metric spaces. Subsequently, we define and explore (α, β, φ, δ)-contractions, a generalization of traditional contractions that allows for a more nuanced understanding of the contraction behavior in extended cone metric spaces. The extended metric space was defined for the first time in 2017, by Kamran et al. (2017). They replaced the constant in the triangle inequality of the metric with a two-variable function and explored various fixed point theorems. In 2022, Das and Bag[4] introduced extended cone metric spaces by incorporating a three-variable map into the third condition of the cone metric. Afterwards, Selko and Sila introduced the concept of extended quasi cone b-metric spaces and demonstrated the Banach contraction within this framework. In 2018, Alqahtani. (2018) and colleagues established several fixed point results for a pair of orbital cyclic functions in an extended metric space. In this paper we prove some fixed point theorems for -orbital-cyclic functions in extended cone metric spaces by using the continuous map φ and a nonnegative constant δ.

On Fixed Point Results for Nonlinear Contractions in Fuzzy Cone Metric Space

The study of Fixed Point Theory in various metric space has been on focus of scientific development for many authors. It has been advanced either by generalizing the contractive inequality or by extending the conditions of metric. Fuzzy metric space has been defined as space in which the distance between elements is not an exact number in difference with metric space. Fixed point Theory is an important framework point of view in fuzzy metric spaces. Many studies have been showed the existence and uniqueness of a fixed point for different type of contractions in these spaces. Nonlinear contractions and their generalizations have been under investigations in several metric spaces. The aim of this paper is the study of fixed points for generalized nonlinear contractions in fuzzy metric space. Our results guarantee the existence and uniqueness of a fixed point for these contractions and extend some known theorems in metric space to fuzzy metric space. As an application of main theorem an example is taken.

On circle patterns and spherical conical metrics

The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the nonuniqueness coming from Möbius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the total geodesic curvature of each circle instead of the cone angles.

Structure-function correlation of retinal photoreceptors in PRPH2-associated central areolar choroidal dystrophy patients assessed by high-resolution scanning laser imaging and microperimetry.

High Magnification Module (HMM™, Heidelberg Engineering, Heidelberg, Germany) imaging is a novel technique, designed to visualize the retina at a cellular level. To assess the potential of HMM™-based metrics as endpoints for future trials, we evaluated correlations between structural HMM™ cone metrics, spectral-domain OCT (SD-OCT, Heidelberg Engineering, Heidelberg, Germany) and retinal sensitivity on microperimetry (MP, MAIA, CenterVue, Padova, Italy) in healthy subjects and p.(Arg142Trp) PRPH2-associated Central Areolar Choroidal Dystrophy (CACD) patients. We projected a default 10° MP grid on composite HMM™ images and performed automated cone density (CD), intercell distance (ICD) and nearest neighbour distance (NND) analysis at stimuli located at 3° and 5° retinal eccentricity. We manually measured intrasubject outer retinal thickness on SD-OCT in absolute and relative scotomas, located outside of focal atrophy. We included 15 CACD patients and five healthy subjects. We found moderate-to-strong correlations of HMM™ metrics and MP sensitivity at 3° eccentricity from the fovea. We found the outer retina at the locations of absolute scotomas to be statistically significant thinner (p = 0.000003, one-sample t-test), as the outer retinal thickness at locations of relative scotomas. Interestingly, HMM™ metrics of these areas did not differ significantly. We found significant correlations between structural photoreceptors metrics on HMM™ imaging and retinal sensitivity on MP in healthy subjects and CACD patients. A multimodal approach, combining SD-OCT, MP and HMM™ imaging, allows for detailed mapping of retinal photoreceptor integrity and restitution potential, important data that could serve as biomarkers in future clinical trials.

Common Fixed Point Theorems of Integral Type Contraction on Cone Metric Spaces and Applications

Common Fixed Point Theorems of Integral Type Contraction on Cone Metric Spaces and Applications

Fuzzy Fixed Point Theorems in Normal Cone Metric Space

In this paper, we proved a few fuzzy fixed point theorems in whole regular cone metric spaces, which can be the generalization of a few current consequences within side the literature.

Almost Contraction Mappings and $(S,T)$-Stability of Jungck Iteration in Cone Metric Spaces over Banach Algebras

In this work, we first introduce almost contraction mappings for a pair of two mappings in cone metric spaces over Banach algebras (CMSBA). Then, we observe that the class of such mappings in this setting contains those of many well known mappings. Finally, we prove some fixed point theorems, and obtain $(S,T)$-stability results of Jungck iterations for some mappings in CMSBA.

On Quasi I-Statistical Convergence of Triple Sequences in Cone Metric Spaces

Fast [12] is credited with pioneering the field of statistical convergence. This topic has been researched in many spaces such as topological spaces, cone metric spaces, and so on (see, for example [19, 21]). A cone metric space was proposed by Huang and Zhang [17]. The primary distinction between a cone metric and a metric is that a cone metric is valued in an ordered Banach space. Li et al. [21] investigated the definitions of statistical convergence and statistical boundedness of a sequence in a cone metric space. Recently, Sakaoğlu and Yurdakadim [29] have introduced the concepts of quasi-statistical convergence. The notion of quasi I-statistical convergence for triple and multiple index sequences in cone metric spaces on topological vector spaces is introduced in this study, and we also examine certain theorems connected to quasi I-statistically convergent multiple sequences. Finally, we will provide some findings based on these theorems.

Adaptive Optics Flood Illumination Ophthalmoscopy in Nonhuman Primates: Findings in Normal and Short-term Induced Detached Retinae

To describe adaptive optics flood illumination ophthalmoscopy (AO-FIO) of the photoreceptor layer in normal nonhuman primates (NHPs) and in the case of a short-term induced retinal detachment (RD). Longitudinal fundamental research study. Four NHPs were used to image normal retinae with AO-FIO (in comparison with 4 healthy humans); 2 NHPs were used to assess the effects of RD. The photoreceptor layer (cone mosaic metrics, including cone density, cone spacing, and cone regularity) was followed with AO-FIO imaging (rtx1, Imagine Eyes) during a surgically induced RD in 2 NHPs using a vehicle solution containing dimethyl sulfoxide, classically used as a chemical solvent. We also performed functional testing of the retina (full-field and multifocal electroretinogram [ERG]). Correlation of cone mosaic metrics (cone density, spacing, and regularity) between normal retinae of NHPs and humans, and cone metrics, power spectrum, and ERG wave amplitudes after RD. Imaging features were very similar in terms of cone reflectivity, cell density, regularity, and spacing values, showing strong positive correlations between NHPs and humans. After RD, AO-FIO revealed several alterations of the cone mosaic slowly recovering during the 3 months after the reattachment, which were not detected functionally by ERG. These results demonstrate by invivo AO-FIO imaging the transient structural changes of photoreceptors after an RD in the primate retina. They also provide an interesting illustration of the AO-FIO potential for investigating photoreceptor toxicity during preclinical studies in NHPs with a high translatability to human studies. Proprietary or commercial disclosure may be found after the references.

Generalized Fixed Points for Four Self-mappings with the Property OWC in CMS

In this research article, we obtained fixed points for Four self – mappings with the property Occasionally Weakly Compatible (OWC) in Cone Metric Spaces (CMS). Our results are generalized, improved some of the results in this article references.