Cone Metric Research Articles

Theta Cone Metric Spaces and Some Fixed Point Theorems

This paper introduces the concept of the theta cone metric, studies its various topological properties, and gives some examples of it. Furthermore, it proves some lemmas and then uses them to give further generalizations of some well-known fixed point theorems. Specifically, Theorem 2 of the paper is a generalization of Reich’s fixed point theorem.

Fixed Point Results for Extensions of Orthogonal Contraction on Orthogonal Cone Metric Space

In this paper, some fixed point results of self mapping which is defined on orthogonal cone metric spaces are given by using extensions of orthogonal contractions. And by taking advantage of these results, the necessary conditions for self mappings on orthogonal cone metric space to have P property are investigated. Also an example is given to illustrate the main results.

Length spectra of flat metrics coming from $q$-differentials

When geometric structures on surfaces are determined by the lengths of curves, it is natural to ask: which curves’ lengths do we really need to know? It is a result of Duchin, Leininger, and Rafi that any flat metric induced by a unit-norm quadratic differential is determined by its marked simple length spectrum. We generalize the notion of simple curves to that of q - simple curves , for any positive integer q , and show that the lengths of q -simple curves suffice to determine a non-positively curved Euclidean cone metric induced by a q -differential.

Vielbein with Mixed Dimensions and Gravitational Global Monopole in the Planar Phase of Superfluid 3He

The planar phase of superfluid $^3$He has Dirac points in momentum space and the analog of Dirac monopole in the real space. Here we discuss the combined effect of Dirac point and Dirac monopole. It is shown that in the presence of the monopole the effective metric acting on Dirac fermions corresponds to the metric produced by the global monopole in general relativity: it is the conical metric. Another consequence is that the primary variable, which gives rise to the effective metric, is the unusual vielbein field in the form of the $4\times 5$ matrix, as distinct from the conventional $4\times 4$ matrix of the tetrad field in tetrad gravity.

Neutrosophic Fixed Point Theorems and Cone Metric Spaces

Neutrosophic Fixed Point Theorems and Cone Metric Spaces

Common Fixed Point Theorems in Intuitionistic Generalized Fuzzy Cone Metric Spaces

In the present work, we study many fixed point results in intuitionistic generalized fuzzy cone metric space. Precisely, we prove new common fixed point theorems for three self mappings that do not require any commutativity or continuity but a generalized contractive condition. Our results are generalizations for many results in the literature. Some examples are given to support these results.

Asymptotic behavior at the isolated singularities of solutions of some equations on singular manifolds with conical metrics

We present the sharp characterization of the behavior at the isolated singularities of positive solutions of some equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in where is the unit ball. The weights can be singular at x = 0. We present the sharp asymptotic behavior of positive solutions of the weighted elliptic equations at x = 0 and establish expansions of these solutions up to arbitrary orders. Asymptotic behavior at the isolated singularitie of positive solutions of elliptic equations without weights has been studied by many authors. We will obtain new results on the asymptotic behavior at the isolated singularities even for positive solutions of equations without weights in the subcritical case.

Common Fixed Point Theorem for T-Contraction with c-Distance on Cone Metric Spaces

Common Fixed Point Theorem for T-Contraction with c-Distance on Cone Metric Spaces

Bôcher-type theorems for the Poisson’s equation on manifolds with conical metrics

We prove a generalized Bocher-type theorem for the nonhomogeneous Laplace’s equations on singular manifolds with conical metrics. More specifically, we give a sharp characterization of the behavior at isolated singularities of a solution bounded on one side for the equation $$\Delta _g u =f$$ ( $$f \in L_g^q$$ with $$q > \frac{n}{2}$$ ). The main results in this paper imply that a nonnegative solution with conical singularities has certain of the attributes of fundamental solutions.

Cone Metric Spaces over Topological Modules and Fixed Point Theorems for Lipschitz Mappings

In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra.

Holography, matrix factorizations and K-stability

Placing D3-branes at conical Calabi-Yau threefold singularities produces many AdS5/CFT4 duals. Recent progress in differential geometry has produced a technique (called K-stability) to recognize which singularities admit conical Calabi-Yau metrics. On the other hand, the algebraic technique of non-commutative crepant resolutions, involving matrix factorizations, has been developed to associate a quiver to a singularity. In this paper, we put together these ideas to produce new AdS5/CFT4 duals, with special emphasis on non-toric singularities.

The Banach Type Contraction for Mappings on Algebraic Cone Metric Spaces Associated with An Algebraic Distance and Endowed with a Graph

The Banach Type Contraction for Mappings on Algebraic Cone Metric Spaces Associated with An Algebraic Distance and Endowed with a Graph

Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity

We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete.

Cone mosaic characteristics in red-green colour deficiency: a comparative study

ObjectiveTo compare photoreceptor cone characteristics using adaptive optics (AO) technology between congenital colour blind patients and healthy age-matched controls. DesignCross-sectional. Participants21 eyes (21 cases) with congenital red-green colour blindness and 21 eyes of age-matched controls (21 controls), all having a best-corrected visual acuity of 20/20 and normal ophthalmological examination, were recruited. MethodsColour vision testing was done using the Farnsworth–Munsell 100-Hue test and Nagel anomaloscope. Imaging of fovea was done using rtx1 AO retinal camera with a sampling window size of 4° × 4°. Peak cone density, intercone spacing, cone regularity, and cone dispersion were determined using AOdetect Mosaic software. ResultsOverall, mean cone characteristics at 2° off fixation in cases and controls, respectively, were density 26 587 ± 1328/26 448 ± 687 cells/mm2 (p = 0.67), spacing 6.69 ± 0.17/6.75 ± 0.11 μm (p = 0.19), regularity 94.36 ± 2.56/94.83 ± 1.28% (p = 0.46), dispersion 10.22 ± 1.21/11.68 ± 0.79% (p = 0.0001), and hexagonality 50.62 ± 4.21/49.52 ± 2.45% (p = 0.31). No significant difference was noted in various cone metrics except dispersion, which was markedly decreased in colour-deficient individuals. At 4° off fixation, the cone characteristics were not significantly different between cases and controls: density 16 521 ± 1230/16 930 ± 678 cells/mm2 (p = 0.37), spacing 8.60 ± 0.29/8.46 ± 0.17 μm (p = 0.19), regularity 92.68 ± 2.83/92.6 ± 3.05% (p = 0.95), dispersion 13.43 ± 3.48/13.65 ± 2.38% (p = 0.87), and hexagonality 45.22 ± 3.8/45.21 ± 5.5% (p = 0.99). A significant reduction in cone density, regularity, and hexagonality was associated with corresponding increase in spacing and dispersion as we moved from 2° to 4° off fixation in both cases and controls. ConclusionCone photoreceptor characteristics are anatomically well preserved in young colour-deficient individuals except cone dispersion.

ATTENUATION OUTER RETINAL BANDS ON OPTICAL COHERENCE TOMOGRAPHY FOLLOWING MACULAR EDEMA: A Possible Manifestation of Photoreceptor Misalignment.

Macular edema is a common retinal disease which may leave important anatomical and functional sequelaes. Directional fundus imaging consists of comparing on- and off-axis images to reveal angle-dependent reflectance properties of fundus structures, which may be related to misaligned photoreceptors. Here, we analyzed directional optical coherence tomography (OCT) and flood-illumination adaptive optics ophthalmoscopy images to detect evidence of misaligned photoreceptors following macular edema. Transversal, observational study. Nine patients having recovered a normal macular profile after macular edema due to retinal vein occlusion were included. For each patient, a reference OCT scan (i.e., with the incident beam normal to the fovea) was acquired, and off-axis scans were then acquired by laterally shifting the entry pupil. In addition, in four of these eyes, directional adaptive optics ophthalmoscopy documented the directional variations of cone metrics. Included patients comprised two women and seven men (age range, 19-76 years). Reference OCT scans showed patchy attenuation of the cone outer segment tips and to a lesser extent of the inner segment/outer segment lines in all, but two eyes; these. Increased intensity of the cone outer segment tips and inner segment/outer segment lines could be observed on off-axis scans. Accordingly, fusion images showed 66% average reduction of the length of cone outer segment tips attenuation. In two cases, although reference scans showed continuity of outer bands, focal attenuation was evidenced in off-axis images. Directional adaptive optics ophthalmoscopy imaging showed a strong directional variability of cone counts in these areas, ranging from near absence to roughly two-third of reference values. In each case, directional variations of cone counts paralleled those of the reflectance of outer bands. After macular edema, focal attenuations of the inner segment/outer segment and of the cone outer segment tips lines may be present on OCT. These areas may show a strong directional variability by both OCT and adaptive optics ophthalmoscopy, suggesting that misaligned photoreceptor outer segments contribute to such features. The evaluation of outer retinal damage following macular edema should therefore take into account the optical Stiles-Crawford effect to disambiguate missing from misaligned cones.

Bôcher-type results for the fourth and higher order equations on singular manifolds with conical metrics

We obtain the Bocher-type theorems and present the sharp characterization of the asymptotic behavior at the isolated singularities of solutions of some fourth and higher order equations on singular manifolds with conical metrics. It is seen that the equations on singular manifolds with conical metrics are equivalent to weighted elliptic equations in \begin{document}$ B \backslash \{0\} $\end{document} , where \begin{document}$ B \subset \mathbb{R}^N $\end{document} is the unit ball. The weights can be singular at \begin{document}$ x = 0 $\end{document} . We present the sharp asymptotic behavior of nonnegative solutions of the weighted elliptic equations near \begin{document}$ x = 0 $\end{document} and the Liouville-type results for the degenerate elliptic equations in \begin{document}$ \mathbb{R}^N \backslash \{0\} $\end{document} .

Infructescence and samara morphometrics and potential mechanism of samara release in Allocasuarina and Casuarina (Casuarinaceae)

In addition to samara aerodynamics, mechanisms of samara release will contribute to seed dispersal outcomes. Analysis of species descriptions of Australian members of the Casuarinaceae (sheoaks) indicated cone shrinkage to be a possible mechanism enhancing samara release. However, measurement of cones of nine species of Allocasuarina and Casuarina did not support this hypothesis, but outward movement of samaras both in the field and in oven-dried cones indicated that another mechanism was operating. Microscopic examination revealed that the proximal margins of the bracteoles are involute after drying, and it is postulated that incurving of the bracteoles during drying moves samaras outward against gravity enhancing samara release. This study also revealed inconsistencies in cone and samara metrics between field samples (1–3 specimens for 9 species) and the published descriptions. This could lead to identification uncertainties, and indicates the need to revise the descriptions to more fully encompass the variation in sheoaks across their natural and anthropogenic ranges.

Fixed Point Theorems for T-Contractions with c-Distance on Cone Metric Spaces

In this paper, we prove the existence and uniqueness of the fixed point for T-contraction mapping under the concept of c-Distance in cone metric spaces with solid cone. The obtained results extend and generalize well known comparable results in the literature.

Some multi-valued contraction theorems on \(\mathcal{H}\)-cone metric

In this paper we define some new type of multi-valued contraction maps on \(\mathcal{H}\)-cone metric and proved some fixed point and common fixed point theorems in the setting of cone metric spaces.

TITIK TETAP PERSEKUTUAN EMPAT PEMETAAN KONTRAKTIF PADA RUANG METRIK CONE

Huang Long Guang dan Zhang Xian pada tahun 2006 telah memperkenalkan metrik yang lebih baru yang disebut metrik cone. Suatu himpunan tak kosong X yang dilengkapi metrik cone d disebut ruang metrik cone. Suatu pemetaan memiliki titik tetap yang tunggal jika pemetaan tersebut merupakan pemetaan kontraktif. Konsep titik tetap persekutuan di ruang metrik cone juga harus memenuhi beberapa kondisi pemetaan yang bersifat coincidence point, point of coincidence, dan kompatibel lemah. Penelitian ini mengkaji titik tetap persekutuan untuk empat pemetaan yang kontraktif pada ruang metrik cone. Hasil penelitian ini menunjukkan bahwa empat pemetaan S, T, I dan J memiliki titik tetap persekutuan yang tunggal pada ruang metrik cone.Kata Kunci: ruang metrik cone, titik tetap persekutuan, pemetan kontraktif, coincidence point, point of coincidence, kompatibel lemah