Cone Metric Research Articles

Towards coupled coincidence theorems of generalized admissible types of mappings on partial satisfactory cone metric spaces and some applications

<abstract><p>This paper introduces a novel class of generalized $ {\alpha} $-admissible contraction types of mappings in the framework of $ {\theta} $-complete partial satisfactory cone metric spaces and proves the existence and uniqueness of coincidence points for such mappings. In this setting, the topology generated and induced by the partial satisfactory cone metric is associated with semi-interior points rather than interior points of the underlying cone. In addition, some applications of the paper's main coincidence point theorems are given. The results of this paper unify, extend and generalize some previously proved theorems in this generalized setting.</p></abstract>

Qualitative properties of weighted fourth order elliptic problems

Qualitative properties of nonnegative solutions of weighted fourth order elliptic problems arising from the equations on singular manifolds with conical metrics are studied. The weights may be singular in the domains. We obtain the uniqueness of positive radial solution of the Navier boundary value problem in the ball $ B $ and its exact regularity at the singular point $ x = 0 $, which determines the exact regularity of the unique positive radial solution in $ B $. We also establish the Liouville type results for the equation on $ \mathbb{R}^N \backslash \{0\} $.

Existence and Uniqueness of Fixed Points in Cone Metric Spaces for ω-distance

Existence and Uniqueness of Fixed Points in Cone Metric Spaces for ω-distance

Extension of a Unique Solution in Generalized Neutrosophic Cone Metric Spaces

In order to solve issues that arise in various branches of mathematical analysis, such as split feasibility problems, variational inequality problems, nonlinear optimization issues, equilibrium problems, complementarity issues, selection and matching problems, and issues proving the existence of solutions to integral and differential equations, fixed point theory provides vital tools. In this study, we discuss topological structure and several fixed-point theorems in the context of generalized neutrosophic cone metric spaces. In these spaces, the symmetric properties play an important role. We examine the existence and a uniqueness of a solution by utilizing new types of contraction mappings under some circumstances. We provide an example in which we show the existence and a uniqueness of a solution by utilizing our main result. These results are more generalized in the existing literature.

Relative Reshetikhin–Turaev Invariants, Hyperbolic Cone Metrics and Discrete Fourier Transforms I

We propose the Volume Conjecture for the relative Reshetikhin–Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it whose asymptotic behavior is related to the volume and the Chern–Simons invariant of the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring. We prove the conjecture in the case that the ambient 3-manifold is obtained by doing an integral surgery along some components of a fundamental shadow link and the complement of the link in the ambient manifold is homeomorphic to the fundamental shadow link complement, for sufficiently small cone angles. Together with Costantino and Thurston’s result that all compact oriented 3-manifolds with toroidal or empty boundary can be obtained by doing an integral surgery along some components of a suitable fundamental shadow link, this provides a possible approach of solving Chen–Yang’s Volume Conjecture for the Reshetikhin–Turaev invariants of closed oriented hyperbolic 3-manifolds. We also introduce a family of topological operations (the change-of-pair operations) that connect all pairs of a closed oriented 3-manifold and a framed link inside it that have homeomorphic complements, which correspond to doing the partial discrete Fourier transforms to the corresponding relative Reshetikhin–Turaev invariants. As an application, we find a Poisson Summation Formula for the discrete Fourier transforms.

Generalized contraction theorem in M -fuzzy cone metric spaces

This work defines MM-Fuzzy Cone Metric Space, as a new metric space. It also analyzes possible forms of contractive conditions and groups them accordingly to set up generalized contractive conditions for self-mappings defined over MM-fuzzy cone metric spaces. We prove the existence of fixed points of these mappings and exhibit the same through a suitable example.

In vivo assessment of associations between photoreceptors structure and macular perfusion in type 1 diabetes

PurposeTo explore the potential relationships between macular vascular network and different adaptive optics (AO) metrics in patients with type 1 diabetes mellitus (DM1) with no signs (NoDR) or mild non-proliferative...

Hyperbolic cone metrics and billiards

A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that rigidity is a generic property, and parameterize the associated deformation space for any flexible metric. As an application, we parameterize the space of hyperbolic polygons with the same symbolic coding for their billiard dynamics, and prove that generically this parameter space is a point.

Spectral geometry on manifolds with fibered boundary metrics I: Low energy resolvent

We study the low energy resolvent of the Hodge Laplacian on a manifold equipped with a fibered boundary metric. We determine the precise asymptotic behavior of the resolvent as a fibered boundary (aka ϕ-) pseudodifferential operator when the resolvent parameter tends to zero. This generalizes previous work by Guillarmou and Sher who considered asymptotically conic metrics, which correspond to the special case when the fibers are points. The new feature in the case of non-trivial fibers is that the resolvent has different asymptotic behavior on the subspace of forms that are fiberwise harmonic and on its orthogonal complement. To deal with this, we introduce an appropriate ‘split’ pseudodifferential calculus, building on and extending work by Grieser and Hunsicker. Our work sets the basis for the discussion of spectral invariants on ϕ-manifolds.

Toric Sasaki–Einstein metrics with conical singularities

We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi–Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi–Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki–Einstein metrics.

Spin(7) Metrics of Cohomogeneity One with Aloff–Wallach Spaces as Principal Orbits

In this article, we construct two continuous 1-parameter family of non-compact $$\mathrm {Spin}(7)$$ metrics with both chiralities, with the principal orbit an Aloff–Wallach space $$N_{k,l}$$ and the singular orbit $${{\mathbb {C}}}{{\mathbb {P}}}^2$$ . For a generic $$N_{k,l}$$ , metrics constructed are locally asymptotically conical (ALC). For $$N_{1,1}$$ , we construct two continuous 1-parameter families with geometric transition from asymptotically conical (AC) metrics to ALC metrics.

Spherical conical metrics and harmonic maps to spheres

A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions.

Low Energy Limit for the Resolvent of Some Fibered Boundary Operators

For certain Dirac operators $\eth_{\phi}$ associated to a fibered boundary metric $g_{\phi}$, we provide a pseudodifferential characterization of the limiting behavior of $(\eth_{\phi}+k\gamma)^{-1}$ as $k\searrow 0$, where $\gamma$ is a self-adjoint operator anti-commuting with $\eth_{\phi}$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $\eth_{\phi}^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give a pseudodifferential characterization of the inverse of some suspended version of the operator $\eth_{\phi}$. One important ingredient in the proof of our main theorem is that the Dirac operator $\eth_{\phi}$ is Fredholm when acting on suitable weighted Sobolev spaces. This result has been known to experts for some time and we take this as an occasion to provide a complete explicit proof.

Some Common Fixed Point Results in Cone Metric Spaces for Rational Contractions

A very engrossing technique in theory of contractive mapping fixed point. A number of authors have defined contractive type mappings on a cone metric spaces X which are generalization of the well -known Banach contraction, and have the property that each of such mapping has a unique fixed point. The fixed point can always be found by using Picard Iteration, opening with initial choice x₀∈X. In this manuscript, we generalize, extend and improve the result under the assumption of normality of cone for rational expression type contraction mapping in cone metric spaces. The present article is to provide a new alternative proof for two and three mapping and obtain the entity and exclusiveness of common fixed point. The concernment of the present paper to open a new direction of proof to be extended based on the methods of rational type contraction mapping in cone metric spaces of fixed-point theory. The assistance of this article is organized as follows. In section 2, preliminary notes. In this section we recall some standard notations and definitions which we needed. In section 3, the main results of the author are given. In this section we evidence of new results for two and three maps. In section 4, gives brief concluding note of the paper.

Common Fixed Point Theorems forF‐Kannan–Suzuki Type Mappings in TVS‐Valued Cone Metric Space with Some Applications

This research paper generalizes and extends various fixed‐point results that demonstrate common fixed‐point theorems forF‐Kannan–Suzuki type mappings in TVS‐valued cone metric spaces. The results are supported using interpretative exemplifications and applications that include nonlinear fractional as well as two‐point periodic ordinary differential equations.

Short-Term Parafoveal Cone Loss Despite Preserved Ellipsoid Zone in Rod Cone Dystrophy.

PurposeRod–cone dystrophy (RCD) is characterized by centripetal loss of rod followed by cone photoreceptors. In this prospective, observational cohort, we used flood-illumination adaptive optics (AO) imaging to investigate parafoveal cone loss in regions with preserved ellipsoid zone (EZ) in patients with RCD.MethodsEight patients with RCD and 10 age-matched healthy controls underwent spectral-domain optical coherence tomography and AO imaging. The RCD cohort underwent a follow-up examination after 6 months. Cone density (CD) and intercone distance (ICD) measurements were performed at 2° temporal from the fovea. Baseline CD and ICD values were compared between the control and patient groups, and longitudinal changes were calculated in the patient group. Residual EZ span in patients was measured in horizontal foveal B-scans.ResultsBetween the control and patient groups, there was no significant difference in the baseline CD (2094 vs. 1750 cells/deg2, respectively; P = 0.09) and ICD (1.46 vs. 1.62 arcmin, respectively; P = 0.08). Mean CD declined by 198 cells/deg2 (−11.3%; P < 0.01), and mean ICD increased by 0.09 arcmin (+5.6%; P = 0.01) at the 6-month follow-up in the patient group. Mean baseline and follow-up residual EZ spans in the six patients with EZ defect were 3189 µm and 3065 µm, respectively (−3.9%; P = 0.08).ConclusionsAO imaging detected significant parafoveal cone loss over 6-month follow-up even in regions with preserved EZ. Further studies to refine AO imaging protocol and validate cone metrics as a structural endpoint in early RCD are warranted.Translational RelevanceCD and ICD may change prior to EZ span shortening in RCD.

Some Compatible and Weakly-Compatible Four Self-Mapping Results Approach to Nonlinear Integral Equations in Fuzzy Cone Metric Spaces

This paper is aimed at proving some unique common fixed point theorems by using the compatible and weakly-compatible four self-mappings in fuzzy cone metric (FCM) space. We prove the results under the generalized rational contraction conditions in FCM spaces with the help of one self-map are continuous. Furthermore, we prove some rational contraction results with the weaker condition of the self-mapping continuity. Ultimately, our theoretical work has been utilized to prove the existence solution of the two nonlinear integral equations. This is an illustrative application of how FCM spaces can be used in other integral type operators.

You can hear the shape of a billiard table: Symbolic dynamics and rigidity for flat surfaces

We give a complete characterization of the relationship between the shape of a Euclidean polygon and the symbolic dynamics of its billiard flow. We prove that the only pairs of tables that can have the same bounce spectrum are right-angled tables that differ by an affine map. The main tool is a new theorem that establishes that a flat cone metric is completely determined by the support of its Liouville current.

Complete noncompact G2-manifolds from asymptotically conical Calabi–Yau 3-folds

We develop a powerful new analytic method to construct complete noncompact Ricci-flat 7-manifolds, more specifically G2-manifolds, that is, Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction gives the first general analytic construction of complete noncompact Ricci-flat metrics in any odd dimension and establishes a link with the Cheeger–Fukaya–Gromov theory of collapse with bounded curvature. The construction starts with a complete noncompact asymptotically conical Calabi–Yau 3-fold B and a circle bundle M→B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics gϵ on M that collapses with bounded curvature as ϵ→0 to the original Calabi–Yau metric on the base B. The G2-metrics we construct have controlled asymptotic geometry at infinity, so-called asymptotically locally conical (ALC) metrics; these are the natural higher-dimensional analogues of the asymptotically locally flat (ALF) metrics that are well known in 4-dimensional hyper-Kähler geometry. We give two illustrations of the strength of our method. First, we use it to construct infinitely many diffeomorphism types of complete noncompact simply connected G2-manifolds; previously only a handful of such diffeomorphism types was known. Second, we use it to prove the existence of continuous families of complete noncompact G2-metrics of arbitrarily high dimension; previously only rigid or 1-parameter families of complete noncompact G2-metrics were known.

The Green’s function for equations with conic metrics

The Green’s function for equations with conic metrics