Generalized Cone Research Articles

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

We investigate complete minimal submanifolds $$f: M^3\rightarrow \mathbb {H}^n$$ in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in Dajczer et al. (Math Z 287: 481–491, 2017 and Comment Math Helv, to appear, 2017), respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of $$\mathbb {H}^n$$ .

Generalized Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings

In this paper we shall acquaint with generalized cone metric spaces, contractive mappings will be considered and prove some classical fixed point theorems with the help of generalized cone metric space.

Generalized cone metric spaces and fixed point theorems of contractive mappings

In this paper we shall acquaint with generalized cone metric spaces, contractive mappings will be considered and prove some classical fixed point theorems with the help of generalized cone metric space.

Optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps characterized by contingent epiderivative

In this paper, firstly, a new notion of generalized cone convex set-valued map is introduced in real normed spaces. Secondly, a property of the generalized cone convex set-valued map involving the contingent epiderivative is obtained. Finally, as the applications of this property, we use the contingent epiderivative to establish optimality conditions of the set-valued optimization problem with generalized cone convex set-valued maps in the sense of Henig proper efficiency. The results obtained in this paper generalize and improve some known results in the literature.

Surfaces foliated by ellipses with constant Gaussian curvature in Euclidean 3-space

In this paper, we study the surfaces foliated by ellipses in three dimensional Euclidean space $\mathbf{E}^3$. We prove the following results: \textbf{(1)} The surface foliated by an ellipse have constant Gaussian curvature $K$ if and only if the surface is flat, i.e. $K=0$. \textbf{(2)} The surface foliated by an ellipse is a flat if and only if it is a part of generalized cylinder or part of generalized cone.

Coincidence and common fixed point results in generalized cone b-metric spaces under expansive type conditions

In this paper we prove some coincidence point and common fixed point results for two weakly compatible self-mappings satisfying expansive type conditions in the generalized cone b-metric spaces. We also give an example to illustrate the main result.

Microstructural tissue-engineering in the rachis and barbs of bird feathers

Feathers do not have to be especially strong but they do need to be stiff and at the same time resilient and to have a high work of fracture. Syncitial barbule fibres are the highest size-class of continuous filaments in the cortex of the rachis of the feather. However, the rachis can be treated as a generalized cone of rapidly diminishing volume. This means that hundreds of syncitial barbule fibres of the rachis may have to be terminated before reaching the tip – creating potentially thousands of inherently fatal crack-like defects. Here I report a new microstructural architecture of the feather cortex in which most syncitial barbule fibres deviate to the right and left edges of the feather rachis from far within its borders and extend into the barbs, side branches of the rachis, as continuous filaments. This novel morphology adds significantly to knowledge of β-keratin self-assembly in the feather and helps solve the potential problem of fatal crack-like defects in the rachidial cortex. Furthermore, this new complexity, consistent with biology’s robust multi-functionality, solves two biomechanical problems at a stroke. Feather barbs deeply ‘rooted’ within the rachis are also able to better withstand the aerodynamic forces to which they are subjected.

Set-valued minimax programming problems under generalized cone convexity

In this paper, a set-valued minimax programming problem (MP) is considered. The Karush–Kuhn–Tucker (KKT) sufficient optimality conditions are established for the problem (MP) via contingent epiderivative under generalized cone convexity assumptions. Mond–Weir (MWD), Wolfe (WD) and mixed (MD) types duality are constructed for the problem (MP) and desired duality theorems between them are established.

A fixed point theorem for Reich contractions in generalized cone b-metric spaces

A fixed point theorem for Reich contractions in generalized cone b-metric spaces

Clinical and Molecular Characteristics of Childhood-Onset Stargardt Disease

PurposeTo describe the clinical and molecular characteristics of patients with childhood-onset Stargardt disease (STGD).DesignRetrospective case series.ParticipantsForty-two patients who were diagnosed with STGD in childhood at a single institution between January 2001 and January 2012.MethodsA detailed history and a comprehensive ophthalmic examination were undertaken, including color fundus photography, autofluorescence imaging, spectral-domain optical coherence tomography (SD-OCT), and pattern and full-field electroretinograms. The entire coding region and splice sites of ABCA4 were screened using a next-generation, sequencing-based strategy. The molecular genetic findings of childhood-onset STGD patients were compared with those of adult-onset patients.Main Outcome MeasuresClinical, imaging, electrophysiologic, and molecular genetic findings.ResultsThe median ages of onset and the median age at baseline examination were 8.5 (range, 3–16) and 12.0 years (range, 7-16), respectively. The median baseline logarithm of the minimum angle of resolution visual acuity was 0.74. At baseline, 26 of 39 patients (67%) with available photographs had macular atrophy with macular/peripheral flecks; 11 (28%) had macular atrophy without flecks; 1 (2.5%) had numerous flecks without macular atrophy; and 1 (2.5%) had a normal fundus appearance. Flecks were not identified at baseline in 12 patients (31%). SD-OCT detected foveal outer retinal disruption in all 21 patients with available images. Electrophysiologic assessment demonstrated retinal dysfunction confined to the macula in 9 patients (36%), macular and generalized cone dysfunction in 1 subject (4%), and macular and generalized cone and rod dysfunction in 15 individuals (60%). At least 1 disease-causing ABCA4 variant was identified in 38 patients (90%), including 13 novel variants; ≥2 variants were identified in 34 patients (81%). Patients with childhood-onset STGD more frequently harbored 2 deleterious variants (18% vs 5%) compared with patients with adult-onset STGD.ConclusionsChildhood-onset STGD is associated with severe visual loss, early morphologic changes, and often generalized retinal dysfunction, despite often having less severe fundus abnormalities on examination. One third of children do not have flecks at presentation. The relatively high proportion of deleterious ABCA4 variants supports the hypothesis that earlier onset disease is often owing to more severe variants in ABCA4 than those found in adult-onset disease.

Autofocusing circular synthetic aperture sonar imagery using phase corrections modeled as generalized cones.

Circular synthetic aperture sonar (CSAS) is a coherent aperture synthesis technique that utilizes backscattered acoustic information from an encircled scene to generate information rich, high-resolution imagery. The aperture length required for image synthesis is much longer than in its linear synthetic aperture sonar counterpart and can result in challenging phase delay and navigation estimation constraints. Residual uncorrected phase errors manifest as focus aberrations in reconstructed CSAS imagery. This paper demonstrates that phase error in image patches can be approximated as an aspect variant linear phase shift representable as a generalized cone in wave-number space. If the geometry of the generalized cone is known, it can be applied as the spectral phase of an inverse filter for aberration correction. A method is derived for reconstructing the error cone geometry from independent estimates of its local curvatures, which are found via a series of one-dimensional line searches that maximize the focus of CSAS sub-aperture images. This approach is applied to real and simulated CSAS data containing aperture distortions, and the results successfully demonstrate estimation and correction of the underlying focus aberrations.

GUCY2D- or GUCA1A-related autosomal dominant cone-rod dystrophy: is there a phenotypic difference?

To compare the phenotype of patients with heterozygous mutation in GUCY2D or GUCA1A causing autosomal dominant cone or cone-rod dystrophies. Five patients from one family with GUCA1A and nine patients from four families with GUCY2D mutations were included. Psychophysical and electrophysiological examinations were performed to study retinal function. Fundus autofluorescence imaging and spectral domain optical coherence tomography were performed for morphologic characterization. Genetic analysis revealed the mutation c.451C>T (p.L151F) in the GUCA1A family. In the GUCY2D group, c.2512C>T (p.R838C) was the most frequent (2 families), c.2512C>G (p.R838G) and c.2513G>A (p.R838H) were found in one family each. Visual acuity was reduced to 0.04 to 0.7 in GUCA1A and to 0.014 to 0.5 in patients with GUCY2D. Dark adaptation showed elevated thresholds in the GUCY2D group. Scotopic electroretinography revealed a tendency to a more affected rod function in the GUCY2D group. Photopic electroretinography showed residual or absent responses in both groups. Fundus alterations were confined to the macula in both groups. GUCA1A and GUCY2D mutations are both accompanied by similar pattern of generalized cone dysfunction with a tendency to less involvement of the rod photoreceptors and a less severe phenotype in patients with GUCA1A.

Sufficient optimality conditions and duality theorems for set-valued optimization problem under generalized cone convexity

In this paper, a new type of generalized cone convexity is introduced in the case of set-valued optimization problem where the maps involved are contingent epiderivable. It extends the notion of \(\rho -(\eta , \theta )\)-invexity from vector optimization to set-valued optimization. The sufficient Karush–Kuhn–Tucker (KKT) optimality conditions are investigated under the stated assumptions. We also study the duality results of Mond–Weir type (MWD), Wolfe type (WD) and mixed type (Mix D) for weak solutions of a pair of set-valued optimization problems.

Scalarization of $$\epsilon $$ ϵ -Super Efficient Solutions of Set-Valued Optimization Problems in Real Ordered Linear Spaces

In this paper, we investigate the scalarization of $$\epsilon $$ ∈ -super efficient solutions of set-valued optimization problems in real ordered linear spaces. First, in real ordered linear spaces, under the assumption of generalized cone subconvexlikeness of set-valued maps, a dual decomposition theorem is established in the sense of $$\epsilon $$ ∈ -super efficiency. Second, as an application of the dual decomposition theorem, a linear scalarization theorem is given. Finally, without any convexity assumption, a nonlinear scalarization theorem characterized by the seminorm is obtained.

Existence and convergence results for best proximity points in cone metric spaces

In this paper, the author introduces generalized cone proximal $\varphi$-cyclic contraction pairs in cone metric spaces and considers the existence and convergence of best proximity point for a pair in cone metric spaces. His results generalize the corresponding results in [1, 4, 5, 7, 8, 12, 13, 15].

The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

In this paper we study the analytic torsion of a manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of higher order near the singularities. Then we identify the metric anomaly of analytic torsion for a bounded generalized cone at its regular boundary in terms of spectral information of the cross-section. In view of previous computations of analytic torsion on cones, this leads to a detailed geometric identification of the topological and spectral contributions to analytic torsion, arising from the conical singularity. The contribution exhibits a torsion-like spectral invariant of the cross-section of the cone, which we study under scaling of the metric on the cross-section.

$$\epsilon $$ ϵ -Henig proper efficiency of set-valued optimization problems in real ordered linear spaces

The aim of this paper is to investigate \(\epsilon \)-Henig proper efficiency of set-valued optimization problems in linear spaces. Firstly, a new notion of \(\epsilon \)-Henig properly efficient point is introduced in linear spaces. Secondly, scalarization theorems of set-valued optimization problems are established in the sense of \(\epsilon \)-Henig proper efficiency. Finally, under the assumption of generalized cone subconvexlikeness, Lagrange multiplier theorems are obtained. Our results generalize some known results in the literature from topological spaces to linear spaces.

A Longitudinal Study of Stargardt Disease: Clinical and Electrophysiologic Assessment, Progression, and Genotype Correlations

To investigate the clinical and electrophysiologic natural history of Stargardt disease and correlate with the genotype. Cohort study of 59 patients. Clinical history, examination, and electrophysiologic assessment were undertaken in a longitudinal survey. Patients were classified into 3 groups based on electrophysiologic findings, as previously published: Group 1 had dysfunction confined to the macula; Group 2 had macular and generalized cone system dysfunction; and Group 3 had macular and both generalized cone and rod system dysfunction. At baseline, there were 27 patients in Group 1, 17 in Group 2, and 15 in Group 3. Amplitude reduction of >50% in the relevant electroretinogram (ERG) component or a peak time shift of >3ms for the 30Hz flicker ERG or bright flash a-wave was considered clinically significant ERG deterioration. Molecular screening of ABCA4 was undertaken. The mean age at baseline was 31.7 years, with the mean follow-up interval being 10.5 years. A total of 22% of patients from Group 1 showed ERG group transition during follow-up, with 11% progressing to Group 2 and 11% to Group 3. Forty-seven percent of patients in Group 2 progressed to Group 3. There was clinically significant ERG deterioration in 54% of all subjects: 22% of Group 1, 65% of Group 2, and 100% of Group 3. At least 1 disease-causing ABCA4 variant was identified in 47 patients. All patients with initial rod ERG involvement demonstrated clinically significant electrophysiologic deterioration; only 20% of patients with normal full-field ERGs at baseline showed clinically significant progression. Such data assist counseling by providing more accurate prognostic information and are also highly relevant in the design, patient selection, and monitoring of potential therapeutic interventions.

A computational model of recovering the 3D shape of a generalized cone

A computational model of recovering the 3D shape of a generalized cone

Scalarization of Set-Valued Optimization Problems with Generalized Cone Subconvexlikeness in Real Ordered Linear Spaces

In real ordered linear spaces, an equivalent characterization of generalized cone subconvexlikeness of set-valued maps is firstly established. Secondly, under the assumption of generalized cone subconvexlikeness of set-valued maps, a scalarization theorem of set-valued optimization problems in the sense of ϵ-weak efficiency is obtained. Finally, by a scalarization approach, an existence theorem of ϵ-global properly efficient element of set-valued optimization problems is obtained. The results in this paper generalize and improve some known results in the literature.