Regular Cone Research Articles

On the energy of a null cone

We derive a formula for the Bondi mass aspect in terms of the asymptotic data of the Bondi–Sachs metric in the affine gauge. We prove the positivity of the total energy of a regular null cone in agreement with a recent result of Chruściel and Paetz.

Canonical Barriers on Convex Cones

On the interior of a regular convex cone K in n-dimensional real space there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the Cheng-Yau metric. The former is associated with a self-concordant logarithmically homogeneous barrier on K, the universal barrier. It is invariant with respect to the unimodular automorphism subgroup of K and is compatible with the operation of taking product cones, but in general it does not behave well under duality. Here we introduce a barrier associated with the Cheng-Yau metric, the canonical barrier. It shares with the universal barrier the invariance, existence, and uniqueness properties and is compatible with the operation of taking product cones, but in addition is well behaved under duality. The canonical barrier can be characterized as the convex solution of the partial differential equation log det F” = 2F that tends to infinity as the argument tends to the boundary of K. Its barrier parameter does not exceed the dimension n of the cone. On homogeneous cones both barriers essentially coincide.

Cone photoreceptor definition on adaptive optics retinal imaging

AimsTo quantitatively analyse cone photoreceptor matrices on images captured on an adaptive optics (AO) camera and assess their correlation to well-established parameters in the retinal histology literature.MethodsHigh resolution retinal images...

Retinal Microstructural Changes in Eyes With Resolved Branch Retinal Vein Occlusion: An Adaptive Optics Scanning Laser Ophthalmoscopy Study

To assess macular photoreceptor abnormalities in eyes with resolved branch retinal vein occlusion (BRVO) using adaptive optics scanning laser ophthalmoscopy (AO-SLO). Prospective observational cross-sectional case series. After complete resolution of macular edema and retinal hemorrhage, 21 eyes (21 patients) with BRVO underwent full ophthalmologic examination and imaging with optical coherence tomography (OCT) and a prototype AO-SLO system. Cone density and spatial mosaic organization were assessed using AO-SLO images. Regular parafoveal cone mosaic patterns were clearly visualized with the prototype AO-SLO imaging system in the BRVO-unaffected side. However, in the side of the retina previously affected by the BRVO, cone mosaic patterns were disorganized and dark regions missing wave-guiding cones were apparent. Additionally, retinal capillaries were dilated, no longer had a uniform caliber, and had less direct paths through the retina. In the affected side, parafoveal cone density was significantly decreased, compared with the corresponding retinal area on the unaffected side (P < .001). Furthermore, the hexagonal Voronoi domain ratio and the nearest-neighbor distances were significantly lower than in the unaffected side (P < .05). These parameters were also correlated with photoreceptor layer integrity in the parafovea. After BRVO-associated retinal hemorrhage and macular edema resolved, affected parafoveal cone density decreases and the cone mosaic spatial arrangement is disrupted, becoming more irregular. These cone microstructural abnormalities may extend to parafovea in the BRVO-unaffected side.

Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions ofLp-Growth

LetCbe a regular cone inℝand letTC=ℝ+iC⊂ℂbe a tubular radial domain. LetUbe the convolutor in Beurling ultradistributions ofLp-growth corresponding toTC. We define the Cauchy and Poisson integral ofUand show that the Cauchy integral of Uis analytic inTCand satisfies a growth property. We represent Uas the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation ofU. Also we show that the Poisson integral ofUcorresponding toTCattainsUas boundary value in the distributional sense.

Macular Cone Abnormalities in Retinitis Pigmentosa with Preserved Central Vision Using Adaptive Optics Scanning Laser Ophthalmoscopy

PurposeTo assess macular photoreceptor abnormalities in eyes with retinitis pigmentosa (RP) with preserved central vision using adaptive optics scanning laser ophthalmoscopy (AO-SLO).MethodsFourteen eyes of 14 patients with RP (best-corrected visual acuity 20/20 or better) and 12 eyes of 12 volunteers underwent a full ophthalmologic examination, fundus autofluorescence, spectral-domain optical coherence tomography (SD-OCT), and imaging with a prototype AO-SLO system. Cone density and spatial organization of the cone mosaic were assessed using AO-SLO images.ResultsIn 3 eyes with RP and preserved central vision, cones formed a mostly regular mosaic pattern with small patchy dark areas, and in 10 eyes, the cone mosaic patterns were less regular, and large dark regions with missing cones were apparent. Only one eye with RP demonstrated a normal, regular cone mosaic pattern. In eyes with RP, cone density was significantly lower at 0.5 mm and 1.0 mm from the center of the fovea compared to normal eyes (P<0.001 and 0.021, respectively). At 0.5 mm and 1.0 mm from the center of the fovea, a decreased number of cones had 6 neighbors in eyes with RP (P = 0.002 for both). Greater decrease in cone density was related to disruption of the photoreceptor inner segment (IS) ellipsoid band on SD-OCT images (P = 0.044); however, dark regions were seen on AO-SLO even in areas of continuous IS ellipsoid on SD-OCT. Decreased cone density correlated thinner outer nuclear layer (P = 0.029) and thinner inner segment and outer segment thickness (P = 0.011) on SD-OCT.ConclusionsCone density is decreased and the regularity of the cone mosaic spatial arrangement is disrupted in eyes with RP, even when visual acuity and foveal sensitivity are good. AO-SLO imaging is a sensitive quantitative tool for detecting photoreceptor abnormalities in eyes with RP.

Probabilistic Analysis of the Grassmann Condition Number

We analyze the probability that a random m-dimensional linear subspace of $\mathbb{R}^{n}$ both intersects a regular closed convex cone $C\subseteq\mathbb{R}^{n}$ and lies within distance ? of an m-dimensional subspace not intersecting C (except at the origin). The result is expressed in terms of the spherical intrinsic volumes of the cone C. This allows us to perform an average analysis of the Grassmann condition number [InlineEquation not available: see fulltext.] for the homogeneous convex feasibility problem ?x?C?0:Ax=0. The Grassmann condition number is a geometric version of Renegar's condition number, which we have introduced recently in Amelunxen and Burgisser (SIAM J. Optim. 22(3):1029---1041 (2012)). We thus give the first average analysis of convex programming that is not restricted to linear programming. In particular, we prove that if the entries of $A\in\mathbb {R}^{m\times n}$ are chosen i.i.d. standard normal, then for any regular cone C, we have [InlineEquation not available: see fulltext.]. The proofs rely on various techniques from Riemannian geometry applied to Grassmann manifolds.

Analytic formulas for complete hyperbolic affine spheres

We classify all regular three-dimensional convex cones which possess an automorphism group of dimension at least two, and provide analytic expressions for the complete hyperbolic affine spheres which are asymptotic to the boundaries of these cones. The affine spheres are represented by explicit hypersurface immersions into three-dimensional real space. The generic member of the family of immersions is given by elliptic integrals.

A cone‐rod dystrophy patient with a homozygous RP1L1 mutation

AbstractPurpose To describe a family with a cone‐rod dystrophy patient who has homozygous mutation of the RP1‐like protein 1 (RP1L1) gene.Methods A family including a cone‐rod dystrophy patient underwent detailed ophthalmic clinical evaluations including high resolution cone photorecepor imaging with adaptive optics fundus camera. Mutation screening of the sequence of RP1L1 gene were performed by DNA sequencing analysis in this family membersResults A patient showed a mild reduction of cone and flicker response in full‐field electroretinogram (ERG). Her ERG also showed slight decrease in the amplitude of rod response. IS/OS junction and COST line in SD‐OCT images are severely disturbed. Response of multifocal ERGs (mfERGs) of the patient was severely reduced. A homozygous RP1L1 mutation (c.3628 T&gt;C) was identified in the patient. The mutation c.3628 T&gt;C in exon 4 resulted in the substitution of proline for serine at amino acid position 1210. This mutation was not reported in SNP database. The serine at position 1210 is well conserved among the RP1L1 family in other species. Four out of five computational assessment tools predicted that this mutation is damaging to the protein function. This mutation was not present in 460 control alleles. Family members with heterozygous S1210P mutation showed normal best‐corrected visual acuity (BCVA), SD‐OCT, mfERGs, and focal macular ERGs. Adaptive Optics (AO) images of the patient showed severe reduction of cone density and irregular cone mosaic despite family members with heterozygous S1210P mutation showed normal cone density and regular cone mosaic.Conclusion We have demonstrated a possibility that autosomal recessive cone‐rod dystrophy may be caused by homozygous RP1L1 mutation.

High resolution retinal image analysis of unilateral retinitis pigmentosa using adptive optics

AbstractPurpose To report the findings of en face adaptive optics (AO) fundus imaging in eyes with unilateral retinitis pigmentosa(RP) patient.Methods The both eyes from a unilateral RP patient, 52 y.o. female, was imaged using both an infrared AO camera (rtx1™, Imagine Eyes, France) and spectral‐domain optical coherence tomography (SD‐OCT – Cirrus™, Carl Zeiss Meditec, Germany). Full field scotopic and phototic electroretinograms(ERGs) were recorded according to the ISCEV standard. Normal funduscopic examination, Goldmann visual field and fluorescence angiography(FA) were also examined. The AO images acquired from the unilateral RP patient were analyzed on cone density and then compared to the AO data from the eyes of 21 healthy volunteers with a mean age of 36 years (range 20‐57).Results The unilateral RP patient showed boney spicule changes, vascular attenuation and decreased peripheral visual field in the right eye. FA showed granular hyperfluorescence in the post pole, mottling of RPE and atrophy of choroid capillary in the peripheral retina in the right eye. Her ERGs of the right eye showed non‐recordable in all responses tested. OCT findings in the right eye showed preservation of the photoreceptor layer only in the foveal and parafoveal regions with loss in the more peripheral macula. All these results in her left eye were normal. AO data showed severe reduction of cone density in the right eye and the left eye showed normal cone density and regular cone mosaic.Conclusion In unilateral RP patient, AO showed only one eye was affected even in the cellular level examination. The examinations using AO may be useful in diagnosis and better understanding of pathology and management of unilateral RP cases.

Objective Assessment of Foveal Cone Loss Ratio in Surgically Closed Macular Holes Using Adaptive Optics Scanning Laser Ophthalmoscopy

PurposeTo use adaptive optics scanning laser ophthalmoscopy (AO-SLO) to quantify cone loss ratio in the foveola in order to assess foveal cone status and to investigate relationships between foveal structural abnormalities and visual function in patients with macular hole (MH) after surgery.MethodsWe evaluated 10 normal eyes of 10 healthy volunteers and 19 eyes of 18 patients in whom anatomically successful MH closure had been performed. All subjects underwent a comprehensive ophthalmologic examination that included measurements of spectral-domain optical coherence tomography and AO-SLO.ResultsOn AO-SLO regular cone mosaic was seen in all normal eyes whereas dark regions suggesting cone loss were seen in all eyes after MH repair. Visual acuity was better in eyes without dark regions at the center of the fovea than in eyes with them (P = 0.001). Cone loss ratio in the foveola correlated with postoperative visual acuity (P<0.001), mean foveal sensitivity (P = 0.029), thinner inner and outer segments at the center of the fovea (P = 0.002), larger size of the disrupted inner and outer segment junction line (P = 0.018), and cone outer segment tip line (P<0.001). Cone loss ratio in the foveola was significantly greater in eyes that had moderately reflective foveal lesions after surgery (P = 0.006).ConclusionsAO-SLO is a useful means of assessing foveal cone damage objectively and quantitatively. The location and extent of cone damage, especially if it involves the foveola, is an important factor determining visual function after MH surgery.

Bimodal colour pattern of individualPinus halepensis Mill. seeds: a new type of crypsis

Variation in seed traits is a well-known phenomenon affecting plant ecology and evolution. Here we describe, for the first time, a bimodal colour pattern of individual seeds, proposing an adaptive explanation, using Pinus halepensis as a model. Pinus halepensis disperses its seeds either by wind on hot dry days, from regular cones, or after fires, mainly from serotinous cones. Post-dispersal seeds are exposed to strong predation by passerine birds, making crypsis important for seed survival. Individual seeds from non-serotinous cones have a bimodal colour pattern: one side is light brown and the other black, exposing only one colour when lying on the ground. Serotinous cones from most trees have seeds with similar bimodal colour patterns, whereas seeds from serotinous cones of some trees are light brown on both sides. The dark side provides the seed with better crypsis on dark soils, whereas the light-brown side is better adapted to light-coloured soils, and mainly to light-grey ash-covered soil, which is the natural post-fire regeneration niche of P. halepensis. The relative reflection curves of the black and brown seed colours differ, and their calculated relative chromatic distance is 5: meaning that seed-predating passerine birds see them differently, and probably prefer seeds that present a higher contrast against the soil background. We propose that such a bimodal colour pattern of individual seeds is probably an overlooked general phenomenon mainly linked to seed dispersal in post-fire and other heterogeneous environments. © 2013 The Linnean Society of London, Biological Journal of the Linnean Society, 2013, 109, 271–278.

Some Remarks on Stability of Generalized Equations

The paper concerns the computation of the graphical derivative and the regular (Frechet) coderivative of the solution map to a class of generalized equations, where the multivalued term amounts to the regular normal cone to a (possibly nonconvex) set given by C 2 inequalities. Instead of the linear independence qualification condition, standardly used in this context, one assumes a combination of the Mangasarian–Fromovitz and the constant rank qualification conditions. Based on the obtained generalized derivatives, new optimality conditions for a class of mathematical programs with equilibrium constraints are derived, and a workable characterization of the isolated calmness of the considered solution map is provided.

A lower bound on the barrier parameter of barriers for convex cones

Let $${K \subset \mathbb R^n}$$ be a regular convex cone, let $${e_1,\ldots,e_n \in \partial K}$$ be linearly independent points on the boundary of a compact affine section of the cone, and let $${x^* \in K^{0}}$$ be a point in the relative interior of this section. For k = 1, . . . , n, let l k be the line through the points e k and x *, let y k be the intersection point of l k with $${\partial K}$$ opposite to e k , and let z k be the intersection point of l k with the linear subspace spanned by all points e l , l = 1, . . . , n except e k . We give a lower bound on the barrier parameter ν of logarithmically homogeneous self-concordant barriers $${F: K^{0}\to \mathbb R}$$ on K in terms of the projective cross-ratios $${q_k = (e_k,x^*;y_k,z_k)}$$ . Previously known lower bounds by Nesterov and Nemirovski can be obtained from our result as a special case. As an application, we construct an optimal barrier for the epigraph of the $${||\cdot||_{\infty}}$$ -norm in $${\mathbb R^n}$$ and compute lower bounds on the barrier parameter for the power cone and the epigraph of the $${||\cdot||_p}$$ -norm in $${\mathbb R^2}$$ .

Fragility of Surface States and Robustness of Topological Order inBi2Se3against Oxidation

Topological surface states are protected against local perturbations, but this protection does not extend to chemical reaction over the whole surface, as demonstrated by theoretical studies of the oxidation of Bi(2)Se(3) and its effects on the surface spin polarization and current. While chemisorption of O(2) largely preserves the topological surface states, reaction with atomic O removes the original surface states and yields two new sets of surface states. One set forms a regular Dirac cone but is topologically trivial. The other set, while topologically relevant, forms an unusual rounded Dirac cone. The details are governed by the hybridization interaction at the interface.

Theorems of the Alternative for Inequality Systems of Real Polynomials

In this paper, we establish theorems of the alternative for inequality systems of real polynomials. For the real quadratic inequality system, we present two new results on the matrix decomposition, by which we establish two theorems of the alternative for the inequality system of three quadratic polynomials under an assumption that at least one of the involved forms be negative semidefinite. We also extend a theorem of the alternative to the case with a regular cone. For the inequality system of higher degree real polynomials, defined by even order tensors, a theorem of the alternative for the inequality system of two higher degree polynomials is established under suitable assumptions. As a byproduct, we give an equivalence result between two statements involving two higher degree polynomials. Based on this result, we investigate the optimality condition of a class of polynomial optimization problems under suitable assumptions.

A Coordinate-Free Condition Number for Convex Programming

We introduce and analyze a natural geometric version of Renegar's condition number $\mathcal{R}$ for the homogeneous convex feasibility problem associated with a regular cone $C\subseteq\mathbb{R}^n$. Let ${\rm Gr}_{n,m}$ denote the Grassmann manifold of $m$-dimensional linear subspaces of $\mathbb{R}^n$, and consider the projection distance $d_p(W_1,W_2) := \| \Pi_{W_1} - \Pi_{W_2}\|$ (spectral norm) between $W_1,W_2\in {\rm Gr}_{n,m}$, where $\Pi_{W_i}$ denotes the orthogonal projection onto $W_i$. We call $\mathscr{C}(W) := \max\{ d_p(W,W')^{-1} \mid W' \in \Sigma_m\}$ the Grassmann condition number of $W\in{\rm Gr}_{n,m}$, where the set of ill-posed instances $\Sigma_m\subset{\rm Gr}_{n,m}$ is defined as the set of linear subspaces touching $C$. We show that if $W ={\rm im}(A^T)$ for a matrix $A\in\mathbb{R}^{m\times n}$, then $\mathscr{C}(W) \le \mathcal{R}(A) \le \mathscr{C}(W)\, \kappa(A)$, where $\kappa(A) =\|A\| \|A^\dagger\|$ denotes the matrix condition number. This extends work by Belloni and Freund in [Math. Program. Ser. A, 119 (2009), pp. 95--107]. Furthermore, we show that $\mathscr{C}(W)$ can also be characterized in terms of the Riemannian distance metric on ${\rm Gr}_{n,m}$. This differential geometric characterization of $\mathscr{C}(W)$ is the starting point of the sequel [D. Amelunxen and P. Bürgisser, Probabilistic analysis of the Grassman condition number, preprint, http:arxiv.org/abs/1112.2603, 2011] to this paper, where the first probabilistic analysis of Renegar's condition number for an arbitrary regular cone $C$ is achieved.

Rim curvature anomaly in thin conical sheets revisited

This paper revisits one of the puzzling behaviors in a developable cone (d-cone), the shape obtained by pushing a thin sheet into a circular container of radius R by a distance η. The mean curvature was reported to vanish at the rim where the d-cone is supported. We investigate the ratio of the two principal curvatures versus sheet thickness h over a wider dynamic range than was used previously, holding R and η fixed. Instead of tending toward 1 as suggested by previous work, the ratio scales as (h/R)(1/3). Thus the mean curvature does not vanish for very thin sheets as previously claimed. Moreover, we find that the normalized rim profile of radial curvature in a d-cone is identical to that in a "c-cone" which is made by pushing a regular cone into a circular container. In both c-cones and d-cones, the ratio of the principal curvatures at the rim scales as (R/h)(5/2)F/(YR(2)), where F is the pushing force and Y is the Young's modulus. Scaling arguments and analytical solutions confirm the numerical results.

Nonlinear Coupled Hydrostatics of Arctic Conical Platforms

The increased exploration of deeper Arctic waters motivates the designs of new floating structures to operate under harsh Arctic conditions. Based on several model tests and investigations, structures with conical sections at the waterline have been shown to be a good design for waters where drifting ice is present, because the approaching ice fails in bending, which induces smaller loads than a crushing failure of ice. However, in most Arctic waters ice features are only present during part of the year and a large portion of the operation time of these structures will be in open water. Therefore, the floating structures must perform well in both conditions.Conical sections at the waterline will induce nonlinear coupling in the hydrostatic restoring forces and moments. It is important to understand how this affects the behavior in both ice and open water conditions. In order to investigate the nonlinear coupled hydrostatic restoring forces, an exact analytic expression for the metacentric height of a regular cone is presented. This is further used to develop an exact analytic expression for the hydrostatic restoring forces and moments for any body whose waterline intersects the frustum of a cone. A platform of the shallow draft-type, the platform type for which exact hydrostatics is most important, is used as a basis for the discussion and the effect of the coupled nonlinear restoring forces is illustrated by comparison to a model test performed in both open water and ice conditions.

The existence of fixed points of left-continuous monotone operators in spaces with a regular cone

In theorems on the existence of a fixed point of an operator the latter is usually assumed to be continuous. In this paper we prove a theorem with sufficient conditions for the existence of a fixed point of an operator which is not necessarily continuous (possibly it is left-continuous). The obtained theorem with the use of regular cones is applied for proving the existence of a fixed point of a nonlinear integral operator. We give an example illustrating the theorem.