Class Of Cones Research Articles

Variability in oil droplet numbers and proportions in the retinas of anole lizards

Colored oil droplets are a common feature in the cones of almost all diurnal lizards. Using white light microscopy, three chromotypes of droplet can be identified in anoles – yellow, green and colorless. These droplets have been associated with different cone classes using microspectrophotometry. The principal member of double cones contain a yellow droplet while the accessory member contains a diffuse yellow pigment. Both members contain the LWS visual pigment. One class of large single cone contains a yellow droplet and the LWS visual pigment. The two remaining classes of large single cone contain the green droplet and either the LWS or MWS visual pigment. As such, by noting the distribution and numbers of the different droplet chromotypes, information about cone distribution, type, and number can be deduced microscopically. Retinas from three anole species (A. cristatellus, A. sagrei, and A. carolinensis) were isolated, flattened and oriented using the two foveae as landmarks. A 19-gauge needle was used to punch out six full-thickness retinal disks from identical retinal regions from both eyes of the three species. The different oil droplets were microscopically identified by color and counted. In all three species at all six retinal areas, the colorless droplets associated with the UVS and SWS single cones represented approximately 10% of the total droplets counted. However, the proportions of the yellow and green droplets were significantly different between the species. For A. cristatellus, 80% of the droplets were green while 10% were yellow. For A. sagrei approximately 85% of the droplets were yellow while only 5% were green. For A. carolinensis 50% of the droplets were yellow while 40% were green. The possible significance of these variable proportions is discussed in terms of possible effects on color vision and ecotype of the three anoles.

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

The structure of stable codimension one integral varifolds near classical cones of density $$Q+1/2$$

The structure of stable codimension one integral varifolds near classical cones of density $$Q+1/2$$

Embedding Ontologies in the Description Logic ALC by Axis-Aligned Cones

This paper is concerned with knowledge graph embedding with background knowledge, taking the formal perspective of logics. In knowledge graph embedding, knowledge— expressed as a set of triples of the form (a R b) (“a is R-related to b”)—is embedded into a real-valued vector space. The embedding helps exploiting geometrical regularities of the space in order to tackle typical inductive tasks of machine learning such as link prediction. Recent embedding approaches also consider incorporating background knowledge, in which the intended meanings of the symbols a, R, b are further constrained via axioms of a theory. Of particular interest are theories expressed in a formal language with a neat semantics and a good balance between expressivity and feasibility. In that case, the knowledge graph together with the background can be considered to be an ontology. This paper develops a cone-based theory for embedding in order to advance the expressivity of the ontology: it works (at least) with ontologies expressed in the description logic ALC, which comprises restricted existential and universal quantifiers, as well as concept negation and concept disjunction. In order to align the classical Tarskian Style semantics for ALC with the sub-symbolic representation of triples, we use the notion of a geometric model of an ALC ontology and show, as one of our main results, that an ALC ontology is satisfiable in the classical sense iff it is satisfiable by a geometric model based on cones. The geometric model, if treated as a partial model, can even be chosen to be faithful, i.e., to reflect all and only the knowledge captured by the ontology. We introduce the class of axis-aligned cones and show that modulo simple geometric operations any distributive logic (such as ALC) interpreted over cones employs this class of cones. Cones are also attractive from a machine learning perspective on knowledge graph embeddings since they give rise to applying conic optimization techniques.

A Bott periodicity theorem for ℓp-spaces and the coarse Novikov conjecture at infinity

We formulate and prove a Bott periodicity theorem for an ℓp-space (1≤p<∞). For a proper metric space X with bounded geometry, especially for a coarsely connected space, we introduce a version of K-homology at infinity and the Roe algebra at infinity and show that to prove the coarse Novikov conjecture, it suffices to prove the coarse assembly map at infinity is an injection. As a result, we show that the coarse Novikov conjecture holds for any metric space with bounded geometry which admits a fibred coarse embedding into an ℓp-space. These include all box spaces of a residually finite hyperbolic group, and a large class of warped cones of a compact metric space with an action by a hyperbolic group.

Hyperbolic cones for casual diffeomorphisms

This paper begins with introducing a new class of diffeomorphisms on pseudo-Riemannian manifolds which are called casual diffeomorphisms. We deduce a new pseudo-Riemannian metric via a casual diffeomorphism. We introduce a class of cones for a diffeomorphism, and we prove the existence of such cones for each p in a hyperbolic set of a casual diffeomorphism. As an example we find a hyperbolic casual diffeomorphism on a Schwarzschild pseudo-Riemannian manifold, and show that the whole manifold is a hyperbolic set for it.

PySilSub: An open-source Python toolbox for implementing the method of silent substitution in vision and nonvisual photoreception research.

The normal human retina contains several classes of photosensitive cell-rods for low-light vision, three cone classes for daylight vision, and intrinsically photosensitive retinal ganglion cells (ipRGCs) expressing melanopsin for non-image-forming functions, including pupil control, melatonin suppression, and circadian photoentrainment. The spectral sensitivities of the photoreceptors overlap significantly, which means that most lights will stimulate all photoreceptors to varying degrees. The method of silent substitution is a powerful tool for stimulating individual photoreceptor classes selectively and has found much use in research and clinical settings. The main hardware requirement for silent substitution is a spectrally calibrated light stimulation system with at least as many primaries as there are photoreceptors under consideration. Device settings that will produce lights to selectively stimulate the photoreceptor(s) of interest can be found using a variety of analytic and algorithmic approaches. Here we present PySilSub (https://github.com/PySilentSubstitution/pysilsub), a novel Python package for silent substitution featuring flexible support for individual colorimetric observer models (including human and mouse observers), multiprimary stimulation devices, and solving silent substitution problems with linear algebra and constrained numerical optimization. The toolbox is registered with the Python Package Index and includes example data sets from various multiprimary systems. We hope that PySilSub will facilitate the application of silent substitution in research and clinical settings.

Automorphisms of Rank-One Generated Hyperbolicity Cones and Their Derivative Relaxations

A hyperbolicity cone is said to be rank-one generated (ROG) if all its extreme rays have rank 1, where the rank is computed with respect to the underlying hyperbolic polynomial. This is a natural class of hyperbolicity cones which are strictly more general than the ROG spectrahedral cones. In this work, we present a study of the automorphisms of ROG hyperbolicity cones and their derivative relaxations. One of our main results states that the automorphisms of the derivative relaxations are exactly the automorphisms of the original cone fixing a certain direction. As an application, we completely determine the automorphisms of the derivative relaxations of the nonnegative orthant and of the cone of positive semidefinite matrices. More generally, we also prove relations between the automorphisms of a spectral cone and the underlying permutation-invariant set, which might be of independent interest.

Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones—the root-determinant cones and the matrix monotone derivative cones—we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open-source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.Funding: This work was supported by Office of Naval Research [Grant N00014-18-1-2079] and the National Science Foundation [Grant OAC-1835443].

Foveal Cone Structure in Patients With Blue Cone Monochromacy.

Blue cone monochromacy (BCM) is a rare inherited cone disorder in which both long- (L-) and middle- (M-) wavelength sensitive cone classes are either impaired or nonfunctional. Assessing genotype-phenotype relationships in BCM can improve our understanding of retinal development in the absence of functional L- and M-cones. Here we examined foveal cone structure in patients with genetically-confirmed BCM, using adaptive optics scanning light ophthalmoscopy (AOSLO). Twenty-three male patients (aged 6-75 years) with genetically-confirmed BCM were recruited for high-resolution imaging. Eight patients had a deletion of the locus control region (LCR), and 15 had a missense mutation-Cys203Arg-affecting the first two genes in the opsin gene array. Foveal cone structure was assessed using confocal and non-confocal split-detection AOSLO across a 300 × 300 µm area, centered on the location of peak cell density. Only one of eight patients with LCR deletions and 10 of 15 patients with Cys203Arg mutations had analyzable images. Mean total cone density for Cys203Arg patients was 16,664 ± 11,513 cones/mm2 (n = 10), which is, on average, around 40% of normal. Waveguiding cone density was 2073 ± 963 cones/mm2 (n = 9), which was consistent with published histological estimates of S-cone density in the normal eye. The one patient with an LCR deletion had a total cone density of 10,246 cones/mm2 and waveguiding density of 1535 cones/mm2. Our results show that BCM patients with LCR deletions and Cys203Arg mutations have a population of non-waveguiding photoreceptors, although the spectral identity and level of function remain unknown.

Learning with cone-based geometric models and orthologics

Recent approaches for knowledge-graph embeddings aim at connecting quantitative data structures used in machine learning to the qualitative structures of logics. Such embeddings are of a hybrid nature, they are data models that also exhibit conceptual structures inherent to logics. One motivation to investigate embeddings is to design conceptually adequate machine learning (ML) algorithms that learn or incorporate ontologies expressed in some logic. This paper investigates a new approach to embedding ontologies into geometric models that interpret concepts by geometrical structures based on convex cones. The ontologies are assumed to be represented in an orthologic, a logic with a full (ortho)negation. As a proof of concept this cone-based embedding was implemented within two ML algorithms for weak supervised multi-label learning. Both algorithms rely on cones but the first addresses ontologies expressed in classical propositional logic whereas the second addresses a weaker propositional logic, namely a weak orthologic that does not fulfil distributivity. The algorithms were evaluated and showed promising results that call for investigating other (sub)classes of cones and developing fine-tuned algorithms based on them.

Performance enhancements for a generic conic interior point algorithm

In recent work, we provide computational arguments for expanding the class of proper cones recognized by conic optimization solvers, to permit simpler, smaller, more natural conic formulations. We define an exotic cone as a proper cone for which we can implement a small set of tractable (i.e. fast, numerically stable, analytic) oracles for a logarithmically homogeneous self-concordant barrier for the cone or for its dual cone. Our extensible, open-source conic interior point solver, Hypatia, allows modeling and solving any conic problem over a Cartesian product of exotic cones. In this paper, we introduce Hypatia’s interior point algorithm, which generalizes that of Skajaa and Ye (Math. Program. 150(2):391–422, 2015) by handling exotic cones without tractable primal oracles. To improve iteration count and solve time in practice, we propose four enhancements to the interior point stepping procedure of Skajaa and Ye: (1) loosening the central path proximity conditions, (2) adjusting the directions using a third order directional derivative barrier oracle, (3) performing a backtracking search on a curve, and (4) combining the prediction and centering directions. We implement 23 useful exotic cones in Hypatia. We summarize the complexity of computing oracles for these cones and show that our new third order oracle is not a bottleneck. From 37 applied examples, we generate a diverse benchmark set of 379 problems. Our computational testing shows that each stepping enhancement improves Hypatia’s iteration count and solve time. Altogether, the enhancements reduce the geometric means of iteration count and solve time by over 80% and 70% respectively.

Area-minimizing cones over products of Grassmannian manifolds

This paper is the continuation of the previous one Jiao and Cui (Area-Minimizing Cones Over Grassmannian Manifolds. J. Geom. Anal. 32, 224 (2022). https://doi.org/10.1007/s12220-022-00963-7), where we re-proved the area-minimization of cones over Grassmannians of n-planes \(G(n,m;{\mathbb {F}})({\mathbb {F}}={\mathbb {R}},{\mathbb {C}},{\mathbb {H}})\), Cayley plane \({\mathbb {O}}P^2\) from the point view of Hermitian orthogonal projectors, and gave area-minimizing cones associated to oriented real Grassmannians \({\widetilde{G}}(n,m;{\mathbb {R}})\) by using Lawlor’s Curvature Criterion Lawlor (Mem Amer Math Soc 91(446), 1991). Here, we make a further step on showing that the cones, of dimension no less than \({\mathbf {8}}\), over minimal products of \(G(n,m;{\mathbb {F}})\) are area-minimizing. Moreover, those cones are very similar to the classical cones over products of spheres, and for the critical situation—the cones of dimension \({\mathbf {7}}\) Lawlor (Mem Amer Math Soc 91(446), 1991), we gain more area-minimizing cones by carefully computing the Jacobian \(inf_{v}det(I-tH^{v}_{ij})\). Certain minimizing cones among them had been found from the perspective of R-spaces Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), or isoparametric theory Tang and Zhang (J Differ Geom 115(2):367–393, 2020) recently, and the generic ones in our results are completely new. We also prove that the cones over minimal product of general \({\widetilde{G}}(n,m;{\mathbb {R}})\) are area-minimizing, it can be seen as generalized results for some \({\widetilde{G}}(2,m;{\mathbb {R}})\) shown in Ohno and Sakai (Josai Math Monogr 13:69–91, 2021), Tang and Zhang (J Differ Geom 115(2):367–393, 2020).

SONC optimization and exact nonnegativity certificates via second-order cone programming

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.

Solving Natural Conic Formulations with Hypatia.jl

Many convex optimization problems can be represented through conic extended formulations (EFs) using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. However, EFs are often significantly larger and more complex than equivalent conic natural formulations (NFs) represented using the much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement easily computable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. Our goal is to establish whether a generic conic interior point solver supporting NFs can outperform an advanced conic solver specialized for EFs across a variety of applied problems. We introduce Hypatia, a highly configurable open-source conic primal-dual interior point solver written in Julia and accessible through JuMP. Hypatia has a generic interface for exotic cones, some of which we define here. For seven applied problems, we introduce NFs using these cones and construct EFs that are necessarily larger and more complex. Our computational experiments demonstrate the advantages, especially in terms of solve time and memory usage, of solving the NFs with Hypatia compared with solving the EFs with either Hypatia or MOSEK 9.

Modeling chromatic contrast sensitivity across different background colors and luminance

In this research we compare chromatic contrast sensitivity models for two separate datasets and for the pooled dataset. They were obtained from two studies employing a very similar experimental paradigm. The data represent threshold visibilities of chromatic Gabor patterns varying in spatial frequency, background chromaticity, direction of color modulation and luminance, at constant stimulus size. Using the extended data set, we reconfirm our previously reported finding that a model based on coloropponent contrast signals is an improvement over a cone contrast model. However, when linear background scaling in classic cone contrast is replaced by nonlinear background scaling, an improvement of almost similar size is obtained. The results of this study can be of interest for the development of vision models employing the processing of spatio-chromatic information.

Morphometric classification and spatial distribution of Philippine volcanoes

The Philippine Island Arc has a large number of volcanoes with diverse morphologies, making it an ideal location to study the factors controlling the morphology and spatial distribution of island arc volcanoes. We have identified 731 volcanic edifices using the SRTM 30 m digital elevation model, and computed their quantitative morphology using the MORVOLC algorithm. Hierarchical classification by principal component (PC) analysis distinguishes four volcano types: small flat cones, small steep cones, large cones, and massifs, with mean volumes of 0.2 km3 (<6.2 km3), 0.4 km3 (<9 km3), 29 km3 (0.15–178 km3), 267 km3 (76–675 km3), mean heights of 125 m (16–721 m), 260 m (53–971 m), 842 m (59–2313 m), 1533 m (1012–2175 m), and mean slopes of 13° (3–21°), 22° (14–37°), 15° (3–28°), 15° (11–22°), respectively. This classification is based mainly on their size and irregularity (PC1) and steepness (mean slope and height/basal width ratio; PC2), and to a lesser extent on the size of the summit region and edifice truncation (PC3) and edifice elongation (PC4). These morphological volcano classes represent stages along an evolutionary trend. The small flat cones are mostly monogenetic, whereas the small steep cone class represents an early growth stage. Some can develop into large polygenetic cones while a few can further grow laterally into massifs, both of which are preferentially found on thickened crust. There is a trend towards more silicic compositions from small to large cones, perhaps due to larger edifice loads preventing mafic dykes from reaching the surface. The distribution and alignment of the edifices within volcanic fields seems to be influenced by both regional and local stress fields and pre-existing structures.

Photosensitive ganglion cells: A diminutive, yet essential population.

Our visual system has evolved to provide us with an image of the scene that surrounds us, informing us of its texture, colour, movement, and depth with an enormous spatial and temporal resolution, and for this purpose, the image formation (IF) dedicates the vast majority of our retinal ganglion cell (RGC) population and much of our cerebral cortex. On the other hand, a minuscule proportion of RGCs, in addition to receiving information from classic cone and rod photoreceptors, express melanopsin and are intrinsically photosensitive (ipRGC). These ipRGC are dedicated to non-image-forming (NIF) visual functions, of which we are unaware, but which are essential for aspects related to our daily physiology, such as the timing of our circadian rhythms and our pupillary light reflex, among many others. Before the discovery of ipRGCs, it was thought that the IF and NIF functions were distinct compartments regulated by different RGCs, but this concept has evolved in recent years with the discovery of new types of ipRGCs that innervate subcortical IF regions, and therefore have IF visual functions. Six different types of ipRGCs are currently known. These are termed M1-M6, and differ in their morphological, functional, molecular properties, central projections, and visual behaviour responsibilities. A review is presented on the melanopsin visual system, the most active field of research in vision, for which knowledge has grown exponentially during the last two decades, when RGCs giving rise to this pathway were first discovered.

Wave Patterns of Gravity–Capillary Waves from Moving Localized Sources

We study wave patterns of gravity–capillary waves from moving localized sources within the classic setup of the problem of ship wakes. The focus is on the co-existence of two wave systems with opposite signatures of group velocity relative to the localized source. It leads to the problem of choice of signs for phase functions of the gravity (“slow”) and capillary (“fast”) branches of the dispersion relation: the question generally ignored when constructing phase patterns of the solutions. We detail characteristic angles of the wake patterns: (i) angle of demarcation of gravity and capillary waves—“the phase Mach” cone, (ii) angle of the minimal group velocity of gravity–capillary waves—“the group Mach” cone, (iii, iv) angles of cusps of isophases that appear after a threshold current speed. The outer cusp cone is naturally associated with the classic cone of Kelvin for pure gravity waves. The inner one results from the effect of capillarity and tends to the “group Mach” pattern at high speeds of current. Amplitudes of the wave patterns are estimated within the recently proposed approach of reference functions for the problem of propagation of packets of linear dispersive waves. The effect of shape is discussed for elliptic reference sources.

Quantifying Color Vision Changes Associated With Cataracts Using Cone Contrast Thresholds.

PurposeTo evaluate effects of age and simulated and real cataractous changes on color vision as measured by the high-definition cone contrast test (CCT).MethodsTwenty-four healthy volunteers from two cohort studies performed CCT using best-corrected visual acuity, filters, mydriasis, and pinhole correction. Retrospective cross-sectional study of patients seen in eye clinics evaluated the relationship between age and color vision, and age and lens status in 355 eyes. Last, 25 subjects underwent CCT before and after cataract surgery.ResultsCCT scores were most reliable in the nonmydriatic condition without pinhole correction. Progressively dense brown filters produced small decreases in S-cone sensitivity. Linear regression analysis of phakic subjects showed a decline for all cone classes with age. Rate of decline was greater for S-cones (slope = −1.09; 95% confidence interval [CI], −1.30 to 0.86) than M-cones (slope = −0.80; 95% CI, −1.03 to −0.58) and L-cones (slope = −0.66; 95% CI, −0.88 to −0.44). CCT scores increased for S-cones but reduced for L- and M-cones in pseudophakic subjects compared with phakic patients. CCT scores after cataract surgery increased for S-cones, M-cones, and L-cones by 33.0 (95% CI, 8.6 to 57.4), 24.9 (95% CI, 3.8 to 46.0), and 22.0 (95% CI, −3.2 to 47.3), respectively.ConclusionsCCT assessment allows for clinically practical quantitation of color and contrast vision improvement after cataract surgery and aging patients who note poor vision despite good visual acuity.Translational RelevanceCCT testing, which quantifies hereditary and acquired color deficiency, can also quantify the degree of cataract severity and, combined with other parameters, can provide more precise guidance for cataract extraction to optimize patient care.