Properties Of Cones Research Articles

From grids to pseudo-grids of lines: Resolution and seminormality

Over an infinite field [Formula: see text], we investigate the minimal free resolution of some configurations of lines. We explicitly describe the minimal free resolution of complete grids of lines and obtain an analogous result about the so-called complete pseudo-grids. Moreover, we characterize the total Betti numbers of configurations that are obtained posing a multiplicity condition on the lines of either a complete grid or a complete pseudo-grid. Finally, we analyze when a complete pseudo-grid is seminormal, differently from a complete grid. The main tools that have been involved in our study are the mapping cone procedure and properties of liftings, of pseudo-liftings and of weighted ideals. Although complete grids and pseudo-grids are hypersurface configurations and many results about such type of configurations have already been stated in literature, we give new contributions, in particular about the maps of the resolution.

Constraint Qualifications for Karush–Kuhn–Tucker Conditions in Multiobjective Optimization

The notion of a normal cone of a given set is paramount in optimization and variational analysis. In this work, we give a definition of a multiobjective normal cone, which is suitable for studying optimality conditions and constraint qualifications for multiobjective optimization problems. A detailed study of the properties of the multiobjective normal cone is conducted. With this tool, we were able to characterize weak and strong Karush–Kuhn–Tucker conditions by means of a Guignard-type constraint qualification. Furthermore, the computation of the multiobjective normal cone under the error bound property is provided. The important statements are illustrated by examples.

The Geometric Properties of a Class of Nonsymmetric Cones

Geometric methods are important for researching the differential properties of metric projectors, sensitivity analysis, and the augmented Lagrangian algorithm. Sun [3] researched the relationship among the strong second-order sufficient condition, constraint nondegeneracy, B-subdifferential nonsingularity of the KKT system, and the strong regularity of KKT points in investigating nonlinear semidefinite programming problems. Geometric properties of cones are necessary in studying second-order sufficient condition and constraint nondegeneracy. In this paper, we study the geometric properties of a class of nonsymmetric cones, which is widely applied in optimization problems subjected to the epigraph of vector k-norm functions and low-rank-matrix approximations. We compute the polar, the tangent cone, the linear space of the tangent cone, the critical cone, and the affine hull of this critical cone. This paper will support future research into the sensitivity and algorithms of related optimization problems.

Small cap decouplings

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $$\mathbb {R}^3$$ . This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

Targeting of the NRL Pathway as a Therapeutic Strategy to Treat Retinitis Pigmentosa.

Retinitis pigmentosa (RP) is an inherited retinal dystrophy (IRD) with a prevalence of 1:4000, characterized by initial rod photoreceptor loss and subsequent cone photoreceptor loss with accompanying nyctalopia, visual field deficits, and visual acuity loss. A diversity of causative mutations have been described with autosomal dominant, autosomal recessive, and X-linked inheritance and sporadic mutations. The diversity of mutations makes gene therapy challenging, highlighting the need for mutation-agnostic treatments. Neural leucine zipper (NRL) and NR2E3 are factors important for rod photoreceptor cell differentiation and homeostasis. Germline mutations in NRL or NR2E3 leads to a loss of rods and an increased number of cones with short wavelength opsin in both rodents and humans. Multiple groups have demonstrated that inhibition of NRL or NR2E3 activity in the mature retina could endow rods with certain properties of cones, which prevents cell death in multiple rodent RP models with diverse mutations. In this review, we summarize the literature on NRL and NR2E3, therapeutic strategies of NRL/NR2E3 modulation in preclinical RP models, as well as future directions of research. In summary, inhibition of the NRL/NR2E3 pathway represents an intriguing mutation agnostic and disease-modifying target for the treatment of RP.

Solvability for Two Forms of Nonlinear Matrix Equations

In this paper, we study nonlinear matrix equations $$\begin{aligned} X^p=A+\sum \limits _{i=1}^m M_i^T(X\#B)M_i \end{aligned}$$ and $$\begin{aligned} X^p=A+\sum \limits _{i=1}^j M_i^T(X\#B)M_i+\sum \limits _{i=j+1}^m M_i^T(X^{-1}\#B)M_i, \end{aligned}$$ where p, m, j are positive integers, $$1\le j\le m$$ , A, B are $$n\times n$$ positive definite matrices and $$M_i(i=1,2,3,\ldots ,m)$$ are $$n\times n$$ nonsingular real matrices. Based on some fixed point theorems for monotone and mixed monotone operators in ordered Banach spaces and some properties of cone, we prove that these equations always have a unique positive definite solution. In addition, an iterative sequence can be given to approximate the unique positive definite solution by employing a multi-step stationary iterative method.

On the inner cone property for convex sets in two-step Carnot groups, with applications to monotone sets

In the setting of two-step Carnot groups we show a “cone property” forhorizontally convex sets. Namely, we prove that, given a horizontally convex set C,a pair of points P ¬ C and Q ¬ int(C), both belonging to a horizontal line , thenan open truncated subRiemannian cone around and with vertex at P is containedin C.We apply our result to the problem of classification of horizontally monotone setsin Carnot groups. We are able to show that monotone sets in the direct product H×Rof the Heisenberg group with the real line have hyperplanes as boundaries.

Free fall water entry of a cone in three degrees of freedom

The hydrodynamic problem of a three dimensional cone entering water in three degrees of freedom is analysed based on the incompressible velocity potential theory. The effect of horizontal motion and rotational motion on the entry process is investigated. The problem is solved through the boundary element method in the time domain. Two physical coordinate systems are constructed. One is the body-fixed system for the solution of fluid velocities and free surface tracking, and the other is the space-fixed system for equations of motion. To match the extremely small wetted surface at the initial stage, the former frame is stretched based on the ratio of physical coordinates and vertical travelled distance. The auxiliary function method is used to decouple the nonlinear mutual dependence between fluid loads and the body motion. Detailed results for cone motion, impact pressure and free surface are provided, and in-depth discussion on the effect of the horizontal motion and rotational motion is given.

Alternative Fuels from Forestry Biomass Residue: Torrefaction Process of Horse Chestnuts, Oak Acorns, and Spruce Cones

The global energy system needs new, environmentally friendly, alternative fuels. Biomass is a good source of energy with global potential. Forestry biomass (especially wood, bark, or trees fruit) can be used in the energy process. However, the direct use of raw biomass in the combustion process (heating or electricity generation) is not recommended due to its unstable and low energetic properties. Raw biomass is characterized by high moisture content, low heating value, and hydrophilic propensities. The initial thermal processing and valorization of biomass improves its properties. One of these processes is torrefaction. In this study, forestry biomass residues such as horse chestnuts, oak acorns, and spruce cones were investigated. The torrefaction process was carried out in temperatures ranging from 200 °C to 320 °C in a non-oxidative atmosphere. The raw and torrefied materials were subjected to a wide range of tests including proximate analysis, fixed carbon content, hydrophobicity, density, and energy yield. The analyses indicated that the torrefaction process improves the fuel properties of horse chestnuts, oak acorns, and spruce cones. The properties of torrefied biomass at 320 °C were very similar to hard coal. In the case of horse chestnuts, an increase in fixed carbon content from 18.1% to 44.7%, and a decrease in volatiles from 82.9% to 59.8% were determined. Additionally, torrefied materials were characterized by their hydrophobic properties. In terms of energy yield, the highest value was achieved for oak acorns torrefied at 280 °C and amounted to 1.25. Moreover, higher heating value for the investigated forestry fruit residues ranged from 24.5 MJ·kg−1 to almost 27.0 MJ·kg−1 (at a torrefaction temperature of 320 °C).

A note on primal-dual stability in infinite linear programming

In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces.

Expansion of Magnetic Aromaticity Criteria to Multilayer Structures: Magnetic Response and Spherical Aromaticity of Matryoshka-Like Cluster [Sn@Cu12 @Sn20 ]12.

Structural characterization of the discrete [Sn@Cu12 @Sn20 ]12- cluster exposed a fascinating architecture composed of three concentric structural layers in which an endohedral Sn atom is enclosed in a Cu12 icosahedron, which in turn is embedded in an Sn20 dodecahedron. Herein, the possibility of sustaining aromatic behavior for this prototypical multilayered species was evaluated, in order to extend this concept to more complex clusters on the basis of magnetic response and bonding analysis by the AdNDP approach. This revealed characteristic features of spherical aromatics, given by the ability to sustain the shielding cone property, similar to archetypal aromatics. The favorable bonding pattern in the [Sn@Cu12 @Sn20 ]12- cluster fulfills the 2(N+1)2 Hirsch rule for aromaticity; thus, the cluster could be regarded as a first member of aromatic multilayered structures. The set of four 13c-2e aromatic bonds that was identified in the internal SnCu12 structure results in spherical aromatic character of this multilayered cluster. This insight builds a bridge between the traditional concept of Hückel's aromaticity and the aromaticity of complex and stable 3D systems that may be explored on the basis of magnetic response and bonding analysis. It also may open a way to novel findings in bottled clusters displaying aromatic behavior in multilayer structures, which are of great interest for inorganic nano- and material sciences due to their unprecedented stability.

Local existence and uniqueness of increasing positive solutions for non-singular and singular beam equation with a parameter

This paper is concerned with a class of beam equations with a parameter. By using the fixed point theorems of mixed monotone operator and the properties of cone, we study the non-singular and singular case, respectively, and obtain the sufficient conditions which guarantee the local existence and uniqueness of increasing positive solutions. Also, we present an iterative algorithm that converges to the solution. Moreover, we get some pleasant properties of the solutions for the boundary value problem dependent parameter. At last, two examples are given to illustrate the main results.

Bishop-Phelps Theorem for Normed Cones

Introduction In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. ‎An abstract cone is analogous to a real vector space‎, ‎except that we take as the set‎ ‎of scalars‎. ‎ In 2004, O‎. ‎Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach theorems in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps theorem is a fundamental theorem in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps theorem states that “the set of support functionals for a closed bounded convex subset of a real Banach space X, is norm dense in and the set of support points of is dense in the boundary of ". Indeed, E. Bishop and R. R. Phelps answer a question posed by ‎Victor Klee in 1958. We give an analogue to the normed cones‎, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type Theorem for normed cones../files/site1/files/52/5.pdf

Shielding cone behavior in the spherical aromatic He@C606−: origin of the record for the most shielded encapsulated 3He nucleus and comparison to He@C706−

The textbook concept of shielding cone is one of the characteristic properties for planar aromatic species, which explains the nuclear shielding of neighbor molecules located above and below the molecular plane. Here, we explore its resemblance in spherical aromatic fullerenes, where particularly for C606-, it explains the record for the highest shielding observed via 3He-NMR for the endohedral 3He@C606- species. Our results compare the behavior for He@C60 and He@C606- denoting relevant changes in their magnetic response behavior, where an induced shielding cone was observed for the latter with its long-range shielding character. In contrast to planar aromatics which give rise to a shielding cone property only for a perpendicularly oriented field, it was found that the shielding cone in spherical aromatic fullerenes occurs to any orientation. Thus, the orientation dependence behavior of the shielding cone is accounted, unraveling a characteristic shielding cone pattern in He@C606- which further rationalizes its record on the most shielded He nucleus, upon rotation or tumbling from the aromatic ring about the applied field. For the C70 case, the opposite situation has been characterized via 3He-NMR experiments, which is explained in terms of the deshielding/shielding region inside the C70 cage. Graphical abstract.

Notes on Constraint Qualifications for Second-Order Optimality Conditions

In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” first-order Abadie constraint qualification. In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set.

In Vitro Comparison of Three Dimensional Cone Beamed Dental Tomography with Intraoral Radiography in Detection of Dental Tomography with Intraoral Radiography in Detection of Dental Root Factures Root Factures

The precise diagnosis of dental root fractures in clinical practice is quite difficult. The aim of this study is to compare the results of three dimensional cone beam computed tomography (CBCT) and conventional intraoral radiography images in the diagnosis of dental root fractures. 50 maxillar central teeth with healthy roots were shot in a single blow in laboratory environment for cracks were studied for 10 teeth and horizontal root fractures created for 40 teeth. Parts of root fractures were sticked together by forming root fractures in five groups as crack, fracture without gap and fracture with 0,2 mm, 0,4 mm and 0,6 mm gaps. Images of all teeth were taken with CBCT and conventional intraoral radiography. Accurate diagnosis averages and positive predictive value test calculated through the data taken from the evaluation results of 30 dentists. Generally the results showed that images taken by the CBCT are better than images taken by the periapical radiography which obtained with traditional methods for the diagnosis of root fractures and cracks. The results also showed that the quality of the images increased as the voxel of the CBCT images decreased. It is shown that images taken by the CBCT provides better and clearer than the images taken by the traditional intraoral radiography in order to diagnose root fractures and cracks.

Circadian clock regulation of cone to horizontal cell synaptic transfer in the goldfish retina.

Although it is well established that the vertebrate retina contains endogenous circadian clocks that regulate retinal physiology and function during day and night, the processes that the clocks affect and the means by which the clocks control these processes remain unresolved. We previously demonstrated that a circadian clock in the goldfish retina regulates rod-cone electrical coupling so that coupling is weak during the day and robust at night. The increase in rod-cone coupling at night introduces rod signals into cones so that the light responses of both cones and cone horizontal cells, which are post-synaptic to cones, become dominated by rod input. By comparing the light responses of cones, cone horizontal cells and rod horizontal cells, which are post-synaptic to rods, under dark-adapted conditions during day and night, we determined whether the daily changes in the strength of rod-cone coupling could account entirely for rhythmic changes in the light response properties of cones and cone horizontal cells. We report that although some aspects of the day/night changes in cone and cone horizontal cell light responses, such as response threshold and spectral tuning, are consistent with modulation of rod-cone coupling, other properties cannot be solely explained by this phenomenon. Specifically, we found that at night compared to the day the time course of spectrally-isolated cone photoresponses was slower, cone-to-cone horizontal cell synaptic transfer was highly non-linear and of lower gain, and the delay in cone-to-cone horizontal cell synaptic transmission was longer. However, under bright light-adapted conditions in both day and night, cone-to-cone horizontal cell synaptic transfer was linear and of high gain, and no additional delay was observed at the cone-to-cone horizontal cell synapse. These findings suggest that in addition to controlling rod-cone coupling, retinal clocks shape the light responses of cone horizontal cells by modulating cone-to-cone horizontal cell synaptic transmission.

Synthesis of a Carbon Nanocone by Cascade Annulation.

More than 50 years have passed since the first observation of graphitic cones in the pyrolysis of carbon. However, to date there has been no report in the literature on the synthesis of such carbon allotropes. Here we present the first synthesis of a carbon nanocone, which comprises a pentagon encircled by 30 hexagons, by means of a palladium-catalyzed cross-coupling reaction. In this synthetic approach, 15 C-C bonds were constructed from a cone-shaped aromatic scaffold, corannulene, and five naphthalene dicarboximide moieties through a cascade of [3 + 3] and [4 + 2] annulations. The conical geometry of the first synthetic carbon nanocone was confirmed by X-ray crystallography. The optical and electronic properties of this graphitic cone were elucidated by UV/vis and fluorescence spectroscopy and cyclic voltammetry.

Spreading in a cone for the Fisher-KPP equation

In this paper we consider the spreading phenomena in the Fisher-KPP equation in a high dimensional cone with Dirichlet boundary condition. We show that any solution starting from a nonnegative and compact supported initial data spreads and converges to the unique positive steady state. Moreover, the asymptotic spreading speeds of the front in all directions pointing to the opening are c0 (which is the minimal speed of the traveling wave solutions of the 1-dimensional Fisher-KPP equation). Surprisingly, they do not depend on the shape of the cone, the propagating directions and the boundary condition.

Cones generated by random points on half-spheres and convex hulls of Poisson point processes

Let $$U_1,U_2,\ldots $$ be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We show that, as $$n\rightarrow \infty $$ , the f-vector of the $$(d+1)$$ -dimensional convex cone $$C_n$$ generated by $$U_1,\ldots ,U_n$$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f-vector of $$C_n$$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $$C_n$$ can be expressed through the expected f-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of Barany et al. (Random Struct Algorithms 50(1):3–22, 2017. https://doi.org/10.1002/rsa.20644 ). Our approach is based on the observation that the random cone $$C_n$$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $$\Vert x\Vert ^{-(d+\gamma )}$$ , where $$\gamma =1$$ . We compute the expected number of facets, the expected intrinsic volumes and the expected T-functional of this random convex hull for arbitrary $$\gamma >0$$ .