Regular Cone Research Articles

Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $\Omega$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones.

Best proximity results in regular cone metric spaces

First, we define the notion of distance between two subsets in regular cone metric spaces. Then, we establish some conditions which guarantee the existence of best proximity points for cyclic contraction mappings on regular cone metric spaces.

Common fixed points in cone metric spaces for [formula omitted]-pairs

In this paper we introduce some contractive conditions of Meir–Keeler type for two mappings, called f - M K -pair mappings and f - C J M -pair (from Ciric, Jachymski, and Matkowski) mappings, in the framework of regular cone metric spaces and we prove theorems which guarantee the existence and uniqueness of common fixed points. We give also a fixed point result for a multivalued mapping that satisfies a contractive condition of Meir–Keeler type. These results extend and generalize some recent results from the literature. To conclude the paper, we extend our main result to non-regular cone metric spaces by using the scalarization method of Du.

High-Resolution Imaging of Resolved Central Serous Chorioretinopathy Using Adaptive Optics Scanning Laser Ophthalmoscopy

To compare pathologic changes in photoreceptors in eyes with resolved central serous chorioretinopathy (CSC) seen on high-resolution images obtained by adaptive optics scanning laser ophthalmoscopy (AO SLO) with visual acuity (VA) and findings on spectral-domain optical coherence tomography (SD OCT). Observational case series. Forty-five eyes of 38 patients with resolved CSC and 20 normal eyes of 20 volunteer subjects. All patients underwent a full ophthalmologic examination, SD OCT, and imaging with an original prototype AO SLO system fabricated using liquid crystal-on-silicon technology. Cone mosaic patterns and cone density on AO SLO images and VA in eyes with CSC. In normal eyes, AO SLO images showed a regular photoreceptor mosaic pattern and average cone densities 0.2, 0.5, and 1.0 mm from the central fovea of 67,900, 33,320, and 14,450 cones/mm(2). In eyes with CSC, cone densities were significantly lower at each distance from the central fovea (P = 0.009 at 0.2 mm, P = 0.007 at 0.5 mm, and P = 0.004 at 1.0 mm), and 2 distinct cone mosaic patterns were seen. Group 1 CSC eyes had regular cone mosaic patterns with small dark regions. Group 2 CSC eyes had irregular mosaic patterns with large dark regions. Compared with group 1, group 2 had significantly lower average cone density and worse average logarithm of the minimum angle of resolution (logMAR) VA (P<0.001). Mean cone density in eyes with disruptions in the photoreceptor inner and outer segment (IS/OS) junction or in the intermediate line on SD OCT images was significantly lower than that in eyes with an intact IS/OS junction or intermediate line (P<0.001 for both). Cone density 0.2 mm from the central fovea correlated with logMAR VA and mean foveal thickness (1-mm diameter area) measured on SD OCT images (P<0.001 for both). Adaptive optics SLO images showed abnormal cone mosaic patterns and reduced cone densities in eyes with resolved CSC, and these abnormalities were associated with VA loss, suggesting that AO SLO is a useful means to detect and measure cone abnormalities associated with VA loss in these eyes.

Patterned procedure for template-synthesis and microstructural characterization of copper nanowires

Highly ordered circular patterns of copper nanowire arrays were successfully deposited into designed anodic aluminum oxide templates. High-resolution transmission electron microscopy was used to study the microstructure of these Cu nanostructures. The results showed that the growth orientation of the copper nanowires was along [220] direction, and the broken orientation were along [2̄02̄] and [02̄2] directions, respectively. Regular cones were formed at the broken end of nanowires. Bent nanowires were also observed, this means that the copper nanowires have good mechanical properties when applied external force. Chemical analysis has been performed on Cu nanowires using electron energy-loss spectroscopy.

An Efficient Rescaled Perceptron Algorithm for Conic Systems

The classical perceptron algorithm is an elementary row-action/relaxation algorithm for solving a homogeneous linear inequality system Ax &gt; 0. A natural condition measure associated with this algorithm is the Euclidean width τ of the cone of feasible solutions, and the iteration complexity of the perceptron algorithm is bounded by 1/τ2 [see Rosenblatt, F. 1962. Principles of Neurodynamics. Spartan Books, Washington, DC]. Dunagan and Vempala [Dunagan, J., S. Vempala. 2007. A simple polynomial-time rescaling algorithm for solving linear programs. Math. Programming 114(1) 101–114] have developed a rescaled version of the perceptron algorithm with an improved complexity of O(n ln (1/τ)) iterations (with high probability), which is theoretically efficient in τ and, in particular, is polynomial time in the bit-length model. We explore extensions of the concepts of these perceptron methods to the general homogeneous conic system Ax ∈ int K, where K is a regular convex cone. We provide a conic extension of the rescaled perceptron algorithm based on the notion of a deep-separation oracle of a cone, which essentially computes a certificate of strong separation. We show that the rescaled perceptron algorithm is theoretically efficient if an efficient deep-separation oracle is available for the feasible region. Furthermore, when K is the cross-product of basic cones that are either half-spaces or second-order cones, then a deep-separation oracle is available and, hence, the rescaled perceptron algorithm is theoretically efficient. When the basic cones of K include semidefinite cones, then a probabilistic deep-separation oracle for K can be constructed that also yields a theoretically efficient version of the rescaled perceptron algorithm.

Fluorescence Measurements in Contact Lenses With a Novel Confocal Microscope

The spatial distribution of macular birefringence was modeled to examine the contribution from the foveal Henle fiber layer, particularly cone axons. The model was tested in 20 normal subjects, age 17–55 yr. Phase retardance due to Henle fibers was modeled for rings increasing in radius around the fovea, using a sinewave of two periods (2f). The 2f sinewave amplitude increased linearly with eccentricity for each individual, (p < 0.004) in 19 of 20 subjects. A good fit to linearity implies regular cone distribution and radial symmetry, and the uniformly excellent fits indicate no effect of age in our sample. The peak of the 2f sinewave amplitude varied across subjects from 1.06 to 2.46 deg. An increasingly eccentric peak with increasing age would indicate a relative decrease of cone axons in the central fovea, but the location of the peak was not associated with age for our sample, which did not include elderly subjects.

Strong violet luminescence from ZnO nanocrystals grown by the low-temperature chemical solution deposition

The luminescence properties of zinc oxide (ZnO) nanocrystals grown from solution are reported. The ZnO nanocrystals were characterized by scanning electron microscopy, X-ray diffraction, cathodo- and photoluminescence (PL) spectroscopy. The ZnO nanocrystals have the same regular cone form with the average sizes of 100–500 nm. Apart from the near-band-edge emission around 381 nm and a weak yellow–orange band around 560–580 nm at 300 K, the PL spectra of the as-prepared ZnO nanocrystals under high-power laser excitation also showed a strong defect-induced violet emission peak in the range of 400 nm. The violet band intensity exhibits superlinear excitation power dependence while the UV emission intensity is saturated at high excitation laser power. With temperature raising the violet peak redshifts and its intensity increases displaying unconventional negative thermal quenching behavior, whereas intensity of the UV and yellow–orange bands decreases. The origin of the observed emission bands is discussed.

Fixed points of multifunctions on regular cone metric spaces

Each metric space is a regular cone metric space. We shall extend a result about Meir–Keeler type contraction mappings on metric spaces to regular cone metric spaces. Also, we shall give some results about fixed point of weakly uniformly strict p -contraction multifunctions on regular cone metric spaces.

Varicones and Growth Cones: Two Neurite Terminals in PC12 Cells

The rat adrenal pheochromocytoma PC12 cell line is one of the traditional models for the study of neurite outgrowth and growth cone behavior. To clarify to what extent PC12 neurite terminals can be compared to neuronal growth cones, we have analyzed their morphology and protein distribution in fixed PC12 cells by immunocytochemistry. Our results show that that PC12 cells display a special kind of neurite terminal that includes a varicosity in close association with a growth cone. This hybrid terminal, or “varicone”, is characterized by the expression of specific markers not typically present in neuronal growth cones. For example, we show that calpain-2 is a specific marker of varicones and can be detected even before the neurite develops. Our data also shows that a fraction of PC12 neurites end in regular growth cones, which we have compared to hippocampal neurites as a control. We also report the extraordinary incidence of varicones in the literature referred to as “growth cones”. In summary, we provide evidence of two different kinds of neurite terminals in PC12 cells, including a PC12-specific terminal, which implies that care must be taken when using them as a model for neuronal growth cones or neurite outgrowth.

Order properties of spaces of non-absolutely integrable vector-valued functions and applications to differential equations

We prove that if the space $Y$ of Henstock-Lebesgue integrable functions from a compact real interval to a Banach space $E$ is normed by the Alexiewicz norm and ordered by a regular cone $E_+$, then the $E_+$-valued functions of $Y$ form a regular order cone of $Y$. This property is shown to hold also when $Y$ is the space of Henstock-Kurzweil integrable functions if $E$ is weakly sequentially complete and $E_+$ is normal. As an application of the obtained results we prove existence and comparison results for least and greatest solutions of nonlocal implicit initial-value problems of discontinuous functional differential equations containing non-absolutely integrable functions.

Alternative Theorems for Quadratic Inequality Systems and Global Quadratic Optimization

We establish alternative theorems for quadratic inequality systems. Consequently, we obtain Lagrange multiplier characterizations of global optimality for classes of nonconvex quadratic optimization problems. We present a generalization of Dine's theorem to a system of two homogeneous quadratic functions with a regular cone. The class of regular cones are cones K for which $(K\cup-K)$ is a subspace. As a consequence, we establish a generalization of the powerful S-lemma, which paves the way to obtain a complete characterization of global optimality for a general quadratic optimization model problem involving a system of equality constraints in addition to a single quadratic inequality constraint. We then present an alternative theorem for a system of three nonhomogeneous inequalities by way of establishing the convexity of the joint-range of three homogeneous quadratic functions using a regular cone. This yields Lagrange multiplier characterizations of global optimality for classes of trust-region type problems with two inequality constraints. Finally, we establish an alternative theorem for systems involving an arbitrary finite number of quadratic inequalities involving Z-matrices, which are matrices with nonpositive off diagonal elements, and present necessary and sufficient conditions for global optimality for classes of nonconvex inequality constrained quadratic optimization problems.

Adaptive Optics Scanning Laser Ophthalmoscopy Images in a Family with the Mitochondrial DNA T8993C Mutation

This study was designed to assess the effect of mitochondrial DNA (mtDNA) mutation T8993C on cone structure in a family expressing neurogenic muscle weakness, ataxia, and retinitis pigmentosa (NARP) syndrome. Five family members were studied, using clinical examination, nerve conduction studies, perimetry, optical coherence tomography (OCT) measures of central retinal thickness, and electroretinography. High-resolution images of cone structure using adaptive optics scanning laser ophthalmoscopy (AOSLO) were obtained in four subjects with stable fixation. Cone spacing was compared to 18 age-similar normal subjects and converted to z-scores at each location where unambiguous cones were identified. Tissue levels of T8993C mutant heteroplasmy in blood and hair follicles were quantified using real-time allele-refractory mutations system (ARMS) quantitative polymerase chain reaction (qPCR). Subjects expressing the T8993C mutation showed varying levels of disease severity. The subject with the lowest mutant load (42%-54%) showed no neurologic or retinal abnormalities. The remaining four subjects with over 77% mutant load all expressed severe neurologic and/or retinal abnormalities. AOSLO images revealed three patterns of cone spacing: pattern 1, normal; pattern 2, increased cone spacing within a contiguous cone mosaic; and pattern 3, patchy cone loss with increased cone spacing. Visual function was most severely affected in pattern 3. High levels of T8993C mutant load were associated with severe neurologic or visual dysfunction, while lower levels caused no detectable abnormalities. Visual function was better in patients with a contiguous and regular cone mosaic. Patients expressing high levels of the mtDNA T8993C mutation show abnormal cone structure, suggesting normal mitochondrial DNA is necessary for normal waveguiding by cones.

BOUNDARY VALUE PROBLEMS VIA AN INTERMEDIATE VALUE THEOREM

AbstractWe use an intermediate value theorem for quasi-monotone increasing functions to prove the existence of the smallest and the greatest solution of the Dirichlet problemu″ +f(t,u) = 0,u(0) = α,u(1) = β between lower and upper solutions, wheref:[0,1] ×E→Eis quasi-monotone increasing in its second variable with respect to a regular cone.

Cytotoxicity evaluation of Active GP and Resilon cones in vitro

This study was done to evaluate the cytotoxicity of Activ GP and Resilon cones in an in vitro cell culture system. Gutta-percha (GP), Activ GP, and Resilon cones were tested in this study. L929 cells were seeded into 96-well plates at 3 x 10(4) cells/well. In one set of experiments, 2-mm segments cut from the tip of GP and Resilon cones were placed into the cell culture wells and incubated for 1, 2, and 3 days. In another set of experiments, 2 20-mm segments of GP, Activ GP, and Resilon cones were incubated in 2 mL cell culture medium for 1 week. Then 100 microL elutes were tested for 24 and 48 h. Cell viability was evaluated by MTT assay. Data were analyzed using 1-way analysis of variance. When GP, Activ GP, and Resilon segments were placed into cell cultures, cell viability in the Resilon group was significantly greater than in the GP and Activ GP groups at any test time. There was no cell viability difference between the Activ GP and GP groups. When the elutes of GP, Activ GP, and Resilon was placed into cell cultures, the results were the same as using segments of the tested material. The cytotoxicity of GP and Activ GP is greater than that of the Resilon cone. There was no cell viability difference between Activ GP and regular GP. Resilon has better biocompatibility than regular GP and Activ GP cones. The cytotoxicity of Activ GP is similar to that of regular GP.

Spatial distribution of macular birefringence associated with the Henle fibers

The spatial distribution of macular birefringence was modeled to examine the contribution from the foveal Henle fiber layer, particularly cone axons. The model was tested in 20 normal subjects, age 17–55 yr. Phase retardance due to Henle fibers was modeled for rings increasing in radius around the fovea, using a sinewave of two periods (2f). The 2f sinewave amplitude increased linearly with eccentricity for each individual, ( p < 0.004) in 19 of 20 subjects. A good fit to linearity implies regular cone distribution and radial symmetry, and the uniformly excellent fits indicate no effect of age in our sample. The peak of the 2f sinewave amplitude varied across subjects from 1.06 to 2.46 deg. An increasingly eccentric peak with increasing age would indicate a relative decrease of cone axons in the central fovea, but the location of the peak was not associated with age for our sample, which did not include elderly subjects.

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ1(w), . . . , Δr(w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δk(w)/Δk-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

On functional improper Volterra integral equations and impulsive differential equations in ordered Banach spaces

In this paper we derive existence and comparison results for discontinuous improper functional integral equations of Volterra type in an ordered Banach space which has a regular order cone. For this purpose we prove Dominated and Monotone Convergence Theorems for improper integrals. The obtained results are then applied to first-order impulsive differential equations. Concrete examples are also solved by using symbolic programming.

Acuity and contrast sensitivity of the bluegill sunfish and how they change during optic nerve regeneration

Spatial vision was studied in the bluegill sunfish, Lepomis macrochirus (9.5-14 cm standard length) to assess the limitations imposed by the optics of the eye, the retinal receptor spacing and the retinotectal projection during regeneration. Examination of images formed by the dioptric elements of the eye showed that spatial frequencies up to 29 c/ degrees could be imaged on the retina. Cone spacing was measured in the retina of fresh, intact eyes. The spacing of rows of double cones predicted 3.4 c/degrees as the cutoff spatial frequency; the spacing between rows of single and double cones predicted 6.7 c/degrees. Contrast sensitivity functions were obtained psychophysically in normals and fish with one regenerating optic nerve. Fish were trained to orient to gratings (mean luminance = 25 cd/m2) presented to either eye. In normals, contrast sensitivity functions were similar in shape and bandwidth to those of other species, peaking at 0.4 c/degrees with a minimum contrast threshold of 0.03 and a cutoff at about 5 c/degrees, which was within the range predicted by cone spacing. Given that the optical cutoff frequency exceeds that predicted by cone spacing, it is possible that gratings could be detected by aliasing with the bluegill's regular cone mosaic. However, tests with high contrast gratings up to 15 c/ degrees found no evidence of such detection. After crushing one optic nerve in three trained sunfish, recovery of visual avoidance, dorsal light reflex and orienting to gratings, were monitored over 315 days. At 64-69 days postcrush, responses to gratings reappeared, and within 2-5 days contrast sensitivity at low (0.15 c/degrees) and medium (1.0 c/ degrees) spatial frequencies had returned to normal. At a high spatial frequency (2.93 c/degrees) recovery was much slower, and complete only in one fish.

Photo- and Cathodoluminescence Studies of ZnO-Filled Opal Nanocomposites

The transmittance, reflectance, photo and cathodoluminescence spectra of a ZnO-filled opal nanocomposite, as well as the spectra of ZnO nanoparticles deposited on sapphire and glass substrates, were investigated. The ZnO nanoparticles have basically a (002) orientation and a regular cone form. They exhibit a strong exciton emission at 380 nm and relatively low green emission at 565 nm. In the ZnO-filled opal nanocomposite a “red shift” of the photonic band gap position in comparison with the bare artificial opal and its eect on the photoluminescence spectra are observed. Quenching of the excitonic emission of the ZnO nanoparticles grown in the opal voids is found and it is assigned to an energy transfer from the ZnO nanoparticles to opal matrix defect centers.