Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.

Shape Preserving Piecewise KNR Fractional Order Biquadratic C 2 Spline

In a recent article, a piecewise cubic fractional spline function is developed which produces C 1 continuity to given data points. In the present paper, an interpolant continuity class C 2 is preserved which gives visually pleasing piecewise curves. The behavior of the resulting representations is analyzed intrinsically with respect to variation of the shape control parameters t and s. The data points are restricted to be strictly monotonic along real line.

Efficacy of fovea-sparing internal limiting membrane peeling for epiretinal membrane foveoschisis

Purpose: To investigate the outcomes of vitrectomy with fovea-sparing internal limiting membrane (ILM) peeling (FSIP) for epiretinal membrane foveoschisis based on new optical coherence tomography definitions. Methods: 27 eyes of 28 patients (67.2 ± 10.5 years old) who underwent vitrectomy with FSIP without gas tamponade for epiretinal membrane foveoschisis were included. All patients underwent follow-up examinations for at least 12 months. In the FSIP technique, the ILM is peeled off in a donut shape, preserving the foveal ILM. The logarithm of the minimal angle of resolution best-corrected visual acuity (logMAR BCVA), central macular thickness (CMT), and surgical complications were examined. Results: The BCVA at 12 months improved significantly from baseline (p < 0.001). Baseline ellipsoid zone defects were found in 3 eyes (10%), and all defective eyes had recovered at 12 months. CMT decreased significantly from baseline (p < 0.001). Acute macular edema, full-thickness macular hole, and recurrence of epiretinal membrane were not observed during follow-up. Conclusion: FSIP achieved good visual outcome and retinal morphological change. Moreover, FSIP might avoid acute macular edema in epiretinal membrane foveoschisis surgery.

Estimating Gini Coefficient from Grouped Data Based on Shape-Preserving Cubic Hermite Interpolation of Lorenz Curve

The Lorenz curve and Gini coefficient are widely used to describe inequalities in many fields, but accurate estimation of the Gini coefficient is still difficult for grouped data with fewer groups. We proposed a shape-preserving cubic Hermite interpolation method to approximate the Lorenz curve by maximizing or minimizing the strain energy or curvature variation energy of the interpolation curve, and a method to estimate the Gini coefficient directly from the coefficients of the interpolation curve. This interpolation method can preserve the essential requirements of the Lorenz curve, i.e., non-negativity, monotonicity, and convexity, and can estimate the derivatives at intermediate points and endpoints at the same time. These methods were tested with 16 grouped quintiles or unequally spaced datasets, and the results were compared with the true Gini coefficients calculated with all census data and results estimated with other methods. Results indicate that the maximum strain energy interpolation method generally performs the best among different methods, which is applicable to both equally and unequally spaced grouped datasets with higher precision, especially for grouped data with fewer groups.