On the Automorphism Groups of some AG-Codes Based on Ca;b Curves

*Partially supported by NATO.

The Space of Differences of Convex Functions on [0, 1]

∗Participant in Workshop in Linear Analysis and Probability, Texas A & M University, College Station, Texas, 2000. Research partially supported by the Edmund Landau Center for Research in Mathematical Analysis and related areas, sponsored by Minerva Foundation (Germany).

On The Brill-noether Theory of Spanned Vector Bundles on Smooth Curves

Here we study the integers (d, g, r) such that on a smooth projective curve of genus g there exists a rank r stable vector bundle with degree d and spanned by its global sections.

Abstract Subdifferential Calculus and Semi-Convex Functions

∗ The work is partially supported by NSFR Grant No MM 409/94.

On the Difference of 4-Gonal Linear Systems on some Curves

Let C = (C, g^1/4 ) be a tetragonal curve. We consider the scrollar invariants e1 , e2 , e3 of g^1/4 . We prove that if W^1/4 (C) is a non-singular variety, then every g^1/4 ∈ W^1/4 (C) has the same scrollar invariants.

On Typical Compact Convex Sets in Hilbert Spaces

Let E be an infinite dimensional separable space and for e ∈ E and X a nonempty compact convex subset of E, let qX(e) be the metric antiprojection of e on X. Let n ≥ 2 be an arbitrary integer. It is shown that for a typical (in the sence of the Baire category) compact convex set X ⊂ E the metric antiprojection qX(e) has cardinality at least n for every e in a dense subset of E.

Stability of Supporting and Exposing Elements of Convex Sets in Banach Spaces

* This work was supported by the CNR while the author was visiting the University of Milan.

On Uniformly Convex and Uniformly Kadec-Klee Renomings

We give a new construction of uniformly convex norms with a power type modulus on super-reflexive spaces based on the notion of dentability index. Furthermore, we prove that if the Szlenk index of a Banach space is less than or equal to ω (first infinite ordinal) then there is an equivalent weak* lower semicontinuous positively homogeneous functional on X* satisfying the uniform Kadec-Klee Property for the weak*-topology (UKK*). Then we solve the UKK or UKK* renorming problems for Lp(X) spaces and C(K) spaces for K scattered compact space.

Exploring the optimum step size for defocus curves

Purpose: To evaluate the effect of reducing the number of visual acuity measurements made in a defocus curve on the quality of data quantified. Setting: Midland Eye, Solihull, United Kingdom. Design: Evaluation of a technique. Methods: Defocus curves were constructed by measuring visual acuity on a distance logMAR letter chart, randomizing the test letters between lens presentations. The lens powers evaluated ranged between +1.50 diopters (D) and -5.00 D in 0.50 D steps, which were also presented in a randomized order. Defocus curves were measured binocularly with the Tecnis diffractive, Rezoom refractive, Lentis rotationally asymmetric segmented (+3.00 D addition [add]), and Finevision trifocal multifocal intraocular lenses (IOLs) implanted bilaterally, and also for the diffractive IOL and refractive or rotationally asymmetric segmented (+3.00 D and +1.50 D adds) multifocal IOLs implanted contralaterally. Relative and absolute range of clear-focus metrics and area metrics were calculated for curves fitted using 0.50 D, 1.00 D, and 1.50 D steps and a near add-specific profile (ie, distance, half the near add, and the full near-add powers). Results: A significant difference in simulated results was found in at least 1 of the relative or absolute range of clear-focus or area metrics for each of the multifocal designs examined when the defocus-curve step size was increased (P<.05). Conclusion: Faster methods of capturing defocus curves from multifocal IOL designs appear to distort the metric results and are therefore not valid. Financial Disclosure: No author has a financial or proprietary interest in any material or method mentioned. © 2013 ASCRS and ESCRS.

ROC Curves within the Framework of Neural Network Assembly Memory Model: Some Analytic Results

On the basis of convolutional (Hamming) version of recent Neural Network Assembly Memory Model (NNAMM) for intact two-layer autoassociative Hopfield network optimal receiver operating characteristics (ROCs) have been derived analytically. A method of taking into account explicitly a priori probabilities of alternative hypotheses on the structure of information initiating memory trace retrieval and modified ROCs (mROCs, a posteriori probabilities of correct recall vs. false alarm probability) are introduced. The comparison of empirical and calculated ROCs (or mROCs) demonstrates that they coincide quantitatively and in this way intensities of cues used in appropriate experiments may be estimated. It has been found that basic ROC properties which are one of experimental findings underpinning dual-process models of recognition memory can be explained within our one-factor NNAMM.

Structuring of Ranked Models

Prognostic procedures can be based on ranked linear models. Ranked regression type models are designed on the basis of feature vectors combined with set of relations defined on selected pairs of these vectors. Feature vectors are composed of numerical results of measurements on particular objects or events. Ranked relations defined on selected pairs of feature vectors represent additional knowledge and can reflect experts' opinion about considered objects. Ranked models have the form of linear transformations of feature vectors on a line which preserve a given set of relations in the best manner possible. Ranked models can be designed through the minimization of a special type of convex and piecewise linear (CPL) criterion functions. Some sets of ranked relations cannot be well represented by one ranked model. Decomposition of global model into a family of local ranked models could improve representation. A procedures of ranked models decomposition is described in this paper.

Applications of the Owa-Srivastava Operator to the Class of K-Uniformly Convex Functions

2000 Mathematics Subject Classification: Primary 30C45, 26A33; Secondary 33C15

Development and Implementation of NURBS Models of Quadratic Curves and Surfaces

This article goes into the development of NURBS models of quadratic curves and surfaces. Curves and surfaces which could be represented by one general equation (one for the curves and one for the surfaces) are addressed. The research examines the curves: ellipse, parabola and hyperbola, the surfaces: ellipsoid, paraboloid, hyperboloid, double hyperboloid, hyperbolic paraboloid and cone, and the cylinders: elliptic, parabolic and hyperbolic. Many real objects which have to be modeled in 3D applications possess specific features. Because of this these geometric objects have been chosen. Using the NURBS models presented here, specialized software modules (plug-ins) have been developed for a 3D graphic system. An analysis of their implementation and the primitives they create has been performed.

Quadratic Time Computable Instances of MaxMin and MinMax Area Triangulations of Convex Polygons

We consider the problems of finding two optimal triangulations of a convex polygon: MaxMin area and MinMax area. These are the triangulations that maximize the area of the smallest area triangle in a triangulation, and respectively minimize the area of the largest area triangle in a triangulation, over all possible triangulations. The problem was originally solved by Klincsek by dynamic programming in cubic time [2]. Later, Keil and Vassilev devised an algorithm that runs in O(n^2 log n) time [1]. In this paper we describe new geometric findings on the structure of MaxMin and MinMax Area triangulations of convex polygons in two dimensions and their algorithmic implications. We improve the algorithm’s running time to quadratic for large classes of convex polygons. We also present experimental results on MaxMin area triangulation.

Interactive 3D Visualization of Bézier Curves using Java Open Graphics Library (JOGL)

We present a new program tool for interactive 3D visualization of some fundamental algorithms for representation and manipulation of Bézier curves. The program tool has an option for demonstration of one of their most important applications - in graphic design for creating letters by means of cubic Bézier curves. We use Java applet and JOGL as our main visualization techniques. This choice ensures the platform independency of the created applet and contributes to the realistic 3D visualization. The applet provides basic knowledge on the Bézier curves and is appropriate for illustrative and educational purposes. Experimental results are included.