Algebraic Computations with Hausdorff Continuous Functions

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.

On Representations of Algebraic Polynomials by Superpositions of Plane Waves

* The author was supported by NSF Grant No. DMS 9706883.

Generalization of Ehrlich-Kjurkchiev Method for Multiple Roots of Algebraic Equations

In this paper a new method which is a generalization of the Ehrlich-Kjurkchiev method is developed. The method allows to find simultaneously all roots of the algebraic equation in the case when the roots are supposed to be multiple with known multiplicities. The offered generalization does not demand calculation of derivatives of order higher than first simultaneously keeping quaternary rate of convergence which makes this method suitable for application from practical point of view.

Biomaterial interactions with compromised tissue surfaces

This thesis is concerned with the nature of biomaterial interactions with compromised host tissue sites. Both ocular and dermal tissues can be wounded, following injury, disease or surgery, and consequently require the use of a biomaterial. Clear analogies exist between the cornea/tear film/contact lens and the dermal wound bed/wound fluid/skin adhesive wound dressing. The work described in this thesis builds upon established biochemistry to examine specific aspects of the interaction of biomaterials with compromised ocular and dermal tissue sites, with a particular focus on the role of vitronectin. Vitronectin is a prominent cell adhesion glycoprotein present in both tear fluid and wound fluid, and has a role in the regulation and upregulation of plasmin. The interaction of contact lenses with the cornea was assessed by a novel on-lens cell-based vitronectin assay technique. Vitronectin mapping showed that vitronectin-mediated cell adhesion to contact lens surfaces was due to the contact lens-corneal mechanical interaction rather than deposition out of the tear film. This deposition is associated predominantly with the peripheral region of the posterior contact lens surface. The locus of vitronectin deposition on the contact lens surface, which is affected by material modulus, is potentially an important factor in the generation of plasmin in the posterior tear film. Use of the vitronectin mapping technique on ex vivo bandage contact lenses revealed greater vitronectin-mediated cell adhesion to the contact lens surfaces in comparison to lenses worn in the healthy eye. The results suggest that vitronectin is more readily deposited from the impaired corneal tissue bed than the intact healthy tissue bed. Significantly, subjects with a deficient tear film were found to deposit high vitronectin-mediated cell adhesion levels to the BCL surface, thus highlighting the influence of the contact lens-tissue interaction upon deposition. Biomimetic principles imply that adhesive materials for wound applications, including hydrogels and hydrocolloids, should closely match the surface energy parameters of skin. The surface properties of hydrocolloid adhesives were found to be easily modified by contact with siliconised plastic release liners. In contrast, paper release liners did not significantly affect the adhesive surface properties. In order to characterise such materials in the actual wound environment, which is an extremely challenging task, preliminary considerations for the design of an artificial wound fluid model from an animal serum base were addressed.

On The Critical Points of some Iteration Methods for Solving Algebraic Equations. Global Convergence Properties

In this work we give su±cient conditions for k-th approximations of the polynomial roots of f(x) when the Maehly{Aberth{Ehrlich, Werner-Borsch-Supan, Tanabe, Improved Borsch-Supan iteration methods fail on the next step. For these methods all non-attractive sets are found. This is a subsequent improvement of previously developed techniques and known facts. The users of these methods can use the results presented here for software implementation in Distributed Applications and Simulation Environ- ments. Numerical examples with graphics are shown.

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.

Equimultiple Locus of Embedded Algebroid Surfaces and Blowing–up in Characteristic Zero

2000 Mathematics Subject Classification: 14B05, 32S25.

Linearly Normal Curves in P^n

2000 Mathematics Subject Classification: 14H45, 14H50, 14J26.

Projectively Normal Line Bundles on K-Gonal Curves and Rational Surfaces

2000 Mathematics Subject Classification: 14H50.

On the Algebraic Properties of Convex Bodies

AMS subject classification: 52A01, 13C99.

An Invariant Theory of Surfaces in the Four-Dimensional Euclidean or Minkowski Space

2010 Mathematics Subject Classification: 53A07, 53A35, 53A10.

An adaptive hierarchical approach to the extraction of high resolution medial surfaces

We introduce a novel algorithm for medial surfaces extraction that is based on the density-corrected Hamiltonian analysis. The approach extracts the skeleton directly from a triangulated mesh and adopts an adaptive octree-based approach in which only skeletal voxels are refined to a lower level of the hierarchy, resulting in robust and accurate skeletons at extremely high resolution. The quality of the extracted medial surfaces is confirmed by an extensive set of experiments. © 2012 IEEE.

On the Critical Points of Kyurkchiev’s Method for Solving Algebraic Equations

This paper is dedicated to Prof. Nikolay Kyurkchiev on the occasion of his 70th anniversary This paper gives sufficient conditions for kth approximations of the zeros of polynomial f (x) under which Kyurkchiev’s method fails on the next step. The research is linked with an attack on the global convergence hypothesis of this commonly used in practice method (as correlate hypothesis for Weierstrass–Dochev’s method). Graphical examples are presented.

A Characterization Theorem for the K-functional Associated with the Algebraic Version of Trigonometric Jackson Integrals

2000 Mathematics Subject Classification: 41A25, 41A36.

A Note on the L2-norm of the Second Fundamental Form of Algebraic Manifolds

2000 Mathematics Subject Classification: 53C42, 53C55.