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

2000 Mathematics Subject Classification: 14H50.

Rotation effect on geodesic and zonal flow modes in tokamak plasmas with isothermal magnetic surfaces

The electrostatic geodesic mode oscillations are investigated in rotating large aspect ratio tokamak plasmas with circular isothermal magnetic surfaces. The analysis is carried out within the magnetohydrodynamic model including heat flux to compensate for the non-adiabatic pressure distribution along the magnetic surfaces in plasmas with poloidal rotation. Instead of two standard geodesic modes, three geodesic continua are found. The two higher branches of the geodesic modes have a small frequency up-shift from ordinary geodesic acoustic and sonic modes due to rotation. The lower geodesic continuum is a newzonal flowmode (geodesic Doppler mode) in plasmas with mainly poloidal rotation. Limits to standard geodesic modes are found. Bifurcation of Alfven continuum by geodesic modes at the rational surfaces is also discussed. Due to that, the frequency of combined geodesic continuum extends from the poloidal rotation frequency to the ion-sound band that can have an important role in suppressing plasma turbulence.

Finitude (Mallee:Time)

An interactive installation with full body interface, digital projection, multi-touch sensitive screen surfaces, interactive 3D gaming software, motorised dioramas, 4.1 spatial sound & new furniture forms - investigating the cultural dimensions of sustainability through the lens of 'time'. “Time is change, time is finitude. Humans are a finite species. Every decision we make today brings that end closer, or alternatively pushes it further away. Nothing can be neutral”. Tony Fry DETAILS: Finitude (Mallee:Time) is a major new media/sculptural hybrid work premiered in 2011 in version 1 at the Ka-rama Motel for the Mildura Palimpsest #8 ('Collaborators and Saboteurs'). Each participant/viewer lies comfortably on their back on the double bed of Room 22. Directly above them, supported by a wooden structure, not unlike a house frame, is a semi-transparent Perspex screen that displays projected 3D imagery and is simultaneously sensitive to the lightest of finger touches. Depending upon the ever changing qualities of the projected image on this screen the participant can see through its surface to a series of physical dioramas suspended above, lit by subtle LED spotlighting. This diorama consists of a slowly rotating series of physical environments, which also include several animatronic components, allowing the realtime composition of whimsical ‘landscapes’ of both 'real' and 'virtual' media. Through subtle, non-didactic touch-sensitive interactivity the participant then has influence over both the 3D graphic imagery, the physical movements of the diorama and the 4.1 immersive soundscape, creating an uncanny blend of physical and virtual media. Five speakers positioned around the room deliver a rich interactive soundscape that responds both audibly and physically to interactions. VERSION 1, CONTEXT/THEORY: Finitude (Mallee: Time) is Version 1 of a series of presentations during 2012-14. This version has been inspired through a series of recent visits and residencies in the SW Victoria Mallee country. Further drawing on recent writings by post colonial author Paul Carter, the work is envisaged as an evolving ‘personal topography’ of place-discovery. By contrasting and melding readily available generalisations of the Mallee regions’ rational surfaces, climatic maps and ecological systems with what Carter calls “a fine capillary system of interconnected words, places, memories and sensations” generated through my own idiosyncratic research processes, Finitude (Mallee Time) invokes a “dark writing” of place through outside eyes - an approach that avoids concentration upon what 'everyone else knows', to instead imagine and develop a sense how things might be. This basis in re-imagining and re-invention becomes the vehicle for the work’s more fundamental intention - as a meditative re-imagination of 'time' (and region) as finite resources: Towards this end, every object, process and idea in the work is re-thought as having its own ‘time component’ or ‘residue’ that becomes deposited into our 'collective future'. Thought this way Finitude (Mallee Time) suggests the poverty of predominant images of time as ‘mechanism’ to instead envisage time as a plastic cyclical medium that we can each choose to ‘give to’ or ‘take away from’ our future. Put another way - time has become finitude.

Topology of toroidal helical fields in non-circular cross-sectional tokamaks

The ordinary differential magnetic field line equations are solved numerically; the tokamak magnetic structure is studied on Hefei Tokamak-7 Upgrade (HT-7U) when the equilibrium field with a monotonic q-profile is perturbed by a helical magnetic field. We find that a single mode (m, n) helical perturbation can cause the formation of islands on rational surfaces with q = m/n and q = (m +/- 1, +/- 2, +/- 3,...)/n due to the toroidicity and plasma shape (i.e. elongation and triangularity), while there are many undestroyed magnetic surfaces called Kolmogorov-Arnold-Moser (KAM) barriers on irrational surfaces. The islands on the same rational surface do not have the same size. When the ratio between the perturbing magnetic field B-r(r) and the toroidal magnetic field amplitude B(phi)0 is large enough, the magnetic island chains on different rational surfaces will overlap and chaotic orbits appear in the overlapping area, and the magnetic field becomes stochastic. It is remarkable that the stochastic layer appears first in the plasma edge region.

Linearly Normal Curves in P^n

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

Modeling corneal surfaces with rational functions for high-speed videokeratoscopy data compression

High-speed videokeratoscopy is an emerging technique that enables study of the corneal surface and tear-film dynamics. Unlike its static predecessor, this new technique results in a very large amount of digital data for which storage needs become significant. We aimed to design a compression technique that would use mathematical functions to parsimoniously fit corneal surface data with a minimum number of coefficients. Since the Zernike polynomial functions that have been traditionally used for modeling corneal surfaces may not necessarily correctly represent given corneal surface data in terms of its optical performance, we introduced the concept of Zernike polynomial-based rational functions. Modeling optimality criteria were employed in terms of both the rms surface error as well as the point spread function cross-correlation. The parameters of approximations were estimated using a nonlinear least-squares procedure based on the Levenberg-Marquardt algorithm. A large number of retrospective videokeratoscopic measurements were used to evaluate the performance of the proposed rational-function-based modeling approach. The results indicate that the rational functions almost always outperform the traditional Zernike polynomial approximations with the same number of coefficients.

Rational design of 3D dendritic TiO2 nanostructures with favorable architectures

Controlling the morphology and size of titanium dioxide (TiO2) nanostructures is crucial to obtain superior photocatalytic, photovoltaic, and electrochemical properties. However, the synthetic techniques for preparing such structures, especially those with complex configurations, still remain a challenge because of the rapid hydrolysis of Ti-containing polymer precursors in aqueous solution. Herein, we report a completely novel approach-three- dimensional (3D) TiO2 nanostructures with favorable dendritic architectures-through a simple hydrothermal synthesis. The size of the 3D TiO2 dendrites and the morphology of the constituent nano-units, in the form of nanorods, nanoribbons, and nanowires, are controlled by adjusting the precursor hydrolysis rate and the surfactant aggregation. These novel configurations of TiO2 nanostructures possess higher surface area and superior electrochemical properties compared to nanoparticles with smooth surfaces. Our findings provide an effective solution for the synthesis of complex TiO2 nano-architectures, which can pave the way to further improve the energy storage and energy conversion efficiency of TiO 2-based devices.

Interior Medial Axis Transform computation of 3D objects bound by free-form surfaces

This paper presents an algorithm for generating the Interior Medial Axis Transform (iMAT) of 3D objects with free-form boundaries. The algorithm proposed uses the exact representation of the part and generates an approximate rational spline description of the iMAT. The algorithm generates the iMAT by a tracing technique that marches along the object's boundary. The level of approximation is controlled by the choice of the step size in the tracing procedure. Criteria based on distance and local curvature of boundary entities are used to identify the junction points and the search for these junction points is done in an efficient way. The algorithm works for multiply-connected objects as well. Results of the implementation are provided. (C) 2010 Elsevier Ltd. All rights reserved.

Surface Reconstruction from Multiple Views using Rational B-Splines

A method for reconstructing 3D rational B-spline surfaces from multiple views is proposed. The method takes advantage of the projective invariance properties of rational B-splines. Given feature correspondences in multiple views, the 3D surface is reconstructed via a four step framework. First, corresponding features in each view are given an initial surface parameter value (s; t), and a 2D B-spline is fitted in each view. After this initialization, an iterative minimization procedure alternates between updating the 2D B-spline control points and re-estimating each feature's (s; t). Next, a non-linear minimization method is used to upgrade the 2D B-splines to 2D rational B-splines, and obtain a better fit. Finally, a factorization method is used to reconstruct the 3D B-spline surface given 2D B-splines in each view. This surface recovery method can be applied in both the perspective and orthographic case. The orthographic case allows the use of additional constraints in the recovery. Experiments with real and synthetic imagery demonstrate the efficacy of the approach for the orthographic case.

Surface Reconstruction from Multiple Views using Rational B-Splines and Knot Insertion

A method for reconstruction of 3D rational B-spline surfaces from multiple views is proposed. Given corresponding features in multiple views, though not necessarily visible in all views, the surface is reconstructed. First 2D B-spline patches are fitted to each view. The 3D B-splines and projection matricies can then be extracted from the 2D B-splines using factorization methods. The surface fit is then further refined via an iterative procedure. Finally, a hierarchal fitting scheme is proposed to allow modeling of complex surfaces by means of knot insertion. Experiments with real imagery demonstrate the efficacy of the approach.

Some Attempts in the Rational Design of Heterogeneous Catalysts Using Density Functional Theory Calculations

Heterogeneous catalysis is of great importance both industrially and academically. Rational design of heterogeneous catalysts is highly desirable, and the computational screening and design method is one of the most promising approaches for rational design of heterogeneous catalysts. Herein, we review some attempts towards the rational catalyst design using density functional theory from our group. Some general relationships and theories on the activity and selectivity are covered, such as the Brønsted–Evans–Polanyi relation, volcano curves/surfaces, chemical potentials, optimal adsorption energy window and energy descriptor of selectivity. Furthermore, the relations of these relationships and theories to the rational design are discussed, and some examples of computational screening and design method are given.

Conchoid surfaces of spheres

The conchoid of a surface F with respect to given xed point O is roughly speaking the surface obtained by increasing the radius function with respect to O by a constant. This paper studies conchoid surfaces of spheres and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in R3 and R4. Moreover we point to remarkable geometric properties of these surfaces and their construction.

Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces.

Rational engineering of a miniprotein that reproduces the core of the CD4 site interacting with HIV-1 envelope glycoprotein

Protein–protein interacting surfaces are usually large and intricate, making the rational design of small mimetics of these interfaces a daunting problem. On the basis of a structural similarity between the CDR2-like loop of CD4 and the β-hairpin region of a short scorpion toxin, scyllatoxin, we transferred the side chains of nine residues of CD4, central in the binding to HIV-1 envelope glycoprotein (gp120), to a structurally homologous region of the scorpion toxin scaffold. In competition experiments, the resulting 27-amino acid miniprotein inhibited binding of CD4 to gp120 with a 40 μM IC50. Structural analysis by NMR showed that both the backbone of the chimeric β-hairpin and the introduced side chains adopted conformations similar to those of the parent CD4. Systematic single mutations suggested that most CD4 residues from the CDR2-like loop were reproduced in the miniprotein, including the critical Phe-43. The structural and functional analysis performed suggested five additional mutations that, once incorporated in the miniprotein, increased its affinity for gp120 by 100-fold to an IC50 of 0.1–1.0 μM, depending on viral strains. The resulting mini-CD4 inhibited infection of CD4+ cells by different virus isolates. Thus, core regions of large protein–protein interfaces can be reproduced in miniprotein scaffolds, offering possibilities for the development of inhibitors of protein–protein interactions that may represent useful tools in biology and in drug discovery.

Complex Hyperbolic Surfaces of Abelian Type

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.