Regularity criteria for the topology of algebraic curves and surfaces

In this paper, we consider the problem of analysing the shape of an object defined by polynomial equations in a domain. We describe regularity criteria which allow us to determine the topology of the implicit object in a box from information on the boundary of this box. Such criteria are given for planar and space algebraic curves and for algebraic surfaces. These tests are used in subdivision methods in order to produce a polygonal approximation of the algebraic curves or surfaces, even if it contains singular points. We exploit the representation of polynomials in Bernstein basis to check these criteria and to compute the intersection of edges or facets of the box with these curves or surfaces. Our treatment of singularities exploits results from singularity theory such as an explicit Whitney stratification or the local conic structure around singularities. A few examples illustrate the behavior of the algorithms

Transonic airfoil flowfield analysis using Cartesian coordinates

A numerical technique for analyzing transonic airfoils is presented. The method employs the basic features of Jameson's iterative solution for the full potential equation, except that Cartesian coordinates are used rather than a grid which fits the airfoil, such as the conformal circle-plane or 'sheared parabolic' coordinates which were used previously. Comparison with previous results shows that it is not necessary to match the computational grid to the airfoil surface, and that accurate results can be obtained with a Cartesian grid for lifting supercritical airfoils

Problems in Cladistic Classification: Higher-Level Relationships in Land Plants

Recent cladistic analyses of green plants recognize an extensive hierarchical series of relatively well-supported monophyletic groups. Translating this hierarchical pattern of relationships into a usable and informative written classification is important for purposes of scientific communication, research and teaching. However, in the context of the Linnean hierarchy, as manifested in the current International code of Botanical Nomenclature (ICBN), effecting this translation confronts substantial practical difficulties--especially the proliferation of hierarchical levels. These problems are exacerbated by the current emphasis of the ICBN on a hierarchy in which different ranks have different formal rank-based endings. These difficulties could be ameliorated by de-emphasizing the importance of ranks in the ICBN and relaxing the constraints on how they are treated, especially at the higher taxonomic levels. Modifications are needed that permit a more straightforward integration of systematic knowledge and botanical nomenclature, and at the same time foster increased stability in the association between names and the groups of organisms that they designate

High rank tensor and spherical harmonic models for diffusion MRI processing

Diffusion tensor imaging (DTI) is a non-invasive quantitative method of characterizing tissue micro-structure. Diffusion imaging attempts to characterize the manner by which the water molecules within a particular location move within a given amount of time. Measurement of the diffusion tensor (D) within a voxel allows a macroscopic voxel-averaged description of fiber structure, orientation and fully quantitative evaluation of the microstructural features of healthy and diseased tissue.;The rank two tensor model is incapable of resolving multiple fiber orientations within an individual voxel. This shortcoming of single tensor model stems from the fact that the tensor possesses only a single orientational maximum. Several authors reported this non-mono-exponential behavior for the diffusion-induced attenuation in brain tissue in water and N-Acetyl Aspartate (NAA) signals, that is why the Multi-Tensor, Higher Rank Tensor and Orientation Distribution Function (ODF) were introduced.;Using the higher rank tensor, we will propose a scheme for tensor field interpolation which is inspired by subdivision surfaces in computer graphics. The method applies to Cartesian tensors of all ranks and imposes smoothness on the interpolated field by constraining the divergence and curl of the tensor field. Results demonstrate that the subdivision scheme can better preserve anisotropicity and interpolate rotations than some other interpolation methods. As one of the most important applications of DTI, fiber tractography was implemented to study the shape geometry changes. Based on the divergence and curl measurement, we will introduce new scalar measures that are sensitive to behaviors such as fiber bending and fanning.;Based on the ODF analysis, a new anisotropy measure that has the ability to describe multi-fiber heterogeneity while remaining rotationally invariant, will be introduced, which is a problem with many other anisotropy measures defined using the ODF. The performance of this novel measure is demonstrated for data with varying Signal to Noise Ratio (SNR), and different material characteristics

A direct approach to computer modelling of fluids

Conventional approaches to Computational Fluid Dynamics (CFD) are highly mathematical in content and presentation, and physical interpretation of the algorithms can often be obscure. This is believed to inhibit advances in the CFD field and the importance of such advances for Naval Architecture, as a particular application, is discussed. As a possible alternative to conventional methods, a "direct" approach to computer modelling of fluids is proposed where all the algorithms involved are "physically transparent" in that they avoid intermediate mathematical interpretations. Rules for the development of such a model are formulated, and a programming strategy, which advocates modularising the algorithms to reflect the cause and effect mechanisms in real fluids, is outlined. The principles of the direct modelling approach are demonstrated in the development of a computer program for 2-dimensional, incompressible, inviscid flows. The technique requires that the total pressure in a flow is decomposed into two principal components, the temporal pressure and the convective pressure, associated respectively with the temporal and convective accelerations of the fluid. The model incorporates a numerically "explicit" pressure spreading algorithm for determining the temporal pressure and acceleration responses to external disturbances. The actual compressibility of the "incompressible" fluid is modelled via the bulk modulus. Convective pressure is synthesised as flow develops by accounting for the small spatial variations in the fluid's density associated with the temporal pressure field. Simple internal flows, and the acceleration of bodies at or near a free-surface, have been modelled successfully. Flows with a finite free-surface distortion or system geometry change will require the incorporation of grid re-generation algorithms for the spatial discretisation. Routes for future developments, including viscous modelling, are discussed. Apart from potential advantages for CFD, the direct approach should benefit general fluid dynamics education since the concepts involved promote a better understanding of fluid behaviour