Stable Mappings and Their Singularities

Golubitsky, Martin; Guillemin, Victor

doi:10.1007/978-1-4615-7904-5

Cited by 1,237 publications

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“…In his paper [27], Montaldi gives the following characterization of the notion of contact by using the terminology of singularity theory (see [8] for the definition of K-equivalence):…”

Section: Height Functions and Contacts With Hyperplanesmentioning

confidence: 99%

Global properties of codimension two spacelike submanifolds in Minkowski space

Izumiya

Nuño–Ballesteros

Fuster

2009

Advg

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Abstract. We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplanes and hyperquadrics in order to get some global information on them. As a consequence we obtain new versions of Carathéodory's and Loewner's conjectures on spacelike surfaces in 4-dimensional Minkowski space and 4-flattenings theorems for closed spacelike curves in 3-dimensional Minkowski space.

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“…In his paper [27], Montaldi gives the following characterization of the notion of contact by using the terminology of singularity theory (see [8] for the definition of K-equivalence):…”

Section: Height Functions and Contacts With Hyperplanesmentioning

confidence: 99%

Global properties of codimension two spacelike submanifolds in Minkowski space

Izumiya

Nuño–Ballesteros

Fuster

2009

Advg

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“…This choice of name is justified because Morse functions are dense in C ∞ (M), the class of smooth functions on the manifold [31,44]. In other words, for every smooth function there is an arbitrarily small perturbation that makes it a Morse function.…”

Section: Smooth Maps On Manifoldsmentioning

confidence: 99%

Isocontour based Visualization of Time-varying Scalar Fields

Mascarenhas

Snoeyink

2009

Mathematics and Visualization

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Summary. Time-varying scalar fields are produced by measurements or simulation of physical processes over time, and must be interpreted with the assistance of computational tools. A useful tool in interpreting the data is graphical visualization, often through level sets, or isocontours of a continuous function derived from the data. In this paper we survey isocontour based visualization techniques for time-varying scalar fields. We focus on techniques that aid selection of meaningful isocontours, and algorithms to extract chosen isocontours. IntroductionPhysical processes that are measured over time, or that are modeled and simulated on a computer, can produce large amounts of time-varying data that must be interpreted with the assistance of computational tools. Such data arises in a wide variety of studies including computational fluid dynamics [15], oceanography [6], medical imaging [60], and climate modeling [46]. The data typically consists of finitely many points in space-time and measured or computed values for each point. Often the values are scalar, with perhaps several scalar values for each sample point. E.g: Pressure, temperature, density. A study of motion or velocity of some kind will result in vector-valued data. In this paper, we focus on scalar-valued data, often called scalar fields. Irrespective of how the points are sampled, we can connect them into a mesh and interpolate the values to obtain a continuous function over the entire domain. Piecewise-linear interpolation is common for large amounts of data, because of its relative ease; multilinear interpolation is also used for regular grids.The goal is to understand the data, usually by exploring it for important features. A medical researcher might be interested in tumors, while a climatologist might be interested in regions of high pressure. Because humans possess a highly developed visual system, transforming the data into images and movies that can be displayed, and providing the scientist with tools to control them can be a powerful visualization method [47]. Popular techniques employed to create such visualizations are direct volume rendering, slicing, and isocontouring. Direct volume rendering employs two classes of algorithms to display all the data: imagespace projection and volume-space projection. In image-space projection algorithms we cast

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“…This prompts a closer examination of the critical manifolds of (1.1) and the singularities of this projection. Sard's theorem [14] states that the regular values of the map f : R m × R n → R m are a set of full measure if f is C n . If 0 is a regular value, then the implicit function theorem implies that the critical manifold C is indeed an n-dimensional manifold.…”

Section: Normal Hyperbolicity and Slow Manifoldsmentioning

confidence: 99%

“…To deal systematically with this issue, we turn to singularity theory and bifurcation theory. Singularity theory [14] gives a set of tools for classifying the singularities of smooth maps up to smooth coordinate changes. Arnold et al [2] discuss application of singularity theory to the critical manifolds of generic slow-fast systems having two slow variables.…”

Section: Singularities Folds and Jumpsmentioning

confidence: 99%

Bifurcations of relaxation oscillations

Guckenheimer

2004

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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We are still far from a comprehensive theory of bifurcation in dynamical systems with multiple time scales. However, systematic application of geometric methods and study of examples have produced descriptions of varied phenomena. These lectures present a selective summary of some of what has been discovered, concentrating on periodic orbits called relaxation oscillations. We focus upon key aspects of the phenomena, avoiding mathematical details and making the analysis as simple as possible. The dynamics of reduced systems plays a central role in our discussion.

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Stable Mappings and Their Singularities

Cited by 1,237 publications

References 0 publications

Global properties of codimension two spacelike submanifolds in Minkowski space

Global properties of codimension two spacelike submanifolds in Minkowski space

Isocontour based Visualization of Time-varying Scalar Fields

Bifurcations of relaxation oscillations

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