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	Embedding
		In mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
		In general topology: an embedding is a homeomorphism onto its image.
		In other words embedding is diffeomorphism to its image (in particular image of embedding is a submanifold).
	www.guajara.com /wiki/en/wikipedia/e/em/embedding.html (276 words)

	Whitney embedding theorem - Wikipedia, the free encyclopedia
		In mathematics, particularly in differential topology, the Whitney embedding theorem states that any smooth (and second-countable) m-dimensional manifold can be smoothly embedded in Euclidean 2m-space.
		The fact that this is the strongest theorem for the maximum number of dimensions it takes to smoothly embed such manifolds is apparent in the fact that the real projective space of dimension m cannot be embedded into Euclidean (2m − 1)-space.
		The Whitney trick is used to prove the h-cobordism theorem; it also shows that two oriented submanifolds of complementary dimensions in a simply connected manifold of dimension
	en.wikipedia.org /wiki/Whitney_embedding_theorem (233 words)

	Attractor Reconstruction - Scholarpedia
		The Whitney Embedding Theorem (Whitney 1936) holds that a generic map from an n-manifold to 2n+1 dimensional Euclidean space is an embedding: the image of the n-manifold is completely unfolded in the larger space.
		The contribution of the Takens Embedding Theorem (Takens 1981) was to show that the same goal could be reached with a single measured quantity.
		An embedding theorem for skew systems (Stark 1999) explores extensions of the methodology when one part of a system is driving another, and only the latter can be observed.
	www.scholarpedia.org /article/Attractor_Reconstruction (1117 words)

	Hassler Whitney Summary
		Whitney received his first degree in mathematics at Yale, then went on to earn a doctorate at Harvard in 1932 with a dissertation entitled "The Coloring of Graphs." He taught at Harvard until 1952, when he accepted an appointment at the Institute for Advanced Study at Princeton, where he remained until his retirement in 1977.
		Whitney was awarded the National Medal of Science (1976), the Wolf Prize (1983), and the Steele Prize (1985).
		Hassler Whitney was the son of New York Supreme Court Justice Edward Baldwin Whitney and Josepha (Newcomb) Whitney, and the grandson of Yale University Professor of Ancient Languages William Dwight Whitney and Connecticut Governor and US Senator Roger Sherman Baldwin, and the great-great-grandson of American founding father Roger Sherman.
	www.bookrags.com /Hassler_Whitney (626 words)

	Embedding
		In mathematics, an embedding (or imbedding) is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
		An isometric embedding is a smooth embedding f : M → N which preserves the metric in the sense that g is equal to the pullback of h by f, i.e.
		In field theory, an embedding of a field E in a field F is a ring homomorphism σ : E → F.
	articles.gourt.com /?article=embedding (620 words)

	Could any curved space be a cut in a higher-dimensional flat space ?
		The problem of embedding an ND (pseudo-) Riemannian manifold in a Ricci-flat space of one higher dimension was taken up again by Magaard.
		For the sake of clarification: the Campbell-Magaard theorem is for local embeddings, while Clarke's theorem is for global embeddings.
		Certain special cases may be handled on an ad-hoc basis more simply with an embedding diagram, and embedding diagrams are used with some limited amount of success to attempt to explain why space-time curvature is equivalent to a force as a pedagogical tool.
	www.physicsforums.com /showthread.php?t=98286 (1619 words)

	Nash embedding theorem - Wikipedia, the free encyclopedia
		The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into R
		A local embedding theorem is much simpler and can be proved using the implicit function theorem of advanced calculus.
		The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the Nash-Moser theorem and Newton's method with postconditioning (see ref.).
	en.wikipedia.org /wiki/Nash_embedding_theorem (536 words)

	UCSC General Catalog 2006-08 - Programs and Courses
		Starting with the fundamental theorem of calculus and related techniques, the integral of functions of a single variable is developed and applied to problems in geometry, probability, physics, and differential equations.
		Surfaces of constant curvature; the theorems of Bonnet and Hadamard.
		Manifolds I. Definition of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem, vector fields, flows, and Lie bracket, Frobenius's theorem.
	reg.ucsc.edu /catalog/html/programs_courses/mathCourses.htm (3953 words)

	Differential topology
		Existence of proper functions and Whitney embedding theorem in the non-compact case.
		Hopf’s theorem (the sum of indices inside a ball).
		Transversality theorem as a generalization of Sard’s theorem.
	www.math.metu.edu.tr /~serge/courses/541-2004/541-2004.html (160 words)

	Courses in the Department of Mathematics
		Hartogs’ Theorem), a deeper study of Riemann surfaces, the uniformization theorem, the Dirichlet problem in higher dimensions, differential equations in a complex domain and the Riemann-Hilbert problem, Hardy spaces.
		Inverse and implicit function theorems, transversality, Sard’s theorem and the Whitney embedding theorem.
		Geodesics and the associated variational formalism (formulas for the 1st and 2nd variation of length), the exponential map, completeness, and the influence of curvature on the structure of a manifold (positive versus negative curvature).
	catalogs.uchicago.edu /divisions/math-courses.html (2661 words)

	2006-2007 Course Register
		Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus.
		Theory of fibre bundles and classifying spaces, fibrations, spectral sequences, obstruction theory, Postnikov towers, transversality, cobordism, index theorems, embedding and immersion theories, homotopy spheres and possibly an introduction to surgery theory and the general classification of manifolds.
		Analytic spaces, Stein spaces, approximation theorems, embedding theorems, coherent analytic sheaves, Theorems A and B of Cartan, applications to the Cousin problems, and the theory of Banach algebras, pseudoconvexity and the Levi problems.
	www.upenn.edu /registrar/register/math.html (4476 words)

	MATHEMATICS EDUCATION
		Harmonic functions, mean-value theorem, maximum principle, Green's representation for the solution of the Dirichlet problem for Laplace's equation; Poisson's equations and the Poisson formula; statement and proof of the existence theorem for general second-order elliptic operators, generalized maximum principles; Sobolev spaces.
		Manifolds, differential structures, tangent bundles, embeddings, immersions, inverse function theorem, Morse-Sard theorem, transversality, Borsuk-Ulam theorem, vector bundles, Euler characteristics, Morse theory, Stokes theorem, Gauss-Bonnet theorem, Whitney embedding theorem.
		Implicit function theorem, manifolds and transversality, Newton polygons, Lyapunov center theorem, variational methods, Ljusternik-Schnirelman theory, mountain-pass theorem, bifurcations with one-dimensional null-spaces, Morse theory and global bifurcations, geometric theory of partial differential equations.
	www.uncc.edu /gradmiss/catalog/MathEd.htm (3321 words)

	Theoretical Foundation - Takens' Embedding Theorems
		The theorem considers a time series which is sampled in regular time intervals as measurements of the observable
		the theorem does not hold, and the probability of choosing ``accidentally'' one of the elements of this subset is zero.
		The embedding process which gives this result is summarized in Fig.
	www.m-engel.de /ess_ver2-html.v2/node5.html (1323 words)

	[No title]
		3) a Poincare version of the Whitney embedding theorem (settling a question of Levitt) 4) the existence of diagonal Poincare embeddings (in the 1-connected case).
		The theorem says that a (2k - n + 2)-connected map f: K^k --> X^n (from a finite complex of dimension k to an n-dimensional Poincare space) is the underlying map of a Poincare embedding, provided also that k
		Theorem A of this paper says that f immerses if and only if the pullback of the Spivak normal fibration of X is stable fiber homotopy equivalent to the Spivak normal fibration of M. Also included is a new homotopy theoretic proof (using equivariant duality) of the existence and uniqueness theorems for the Spivak fibration.
	www.lehigh.edu /~dmd1/h123 (1314 words)

	PlanetMath: embedding
		This should really be restated as the map
		A celebrated theorem of Whitney states that every
		This is version 4 of embedding, born on 2004-12-11, modified 2004-12-11.
	planetmath.org /encyclopedia/WhitneysTheorem.html (53 words)

	Embedding
		In the geometry of manifolds, a manifold M given abstractly is considered as a candidate to be embedded in Euclidean space of given dimension n (at least dim M, naturally: see invariance of domain).
		That means we look for a submanifold of n-dimensional Euclidean space that is at least homeomorphic to M.
		In domain theory, an embedding is a complete partial order F in [X -> Y] is an embedding if
	www.teachersparadise.com /ency/en/wikipedia/e/em/embedding.html (240 words)

	Math Courses
		MATH 20500 is concerned with line and surface integrals, and the theorems of Green, Gauss, and Stokes.
		In MATH 31000, the following subjects are studied: saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero.
		-spaces, Fubini's theorem, differentiation, Fourier transforms, locally convex spaces, weak topologies, and convexity; compact operators; spectral theorem and integral operators; Banach algebras and general spectral theory; Sobolev spaces and embedding theorems; Haar measure; and Peter-Weyl theorem, holomorphic functions, Cauchy's theorem, harmonic functions, maximum modulus principle, meromorphic functions, conformal mapping, and analytic continuation.
	www.math.uchicago.edu /undergrad/ucourses01.html (2991 words)

	Denis R.
		Its treatment encompasses a general study of surgery, laying a solid foundation for further study and greatly simplifying the classification of surfaces.
		Its subjects include topological spaces and properties, some advanced calculus, differentiable manifolds, orientability, submanifolds and an embedding theorem, and tangent spaces.
		A classic exposition of the branch of mathematical logic known as category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers.
	www.yurinsha.com /396/p2.htm (590 words)

	Gauging the number of variables in a description of brain processes: levels of abstraction
		Pair-wise dependence between variables of a system can be investigated by applying the Takens embedding theorem to combinations of two observables (= observed variables).
		The embedding methods were introduced to characterize attractors for chaotic systems, that as a rule should have a fractional dimensionality.
		If this level is n, the Whitney embedding theorem (Kantz and Schreiber 1997, p126) suggests that maximally 2n+1 observables are needed as global coordinates.
	www.home.zonnet.nl /dekker.aj/chapter3.htm (5972 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Indeed, a quasiperiodic flow on a torus by an arbitrary small perturbation can be made a stable flow with a finite number of stable periodic motions.
		There is no natural measure on an infinite-dimensional space by theorem of Sudakov and one has to suggest how to define notion of probability one.
		Those results are Thom's transversality theorem, Whitney's embedding theorem, Mather's stability theorem, and Kupka-Smale's theorem.
	www.math.princeton.edu /~kaloshin/undergr.html (378 words)

	METU MATHEMATICS DEPARTMENT
		Compact operators, compact operators in Hilbert Spaces, Banach Algebras, The spectral theorem for normal operators, unbounded operators between Hilbert spaces, the spectral theorem for unbounded self-adjoint operators, self-adjoint operators, self-adjoint extensions.
		Fields, field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure and normality, the Galois group of a polynomial, finite fields.
		The aim of this course is to test the knowledge of the student in the basic areas of mathematics.
	www.math.metu.edu.tr /courses/graduate.shtml (2274 words)

	A New Approach for Dimensionality Reduction: Theory and Algorithms - Broomhead, Kirby (ResearchIndex)
		Abstract: This paper applies Whitney's embedding theorem to the data reduction problem and introduces a new approach motivated in part by the (constructive) proof of the theorem.
		The notion of a good projection is introduced which involves picking projections of the high-dimensional system that are optimized such that they are easy to invert.
		We examine this quantity for the face data base in Section 4, taking P as the KL basis.
	citeseer.ist.psu.edu /broomhead98new.html (589 words)

	David Guarrera's Worldsheet: March 2005
		You may be familiar with the Whitney Embedding Theorem, which states that any manifold can be embedded in R^n for some high enough n.
		There is a stronger theorem, the so called Nash Embedding Theorem, that says that any manifold with Riemmanian metric g may be embedded in R^m for high enough m such that the inherited or "pullback" metric from R^m is exactly g.
		In geometry courses, it is often emphasized that a manifold need not be embedded in R^n, has an existence and topology of it's own, etc. But the above, at least for me, provides a very nice picture of why we use the Levi-Civita connection.
	web.mit.edu /guarrera/www/2005_03_01_dgfactor_archive.html (2477 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Seminar in Geometric Methods for High-Dimensional Data DATE: 2/3/2005 TIME: 3:15 pm in Weber 117 SPEAKER: Michael Kirby (CSU Mathematics) TITLE: Whitney's Theorem for Data Reduction: From theory to algorithms ABSTRACT: Whitney's (easy) embedding theorem states that manifolds of dimension m can be embedded in a Euclidean space of dimension 2m+1.
		The proof of this theorem is constructive suggesting an algorithm for data reduction where the data is sampled from a manifold.
		In this talk we present several ideas for implementing Whitney's theorem on real data.
	www.math.colostate.edu /~kley/geometric/kirby.txt (100 words)

	Fractal Dimension
		In fact, this is just a different measure of dimension, called the embedding dimension: a set has embedding dimension n if n is the smallest integer for which it can be embedded into
		Thus, the embedding dimension of a plane is 2, the embedding dimension of a sphere is 3, and the embedding dimension of a klein bottle is 4, even though they all have (topological) dimension two.
		A famous theorem (the Whitney embedding theorem) says that if a manifold has topological dimension n, its embedding dimension is at most 2n.
	www.math.sunysb.edu /~shafikov/teaching/fall01/mat331/Book/Fractal_Dimension.html (1323 words)

	Face Processing with Whitney Reduction Networks (ResearchIndex)
		Abstract: This paper investigates the application of the Whitney Reduction Network (WRN) to the lowdimensional characterization of digital images human faces.
		Motivated by Whitney's Embedding theorem from differential topology the WRN provides a nonlinear parameterization of m dimensional manifolds.
		1 The whitney reduction network: a method for computing autoas..
	citeseer.ist.psu.edu /54126.html (320 words)

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