The embedding property is a consequence of a result known as Desargues ' theorem.

For instance the Higman embedding theorem can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem.

In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules ( Mitchell's embedding theorem ).

A simplified proof of the second Nash embedding theorem was obtained by who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied.

In particular, as follows from the Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C < sup > 1 </ sup >- embedding into an arbitrarily small neighborhood in 2m-dimensional Euclidean space.

The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into R < sup > n </ sup >.

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.

The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough.

Nash embedding theorem ).

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

Alternatively, one may invoke Mitchell's embedding theorem.

The proof will still apply to any ( small ) abelian category because of Mitchell's embedding theorem, which states that any small abelian category can be represented as a category of modules over some ring.

In fact, as follows from the Nash embedding theorem, all Riemannian manifolds can be realized this way.

* Nash embedding theorem

However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of T < sub > p </ sub > M on which the exponential map is an embedding ( i. e., the exponential map is a local diffeomorphism ).

* The Kodaira embedding theorem gives a criterion for a kähler manifold to be projective.

" He proved important results in this area such as the biholomorphic embedding theorem for a Stein manifold as a closed submanifold in, and a new proof of Remmert's proper mapping theorem.

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a formula p with parameters in M, such that p ( x ) is true and the set

This vision includes establishing a University College to foster both in-depth and wide-interest, society-interest driven education for upcoming engineers ; establishing a combined Graduate School to manage the graduate programs ; an increase of the student body by 50 percent ; a 50 percent increase in the number of annual Ph. D graduations ; an increase of knowledge valorisation to a campus-wide score of 4. 2 ; increasing the international position of the university to within the top-100 universities ; and increasing the embedding of the university within the city and the Brainport region by transforming the campus into a high-grade science park with laboratories, housing facilities for 700 students and researchers and supporting facilities.

Bombieri is also known for his pro bono service on behalf of the mathematics profession, e. g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts ( like the papers of John Nash on embedding Riemannian manifolds and of Per Enflo on the invariant subspace problem ).

The draft expired after six months, but was notable for its acknowledgement of the NCSA Mosaic browser's custom tag for embedding in-line images, reflecting the IETF's philosophy of basing standards on successful prototypes.

After the tissues have been dehydrated, cleared, and infiltrated with the embedding material, they are ready for external embedding.

IPv6 also provides for new multicast implementations, including embedding rendezvous point addresses in an IPv6 multicast group address, which simplifies the deployment of inter-domain solutions.

* RDFa, attribute level extensions to XHTML, for embedding metadata

* The embedding relation for countable total orderings.

The majority of the efforts have looked at embedding spatial database indices such as the Space Filling Curves ( SFCs ) including the Hilbert curves, Z-curves, k-d tree, MX-CIF Quad tree and R *- tree for managing, routing, and indexing of complex Grid resource query objects over DHT networks.

Other uses for SVG include embedding for use in word processing ( e. g. with LibreOffice ) and desktop publishing ( e. g., Scribus ), plotting graphs ( e. g., gnuplot ), and importing paths ( e. g., for use in GIMP or Blender ).

It defines markup for timing, layout, animations, visual transitions, and media embedding, among other things.

One of the areas that improves steganographic robustness is usage of a key scheme for embedding messages.

I regard ' Hungarian ' ( embedding an abbreviated version of a type in a variable name ) a technique that can be useful in untyped languages, but is completely unsuitable for a language that supports generic programming and object-oriented programming — both of which emphasize selection of operations based on the type an arguments ( known to the language or to the run-time support ).

CoSGOP is derived from goal-oriented planning ( Gesellschaft für Technische Zusammenarbeit-GTZ 1988 ), which was oriented towards the elaboration and implementation of projects based on a logical framework, which was useful for embedding a specific project in a wider development frame and defining its major elements.

Planner was invented for the purposes of the procedural embedding of knowledge 1971 and was a rejection of the resolution uniform proof procedure paradigm 1965, which

An inline link displays remote content without the need for embedding the content.

A separate invisible hot area interface allows for swapping skins or labels within the linked hot areas without repetitive embedding of links in the various skin elements.

Then for arbitrary ε > 0 there is an embedding ( or immersion ) ƒ < sub > ε </ sub >: M < sup > m </ sup > → R < sup > n </ sup > which is

These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the ambient space ( because the embedding is biholomorphic ).

Bernhard Riemann extended Gauss's theory to higher dimensional spaces called manifolds in a way that also allows distances and angles to be measured and the notion of curvature to be defined, again in a way that was intrinsic to the manifold and not dependent upon its embedding in higher-dimensional spaces.

Kähler manifolds find important applications in the field of algebraic geometry where they represent generalizations of complex projective algebraic varieties via the Kodaira embedding theorem.

The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said ( rather surprisingly ) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifold extrinsically defined as submanifolds of Euclidean space.

All compact topological manifolds can be embedded into for some n, by the Whitney embedding theorem.

One may also define graph families by the presence or absence of more complex knots and links in their embeddings, or by linkless embedding in three-dimensional manifolds other than Euclidean space.

* Kodaira embedding theorem: for compact complex manifolds, ampleness and positivity coincide.

Another simple procedure for triangulating differentiable manifolds was given by Hassler Whitney in 1957, based on his embedding theorem.

Abstract manifolds have a canonical tangent bundle, but do not have a normal bundle: only an embedding ( or immersion ) of a manifold in another yields a normal bundle.