In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety.

However, when the steganographic robustness is increased a bandwidth of the whole embedding system is decreased.

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

The term " institutionalization " is widely used in social theory to refer to the process of embedding something ( for example a concept, a social role, a particular value or mode of behavior ) within an organization, social system, or society as a whole.

** A Fuchsian group is sometimes considered as a special case of a Kleinian group, by embedding the hyperbolic plane isometrically into three dimensional hyperbolic space and extending the group action on the plane to the whole space.

The first part comprises the embedding of a concept in the world of concepts as a whole, i. e. the totality of all relations to other concepts.

We can avoid such problems by embedding the sphere in three-dimensional space and parameterizing it with three Cartesian coordinates, placing the north pole at, the south pole at, and the equator at,.

where gives the standard embedding the ( n − 2 )- sphere in R < sup > n − 1 </ sup >.

Much as elliptical and hyperbolic spaces can be visualized by an isometric embedding in a flat space of one higher dimension ( as the sphere and pseudosphere respectively ), anti de Sitter space can be visualized as the Lorentzian analogue of a sphere in a space of one additional dimension.

Some authors define anti de Sitter space as equivalent to the embedded sphere itself, while others define it as equivalent to the universal cover of the embedding.

A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S < sup > 2 </ sup > in R < sup > 3 </ sup > may fail to " separate the space cleanly ", unless an extra condition of tameness is used to suppress possible wild behaviour.

The Alexander horned sphere is a wild embedding of a sphere into space, discovered by.

It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:

By considering only the points of the tori that are not removed at some stage, an embedding results of the sphere with a Cantor set removed.

Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R < sup > 3 </ sup >.

The solid Alexander horned sphere is an example of a crumpled cube ; i. e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere.

Another example, also found by Alexander, is Antoine's horned sphere, which is based on Antoine's necklace, a pathological embedding of the Cantor set into the 3-sphere.

More generally, one can define an n-linked embedding for any n to be an embedding that contains an n-component link that cannot be separated by a topological sphere into two separated parts ; minor-minimal graphs that are intrinsically n-linked are known for all n.

Conversely, as Lefschetz shows, whenever a graph G has a set of cycles with this property, they necessarily form the face cycles of an embedding of the graph onto the sphere.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

The Levi-Civita notions of intrinsic derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding, since the definition of the Christoffel symbols make sense in any Riemannian manifold.

The eigenfunctions of the Laplace – Beltrami operator on the manifold serve as the embedding dimensions, since under mild conditions this operator has a countable spectrum that is a basis for square integrable functions on the manifold ( compare to Fourier series on the unit circle manifold ).

In 1963, his seminal article The position of embedding transformations in a Grammar introduced the transformational cycle, which has been a foundational insight for theories of syntax since that time.

Most importantly, an embedding is merely a shape, while a potential plot has a distinguished " downward " direction ; thus turning a gravity well " upside down " ( by negating the potential ) turns the attractive force into a repulsive force, while turning a Schwarzschild embedding upside down ( by rotating it ) has no effect, since it leaves its intrinsic geometry unchanged.

He chose this land at the banks of Sarasvati River ( since dried up before 1900BCE ) for embedding spirituality with 8 virtues: austerity ( tapas ), truth ( satya ), forgiveness ( kshama ), kindness ( daya ), purity ( sucha ), charity ( dana ), yagya and brahmacharya.

However, since every compact manifold can be embedded in, by the Whitney embedding theorem, every manifold admits a normal bundle, given such an embedding.

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time ( a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a ' space-like surface ') and draw an " embedding diagram " depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an " Einstein – Rosen bridge ".

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time ( a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a ' space-like surface ') and draw an " embedding diagram " depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an " Einstein-Rosen bridge " or Schwarzschild wormhole.

An algorithm may learn an internal model of the data, which can be used to map points unavailable at training time into the embedding in a process often called out-of-sample extension.

KPCA has an internal model, so it can be used to map points onto its embedding that were not available at training time.

Finally, it uses an eigenvector-based optimization technique to find the low-dimensional embedding of points, such that each point is still described with the same linear combination of its neighbors.

They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.

Witsenhausen ( 1974 ) conjectures that the maximum sum of squared distances, among n points with unit diameter in R < sup > d </ sup >, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.

General position is a property of configurations of points, or more generally other subvarieties ( lines in general position, so no three concurrent, and the like ) – it is an extrinsic notion, which depends on an embedding as a subvariety.

An important example of this type comes from computational geometry: the duality for any finite set S of points in the plane between the Delaunay triangulation of S and the Voronoi diagram of S. As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual.

Effective divisors D on C consisting of distinct points have a linear span in the canonical embedding with dimension directly related to that of the linear system in which they move ; and with some more discussion this applies also to the case of points with multiplicities.

An embedding of a graph into three-dimensional space consists of a mapping from the vertices of the graph to points in space, and from the edges of the graph to curves in space, such that each endpoint of each edge is mapped to an endpoint of the corresponding curve, and such that the curves for two different edges do not intersect except at a common endpoint of the edges.

If a graph has a linkless or flat embedding, then modifying the graph by subdividing or unsubdividing its edges, adding or removing multiple edges between the same pair of points, and performing Y-Δ transforms that replace a degree-three vertex by a triangle connecting its three neighbors or the reverse all preserve flatness and linklessness.

Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as " sending " points of to points of and hence of embedding within despite the fact that the function need not be one-to-one.

* Bordiga surfaces: A degree 6 embedding of the projective plane into P < sup > 4 </ sup > defined by the quartics through 10 points in general position.

Mathematically this invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors and multiplying time by the square root of ; or by keeping time a real quantity and embedding the vectors in a Minkowski space.

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. g = f * h. Explicitly, for any two tangent vectors

In general, for an algebraic category C, an embedding between two C-algebraic structures X and Y is a C-morphism e: X → Y which is injective.

VHS Hi-Fi audio is achieved by using audio frequency modulation ( AFM ), modulating the two stereo channels ( L, R ) on two different frequency-modulated carriers and embedding the combined modulated audio signal pair into the video signal.

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

A toroidal embedding of K < sub > 3, 3 </ sub > may be obtained by replacing the crossing by a tube, as described above, in which the two holes where the tube connects to the plane are placed along one of the crossing edges on either side of the crossing.

While DDE was limited to transferring limited amounts of data between two running applications, OLE was capable of maintaining active links between two documents or even embedding one type of document within another.

Polyacrylamide gel ( PAG ) had been known as a potential embedding medium for sectioning tissues as early as 1964, and two independent groups employed PAG in electrophoresis in 1959.

HTML5 adds two new tags to the HTML standard: < video > and < audio > for direct embedding of video and audio content into a web page.

Non-linear methods can be broadly classified into two groups: those that provide a mapping ( either from the high dimensional space to the low dimensional embedding or vice versa ), and those that just give a visualisation.

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

As an isotopy version of his embedding result, Haefliger proved that if is a compact-dimensional-connected manifold, then any two embeddings of into are isotopic provided.

All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.

The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge.

In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph have nonzero linking number.