In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is given by some injective and structure-preserving map f : X ¿ Y. The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which X and Y are instances.
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Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols.
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Graph embedding
In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface ¿ is a representation of on ¿ in which points of ¿ are associated to vertices and simple arcs (homeomorphic images of) are associated to edges in such a way that: the endpoints of the arc associated to an edge are the points associated to the end vertices of, no arcs include points associated with other vertices, two arcs never intersect at a point which is interior to either of the arcs.
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Semidefinite programming
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space, i.e. , a spectrahedron. Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons.
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Connectivity (graph theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) which need to be removed to disconnect the remaining nodes from each other. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its robustness as a network.
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Topological property
In topology and related areas of mathematics a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
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Network theory
Network theory is an area of computer science and network science and part of graph theory. It has application in many disciplines including statistical physics, particle physics, computer science, biology, economics, operations research, and sociology. Network theory concerns itself with the study of graphs as a representation of either symmetric relations or, more generally, of asymmetric relations between discrete objects.
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Gramian matrix
In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by . An important application is to compute linear independence: a set of vectors is linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram.
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