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A facet of a polytope is the set of its points which satisfy an essential defining inequality of the polytope with equality. If the polytope is d-dimensional, then its facets are (d − 1)-dimensional. For any graph G, the facets of MP(G) are given by the following inequalities: [1]: 275–279 x ≥ 0 E
The simplest nontrivial case of the clique-finding problem is finding a triangle in a graph, or equivalently determining whether the graph is triangle-free. In a graph G with m edges, there may be at most Θ(m 3/2) triangles (using big theta notation to indicate that this bound is tight).
In the context of metric measure spaces, the definition of a Poincaré inequality is slightly different.One definition is: a metric measure space supports a (q,p)-Poincare inequality for some , < if there are constants C and λ ≥ 1 so that for each ball B in the space, ‖ ‖ () ‖ ‖ ().
satisfying the Kraft inequality, we can construct a prefix code as follows. Define the i th codeword, C i , to be the first ℓ i {\displaystyle \ell _{i}} digits after the radix point (e.g. decimal point) in the base r representation of
Two-dimensional linear inequalities are expressions in two variables of the form: + < +, where the inequalities may either be strict or not. The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane. [2]
Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the ...
As in the decision case, a problem in the #CSP is defined by a set of relations. Each problem takes a Boolean formula as input and the task is to compute the number of satisfying assignments. This can be further generalized by using larger domain sizes and attaching a weight to each satisfying assignment and computing the sum of these weights.
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).