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In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the generators representing the identity element of the group) in terms of the length of that relation (see pp. 79–80 in [1]).
For graphs of constant arboricity, such as planar graphs (or in general graphs from any non-trivial minor-closed graph family), this algorithm takes O (m) time, which is optimal since it is linear in the size of the input. [18] If one desires only a single triangle, or an assurance that the graph is triangle-free, faster algorithms are possible.
An acyclic graph is a forest, but connectedness is usually assumed; as a result, the condition that is usually considered is that primal graphs are trees. This property of tree-like constraint satisfaction problems is exploited by decomposition methods , which convert problems into equivalent ones that only contain binary constraints arranged ...
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 ...
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. Visualizing convexity and Jensen's inequality. In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.
Typically, A is some subset of the Euclidean space, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The domain A of f is called the search space or the choice set , while the elements of A are called candidate solutions or feasible solutions .
The quantile is the unique function satisfying the Galois inequalities Q ( p ) ≤ x {\displaystyle Q(p)\leq x} if and only if p ≤ F ( x ) . {\displaystyle p\leq F(x).} If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have
In mathematical analysis, the Minkowski inequality establishes that the L p spaces are normed vector spaces.Let be a measure space, let < and let and be elements of (). Then + is in (), and we have the triangle inequality