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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]
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, ‖ ‖ () ‖ ‖ ().
Hardy's inequality is an inequality in mathematics, named after G. H. Hardy.. Its discrete version states that if ,,, … is a sequence of non-negative real numbers, then for every real number p > 1 one has
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).
Several graph-theoretic concepts are related to each other via complementation: The complement of an edgeless graph is a complete graph and vice versa. Any induced subgraph of the complement graph of a graph G is the complement of the corresponding induced subgraph in G. An independent set in a graph is a clique in the complement graph and vice ...
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
For four points in order around a circle, Ptolemy's inequality becomes an equality, known as Ptolemy's theorem: ¯ ¯ + ¯ ¯ = ¯ ¯. In the inversion-based proof of Ptolemy's inequality, transforming four co-circular points by an inversion centered at one of them causes the other three to become collinear, so the triangle equality for these three points (from which Ptolemy's inequality may ...
In case there are several permutations with this property, let σ denote one with the highest number of integers from {, …,} satisfying = (). We will now prove by contradiction , that σ {\displaystyle \sigma } has to keep the order of y 1 , … , y n {\displaystyle y_{1},\ldots ,y_{n}} (then we are done with the upper bound in ( 1 ), because ...