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The optimal constant C in the Poincaré inequality is sometimes known as the Poincaré constant for the domain Ω. Determining the Poincaré constant is, in general, a very hard task that depends upon the value of p and the geometry of the domain Ω. Certain special cases are tractable, however.
The feasible regions of linear programming are defined by a set of inequalities. In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. [1] It is used most often to compare two numbers on the number line by their size.
The bounds these inequalities give on a finite sample are less tight than those the Chebyshev inequality gives for a distribution. To illustrate this let the sample size N = 100 and let k = 3. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean.
Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
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.
Be careful, in this application, Garding's Inequality seems useless here as the final result is a direct consequence of Poincaré's Inequality, or Friedrich Inequality. (See talk on the article). As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L 2 (Ω) the Poisson equation
Young's inequality has an elementary proof with the non-optimal constant 1. [ 4 ] We assume that the functions f , g , h : G → R {\displaystyle f,g,h:G\to \mathbb {R} } are nonnegative and integrable, where G {\displaystyle G} is a unimodular group endowed with a bi-invariant Haar measure μ . {\displaystyle \mu .}
In mathematics, an inequation is a statement that an inequality holds between two values. [1] [2] It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are: