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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:
Variable elimination and the simplex algorithm are used for solving linear and polynomial equations and inequalities, and problems containing variables with infinite domain. These are typically solved as optimization problems in which the optimized function is the number of violated constraints.
Redundant constraint can be identified by solving a linear program as follows. Given a linear constraints system, if the -th inequality is satisfied for any solution of all other inequalities, then it is redundant. Similarly, STIs refers to inequalities that are implied by the non-negativity of information theoretic measures and basic ...
When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality. The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have ...
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 original such inequality, for functions of two real variables, was introduced by Ladyzhenskaya in 1958 to prove the existence and uniqueness of long-time solutions to the Navier–Stokes equations in two spatial dimensions (for smooth enough initial data). There is an analogous inequality for functions of three real variables, but the ...
In complex analysis, a complex domain (or simply domain) is any connected open subset of the complex plane C. For example, the entire complex plane is a domain, as is the open unit disk, the open upper half-plane, and so forth. Often, a complex domain serves as the domain of definition for a holomorphic function.
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution