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In mathematics, Farkas' lemma is a solvability theorem for a finite system of linear inequalities. It was originally proven by the Hungarian mathematician Gyula Farkas . [ 1 ] Farkas' lemma is the key result underpinning the linear programming duality and has played a central role in the development of mathematical optimization (alternatively ...
Bernstein inequalities (probability theory) Boole's inequality; Borell–TIS inequality; BRS-inequality; Burkholder's inequality; Burkholder–Davis–Gundy inequalities; Cantelli's inequality; Chebyshev's inequality; Chernoff's inequality; Chung–Erdős inequality; Concentration inequality; Cramér–Rao inequality; Doob's martingale inequality
However, the elimination process results in a new system that possibly contains more inequalities than the original. Yet, often some of the inequalities in the reduced system are redundant. Redundancy may be implied by other inequalities or by inequalities in information theory (a.k.a. Shannon type inequalities).
If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system. These are exactly the properties required for the solution set to be a linear subspace of R n. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.
Homogeneous system: Homogeneous system of linear algebraic equations; System of homogeneous differential equations. System of homogeneous first-order differential ...
Consider the system of linear equations: L i = 0 for 1 ≤ i ≤ M, and variables X 1, X 2, ..., X N, where each L i is a weighted sum of the X i s. Then X 1 = X 2 = ⋯ = X N = 0 is always a solution. When M < N the system is underdetermined and there are always an infinitude of further solutions.
Inequalities within the field of mathematics. See also: Category:Social inequality; Category:Economic inequality; Subcategories. This category has the following 4 ...
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]