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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.
Farkas' lemma. 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 ...
Linear inequality. In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality: [1] < less than. > greater than. ≤ less than or equal to. ≥ greater than or equal to. ≠ not equal to.
Statement. Muirhead's inequality states that [a] ≤ [b] for all x such that xi > 0 for every i ∈ { 1, ..., n } if and only if there is some doubly stochastic matrix P for which a = Pb. Furthermore, in that case we have [a] = [b] if and only if a = b or all xi are equal. The latter condition can be expressed in several equivalent ways; one of ...
Fourier–Motzkin elimination. Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions. The algorithm is named after Joseph Fourier [1] who proposed the method in 1826 and Theodore Motzkin who re-discovered it in 1936.
Bhatia–Davis inequality, an upper bound on the variance of any bounded probability distribution. 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.
Cumulative inequality theory. Cumulative inequality theory or cumulative disadvantage theory is the systematic explanation of how inequalities develop. The theory was initially developed by Merton in 1988, [1] who studied the sciences and prestige. He believed that recognition from peers, and from published research in the scientific field ...
Stanley's reciprocity theorem. In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any rational cone (defined below) and the generating function of the cone's interior.
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