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This can be concisely written as the matrix inequality , where A is an m×n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. [citation needed] In the above systems both strict and non-strict inequalities may be used. Not all systems of linear inequalities have solutions.
Jensen's inequality generalizes the statement that a secant line of a convex function lies above its graph. 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. It was proved by Jensen in 1906, [1] building on an ...
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.
Cauchy–Schwarz inequality. The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) [1][2][3][4] is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces (p ≥ 1), and inner product spaces.
Proof [2]. Since + =, =. A graph = on the -plane is thus also a graph =. From sketching a visual representation of the integrals of the area between this curve and the axes, and the area in the rectangle bounded by the lines =, =, =, =, and the fact that is always increasing for increasing and vice versa, we can see that upper bounds the area of the rectangle below the curve (with equality ...
Bernoulli's inequality. An illustration of Bernoulli's inequality, with the graphs of and shown in red and blue respectively. Here, In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of . It is often employed in real analysis. It has several useful variants: [1]
In economics, the Gini coefficient (/ ˈdʒiːni / JEE-nee), also known as the Gini index or Gini ratio, is a measure of statistical dispersion intended to represent the income inequality, the wealth inequality, or the consumption inequality [3] within a nation or a social group. It was developed by Italian statistician and sociologist Corrado ...