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This characterization is used to specify intervals by mean of interval notation, which is described below. An open interval does not include any endpoint, and is indicated with parentheses. [2] For example, (,) = {< <} is the interval of all real numbers greater than 0 and less than 1.
Instead, the inequalities must be solved independently, yielding x < 1 / 2 and x ≥ −1 respectively, which can be combined into the final solution −1 ≤ x < 1 / 2 . Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the logical conjunction of the inequalities ...
It is conceptually one of the simplest inequality indices used in econometrics. A more frequently encountered inequality measure is the Gini coefficient which is based on the summation, over all income-ordered population-percentiles, of the cumulative income up to each percentile. That sum is divided by the maximum value that it could have (its ...
The inequality was first proven by Grönwall in 1919 (the integral form below with α and β being constants). [1] Richard Bellman proved a slightly more general integral form in 1943. [2] A nonlinear generalization of the Grönwall–Bellman inequality is known as Bihari–LaSalle inequality. Other variants and generalizations can be found in ...
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:
Both notations are now used in mathematics, although Iverson's notation will be followed in this article. In some sources, boldface or double brackets x are used for floor, and reversed brackets x or ]x[for ceiling. [7] [8] The fractional part is the sawtooth function, denoted by {x} for real x and defined by the formula {x} = x − ⌊x⌋ [9]
In calculus, and especially multivariable calculus, the mean of a function is loosely defined as the average value of the function over its domain.In one variable, the mean of a function f(x) over the interval (a,b) is defined by: [1]
Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2] It is similar to, but incomparable with, one of Bernstein's inequalities.