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A power inequality is an inequality containing terms of the form a b, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: For any real x, +.
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
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
In mathematics real is used as an adjective, meaning that the underlying field is the field of the real numbers (or the real field). For example, real matrix, real polynomial and real Lie algebra. The word is also used as a noun, meaning a real number (as in "the set of all reals").
For example, a linearly ordered ... that is, the following inequality ... there are no positive, infinitesimal real numbers. The Archimedean property of real numbers ...
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: <
Notice that the a-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if + + =. In the general case, one can consider instead [] / (+ +), which is called a Muirhead mean. [1] Examples
The prototypical apartness relation is that of the real numbers: two real numbers are said to be apart if there exists (one can construct) a rational number between them. In other words, real numbers x {\displaystyle x} and y {\displaystyle y} are apart if there exists a rational number z {\displaystyle z} such that x < z < y {\displaystyle x<z ...