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Cauchy–Schwarz inequality (Modified Schwarz inequality for 2-positive maps [27]) — For a 2-positive map between C*-algebras, for all , in its domain, () ‖ ‖ (), ‖ ‖ ‖ ‖ ‖ ‖. Another generalization is a refinement obtained by interpolating between both sides of the Cauchy–Schwarz inequality:
For instance, to solve the inequality 4x < 2x + 1 ≤ 3x + 2, it is not possible to isolate x in any one part of the inequality through addition or subtraction. 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 .
The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is ...
The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0, i.e. 0 ...
The absolute difference of two real numbers and is given by , the absolute value of their difference. It describes the distance on the real line between the points corresponding to and . It is a special case of the L p distance for all and is the standard metric used for both the set of rational numbers and their completion, the set of real ...
Hölder's inequality. In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of Lp spaces. Hölder's inequality— Let (S, Σ, μ) be a measure space and let p, q ∈[1, ∞] with 1/p + 1/q = 1.
In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of a random variable (with finite variance) from its mean. More specifically, the probability that a random variable deviates from its mean by more than is at most , where is any positive ...
Landau's inequality provides an upper bound for the absolute values of the product of the roots that have an absolute value greater than one. This inequality, discovered in 1905 by Edmund Landau, [9] has been forgotten and rediscovered at least three times during the 20th century. [10] [11] [12]
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