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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 ...
In metric geometry, the discrete metric takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the Hamming distance, which is important in coding and information theory. In the field of real or complex numbers, the distance of the discrete metric ...
is used. This well-known method was published by the German mathematician Wilhelm Kutta in 1901, after Karl Heun had found a three-step one-step method of order 3 a year earlier. [19] The construction of explicit methods of even higher order with the smallest possible number of steps is a mathematically quite demanding problem.
It is considered one of the most important and widely used inequalities in mathematics. [5] Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy .
Azuma's inequality; 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
Cheng's eigenvalue comparison theorem (Riemannian geometry) Chern–Gauss–Bonnet theorem (differential geometry) Chevalley's structure theorem (algebraic geometry) Chevalley–Shephard–Todd theorem (finite group) Chevalley–Warning theorem (field theory) Chinese remainder theorem (number theory) Choi's theorem on completely positive maps ...
The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
Triangle inequalities (8 P) Pages in category "Geometric inequalities" The following 30 pages are in this category, out of 30 total.