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The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
The numerical solution to the linear test equation decays to zero if | r(z) | < 1 with z = hλ. The set of such z is called the domain of absolute stability. In particular, the method is said to be absolute stable if all z with Re(z) < 0 are in the domain of absolute stability. The stability function of an explicit Runge–Kutta method is a ...
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
the value group or valuation group Γ v = v(K ×), a subgroup of Γ (though v is usually surjective so that Γ v = Γ); the valuation ring R v is the set of a ∈ K with v ( a ) ≥ 0, the prime ideal m v is the set of a ∈ K with v ( a ) > 0 (it is in fact a maximal ideal of R v ),
The solution is the weighted average of six increments, where each increment is the product of the size of the interval, , and an estimated slope specified by function f on the right-hand side of the differential equation.
For any given n ≥ 1, among the polynomials of degree n with leading coefficient 1 (monic polynomials): = is the one of which the maximal absolute value on the interval [−1, 1] is minimal. This maximal absolute value is: 1 2 n − 1 {\displaystyle {\frac {1}{2^{n-1}}}} and | f ( x ) | reaches this maximum exactly n + 1 times at: x = cos ...
Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because d d x ( ln | u | ) = u ′ u {\displaystyle {\tfrac {d}{dx}}(\ln |u|)={\tfrac {u'}{u}}} , which justifies taking the absolute ...
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.