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  2. Ostrowski's theorem - Wikipedia

    en.wikipedia.org/wiki/Ostrowski's_theorem

    The following proof follows the one of Theorem 10.1 in Schikhof (2007). Let | | be an absolute value on the rationals. We start the proof by showing that it is entirely determined by the values it takes on prime numbers.

  3. Absolute value - Wikipedia

    en.wikipedia.org/wiki/Absolute_value

    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 ...

  4. Absolute value (algebra) - Wikipedia

    en.wikipedia.org/wiki/Absolute_value_(algebra)

    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 ((), ()) = = ((), ()).

  5. Complex number - Wikipedia

    en.wikipedia.org/wiki/Complex_number

    The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument).

  6. Weak derivative - Wikipedia

    en.wikipedia.org/wiki/Weak_derivative

    Usually, the existence of multiple solutions is not a problem, since functions are considered to be equivalent in the theory of L p spaces and Sobolev spaces if they are equal almost everywhere. The characteristic function of the rational numbers 1 Q {\displaystyle 1_{\mathbb {Q} }} is nowhere differentiable yet has a weak derivative.

  7. Linear recurrence with constant coefficients - Wikipedia

    en.wikipedia.org/wiki/Linear_recurrence_with...

    a term with real characteristic roots converges to 0 as t grows indefinitely large if the absolute value of the characteristic root is less than 1. If the absolute value equals 1, the term will stay constant as t grows if the root is +1 but will fluctuate between two values if the root is −1. If the absolute value of the root is greater than ...

  8. Homogeneous function - Wikipedia

    en.wikipedia.org/wiki/Homogeneous_function

    The absolute value of a complex number is a ... only positive values, one gets a positively ... of a given degree are exactly the solution of a specific ...

  9. Absolutely integrable function - Wikipedia

    en.wikipedia.org/wiki/Absolutely_integrable_function

    Then | | + + + + + | | so | | + + + + + | | This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for ...