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If ?(x) is irrational, then x is either algebraic of degree greater than two, or transcendental. The question-mark function has fixed points at 0, 1 / 2 and 1, and at least two more, symmetric about the midpoint. One is approximately 0.42037. [6]
Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square of 3 may be written as 3 2, which is the number 9. In some cases when superscripts are not available, as for instance in programming languages or plain text files, the notations x ^2 ( caret ) or x **2 may be used in place of x 2 .
If X has a standard uniform distribution, then by the inverse transform sampling method, Y = − λ −1 ln(X) has an exponential distribution with (rate) parameter λ. If X has a standard uniform distribution, then Y = X n has a beta distribution with parameters (1/n,1). As such, The Irwin–Hall distribution is the sum of n i.i.d. U(0,1 ...
Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., t 2 = ε 2 t, t 3 = ε 3 t, etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996 ; Bender & Orszag 1999 ).
The space diagonal of the unit cube is √ 3. Distances between vertices of a double unit cube are square roots of the first six natural numbers, including the square root of 3 (√7 is not possible due to Legendre's three-square theorem) This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio √ 3.
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