Function question mark

The function question mark is, in Mathématiques, a function, noted $?\; \backslash \; left\; (X\; \backslash \; right)$.

This function was defined by Hermann Minkowski in 1904 in order to create a application continues of the whole of the irrational numbers quadratic of the interval $\backslash \; left]\; 0,1\; \backslash \; right\; towards\; the\; whole\; of\; [[the\; rational\; Number|rational\; numbers]\; dyadic\; of\; the\; same\; interval.\; The\; current\; current\; definition\; was\; posed\; byArnaud\; Denjoyin\; 1938.$

Definition

Either $x$ a Real number and $x\_1,\; x\_2,\; \backslash \; ldots$ its representation in Fraction continues. One poses:

$\{\backslash \; rm?\}(X)\; =\; x\_0\; +\; \backslash \; sum\_\; \{k=1\}\; ^\; \backslash \; infty\; \backslash \; frac\; \{(-\; 1)\; ^\; \{k+1\}\}\; \{2^\; \{x\_1\; +\; \backslash \; cdots\; +\; x\_k-1\}\}$

This definition is legitimate. In the case of an irrational number, the series always converges. In the case of a rational number, its continued function is limited to $x\_1,\; x\_2,\; \backslash \; ldots,\; x\_n$ and two successive terms of the series for $k\; \backslash \; Ge\; n$ are cancelled; it is then possible to write:

$\{\backslash \; rm?\}(X)\; =\; x\_0\; +\; \backslash \; sum\_\; \{k=1\}\; ^\; \{n-1\}\; \backslash \; frac\; \{(-\; 1)\; ^\; \{k+1\}\}\; \{2^\; \{x\_1\; +\; \backslash \; cdots\; +\; x\_k-1\}\}$

Examples

• $?\backslash \; left\; (0\; \backslash \; right)\; =\; 0$
• $\backslash \; frac\; \{1\}\; \{3\}\; =\; \backslash \; left$: $?\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{3\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{4\}$
• $\backslash \; frac\; \{177\}\; \{233\}\; =\; \backslash \; left$: $?\; \backslash \; left\; (\backslash \; frac\; \{177\}\; \{233\}\; \backslash \; right)\; =\; \backslash \; frac\; \{7193\}\; \{8192\}$
• $?\; \backslash \; left\; (1\; \backslash \; right)\; =\; 1$
• $\backslash \; sqrt\; \{2\}\; =\; \backslash \; left$: $?\; \backslash \; left\; (\backslash \; sqrt\; \{2\}\; \backslash \; right)\; =\; \backslash \; frac\; \{7\}\; \{5\}$

Properties

• the function question mark is strictly increasing.
• It is absolutely continuous.
• $?\; \backslash \; left\; (x+1\; \backslash \; right)\; =?\; \backslash \; left\; (X\; \backslash \; right)\; +1$
• If $x$ is a rational number, $?\; \backslash \; left\; (X\; \backslash \; right)$ is a dyadic rational number.
• If $x$ is a quadratic irrational number, its continued fraction is periodic and $?\; \backslash \; left\; (X\; \backslash \; right)$ is rational not-dyadic.
• If $\backslash \; frac\; \{p\}\; \{Q\}$ and $\backslash \; frac\; \{p\text{'}\}\; \{q\text{'}\}$ is two irreducible fractions such as $\backslash \; left|pq\text{'}-p\text{'}\; Q\; \backslash \; right|=1$ (two successive elements of a Continuation of Farey), $?\; \backslash \; left\; (\backslash \; frac\; \{p+p\text{'}\}\; \{q+q\text{'}\}\; \backslash \; right)\; =\; \backslash \; frac\; \{1\}\; \{2\}\; \backslash \; left\; (?\; \backslash \; left\; (\backslash \; frac\; \{p\}\; \{Q\}\; \backslash \; right)\; +?\; \backslash \; left\; (\backslash \; frac\; \{p\text{'}\}\; \{q\text{'}\}\; \backslash \; right)\; \backslash \; right)$
• the function question mark is a particular case of the curves fractales of of Rham.

See too

Internal bonds

• Theory of the numbers

External bonds

• '' Minkowski' S Question Mark Function '' ( MathWorld )

References

• H. Minkowski, Verhandlungen of the III internationalen mathematiker-kongresses in heidelberg , (1904) Berlin.
• A. Denjoy, On a real function of Minkowski , J. Maths. Pure Appl. 17 (1938) p105-151.
• Biblioni, L., Paradise, J., Viader, P., have New Light one Minkowski' S? (X) Does Function , Newspaper off Number Theory, 73 (1998), 212-227
• Biblioni, L., Paradise, J., Viader, Derivative P., The off Minkowski' S Singular Function , Newspaper off Mathematical Analysis and Applications, (2001) 107-125
• Conley, Randolph Mr. have Survey off the Minkowski? (X) Function , Master' S Thesis, West Virginia University, (2003)
• Vepstas, Linas, The Minkowski Question Mark and the Modular Group SSL (2, Z) , (2004)
• Vepstas, Linas, Modular Fractal Measures , (2004).

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