mapy-i-podróże: Minkowski's question-mark function

Evidence: 100 • authority: 72

In mathematics, Minkowski's question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. It also maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.

Evidence: 59 • authority: 88

THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION. Abstract The Minkowski question mark function ?( x ) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?( x ). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p -adic distribution of rationa…

Evidence: 52 • authority: 88

Stern–Brocot tree. Abstract Christoffel words have a strong connection with continued fractions of rational numbers, as was recognized by Christoffel. They are in some sense equivalent to them, being a way to encode the expansion of these numbers into continued fractions. The Stern–Brocot tree is another way to compute continued fractions. Among other things in this chapter, we show the equivalence between semi-convergents of rational numbers, nodes in the Stern–Brocot tree, palindromic prefixes of central words, and points on the frontier of the convex hull of some subset of $ {\mathbb Z}^2 $ associated with a rational number. The Raney tree (or Calkin–Wilf tree) is another specialization…

Evidence: 52 • authority: 88

Stern–Brocot Tree. Christoffel words have a strong connection with continued fractions of rational numbers, as was recognized by Christoffel. They are in some sense equivalent to them, being a way to encode the expansion of these numbers into continued fractions. The Stern–Brocot tree is another way to compute continued fractions. Among other things in this chapter, equivalence is shown between semi-convergents of rational numbers, nodes in the Stern–Brocot tree, palindromic prefixes of centralwords, points on the frontier of the convex hull of some subset of Z 2 associatedwith a rational number, and special points relative to a half-line of rational slope.

Evidence: 38 • authority: 72

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belon…

Evidence: 38 • authority: 80

Cluster duality between Calkin–Wilf tree and Stern–Brocot tree

Evidence: — • authority: 72

A simple or regular continued fraction is a continued fraction with numerators all equal to one, and denominators built from a sequence { a i } {\displaystyle \{a_{i}\}} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like a 0 + 1 a 1 + 1 a 2 + 1 ⋱ + 1 a n {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}}} or an infinite continued fraction like a 0 + 1 a 1 + 1 a 2 + 1 ⋱ . {\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots }}}}}}.} Typically, such a continued fraction is obtained through a recursive process which starts by representing a number as…

Evidence: — • authority: 72

In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as a + b c d , {\displaystyle {a+b{\sqrt {c}} \over d},} for integers a, b, c, d;…

Evidence: — • authority: 72

In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number. The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always p…

Evidence: — • authority: 80

On numbers badly approximable by dyadic rationals