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Minkowski's question-mark function

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In mathematics, the Minkowski question-mark function (or the slippery devil's staircase), denoted by, is a function possessing various unusual fractal properties, defined by. [1]

53 relations: Absolute continuity, Algebraic number, Algorithm, Arnaud Denjoy, Binary number, C (programming language), Cambridge, Cambridge University Press, Cantor function, Continued fraction, De Rham curve, Derivative, Determinant, Division by two, Dyadic rational, Even and odd functions, Farey sequence, Fixed point (mathematics), Floor and ceiling functions, Fractal, Function (mathematics), Gibbs measure, Graph of a function, Inverse function, Irrational number, Irreducible fraction, John Horton Conway, Linear fractional transformation, Loop invariant, Loop variant, Mathematics, Mathematische Annalen, Measure (mathematics), Modular group, Monoid, Monotonic function, Multifractal system, Periodic function, Point reflection, Pompeiu derivative, Quadratic irrational number, Raphaël Salem, Rational number, Recursion, Run-length encoding, Self-similarity, Springer Science+Business Media, Stern–Brocot tree, Transactions of the American Mathematical Society, Transcendental number, ..., Unit interval, University of Nottingham, West Virginia University. Expand index (3 more) »

Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

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Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

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Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

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Arnaud Denjoy

Arnaud Denjoy (1884–1974) was a French mathematician.

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Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

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C (programming language)

C (as in the letter ''c'') is a general-purpose, imperative computer programming language, supporting structured programming, lexical variable scope and recursion, while a static type system prevents many unintended operations.

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Cambridge

Cambridge is a university city and the county town of Cambridgeshire, England, on the River Cam approximately north of London.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Cantor function

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous.

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Continued fraction

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.

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De Rham curve

In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Division by two

In mathematics, division by two or halving has also been called mediation or dimidiation.

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Dyadic rational

In mathematics, a dyadic fraction or dyadic rational is a rational number whose denominator, when the ratio is in minimal (coprime) terms, is a power of two, i.e., a number of the form \frac where a is an integer and b is a natural number; for example, 1/2 or 3/8, but not 1/3.

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Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

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Farey sequence

In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

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Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function.

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Floor and ceiling functions

In mathematics and computer science, the floor function is the function that takes as input a real number x and gives as output the greatest integer less than or equal to x, denoted \operatorname(x).

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Fractal

In mathematics, a fractal is an abstract object used to describe and simulate naturally occurring objects.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Gibbs measure

In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics.

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Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

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Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

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Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

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Irreducible fraction

An irreducible fraction (or fraction in lowest terms or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and -1, when negative numbers are considered).

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John Horton Conway

John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.

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Linear fractional transformation

In mathematics, the phrase linear fractional transformation usually refers to a Möbius transformation, which is a homography on the complex projective line P(C) where C is the field of complex numbers.

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Loop invariant

In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration.

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Loop variant

In computer science, a loop variant is a mathematical function defined on the state space of a computer program whose value is monotonically decreased with respect to a (strict) well-founded relation by the iteration of a while loop under some invariant conditions, thereby ensuring its termination.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Mathematische Annalen

Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.

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Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.

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Modular group

In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.

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Monoid

In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.

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Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

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Multifractal system

A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed.

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Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

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Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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Pompeiu derivative

In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set.

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Quadratic irrational number

In mathematics, a quadratic irrational number (also known as a quadratic irrational, a quadratic irrationality or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the set of rational numbers.

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Raphaël Salem

Raphaël Salem (Greek: Ραφαέλ Σαλέμ; November 7, 1898 in Saloniki, Ottoman Empire (now Thessaloniki, Greece) – June 20, 1963 in Paris, France), was a Greek mathematician after whom are named the Salem numbers and whose widow founded the Salem Prize.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Recursion

Recursion occurs when a thing is defined in terms of itself or of its type.

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Run-length encoding

Run-length encoding (RLE) is a very simple form of lossless data compression in which runs of data (that is, sequences in which the same data value occurs in many consecutive data elements) are stored as a single data value and count, rather than as the original run.

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Self-similarity

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts).

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Stern–Brocot tree

In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond one-for-one to the positive rational numbers, whose values are ordered from the left to the right as in a search tree.

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Transactions of the American Mathematical Society

The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.

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Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

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Unit interval

In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.

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University of Nottingham

The University of Nottingham is a public research university in Nottingham, United Kingdom.

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West Virginia University

West Virginia University (WVU) is a public, land-grant, space-grant, research-intensive university in Morgantown, West Virginia, United States.

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? function, ?(x), Conway box function, Minkowski question mark, Minkowski question mark function, Minkowski question-mark function, Minkowski's question mark function, Question mark function, Slippery devil's staircase.

References

[1] https://en.wikipedia.org/wiki/Minkowski's_question-mark_function

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