Cantor functionIn mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow.
De Rham curveIn mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham. The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all special cases of the general de Rham curve. Consider some complete metric space (generally 2 with the usual euclidean distance), and a pair of contracting maps on M: By the Banach fixed-point theorem, these have fixed points and respectively.
Dense orderIn mathematics, a partial order or total order < on a set is said to be dense if, for all and in for which , there is a in such that . That is, for any two elements, one less than the other, there is another element between them. For total orders this can be simplified to "for any two distinct elements, there is another element between them", since all elements of a total order are comparable. The rational numbers as a linearly ordered set are a densely ordered set in this sense, as are the algebraic numbers, the real numbers, the dyadic rationals and the decimal fractions.
Blancmange curveIn mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a Blancmange pudding. It is a special case of the more general de Rham curve; see also fractal curve.
Dyadic rationalIn 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.
Modular groupIn mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.
Continued fractionIn 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. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression.
Rational numberIn mathematics, a rational number is a number that can be expressed as the quotient or fraction \tfrac p q of two integers, a numerator p and a non-zero denominator q. For example, \tfrac{-3}{7} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold \Q. A rational number is a real number.
