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Misiurewicz Point
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By analogy, the term ''Misiurewicz point'' is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic. This term makes less sense for maps in greater generality that have more than one free critical point because some critical points might be periodic and others not. These points are named after the Polish-American mathematician Michał Misiurewicz, who was the first to study them. Mathematical notation A parameter c is a Misiurewicz point M_ if it satisfies the equations: :f_c^(z_) = f_c^(z_) and: :f_c^(z_) \neq f_c^(z_) so: :M_ = c : f_c^(z_) = f_c^(z_) where: * z_ is a critical point of f_c, * k and n are positive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X either ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraction (mathematics)
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Numbers
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fraction (mathematics)
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A ''common'', ''vulgar'', or ''simple'' fraction (examples: \tfrac and \tfrac) consists of a numerator, displayed above a line (or before a slash like ), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not ''common'', including compound fractions, complex fractions, and mixed numerals. In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction , the numerator 3 indicates that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). 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 , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turn (geometry)
A turn is a unit of plane angle measurement equal to radians, 360 degrees or 400 gradians. Subdivisions of a turn include half-turns, quarter-turns, centiturns, milliturns, etc. The closely related terms ''cycle'' and ''revolution'' are not equivalent to a turn. Subdivisions A turn can be divided in 100 centiturns or milliturns, with each milliturn corresponding to an angle of 0.36°, which can also be written as 21′ 36″. A protractor divided in centiturns is normally called a "percentage protractor". Binary fractions of a turn are also used. Sailors have traditionally divided a turn into 32 compass points, which implicitly have an angular separation of 1/32 turn. The ''binary degree'', also known as the ''binary radian'' (or ''brad''), is turn. The binary degree is used in computing so that an angle can be represented to the maximum possible precision in a single byte. Other measures of angle used in computing may be based on dividin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branching Theorem
In mathematics, the branching theorem is a theorem about Riemann surfaces. Intuitively, it states that every non-constant holomorphic function is locally a polynomial. Statement of the theorem Let X and Y be Riemann surfaces, and let f : X \to Y be a non-constant holomorphic map. Fix a point a \in X and set b := f(a) \in Y. Then there exist k \in \N and charts \psi_ : U_ \to V_ on X and \psi_ : U_ \to V_ on Y such that * \psi_ (a) = \psi_ (b) = 0; and * \psi_ \circ f \circ \psi_^ : V_ \to V_ is z \mapsto z^. This theorem gives rise to several definitions: * We call k the '' multiplicity '' of f at a. Some authors denote this \nu (f, a). * If k > 1, the point a is called a ''branch point'' of f. * If f has no branch points, it is called ''unbranched''. See also unramified morphism In algebraic geometry, an unramified morphism is a morphism f: X \to Y of schemes such that (a) it is locally of finite presentation and (b) for each x \in X and y = f(x), we have that # The residue f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
External Ray
An external ray is a curve that runs from infinity toward a Julia or Mandelbrot set. Although this curve is only rarely a half-line (ray) it is called a ray because it is an image of a ray. External rays are used in complex analysis, particularly in complex dynamics and geometric function theory. History External rays were introduced in Douady and Hubbard's study of the Mandelbrot set Types Criteria for classification : * plane : parameter or dynamic * map * bifurcation of dynamic rays * Stretching * landing plane External rays of (connected) Julia sets on dynamical plane are often called dynamic rays. External rays of the Mandelbrot set (and similar one-dimensional connectedness loci) on parameter plane are called parameter rays. bifurcation Dynamic ray can be: * bifurcated = branched = broken * smooth = unbranched = unbroken When the filled Julia set is connected, there are no branching external rays. When the Julia set is not connected then some external r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tan Lei
Tan Lei (; 1963–2016) was a mathematician specialising in complex dynamics and functions of complex numbers. She is most well-known for her contributions to the study of the Mandelbrot set and Julia set. Career After gaining her PhD in Mathematics in 1986 at University of Paris-Sud, Orsay, Tan worked as an assistant researcher in Geneva. She then conducted postdoctoral projects at the Max Planck Institute for Mathematics and University of Bremen until 1989, when she was made a lecturer at Ecole Normale Superieure de Lyon in France. Tan held a research position at University of Warwick from 1995 to 1999, before becoming a senior lecturer at Cergy-Pontoise University. She was made professor at University of Angers in 2009. Mathematical work Tan obtained important results about the Julia and Mandelbrot sets, in particular investigating their fractality and the similarities between the two.''Local properties of The Mandelbrot set M, Similarity between M and Julia sets'', Proc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-similarity
__NOTOC__ 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). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole. For instance, a side of the Koch snowflake is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape. The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a counterexample, whereas any portion of a straight line may resemble the whole, further detail is not revealed. A time developing phenomenon is said to exhibit self-similarity if the numerical v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior (topology)
In mathematics, specifically in general topology, topology, the interior of a subset of a topological space is the Union (set theory), union of all subsets of that are Open set, open in . A point that is in the interior of is an interior point of . The interior of is the Absolute complement, complement of the closure (topology), closure of the complement of . In this sense interior and closure are Duality_(mathematics)#Duality_in_logic_and_set_theory, dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary (topology), boundary. The interior, boundary, and exterior of a subset together partition of a set, partition the whole space into three blocks (or fewer when one or more of these is empty set, empty). Definitions Interior point If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
