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Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product ^ of the discrete two-point space \underline 2 . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set-theoretic Limit
In mathematics, the limit of a sequence of Set (mathematics), sets A_1, A_2, \ldots (subsets of a common set X) is a set whose elements are determined by the sequence in either of two equivalent ways: (1) by upper and lower bounds on the sequence that converge monotonically to the same set (analogous to Limit of a sequence, convergence of real-valued sequences) and (2) by convergence of a sequence of indicator functions which are themselves Real number, real-valued. As is the case with sequences of other objects, convergence is not necessary or even usual. More generally, again analogous to real-valued sequences, the less restrictive limit infimum and limit supremum of a set sequence always exist and can be used to determine convergence: the limit exists if the limit infimum and limit supremum are identical. (See below). Such set limits are essential in Measure (mathematics), measure theory and probability. It is a common misconception that the limits infimum and supremum describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Progression
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' is the common ratio and ''a'' is the initial value. The sum of a geometric progression's terms is called a '' geometric series''. Properties The ''n''th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Geometric sequences satisfy the linear recurrence relation :a_n = r\,a_ f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benoit Mandelbrot
Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of physical phenomena and "the uncontrolled element in life". He referred to himself as a "fractalist" and is recognized for his contribution to the field of fractal geometry, which included coining the word "fractal", as well as developing a theory of "roughness and self-similarity" in nature. In 1936, at the age of 11, Mandelbrot and his family emigrated from Warsaw, Poland, to France. After World War II ended, Mandelbrot studied mathematics, graduating from universities in Paris and in the United States and receiving a master's degree in aeronautics from the California Institute of Technology. He spent most of his career in both the United States and France, having dual French and American citizenship. In 1958, he began a 35-year career ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Fractal Geometry Of Nature
''The Fractal Geometry of Nature'' is a 1982 book by the Franco-American mathematician Benoît Mandelbrot. Overview ''The Fractal Geometry of Nature'' is a revised and enlarged version of his 1977 book entitled ''Fractals: Form, Chance and Dimension'', which in turn was a revised, enlarged, and translated version of his 1975 French book, ''Les Objets Fractals: Forme, Hasard et Dimension''. '' American Scientist'' put the book in its one hundred books of 20th century science. As technology has improved, mathematically accurate, computer-drawn fractals In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ... have become more detailed. Early drawings were low-resolution black and white; later drawings were higher resolution and in color. Many examples were created by programmers working w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory is that a binary tree is a triple , where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton (a single–element set) containing the root. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. In ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Numeral System
A ternary numeral system (also called base 3 or trinary) has 3 (number), three as its radix, base. Analogous to a bit, a ternary numerical digit, digit is a trit (trinary digit). One trit is equivalent to binary logarithm, log2 3 (about 1.58496) bits of Units of information, information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary numeral system, binary. For example, decimal 365 (number), 365 or senary corresponds to binary (nine bits) and to ternary (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Set Binary Tree
A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. In formal Jewish worship, a cantor is a person who sings solo verses or passages to which the choir or congregation responds. Overview The term itself is derived from the Latin for "singer", though it is frequently used to translate a range of equivalent terms in other languages, such as for the leader of singing on a traditional chundan vallam or snake boat of Kerala. Role of the cantor Judaism In Judaism, a cantor sings and leads congregants in prayer in religious services. They are also sometimes called a ''huzzan'', which is the Hebrew term for the role. They often come from a long line of cantors in their family; born with a natural gift of singing with extensive vocal range. The role of professional cantor is especially c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal String
An ordinary fractal string \Omega is a bounded, open subset of the real number line. Such a subset can be written as an at-most-countable union of connected open intervals with associated lengths \mathcal=\ written in non-increasing order; we also refer to \mathcal as a fractal string. For example, \mathcal=\left\ is a fractal string corresponding to the Cantor set. A fractal string is the analogue of a one-dimensional "fractal drum," and typically the set \Omega has a boundary \partial\Omega which corresponds to a fractal such as the Cantor set. The heuristic idea of a fractal string is to study a (one-dimensional) fractal using the "space around the fractal." It turns out that the sequence of lengths \mathcal of the set itself is "intrinsic," in the sense that the fractal string \mathcal itself (independent of a specific geometric realization of these lengths as corresponding to a choice of set \Omega) contains information about the fractal to which it corresponds. For eac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
