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Universal Chord Theorem
In mathematical analysis, the universal chord theorem states that if a function ''f'' is continuous on 'a'',''b''and satisfies f(a) = f(b) , then for every natural number n, there exists some x \in ,b such that f(x) = f\left(x + \frac\right) . History The theorem was published by Paul Lévy in 1934 as a generalization of Rolle's theorem. Statement of the theorem Let H(f) = \ denote the chord set of the function ''f''. If ''f'' is a continuous function and h \in H(f) , then \frac h n \in H(f) for all natural numbers ''n''. Case of ''n'' = 2 The case when ''n'' = 2 can be considered an application of the Borsuk–Ulam theorem to the real line. It says that if f(x) is continuous on some interval I = ,b with the condition that f(a) = f(b) , then there exists some x \in ,b such that f(x) = f\left(x + \frac\right) . In less generality, if f : ,1\rightarrow \R is continuous and f(0) = f(1) , then there exists x \in \left ,\frac\right/math> that satisfies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chord Graph
Chord or chords may refer to: Art and music * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord, a chord played on a guitar, which has a particular tuning * The Chords (British band), 1970s British mod revival band * The Chords (American band), 1950s American doo-wop group * ''The Chord'' (painting), a c.1715 painting by Antoine Watteau * Andrew Chord, a comic book character who is the former mentor of the New Warriors Mathematics * Chord (geometry), a line segment joining two points on a curve * Chord (graph theory), an edge joining two nonadjacent nodes in a cycle People * Chord Overstreet, American actor and musician * Chords (musician), a Swedish hiphop/reggae artist Programming * Chord (concurrency), a concurrency construct in some object-oriented programming languages * Chord (peer-to-peer), a peer-to-peer protocol and algorithm for distributed hash tables (DHT) Science and technology * Chord (astronomy), a line cross ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Lévy (mathematician)
Paul Pierre Lévy (15 September 1886 – 15 December 1971) was a French mathematician who was active especially in probability theory, introducing fundamental concepts such as local time, stable distributions and characteristic functions. Lévy processes, Lévy flights, Lévy measures, Lévy's constant, the Lévy distribution, the Lévy area, the Lévy arcsine law, and the fractal Lévy C curve are named after him. Biography Lévy was born in Paris to a Jewish family which already included several mathematicians. His father Lucien Lévy was an examiner at the École Polytechnique. Lévy attended the École Polytechnique and published his first paper in 1905, at the age of nineteen, while still an undergraduate, in which he introduced the Lévy–Steinitz theorem. His teacher and advisor was Jacques Hadamard. After graduation, he spent a year in military service and then studied for three years at the École des Mines, where he became a professor in 1913. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rolle's Theorem
In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one point, somewhere between them, at which the slope of the tangent line is zero. Such a point is known as a stationary point. It is a point at which the first derivative of the function is zero. The theorem is named after Michel Rolle. Standard version of the theorem If a real-valued function is continuous on a proper closed interval , differentiable on the open interval , and , then there exists at least one in the open interval such that f'(c) = 0. This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem. History Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borsuk–Ulam Theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an ''n''-sphere into Euclidean ''n''-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Formally: if f: S^n \to \R^n is continuous then there exists an x\in S^n such that: f(-x)=f(x). The case n=1 can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature. The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case. The case n=2 is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal barometric pressures, assuming that both parameters vary continuously in space. The Borsuk–Ulam theorem has several equivalent statements in terms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider the closed interval I = ,b/math> ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
