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Grünwald–Letnikov Derivative
In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 millio ... in 1868. Constructing the Grünwald–Letnikov derivative The formula :f'(x) = \lim_ \frac for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be: :\beginf''(x)&=\lim_\frac\\&=\lim_\frac\end Assuming that the ''h'' 's converge synchronously, this simplifies to: : = \lim_ \frac which can be justified rigorously by the mean value theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, 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). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D :D f(x) = \frac f(x)\,, and of the integration operator J The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, such as identities. :J f(x) = \int_0^x f(s) \,ds\,, and developing a calculus for such operators generalizing the classical one. In this context, the term ''powers'' refers to iterative application of a linear operator D to a function f, that is, repeatedly composing D with itself, as in D^n(f) = (\underbrace_n)(f) = \underbrace_n (f)\cdots))). For example, one may ask for a meaningful interpretation of :\sqrt = D^\frac12 as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied ''twice'' to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anton Karl Grünwald , the championship trophy of the Swedish ju ...
Anton may refer to: People * Anton (given name), including a list of people with the given name * Anton (surname) Places * Anton Municipality, Bulgaria ** Anton, Sofia Province, a village * Antón District, Panama ** Antón, a town and capital of the district * Anton, Colorado, an unincorporated town * Anton, Texas, a city * Anton, Wisconsin, an unincorporated community *River Anton, Hampshire, United Kingdom Other uses * Case Anton, codename for the German and Italian occupation of Vichy France in 1942 * Anton (computer), a highly parallel supercomputer for molecular dynamics simulations * ''Anton'' (1973 film), a Norwegian film * ''Anton'' (2008 film), an Irish film *Anton Cup The Anton Cup is the championship trophy of the Swedish junior hockey league, J20 SuperElit. The trophy was donated by Anton Johansson, chairman of the Swedish Ice Hockey Association between 1924 and 1948, in 1952, as an award for Sweden's top-ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prague
Prague ( ; cs, Praha ; german: Prag, ; la, Praga) is the capital and List of cities in the Czech Republic, largest city in the Czech Republic, and the historical capital of Bohemia. On the Vltava river, Prague is home to about 1.3 million people. The city has a temperate climate, temperate oceanic climate, with relatively warm summers and chilly winters. Prague is a political, cultural, and economic hub of central Europe, with a rich history and Romanesque architecture, Romanesque, Czech Gothic architecture, Gothic, Czech Renaissance architecture, Renaissance and Czech Baroque architecture, Baroque architectures. It was the capital of the Kingdom of Bohemia and residence of several Holy Roman Emperors, most notably Charles IV, Holy Roman Emperor, Charles IV (r. 1346–1378). It was an important city to the Habsburg monarchy and Austro-Hungarian Empire. The city played major roles in the Bohemian Reformation, Bohemian and the Protestant Reformations, the Thirty Year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksey Letnikov
Aleksey Vasilyevich Letnikov (russian: Алексéй Васи́льевич Лéтников, link=no, 1837–1888) was a Russian mathematician. After graduating from the Konstantinovsky Land-Surveying Institute (russian: Константиновский Межевой Институт) in Moscow, Letnikov attended classes at Moscow University and the Sorbonne. In 1860 he became an Instructor of Mathematics at the Konstantinovsky Institute. He obtained the degrees of Master and Ph.D. from Moscow University in 1868 and 1874 respectively. In 1868 Letnikov became a professor at the Imperial Moscow Technical School and from 1879 to 1880 was an Inspector at the school. From 1883 he was the principal of the Aleksandrov Commercial School (russian: Александровское коммерческое училище, currently The State University of Managementbr>and from 1884 he was a Corresponding Member of the Russian Academy of Sciences. His most renowned contribution to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moscow
Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million residents within the city limits, over 17 million residents in the urban area, and over 21.5 million residents in the metropolitan area. The city covers an area of , while the urban area covers , and the metropolitan area covers over . Moscow is among the world's largest cities; being the most populous city entirely in Europe, the largest urban and metropolitan area in Europe, and the largest city by land area on the European continent. First documented in 1147, Moscow grew to become a prosperous and powerful city that served as the capital of the Grand Duchy that bears its name. When the Grand Duchy of Moscow evolved into the Tsardom of Russia, Moscow remained the political and economic center for most of the Tsardom's history. Whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Value Theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, the theorem states that if f is a continuous function on the closed interval , b/math> and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that the tangent at c is parallel to the secant line through the endpoints \big(a, f(a)\big) and \big(b, f(b)\big), that is, : f'(c)=\frac. History A special case of this theorem for inverse interpolation of the sine was first described by Parameshvara (1380–1460), from the Kerala School of Astronomy and Mathematics in India, in his comme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
