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Identric Mean
The identric mean of two positive real numbers ''x'', ''y'' is defined as: : \begin I(x,y) &= \frac\cdot \lim_ \sqrt xi-\eta\\ pt&= \lim_ \exp\left(\frac-1\right) \\ pt&= \begin x & \textx=y \\ pt\frac \sqrt -y& \text \end \end It can be derived from the mean value theorem by considering the secant of the graph of the function x \mapsto x\cdot \ln x. It can be generalized to more variables according by the mean value theorem for divided differences. The identric mean is a special case of the Stolarsky mean. See also * Mean * Logarithmic mean In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass trans ... References {{DEFAULTSORT:Identric Mean Means ... [...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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...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 commentari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciprocal) trigonometric function of the cosine * the secant method, a root-finding algorithm in numerical analysis, based on secant lines to graphs of functions * a secant ogive Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ... in nose cone design {{mathdab sr:Секанс ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Value Theorem For Divided Differences
In mathematical analysis, the mean value theorem for divided differences generalizes the mean value theorem to higher derivatives. Statement of the theorem For any ''n'' + 1 pairwise distinct points ''x''0, ..., ''x''''n'' in the domain of an ''n''-times differentiable function ''f'' there exists an interior point : \xi \in (\min\,\max\) \, where the ''n''th derivative of ''f'' equals ''n'' ! times the ''n''th divided difference at these points: : f _0,\dots,x_n= \frac. For ''n'' = 1, that is two function points, one obtains the simple mean value theorem. Proof Let P be the Lagrange interpolation polynomial for ''f'' at ''x''0, ..., ''x''''n''. Then it follows from the Newton form of P that the highest term of P is f _0,\dots,x_nx-x_)\dots(x-x_1)(x-x_0). Let g be the remainder of the interpolation, defined by g = f - P. Then g has n+1 zeros: ''x''0, ..., ''x''''n''. By applying Rolle's theorem first to g, then to g', a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stolarsky Mean
In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975. Definition For two positive real numbers ''x'', ''y'' the Stolarsky Mean is defined as: : \begin S_p(x,y) & = \lim_ \left(\right)^ \\0pt& = \begin x & \textx=y \\ \left(\right)^ & \text \end \end Derivation It is derived from the mean value theorem, which states that a secant line, cutting the graph of a differentiable function f at ( x, f(x) ) and ( y, f(y) ), has the same slope as a line tangent to the graph at some point \xi in the interval ,y/math>. : \exists \xi\in ,y f'(\xi) = \frac The Stolarsky mean is obtained by : \xi = f'^\left(\frac\right) when choosing f(x) = x^p. Special cases *\lim_ S_p(x,y) is the minimum. *S_(x,y) is the geometric mean. *\lim_ S_p(x,y) is the logarithmic mean. It can be obtained from the mean value theorem by choosing f(x) = \ln x. *S_(x,y) is the power mean with exponent \frac. *\lim_ S_p(x,y) is the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set. For a data set, the ''arithmetic mean'', also known as "arithmetic average", is a measure of central tendency of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers ''x''1, ''x''2, ..., x''n'' is typically denoted using an overhead bar, \bar. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the ''sample mean'' (\bar) to distinguish it from the mean, or expected value, of the underlying distribution, the ''population mean'' (denoted \mu or \mu_x).Underhill, L.G.; Bradfield d. (1998) ''Introstat'', Juta and Company Ltd.p. 181/ref> Outside probability and statistics, a wide range of other notions of mean are o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Mean
In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer. Definition The logarithmic mean is defined as: :\begin M_\text(x, y) &= \lim_ \frac \\ pt &= \begin x & \textx = y ,\\ \frac & \text \end \end for the positive numbers x, y. Inequalities The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent one-third but larger than the geometric mean, unless the numbers are the same, in which case all three means are equal to the numbers. : \sqrt \leq \frac\leq \left(\frac2\right)^3 \leq \frac \qquad \text x > 0 \text y > 0. Toyesh Prakash Sharma generalizes the arithmetic logarithmic geometric mean inequality for any n belongs to the whole number as : \sqrt (\ln(\sqrt))^ (\ln(\sqrt)+n)\leq \frac\leq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
