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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 A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ... * Logarithmic mean 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 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] |
Mean Value Theorem
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant line, 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 (mathematics), interval starting from local hypotheses about derivatives at points of the interval. 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 commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant intersects the circle at exactly two points. A Chord (geometry), chord is the line segment determined by the two points, that is, the interval (mathematics), interval on the secant whose ends are the two points. Circles A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a ''secant line'', at one point a ''tangent line'' and at no points an ''exterior line''. A ''chord'' is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in Euclid's Elements, ... [...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 In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc (geometry), arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant lin .... Proof Let P be the Lagrange interpolation polynomial for ''f'' at ''x''0, ..., ''x''''n''. Then it follows from the Newto ... [...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 and y the Stolarsky Mean is defined as: : S_p(x,y) = \left \{ \begin{array}{l l} x, & \text{if }x=y, \\ \left({\frac{x^p-y^p}{p (x-y)\right)^{1/(p-1)}, & \text{otherwise}. \end{array} \right . 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{f(x)-f(y)}{x-y} The Stolarsky mean is obtained by : \xi = \left '\right{-1}\left(\frac{f(x)-f(y)}{x-y}\right) when choosing f(x) = x^p. Special cases *\lim_{p\to -\infty} S_p(x,y) is the minimum. *S_{-1}(x,y) is the geometric mean. *\lim_{p\to 0} S_p(x,y) is the logarithmic mean. It can be obtained from the mean val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean
A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statistics. Each attempts to summarize or typify a given group of data, illustrating the magnitude and sign of the data set. Which of these measures is most illuminating depends on what is being measured, and on context and purpose. The ''arithmetic mean'', also known as "arithmetic average", is 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 numbers are from observing a sample of a larger group, the arithmetic mean is termed the '' sample mean'' (\bar) to distinguish it from the group mean (or expected value) of the underlying distribution, denoted \mu or \mu_x. Outside probability and statistics, a wide rang ... [...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 by : L(x, y) = \left \{ \begin{array}{l l} x, & \text{if }x = y,\\ \dfrac{x - y}{\ln x - \ln y}, & \text{otherwise}, \end{array} \right . for x, y \in \mathbb{R}. Inequalities The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal. More precisely, for x, y \in \mathbb{R} with x \neq y, we have \frac{2xy}{x + y} \leq \sqrt{x y} \leq \frac{x - y}{\ln x - \ln y} \leq \frac{x + y}{2} \leq \left(\frac{x^2+y^2}2\right)^{1/2}. Sharma sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
