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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] |
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] |
Minimum
In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative'' extrema) or on the entire domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued function ''f'' defined on a domain ''X'' has a global (or absolute) maximum point at ''x''∗, if for all ''x'' in ''X''. Similarly, the function has a global (or absolute) minimum point at ''x' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. 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. The process of finding a derivative is called differentiation. There are multiple different notations for differentiation. '' Leibniz notation'', named after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas ''prime notation'' is written by adding a prime mark. Higher order notations represent repeated differentiation, and they are usually denoted in Leib ... [...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] |
Maximum
In mathematical analysis, the maximum and minimum of a function (mathematics), function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given Interval (mathematics), range (the ''local'' or ''relative'' extrema) or on the entire domain of a function, domain (the ''global'' or ''absolute'' extrema) of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory, the maximum and minimum of a set (mathematics), set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. Definition A real-valued Function (mathematics), function ''f'' defined on a Domain of a function, domain ''X'' has a global (or absolute) m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Mean
In mathematics, the root mean square (abbrev. RMS, or rms) of a set of values is the square root of the set's mean square. Given a set x_i, its RMS is denoted as either x_\mathrm or \mathrm_x. The RMS is also known as the quadratic mean (denoted M_2), a special case of the generalized mean. The RMS of a continuous function is denoted f_\mathrm and can be defined in terms of an integral of the square of the function. In estimation theory, the root-mean-square deviation of an estimator measures how far the estimator strays from the data. Definition The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In the case of a set of ''n'' values \, the RMS is : x_\text = \sqrt. The corresponding formula for a continuous function (or waveform) ''f''(''t'') defined over the interval T_1 \le t \le T_2 is : f_\text = \sqrt , and the RMS for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Mean
In mathematics and statistics, the arithmetic mean ( ), arithmetic average, or just the ''mean'' or ''average'' is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a Survey (statistics), survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps to distinguish it from other types of means, such as geometric mean, geometric and harmonic mean, harmonic. Arithmetic means are also frequently used in economics, anthropology, history, and almost every other academic field to some extent. For example, per capita income is the arithmetic average of the income of a nation's Human population, population. While the arithmetic mean is often used to report central tendency, central tendencies, it is not a robust statistic: it is greatly influenced by outliers (Value (mathematics), values much larger or smaller than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Power Mean
Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Mathematics, science and technology Computing * IBM POWER (software), an IBM operating system enhancement package * IBM POWER architecture, a RISC instruction set architecture * Power ISA, a RISC instruction set architecture derived from PowerPC * IBM Power microprocessors, made by IBM, which implement those RISC architectures * Power.org, a predecessor to the OpenPOWER Foundation Mathematics * Exponentiation, "''x'' to the power of ''y''" * Power function * Power of a point * Statistical power Physics * Magnification, the factor by which an optical system enlarges an image * Optical power, the degree to which a lens converges or diverges light Social sciences and politics * Economic power, encompassing several concepts that economists use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean of numbers is the Nth root, th root of their product (mathematics), product, i.e., for a collection of numbers , the geometric mean is defined as : \sqrt[n]. When the collection of numbers and their geometric mean are plotted in logarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking the natural logarithm of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using the exponential function , :\sqrt[n] = \exp \left( \frac \right). The geometric mean of two numbers is the square root of their product, for example with numbers and the geometric mean is \textstyle \sqrt = The geometric mean o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...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] |
