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Heinz Mean
In mathematics, the Heinz mean (named after E. Heinz) of two non-negative real numbers ''A'' and ''B'', was defined by Bhatia as: :\operatorname_x(A, B) = \frac, with 0 ≤ ''x'' ≤ . For different values of ''x'', this Heinz mean interpolates between the arithmetic (''x'' = 0) and geometric (''x'' = 1/2) means such that for 0 < ''x'' < : : The Heinz means appear naturally when symmetrizing -divergences. It may also be defined in the same way for , and satisfies a similar interpolation formula.. See also * |
Erhard Heinz
Erhard Heinz (30 April 1924, Bautzen – 29 December 2017, Göttingen) was a German mathematician known for his work on partial differential equations, in particular the Monge–Ampère equation. He worked as professor in Stanford, Munich and from 1966 until his retirement 1992 at the University of Göttingen. Heinz obtained his PhD in 1951 under the supervision of Franz Rellich at the University of Göttingen. His most important scientific work deals with the existence and regularity theory of systems of non-linear partial differential equations, with applications to differential geometry and mathematical physics. He obtained important results in the theory of surfaces with prescribed mean curvature, in particular of minimal surfaces, for the Weyl embedding problem, and for systems of Monge-Ampère type. In 1994 he was awarded the Cantor medal. His doctoral students include Hans Wilhelm Alt, Wolf von Wahl, Willi Jäger, Helmut Werner, Reinhold Böhme, Friedrich Tomi, and Fried ... [...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] |
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] |
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] |
Positive Semidefinite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to ... [...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] |
Muirhead's Inequality
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions ''a''-mean For any real vector :a=(a_1,\dots,a_n) define the "''a''-mean" 'a''of positive real numbers ''x''1, ..., ''x''''n'' by : \frac\sum_\sigma x_^\cdots x_^, where the sum extends over all permutations σ of . When the elements of ''a'' are nonnegative integers, the ''a''-mean can be equivalently defined via the monomial symmetric polynomial m_a(x_1,\dots,x_n) as : = \frac m_a(x_1,\dots,x_n), where ℓ is the number of distinct elements in ''a'', and ''k''1, ..., ''k''ℓ are their multiplicities. Notice that the ''a''-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if a_1+\cdots+a_n=1. In the general case, one can consider instead , which is called a Muirhead mean.Bullen, P. S. Handbook of means an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality Of Arithmetic And Geometric Means
Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of income ** Wealth inequality, an unequal distribution of wealth ** Spatial inequality, the unequal distribution of income and resources across geographical regions ** International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group ** Gender inequality, unequal treatment or perceptions due to gender ** Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between different neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
