In mathematics, the arithmetic–geometric mean of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means:
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite set of real numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers a1, a2, ..., an, the geometric mean is defined as
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic feature of the median in describing data compared to the mean is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of the center. Median income, for example, may be a better way to describe center of the income distribution because increases in the largest incomes alone have no effect on median. For this reason, the median is of central importance in robust statistics.
In commutative algebra, the prime spectrum of a ring R is the set of all prime ideals of R, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings .
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In probability theory, Chebyshev's inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/k2 of the distribution's values can be k or more standard deviations away from the mean. The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] or in terms of two positive parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of the distribution.
In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder.
In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function . It is also called Kolmogorov mean after Soviet mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means.
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices.
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc.
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.
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.
In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p = 2.
In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.
In mathematics of special functions, the Neuman–Sándor meanM, of two positive and unequal numbers a and b, is defined as:
