In
mathematics, a function ''f'' of ''n'' variables
:''x''
1, ..., ''x''
''n''
leads to a Chisini mean ''M'' if for every vector <''x''
1, ..., ''x''
''n''>, there exists a unique ''M'' such that
:''f''(''M'',''M'', ..., ''M'') = ''f''(''x''
1,''x''
2, ..., ''x''
''n'').
The
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
,
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
,
geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
,
generalised,
Heronian and
quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. ''Quadratus'' is Latin for ''square''.
Mathematics ...
means are all Chisini means, as are their weighted variants.
They were introduced by
Oscar Chisini in 1929.
See also
*
Fréchet mean 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 ...
*
Generalized mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).
...
*
Jensen's inequality
In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier ...
*
Quasi-arithmetic mean
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 f. It is ...
*
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)
& ...
References
*Chisini, O. "Sul concetto di media." Periodico di Matematiche 4, 106–116, 1929.
Mathematical analysis
Means
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