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The Möbius function () is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated Moebius) in 1832. [ i ] [ ii ] [ 2 ] It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula .
For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function; I ∗ 1 = σ 1 = σ, the divisor function; If the starting function is the Möbius function itself, the list of functions is: μ, the Möbius ...
The Smith chart, used by electrical engineers for analyzing transmission lines, is a visual depiction of the elliptic Möbius transformation Γ = (z − 1)/(z + 1). Each point on the Smith chart simultaneously represents both a value of z (bottom left), and the corresponding value of Γ (bottom right), for |Γ |<1.
An example of such linear fractional transformation is the Cayley transform, which was originally defined on the 3 × 3 real matrix ring. Linear fractional transformations are widely used in various areas of mathematics and its applications to engineering, such as classical geometry , number theory (they are used, for example, in Wiles's proof ...
As functions of , these are examples of Möbius transformations, which under composition of functions form the Mobius group PGL(2, Z). The six transformations form a subgroup known as the anharmonic group, again isomorphic to S 3. They are the torsion elements (elliptic transforms) in PGL(2, Z).
Mertens function to n = 10 000 Mertens function to n = 10 000 000. In number theory, the Mertens function is defined for all positive integers n as = = (), where () is the Möbius function. The function is named in honour of Franz Mertens.
In graph theory, the Möbius ladder M n, for even numbers n, is formed from an n-cycle by adding edges (called "rungs") connecting opposite pairs of vertices in the cycle. It is a cubic, circulant graph, so-named because (with the exception of M 6 (the utility graph K 3,3), M n has exactly n/2 four-cycles [1] which link together by their shared edges to form a topological Möbius strip.
for some real-valued functions a and b depending, respectively, on x and y. Conversely, given any such pair of real-valued functions, there exists a vector field X satisfying 1. and 2. Hence the Lie algebra of infinitesimal symmetries of the conformal structure, the Witt algebra, is infinite-dimensional.