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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.
The formula is also correct if f and g are functions from the positive integers into some abelian group (viewed as a Z-module). In the language of Dirichlet convolutions, the first formula may be written as = where ∗ denotes the Dirichlet convolution, and 1 is the constant function 1(n) = 1. The second formula is then written as
[117] [118] Modern musical groups taking their name from the Möbius strip include American electronic rock trio Mobius Band [119] and Norwegian progressive rock band Ring Van Möbius. [120] Möbius strips and their properties have been used in the design of stage magic. One such trick, known as the Afghan bands, uses the fact that the Möbius ...
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.
A hyperboloid of one sheet, which is a surface of revolution contains a pencil of circles which is mapped onto a pencil of circles. A hyperboloid of one sheet contains additional two pencils of lines, which are mapped onto pencils of circles. The picture shows one such line (blue) and its inversion.
Republican lawmakers on Tuesday grilled FEMA Administrator Deanne Criswell on her agency's handling of hurricane recovery efforts, including about allegations of political bias. The two House ...
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. [2]
The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions g 1 , g 2 , g 3 , g 4 such that each g i is the inverse of f i .