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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
[9] [10] [11] A path along the edge of a Möbius strip, traced until it returns to its starting point on the edge, includes all boundary points of the Möbius strip in a single continuous curve. For a Möbius strip formed by gluing and twisting a rectangle, it has twice the length of the centerline of the strip.
The Möbius–Kantor configuration. In geometry, the Möbius–Kantor configuration is a configuration consisting of eight points and eight lines, with three points on each line and three lines through each point.
A knot is created by beginning with a one-dimensional line segment, wrapping it around itself arbitrarily, and then fusing its two free ends together to form a closed loop. [3] Mathematically, we can say a knot K {\displaystyle K} is an injective and continuous function K : [ 0 , 1 ] → R 3 {\displaystyle K\colon [0,1]\to \mathbb {R} ^{3 ...
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.
In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity.It is also called the inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry.
The Möbius–Kantor graph is a subgraph of the four-dimensional hypercube graph, formed by removing eight edges from the hypercube. [1] Since the hypercube is a unit distance graph, the Möbius–Kantor graph can also be drawn in the plane with all edges unit length, although such a drawing will necessarily have some pairs of crossing edges.
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