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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 ...
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 ...
Examples of this trope include Martin Gardner ' s "No-Sided Professor" (1946), Armin Joseph Deutsch ' s "A Subway Named Mobius" (1950) and the film Moebius (1996) based on it. An entire world shaped like a Möbius strip is the setting of Arthur C. Clarke 's "The Wall of Darkness" (1946), while conventional Möbius strips are used as clever ...
In this example the fact that the Legendre identity is derived from the Sieve of Eratosthenes is clear: the first term is the number of integers below X, the second term removes the multiples of all primes, the third term adds back the multiples of two primes (which were miscounted by being "crossed out twice") but also adds back the multiples ...
More mysteries surround the Möbius strip. For example, this conjecture only proves the minimum size of a paper Möbius with one twist—not three or more. For such a simple shape, it’s amazing ...
Caroline Fenton, Jason Fitz & Adam Breneman break down Oregon's win vs. Wisconsin, the Georgia Bulldogs' takedown of the Tennessee Volunteers and more.
St. Helens High School principal Katy Wagner was charged with two counts of felony criminal mistreatment after two teachers under her supervision were arrested for a nearly 10-year sex crime spree.