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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 previous formula arises in the special case of the constant function α(n) = 1, whose Dirichlet inverse is α −1 (n) = μ(n). A particular application of the first of these extensions arises if we have (complex-valued) functions f ( n ) and g ( n ) defined on the positive integers, with
In August 2020, the U.S. Food and Drug Administration (FDA) became aware of nitrosamine impurities in certain samples of rifampin. [62] The FDA and manufacturers are investigating the origin of these impurities in rifampin, and the agency is developing testing methods for regulators and industry to detect the 1-methyl-4-nitrosopiperazine (MNP ...
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In terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion–exclusion principle.Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem.
Möbius was born in Schulpforta, Electorate of Saxony, and was descended on his mother's side from religious reformer Martin Luther. [2] He was home-schooled until he was 13, when he attended the college in Schulpforta in 1803, and studied there, graduating in 1809.
In mathematical physics, the primon gas or Riemann gas [1] discovered by Bernard Julia [2] is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee-Yang theorem.
Rifampin rapidly kills fast-dividing bacilli strains as well as "persisters" cells, which remain biologically inactive for long periods of time that allow them to evade antibiotic activity. [7] In addition, rifabutin and rifapentine have both been used against tuberculosis acquired in HIV-positive patients.