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This is a general situation in order theory: A given order can be inverted by just exchanging its direction, pictorially flipping the Hasse diagram top-down. This yields the so-called dual, inverse, or opposite order. Every order theoretic definition has its dual: it is the notion one obtains by applying the definition to the inverse order.
This last example shows that a set that is intuitively "nearly sorted" can still have a quadratic number of inversions. The inversion number is the number of crossings in the arrow diagram of the permutation, [ 6 ] the permutation's Kendall tau distance from the identity permutation, and the sum of each of the inversion related vectors defined ...
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the density of the Earth from measurements of its gravity field. It is called an inverse problem because ...
Examples of inversion of circles A to J with respect to the red circle at O. Circles A to F, which pass through O, map to straight lines. Circles G to J, which do not, map to other circles. The reference circle and line L map to themselves. Circles intersect their inverses, if any, on the reference circle.
As noted above, the inverse with respect to a circle of a curve of degree n has degree at most 2n.The degree is exactly 2n unless the original curve passes through the point of inversion or it is circular, meaning that it contains the circular points, (1, ±i, 0), when considered as a curve in the complex projective plane.
Many specific examples are given in the article on multiplicative functions. The theorem follows because ∗ is (commutative and) associative, and 1 ∗ μ = ε , where ε is the identity function for the Dirichlet convolution, taking values ε (1) = 1 , ε ( n ) = 0 for all n > 1 .