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ggplot2 is an open-source data visualization package for the statistical programming language R.Created by Hadley Wickham in 2005, ggplot2 is an implementation of Leland Wilkinson's Grammar of Graphics—a general scheme for data visualization which breaks up graphs into semantic components such as scales and layers. ggplot2 can serve as a replacement for the base graphics in R and contains a ...
A cut or split is trivial when one of its two sides has only one vertex in it; every trivial cut is a split. A graph is said to be prime (with respect to splits) if it has no nontrivial splits. [2] Two splits are said to cross if each side of one split has a non-empty intersection with each side of the other split.
If a graph is both a split graph and an interval graph, then its complement is both a split graph and a comparability graph, and vice versa. The split comparability graphs, and therefore also the split interval graphs, can be characterized in terms of a set of three forbidden induced subgraphs. [7] The split cographs are exactly the threshold ...
Split extensions are very easy to classify, because an extension is split if and only if the group G is a semidirect product of K and H. Semidirect products themselves are easy to classify, because they are in one-to-one correspondence with homomorphisms from H → Aut ( K ) {\displaystyle H\to \operatorname {Aut} (K)} , where Aut( K ) is ...
It can be used for line or line-segment clipping against a rectangular window, as well as against a convex polygon. The algorithm is based on classifying a vertex of the clipping window against a half-space given by a line p: ax + by + c = 0. The result of the classification determines the edges intersected by the line p. The algorithm is ...
For a left split sequence, the map t × r: B → A × C gives an isomorphism, so B is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection C → A × C gives an injection C → B splitting r (2.).
In particular, the multiplicative group G m is the group GL(1), and so its group G m (k) of k-rational points is the group k* of nonzero elements of k under multiplication. Another reductive group is the special linear group SL ( n ) over a field k , the subgroup of matrices with determinant 1.
The symmetry group of a cube is the internal direct product of the subgroup of rotations and the two-element group {−I, I}, where I is the identity element and −I is the point reflection through the center of the cube. A similar fact holds true for the symmetry group of an icosahedron. Let n be odd, and let D 4n be the dihedral group of ...