Kristine Adams Un Official Fan Site Tribute

Kristine Adams : This Is An Un Official Fan Site Tribute
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Movie Title Year Distributor Notes Rev Formats Home Made Couples 20 2012 Homemade Media DRO Homegrown Video 768: The Hardon Riders of Porn 2009 Pure Play Media Facial DRO Kristine Adams Twins 2014 97% Amateurs MastOnly O Pregnant Kink.Com 7 2014 Evasive Angles DRO Stroke Suck and Tease 18 2012 Sticky Video BJOnly Facial DRO White Chick Black Dick 8 2012 AMVC Facial subspace of Rn+1 for which the coordinates sum to 0, and let F be the set of vectors in E of length v2 and which are integer vectors, i.e. have integer coordinates in Rn+1. Such a vector must have all but two coordinates equal to 0, one coordinate equal to 1, and one equal to –1, so there are n2 + n roots in all. One choice of simple roots expressed in the standard basis is: ai = ei – ei+1, for 1 = i = n. The reflection si through the hyperplane perpendicular to ai is the same as permutation of the adjacent i-th and (i + 1)-th coordinates. Such transpositions generate the full permutation group. For adjacent simple roots, si(ai+1) = ai+1 + ai = si+1(ai) = ai + ai+1, that is, reflection is equivalent to adding a multiple of 1; but reflection of a simple root perpendicular to a nonadjacent simple root leaves it unchanged, differing by a multiple of 0.
The An root lattice – that is, the lattice generated by the An roots – is most easily described as the set of integer vectors in Rn+1 whose components sum to zero. The A2 root lattice is the vertex arrangement of the triangular tiling. The A3 root lattice is known to crystallographers as the face-centered cubic (or cubic close packed) lattice.[30]. It is the vertex arrangement of the tetrahedral-octahedral honeycomb. The A3 root system (as well as the other rank-three root systems) may be modeled in the Zometool Construction set.[31] In general, the An root lattice is the vertex arrangement of the n-dimensional simplectic honeycomb. Bn Simple roots in B4 e1 e2 e3 e4 a1 1 -1 0 0 a2 0 1 -1 0 a3 0 0 1 -1 a4 0 0 0 1 Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-3.pngDyn2-node n3.pngDyn2-4b.pngDyn2-nodeg n4.png Let E = Rn, and let F consist of all integer vectors in E of length 1 or v2. The total number of roots is 2n2. One choice of simple roots is: ai = ei – ei+1, for 1 = i = n – 1 (the above choice of simple roots for An-1), and the shorter root an = en. The reflection sn through the hyperplane perpendicular to the short root an is of course simply negation of the nth coordinate. For the long simple root an-1, sn-1(an) = an + an-1, but for reflection perpendicular to the short root, sn(an-1) = an-1 + 2an, a difference by a multiple of 2 instead of 1. The Bn root lattice – that is, the lattice generated by the Bn roots – consists of all integer vectors. B1 is isomorphic to A1 via scaling by v2, and is therefore not a distinct root system. Cn Root system B3, C3, and A3=D3 as points within a cube and octahedron

Simple roots in C4 e1 e2 e3 e4 a1 1 -1 0 0 a2 0 1 -1 0 a3 0 0 1 -1 a4 0 0 0 2 Dyn2-nodeg n1.pngDyn2-3.pngDyn2-nodeg n2.pngDyn2-3.pngDyn2-nodeg n3.pngDyn2-4a.pngDyn2-node n4.png Let E = Rn, and let F consist of all integer vectors in E of length v2 together with all vectors of the form 2?, where ? is an integer vector of length 1. The total number of roots is 2n2. One choice of simple roots is: ai = ei – ei+1, for 1 = i = n – 1 (the above choice of simple roots for An-1), and the longer root an = 2en. The reflection sn(an-1) = an-1 + an, but sn-1(an) = an + 2an-1. The Cn root lattice – that is, the lattice generated by the Cn roots – consists of all integer vectors whose components sum to an even integer. C2 is isomorphic to B2 via scaling by v2 and a 45 degree rotation, and is therefore not a distinct root system. Dn Simple roots in D4 e1 e2 e3 e4 a1 1 -1 0 0 a2 0 1 -1 0 a3 0 0 1 -1 a4 0 0 1 1 DynkinD4 labeled.png Let E = Rn, and let F consist of all integer vectors in E of length v2. The total number of roots is 2n(n – 1). One choice of simple roots is: ai = ei – ei+1, for 1 = i < n - 1 (the above choice of simple roots for An-1) plus an = en + en-1. Reflection through the hyperplane perpendicular to an is the same as transposing and negating the adjacent n-th and (n – 1)-th coordinates. Any simple root and its reflection perpendicular to another simple root differ by a multiple of 0 or 1 of the second root, not by any greater multiple. The Dn root lattice – that is, the lattice generated by the Dn roots – consists of all integer vectors whose components sum to an even integer. This is the same as the Cn root lattice. The Dn roots are expressed as the vertices of a rectified n-orthoplex, Coxeter-Dynkin diagram: CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png...CDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png. The 2n(n-1) vertices exist in the middle of the edges of the n-orthoplex. D3 coincides with A3, and is therefore not a distinct root system. The 12 D3 root vectors are expressed as the vertices of CDel node.pngCDel split1.pngCDel nodes 11.png, a lower symmetry construction of the cuboctahedron. D4 has additional symmetry called triality. The 24 D4 root vectors are expressed as the vertices of CDel node.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png, a lower symmetry construction of the 24-cell. E6, E7, E8 E6Coxeter.svg 72 vertices of 122 represent the root vectors of E6 (Green nodes are doubled in this E6 Coxeter plane projection) E7Petrie.svg 126 vertices of 231 represent the root vectors of E7 E8 graph.svg 240 vertices of 421 represent the root vectors of E8 DynkinE6AltOrder.svg DynkinE7AltOrder.svg DynkinE8AltOrder.svg The E8 root system is any set of vectors in R8 that is congruent to the following set: {\displaystyle D_{8}\cup \left\{{\frac {1}{2}}\left(\sum _{i=1}^{8}\varepsilon _{i}\mathbf {e} _{i}\right):\varepsilon _{i}=\pm 1,\,\varepsilon _{1}\cdots \varepsilon _{8}=+1\right\}.}{\displaystyle D_{8}\cup \left\{{\frac {1}{2}}\left(\sum _{i=1}^{8}\varepsilon _{i}\mathbf {e} _{i}\right):\varepsilon _{i}=\pm 1,\,\varepsilon _{1}\cdots \varepsilon _{8}=+1\right\}.} The root system has 240 roots. The set just listed is the set of vectors of length v2 in the E8 root lattice, also known simply as the E8 lattice or G8. This is the set of points in R8 such that: all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed), and the sum of the eight coordinates is an even integer. Thus, {\displaystyle E_{8}=\left\{\alpha \in \mathbb {Z} ^{8}\cup (\mathbb {Z} +{\tfrac {1}{2}})^{8}:|\alpha |^{2}=\sum \alpha _{i}^{2}=1,\,\sum \alpha _{i}\in 2\mathbb {Z} .\right\}}{\displaystyle E_{8}=\left\{\alpha \in \mathbb {Z} ^{8}\cup (\mathbb {Z} +{\tfrac {1}{2}})^{8}:|\alpha |^{2}=\sum \alpha _{i}^{2}=1,\,\sum \alpha _{i}\in 2\mathbb {Z} .\right\}} The root system E7 is the set of vectors in E8 that are perpendicular to a fixed root in E8. The root system E7 has 126 roots. The root system E6 is not the set of vectors in E7 that are perpendicular to a fixed root in E7, indeed, one obtains D6 that way. However, E6 is the subsystem of E8 perpendicular to two suitably chosen roots of E8. The root system E6 has 72 roots. Simple roots in E8: even coordinates 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 1 0 -½ -½ -½ -½ -½ -½ -½ -½ An alternative description of the E8 lattice which is sometimes convenient is as the set G'8 of all points in R8 such that all the coordinates are integers and the sum of the coordinates is even, or all the coordinates are half-integers and the sum of the coordinates is odd. The lattices G8 and G'8 are isomorphic; one may pass from one to the other by changing the signs of any odd number of coordinates. The lattice G8 is sometimes called the even coordinate system for E8 while the lattice G'8 is called the odd coordinate system. One choice of simple roots for E8 in the even coordinate system with rows ordered by node order in the alternate (non-canonical) Dynkin diagrams (above) is: ai = ei – ei+1, for 1 = i = 6, and a7 = e7 + e6 (the above choice of simple roots for D7) along with {\displaystyle \mathbf {\alpha } _{8}=\mathbf {\beta } _{0}=-{\frac {1}{2}}\left(\sum _{i=1}^{8}e_{i}\right)=(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).}{\displaystyle \mathbf {\alpha } _{8}=\mathbf {\beta } _{0}=-{\frac {1}{2}}\left(\sum _{i=1}^{8}e_{i}\right)=(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2,-1/2).} Simple roots in E8: odd coordinates 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 -½ -½ -½ -½ -½ ½ ½ ½ One choice of simple roots for E8 in the odd coordinate system with rows ordered by node order in alternate (non-canonical) Dynkin diagrams (above) is: ai = ei – ei+1, for 1 = i = 7 (the above choice of simple roots for A7) along with a8 = ß5, where ßj = {\displaystyle \textstyle {\frac {1}{2}}(-\textstyle \sum _{i=1}^{j}e_{i}+\textstyle \sum _{i=j+1}^{8}e_{i}).}{\displaystyle \textstyle {\frac {1}{2}}(-\textstyle \sum _{i=1}^{j}e_{i}+\textstyle \sum _{i=j+1}^{8}e_{i}).} (Using ß3 would give an isomorphic result. Using ß1,7 or ß2,6 would simply give A8 or D8. As for ß4, its coordinates sum to 0, and the same is true for a1...7, so they span only the 7-dimensional subspace for which the coordinates sum to 0; in fact –2ß4 has coordinates (1,2,3,4,3,2,1) in the basis (ai).) Since perpendicularity to a1 means that the first two coordinates are equal, E7 is then the subset of E8 where the first two coordinates are equal, and similarly E6 is the subset of E8 where the first three coordinates are equal. This facilitates explicit definitions of E7 and E6 as: E7 = {a ? Z7 ? (Z+½)7 : ?ai2 + a12 = 2, ?ai + a1 ? 2Z}, E6 = {a ? Z6 ? (Z+½)6 : ?ai2 + 2a12 = 2, ?ai + 2a1 ? 2Z} Note that deleting a1 and then a2 gives sets of simple roots for E7 and E6. However, these sets of simple roots are in different E7 and E6 subspaces of E8 than the ones written above, since they are not orthogonal to a1 or a2. F4 Simple roots in F4 e1 e2 e3 e4 a1 1 -1 0 0 a2 0 1 -1 0 a3 0 0 1 0 a4 -½ -½ -½ -½ Dyn2-node n1.pngDyn2-3.pngDyn2-node n2.pngDyn2-4b.pngDyn2-nodeg n3.pngDyn2-3.pngDyn2-nodeg n4.png 48-root vectors of F4, defined by vertices of the 24-cell and its dual, viewed in the Coxeter plane For F4, let E = R4, and let F denote the set of vectors a of length 1 or v2 such that the coordinates of 2a are all integers and are either all even or all odd. There are 48 roots in this system. One choice of simple roots is: the choice of simple roots given above for B3, plus a4 = – {\displaystyle \textstyle {\frac {1}{2}}\sum _{i=1}^{4}e_{i}}\textstyle {\frac {1}{2}}\sum _{i=1}^{4}e_{i}. The F4 root lattice – that is, the lattice generated by the F4 root system – is the set of points in R4 such that either all the coordinates are integers or all the coordinates are half-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions. G2 Simple roots in G2 e1 e2 e3 a1 1 -1 0 ß -1 2 -1 Dyn2-nodeg n1.pngDyn2-6a.pngDyn2-node n2.png The root system G2 has 12 roots, which form the vertices of a hexagram. See the picture above. One choice of simple roots is: (a1, ß = a2 – a1) where ai = ei – ei+1 for i = 1, 2 is the above choice of simple roots for A2. The G2 root lattice – that is, the lattice generated by the G2 roots – is the same as the A2 root lattice. The root poset Hasse diagram of E6 root poset with edge labels identifying added simple root position The set of positive roots is naturally ordered by saying that {\displaystyle \alpha \leq \beta }\alpha \leq \beta if and only if {\displaystyle \beta -\alpha }\beta -\alpha is a nonnegative linear combination of simple roots. This poset is graded by {\displaystyle \deg {\big (}\sum _{\alpha \in \Delta }\lambda _{\alpha }\alpha {\big )}=\sum _{\alpha \in \Delta }\lambda _{\alpha }}{\displaystyle \deg {\big (}\sum _{\alpha \in \Delta }\lambda _{\alpha }\alpha {\big )}=\sum _{\alpha \in \Delta }\lambda _{\alpha }}, and has many remarkable combinatorial properties, one of them being that one can determine the degrees of the fundamental invariants of the corresponding Weyl group from this poset.[32] The Hasse graph is a visualization of the ordering of the root poset

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