Exact sequences, representations

Proof. The kernel of a morphism of group schemes is representable. Consider the morphism of S-schemes G Ker( ). For any closed point s e S the morphism Gg Ker(V )s is flat as it is a surjection of group schemes over a field. Hence we can apply the criterion of flatness per fibre (EGA IV 11.3.11) to see that the morphism G -+ Ker( ) is flat and that Ker( ) is flat over S, Thus ip is admissible and as Im(< ) = Ker( ) follows from the above, the sequence is exact and (p is admissible also, ... 

The latter is involved, for example, in the reduction of aromatics to their dihydro derivatives, of quinones to hydroquinones, carbonyls to alcohols. A priori, there are six plausible different possibilities of transferring two electrons and two protons, depending on the exact order of the steps. These six paths involve the articipation of seven different intermediates A , A , AH", AH , AH ", AH2 ,and AH. It is thus seen that the discussion of the possible mechanism(s) may rapidly become an overwhelming task, particularly when more steps are involved, as in the 4e -f- 4H " sequence in Eq. (126). Thus in order to discuss the exact route followed, a convenient representation of the different possible schemes is highly desirable. Such a representation of all possible paths is given by square-schemes diagrams [101], such as that shown in Scheme 8 for the A/AH2 reduction. The horizontal... 

Fig. 8.1 Schematic representation of the biogeochemical zonation in marine sediments. The names of the main zones were proposed by Froelich et al. (1979) and Berner (1981, in parenthesis). The depth scale is quasi-logarithmic die exact depths, however, vary strongly and increase from the shelf to the deep sea. The pore water chemistry shows relevant dissolved species. Peak heights and concentration scales are arbitrary. The chemical profiles reflect the depdi sequence of the dominant mineralization processes through which organic matter is oxidized to CO, (Modified from Froelich et al. 1979).

A binary tree is a hierarchic structure with every element having exactly no, one, or two immediate successors. When another structure tree, forest. List) is given a binary tree representation, some of the linked memory allocation needed to store the data it holds becomes available in a form that is more sequential. This is related to a descriptive term about binary trees, the notion of completeness. A binary tree is said to be complete when all the nodes that are present at a level are in sequence, beginning at the left side of the structure, with no gaps. 

Fig. 1. Sequence distribution in a copolymer containing 500 monomer units, of which 300 are of type A and 200 of type B, distributed completely at random (rjr2 = 1). The total number of sequences is 240. Owing to the finite number of units necessarily chosen for the representation, some figures have been rounded. The exact number of sequences of various lengths is given by 48 0.6" for monomer A, and 72 0.4"" for monomer B, n being the number of units contained in the sequence...

---