Maximal chain

An alternative description of the cover algorithm is as follows consider maximal chains of nodes with... 

Proof. By Corollary 2, for all maximal chains of irreducible closed subvarieties ... 

Each closed subset V (p)) is a copy of z, but they have all been pasted together here. The whole set-up is called a surface for 2 reasons 1) all maximal chains of irreducible proper closed subsets have length 2, just as in A. 2) If O is the local ring at a closed point x, then O has Krull dimension 2. In fact, if x = [(p, /)], then its maximal ideal is generated by p and /, and there is no single element g O such that m — / g). 

Thus we see that A Bn) is a triangulation of the n-dimensional imit cube. It is perhaps worth noting that this triangulation does not introduce new vertices, and that the number of maximal simplices is equal to n. In fact, they are naturally indexed by the permutations of n, as are the maximal chains of See the left and the middle parts of Figure 10.16. 

In the poset. F(Z (iT i)) U 0 this acyclic matching can be extended to have only the top-dimensional critical element, since the other one is matched with 0. When considered in Si, these maximal chains consist of n—2 elements hence they correspond to critical simplices of dimension n - 3 in A n ). Case 2. Let S = for tt (1)(2)... (n). The matching rule in this... 

When an acyclic category is shown to be shellable, and aU of its maximal chains have the same number of morphisms, the number of spheres can be found by computing the Mobius function. The latter can often be done by some explicit combinatorial counting procedure. 

Theorem 12.5 can frequently be applied in this situation. When the acyclic category C is graded, one can take any rank selection R. Clearly, the nerve of R is an induced subcomplex of the nerve of C. Furthermore, one can see that any maximal chain of C will intersect f in a maximal chain hence the conditions of Theorem 12.5 are satisfied. We can therefore conclude the following proposition. 

Naturally, in order to be able to decide imiquely which of the simplices we should take first, we require that no two maximal simplices receive the same string of labels. Observe that this actually implies that in every interval the maximal chains can be lexicographically ordered as well, and no two chains will receive the same string of labels. 

There is almost no difference between considering pure posets (i.e., posets in which maximal chains all have the same length) and the nonpure ones, so we will not make any distinction. However, there is one additional condition, which we always require to be satisfied, whenever we are labeling a nonpure poset. 

Remark 12.7. The additional condition for the nonpure poset is needed in order to make sure that we avoid the following peculiar situation. It may happen that there are two maximal chains c and d differing from each other... 

For any interval x,t, any maximal chain c in [x,t] such that c is not lexicographically least in there exist elements y,q,z c such that... 

In the shelling order induced by a LEX-labeling, the mediocre maximal chains correspond to the spanning simplices, and can therefore be used to deliver a basis for the cohomology H A P) Z). 

Let P be a lexshellable poset, and let A be a LEX-labehng of edges of P. We will prove that the lexicographic ordering of the maximal chains in P gives a shelling order on the maximal simplices of A P). 

The main difference in the acyclic category case is that the notion of the interval is replaced with the notion of the morphism. Accordingly, a maximal chain in the interval is replaced with the composable sequence of indecomposable morphisms, which together compose to yield the corresponding morphism. To abbreviate our language, we say that a maximal composable sequence of morphisms (mi,..., nik) is in m if it composes to m. 

Change the circular order of substituents about chain atoms to maximize chain lengths. 

The fact that there are a finite number of non-wandering trajectories in Morse-Smale systems implies that any chain has a finite length which does not exceed the total number of non-wandering trajectories. Moreover, a maximal chain can end only at a stable equilibrium state or a periodic orbit. 

Hence, a plot of the inverse number average degree of polymerization, Pn against the rate of polymerization Rp, - the rate of polymerization can be easily varied by the concentration of the initiator - yields the monomer chain transfer constant Cm as intercept of a linear plot. The value of Cm constitutes a limit for the maximum number average degree of polymerization, P , as transfer to monomer cannot be avoided. Methyl methacrylate, for instance, has a monomer chain transfer constant of about Cm = 5 x 10 at 60 °C, leading to a maximal chain length of about 20 000, whereas in a radical polymerization of vinyl acetate with Cm = 2 x 10 " at 60 °C, the limit is already reached at 5000. 

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