The Main Theorem

Let v be an element in xV. Then, by definition, there exists an element K in /C such that v 6 x(K). Thus, by induction, Xv x(K) = xU k) This shows that X extends x.x- 

In this section, we look at spherical Coxeter sets containing at least three elements none of them thin. We shall apply Proposition 12.2.7 in order to prove that closed subsets of S generated by such Coxeter sets are faithfully embedded in S. We also establish the corresponding recognition theorem. 

Let us denote by G the Schur group of L) with respect to y. Then Gyz acts transitively on zl. 

Let v and w be elements in zl. We have to find an element in Gyz with vg = w. Without loss of generality we may assume that s = l(L  

Let us denote by x the map on xV U -c which is the identity on xV and maps v to w. Then x is faithful. Thus, as L is assumed to have no thin element, we obtain from Proposition 12.2.7 an element g in G which is the identity on xV and maps v to w. 

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The main theorem. In order to combine the results for H and into one expression we recall that... 

In the sequel we will see that the proposition holds for an arbitrary permutation group and we will refer to it as the theorem or the main theorem. 

The main theorem, stated in Sec. 16 and proved in Sec. 19, combined with the proposition of Sec. 25 yields the following proposition Tvfo permutation groups are combinatorially equivalent if and only if they have the same cycle index. 

Making use of the relationship discussed above, "the number of noncongruent planted trees equals the number of nonequivalent configurations of three planted trees", of the generating function and the main theorem of Chapter 1 (Sec. 16) and taking the special case n 0 into account, we establish for each of the three situations an equation ... 

In the last four formulas, n stands for the number of vertices of degree 4 the number of vertices of this tree is 3 + 2. For odd , both sides of the five formulas may be assigned the value 0. With this agreement they hold for all , = , 2, 3,.... Relation (2.47) is derived by means of the main theorem of Chapter 1, applied to the special case of... 

We note that chemical substitution of a radical into a basic compound corresponds (in the sense of the main theorem of Chapter 1) to the algebraic substitution of the generating function into the cycle index of the group of the basic compound. 

In detail, we find the series (3.4) according to the main theorem of Chapter 1 by setting... 

The example demonstrates that the concepts in chemistry rely heavily on notions from group theory, specifically the concept, introduced in Sec. 11, of the equivalence of configurations with respect to a permutation group. The cycle index and the main theorem of Sec. 16 play a role. 

A stationary scheme. The main theorem on the convergence of iterations. Quite often, the iteration schemes such as... 

C. Definitions of subduction, induction, and regular induction are important, as is the main theorem on induction embodied in Eq. (34), but not its proof. 

The following two theorems are the main theorems about semisimple associative rings with 1. They are due to Emil Artin cf. [2]. Less general versions have been given earlier by Joseph Wedderburn cf. [39 Theorem 10] and [39 Theorem 17]. 

We return now to the discussion of Section 3. A criterion for the stability or instability of the rest point had been obtained in the form (3.4). If the rest point is locally asymptotically stable, it is possible that there could still be limit cycles surrounding it. The following arguments show that this cannot happen. These arguments are very detailed and very tedious the reader who is not interested in the technique might be well advised to skip to the statement of the main theorem. 

Armed with the preceding discussion, we can now state the main theorem of this chapter. It shows that, in contrast to the basic chemostat, coexistence can occur if the competition is at a higher trophic level. (We remind the reader of the general assumption of hyperbolicity of limit cycles.)... 

In this appendix, basic theorems on differential inequalities are stated and interpreted. The main theorem is usually attributed to Kamke [Ka] but the work of Muller [Mii] is prior. A more general version due to Burton and Whyburn [BWh] is also needed. We follow the presentation in Coppel [Co, p. 27] and Smith [S2 S6j. The nonnegative cone in R", denoted by R , is the set of all n-tuples with nonnegative coordinates. One can define a partial order on R" by < x if x—R". Less formally, this is true if and only if < x, for ail i. We write x < if x, < )>/ for all i. The same notation will be used for matrices with a similar meaning. 

Before stating the main theorem of this section, one must decide how large the tuples g of generators are for which someone tries to find exp -collisions. For pL = 2, the proof is simple (see Case 1 below) and was known in [B0CP88]. 

We can now formulate the main theorem which allows the construc-... 

It is harder to prove the main theorem of elimination theory ... 

This induces an -module structure in v - The reader can check that the compatibility demanded between fa and fa over Ui fl Uj is exactly what is needed to insure that the 2 -module structures that we get on u.ru. are the same. The main theorem in this direction is that every locally free ox-module arises as the sheaf of sections of a unique vector bundle (up to isomorphism). 

Finally, forms (IV.) and (V.) of the Main Theorem are even deeper. (V.) is a much more global statement since the properness of / is involved. There is a cohomological proof, due to Grothendieck (cf. EGA, Ch. Ill, 4.3) and a proof using a combination of projective techniques and completions, due to Zariski. As... 

Now we come to the main theorem of this Appendix concerning the representation of scalar, vector, and tensor linear isotropic functions of vectors and tensors (scalars as independent variables play the role of parameters). 

After integral transformation of equation (4.42), taking into account equation (4.44) at T = To, according to the main theorem we obtain ... 

The first step in the theory is to provide a canonical decomposition of a KMS state into its extremal KMS components. Rather than stating the main theorem first, let us try to gain some feeling on how things should (and do ) go in a simplified case. 

When the considered acyclic categories have terminal and/or initial objects, there exist further subdivision results that are sometimes useful. Both results are corollaries of the main Theorem 10.21. 

The Main Theorem of Discrete Morse Theory for CW Complexes... 

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