Main Theorem

Theorem 8.1 Assume that G has some fixed bonds. Then there is an edge cut R of type 1 or a standard combination Rp Rg of edge cuts of type 2 in G, satisfying — 

such that e is an M-double bond. If e is an M-double bond too, then by Lemma 8.1 all the edges e, ., e are fixed single bonds, where e is on C or C . Thus e, e, . , e is an edge cut consisting of fixed single bonds. If e is an M-dngle bond, then e is an M-double bond. We consider the following two subcases. 

Case 2. Edge e is a fixed double bond. This case can be dealt with as Subcase 1.1. 

We have proved that a fixed single bond on one of the perimeters (C or C ) determines an edge cut consisting of fixed single bonds including the given fixed single bond. 

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Haupt-satz, m. fundamental principle or law main theorem axiom, -schalter, m. main (or master) switch, -schlagader, /. main artery aorta, -schluss, m. (Elec.) main circuit. -schliissel, m. master key. -schnitt, m. principal section, -schwlngung, /. principal vibration, -serie, /. principal series, -tlcherung,/. (Elec.) main fuse, main cutout. 

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. 

Polya s paper, translated here for the first time, was a landmark in the history of combinatorial analysis. It presented to mathematicians a unified technique for solving a wide class of combinatorial problems " a technique which is summarized in Polya s main theorem, the "Hauptsatz" of Section 16 of his paper, which will here be referred to as "Polya s Theorem". This theorem can be explained and expounded in many different ways, and at many different levels, ranging from the down-to-earth to highly abstract. It will be convenient for future reference to review the essentials of Polya s Theorem, and to this end I offer the following, rather mundane, way of looking at the type of problem to which the theorem applies and the way that it provides a solution. 

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

As XW is smooth, p is an isomorphism over U(i, 2)(X) by Zariski s main theorem [Hartshorne (2), V. 5.2]. Now we show (1). As p is an isomorphism over Ai3i Z(lt2)(X), it is enough to prove the smoothness at the points of E = ni(E). Le Barz has given analytic local coordinates around any point e E and so proved the smoothness of H3(X). To simplify notations we will assume that the dimension of A is 3. The argument for general dimension d is completely analogous, only more difficult to write down. Now let... 

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. 

To prove our main theorem, establishing that the existence of a solution of the spectral optimization problem is equivalent to the existence of a solution of the... 

The proof of Glauberman s theorem requires Richard Brauer s Second Main Theorem on modular representations of finite groups cf. [5]. A proof of Glauberman s theorem is beyond the scope of this monograph. 

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-... 

N.B. Au lieu d invoquer 8.5 on peut aussi invoquer le "Main Theorem" de Zariski, qui implique directement que u est tme immersion ouVerte, done un isomorphisme. ... 

L EMain Theorem" appliquE au morphisme birationnel y X —> G, compte tenu que G est normal. L Equivalence de ces conditions avec (iv) rEsuite aussitftt de la derniEre assertion de 2.1 caractErisant 1 ensemble des Elements de X en lesquels y est Etale ... 

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