Proof of Corollary

Cutting Q along this torus and applying this diflFeomorphism, we obviously make coincident the two above-mentioned coordinate systems. Continuing this process successively, we extend this unique coordinate system onto all other bricks . 

The process stops at the moment when the surface Q under surgery becomes a direct product xS, It is clear that under this procedure the integral / changes and becomes a certain smooth function on the direct product, which is constant on the multiplier and Corollary 2.1.5 follows. 

10 Topological Obstacles for Smooth IntegrahUity and Graphlike Manifolds. Not 

Each Three-Dimensional Manifold Can Be Realized as a Constant-Energy Manifold of an Integrahle System 

Now we will prove Corollary 2.1.4. We will use the results of Waldhausen presented in [184]. This work was first kindly mentioned to me by Zieschang. Waldhausen examined a special class of three-dimensional manifolds W which he called graphlike manifolds (Graphenmannigfaltigkeiten). They are defined in the following way. In a manifold W there must exist a family T of non-intersecting tori T, discarding which we obtain a manifold each of whose connected components is fibred with circle 5 as typical fibre over a two-dimensional manifold (maybe, with a boundary). 

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Proof. One can follow exactly the inductive proof of Corollary 4 above, using k [Xo,..., Xn] instead of the affine coordinate ring. In case Y = 0, the last step is slightly different. By induction, we have f, ..., fr such that... 

The statement of the proposition follows easily from these properties, the construction of the functor O and the definition of the class B given in the proof of Corollary 2.18. 

We will not detail further the first point since it suffices to have a look at the proof of Corollary 8.7 to realize that one can select a subsequence having the desired property. We turn instead to the second issue and we will discuss it in an informal way we develop the idea set forth in [Giacomin... 

This corollary is the starting point for most of our proofs of undecidability. The general strategy will be to show that a property P of schemes is not partially decidable by finding an algorithm to transform each scheme S into a two-tape oneway deterministic finite state acceptor M(S) such that L (M(S)) O D = if and... 

This completes the proof of the second lemma. A corollary is that for a time-independent solution either all components are nonnegative, or all non-positive. For a stationary probability distribution one has, of course, psn 0, because C— 1. 

We keep the notation of the proof of (41). The hypothesis of the corollary implies that the surjective homomorphism p. PAM is already given. Hence the flatness of PA/ is equivalent to the existence of the surjection q pA - R, and this is another way to formulate what has to be proved. 

Proof. From Corollary 3.5.4(11) we know that, for each closed subset T of (L), there exists a subset K of L such that (K) = T. This shows that the map in question is surjective. 

The goal of this section is the proof of Theorem 11.4.6. In this theorem, we focus on faithful maps from certain subsets Y of X to X which extend faithfully to a subset of X containing Y and one additional element. The main idea of the proof of this theorem is the use of Corollary 11.4.3 in the proof of Proposition 11.4.5. 

Proof. By Corollary 11.4.9, A -compatibility is reflexive. That iF-compatibi-lity is symmetric follows immediately from the definition of iF-compatibility. Let us prove transitivity. 

We hope the reader will appreciate the elegance and simplicity of the arguments supporting Theorem 3.2, which are based on the LaSalle corollary. In particular, a linearized stability analysis about each of the rest points of (3.3), required in Chapter 1, was completely avoided. A careful reading of the proof of Theorem 3.2 reveals that assumption (iii) on f is not crucial to the proof we will have more to say about this later. Finally, it should be noted that the assumption (iv) on f can be relaxed somewhat. It can be weakened to requiring only that f be locally Lipschitz continuous... 

The reader will have noticed that the Liapunov function used in the proof of the theorem was not obvious on either biological or mathematical grounds. Its discovery by Hsu greatly simplified and extended earlier arguments given in [HHW]. This is typical of applications of the LaSalle corollary. Considerable ingenuity, intuition, and perhaps luck are required to find a Liapunov function. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

A subgroup H is maximal if there are no intermediate subgroups between H and G in the branching scheme of G. A proof of this corollary is presented in the Appendix. 

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

On first acquaintance, [Del appears to offer a neat way to cut through the complexity—a direct abstract proof of the existence of / , with indications about how to derive the concrete special situations (which, after all, motivate and enliven the abstract formalism). Such an impression is bolstered by Verdier s paper [V j. Verdier gives a reasonably short proof of the flat base change theorem, sketches some corollaries (for example, the flnite tor-dimension case is treated in half a page [ibid., p. 396], as is the smooth case [ibid., pp. 397-398]), and states in conclusion that all the results of [H], except the theory of dualizing and residual complexes, are easy consequences of the existence theorem. In short, Verdier s concise summary of the main features, together with some background from [H] and a little patience, should suffice for most users of the duality machine. 

Proof. — It was shown in [18, Corollary 2.7] that the triple (W, G, F,) defines a closed model structure on K ShjfT) in the sense of Quillen. The proof of existence of factorizations given in [18] shows that they are functorial and therefore the stronger axioms which we use are satisfied. 

Proof. The fact that the second condition implies the first is a particular case of Corollary 2.25. To show that the first one implies the second it is sufficient in view of I. mma 2.8 to verify that if we have a morphism i in C D Wa then the morphism Hom, Hom, % , ) is a trivial fibration. This follows from Lemma... 

More generally, it may be shown for the three-dimenrional case that whenever and differ in any one transformation property, then the overlap between them vanishes. Thus if quircs the factor under a rotation through a = 2x/5 radians and is even with respect to inversion (f.c., it is of efg type), and if is multiplied by e " under the rotation but may also be even (of species e g), then those two orbits do not overlap at all. The proof of this theorem requires a more acting notation, but essentially no diff ent arguments from those which we used in the one-dimensional case above. As an important corollary to this result, it follows that unless the transformation properties of two orbitals are identical, these are not suitable for the formation of electron-pair bonds. This limits the posribilities of binding very considerably and eases the subsequent discusrion. 

From Theorem 2.2.3 we immediately obtain an important corollary, namely, the proof of Claim 2.1.2 concerning the possibility to realise any three-dimensional manifold of class (Q) (that is, gluing an arbitrary number of elementary manifolds of types I, II, III) in the form of an isoenergy surface of some integrable Hamiltonian system on an appropriate symplectic manifold M. In other words, from Theorem 2.2.3 there immediately follows the equality (Q) = (H). 

The proof of Theorem 5.4.1 is carried out by direct calculations using the formulae for the Poisson bracket of functions in the canonical symplectic structure of the space T U. This theorem suggests an interesting corollary. 

The proof of Hurewicz theorem is outside the scope of this book. Instead, we formulate some useful corollaries for future reference. For both statements, assume that X is a CW complex, and that A and B are CW subcomplexes, such that X = A U B. 

The most important corollary of Theorem 2 is this no nuclear configuration can represent a transition state of a given reaction if the rotation about the third or a higher odd-order axes transforms the reactants into the products. Without attending to the proof of this statement, we illustrate it with some examples. 

For a proof of Theorem A.2 see [Bingham et al. (1987), Theorem 1.7.1 and Corollary 8.1.7]. We stress however that the two theorems to which we have just referred, called Tauberian theorem, give also the converse implications and Tauberian precisely refers to the converse statements. Theorem A.2 just gives the asymptotic behavior of the Laplace transform, given the asymptotic behavior of the function and it is appropriate to refer to it as an Abelian theorem. 

A corollary is that a demonstration of the Schild equation holding over a small range of concentrations should not be taken as proof that the action of an antagonist is competitive. Clearly, as wide as practicable a range of antagonist concentrations should be tested, especially if there is evidence for the presence of spare receptors. 

Each chapter is numbered separately and theorems, lemmas, propositions and corollaries are numbered consecutively within each chapter thus Theorem 6.7 is the 7th numbered result, but not necessarily the 7th theorem, in Chapter VI. The end of a proof is indicated by the symbol ... 

Corollary 1.3. The stack S(g,p) is a separated algebraic stack of finite type over Spec(Z). Proof. We know that Ag is a separated algebraic stack of finite type over Spec(Z) (see for example [FC] I 4.11). However, it is easy to see that any stack which allows a representable morphism to an algebraic stack is an algebraic stack itself. Hence S(g,p) is an algebraic stack by Proposition 1.2. In the same manner we see that S(g>p) is separated and of finite type over Spec(Z). O... 

Proof. This is an immediate consequence of Lemma 9.5.1 and Corollary 8.6.5. 

The statement formulated here as Corollary 14.2 was first conjectured by Cyvin and the present author [104]. It played a significant role in the construction and enumeration of normal benzenoids [3, 104]. Its formal proof is given in [105]. 

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