The Extension Theorem

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. 

Theorem 11.4.6 plays a crucial role in the proof of Lemma 12.3.1. 

Let us denote by R the set of all elements in (L) which do not satisfy the equation in question. By way of contradiction, we assume that R is not empty. We pick an element r in R, and we do this in such a way that r) is as small as possible. 

Proposition 11.4.4 Let x be an element inD(y,z). Then, each faithful map X from y U zV to X extends faithfully to a bijective map from x,y JzV 

Let p (respectively q) denote the uniquely determined element in (L) which satisfies x G yp (respectively z xq). Then z G ypq. Thus, there exists an element s in pq such that z G ys. 

Now the claim follows from the fact that K has been chosen arbitrarily in AS. 

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Mathematically equation (A2.1.25) is the direct result of the statement that U is homogeneous and of first degree in the extensive properties S, V and n.. It follows, from a theorem of Euler, that... 

The use of Polya s Theorem in a specialized context such as the above, has led to the extension of the theorem along certain useful lines. One such derivation pertains to the situation where the boxes are not all filled from the same store of figures. More specifically, the boxes are partitioned into a number of subsets, and there is a store of figures peculiar to each subset. To make sense of this we must assume that no two boxes in different subsets are in the same orbit of the group in question. A simple extension of Polya s Theorem enables us to tackle problems of this type. Instead of the cycle index being a function of a single family of variables, the 5j, we have other families of variables, one for each subset. An example from chemical enumeration will make this clear. 

It must be observed that the formal proof of the theorem depends on the possibility of returning to the initial state along at least one path such as (). The extension of the theorem to vital processes, phosphorescence, and radioactive changes, which have not yet been reversed, must therefore be regarded as inductive, although highly probable. 

The extensive thermodynamic variables are homogeneous functions of degree one in the number of moles, and Euler s theorem can be used to relate the composition derivatives of these variables. 

In fact, the Valley theorem is a simple extension of the Cusp theorem. However, the Cusp theorem provides only a local information (for r=0), while the Valley theorem... 

G.Berthier, "The three theorems of the Hartree-Fock theory and their extensions ", pp. 91-102 in the same reference as 3. 

Another remarkable point is the appearance in [Q(t0)Yfirst time when n = 4 (we cannot have two 6LW) with no particle in common if we do not have at least four particles), but also exist to higher orders in the concentration. Their evaluation necessitates some delicate mathematical manipulations (application of the factorization theorem) but the extension of this technique to the higher-order terms of the virial expansion does not seem to pose any new problem. 

For a fluid flow, of course, one uses the Reynolds transport theorem to establish the relationship between a system (where the momentum balance applies directly) and a control volume (through which fluid flows). In terms of Eq. 3.2, the extensive variable N is the momentum vector P = mV and the intensive variable tj is the velocity vector V. Thus the fundamental approach yields the following vector equation... 

Thus while the Jahn-Teller theorem is generally invoked to account for the distorted octahedral geometries of the copper(II) ion, in terms of the first-order vibronic coupling the extension of the coupling to some second-order effects also allows some rationalization of the tetrahedral and five-coordinate geometries of the copper(II) ion.1063-1067... 

As the Gibbs energy is a first-order homogenous function of the extensive variables A7 and n. the application of Euler s theorem yields... 

In the present case, the extension of the scalar counting rules for n(S) to symmetry theorems is straightforwardly achieved by replacing n(S), n(v), n(e) n( > ) by the permutation representations F,T(S), r,T(v), r,T(e) and ra(vj) [13]. A permutation representation /j/v), of a set of objects x has character y(R) under operation (R) of the symmetry group of the undistorted framework, where x(R) is equal to the number of objects wnshifted under operation R. The subscript a is often dropped if there is no danger of confusion. With these replacements, equation (4) becomes... 

The goal of the last of the four sections of this chapter is the proof of a specific extension theorem (Theorem 11.4.6) for Coxeter sets. 

In the first section of this chapter, we focus on specific characteristics of spherical Coxeter sets such as maximal elements and conjugation. The second section is devoted to an extension theorem for spherical Coxeter sets. Our approach to this theorem (which follows the line of [46]) is partially inspired by a geometrical reasoning provided by Jacques Tits in [37],... 

As an extension of Noether s theorem to quantum mechanics, the hypervirial theorem [101] derives conservation laws from invariant transformations of the theory. Consider a unitary transformation of the Schrodinger equation, U(H — F)T = U(H — = 0, and assume the variational Hilbert space closed under a... 

Hence, the extremum L(x) is the point of minimum. Thus, the problem of entropy maximization is transformed into the problem of heat minimization and the Kirchhoff and Prigogine theorems result from the extension of the second law to the passive isothermal circuits. The graphical interpretation of problem (21) is given in Figure 3b. 

Duhem s theorem is another rule, similar to the phase rule, but less celebratec It applies to closed systems for which the extensive state as well as the intensiv state of the system is fixed. The state of such a system is said to be completel determined and is characterized not only by the 2 + (iV—l)ir intensive phase rule variables but also by the it extensive variables represented by the masse (or mole numbers) of the phases. Thus the total number of variables is... 

It is easy to apply the Ostrogradsky theorem for deriving the entropy fluxes and the thermodynamic forces that initiate these fluxes. In fact, the balance of entropy (it also is an extensive parameter, S = ps) is... 

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