The c theorem

We have seen in eq. (V.31) that the bulk modulus of the gel should scale like c/N)T, A similar scaling law should also hold for the shear modulus, which is more easily accessible to experiment. (In what follows, since we are interested only in scaling properties, we use the same symbol E for both.) It is possible to test eq. (V.31) by varying either the quality of the solvent (i.e., v) or the length of the chain (i.e.,N). Recall that c, v,and N are always linked by the c theorem [eq. (V.25)]. 

The quantitative aspects of the swelling equilibrium of mesogels are also distinctive. The equilibrium swelling of simple gels is often rationalized in terms of the c theorem [17]. Denoting the number of monomers in the chain segments between trifunctional crosslinks by N, this theorem states that the monomer concentration in a gel in equilibrium with a reservoir of good solvent Ce is... 

This higher scattering at low q for gels than for solution is not in agreement with the chemical theories or the c theorem. It could be related to a heterogeneous structure, for... 

This can be generalized to the following fundamental rule of doping. Assuming that the doping ion is introduced irreversibly (with a concentration C) and all other defects are in local equilibrium, the following statement applies for simple defect chemistry (referred to, in what follows, as the C theorem ) ... 

This result is in good agreement with the conclusion of the c theorem. In Fig. 2 are also reported the theoretical variations Rg = f() calculated from the affine deformation and the phantom network models, respectively. For the latter model, the memory term has been taken equal to unity, since the dimensions of the elastic chains are unperturbed in the dry state. It is seen from Fig. 2 that Rg does not show any appreciable dependence on the volume fraction in the range 0.05 < < 0.2 and its variation remains significantly... 

This result expresses the c theorem. It implies that in a good solvent and at the swelling equilibrium, the end-to-end distance V(b ) of a network drain of the gel scales with N like the end-to-end distance Rp of a free macromolecule of the same molecular weight in the same solvent. 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Note that, by the imbedding theorems, there exists a constant c > 0 such that... 

To simplify the notations we do not indicate the dependence of the solutions on the parameters s, 5. Our aim is first to prove the existence of solutions to (5.185)-(5.188) and next to justify the passage to limits as c, 5 —> 0. A priori estimates uniform with respect to s, 5 are needed to study the passage to the limits, and we shall derive all the necessary inequalities while the existence theorem is proved. 

Stresses are usually related to strains through an effective modulus. If the components of stress are nondimensionalized by a suitable scalar modulus c, then they are also of order c. Using (A.94), (A.lOl), and the binomial theorem in (A.39), the relation between the normalized spatial stress s = s/c and the normalized referential stress S = S/c becomes... 

The reason for this complication of the theory is evident the truncated set may contain certain variable parameters, and, if these are carefully adjusted to render the best possible description of a specific state, they may become rather unsuitable for the description of another state. According to Section II.C(3), a truncated set should, e.g., always contain a scale factor as a variable parameter and, if this quantity is fitted to the ground state, it may give a basic set which is rather "out of scale for even the first excited state. Since the virial theorem is not satisfied for this state, the corresponding total energy may be comparatively poorly reproduced. This implies that in treating excited states, it is desirable to have reliable criteria for the accuracy of both energies and wave functions. 

In the case the calculations are based on a truncated set Wlf 2,. . . containing adjustable parameters, the A splitting is of particular importance, since it permits the investigator to use different values of these parameters for different eigenvalues Xk— the relation III.95 will anyway be valid. The scale factor rj is such a parameter, and the results in Section II.C(3) and III.D(lb) show that, by means of the A splitting, it is now possible to get the virial theorem exactly fulfilled for at least one of the eigenfunctions associated with each Xk. 

These results are sometimes interpreted as a converse to the coding theorem. That is, we have shown that the mutual information per channel symbol between a source and destination is limited by the capacity of the channel. The problem is that we have not demonstrated any relationship between error probability and source rate when the source rate is greater than the mutual information. Unfortunately, this is not as trivial to obtain as it might appear. In the next section, we find a lower bound to Hie probability of error when the source rate is greater than C. 

In this section, we shall state the final theorem necessary to give unity to the previous results if RT < CT, then reliable communication is possible with as small an error probability as desired. This remarkable theorem was first stated and essentially proved in 1948 by C. E. Shannon.10 The first rigorous proof of the theorem was given by Feinstein,11 and a number of other proofs and generalizations have subsequently been given. 

The Pythagorean theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sums of the squares of the other two sides. That is, if the hypotenuse is c and the other two sides arc a and b, then a1 + b2 = c2. 

It is worth noting here that on the square grid (h = h. = h) this condition is automatically fulfilled. A proper choice of (p guarantees the sixth order of accuracy of scheme (9) on any such grid. Convergence of scheme (9) with the fourth order in the space C can be established without concern of condition (11). An alternative way of covering this is to construct an a priori estimate for A z p and then apply the embedding theorem (see Section 4). 

T-ZAlfA) appears in the eq.(lO) multiplied by < i i > when natural orbitals are used. Thus, if < i i > is small (< i i >very small volume around the nucleus of A. In the remaining part of the volume occupied by the molecular system the description of this orbital cannot be deduced from the Valley theorem. Therefore, we will consider here only strongly occupied orbitals with < > 1 or < i i > 2. 

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