Lagrangian theory

P. G. De Gennes. Exponents for the excluded volume problem as derived by the Wilson method. Phys Lett 38A 339, 1972 J. des Cloiseaux. The Lagrangian theory of polymer solutions at intermediate concentrations. J Phys 26 281-291, 1975. 

Relativistic Lagrangian theories 10.3.3 Nonabelian gauge symmetries... 

Relativistic Lagrangian theories implying the gauge field equations... 

In the preceding sections, we have seen that the size of a large chain is characterized by an exponent v which, for each type of chain, has a universal value. We shall now define another exponent y whose real importance has been recognized only recently by the polymerists, but which plays an essential role in the Lagrangian theory of polymers. This exponent appears when one tries to characterize the asymptotic behaviour of the number of configurations. 

However, we must note that none of these theories predicted correct scaling laws for long polymers. It was only in 1975 that the application of renormalized Lagrangian theories to polymer solutions, led to a correct description of these systems and provided a means for accurately calculating some of their properties. 

Moreover, these results remain valid when the two-body repulsive interaction between polymer segments diminishes. In particular, de Gennes pointed out in 1975 that for polymer solutions, Flory s 0 point (T = TF) is a tri-critical point and must be treated as such. Thus, the Lagrangian theory can be used not only to describe the behaviour of solutions in the repulsive domain where excluded volume is dominant, but also in the attractive domain. 

First, let us consider field theory. If the Lagrangian theory defined in Chapter 11 is a genuine field theory, as we believe, the Green s function has a pole for k2 = - m2 (where m can be considered as the mass of the particle which the field describes) and also a cut for more negative values of k1. Thus, for large values of r, we must attribute a dominant role to this pole. At the pole, O(x) vanishes, and in the vicinity of this point we may set... 

CPT theorem - A theorem in particle physics which states that any local Lagrangian theory that is invariant under proper... 

The optimization problem in Eq. (5.146) is a standard situation in optimization, that is, minimization of a quadratic function with linear constraints and can be solved by applying Lagrangian theory. From this theory, it follows that the weight vector of the decision function is given by a linear combination of the training data and the Lagrange multiplier a by... 

CPT theorem - Atheorem in particle physics which states that any local Lagrangian theory that is invariant under proper Lorentz transformations is also invariant under the combined operations of charge conjugation, C, space inversion, P, and time reversal, T, taken in any order. 

It was de Gennes (1972) who first introduced the Lagrangian theory to interpret the behavior of polymer solutions. Although des Cloizeaux showed the similarity between his and Flory s interpretations of the osmotic pressure, de Gennes pointed out that des Cloizeaux interpretation is superior to Flory s. To compare the two theories, de Gennes first translated the Flory lattice theory into his language. Let represent the fraction of lattice sites occupied by the monomers. Then cubic centimeters and is the volume of the unit needed in the cubic lattice. Flory s equation of osmotic pressure [Eq. (9.20)] can be expressed in the form... 

APPENDIX E GREEN S FUNCTION 217 APPENDIX D LAGRANGIAN THEORY... 

Lagrangian theory is basically the variational principle. There are several forms to express this principle. The simplest one, also the earUest one, is the principle of action. 

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