A presentation of the Fermi, Gupta–Bleuler, and radiation gauge methods for quantizing the free electromagnetic field is given in the Weyl algebra formalism for quantum field theory first introduced by Segal. The abstract Weyl algebra of the vector potential is defined using the formalism of Manuceau. Then the Fermi and Gupta–Bleuler methods are given as schemes for constructing representations of the algebra. The algebra of the physical photons is shown to be a factor algebra of a certain subalgebra of the original algebra of the vector potential. In this formalism, the application of the supplementary condition in the Fermi method, and the supplementary condition and indefinite metric in the Gupta–Bleuler method, can be interpreted as the means by which a representation of this factor algebra is obtained. The Weyl algebra of the physical photons is the Weyl algebra associated with the radiation gauge method. It is also shown that in the Fock representation of the Weyl algebra given by the Fermi method, automorphisms of the algebra corresponding to Lorentz transformations cannot always be implemented by unitary transformations. This leads us to construct a new representation of the Weyl algebra which provides a covariant representation for the vector potential.
The usefulness of the genetic algorithm (ga) as judged by numerous applications in engineering and other contexts cannot be questioned. However, to make the application successful, often considerable effort is needed to customise the ga to suit the problem or class of problems under consideration. Perhaps the most basic decision which the designer of a ga makes, is whether to use binary or real coding. If the variable of the parameter space of an optimisation problem is continuous, a real coded ga is possibly indicated. Real numbers have a floating-point representation on a computer and the decision space is always discretised; it is not immediately evident that real coding should be the preferred method for encoding this particular problem. We re-visit this, and other decisions, which ga designers need to make. We present simulations on a standard test function, which show the result that no one ga performs best on every test problem. Perhaps the initial choice to code a problem using a real or binary coding is a false dichotomy. What counts are the algorithms for implementing
Carlson has shown that if the predicted price in the linear cobweb model is taken as the average of all previous actual prices, then stability results independently of parameter values provided only that the demand-curve gradient is less than that of the supply curve. This result has subsequently been generalised by Manning and by Holmes and Manning. We investigate the robustness of their results.
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