Classical Distribution Theory Research Articles

A note on the Penrose junction conditions

Impulsive pp-waves are commonly described either by a distributional spacetime metric or, alternatively, by a continuous one. The transformation T relating these forms clearly has to be discontinuous, which causes two basic problems. First, it changes the manifold structure and second, the pullback of the distributional form of the metric under T is not well defined within classical distribution theory. Nevertheless, from a physical point of view both pictures are equivalent. In this work, after calculating T as well as the `Rosen' form of the metric in the general case of a pp-wave with arbitrary wave profile we give a precise meaning to the term `physically equivalent' by interpreting T as the distributional limit of a suitably regularized sequence of diffeomorphisms. Moreover, it is shown that T provides an example of a generalized coordinate transformation in the sense of Colombeau's generalized functions.

Multiplication and mixed differentiation in generalized Fock spaces

The classical distribution theory founded by Laurent Schwartz [26] has an intrinsic multiplication problem. He indicates this in his paper [27]. This problem is clearly demonstrated by selecting two particular distributions and attempting to define a multiplication that enjoys commutative, associative, and unit identity properties. This process leads to the contradiction, 1 = eP I [lo]. A new generalized function theory developed by J. Colombeau [4-91 provides a solution to the multiplication problem and also develops a general integration theory. T. Todorov reexamines the new generalized function theory in the spirit of non-denumerable sequences termed ultrapowers [29]. The L. Schwartz distribution theory was reexamined in the spirit of fundamental sequences [ 16,333. However, there are new considerations in the theory of new generalized function theory requiring an algebraic notion of a particular ideal contained in a specified test function space. A brief review of these constructions can be found in Section 7 and a comprehensive review can be found in Ref. [25]. A principle application for our development will be modeling a physical system consisting of an “infinite” number of particles. A system of n-particles termed bosoms or fermions has previously been modeled by n-dimensional symmetric or antisymmetric functions. When this system is enlarged to an infinite number of particles, the model for the state space vector has the form,

Edward West and the Classical Theory of Distribution and Growth

Edward West's Essay on the Application of Capital to Land is rightly famous for its treatment of diminishing returns and rent. It is not generally recognized that West also set out a model of dynamic equilibrium between the growth of population and capital, and of the simultaneous determination of wage and profit rates. Such a model has been attributed to David Ricardo, but West, unlike Ricardo, stated it explicitly, and did so before any of Ricardo's writings on distribution and accumulation had been published. He should be given credit for it. Copyright 1988 by The London School of Economics and Political Science.

ZUR VERTEILUNGSTHEORETISCHEN KRITIK AN KEYNES

SUMMARY Keynes paid no attention to the problem of distribution. Neo-Keynesians investigated the dependency of alternative distributions on investment, but did not explain total profits. Growing accumulation of capital and constant distribution cause a decline of profitability. To preserve private investments an increase of total profits would be necessary. However wages are politically given and an increase of government expenditure is limited in its effects because of growing public supply. Distribution is in any case only a precondition for the propensity to invest. According to Keynes investments determine the economic process. A theoretical explanation of distribution is only possible by an explanation of the investment process, which depends on individual profitability and total profits. The theory of distribution has to be a profit theory. By this Keynesian Theory includes a reversal of the classical distribution theory, which starts from the wages.