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Weak Derivatives
Published in Kenneth Kuttler, Modern Analysis, 2017
Then we define the Sobolev space W1p(U) to be the closure of Cc∞(U¯) in X1p(U) where Cc∞(U¯) is defined to be restrictions of all functions in Cc∞(ℝn) to U. Show that this definition of weak derivative is well defined and that X1p(U) is a reflexive Banach space. Hint: To do this, show the operator u → u,i· is a closed operator and that X1p(U) can be considered as a closed subspace of Lp(U)n+1. Show that in general the product of reflexive spaces is reflexive and then recall the Problem 7 of Chapter 6 which states that a closed subspace of a reflexive Banach space is reflexive. Thus, conclude that W1p(U) is also a reflexive Banach space.
NPV approach to material requirements planning theory – a 50-year review of these research achievements
Published in International Journal of Production Research, 2019
The servomechanism approach to the production and inventory control in the Sobolev space, where mostly robustly perturbed delays have been investigated, could be found in the papers of Ludvik Bogataj (1990a, 1990b, 1991). These studies have upgraded the initial results of Bogataj and Pritchard (1978). Some applied results of inventory control can also be found in Bogataj and Bogataj (1990, 1992). These results highlighted the simplicity of modelling time-lag perturbations in production-inventory systems in the space of complex variables, as Simon suggested, if there is no dynamic of robust perturbations. But as described in Ludvik Bogataj’s articles, sometimes the robust perturbations in the Sobolev space could be studied more analytically and without approximations, and also perturbed functions in the Laplace-transformed equations could be analysed better. Namely, in many problems of mathematical physics and calculation of variations, it is not enough to deal with the classical solutions of differential equations. It is necessary to introduce the notion of weak derivatives and work in Sobolev spaces. This approach is also useful when studying the lead-time and other time delays in production-inventory systems.