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By Terlisten D.

In army expertise, as in nature, for each motion, i.e. a brand new improvement, there follows a response, a counter-measure opposed to the hot expertise. precisely this occurred within the German military touching on anti airplane defence. starting with the German French warfare of 1870/71. the armies used balloons as artillery reconnaissance to find objectives for shelling the adversarial entrance zone. As a defence to this reconnaissance the German military, governed through an Emperor this time, built a so-called ''Ballonabwehrkanone'' (balloon-defence-weapon), abbreviated ''ВАК''. They have been made cellular through being developed on horse-drawn platform and as such have been an early self-propelled weapon. these rudimentary self-propelled wagons have been in fact very gradual and in simple terms of constrained use, in general close to roads...

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32 1. Mathematical preliminaries In our presentation of the matching principle we have described how it can be applied to functions which involve and exponential terms (or functions which can be expanded in terms of these). However, when we apply this same procedure to logarithmic functions, we encounter a difficulty which requires a careful adjustment of the matching principle. 9 MATCHING WITH LOGARITHMIC TERMS To see that we have a problem, let us consider an appropriate example, expand and then attempt to match in the way described above.

This example prompts a number of additional and important observations. e. expansion A breaks down, producing a scaling used to obtain expansion B, but B does not necessarily break down to recover the scaling used in A. 53) does not. 55). 53) and thereby make plain the nature of the breakdown; this will prove to be a useful adjunct in some of our later work. 52), for x = O(l) as arises from the exponential term. 5). e. . ) or, which is the usual practice, simply to state which terms we will retain.

5 ASYMPTOTIC EXPANSIONS WITH A PARAMETER We now introduce functions, which depend on a parameter and are to be expanded as Here, x may be either a scalar or a vector (although our early examples will involve only scalars). In the case of vectors, we might write (in longhand) note that commas separate the variables, but that a semicolon is used to separate the parameter. As we shall see, it does not much matter in this work if the function we (eventually) seek is a solution of an ordinary differential equation (x is a scalar) or a solution of a partial differential equation (x is a vector): the techniques are essentially the same.

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Flakpanzer IV Wirbelwind (Sd.Kfz. 161.4) & Ostwind by Terlisten D.

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