By Stavros Busenberg, Bruno Forte, Hendrik K. Kuiken (auth.), Vincenzo Capasso, Antonio Fasano (eds.)
The 1990 CIME path on Mathematical Modelling of commercial techniques got down to illustrate a few advances in questions of business arithmetic, i.e.of the functions of arithmetic (with all its "academic" rigour) to real-life difficulties. The papers describe the genesis of the versions and illustrate their appropriate mathematical features. one of the themesdealt with are: thermally managed crystal progress, thermal behaviour of a high-pressure gas-discharge lamp, the sessile-drop challenge, etching strategies, the batch-coil- annealing technique, inverse difficulties in classical dynamics, photo illustration and dynamical structures, scintillation in rear projections monitors, identity of semiconductor properties,pattern popularity with neural networks. CONTENTS: H.K. Kuiken: Mathematical Modelling of business Processes.- B. strong point: Inverse difficulties in arithmetic for Industry.- S. Busenberg: Case experiences in commercial Mathematics.
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Extra info for Mathematical Modelling of Industrial Processes: Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Bari, Italy, Sept. 24–29, 1990
Sample text
29) The amazing result of Lighthill's transformation is that the velocity field no longer appears explicitly in the problem definition of Eqs. (26)-(29). It is hidden in the transformation of Eqs. (22)-(25). Therefore, we can solve this problem without having to know the velocity field explicitly. Only when we need explicit numerical values, do we have to give the velocity field also. The problem defined by (26)-(29) is classical. It is easily solved by means of the Laplace transform. ~K. (30) where K is an arbitrary constant, leaves the problem definition unchanged.
In what follows we shall assume that we gravity a JJJJ 7//////////////// Figure 3: Sessile drop. Case a: Non-wetting, Case b: Wetting. have to do with the case depicted by Fig. 3a, in which the drop has its largest width somewhere above the horizontal surface, not on it. We can always achieve this by choosing the right substrate material. As an example think of the surface of a car which has not been polished for quite a while. Rain falling on it will have a tendency to spread out. Once the car has had its polish, drops such as those of Fig.
Using the earlier result for Qo we can now write down behaviour Ti~er = 1 - Yi Ti ---~ ~ - ~i { ( " 1 - 1)Z + 4 ( 1 - ln2)ln2 + 2 } + O ( ~ i ) = 1 - Y + ~ { (11 - H , ) Y + 2 1 n 2 } + O (1) ~i (54) 39 The outer solution is characterized by the exponential smallness of the second term on the left of equation (23). Therefore, if we write T = Touter = 1 - Y + --T1 + O ~ ' (55) we find on account of Eq. (25) T1 = A1(1 - Y ) , (56) where A1 is some constant which will be determined presently. Comparing (54) and (55) we see that the inner and the outer expansion match perfectly when Ai = 2 in 2 (57) and Hi= 1+21n2 .