By G. Chartrand, S. F. Kapoor
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Extra resources for The Many Facets of Graph Theory
Where N2, ... , Nk are certain constants. 11 Dimensional Analysis 39 This would mean that the dimension of al was equal to the product of powers of the dimensions of a2, ... , ak, which contradicts the assumption that the dimensions of the quantities at, ... , ak are independent. 16) has a solution. 16) would be a linear combination of the others, so that (cf. the argument above), the dimension of anyone of the quantities ai, ... ak could be written as a product of the powers of the dimensions of the others, and this would contradict the assumption.
7). 22) for T < TJ = v'8(Qltt)1/4, and by h = 0 for T ~ rJ. I. are the water density and dynamic viscosity, 9 is the gravitational acceleration, and r is the distance from the well of the point at which the observation is made. 24) such that the spatial distribution of the ground-water mound height, o. 7. (a) A ground-water mound is formed rapidly through a well of small radius. (b) If capillary forces are negligible the mound volume stays constant as the mound height decreases. (c) When part of the water is retained by capillary forces the mound volume decreases with time.
Therefore, with the help of self-similar solutions ra-searchers have attempted to envisage the characteristic properties of new phenomena. Self-similar solutions have also served as standards in evaluating approximate methods for solving more complicated problems. The appearance of computers changed the general attitude toward self-similar solutions but did not decrease the interest in them. Previously it had been considered that the reduction of partial to ordinary t The fact that we identify one of the independent variables with time significance.