By Ladislav NebeskyÌ
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Extra info for Algebraic properties of trees
36) The parameter r. 4 Self-simiLar- solutions as intermediate asymptotics 21 able, but has the form ~1/2 U ~1 = pg (~t)1/2 ~ (e, 11, -;) . 37) Just as for ~1 = ~, we are interested in the behaviour of the solution for small 11. 37). 38) where cp is a finite quantity and a is a constant that depends only on Itl/~; it is non-zero for ~1 :F It, but equal to zero for ~1 = It. If one tries to pass to the limit 11 -. , one gets an empty relation. , the time as large or r. as small. 39) shows that the representation for large times is given not by a solution of linear-source type but by another self-similar solution.
5) We now note that by virtue of the equivalence of systems within a given class, we may assume that system 1 is the original system of the class, without altering the class of the systems of units. In this case, system 2 can be obtained from the new original system (system 1) by decreasing the fundamental units by factors of ~/Lb M2/Ml and T2/Tb respectively. Consequently, the numerical value 42 of the quantity under discussion in the second system of units, is, by the definition of the dimension function, a2 = 414>(L2/ Lit M2/M l, T2/Tt}.
It is customary (following a suggestion of Maxwell) to denote the dimension of a quantity tfJ by [tfJ]. We emphasize that the dimension of a given physical quantity is different in different classes of systems of units. For example, the dimension of density p in the LMT class is [p] = M L -3; in the LFT class, it is [p] = L -4 FT2. Quantities whose numerical values are identical in all systems oj units within a given class are called dimensionless; clearly, the dimension function is equal to unity for a dimensionless quantity.