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Algebraic systems can be examined with maxdimsystems
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In the below example, inequations are given in the Pivots list, and the dimension is 2
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This next example has a constraint on the classvars present in the result
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In each of the above examples maxdimsystems picked out the maximum dimensional case. This is in contrast to rifsimp, with its casesplit option, which obtains all cases.
For the second example, computation with rifsimp using casesplit gives 2-d, 1-d and 0-d cases, of which the 2-d case is the maximal case from the above example.
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from which the caseplot with initial data counts could be viewed by issuing the command:
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One example of practical use of maxdimsystems is the determination of the forms of (where is not identically zero) for which the ode has the maximum dimension.
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where the above is an overdetermined system of PDE in the infinitesimal symmetries .
So we compute the maximum dimensional case, which is not displayed as it is quite large:
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from which we see that we have one case, of dimension 8, and it occurs when the displayed ODE in is satisfied. This classical result is also explored in the ISSAC Proc. article listed in the references (the primary reference for this command).
The timing above should be compared to the rifsimp full case analysis that takes significantly longer, and has a great number of lower dimensional cases:
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Consider the nonlinear Heat equation
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The overdetermined system for it's symmetries is given by
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At the same time as determining maximum dimensional cases, one can request, through a partitioning of the maxdimvars, that the dependent variables be ranked in a specific manner. To always isolate derivatives of in terms of the other dependent variables, one could specify maxdimvars=[[U],[X,T]]. For this problem we would obtain
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Alternatively we could try to obtain PDE only in and the classifying function by an elimination ranking:
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