For example, theories of space-time can be formulated in one of two mathematically equivalent ways: as an ontology of points, with spatiotemporal regions being defined as sets of points; or as an ontology of regions, with points being defined as convergent sets of regions.
It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of quantifying the extent of nonproperness of the action.
Letting R be an arbitrary ring of coefficients, the equivariant cohomology of X with respect to the action of G is defined in one of the following equivalent ways.
In the presence of bivalence, these would all be equivalent ways of saying the same thing.
Thus there are two equivalent ways of decomposing the variation in accessibility, depending on the choice of reference periods for the counterpart reference period t of the output population and input infrastructure indices: ( {PA}_i^t ) and ( {IA}_i^t ), t = 0, 1.
It is likely that part of what is causing confusion is that there are a number of non-equivalent ways of drawing the distinction, some of which are better suited to certain subject areas than others.
So far we have been speaking informally, but there are two equivalent ways of developing definite descriptions ideas formally.
The standard theory usually maintains that "If p, q" and "p only if q" are equivalent ways of expressing the truth-functional "p ⊃ q".
If the native structure allows several equivalent ways of recruiting a productive nucleus the folding reaction is free to redistribute within these topological constraints.
In trying to show that there is an ambiguity infecting "if -sentences if -sentencesrinics havEnglished on two docritics thaveregard as mistaken: focusedthat there is a recipronitwobetween necessary andoctrinesentheynditions, and, second, that "if p, q" and "p only if q" aregardivasent ways of saying the samistaken.
