The chapters of this volume all have their own level of
presentation. The topics have been chosen based on the active
research interest associated with them. Since the interest in some
topics is older than that in others, some presentations contain
fundamental definitions and basic results while others relate very
little of the elementary theory behind them and aim directly toward
an exposition of advanced results. Presentations of the latter sort
are in some cases restricted to a short survey of recent results
(due to the complexity of the methods and proofs themselves). Hence
the variation in level of presentation from chapter to chapter only
reflects the conceptual situation itself. One example of this is
the collective efforts to develop an acceptable theory of
computation on the real numbers. The last two decades has seen at
least two new definitions of effective operations on the real
numbers.
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