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Encyclopedia > Short map

In mathematics, a short map is a function f from a metric space X to another metric space Y such that for any we have

.

Here dX and dY denote metrics on X and Y, respectively. In other words, f is short iff it is 1-Lipschitz.


One can say that f is strictly short if the inequality is always strict. Then a contraction mapping is strictly short, but not conversely (even with X = Y).


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National Map (8823 words)
Map 1 also identifies a number of distinct and important dialect areas in the Eastern United States, which were clearly set out in the work of the Linguistic Atlas (Kurath and McDavid 1961): Eastern New England; New York City; and the Mid-Atlantic coastal area encompassing Philadelphia, Wilmington and Baltimore.
A remarkable finding of Map 1 is that the major phonological boundaries of the U.S. as determined by new and vigorous sound changes which arose in the 20th century coincide with the major lexical boundaries based on vocabulary wjocj was largely set in the early settlement periods.
Map 1 therefore identifies all of the NCS speakers as those for whom the difference in the F2 of /e/ and /o/ is less than 375 Hz.
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