The best solution to this problem is to avoid the use of technical
vocabulary and use plain English and unambiguous formulas.
It is quite easy to see how the term 'strictly monotonic' gets
abreviated to 'monotonic' when used by non-mathematicians.
I am pretty sure that if we started using the terms 'surjection',
'bijection' &ct. instead of 'one to one', 'one to many' we would end up
with similar confusion.
From: Tom.Petch [mailto:sisyphus(_at_)dial(_dot_)pipex(_dot_)com]
Sent: Monday, February 20, 2006 9:44 AM
To: Yaakov Stein
Subject: Re: 'monotonic increasing'
----- Original Message -----
From: "Yaakov Stein" <yaakov_s(_at_)rad(_dot_)com>
To: "Tom.Petch" <sisyphus(_at_)dial(_dot_)pipex(_dot_)com>; "Elwyn Davies"
Cc: "ietf" <ietf(_at_)ietf(_dot_)org>
Sent: Sunday, February 19, 2006 7:10 AM
Subject: RE: 'monotonic increasing'
Actually, even mathematicians don't agree on the wording here.
In analysis we commonly talk about monotonic functions, which
can be either monotonically increasing ( x <= y => f(x) <=
f(y) ) or monotonically decreasing ( x <= y => f(x) >= f(y) ).
Since analysis deals with continuous entities, the
distinction of nondecreasing vs. increasing is usually not
important, and thus not worried about. However physicists
tend to make the distinction by saying "nondecreasing".
In dealing with sequences, the distinction is almost
universally made between nondecreasing ( x_n <= x_n+1) and
increasing (x_n < x_n+1) although the European school prefers
stressing the difference by using the word "strictly" instead.
To make things more confusing, in order theory (where you
would expect the wording to be the tightest) the wording used
is monotone (for "increasing") and antitone (for "decreasing").
Of course there the distinction between nondecreasing and
increasing is not important since the description is of a
(partial) order relation, and if that relation includes
equality as a special case than you get one variety, while if
not you get the other.
Beautiful (as mathematics always is).
But just to be clear, if you saw a reference to 'monotonic
increasing' in an American journal, say of applied
mathematics, would you be sure you understood what was meant?
And if so, would that be S_i+1 >= S_i U+2200 i or S_i+1 >
S_i U+2200 i?
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