2021年6月11日 · The set of slides you link to speaks of "monotone disjunctions" in a particular AI context, and the terse woirding "no negations" makes sense only in this implicit context. It is common in this area to consider disjunctive clauses of the form $$ x \lor y \lor \neg z \lor w \lor \cdots \lor \neg u $$ -- that is, a disjunction of input variables ...

2016年4月17日 · This largely echoes what was said by Thomas above, but, taking monotone as a term referring to the behavior of a function over the whole domain, one does not need to say "I'm used to." That said, one should always be clear on what definitions are being used as consistency is not a human's strong point.

2019年1月13日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

The critical complexity of all (monotone) Boolean functions and monotone graph properties. Information and Control, 67:212-222, 1985. The only important result that comes to mind using the other definition is this paper but the proof appears to go through using either definition.

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Even without the assumption of continuity, a monotone function on $\mathbb{R}$ is differentiable except on a set of measure $0$ (and it can have only countably many discontinuities). This is mentioned on Wikipedia , and proofs can be found in books on measure theory such as Royden or Wheeden and Zygmund.

Note that there are two (very similar) cases, monotone non-decreasing and monotone non-increasing. In what follows, we deal with monotone non-decreasing. It is useful to treat limits from the left and limits from the right separately.

Note that the Monotone Convergence Theorem applies regardless of whether the above interpretations: a non-decreasing (or strictly increasing) sequence converges if it is bounded above, and a non-increasing (or strictly decreasing) sequence converges if it is bounded below.

2020年4月5日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.