phrases - "at all the vertices", what does this mean?


My professor has written a statement like this:



function is non-negative at all the vertices of the structure S and positive at some vertex



for a publication. It is a peer-reviewed publication so it should not have mistakes but I find at least two different meanings for this statement:




  1. Does it mean "function is non-negative at all of the vertices in the structure S and positive at some vertex"?




  2. Or does it mean "the function is not non-negative at all in the vertices of the structure S"?





Answer



It means "[The] function is non-negative at all of the vertices in the structure S and positive at some vertex". The of isn't necessary though it may help to clarify the meaning.


There may be some confusion between the phrases not negative at all and non-negative at all. Some examples may help to clarify what I mean. Note that these sentences do not have the same meaning as your original sentence.


This sentence asserts that the function is never negative.



The function is not negative at all.



It could also be written as



The function is not at all negative.



The phrase at all serves to emphasize not.


However in your sentence the phrase at all refers to the set of vertices. It could be written as



The function is not negative at all the vertices ...



(Though, as jwpat points out, there are better ways to say this.)


If it were reordered it would not make sense. Thus at all is not serving to emphasize not.



*The function is not at all negative the vertices ...



This question discusses the difference between at all and not at all.


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