maximal vs. maximum

[Sandsoppa]

Thanks for all of your excellent input.

While I search more online, the things become a little bit more complicated, but interesting.

Here are two findings: one is a paragraph talking about "minimal", I copied to the end of this post for your reference.

Another is the "maximal". In most of the dictionaies, the synonyms or definition of "maximal" include often "maximum", however, not in another way around when the maximum is defined as an adjective.

My ongoing questions are: If any of you experts sees the following example sentences [I copied from American Heritage Dictionary], and if the "maximal" is replaced by "maximum", what the difference will be?

The maximal and minimal values are reached with full loaded and empty girder. [Jour. Franklin Inst., CXXVI. 240.]

A maximal muscular clench was recorded on a dynamometer. [Amer. Jour. Psychol., I. 191.]

Thanks for any comments!

Sand

Usage Note: Minimal and minimize come from the Latin adjective minimus, "least, smallest," and people therefore use minimal to refer to the smallest possible amount, as in The amplifier reduces distortion to the minimal level that can be obtained with present technologies. In recent years, however, people have begun to use minimal more loosely to refer to a small amount, as in If you would just put in a minimal amount of time on your homework, I am sure your grades would improve. Language critics have objected to this usage, but it is fairly common. In an earlier survey, the Usage Panel was asked what minimal meant in the sentence Alcohol has a particularly unpleasant effect on me when I have a minimal amount of food in my stomach. Under the strict interpretation of minimal, this sentence should mean only "Alcohol has an unpleasant effect when I have eaten nothing." If the looser interpretation is allowed, however, the sentence can also mean " ... when I have eaten a bit." Twenty-nine percent of the Panel held to the strict interpretation (that is, "eaten nothing"); 34 percent said that it could have only the looser meaning (that is, "eaten a bit"); and 37 percent said that it could have either meaning. Thus, 71 percent allowed the looser sense of minimal, so it should be considered acceptable, at least in nontechnical use. • The verb minimize has undergone a similar extension of meaning. In its strict sense it means "to reduce to the smallest possible level," but quite often the context requires us to interpret what the smallest possible level might be. Thus when a manager announces that The company wants to minimize the risk of accidents to line workers, we naturally think that the company plans to reduce the risk to the smallest level after considerations of efficiency and cost are taken into account, not that risks are to be reduced to the lowest level regardless of disruptions and cost. People also use minimize more loosely to mean "to make appear to be of little importance; play down," as in The President tried to minimize the problems posed by the nation's trade imbalance. This sense is well established.

[Steven0103]

"To know, is to know that you know nothing. That is the meaning of true knowledge." - Confucius

You know nothing. (^0^)

[Warrence]

Hi,

the last message was long ago, but maybe someone else still needs the answer, which can be found here:
http://forums.xkcd.com/viewtopic.php?f=17&t=63907

In short: something "maximal" cannot be enlarged; Something "maximum" is the biggest instance of its kind (Analogously "minimal"/"minimum"). This distinction is very special, as someone mentioned earlier, that's possibly the reason why there hasn't been a useful answer yet. I can only make up examples from graph theory, e.g. the following.

"T is a maximal subtree of G" means that T is no longer a tree if any edge from G (and not already in T) is added to T, so T cannot be enlarged.

"T is a maximum subtree of G" means that there is no subtree in G that has more edges and vertices than T (there may be other maximum subtrees in G, with the same size as T), so T is the "biggest".

Cite from d271828b found here http://forums.xkcd.com/viewtopic.php?f=17&t=63907#p2303175
"I had a professor who talked about a maximal burrito-- it wasn't the best burrito that could be made, but there was nothing that could be added to that particular burrito to make it better!"

Hope that helps.