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In school, I learnt that we usually put an indefinite article in front of some adjective-modified uncountable nouns to refer to an example of the uncountable noun.

for example:

I have a great love for music.

He has a limited knowledge of French.

However, not all the uncountable nouns can have an indefinite article in front of them.

for example:

A Portuguese weather is not my interested subject.

Regarding "concern" and "respect", I usually put a "a/an" in front of them?

for example:

1. Don't show an indifferent concern over the current affairs.

2. Please show a great respect to your teachers.

However, today I have found both "concern" & "respect" used with no articles in front of their adjectives.

for example:

1. German coach has said that Argentina team is a great team, we should play against them with great respect.

I am wondering why there is no indefinite article in front of the "respect" above.

I am hoping for some advice on this facet. Thank you.
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In school, I learnt that we usually put an indefinite article in front of some adjective-modified uncountable nouns to refer to an example of the uncountable noun.-- Not 'usually' at all, but it can often be done.

However, not all the uncountable nouns can have an indefinite article in front of them.-- That's right; some sound awkward or impossible.

A Portuguese weather is not an interesting subject for me.-- But 'A Dutch uncle is better than no uncle at all'. The problem with your particular example is that the adjective is too specific.

Regarding "concern" and "respect", I usually put a "a/an" in front of them -- I don't think that is a good idea.

1. Don't show an indifferent concern over the current affairs.-- The article is OK here.

2. Please show a great respect to your teachers.-- It is not OK here

However, today I have found both "concern" & "respect" used with no articles in front of their adjectives.-- Yes indeed, it is quite common.

I am wondering why there is no indefinite article in front of the "respect" above.-- 'Respect' just resists this sort of 'countabilitization' more than some other non-count nouns. 'Great' + 'respect' are also very highly collocated as here.
Mister MicawberI am wondering why there is no indefinite article in front of the "respect" above.-- 'Respect' just resists this sort of 'countabilitization' more than some other non-count nouns. 'Great' + 'respect' are also very highly collocated as here.

Could you consult me about the general rules of indefinite articles in front of adjective-motified uncountable nouns?

I know the rules are not specific but general ones will do.

Thank you.
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The rule is just what you have already stated (but as I have amended it):

"...we can sometimes put an indefinite article in front of some adjective-modified uncountable nouns to refer to an example of the uncountable noun."

There are no further 'rules' that I know of beyond usage and reason.
Mister MicawberBut 'A Dutch uncle is better than no uncle at all'

what does it mean? is it a saying or proverb?

Thank you.
No, I just made it up as an example sentence. Catchy though, isn't it?
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yes, it is quite catchy, but I have another thought.

does it imply that something wrong with "Dutch uncle"?

just kidding.

By the way,

To me, Portuguese weather will do already in the previous sentence, because there is only one Portuguese weather in general. However, I know you have your reason to put a "a" in front of it, May I have the privilege to know the reason?

Thank you so much.
No, the original post is right-- 'Portuguese weather' cannot take the indefinite article in any normal situation that I can conceive.