Learn English – Can we use “with no” rather than “without”

A. The official from X association

The person that I am talking about is coming from that association. I probably expect that person to act, while they are here, on behalf of that organisation.

B. The official for X association

That is one of our own people, not a member of, or acting for that association. It it the person, that for us, keeps in contact and deals with that association.

C. The official with X association

More or less the same as A), but in this case it is certainly true that the person is here in their role as member or representative of the association. You might hear this phrase on a news-item, if someone is called upon as an expert: "And to answer some questions about X, here is ms. Y, who is with the X association."

D. The official at X association

The person is working for, or physically at the premises of X association, but I am not necessarily talking to or otherwise communication with them.

In the context of a scientific paper, I would read 'associated with' as meaning 'related to in some way'. 'Corresponding to' on the other hand is more strictly defined - it implies a one-to-one relationship. x and y values on a graph would be a good example - for every value of x, there is a corresponding value y.

In your example:

group 'T' which has fields T₁, T₂, … T_n.

Neither seems to be the case. I would suggest something along the lines of:

...group 'T' which is comprised of fields T₁, T₂, … T_n.

...group 'T' which is composed of fields T₁, T₂, … T_n.

...group 'T' which ranges from T₁, T₂, … T_n.