Why does these sentences could omit relative pronoun as a subject pronoun?

Question

These sentences have perplexed me about leaving relative pronoun in relative clause. I learn that we can not omit relative pronoun when this pronoun is subject. So why in these two sentences, both pronouns play the role as the subjects, but they all are omitted and changed into -ing clause?

1.We are confident that all the doctors WORKING in the hospital that we represent know that they have acquired the competencies to care for critically ill patients.

2. We are aware that the organization has recently circulated a letter to its members PROPOSING to change its name to include our title.

I think "doctors" and "a letter" are subjects so I think there are "who work" and "that propose".

These sentences are written by me. Are they correct if I omit the relative pronouns?

3. I came across a Twitter WRITTEN by a trainee doctor that share about his way to prepare for the exam.

4. Here are some of my learning sources INCLUDING math, history and physics which I have collected for a period of time.

Thank you a lot for your help.

Answer 1

These are not analyzed as relative clauses since there is no possibility of them containing a relative phrase:

*People who living near the site, etc.

Past-participial modifiers are bare passives, as evident from the admissibility of a by phrase in internalised complement function. Gerund-participials can be active or passive. Passive gerund-participials contrast with the past-participials in aspectuality as progressive vs non-progressive, but with actives, the progressive vs non-progressive distinction is lost (CaGEL p1265).

So in (1), doctors working in the hospital could have two interpretations: doctors who work in the hospital or doctors who are working in the hospital.

In (2), the relevant interpretation would be a letter that proposes...

(3) is a bare passive as it includes neither be nor get.

(4) would be interpreted as sources which include...

The use of the non-finite clauses as modifiers often avoids awkward phrasing as relative clauses pile up. For example, in (4) there is already a relative clause which I have collected for a period of time. If both modifiers were relative clauses we'd end up with something like:

Here are some of my learning sources which include math, history and physics, and which I have collected for a period of time.

or

Here are some of my learning sources, which include math, history, and physics, that I have collected for a period of time.

Not as elegant as the version you came up with in (4).

Answer 2

You're not omitting anything, but using a different kind of clause to modify the nouns in question (“doctors”, “members” etc.).

Those nouns are now being modified, not by a relative clause, but by a non-finite clause where there is no relative pronoun involved.

You may encounter the term 'reduced relative clause' for the non-finite clauses, but ignore it; it’s a misnomer. They are not a kind of relative clause but a distinct kind of non-relative clause, either a gerund-participial or a past-participial.

The two kinds of clause, relative and non-finite, are semantically similar (cf. "all the doctors who work in the hospital"), but the similarity ends there.