Grammatical complements for "allow"

Are any of these verb phrases grammatical?

• allows the user of modeling and resolving
• allows the user to model and resolve

Which version of the following sentence is correct/better?

• In particular, it allows to acquire a signal using a sample rate significantly lower than the one dictated by the Nyquist criterion.
• In particular, it allows acquiring a signal using a sample rate significantly lower than the one dictated by the Nyquist criterion.

I searched Google Scholar for "allows to acquire" and "allows acquiring", and it seems people use both more or less with the same frequency.

The semantics of the verb allow meaning "permit" has three arguments, making it a trivalent verb. Using the linguistic terminology for thematic relations, there is the entity that is granting the permission (the agent), the entity that receives the permission (the patient), and the thing that is permitted (the theme).

The verb allow can be used in three different syntactic constructions which explicitly indicate what is allowed (the theme). Here are the three constructions with examples from the Corpus of Contemporary American English:

(a) with a gerund complement indicating what is allowed: allow [gerund phrase]

the president last month unveiled plans to allow [drilling in Atlantic waters from Virginia to mid-Florida]

Sets up Web portals that standardize policies and allow [purchasing across state lines]

Note that in this construct, the verb does not include an object indicating the entity which is being allowed, nor is the gerund complement introduced with of, as in the first example from the question. This would be grammatical, though:

Allows modeling and resolving

(b) with just a noun phrase object indicating what is allowed: allow [noun phrase]

and are working with the school administrators to allow [use of personal devices in our schools].

The learning commons has extended its hours of operation to allow [more access to its physical space].

(c) with a noun phrase object (indicating the patient) and an infinitive complement (indicating the theme): allow [noun phrase] [infinitive phrase]

This is the only construct where all three arguments to the verb can be expressed together.

we're not going to allow [them] [to disrupt this debate and take the focus off the real issue here].

Certainly parents who allow [their children] [to consume violent entertainment products] do share some of the responsibility.

These tools and resources allow [you] [to connect, communicate, collaborate, and create].

Note that the recipient of the allowance (the patient) isn't always a person or person-like entity:

Only nine states allow [life-sustaining treatment] [to be withheld from patients in a persistent vegetative state].

Note also that this constructions requires both a direct object and an infinitive complement. The verb allow cannot be used with just an infinitive complement; that is, constructions like "allow to [verb]" are ungrammatical. The following examples which I found via Google are clearly written by non-native speakers and are not grammatical:

* What smartphone allows to connect my computer to it for surfing ?

* Image drag and drop allows to create executable files

* Will java allow to use functional interfaces as methods?

Only the second is correct as written.

However, "allows us to acquire", "allows one to acquire", "allows the system to acquire" and so on are all also valid.