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The phrase "a first application for" is correct and usable in written English.
It can be used when referring to the initial submission of a request or proposal, often in contexts like job applications, patents, or grants.
Example: "She submitted a first application for the research grant last week."
Alternatives: "an initial application for" or "a preliminary application for".
Exact(4)
The theoretical results suggest to Chen that a first application for magnon cooling may be for scientists working on projects that require wireless cooling at extremely low temperatures.
There, a first application for habeas corpus had been denied, after hearing, by one District Court, and the denial was affirmed by the Court of Appeals.
In[47] we presented a first application for face detection.
A first application for a semiquantitative screening of multiple mycotoxins in various food (including six nut samples) was published in 2010 [ 25].
Similar(56)
A fourth application, for Lázaro Cárdenas, a Pacific port further south, is expected soon.
Lockheed selected a second application for computer-aided design (CAD), the French firm Dassault System's CATIA.
Last month, Fritz handed in a second application for EU funds to extend the project until 2027.
The cost would be somewhere between 2 million and 4 million euros ($3.12 million to $6.24 million) more — which would require a second application for European Union funds.
Indeed, BASF has a second application for Amflora pending that would allow the potato residue after starch extraction to be used as animal feed.
Corporation Counsel Joel Asarch said the same consortium had submitted a second application for a Cruise to Nowhere, but that it, too, would most likely be denied.
On his way to Rome to make a second application for the pallium, Malachy died at Clairvaux in the arms of Bernard.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com