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"a first shortcoming" is a correct and usable phrase in written English.
It can be used in a sentence when describing a specific flaw or weak point that is the first in a series or succession. Example: Despite being highly praised for his leadership skills, John's first shortcoming as a manager was his lack of attention to detail.
Exact(2)
A first shortcoming of Dfdist is the possibility of detecting false positives (i.e. committing type-I errors) [ 33, 34].
Regarding study limitations, a first shortcoming of the work is related to the notion of work economic sector and operationalization of it that were used.
Similar(58)
A third shortcoming, say critics, is that the system cannot address the problems faced by American schools.
A second shortcoming in existing activism literature is that tests of post-activism changes in operating performance typically do not adequately control for natural trends accounting metrics.
A third shortcoming of the Lords' conclusions, and indeed most analysis in this area, is that all immigrants are lumped together.
A third shortcoming is that, after an adjustment period, wages and contracts are more likely to adjust more frequently than they would with a 2% inflation target, making monetary policy less effective.
From the endogenous weighting system follows a second shortcoming.
A second shortcoming is that the gradual-morphological model leads people to view eye evolution as a single, linear path from simple to complex eye.
A third shortcoming is the limited utilization of domain ontologies in current retrieval methods.
A second shortcoming is that we included patients with different etiologies for ALI/ARDS.
The fact that the amount of goods being transferred could not be quantified is a second shortcoming.
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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