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The phrase "a specified majority of" is correct and usable in written English.
It can be used when referring to a particular percentage or number of votes or support required for a decision or action to be taken.
Example: "The proposal will only pass if it receives a specified majority of the votes from the committee members."
Alternatives: "a defined majority of" or "a designated majority of".
Exact(2)
These clauses are intended to avoid an Argentina-style holdout problem by binding all bondholders to a deal agreed to by a specified majority of bondholders.
Notably, the market's shift to the new wording of the rankings clause, as well as the wider adoption of "collective action clauses" (which limit the power of "holdouts" who decline a settlement favored by a specified majority of bondholders), is likely to further limit the impact of the earlier decisions.
Similar(58)
Mr Rhodes was instrumental in cajoling international banks to reach deals with a series of indebted governments and in pushing for "collective action clauses" that bind all creditors to follow the will of a specified majority.
A bid contracts to win a specified number of odd tricks with a specified trump or at no trump.
Where the type of personality disorder was specified, the majority of studies investigated borderline or anti-social personality disorders.
In cases in which the reason was specified, the majority of patients who discontinued infliximab due to elective reasons attributed the discontinuation to the wish (patient's or physician's) to switch to subcutaneous TNF blockers.
The sources of cost and outcome data were clearly specified in the majority of the analyses (93% for costs and 87% for outcomes).
Moreover, the potential impact of ADHD on treatment response is unknown; ADHD status has not been specified in the majority of the above studies.
The source of cost and outcome data was clearly specified in the majority of studies (80% for costs and 97% for outcomes).
Two of the 17 studies specified that the majority of preschoolers' screen-viewing behaviors in care were sedentary, [ 43, 51] with very little physical activity being accumulated while engaging in such activities.
When multiple measures of the same variable were available, as for vital signs, the measure introduced into the prediction rule was the first available one in four studies, the most abnormal one in five, and not specified in the majority of studies.
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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