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added link to actual question
Stephan Kolassa
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My inclination would be to create a new question, rather than using the omnibus question. A special-purpose question allows the space to go into more detail, gives other people a chance to write competing answers, makes it much easier to find the answer (someone searching for this information is unlikely to look under "How does the admissions process work for US Ph.D. programs, particularly for weak or borderline students?"), and makes it clear to people why their question is a duplicate.

I see at least three different aspects of this general question:

  1. If I've taken plenty of advanced courses in X in the process of completing a degree in another field, can I apply to graduate school in X?

  2. What if I haven't taken many courses in X, but I have acquired a good grasp of X through self-study and working in a related field?

  3. What if I've never studied X, but I have done very well in an unrelated field? Could I be admitted to graduate school in X on the basis of general intellectual promise, and then make up any missing background after enrollment?

I think all three could be addressed in the same question/answer.