Dangling participles

A common problem in academic writing is long introductory clauses. Stylistically, these tend to be repetitive and to be used as brief summaries that are really needed and can confuse. That is particularly true when they create danglers, which are grammatically incorrect.
Here is  an explantion of the problem from The Chicago Manuel of Style:
5.115: Dangling participles Chapter Contents / Grammar / Verbs / Participles and Gerunds Both participles and gerunds are subject to dangling. A participle that has no syntactic relationship with the nearest subject is called a dangling participleor just a dangler. In effect, the participle ceases to function as a modifier and functions as a kind of preposition. Often the sentence is illogical, ambiguous, or even incoherent, as in Frequently used in early America, experts suggest that shaming is an effective punishment (used does not modify the closest noun, experts; it modifies shaming), or Being a thoughtful mother, I believe Meg gives her children good advice (the writer at first seems to be attesting to his or her own thoughtfulness rather than Meg’s). Recasting the sentence so that the misplaced modifier is associated with the correct noun is the only effective cure {experts suggest that shaming, often used in early America, is an effective punishment}. But rewording to avoid the participle or gerund may be preferable {I believe that because Meg is a thoughtful mother, she gives her children good advice}. Using passive voice in an independent clause can also produce a dangler. In Finding that the questions were not ambiguous, the exam grades were not changed, the participle finding“dangles” because there is no logical subject to do the finding. The sentence can be corrected by using active voice instead of passive, so that the participle precedes the noun it modifies {finding that the questions were not ambiguous, the teacher did not change the exam grades}. Quite often writers will use it or there as the subject of the independent clause after a participial phrase, thereby producing a dangler without a logical subject, as in Reviewing the suggestions, it is clear that no consensus exists. (A possible revision: Our review of the suggestions shows that no consensus exists.)