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The phrase "mutual influence of" is correct and usable in written English.
You can use it to describe two or more entities that influence one another. For example, "The mutual influence of economic and social conditions affects the development of a city."
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There was clearly mutual influence of Indo-Aryan languages at an early time, together with movement of groups of speakers (compare the position of Pahari).
John Quincy Adams put it nicely when he observed that "the mutual influence of these two mighty minds on each other is a phenomenon, like the invisible and mysterious movements of the magnet in the physical world".
Mutual influence of ions was analyzed in multicomponent systems.
Instead, SNA emphasizes the mutual influence of structure and social connections.
Furthermore, the mutual influence of attributes is also indicated on the same matrix.
The mutual influence of both outer flow non-uniformity and blade oscillations has been investigated.
The boundary conditions used accounted for the mutual influence of the individual points.
These small but sharp singularities suffer due to the mutual influence of their adjacent high energy transitions [20], e.g., Figure 16 explains this phenomenon, where small edges are suppressed due to the mutual influence of its adjacent sharp transition.
Sarcletti et al. researched the mutual influence of surface energy and substrate temperature on the mobility in organic semiconductors [23].
With the combination of different enterprise design documents, a mutual influence of the chain system has been formed.
Moreover, it determines the establishment of mutual influence of nanomaterial-to-cell, cell-to-cell, and cell-to-substrate.
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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