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In this music for the violin, the strange sign on top of both E-flat on measure 74 confused me, as I have no idea what it is, and it is not exclusive to just E-flats, as it is also present in an F and a B previously in the sheet-music.
The Piece is Arlington Sketches. What can it mean?
The arrows could refer to the finger positions. The first E-flat is low 1st finger. The second one low 4th. If the F you mention is on the E-string it is low 1st, if it is on the D-string it is low 2nd. And so forth.
It is common in violin sheet music to indicate low finger positions with down arrows, especially in music with pedagogical intention.
EDIT: Now that the OP has posted another screenshot it is even more clear that it is in fact low fingering that is the matter as I described above. All those notes which have a down arrow are low finger positions on the violin.
Beginners of violin playing are used to learn first little melodies in major on the E, A and D string: the motifs like DoReMiDo will be (analog to Guitar frets) 0,2,4,0.
The arrows above F, Bb and Eb are assigning that the finger position is now lower compared by F#, B and E.
So I agree with Lars Peter Schultz, additionally trying to show the pedagogical intention as reminder of the natural F and the flattened B and E.
The down- or up-arrow is often pencilled in by performers to indicate intonation (as in, "this note should be played lower [higher] in pitch than feels natural"). I've never seen it actually used in printed music, but that would be consistent with the context of those four notes in the example.
It is probably a reminder of the notation rule that a flat at the same pitch lasts for the whole measure, but any other octaves of the same note have the pitch indicated with their own accidentals.
I suggest it's telling you which string to use. In the passage that starts D up to A, you'd play those on the open D and A strings.
For the following descending G F E you could then switch back to the D string - but you'd quickly need to change down to the G string to play the C.
So in the first passage the down arrow means that instead of the D string, use the G string for the next few notes. You can then play the descending passage G F E C without needing to change string part way through.
In general, the down arrow means "skip a string" to play the next note.
