I have received from the author a reprint of his paper
I.A. Stepanov, Thermodynamics of substances with negative thermal expansion and negative compressibility, Journal of Non-Crystalline Solids 356 (2010) 1168-1172
where it is claimed:
It is shown that for substances with positive thermal expansion and positive compressibility, and for substances with negative thermal expansion and negative compressibility, δQ = dU + PdV, but for substances with positive thermal expansion and negative compressibility, and for substances with negative thermal expansion and positive compressibility, δQ = dU − PdV.
Igor tries to prove above for quite awhile, see for example my review on his previous paper (in Russian)
yet I have not mentioned the progress. The prove is based on Eq (22) where the author assumes that (dS/dV)_U = (dS/dV)_T, however it has not been explained how the author derives this equation from the correct equation that is written directly below Eq (22). The only explanation in the paper, though repeated many times througout the text, is that when U is constant then T is constant. Yet, such a statement makes no difference. Let us consider a function in two variables z(x, y). Its differential is expressed as
dz = (dz/dx)_y dx + (dz/dy)_x dy
Let us suppose that we say z is constant and y is constant. Does this imply that (dz/dx)_y is equal to zero in this case? Definitely not. So I can only advise the author to learn infinitesimal calculus properly.