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The freeness of hyperplane arrangements in a three dimensional vector space over finite field is discussed. We prove that if the number of hyperplanes is greater than some bound, then the freeness is determined by the characteristic polynomial.
We have continued, by utilizing Nevanlinna's value distribution theory, our previous studies on the existence or solvability of meromorphic solutions of functional equations with constant coefficients to that of similar types of functional equations with meromorphic (small functions) coefficients. The results obtained are relating to value sharing or unicity of meromorphic functions.
We give an interpretation of the Scharfetter-Gummel (SG) scheme in the theory of semiconductor devices. The key fact is that the SG scheme is based on a harmonic relation between the Green function and the Green matrix for a two-point boundary value problem. An a consequence of our interpretation, we obtain error estimates in $L^r$, $r\in[2,\infty]$, where the coefficient functions may be discontinuous.