In this paper we study the projective normality of certain Artin-Schreier curves Y_f defined over a field F of characteristic p by the equations y^q+y=f(x), q being a power of p and f in F[x] being a polynomial in x of degree m, with (m,p)=1. Many Y_f curves are singular and so, to be precise, here we study the projective normality of appropriate projective models of their normalization.

We present identities of various kinds for generalized q-ApostolBernoulli and Apostol-Euler polynomials and power sums, which resemble q-analogues of formulas from the 2009 paper by Liu and Wang. These formulas are divided into two types: formulas with only q-ApostolBernoulli, and only q-Apostol-Euler polynomials, or so-called mixed formulas, which contain polynomials of both kinds. This can be seen as a logical consequence of the fact that the q-Appell polynomials form a commutative ring. The functional equations for Ward numbers operating on the q-exponential function, as well as symmetry arguments, are essential for many of the proofs. We conclude by finding multiplication formulas for two q-Appell polynomials of general form. This brings us to the q-H polynomials, which were discussed in a previous paper.

A proper edge coloring of a class 2 graph G is minimal if it contains a color class of cardinality equal to the resistance r(G) of G, which is the minimum number of edges that have to be removed from G to obtain a graph which is Δ(G)-edge colorable, where Δ(G) is the maximum degree of G. In this paper using some properties of minimal edge colorings of a class 2 graph and the notion of reflective edge colorings of the direct product of two graphs, we are able to prove that the double graph of a class 2 graph is of class 1. This result, recently conjectured, is moreover extended to some generalized double graphs.

We prove an extension of the Babbage-Enriques-Petri theorem for semi-canonical curves. We apply this to show that the Prym variety of a generic element of a codimension $k$ subvariety of $\kr_g$ is not isogenous to another distinct Prym variety, under some mild assumption on $k$.