In this paper we use duality to construct new classes of Hopf orders in the group algebraKCp3 , where K is a finite extension of Qp and Cp3 denotes the cyclic group of order p3. Included in this collection is a subcollection of Hopf orders which are realizable as Galois groups.

This paper introduces parity versions of familiar graph theoretic results, in particular results related to 2-connectedness. The even and odd circuit connected graphs are characterized. The realizable, even-realizable, alternating-realizable, dual realizable and dual even-realizable graphs are classified.

topological graph (AT-graph): A = (G, R); G = (V, E) is a graph, R ⊆ ( E 2 ) in a topological graph T ... RT = set of crossing pairs of edges AT-graph A is realizable if there exists a topological graph T which is a drawing of G and RT = R. simply realizable ... T is simple rectilinearly realizable ... edges of T are straight-line segments weakly realizable ... RT ⊆ R Abstract topological graph...

A graph G is a 2-tree if G = K3, or G has a vertex v of degree 2, whose neighbors are adjacent, and G − v is a 2-tree. Clearly, if G is a 2-tree on n vertices, then |E(G)| = 2n − 3. A non-increasing sequence π = (d1, . . . , dn) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. Yin and Li (Acta Mathematica Sinica, English Series, 25(2009)795–80...

Let R be a real closed field, P,Q ⊂ R[X1, . . . , Xk] finite subsets of polynomials, with the degrees of the polynomials in P (resp. Q) bounded by d (resp. d0). Let V ⊂ Rk be the real algebraic variety defined by the polynomials in Q and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizabl...