The Diophantine theory studies the rational solutions (X, Y, Z) ∈ Q of equation (1). It is convenient to ignore the trivial solution (0, 0, 0) and to identify solutions if they differ by multiplication by a non-zero scalar. Solutions to (1) are thus viewed as points in the projective plane P2(Q). Let E(Q) ⊂ P2(Q) denote this solution set. More generally, if F is any field, let E(F ) ⊂ P2(F ) be...

tance less than 8ep from the point P, then X and P lie together in a connected sub-set of M every point of which is at a distance of less than e from the point P. The setM is said to be uniformly connected im kleinen if for every positive number e there exists a positive number 68 such that if P1 and P2 are two points of M at a distance apart less than 58 then they lie together in a connected s...

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists F ⊆ V , |F | ≤ k, such that G[V \ F ] is a forest and G[F ] is connected. We show that Connected ...

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. Lu and Guo (submitted for publication) [3] recently introduced strong quasi-Hamiltonianconnectivity of a multipartite tournament D as follows: For any two distinct vertices x and y ofD, there is a pathwith at least one vertex from each partite set ofD from x to y and from y to x. We obtain the definition for...

A graph H has the property MT, if for all graphs G, G is H-free if and only if every minimal (chordal) triangulation of G is H-free. We show that a graph H fulfils property MT if and only if H is edgeless, H is connected and is an induced subgraph of P5, or H has two connected components and is an induced subgraph of 2P3. This completes results of Parra and Scheffler, who have shown that MT hol...

Let M be a compact, connected, C2-smooth and globally minimal hypersurface M in P2(C) which divides the projective space into two connected parts U+ and U. We prove that there exists a side, U or U+, such that every continuous CR function on M extends holomorphically to this side. Our proof of this theorem is a simplification of a result originally due to F. Sarkis.