In this paper we give an application of equivariant moduli spaces to the study of smooth group actions on certain 4-manifolds. A rich source of examples for such actions is the collection of algebraic surfaces (compact and nonsingular) together with their groups of algebraic automorphisms. From this collection, further examples of smooth but generally nonalgebraic actions can be constructed by ...

The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab(P2). We show that Stab†(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab (P2) and show that Stab†(P2) is contractible. Introduction Motivated by the concept of Π-stability conditio...

For a simple path Pr on r vertices, the square of Pr is the graph P2 r on the same set of vertices of Pr , and where every pair of vertices of distance two or less in Pr is connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k, k + 1, . . . , k + q − 1, so that the induc...

For simplicity, we use the following convention: Y is a non empty set, P1, P2 are partitions of Y , A, B are subsets of Y , i is a natural number, x, y, x1, x2, z0 are sets, and X, V , d, t, S1, S2 are sets. The following proposition is true (1) If X ∈ P1 and V ∈ P1 and X ⊆ V, then X = V. Let us consider S1, S2. We introduce S1 ⋐ S2 and S2 ⋑ S1 as synonyms of S1 is finer than S2. We now state s...

Let G = (V, E) be a connected simple graph. A labeling f : V → Z2 induces an edge labeling f∗ : E → Z2 defined by f∗(xy) = f(x) + f(y) for each xy ∈ E. For i ∈ Z2, let vf (i) = |f−1(i)| and ef (i) = |f∗−1(i)|. A labeling f is called friendly if |vf (1)− vf (0)| ≤ 1. The full friendly index set of G consists all possible differences between the number of edges labeled by 1 and the number of edge...

Let X be a continuum, that is a compact, connected, nonempty metric space. The span of X is the least upper bound of the set of real numbers r which satisfy the following conditions: there exists a continuum, C, contained in X × X such that d(x, y) is larger than or equal to r for all (x, y) in C and p1(C) = p2(C), where p1, p2 are the usual projection maps. The following question has been aske...