Let M be a smooth manifold with a circle action, and {P } be the fixed point sets. The problem I want to discuss in this paper is how to get the topological information of one relatively complicated fixed point set, say P 0 , from the other much simpler fixed points. Such problems are interesting in symplectic geometry and geometric invariant theory, especially in the study of moduli spaces. In...

Blowing up a point p in a manifold M builds a new manifold M̂ in which p is replaced by the projectivization of the tangent space TpM . This well–known operation also applies to fixed points of diffeomorphisms, yielding continuous homomorphisms between automorphism groups of M and M̂ . The construction for maps involves a loss of regularity and is not unique at the lowest order of differentiabili...

A partial orthomorphism of a group G (with additive notation) is an injection π : S → G for some S ⊆ G such that π(x) − x 6= π(y) − y for all distinct x, y ∈ S. We refer to |S| as the size of π, and if S = G, then π is an orthomorphism. Despite receiving a fair amount of attention in the research literature, many basic questions remain concerning the number of orthomorphisms of a given group, a...

In this paper, we study pattern matching conditions in the cycle structures of permutations. We develop methods to find the joint distribution of the number of cycles and the number of cycle descents over the set of permutations in the symmetric group Sn which have no cycle occurrences or no consecutive cycle occurrences of a given pattern or set of patterns. Our methods also allow us to refine...

‎inspired by the work of suzuki in‎ ‎[t. suzuki‎, ‎a generalized banach contraction principle that characterizes metric completeness‎, proc‎. ‎amer‎. ‎math‎. ‎soc. ‎136 (2008)‎, ‎1861--1869]‎, ‎we prove a fixed point theorem for contractive mappings‎ ‎that generalizes a theorem of geraghty in [m.a‎. ‎geraghty‎, ‎on contractive mappings‎, ‎proc‎. ‎amer‎. ‎math‎. ‎soc., ‎40 (1973)‎, ‎604--608]‎an...