The fundamental group is an invariant of topological spaces that measures the contractibility of loops. This project studies the fundamental group, its basic properties, some elementary computations, and a resulting theorem: the Fixed Point Theorem of Brouwer, which states that every continuous function mapping the disk to itself has a fixed point.

In Alexandrov spaces of curvature bounded either above (CBA) or below (CBB), we obtain extrinsic curvature bounds on subspaces associated with semiconcave functions. For CBA spaces, we obtain new intrinsic curvature bounds on subspaces. For CBB spaces whose boundary is extrinsically curved, we strengthen Perelman’s concavity theorem for distance from the boundary, deriving corollaries on sharp ...

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph Cm × Pn+1. We distinguish two types of Hamiltonian cycles depending on their contractibility (as Jordan curves) and denote their numbers h m (n) and hm(n). For fixedm, both of them satisfy linear homogeneous recurrence relations with constant coefficients. We derive their generating functions and other related result...

The pervasiveness of research agreements between pharmaceutical and biotechnology companies is puzzling, since it is hard to contract on the exact nature of the research activities. A major concern of financing companies is that the R&D firms use their funding to subsidize other projects or substitute one project for another. We develop a model based on the property-rights theory of the firm th...

To any graph G we can associate a simplicial complex ∆(G) whose simplices are the complete subgraphs of G, and thus we say that G is contractible whenever ∆(G) is so. We study the relationship between contractibility and K-nullity of G, where G is called K-null if some iterated clique graph of G is trivial. We show that there are contractible graphs which are not K-null, and that any graph whos...

Abstract Two reduced projective schemes are said to be Cremona equivalent if there is a map that maps one in the other. In this paper I revise some of known results about equivalence and extend main result Mella Polastri (Bull Lond Math Soc 41(1):89–93, 2009) [20] reducible schemes. This allows prove very general contractibility for union rational subvarieties.