We construct virtual fundamental classes in all intersection theories including Chow theory, K-theory and algebraic cobordism for quasi-projective Deligne-Mumford stacks with perfect obstruction prove the pullback formula, torus localization formula cosection principle.

The figure eight is a recently discovered periodic solution to the Newtonian three-body problem in which three equal masses traverse a single closed planar curve which has the form of a figure eight (figure 1). See [Moore], and [CM]. In particular, it has one point of self-intersection, the origin, which divides the eight into two symmetric parts, its two lobes. In [CM] it was proved that each ...

The intersection graph $mathbb{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbb{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the fi...