In this paper we present two algorithms for irreducible decomposition of monomial ideals. We first use staircase structures to study the monomial ideals. We generalize the shifting degrees rule from two variables to three variables and then arbitrary case. With the aid of monomial tree representation, a new algorithm for irreducible decomposition of monomial ideals is provided. For the second method, we associate a monomial ideal with a Scarf complex, which is introduced by Herbert Scarf. Every facet of the Scarf complex corresponds to an irreducible component, and vice versa. Milowski(2004) developed a method to enumerate all the facets based on reverse search by exchanging one monomial at a time. We define a facet graph where nodes are facets and there is an edge from a facet F A to another facet F B if F B can be obtained from F A by exchanging one vertex. We give a new generic deformation called ordinal deformation and prove that the facet graph of an ordinally generic monomial ideal is strongly connected. This yields a simpler and more efficient enumeration algorithm.

We demonstrate a software system that runs on smartphones and allows two or more geographically dispersed participants to collaborate on the solution of mathematical problems. We show how participants can create and join a collaborative session, how they can use a virtual whiteboard for the exchange of geometrical drawings and of mathematical formulas, and how they can communicate using text messages. We demonstrate the turn-management mechanism provided by our system, and we show how the system facilitates crossreferencing within the session.