We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold M is at least the weight of the fundamental group of M . This lower bound is sharp for the 3-manifolds RP, L(3, 1), L(5, 2), S1 × S1 × S1, S2 × S1, S×− S1 and...

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of bigraded $K[x,y]$-module $M$, where $K$ is field. The takes as input short chain complex free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\operatorname{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ mem...

One of the most important computations in algebraic geometry or commutative algebra that a computer algebra system should provide is the computation of finite free resolutions of ideals and modules. Resolutions are used as an aid to understand the subtle nature of modules and are also a basis of further computations, such as computing sheaf cohomology, local cohomology, Ext, Tor, etc. Modern me...

i Conventional distributed adaptive routing strategies usually work well for packet switch ng networks only in the absence of link/node failures. However, they cannot avoid looping messages for an extended period in case of link/node failures. In this paper, we develop a multi-order routing strategy which is loop-free even in the s presence of link/node failures. Unlike most conventional method...