arXiv subjects

Afshan Sadiq

Publications and source records attributed to Afshan Sadiq.

6 recordsLinked to original sources

arXiv2016

Augmented Zagreb Index of Polyhex Nanotubes

Augmented Zagreb Index is a newly defined degree based topological invariant which has been well established for its better correlation properties and is defined as $AZI(G)= \sum_{uv\in E(G)}(\frac{d_G (u)d_G (v)}{d_G (u)+ d_G (v)-2})^3 $, where $E(G)$ is the edge set of graph $G$ and $d(u),\,\,d(v)$ are the degrees of the end vertices $u$ and $v$ of edge $uv$, respectively. It has outperformed many well known degree based topological indices. In this article we give closed formulae for the augmented Zagreb index of arm-chair polyhex and zigzag edge polyhex nanotubes.

math.CO↗

arXiv2014

On primary decomposition of modules

Primary decomposition is a very important tool of commutative algebra and geometry. In this paper we generalized some of the existing algorithms of primary decomposition developed by Eisenbud et al. (cf. [EHV]) for free modules and also filled some gaps by providing proofs of important theorems(2.9, 2.12, 2.13) appeared in [EHV]. All these algorithms are programmed and implemented in {\sc Singular}.

math.AC↗

arXiv2014

An Algorithm to Compute a Primary Decomposition of Modules in Polynomial Rings over the Integers

We present an algorithm to compute the primary decomposition of a submodule $\mathcal{N}$ of the free module $\Z[x_1, \ldots, x_n]^m$. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of $\mathcal{N}$, i.e. the minimal associated primes of the ideal $\Ann(\Z[x_1, \ldots, x_n]^m /\mathcal{N})$ in $\Z[x_1,\ldots,x_n]$ and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in {\sc Singular}.

math.AC↗

arXiv2011

An algorithm for primary decomposition in polynomial rings over the integers

We present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama-Yokoyama resp. Eisenbud-Hunecke-Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in SINGULAR. Examples and timings are given at the end of the article.

math.AC↗

arXiv2010

F4 Algorithm For Euclidean Rings

This short note is the generalization of Faugere F4-algorithm for polynomial rings with coefficients in Euclidean rings. This algorithm computes successively a Groebner basis replacing the reduction of one single s-polynomial in Buchberger's algorithm by the simultaneous reduction of several polynomials.

math.AC↗

arXiv2009

Standard Bases over Rings

The theory of standard bases in polynomial rings with coefficients in a ring R with respect to local orderings is developed. R is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in R.

math.AC↗