On linear resolution of powers of an ideal

In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k},$ and $J_{1}^{k}$ have linear resolution if and only if $k\neq 2.$ The procedures we include in this work is encoded in computer algebra package CoCoA.Comment: 10 pages, to appear in Osaka Journal of Mathematic

AN IMPROVED GENETIC ALGORITHM WITH A LOCAL OPTIMIZATION STRATEGY AND AN EXTRA MUTATION LEVEL FOR SOLVING TRAVELING SALESMAN PROBLEM

The Traveling salesman problem (TSP) is proved to be NP-complete in most cases. The genetic algorithm (GA) is one of the most useful algorithms for solving this problem. In this paper a conventional GA is compared with an improved hybrid GA in solving TSP. The improved or hybrid GA consist of conventional GA and two local optimization strategies. The first strategy is extracting all sequential groups including four cities of samples and changing the two central cities with each other. The second local optimization strategy is similar to an extra mutation process. In this step with a low probability a sample is selected. In this sample two random cities are defined and the path between these cities is reversed. The computation results show that the proposed method also finds better paths than the conventional GA within an acceptable computation time

On Hyper Hoop-algebras

In this paper, we apply the hyper structure theory to hoop-algebras and introduce the notion of (quasi) hyper hoop-algebra which is a generalization of hoop-algebra and investigate some related properties. We also introducethe notion of (weak)filters on hyper hoop-algebras, and give several properties of them. Finally, we characterize the (weak) filter generated by a non-empty subset of a hyper hoop-algebra

On the regularity and defect sequence of monomial and binomial ideals

summary:When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$, with homogeneous maximal ideal $\mathfrak {m}$, it is known that for an ideal $I$ of $S$, the regularity of powers of $I$ becomes eventually a linear function, i.e., ${\rm reg}(I^m)=dm+e$ for $m\gg 0$ and some integers $d$, $e$. This motivates writing ${\rm reg}(I^m)=dm+e_m$ for every $m\geq 0$. The sequence $e_m$, called the \emph {defect sequence} of the ideal $I$, is the subject of much research and its nature is still widely unexplored. We know that $e_m$ is eventually constant. In this article, after proving various results about the regularity of monomial ideals and their powers, we give several bounds and restrictions on $e_m$ and its first differences when $I$ is a primary monomial ideal. Our theorems extend the previous results about $\mathfrak {m}$-primary ideals in the monomial case. We also use our results to obtatin information about the regularity of powers of a monomial ideal using its primary decomposition. Finally, we study another interesting phenomenon related to the defect sequence, namely that of regularity jump, where we give an infinite family of ideals with regularity jumps at the second power