Syzygies, Betti numbers and regularity of cover ideals of certain multipartite graphs

Let $G$ be a finite simple graph on $n$ vertices. Let $J_G \subset K[x_1, \ldots, x_n]$ be the cover ideal of $G$. In this article, we obtain syzygies, Betti numbers and Castelnuovo-Mumford regularity of $J_G^s$ for all $s \geq 1$ for certain classes of graphs $G$.Comment: Title and other minor change

The Marriages of Intellectual Property & Insurance

The marriages of Intellectual Property and Insurance may sound a magical word for all those companies which are playing the game of intellectual property in their business. These companies manage their intellectual property under the guidance of IP management and an IP policy in the company. These companies earn a huge amount of money in this game; but there are also chances for the companies to face the risks ahead. These expected risks can be minimized to some extent, if the companies understand the meaning and the importance of this new term “Intellectual Property Insurance”. In order to make the companies appreciate the depth significance of the term, the paper is written and the paper emphasizes the need of IP insurance in the business, the role of IP insurance in mitigating the risks involved in the business and the framework of IP insurance policy etc.Intellectual Property, Insurance, Intellectual Property Insurance

The Maximum Exposure Problem

Given a set of points P and axis-aligned rectangles R in the plane, a point p in P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k^2) rectangles, we can expose at least Omega(1/k) of the optimal number of points