Colouring the Triangles Determined by a Point Set

Let P be a set of n points in general position in the plane. We study the chromatic number of the intersection graph of the open triangles determined by P. It is known that this chromatic number is at least n^3/27+O(n^2), and if P is in convex position, the answer is n^3/24+O(n^2). We prove that for arbitrary P, the chromatic number is at most n^3/19.259+O(n^2)

Training Management Information System of the Defense Institute of Security Assistance Management: User Satisfaction as a Measure of Its Effectiveness

The purpose of this study was to evaluate the effectiveness of the Training Management System (TMS) installed in the Security Assistance Organizations around the world. User satisfaction was measured as an indicator of the system\u27s effectiveness. In order to provide an objective measurement of the system effectiveness, the following research questions were addressed: (1) What is the system effectiveness regarding the level of product quality provided by TMS? (2) What is the level of involvement and knowledge of TMS user related to the information services function? (3) What is the level of user perceived satisfaction with the staff and services provided by support people of TMS? (4) What is the perceived difference in levels of satisfaction between military and civilian for each of the questions 1, 2 and 3 above? (5) What is the impact of experience with the system on questions 1 to 3 above? User satisfaction was determined to be the best possible measure of system effectiveness and it was measured by administering a user satisfaction survey. The data gathered from this survey was analyzed and that analysis provided the basis for concluding that TMS was meeting the users\u27 needs, but that the system effectiveness could be improved by providing training. Recommendations were offered to the TMS staff support and suggestions for further research were also given

Exotic Ideals in Free Transformation Group C∗C^*-Algebras

Let $\Gamma$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_\eta \Gamma$ be the completion of the convolution algebra $C_c(\Gamma,C(X))$ with respect to a $C^*$-norm $\eta$. A non-zero ideal $J \unlhd C(X) \rtimes_\eta \Gamma$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $\Gamma$ is non-amenable and there is an invariant probability measure on $X$. This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen. Using the Koopman representation and a recent theorem of Elek, we show that when $\Gamma$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $\Gamma$ on $X$ and a $C^*$-norm $\eta$ on $C_c(\Gamma, C(X))$ such that $C(X)\rtimes_\eta\Gamma$ contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen. The opaque and grey ideals in $C(X)\rtimes_\eta \Gamma$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $\Gamma$ is free, however the problem of whether these ideals can be non-zero was left unresolved. We present an example of a free action of $\Gamma$ on a compact Hausdorff space $X$ along with a $C^*$-norm $\eta$ for which these ideals are non-trivial, in particular, they are exotic ideals.Comment: Article is totally rewritten, reorganized, and has a new title (former title: "Exotic Ideals in Represented Free Transformation Groups") Includes some new results. 16 page

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an \'etale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn \'etale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.Comment: Corrected a typo in Lemma 2.14 in the published versio