On Convex Geometric Graphs with no k+1k+1 Pairwise Disjoint Edges

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the extremal examples are completely characterized. For all other values of $k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the $kn$ "longest possible" edges of $CK(n)$, the complete CGG of order $n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $q \leq n-2k$ and that this value of $q$ is optimal. We generalize our discussion to the following question: what is the maximal possible number $f(n,k,q)$ of edges in a CGG on $n$ vertices that does not contain $k+1$ pairwise disjoint edges, and, in addition, admits an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining $f(n,k,q)$ for all relevant values of $n,k$ and $q$.Comment: 17 pages, 9 figure

Characterization of co-blockers for simple perfect matchings in a convex geometric graph

Consider the complete convex geometric graph on $2m$ vertices, $CGG(2m)$, i.e., the set of all boundary edges and diagonals of a planar convex $2m$-gon $P$. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect Matchings in a Convex Geometric Graph], the smallest sets of edges that meet all the simple perfect matchings (SPMs) in $CGG(2m)$ (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$. In this paper we characterize the co-blockers for SPMs in $CGG(2m)$, that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings $M$ in $CGG(2m)$ where all edges are of odd order, and two edges of $M$ that emanate from two adjacent vertices of $P$ never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with $m$, the number of co-blockers grows super-exponentially.Comment: 8 pages, 4 figure

Identification and properties of molecular systems of potential use in solar-pumped lasers

The concepts and computational tools of theortical chemistry are used to investigate molecular properties needed in direct solar-pumped lasers. Compounds of the type RR'CXY, with R and R' organic groups, and X and Y halide atoms were identified as likely candidates because of their highly enhanced absorption coefficients over compounds with a single halide atom. The use of a combination of vibrational excitation followed by electronic excitation to enhance quantum yields at certain wavelengths is indicated. A self-consistent eikonal approximation to state-to-state transitions was tested for CH3I and is useful for other problems involving electronic energy and charge transfer. An approach to calculate potential energy surfaces and transition dipoles was developed which is based on the generation of eigenstates of the nonrelativisitc Hamiltonian followed by incorporation of the spin-orbit coupling by configuration interaction

On Ecological Fallacy and Assessment Errors Stemming From Misguided Variable Selection: Investigating the Effect of Data Aggregation on the Outcome of Epidemiological Study

In behavioral studies, ecological fallacy is a wrong assumption about an individual based on aggregate data for a group. In the present study, the validity of this assumption was tested using both individual estimates of exposure to air pollution and aggregate air pollution data estimated for 1,492 schoolchildren living in the in vicinity of a major coal-fired power station in the Hadera sub-district of Israel. In 1996 and 1999, the children underwent subsequent pulmonary function (PF) tests, and their parents completed a detailed questionnaire on their health status, and housing conditions. The association between children’s PF development and their long-term exposure to air pollution was then investigated in two phases. During the first phase, the average rates of PF change observed in small statistical areas in which the children reside were compared with average levels of air pollution detected in these areas. During the second phase of the analysis, an individual pollution estimate was calculated for each child covered by the survey, using a "spatial join" tool in ArcGIS. While the analysis of aggregate data showed no significant differences in the PF development among the schoolchildren surveyed, the comparison of individual pollution estimates with the results of PF tests detected a significant negative association between changes in PF results and the estimated level of air pollution. As argued, these differences are attributed to the fact that average exposure levels are likely to cause a misclassification bias of individual exposure, as further demonstrated in the study using pattern detection techniques of spatial analysis (local Moran's I and Gettis-Ord statistic). The implications of the results of the analysis for geographical and epidemiological studies are discussed, and recommendations for public health policy are formulated.

Realization of SU(N) Kondo effect in strong magnetic field

In this paper we suggest a realization for the SU(N) Kondo effect, using quantum dots at strong magnetic field. We purpose using edge states of the quantum Hall effect as pseudo spin that interact with multiple quantum dots structures. In the suggested realization one can access each pseudo spin separately and hence may perform a set of experiments that were impossible until now. We focus on the realization of SU(2) and SU(3) Kondo and find a conductivity of 3/4 quantum conductance in the SU(3) case.Comment: 5 pages, 2 figures. Including supplementary materia