Volume Of Convex Hull Research Articles

Comment on “Volumes spanned by random points in the hypercube”

AbstractWe symbolically evaluate a definite integral (due to Dyer, Füredi & McDiarmid 2) in connection with threshold phenomena for convex hull volumes strictly within the unit d‐dimensional hypercube as d → ∞. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

FIRE REDUCES MORPHOSPACE OCCUPATION IN PLANT COMMUNITIES

The two main assembly processes claimed to structure plant communities are habitat filtering and competitive interactions. The set of species growing in fire-prone communities has been filtered in such a way that species without fire-persistence traits have not successfully entered the community. Because plant traits are evolutionarily conserved and fire traits are correlated with other plant traits, communities under high fire frequency should not include all possible trait combinations, and thus the morphospace occupation by species in these communities should be lower than expected by chance (underoccupied). In contrast, communities under low fire frequency would lack the filtering factor, and thus their underoccupation of the morphospace is not expected. We test this prediction by comparing the morphospace occupation by species in communities located in the western Mediterranean Basin, five of them subject to high fire frequency (HiFi) and four to low fire frequency (LowFi). We first compile a set of morphological and functional traits for the species growing on the nine sites, then we compute the morphospace occupation of each site as a convex hull volume, and finally, to assert that our results are not a product of a random branching pattern of evolution, we simulate our traits under a null model of neutral evolution and compare the morphospace occupation of the simulated traits with the results from the empirical data. The results suggest that, as predicted, there is a clear differential morphospace occupation between communities under different fire regimes in such a way that the morphospace is underoccupied in HiFi communities only. The simulation of a neutral evolutionary model does not replicate the observed pattern of differential morphospace occupation, and thus it should be attributed to assembly processes. In conclusion, our results suggest that fire is a strong community assembling process, filtering the species that have fire-persistent traits and thus assembling phenotypically and phylogenetically clustered communities with vacant zones in the morphospace.

NEW MULTIDIMENSIONAL FUNCTIONAL DIVERSITY INDICES FOR A MULTIFACETED FRAMEWORK IN FUNCTIONAL ECOLOGY

Functional diversity is increasingly identified as an important driver of ecosystem functioning. Various indices have been proposed to measure the functional diversity of a community, but there is still no consensus on which are most suitable. Indeed, none of the existing indices meets all the criteria required for general use. The main criteria are that they must be designed to deal with several traits, take into account abundances, and measure all the facets of functional diversity. Here we propose three indices to quantify each facet of functional diversity for a community with species distributed in a multidimensional functional space: functional richness (volume of the functional space occupied by the community), functional evenness (regularity of the distribution of abundance in this volume), and functional divergence (divergence in the distribution of abundance in this volume). Functional richness is estimated using the existing convex hull volume index. The new functional evenness index is based on the minimum spanning tree which links all the species in the multidimensional functional space. Then this new index quantifies the regularity with which species abundances are distributed along the spanning tree. Functional divergence is measured using a novel index which quantifies how species diverge in their distances (weighted by their abundance) from the center of gravity in the functional space. We show that none of the indices meets all the criteria required for a functional diversity index, but instead we show that the set of three complementary indices meets these criteria. Through simulations of artificial data sets, we demonstrate that functional divergence and functional evenness are independent of species richness and that the three functional diversity indices are independent of each other. Overall, our study suggests that decomposition of functional diversity into its three primary components provides a meaningful framework for its quantification and for the classification of existing functional diversity indices. This decomposition has the potential to shed light on the role of biodiversity on ecosystem functioning and on the influence of biotic and abiotic filters on the structure of species communities. Finally, we propose a general framework for applying these three functional diversity indices.

An MEG parameter for estimation of good outcome of focal epilepsy surgery

Abstract Epilepsy surgery is a viable therapy option especially for patients with a well-defined focus. A parameter for estimation of good surgical outcome, acquired during presurgical workup using MEG dipole localizations, is presented. MEG data of 22 surgically treated epilepsy patients were analyzed retrospectively. Data had been recorded during presurgical workup using a 74-channel dual-unit biomagnetometer (Magnes II, 4D-Neuroimaging, CA, USA). Individual MRI was co-registered. Data were inspected visually and analysed using single dipole localization in a BEM volume conductor model (boundary element method), using an automated process (Curry 4.6, Compumedics Neuroscan, El Paso, TX, USA). Resulting localizations were inserted additively into a 3D-volume. This analysis resulted in continuous dipole frequency volumes with local maxima at loci designated by an increased amount of dipole localizations. The volume of the smallest hull, enveloping loci with dipole localization amounts above the 99.9th percentile, was calculated and normalized to the total number of localizations. The resulting convex hull volume (“chv”) values were then compared to surgery outcome. A threshold for good outcome was estimated by calculating the minimum parameter value of patients with Engel outcomes 3 and 4. Lowest chv value of patients with Engel outcomes 3 and 4 was 1.22 mm3/localization. Using this value as threshold for good outcome, 11 patients with outcomes 1 and 2 were under this threshold. The 6 patients with outcomes 3 and 4 yielded results above the threshold, as well as 6 patients with outcomes 1 and 2. This distribution was found to be significant (p

A TRAIT-BASED TEST FOR HABITAT FILTERING: CONVEX HULL VOLUME

Community assembly theory suggests that two processes affect the distribution of trait values within communities: competition and habitat filtering. Within a local community, competition leads to ecological differentiation of coexisting species, while habitat filtering reduces the spread of trait values, reflecting shared ecological tolerances. Many statistical tests for the effects of competition exist in the literature, but measures of habitat filtering are less well-developed. Here, we present convex hull volume, a construct from computational geometry, which provides an n-dimensional measure of the volume of trait space occupied by species in a community. Combined with ecological null models, this measure offers a useful test for habitat filtering. We use convex hull volume and a null model to analyze California woody-plant trait and community data. Our results show that observed plant communities occupy less trait space than expected from random assembly, a result consistent with habitat filtering.

Semi-Online Maintenance of Geometric Optima and Measures

We give the first nontrivial worst-case results for dynamic versions of various basic geometric optimization and measure problems under the semi-online model, where during the insertion of an object we are told when the object is to be deleted. Problems that we can solve with sublinear update time include the Hausdorff distance of two point sets, discrete 1-center, largest empty circle, convex hull volume in three dimensions, volume of the union of axis-parallel cubes, and minimum enclosing rectangle. The decision versions of the Hausdorff distance and discrete 1-center problems can be solved fully dynamically. Some applications are mentioned.