Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.     

An ordering of dependence for contingency tables

A majorization ordering is defined on matrices with the same row and column sums. This ordering is used as an ordering of dependence for contingency tables. Results are derived for maximal and minimal matrices with respect to the majorization ordering. This theory can be used to maximize and minimize Schur concave functions defined over matrices, when there are row and column sum constraints; in this paper, it is applied to the algorithm of Mehta and Patel (1983) for finding the P-value of Fisher's exact test.     

Corrections to Lee's visibility polygon algorithm

We present a modification and extension of the (linear time) visibility polygon algorithm of Lee. The algorithm computes the visibility polygon of a simple polygon from a viewpoint that is either interior to the polygon, or in its blocked exterior (the cases of viewpoints on the boundary or in the free exterior being simple extensions of the interior case). We show by example that the original algorithm by Lee, and a more complex algorithm by El Gindy and Avis, can fail for polygons that wind sufficiently. We present a second version of the algorithm, which does not extend to the blocked exterior case.This work was partially supported by grants from the Central Research Fund of the University of Alberta and the Natural Sciences and Engineering Research Council of Canada.     

A bargaining game model for measuring performance of two-stage network structures

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs), where the internal structures of DMUs are treated as a black-box. Recently DEA has been extended to examine the efficiency of DMUs that have two-stage network structures or processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model not only provides an overall efficiency score for the entire process, but also yields an efficiency score for each of the individual stages. The current paper develops a Nash bargaining game model to measure the performance of DMUs that have a two-stage structure. Under Nash bargaining theory, the two stages are viewed as players and the DEA efficiency model is a cooperative game model. It is shown that when only one intermediate measure exists between the two stages, our newly developed Nash bargaining game approach yields the same results as applying the standard DEA approach to each stage separately. Two real world data sets are used to demonstrate our bargaining game model.     

Barycentric coordinates for convex sets

In this paper we provide an extension of barycentric coordinates from simplices to arbitrary convex sets. Barycentric coordinates over convex 2D polygons have found numerous applications in various fields as they allow smooth interpolation of data located on vertices. However, no explicit formulation valid for arbitrary convex polytopes has been proposed to extend this interpolation in higher dimensions. Moreover, there has been no attempt to extend these functions into the continuous domain, where barycentric coordinates are related to Green’s functions and construct functions that satisfy a boundary value problem. First, we review the properties and construction of barycentric coordinates in the discrete domain for convex polytopes. Next, we show how these concepts extend into the continuous domain to yield barycentric coordinates for continuous functions. We then provide a proof that our functions satisfy all the desirable properties of barycentric coordinates in arbitrary dimensions. Finally, we provide an example of constructing such barycentric functions over regions bounded by parametric curves and show how they can be used to perform freeform deformations.      

Optoelectronic,Aggregation, and Redox Properties of Double-Rotor Boron Difluoride Hydrazone Dyes

We develop the chemistry of boron difluoride hydrazone dyes (BODIHYs) bearing two aryl substituents and explore their properties. The low-energy absorption bands (λ_max=427–464 nm) of these dyes depend on the nature of the N-aryl groups appended to the BODIHY framework. Electron-donating and extended π-conjugated groups cause a redshift, whereas electron-withdrawing groups result in a blueshift. The title compounds were weakly photoluminescent in solution and strongly photoluminescent as thin films (λ_PL=525–578 nm) with quantum yields of up to 18 % and lifetimes of 1.1–1.7 ns, consistent with the dominant radiative decay through fluorescence. Addition of water to THF solutions of the BODIHYs studied causes molecular aggregation which restricts intramolecular motion and thereby enhances photoluminescence. The observed photoluminescence of BODIHY thin films is likely facilitated by a similar molecular packing effect. Finally, cyclic voltammetry studies confirmed that BODIHY derivatives bearing para-substituted N-aryl groups could be reversibly oxidized (E_ox1=0.62–1.02 V vs. Fc/Fc⁺) to their radical cation forms. Chemical oxidation studies confirmed that para-substituents at the N-aryl groups are required to circumvent radical decomposition pathways. Our findings provide new opportunities and guiding principles for the design of sought-after multifunctional boron difluoride complexes that are photoluminescent in the solid state.     

A comparative sonochemical reaction that is independent of the intensity of ultrasound and the geometry of the exposure apparatus

Sonolysis of aqueous solutions of n-alkyl anionic surfactants results in the formation of secondary carbon-centered radicals (-*CH-). The yield of -*CH- depends on the bulk surfactant concentration up to a maximum attainable radical yield (the 'plateau yield') where an increasing surfactant concentration (below the critical micelle concentration) no longer affects the -*CH- yield. In an earlier study it was found that the ratio of -*CH- detected following sonolysis of aqueous solutions of sodium pentane sulfonate (SPSo) to that of sodium dodecyl sulfate (SDS) (i.e. CH(SPSo)/CH(SDS)) depended on the frequency of sonolysis, but was independent of the ultrasound intensity, at the plateau concentrations [J.Z. Sostaric, P. Riesz, Adsorption of surfactants at the gas/solution interface of cavitation bubbles: an ultrasound intensity-independent frequency effect in sonochemistry, J. Phys. Chem. B 106 (2002) 12537-12548]. In the current study, it was found that the CH(SPSo)/CH(SDS) ratio depended only on the ultrasound frequency and did not depend on the geometry of the ultrasound exposure apparatus considered.