Products of mixed covering arrays of strength two

A covering array CA(N;t,k, v is an N × k array such that every N × t subarray contains all t‐tuples from v symbols at least once, where t is the strength of the array. Covering arrays are used to generate software test suites to cover all t‐sets of component interactions. The particular case when t = 2 (pairwise coverage) has been extensively studied, both to develop combinatorial constructions and to provide effective algorithmic search techniques. In this paper, a simple “cut‐and‐paste” construction is extended to covering arrays in which different columns (factors) admit different numbers of symbols (values); in the process an improved recursive construction for covering arrays with t = 2 is derived. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 124–138, 2006     

Microwave-accelerated Claisen rearrangements of allyl tetronates and tetramates

[3,3]-Sigmatropic rearrangements of allyl tetronates and allyl tetramates to give 3-allyltetronic or -tetramic acids, respectively, proceed within 20-60 min under microwave irradiation (300 W, 130-190 °C). Consecutive (homo)sigmatropic [1,5] H-shifts such as oxa-ene reactions are promoted less effectively, which allows the isolation of Claisen intermediates of sigmatropic domino sequences, in contrast to conventional heating.     

Comparison of mass resolution criteria in mass spectrometry

For a single peak, mass spectral resolution can be expressed in terms of peak width or ratio of peak position to peak width. Alternatively, for two equally intense peaks of equal width, resolution can be defined as the minimum peak separation such that the height of the valley between the combined peaks is less than a specified ratio (1%, 10%, 50%, 100%) of the individual (or combined) peak maximum. All these definitions depend on peak shape. Conversion formulae between various mass resolution criteria are presented for each of eight spectral peak shapes: Gaussian, triangular, trapezoidal, Lorentzian (absorption-mode, magnitude-mode, and sine-apodized magnitude-mode), and sinc (absorption-mode and magnitude-mode). From these formulae, mass resolutions based upon different criteria are readily compared for the same or different line shapes.     

Point sets with uniformity properties and orthogonal hypercubes

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs–2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.This author would like to thank the Mathematics Department of the University of Tasmania for its hospitality during his sabbatical when this paper was written. The same author would also like to thank the NSA for partial support under grant agreement # MDA904-87-H-2023.This author's research was supported by a grant from the Commonwealth of Australia through the Australian Research Council.     

On the Formation of a Protic Ionic Liquid in Nature

The practical utility of ionic liquids (ILs) makes the absence (heretofore) of reported examples from nature quite puzzling, given the facility with which nature produces many other types of exotic but utilitarian substances. In that vein, we report here the identification and characterization of a naturally occurring protic IL. It can be formed during confrontations between the ants S. invicta and N. fulva. After being sprayed with alkaloid‐based S. invicta venom, N. fulva detoxifies by grooming with its own venom, formic acid. The mixture is a viscous liquid manifestly different from either of the constituents. Further, we find that the change results as a consequence of formic acid protonation of the N centers of the S. invicta venom alkaloids. The resulting mixed‐cation ammonium formate milieu has properties consistent with its classification as a protic IL.