A lower bound for the optimal crossing-free Hamiltonian cycle problem

Consider a drawing in the plane ofK _n , the complete graph onn vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing ofK _n . If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let (n) represent the maximum number of cfhc's of any drawing ofK _n , and (n) the maximum number of cfhc's of any rectilinear drawing ofK _n . The problem of determining (n) and (n), and determining which drawings have this many cfhc's, is known as the optimal cfhc problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for (n) and (n). In particular, it is shown that (n) is at leastk × 3.2684 ⁿ . We conjecture that both (n) and (n) are at mostc × 4.5 ⁿ .This research, part of which was conducted at Queen's University, was supported by an N.S.E.R.C. postgraduate scholarship.     

Analysis of Colubroidea snake venoms by liquid chromatography with mass spectrometry: evolutionary and toxinological implications

The evolution of the venomous function of snakes and the diversification of the toxins has been of tremendous research interest and considerable debate. It has become recently evident that the evolution of the toxins in the advanced snakes (Colubroidea) predated the evolution of the advanced, front-fanged delivery mechanisms. Historically, the venoms of snakes lacking front-fanged venom-delivery systems (conventionally grouped into the paraphyletic family Colubridae) have been largely neglected. In this study we used liquid chromatography with mass spectrometry (LC/MS) to analyze a large number of venoms from a wide array of species representing the major advanced snake clades Atractaspididae, Colubrinae, Elapidae, Homalopsinae, Natricinae, Psammophiinae, Pseudoxyrhophiinae, Xenodontinae, and Viperidae. We also present the first sequences of toxins from Azemiops feae as well as additional toxin sequences from the Colubrinae. The large body of data on molecular masses and retention times thus assembled demonstrates a hitherto unsuspected diversity of toxins in all lineages, having implications ranging from clinical management of envenomings to venom evolution to the use of isolated toxins as leads for drug design and development. Although definitive assignment of a toxin to a protein family can only be done through demonstrated structural studies such as N-terminal sequencing, the molecular mass data complemented by LC retention information, presented here, do permit formulation of reasonable hypotheses concerning snake venom evolution and potential clinical effects to a degree not possible till now, and some hypotheses of this kind are proposed here. The data will also be useful in biodiscovery.     

Ring strain and its effect on the rate of the general-base catalyzed enolization of cyclobutanone

[reaction: see text] The 3-quinuclidinone-catalyzed (pK(BH) = 7.5) enolization of cyclobutanone (1) in D(2)O at 25 degrees C, I = 1.0 (KCl) was followed by deuterium incorporation, which was determined by (1)H NMR. The second-order rate constant for the buffer-catalyzed deprotonation of 1 was found to be k(B) = 3.3 x 10(-4) M(-1) s(-1), which is compared to rates for acetone and 2-(2'-oxopropyl)benzaldehyde under similar conditions. The data shows that ring strain has very little effect on the energy barrier to deprotonation of 1 vs the unstrained systems.     

Determination of selected elements in bone samples by neutron activation and γ-spectrometry

A non-destructive method for the quantification of eleven elements in bone samples is described. The analytical scheme is based on short (30 s) irradiation with thermal neutrons followed immediately (decay time 10 s) by counting fluorine-20 for 30 s and, after a total waiting time of 150 s, by 10-min γ-spectrometry counting, which give data for Ca, Cl, Mg, Mn, Na, and V. Use of a boron carbide shield for a second set of irradiations with epithermal neutrons permits the additional determination of bromine and strontium and calculation of the contribution of aluminium and phosphorus to the total activity of ₂₈Al.     

Large-scale resonance amplification of optical sensing of volatile compounds with chemoresponsive visible-region diffraction gratings

Micropatterning of the vapochromic charge-transfer salt, [Pt(CNC6H4C10H21)4][Pd(CN)4], on transparent platforms yields transmissive chemoresponsive diffraction gratings. Exposure of the gratings to volatile organic compounds (VOCs) such as chloroform and methanol leads to VOC uptake by the porous material comprising the grating lattice or framework, and a change in the material's complex refractive index, ?. The index change is accompanied by a change in the degree of index contrast between the lattice and the surrounding medium (in this case, air), and a change in the diffraction efficiency of the grating. When a monochromatic light source that is not absorbed by the lattice material is employed as a probe beam, only changes in the real component of ? are sensed. Under these conditions, the grating behaves as a nonselective, but moderately sensitive, sensor for those VOCs capable of permeating the porous lattice material. When a probe color is shifted to a wavelength coincident with the vapochromic charge-transfer transition of the lattice material, the sensor response is selectively amplified by up to 3.5 orders of magnitude, resulting in greatly enhanced sensitivity and some degree of chemical specificity. On the basis of studies at four probe wavelengths, the amplification effect is dominated by resonant changes in the imaginary component of the refractive index. The observed wavelength- and analyte-dependent amplification effects are quantitatively well described by a model that combines a Kramers-Kronig analysis with an effective-medium treatment of dielectric effects.     

Heteropoly acids in the determination of phosphorus by neutron activation analysis

Phosphorus is selectively extracted as molybdovanadophosphoric acid (MVPA) and determined indirectly using⁵²V. Silicon, germanium, arsenic, fluoride and tannic acid do not interfere. The sensitivity for phosphorus was 1421±13 counts⁵²V per ng P ml^–1 with a detection limit of 0.2 ng P ml^–1. The precision is ±10% over the range 0.7±70 ng P ml^–1. The sensitivity is four orders of magnitude greater than for conventional neutron activation analysis (NAA) using the³¹P(n, )²⁸ Al reaction and a SLOWPOKE II reactor. Greater selectivity for phosphorus is obtained than by direct measurement of³²P by beta ray spectroscopy.