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.     

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.     

An electrochemical study of redox indicators : Part I. Substituted chrysoidines

The electrochemical oxidation of various substituted chrysoidines was studied by cyclic voltammetry, to determine which have stable oxidation products. Only 4-hydroxy-chrysoidine has a stable product; the apparent oxidation potential is 0.779 V vs. NHE. 4-Methoxy- and 4-ethoxy-chrysoidine rapidly lose methanol or ethanol, respectively, so that the 4-hydroxychrysoidine wave appears on subsequent scans. All the other chrysoidines studied are irreversible. The results indicate that a hydroxy group in the 4-position is necessary for stability; the 4-alkoxychrysoidines can achieve the stable quinoidal structure by cleavage of the alkoxy group after nucleophilic attack.     

Refinement of borate structures from 11B MAS NMR spectroscopy and density functional theory calculations of 11B electric field gradients

The refinement of borate structures using DFT calculations combined with experimental (11)B quadrupole coupling parameters from solid-state NMR spectroscopy is presented. The (11)B electric field gradient (EFG) tensors, calculated using the WIEN2k software for trigonal and tetrahedral boron sites in a series of model compounds, exhibit a convincing linear correlation with the quadrupole coupling tensor elements, determined from (11)B MAS NMR spectra of the central or satellite transitions. The model compounds include Li(2)B(4)O(7), Mg(2)B(2)O(5), Mg(3)B(2)O(6), NH(4)B(C(6)H(5))(4), and colemanite (CaB(3)O(4)(OH)(3).H(2)O). The (11)B quadrupole moment, Q = 0.0409 +/- 0.0002 barn, derived from the linear correlation, is in excellent agreement with the accepted value for Q((11)B). This demonstrates that DFT (WIEN2k) calculations can provide precise (11)B quadrupole coupling parameters on an absolute scale. On the other hand, DFT calculations based on the reported crystal structures for datolite (CaBSiO(4)(OH)) and danburite (CaB(2)Si(2)O(8)) cannot reproduce the experimental (11)B quadrupole coupling parameters to the same high precision. However, optimization of these structures by minimization of the forces between the atoms (obtained by DFT) results in a significant improvement between the calculated and experimental (11)B quadrupole coupling parameters, which indicates that reliable refinements of the borate structures are obtained by this method. Finally, the DFT calculations also provide important structural information about the sign and orientation of the EFG tensor elements in the crystal frame, a kind of information that cannot be achieved from (11)B NMR experiments on powdered samples.     

A New Family of Nonstoichiometric Layered Rare-Earth Tin Antimonides, RESn(x)()Sb(2) (RE = La, Ce, Pr, Nd, Sm): Crystal Structure of LaSn(0.75)Sb(2)

A new class of nonstoichiometric layered ternary rare-earth tin antimonides, RESn(x)()Sb(2) (RE = La, Ce, Pr, Nd, Sm), has been synthesized through reaction of the elements at 950 degrees C. In the lanthanum series LaSn(x)()Sb(2), tin can be incorporated from a maximum content of x approximately 0.7 or 0.8 to as low as x approximately 0.10. The structure of lanthanum tin diantimonide with the maximum tin content, LaSn(0.75)Sb(2), has been determined by single-crystal X-ray diffraction methods. It crystallizes in the orthorhombic space group -Cmcm with a = 4.2425(5) ?, b = 23.121(2) ?, c = 4.5053(6) ?, and Z = 4. The isostructural rare-earth analogues were characterized by powder X-ray diffraction. The structure of LaSn(0.75)Sb(2) comprises layers of composition "LaSb(2)" in which La atoms are coordinated by Sb atoms in a square-antiprismatic geometry. Between these layers reside chains of Sn atoms distributed over three crystallographically independent sites, each partially occupied at about 20%. The structure of LaSn(0.75)Sb(2) can be regarded as resulting from the excision of RE-Sb and Sb-Sb bonds in the related structures of binary rare-earth diantimonides, RESb(2), and then intercalation of Sn atoms between layers.     

Determination of palladium in different samples by neutron activation after selective preconcentration with α-benzildioxime

A procedure is described for the determination of submicrogram quantities of palladium in sea water, biological and geological materials. Palladium is preconcentrated by coprecipitation with α-benzildioxime at pH 2 in the presence of citric acid followed by neutron activation. The method is highly selective and only traces of other metals are adsorbed on the surface of the precipitate. The instrumental variant of counting of long-lived ¹⁰⁹Pd (t$\text{1}\text{2}$ = 13.6 h) after 16-h irradiation gives a detection limit of 10 ng. A further decrease of the detection limit to 1 ng can be attained by removal of radioactive impurities (mainly ²⁴Na, ⁵⁶Mn and ⁸²Br) after washing the dichloromethane extract of dissolved precipitate with an aqueous solution of citric acid containing inactive carriers of bromide, manganese and sodium ions. Palladium (¹⁰⁹Pd) is finally measured by a NaI(Tl) well type scintillation detector. The method can be applied to most environmental samples.     

Möbius aromaticity in [12]annulene: cis-trans isomerization via twist-coupled bond shifting

Density functional and coupled cluster calculations show that facile thermal configuration change in [12]annulene occurs via a twist-coupled bond-shifting mechanism. The transition state for this process is highly aromatic with M?bius topology. At the CCSD(T)/cc-pVDZ//BH&HLYP/6-311+G** level, the isomerization of tri-trans-[12]annulene 1a (CTCTCT) to its di-trans isomer 2 (CCCTCT) via such a mechanism has a barrier of 18.0 kcal/mol, in good agreement with earlier experiments. Two other aromatic M?bius bond-shifting transition states were located that result in configuration change for other [12]annulene conformers. This mechanism contrasts sharply with diradical configuration change for acyclic polyenes and with planar bond-shifting mechanisms generally assumed for annulenes. This constitutes evidence that neutral M?bius aromatic annulenes play a role in the dynamic processes of neutral [4n]annulenes.