Sums and Differences of Two Cubic Polynomials

 When f(x) is a cubic polynomial with integral coefficients, we show that almost all integers represented as the sum or difference of two values of f(x), with , are thus represented essentially uniquely.     

How Ionization Catalyzes Diels-Alder Reactions

The catalytic effect of ionization on the Diels-Alder reaction between 1,3-butadiene and acrylaldehyde has been studied using relativistic density functional theory (DFT). Removal of an electron from the dienophile, acrylaldehyde, significantly accelerates the Diels-Alder reaction and shifts the reaction mechanism from concerted asynchronous for the neutral Diels-Alder reaction to stepwise for the radical-cation Diels-Alder reaction. Our detailed activation strain and Kohn-Sham molecular orbital analyses reveal how ionization of the dienophile enhances the Diels-Alder reactivity via two mechanisms: (i) by amplifying the asymmetry in the dienophile's occupied π-orbitals to such an extent that the reaction goes from concerted asynchronous to stepwise and thus with substantially less steric (Pauli) repulsion per reaction step; (ii) by enhancing the stabilizing orbital interactions that result from the ability of the singly occupied molecular orbital of the radical-cation dienophile to engage in an additional three-electron bonding interaction with the highest occupied molecular orbital of the diene.     

Growth of embedded ErAs nanorods on (4 1 1)A and (4 1 1)B GaAs by molecular beam epitaxy

The low solubility of Er in GaAs results in the formation of ErAs nanostructures when GaAs is grown with 5–6 at% Er/Ga ratio by molecular beam epitaxy on GaAs surfaces. For growth on the (4 1 1)A GaAs surface, cross-sectional scanning transmission electron microscopy images show the presence of ErAs nanorods embedded in a GaAs matrix extending along the [2 1 1] direction with a spacing of roughly 7 nm and a diameter of roughly 2 nm. Growth on the GaAs (4 1 1)B surface resulted in only nanoparticle formation. Variation of the polarized optical absorption with in-plane polarization angle is consistent with coupling to surface plasmon resonances of the semimetallic nanostructures.     

Synthesis,characterization, and dielectric properties of β-Gd<Subscript>2</Subscript>(MoO<Subscript>4</Subscript>)<Subscript>3</Subscript> thin films prepared by chemical solution deposition

A chemical solution was employed for deposition of gadolinium molybdate [β-Gd₂(MoO₄)₃] thin films. Gadolinium acetylacetonate hydrate {[CH₃COCH = C(O–)CH₃]₃Gd·xH₂O}, molybdenum isopropoxide {Mo[OCH(CH₃)₂]₅}, and acetylacetone were used in synthesis of this molybdate. Thermal gravimetry and differential scanning calorimetry suggested that crystallization of β-Gd₂(MoO₄)₃ occurs at around 480 °C. Phase-pure, orthorhombic β-Gd₂(MoO₄)₃ films were deposited on Pt/Ti/SiO₂/Si(100) substrates. β-Gd₂(MoO₄)₃ films crystallized at 750 °C showed a strong (00l) preferred orientation. The film dielectric constant measured was 10~14 and the dielectric loss was less than 3%. There was no marked signature in the permittivity at the bulk Curie temperature, approximately 159 °C.     

Diet discrimination factors are inversely related to δ15 N and δ13C values of food for fish under controlled conditions

A recent literature review reported negative relationships between diet discrimination factors (DDFs = X_fish – X_food; X = δ¹⁵N or δ¹³C) and the values of δ¹⁵N and δ¹³C in the food of wild organisms but there has been no laboratory‐based confirmation of these relationships to date. Laboratory reared guppies (Poecilia reticulata) fed a series of diets with a range of δ¹³C (?22.9 to ?6.6‰) and δ¹⁵N (6.5 to 1586‰) values were used to magnify diet‐tissue dynamics in order to calculate DDFs once the fish had achieved equilibrium with each of the diets. Values of DDFs range widely for δ¹⁵N (7.1 to ?849‰) and δ¹³C (1.1 to ?7.0‰) and showed a strong negative correlation with the stable isotope value in the food for δ¹⁵N (slope = ?0.59 ± 0.02, r² = 0.95) and δ¹³C (slope = ?0.56 ± 0.02, r² = 0.94). Based on these relationships, the magnitude of DDF change over environmentally relevant values of δ¹⁵N or δ¹³C would be significant and could confound the interpretation of stable isotopes in the environment. Using highly enriched experimental diets, our study adds to a growing number of studies that undermine the consistent trophic enrichment paradigm with results that demonstrate the currently poor mechanistic understanding of how DDFs arise. The results of our study highlight that the magnitude of the stable isotope values in prey must be considered when choosing DDF values. Future laboratory studies should therefore be directed at uncovering the mechanistic basis of DDFs and, like others before, we recommend the determination of diet‐dependent DDFs under laboratory conditions before modeling dietary proportions or calculating trophic positions. Copyright © 2010 John Wiley & Sons, Ltd.     

Predicting the long tail of book sales: Unearthing the power-law exponent

The concept of the long tail has recently been used to explain the phenomenon in e-commerce where the total volume of sales of the items in the tail is comparable to that of the most popular items. In the case of online book sales, the proportion of tail sales has been estimated using regression techniques on the assumption that the data obeys a power-law distribution. Here we propose a different technique for estimation based on a generative model of book sales that results in an asymptotic power-law distribution of sales, but which does not suffer from the problems related to power-law regression techniques. We show that the proportion of tail sales predicted is very sensitive to the estimated power-law exponent. In particular, if we assume that the power-law exponent of the cumulative distribution is closer to 1.1 rather than to 1.2 (estimates published in 2003, calculated using regression by two groups of researchers), then our computations suggest that the tail sales of Amazon.com, rather than being 40% as estimated by Brynjolfsson, Hu and Smith in 2003, are actually closer to 20%, the proportion estimated by its CEO.     

From Metal Binding to Nanoparticle Formation: Monitoring Biomimetic Iron Oxide Synthesis within Protein Cages using Mass Spectrometry

Mass measurements of metal‐mineralized protein cages allowed quantitative examination of the effects of metal‐ion concentration on the final nanoparticle size. Modeling using a kinetic master equation suggests that particle growth involves both a binding phase and a growth phase (see picture; I: relative abundance; LiDps: a DNA binding protein; ⁿFe: number of Fe atoms).

     

Assembly of multilayer films incorporating a viral protein cage architecture

Protein cage architectures such as viral capsids, heat shock proteins, ferritins, and DNA-binding proteins are nanoscale modular subunits that can be used to expand the structural and functional range of composite materials. Here, layer-by-layer (LbL) assembly was used to incorporate cowpea chlorotic mottle virus (CCMV) into multilayer films. Three types of multilayer films were prepared. In the first type, ionic interactions were employed to assemble CCMV into triple layers. In the second type, complementary biological interactions (streptavidin/biotin) were used for this purpose. In a third variation of LbL assembly, complementary biological interactions were employed to produce nanotextured films that exhibit in-plane order over a micron scale without the need to adsorb onto a prepatterned template.     

Photoinduced Rydberg ionization spectroscopy of phenylacetylene: vibrational assignments of the C state of the cation

The photoinduced Rydberg ionization spectrum of the third excited electronic state of phenylacetylene cation was recorded via the origin of the cation ground electronic state. The origin of this state is 17 834 cm(-1) above the ground state of the cation, and the spectrum shows well-resolved vibrational features to the energy of 2200 cm(-1) above this. An assignment of the vibrational structure was made by comparison to calculated frequencies and Franck-Condon factors. From the assignments, and electronic structure considerations, the electronic symmetry of the C state is established to be (2)B(1).     

State-resolved thermalization of singlet and mixed singlet-triplet states of CH2

The role of mixed states in the collision-induced thermalization, intersystem crossing, and reactive loss of CH(2) (~a (1)A1) has been monitored using Doppler-resolved transient frequency modulation absorption spectroscopy. Singlet CH(2) is produced in a hot initial distribution of translation and rotational energy states in the 308 nm photodissociation of ketene in a large excess of argon. Collisions with Ar and ketene cool the translational and rotational degrees of freedom, while depleting the total singlet CH(2) population through reaction and intersystem crossing. Direct monitoring of the time-dependent populations of rotational levels containing mixed singlet and triplet character reveals a rapid interconversion between the two components, but no discernable difference between the kinetics of the pure singlet and mixed states at longer times.