Gilbert N. Lewis and the chemical bond: the electron pair and the octet rule from 1916 to the present day

We describe the development of Lewis's ideas concerning the chemical bond and in particular the concept of the electron pair bond and the octet rule. We show that the concept of the electron pair bond has endured to the present day and is now understood to be a consequence of the Pauli principle. In contrast the octet rule is now regarded as much less important than was originally generally believed, although Lewis himself knew several exceptions and regarded it as less important than what he called the rule of two (the electron pair). The octet rule was more strongly promoted by Langmuir who is also responsible for the term covalent bond. However, many more exceptions to the octet rules than were known to Lewis are now known and the terms hypervalent and hypovalent used to describe such molecules are no longer particularly useful. Today it is realized that bonding electron pairs in many molecules are not as well localized as Lewis believed, nevertheless resonance structures, i.e., plausible alternative Lewis structures, are still often used to describe such molecules. Moreover electrons are not always found in pairs, as for example in linear molecules, which can, however, be satisfactorily described by Linnett's double quartet theory. The electron density distribution in a molecule can now be analyzed using the ELF and other functions of the electron density to show where electron pairs are most probably to be found in a molecule.     

Characterization of high molecular weight polyethylenes using high temperature asymmetrical flow field-flow fractionation with on-line infrared, light scattering, and viscometry detection

High temperature asymmetrical flow field-flow fractionation (HTAF4) coupled to infrared (IR), multi-angle light scattering (MALS), and viscometry (Visc) detection is introduced as a tool for the characterization of high molecular weight polyethylenes. The high molecular weight fraction strongly affects the rheological behaviour and processability of polyethylene materials and can often not be accurately resolved by current technology such as high temperature size-exclusion chromatography (HTSEC). Molecular weight (M), radius of gyration (Rg), and intrinsic viscosity [eta] of linear high density polyethylene (HDPE) and branched low density polyethylene (LDPE) samples are studied in detail by HTAF4 and are compared to HTSEC. HTAF4 showed a better separation and mass recovery than HTSEC for very high molecular weight fractions in HDPE and LDPE samples. As no stationary phase is present in an HTAF4 channel, the technique does not show the typical drawbacks associated with HTSEC analysis of high molecular weight polyethylenes, such as, exclusion effects, shear degradation, and anomalous late elution of highly branched material. HTAF4 is applied to study the relation between the molecular weight and the zero shear viscosity eta0 for high molecular weight HDPE. It was found that the zero shear viscosity values predicted from HTAF4 results are in good qualitative agreement with measured values obtained from dynamic mechanical spectroscopy (DMS) experiments, whereas eta0 values predicted from HTSEC do not show a strong correlation. The low molecular weight cutoff of HTAF4 is approximately 5x10(4) as a result of relatively large pores in the HTAF4 channel membrane. HTAF4 is, therefore, currently not suited to analyze low molecular weight materials.     

Mean2-Bounded Operators on Hilbert Space and Weight Sequences of Positive Operators

We introduce and study the class of mean₂-bounded operators on Hilbert space. These operators, which are characterized by a uniform boundedness condition on the 'quadratic averages' of their powers, are intimately related to operator-valued weight sequences and weighted norm inequalities. Mean₂-bounded operators exhibit a lively interplay with the discrete Hilbert transform in an array of settings. Under appropriate circumstances, mean₂-bounded operators have a rich spectral theory leading to powerful transference properties.     

Distributional control and generalized analyticity

Let S be a strongly continuous, separation-preserving representation of a locally compact abelian group G in L^p(), where 1p<, and is an arbitrary measure. We show that S is uniformly bounded with respect to the L^p-and L-norms if and only if it satisfies a certain boundedness condition for distribution functions. These equivalent conditions facilitate the transference from L^p(G) to L^p() of the a.e. convergence for a wide class of sequences of convolution operators. The result unifies and generalizes various aspects of ergodic theory--in particular, the ergodic singular integral operators and ergodic Hardy spaces.     

Mean-boundedness and Littlewood-Paley for separation-preserving operators

Suppose that is a -finite measure space, , and is a bounded, invertible, separation-preserving linear operator such that the linear modulus of is mean-bounded. We show that has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra' of , and to furnish with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for .

     

Accelerated stochastic simulation of the stiff enzyme-substrate reaction

The enzyme-catalyzed conversion of a substrate into a product is a common reaction motif in cellular chemical systems. In the three reactions that comprise this process, the intermediate enzyme-substrate complex is usually much more likely to decay into its original constituents than to produce a product molecule. This condition makes the reaction set mathematically "stiff." We show here how the simulation of this stiff reaction set can be dramatically speeded up relative to the standard stochastic simulation algorithm (SSA) by using a recently introduced procedure called the slow-scale SSA. The speedup occurs because the slow-scale SSA explicitly simulates only the relatively rare conversion reactions, skipping over occurrences of the other two less interesting but much more frequent reactions. We describe, explain, and illustrate this simulation procedure for the isolated enzyme-substrate reaction set, and then we show how the procedure extends to the more typical case in which the enzyme-substrate reactions occur together with other reactions and species. Finally, we explain the connection between this slow-scale SSA approach and the Michaelis-Menten [Biochem. Z. 49, 333 (1913)] formula, which has long been used in deterministic chemical kinetics to describe the enzyme-substrate reaction.