An identity for the Dedekind eta-function involving two independent complex variables

The authors prove a new identity for the Dedekind eta-functionthat involves third powers of the eta-function, with each ofthe two cubes being a function of a different complex variable.     

Correlation between rheological properties of zinc carboxylate liquids and molecular structure

Using Fourier transform infrared spectroscopy (FTIR), the viscosity of zinc 2-ethylhexanoate liquid has been found to correlate with the intensity of an asymmetric COO stretching resonance at 1632 cm-1. This is consistent with the presence of the zinc carboxylate polymer, catena-2-ethylhexanato-(O,O')-tri-micro-2-ethylhexanato-(O,O')-dizinc(II) as the origin of the viscosity, a conclusion that is further supported by theoretical predictions. Density functional theory has been used to assign the IR spectra of the zinc carboxylate dimer, catena-2-ethylhexanato-(O,O')-di-(tri-micro-2-ethylhexanato-(O,O')-dizinc(II)-formic) acid, and the model of the molecular liquid, micro-4-oxo-hexakis-(micro-2-ethylhexanato)-tetrazinc(II). The predicted spectra indicate that the decreased symmetry of the polymer relative to the zinc 2-ethylhexanoate liquid increases the intensity of the asymmetric carboxylate stretch at 1632 cm-1 and leads to the observed correlation.     

Surface extenders and an optimal pore size promote high dynamic binding capacities of antibodies on cation exchange resins

Increased recombinant protein expression yields and a large installed base of manufacturing facilities designed for smaller bulk sizes has led to the need for high capacity chromatographic resins. This work explores the impact of three pore sizes (with dextran distribution coefficients of 0.4, 0.53, and 0.64), dextran surface extender concentration (11–20 mg/mL), and ligand density (77–138 μmol H+/mL resin) of cation exchange resins on the dynamic binding capacity of a therapeutic antibody. An intermediate optimal pore size was identified from three pore sizes examined. Increasing ligand density was shown to increase the critical ionic strength, while increasing dextran content increased dynamic binding capacity mainly at the optimal pore size and lower conductivities. Dynamic binding capacity as high as 200 mg/mL was obtained at the optimum pore size and dextran content.     

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets of the natural numbers such that the Robin inequality holds for all but finitely many . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.      

The Nicolas and Robin inequalities with sums of two squares

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality σ(n) < e ^γ n log log n holds for every integer n > 5040, where σ(n) is the sum of divisors function, and γ is the Euler–Mascheroni constant. We exhibit a broad class of subsets S{\mathcal {S}} of the natural numbers such that the Robin inequality holds for all but finitely many n ? S{n \in \mathcal {S}} . As a special case, we determine the finitely many numbers of the form n = a ² + b ² that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality n/φ(n) < e ^γ log log n; since σ(n)/n < n/φ(n) for n > 1 our results for the Robin inequality follow at once.     

Representation in the (Artificial) Immune System

Much of contemporary research in Artificial Immune Systems (AIS) has partitioned into either algorithmic machine learning and optimisation, or, modelling biologically plausible dynamical systems, with little overlap between. We propose that this dichotomy is somewhat to blame for the lack of significant advancement of the field in either direction and demonstrate how a simplistic interpretation of Perelson’s shape-space formalism may have largely contributed to this dichotomy. In this paper, we motivate and derive an alternative representational abstraction. To do so we consider the validity of shape-space from both the biological and machine learning perspectives. We then take steps towards formally integrating these perspectives into a coherent computational model of notions such as life-long learning, degeneracy, constructive representations and contextual recognition—rhetoric that has long inspired work in AIS, while remaining largely devoid of operational definition.     

Isotope quantum effects in water around the freezing point

We have measured the difference in electronic structure factors between liquid H(2)O and D(2)O at temperatures of 268 and 273 K with high energy x-ray diffraction. These are compared to our previously published data measured from 279 to 318 K. We find that the total structural isotope effect increases by a factor of 3.5 over the entire range, as the temperature is decreased. Structural isochoric temperature differential and isothermal density differential functions have been used to compare these data to a thermodynamic model based upon a simple offset in the state function. The model works well in describing the magnitude of the structural differences above approximately 310 K, but fails at lower temperatures. The experimental results are discussed in light of several quantum molecular dynamics simulations and are in good qualitative agreement with recent temperature dependent, rotationally quantized rigid molecule simulations.     

Ab initio calculations on the isomerization of alkene radical cations

Ab initio calculations on the isomerization of butene and pentene radical cations indicate that, for all classical ion structures, the lowest barrier for a rearrangement to the most stable ion structure is below the dissociation limit. Isomerizations of linear butene radical cations to the isobutene structure take place via the CH₃CC₂H₅^·+ structure, whereas in the pentene case the connection between linear and branched ion structures proceeds via the 1,2-dimethylcyclopropane radical cation. From the results a qualitative model is derived which suggests that for larger alkene radical cations an isomerization to structures with four alkyl substituents on the double bond may be in close competition with dissociation.     

Optimization of the CHARMM additive force field for DNA: Improved treatment of the BI/BII conformational equilibrium

The B-form of DNA can populate two different backbone conformations: BI and BII, defined by the difference between the torsion angles ε and ζ (BI = ε-ζ < 0 and BII = ε-ζ > 0). BI is the most populated state, but the population of the BII state, which is sequence dependent, is significant and accumulating evidence shows that BII affects the overall structure of DNA, and thus influences protein-DNA recognition. This work presents a reparametrization of the CHARMM27 additive nucleic acid force field to increase the sampling of the BII form in MD simulations of DNA. In addition, minor modifications of sugar puckering were introduced to facilitate sampling of the A form of DNA under the appropriate environmental conditions. Parameter optimization was guided by quantum mechanical data on model compounds, followed by calculations on several DNA duplexes in the condensed phase. The selected optimized parameters were then validated against a number of DNA duplexes, with the most extensive tests performed on the EcoRI dodecamer, including comparative calculations using the Amber Parm99bsc0 force field. The new CHARMM model better reproduces experimentally observed sampling of the BII conformation, including sampling as a function of sequence. In addition, the model reproduces the A form of the 1ZF1 duplex in 75 % ethanol, and yields a stable Z-DNA conformation of duplex (GTACGTAC) in its crystal environment. The resulting model, in combination with a recent reoptimization of the CHARMM27 force field for RNA, will be referred to as CHARMM36.