A probabilistic approach to spectral analysis of growth-fragmentation equations

The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.     

Solubilization of silica nanoparticles in alkanes and poly(α-olefin)s by functionalized polyisobutylene oligomers

Covalent and noncovalent chemical methods that use oligomeric lipophilic agents to solubilize silica nanoparticles in heptane and poly(α-olefin) (EPAO) solvents are described. While only modest solubilization efficiencies are seen with an octadecyl group, a variety of terminally functionalized polyisobutylene (PIB) derivatives are more efficient. Both covalent and noncovalent chemistry was found to be effective. Covalent modification solubilized up to 34 wt% of silica nanoparticles (SiNPs) as stable solutions in heptane or PAOs. Noncovalent modification was however more effective, solubilizing up to 70% of SiNPs in heptane or PAOs. The most successful covalent approach used PIB oligomers containing terminal triethoxysilane groups to covalently modify SiNPs. Alternatively, SiNPs that were first functionalized with amine groups could be solubilized in heptane or PAOs with polyisobutylene containing sulfonic acid groups using acid–base chemistry. Studies of these and other solubilization chemistry was also carried out using fluorescent labels, studies that confirmed the gravimetric analyses of the heptane-solubilized SiNPs. Transmission electron microscopy of a PAO solution of these solutions showed that these SiNPs were present as small aggregates dispersed in the PAOs.     

A Buchwald–Hartwig Protocol to Enable Rapid Linker Exploration of Cereblon E3-Ligase PROTACs**

A palladium-catalysed Buchwald–Hartwig amination for lenalidomide-derived aryl bromides was optimised using high throughput experimentation (HTE). The substrate scope of the optimised conditions was evaluated for a range of alkyl- and aryl- amines and functionalised aryl bromides. The methodology allows access to new cereblon-based bifunctional proteolysis targeting chimeras with a reduced step count and improved yields.     

Tandem Chemoselective Suzuki–Miyaura Cross‐Coupling Enabled by Nucleophile Speciation Control

Control of boronic acid speciation is presented as a strategy to achieve nucleophile chemoselectivity in the Suzuki–Miyaura reaction. Combined with simultaneous control of oxidative addition and transmetalation, this enables chemoselective formation of two C? C bonds in a single operation, providing a method for the rapid preparation of highly functionalized carbogenic frameworks.     

Pressure field due to drag reducing outer layer devices in turbulent boundary layers

The wall static pressure in the vicinity of drag reducing outer layer devices in flat wall turbulent boundary layers has been measured and compared with an inviscid theory. Symmetric and cambered airfoil devices have been examined at small angles of attack and very low chord Reynolds numbers. Airfoil devices impose a sequence of strong favorable and adverse pressure gradients on the boundary layer whose drag is to be reduced. At very small angles of attack (± 2°), this pressure field extends up to about three chord lengths downstream of the trailing edge of an airfoil device. Also examined are the pressures on the upper and lower surfaces of a symmetric airfoil device in the freestream and near the wall. The freestream pressure distribution around an airfoil section is altered by the wall proximity. The relevance of lift enhancement caused by wall proximity to drag reduction has been discussed. The pressure distributions on the flat wall beneath the symmetric airfoil devices are predicted well by the inviscid theory. However, the remaining pressure distributions are predicted only qualitatively, presumably because of strong viscous effects.     

Fatigue-damage evaluation for mild steel incorporating mean stress and overload effects

Successful estimation of the service life of a structure or component which is subjected to a complex history of loading, depends on a suitable cumulative-damage summation technique. A general technique must be capable of predicting the effects on fatigue life of geometry, mean stress or strain, occasional overloads or overstrains, frequency of cycling and environment. As a contribution towards the general solution, this paper describes the utilization of a fatigue-damage summation method which incorporates two of these variables, mean stress and overloads. The method is tested for complex load histories in mild-steel specimens in the intermediate to long-life range. A four-part constant-stress-amplitude testing program was carried out consisting of: (a) tests with a constant mean stress, (b) tests on prestrained specimens, (c) tests with a mean stress applied in one block of cycles, and (d) tests with a mean stress applied in frequent regularly spaced short blocks. The constant-mean-stress results, which include several values of mean stress both tensile and compressive, are reduced to a single curve on a stress-life plot with either of two simple parameters (from Morrow and Smith, et al.) as ordinate. The prestrained specimen tests result in another curve lower than the first, showing the reduction in life due to the few initial cycles of high strain. These two effects are then incorporated into a cumulative-damage summation technique which is based on the well-known Miner's rule. The two curves on the stress-life plot are the foundation for the subsequent summations. The technique assumes that the first application of an overload causing appreciable plastic straining reduces the remaining fatigue life. A significant point is that this assumption apparently holds for both tensile and compressive overloads. The accuracy of this technique is demonstrated for a wide range of stress conditions and loading histories.     

On the Validity of Darcy's Law for Stable High-Concentration Displacements in Granular Porous Media

Recently, it has been suggested that Darcy's Law might not be applicable for modelling miscible, density-dependent flow in porous media. To investigate this, three sets of careful laboratory column experiments were performed on coarse and medium sands, consisting of upward displacement of water by sodium chloride solutions with concentrations ranging from 5 to 200g/l. Data on salt concentrations and water pressures were collected in horizontal transects along the flow direction. Salt concentration data were also collected in the influent and exit lines. The experimental data were analysed using a simplified approach based on Darcy's Law alone, applied with the assumption of a sharp interface. Darcy's Law was used to estimate porous medium permeability by fitting predictions to experimental data. Consistent estimates of permeability were obtained for each set of experiments. The results indicate that Darcy's Law adequately describes high concentration displacements through saturated coarse- and medium-grained porous media.     

The numerical analysis of water and solute movement in scale heterogeneous profiles

A computer based numerical method is presented for the analysis of water and solute movement in unsaturated heterogeneous porous materials. Such a method is necessary since, for those field studies where solute movement is of concern, the soil profiles under consideration are invariably heterogeneous. The numerical analysis is based on a general one-dimensional finite difference soil water flow model which includes a numerical technique combining the concepts of scale heterogeneity with an interpolative soil water hysteresis model. An explicit finite difference solute movement subroutine is incorporated into the unsaturated flow model to describe the transport of nonreactive solutes. A velocity dependent longitudinal dispersion coefficient is used in the solution of the hydrodynamic dispersion equation. The resulting hysteretic scale heterogeneous solute movement model permits the study of solute dynamics during infiltrating and redistribution in realistically complex spatially varying soil profiles. Results are presented for the leaching of both coarse grading to fine and fine grading to coarse sand profiles. Both vertical and horizontal profiles are studied using either a constant flux or a constant concentration input boundary condition. The four cases studied demonstrate the versatility of the numerical method and emphasise the substantial differences in transport behavior that can arise between heterogeneous and homogeneous profiles.Now with BHP Petroleum Pty. Ltd., GPO Box 1911R, Melbourne, Vic. 3001, Australia.     

Vehicular traffic effects on development and yield of corn (maize)

A 52 plot experiment was performed during the growing season of 1977 in a Ste. Rosalie clay soil, using a randomized complete-block design with 13 treatments of machinery traffic within each of four blocks. Three vehicle contact pressures, four numbers of tractor passes, and a control of zero traffic were used to relate the growth and yield variables to wheel traffic and the resulting soil compaction. The number of days required for (a) plant emergence, (b) tasselling, and (c) silking were minimum in moderately compacted plots. The plant growth rate monitored at 41, 54, 68, and 105 days from the seeding time was different from plot to plot. Growth models at different times of the season were derived in terms of the wheel traffic variables, and plant and ear moisture content variation with traffic treatment was investigated. Yield and ear yield increased with increases in machine contact pressure and passes reaching a peak around 500 kPa and dropped off for further increases. The reduction in yield was over 35% in some cases, suggesting that careful traffic planning is essential to obtain better production in agricultural fields. Prediction models were obtained for all the plant growth characteristics in terms of traffic variables. A relation for yield in terms of soil bulk density was established for a dry season.