Notes on plane partitions VI

We consider the problem of plane partitions with given minimal differences between parts along rows and columns. We obtain a formula for the generating function in the two row case.     

The homogeneous projective transformation of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: basil.kouvaritakis{at}eng.ox.ac.uk Algebraic quadratic equations are a special case of a singlegeneralized algebraic quadratic matrix equation (GQME). Hence,the importance of that equation in science and engineering isevident. This paper focus on the study of solutions of thatGQME and a unified framework for the characterization and identificationof solutions at infinity and of finite solutions of generalquadratic algebraic matrix equations is presented. The analysisis based on the concept of homogeneous projective transformationfor general polynomial systems (Morgan, 1986). In addition,a numerical error analysis for the computed solutions is providedfor the assessment of numerical accuracy, stability and conditioningof the computed solutions. The proposed framework is independentof any numerical method and therefore it can be used along withvarious possible numerical methods for the GQME solution, especiallymatrix flow-based algorithms (Chu, 1994) (e.g. continuation/homotopy,Morgan, 1989).     

A unified Monte Carlo treatment of the transport of electromagnetic energy, electrons, and phonons in absorbing and scattering media

The scalar Boltzmann transport equation (BTE) is often applicable to radiative energy transfer, electron-beam propagation, as well as thermal conduction by electrons and phonons provided that the characteristic length of the system is much larger than the wavelength of energy carriers and that certain interference phenomena and the polarization nature of carriers are ignored. It is generally difficult to solve the BTE analytically unless a series of assumptions are introduced for the particle distribution function and scattering terms. Yet, the BTE can be solved using statistical approaches such as Monte Carlo (MC) methods without simplifying the underlying physics significantly. Derivations of the MC methods are relatively straightforward and their implementation can be achieved with little effort; they are also quite powerful in accounting for complicated physical situations and geometries. MC simulations in radiative transfer, electron-beam propagation, and thermal conduction by electrons and phonons have similar simulation procedures; however, there are important differences in implementing the algorithms and scattering properties between these simulations. The objective of this review article is to present these simulation procedures in detail and to show that it is possible to adapt an existing MC computer code, for instance, in radiative transfer, to account for physics in electron-beam transport or phonon (or electronic thermal) conduction by sorting out the differences and implementing the correct corresponding steps. Several simulation results are presented and some of the difficulties associated with different applications are explained.     

Modelling the morphology of migrating bacterial colonies

We present a model which aims at describing the morphology of colonies of Proteus mirabilis and Bacillus subtilis. Our model is based on a cellular automaton which is obtained by the adequate discretisation of a diffusion-like equation, describing the migration of the bacteria, to which we have added rules simulating the consolidation process. Our basic assumption, following the findings of the group of Chuo University, is that the migration and consolidation processes are controlled by the local density of the bacteria. We show that it is possible within our model to reproduce the morphological diagrams of both bacteria species. Moreover, we model some detailed experiments done by the Chuo University group, obtaining a fine agreement.     

Functional derivative for the distribution function of a Λ-nucleon pair in nuclear matter

An exact expression for the functional derivative of the distribution function of a -nucleon pair in nuclear matter is derived. An approximate expression is also derived by means of the Kirkwood superposition approximation. The latter expression is subsequently used to obtain the Euler equation for the correlation functionf(r1) of a -nucleon pair in nuclear matter.     

A Class of Integrable Non-QRT Mappings and Their Deautonomisation

We study classes of mappings which do not belong to the QRT family. We obtain several integrable non-autonomous forms of these mappings extending previous results where only linearisable cases were found. Using our recently introduced method of singularity confinement with full deautonomisation, we analyse a mapping which, while non-integrable, does possess confined singularities and show that our method makes it possible to obtain the exact value of its algebraic entropy.     

Determination of Fenofibric Acid in Human Plasma by LC–MS/MS and LC–UV

A sensitive, high-throughput and economic liquid chromatographic method for determination of fenofibric acid in human plasma was developed and validated by ultraviolet detection and tandem mass spectrometry, then applied in pharmacokinetic study to investigate Lipanthyl™ 200 mg MC bioavailability under food and fasting conditions. Fenofibric acid with 2-chloro fenofibric acid-d6 (internal standard) was extracted from 100 µL of human plasma by acetonitrile in a single extraction step. 25 and 2 µL from supernatant were injected onto ACE column, 50 mm, 5 micron with 4.6 mm inner diameter for LC–UV and 2.1 mm for LC–MS/MS, and both systems were eluted isocratically by water:methanol:formic acid (35:65:0.1, v/v/v), with a constant flow rate of 1 mL min⁻¹. The established calibration curve was linear between 0.05–20 µg mL⁻¹, and the within- and between-day precisions were all below 13 % in both LC–MS/MS and LC–UV systems during validation, and accuracies ranged between 91 and 112 %. Twenty-eight healthy adult subjects participated in this clinical study, and the pharmacokinetic parameters including coefficient of variation were calculated and discussed. A dramatic decrease in C _max and AUC0-72 (3.63- and 1.85-fold, respectively) were observed for Lipanthyl™ MC under fasting conditions with more variable inter subject measurements comparing to the fed state.

     

Viscoelastic properties and interfacial tension of polystyrene–polyethylene blends

The linear viscoelastic properties of polystyrene polyethylene (PS/PE) blends have been investigated in the molten state. For concentrations of the dispersed phase equal to 30 vol %, the blends exhibited a droplet‐matrix morphology with a volume‐average diameter of 5.5 μm for a 70/30 PS/PE blend at 200 °C and 14.7 μm for a 30/70 PS/PE blend at 230 °C. Enhanced elasticity (G′) for both blends, in the terminal zone, compared to the modulus of the matrix (PS and PE, respectively) was observed. This is related to the deformation of the droplets in the matrix phase and hence to the interfacial forces between the blend components. The results for these uncompatibilized blends are shown to be in agreement with the predictions of the emulsion model of Palierne. These predictions were used to obtain the interfacial tension between PS and PE, which was found to be between 2 and 5 mN/m at 200 °C and 4 ± 1 mN/m at 230 °C. Independent interfacial tension measurements using the breaking‐thread method resulted in a value of 4.7 mN/m and 4.1 mN/m at 200 °C and 230 °C for the respective blends. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1359–1368, 2000     

The influence of partial emulsification on coalescence suppression and interfacial tension reduction in PP/PET blends

This study examines how the relative role of coalescence suppression and interfacial tension reduction influence the particle size at various levels of in situ compatibilization. The polymers studied are polyethylene terephthalate (PET) as matrix and a polypropylene (PP) as dispersed phase compatibilized by a triblock copolymer of poly(styrene–hydrogenated butadiene–styrene) (SEBS) grafted with maleic anhydride. The interfacial tension was studied by the breaking‐thread method, and it was used along with the morphology to characterize the emulsification efficacy of the copolymers. By modifying the concentration of MA grafted on the SEBS, different levels of emulsification of the blends were obtained. A comparison of 1/99 and 10/90 PP/PET blends compatibilized by SEBS‐g‐MA allows one to distinguish the relative role of interfacial tension and coalescence suppression in diminishing particle size. It is shown that varying degrees of residual coalescence remain, depending on the level of %MA in the copolymer. A detailed study of the 2%MA system below interfacial saturation was carried out to shed further light on the dependence of coalescence suppression on emulsification level and interfacial coverage. After separating out the contribution of interfacial tension on particle size reduction, it is shown that coalescence suppression for this system increases gradually with areal density of modifier at the interface right up to the region of interfacial saturation. Finally, the interfacial and morphological data were used to test the ability of the Lee and Park model to describe coalescence in polymer blends. Reasonable agreement was found between the parameter c₁, describing the coalescence in that model, and the trends related to residual coalescence from this study. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 939–951, 1999