Études sur l'équation [partial derivative sign]p̳u/[partial derivative sign]xp̳ + a̳[delta]m̳u̳=0,

On t.p. p̳ and m̳ are superscript; a̳ and u̳ are subscript.

Über die partielle Differential-Gleichung [Delta]v+k²v=0 und deren Auftreten in der mathematischen Physik.

Available on demand as hard copy or computer file from Cornell University Library.

Semisolid microstructural evolution of AlSi7Mg alloy during partial remelting

Different as-cast microstructures of an AlSi7Mg alloy were produced by controlling the solidification conditions. The as-cast grain size ranged from 1.4 mm to 160 mum and the morphology varied from dendritic to rosette-like to globular. The as-cast materials were then partially remelted and isothermally held at 580degreesC for microstructure evolution. The final microstructure depended on the initial as-cast microstructure and the isothermal holding time. After partial remelting and isothermal holding, coarse-grained dendritic structures were not able to evolve to a globular structure, while structures with medium sized dendritic grains evolved to a globular structure with a relatively large particle size after a long isothermal holding time. Fine-grained structures evolved to well-rounded globular grains within times ranging front 10 min to 5 min as the dendritic nature of the starting structure diminished. An empirical equation has been established to describe the relationship between the evolved microstructure and the as-cast microstructure. (C) 2003 Elsevier B.V. All rights reserved.

Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Partial regularity of minimizers of a functional involving forms and maps

We discuss the partial regularity of minimizers of energy functionals such as (1)/(p)integral(Omega)[sigma(u)dA(p) + (1)/(2)delu(2p)]dx, where u is a map from a domain Omega is an element of R-n into the m-dimensional unit sphere of Rm+1 and A is a differential one-form in Omega.

Gelatinisation of starch in mixtures of sugars. II. Application of differential scanning calorimetry

Differential scanning calorimetry was used to investigate the effect of mixtures of glucose and fructose, and five types of honeys on starch gelatinisation. At a 1:1 starch:water ratio, glucose generally increased the enthalpy (DeltaH(gel)) and temperatures (T-onset, T-peak and T-end) of gelatinisation more than fructose. Upon mixing, DeltaH(gel) of the low-temperature endotherm decreased in comparison to the sole sugars, but was fairly constant (7.7 +/- 0.33 J/g dry starch). DeltaH(gel) of the high-temperature endotherm increased with the fructose content. For both endotherms, the gelatinisation temperatures were unchanged (CV less than or equal to 3%) for the mixtures. With the honeys (moisture, 14.9-18.0%; fructose, 37.2-44.0%; glucose, 28.3-31.9%) added at 1.1-4.4 g per g dry starch, the enthalpy and temperatures of gelatinisation did not vary significantly (CV less than or equal to 6%). Typical thermograms are presented, and the results are interpreted in the light of the various proposed mechanisms for starch gelatinisation in sugar-water systems, total sugar content and possible sugar-sugar interactions. The thermograms were broader in the presence of the sugars and honeys, and a biphasic character was consistently exhibited. The application of an exponential equation to the gelatinisation temperatures of the starch-honey mixtures revealed an opposing influence of fructose and glucose during gelatinisation. The mechanism of starch gelatinisation may be better understood if techniques could be perfected to quantify breakage and formation of hydrogen bonds in the starch granules, and suggested techniques are discussed. (C) 2004 Elsevier Ltd. All rights reserved.

Specific heat capacity of Australian honeys from 35 to 165C as a function of composition using differential scanning calorimetry

Modulated temperature differential scanning calorimetry was used to investigate the specific heat capacity (C-p) of 10 Australian honeys within the processing and handling temperatures. The values obtained were found to be different from the literature values at certain temperatures, and are not predictable by the additive model. The C-p of each honey exhibited a cubic relationship (P < 0.001) with the temperature (T, C). In addition, the moisture (M, %), fructose (F, %) and glucose (G, %) contents of the honeys influenced their C-p. The following equation (r(2) = 0.92) was proposed for estimating C-p of honey, and is recommended for use in the honey industry and in research: C = 996.7 + 1.4 x 10(-3)T + 5.6 x 10(-5)T(2) - 2.4 x 10(-7)T(3) - 56.5M - 25.8F - 31.0G + 1.5(M * F) + 1.8(M * G) + 0.8(F * G) - 4.6 x 10(-2) (M * F * G).

Stochastic delay differential equations for genetic regulatory networks

Time delay is an important aspect in the modelling of genetic regulation due to slow biochemical reactions such as gene transcription and translation, and protein diffusion between the cytosol and nucleus. In this paper we introduce a general mathematical formalism via stochastic delay differential equations for describing time delays in genetic regulatory networks. Based on recent developments with the delay stochastic simulation algorithm, the delay chemical masterequation and the delay reaction rate equation are developed for describing biological reactions with time delay, which leads to stochastic delay differential equations derived from the Langevin approach. Two simple genetic regulatory networks are used to study the impact of' intrinsic noise on the system dynamics where there are delays. (c) 2006 Elsevier B.V. All rights reserved.

Student Difficulties with units in differential equations in modelling contexts

First-year undergraduate engineering students' understanding of the units of factors and terms in first-order ordinary differential equations used in modelling contexts was investigated using diagnostic quiz questions. Few students appeared to realize that the units of each term in such equations must be the same, or if they did, nevertheless failed to apply that knowledge when needed. In addition, few students were able to determine the units of a proportionality factor in a simple equation. These results indicate that lecturers of modelling courses cannot take this foundational knowledge for granted and should explicitly include it in instruction.

Integration of the Manakov-PMD Equation with Precomputed M(w) Matrices for PMD Simulation

A novel direct integration technique of the Manakov-PMD equation for the simulation of polarisation mode dispersion (PMD) in optical communication systems is demonstrated and shown to be numerically as efficient as the commonly used coarse-step method. The main advantage of using a direct integration of the Manakov-PMD equation over the coarse-step method is a higher accuracy of the PMD model. The new algorithm uses precomputed M(w) matrices to increase the computational speed compared to a full integration without loss of accuracy. The simulation results for the probability distribution function (PDF) of the differential group delay (DGD) and the autocorrelation function (ACF) of the polarisation dispersion vector for varying numbers of precomputed M(w) matrices are compared to analytical models and results from the coarse-step method. It is shown that the coarse-step method achieves a significantly inferior reproduction of the statistical properties of PMD in optical fibres compared to a direct integration of the Manakov-PMD equation.

Towards the in silico identification of class II restricted T-cell epitopes:a partial least squares iterative self-consistent algorithm for affinity prediction

Motivation: The immunogenicity of peptides depends on their ability to bind to MHC molecules. MHC binding affinity prediction methods can save significant amounts of experimental work. The class II MHC binding site is open at both ends, making epitope prediction difficult because of the multiple binding ability of long peptides. Results: An iterative self-consistent partial least squares (PLS)-based additive method was applied to a set of 66 pep- tides no longer than 16 amino acids, binding to DRB1*0401. A regression equation containing the quantitative contributions of the amino acids at each of the nine positions was generated. Its predictability was tested using two external test sets which gave r pred =0.593 and r pred=0.655, respectively. Furthermore, it was benchmarked using 25 known T-cell epitopes restricted by DRB1*0401 and we compared our results with four other online predictive methods. The additive method showed the best result finding 24 of the 25 T-cell epitopes. Availability: Peptides used in the study are available from http://www.jenner.ac.uk/JenPep. The PLS method is available commercially in the SYBYL molecular modelling software package. The final model for affinity prediction of peptides binding to DRB1*0401 molecule is available at http://www.jenner.ac.uk/MHCPred. Models developed for DRB1*0101 and DRB1*0701 also are available in MHC- Pred

Solving Differential Equations by Parallel Laplace Method with Assured Accuracy

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

Existence of Global Solutions to Supercritical Semilinear Wave Equations

∗The author was partially supported by Alexander von Humboldt Foundation and the Contract MM-516 with the Bulgarian Ministry of Education, Science and Thechnology.

Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30