Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

Generalized Convolution Transforms and Toeplitz Plus Hankel Integral Equations

Mathematics Subject Classification: 44A05, 44A35

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Ultra-compact dwarf galaxies: a new class of compact stellar system discovered in the Fornax Cluster

We have used the 2dF spectrograph on the Anglo-Australian Telescope to obtain a complete spectroscopic sample of all objects in the magnitude range, 16.5 < bj < 19.8, regardless of morphology, in an area centred on the Fornax Cluster of galaxies. Among the unresolved targets are five objects which are members of the Fornax Cluster. They are extremely compact stellar systems with scale lengths less than 40 parsecs. These ultra-compact dwarfs are unlike any known type of stellar system, being more compact and significantly less luminous than other compact dwarf galaxies, yet much brighter than any globular cluster.

Noise-free scattering of the quantized electromagnetic field from a dispersive linear dielectric

We study the scattering of the quantized electromagnetic field from a linear, dispersive dielectric using the scattering formalism for quantum fields. The medium is modeled as a collection of harmonic oscillators with a number of distinct resonance frequencies. This model corresponds to the Sellmeir expansion, which is widely used to describe experimental data for real dispersive media. The integral equation for the interpolating field in terms of the in field is solved and the solution used to find the out field. The relation between the ill and out creation and annihilation operators is found that allows one to calculate the S matrix for this system. In this model, we find that there are absorption bands, but the input-output relations are completely unitary. No additional quantum-noise terms are required.

The use of analytical hierarchy process in spatial decision support system for land use management

Geographic information systems give us the possibility to analyze, produce, and edit geographic information. Furthermore, these systems fall short on the analysis and support of complex spatial problems. Therefore, when a spatial problem, like land use management, requires a multi-criteria perspective, multi-criteria decision analysis is placed into spatial decision support systems. The analytic hierarchy process is one of many multi-criteria decision analysis methods that can be used to support these complex problems. Using its capabilities we try to develop a spatial decision support system, to help land use management. Land use management can undertake a broad spectrum of spatial decision problems. The developed decision support system had to accept as input, various formats and types of data, raster or vector format, and the vector could be polygon line or point type. The support system was designed to perform its analysis for the Zambezi river Valley in Mozambique, the study area. The possible solutions for the emerging problems had to cover the entire region. This required the system to process large sets of data, and constantly adjust to new problems’ needs. The developed decision support system, is able to process thousands of alternatives using the analytical hierarchy process, and produce an output suitability map for the problems faced.

Conditional equilibrium constants in multicomponent heterogeneous adsorption: The conditional affinity spectrum

The concept of conditional stability constant is extended to the competitive binding of small molecules to heterogeneous surfaces or macromolecules via the introduction of the conditional affinity spectrum (CAS). The CAS describes the distribution of effective binding energies experienced by one complexing agent at a fixed concentration of the rest. We show that, when the multicomponent system can be described in terms of an underlying affinity spectrum [integral equation (IE) approach], the system can always be characterized by means of a CAS. The thermodynamic properties of the CAS and its dependence on the concentration of the rest of components are discussed. In the context of metal/proton competition, analytical expressions for the mean (conditional average affinity) and the variance (conditional heterogeneity) of the CAS as functions of pH are reported and their physical interpretation discussed. Furthermore, we show that the dependence of the CAS variance on pH allows for the analytical determination of the correlation coefficient between the binding energies of the metal and the proton. Nonideal competitive adsorption isotherm and Frumkin isotherms are used to illustrate the results of this work. Finally, the possibility of using CAS when the IE approach does not apply (for instance, when multidentate binding is present) is explored. © 2006 American Institute of Physics.

Panel Method Formulation for Oscillating Airfoils in Sonic Flow

The mathematical model for two-dimensional unsteady sonic flow, based on the classical diffusion equation with imaginary coefficient, is presented and discussed. The main purpose is to develop a rigorous formulation in order to bring into light the correspondence between the sonic, supersonic and subsonic panel method theory. Source and doublet integrals are obtained and Laplace transformation demonstrates that, in fact, the source integral is the solution of the doublet integral equation. It is shown that the doublet-only formulation reduces to a Volterra integral equation of the first kind and a numerical method is proposed in order to solve it. To the authors' knowledge this is the first reported solution to the unsteady sonic thin airfoil problem through the use of doublet singularities. Comparisons with the source-only formulation are shown for the problem of a flat plate in combined harmonic heaving and pitching motion.

Caractérisation et surexpression des fimbriae de type chaperon-placier de Salmonella enterica sérovar Typhi

Salmonella enterica sérovar Typhi (S. Typhi) est l’agent responsable de la fièvre typhoïde et cause environ 200 000 morts et 27 millions de cas annuellement. C’est un pathogène entérique dont le réservoir est restreint à l’Homme. Les raisons de cette restriction d’hôte sont méconnues et pourraient dépendre de l’expression de facteurs d’adhésion à des étapes importantes au cours de la pathogenèse. L’annotation bioinformatique du génome de S. Typhi identifie 12 fimbriae de type chaperon-placier (FCP), un curli ainsi qu’un pilus de type IV. L’objectif de ce projet de recherche est d’étudier ces systèmes d’adhésion peu caractérisés. D’abord, le niveau d’expression de ces gènes a été évalué dans différentes conditions de culture in vitro en utilisant une approche de gènes rapporteurs. L’expression des 14 systèmes d’adhésion a été détectée. Nos résultats indiquent qu’une carence en fer favorise l’expression des opérons bcf et csg. Indépendamment du fer, l’expression de bcf, csg, pil, sef, sta, stc, stg et sth est influencée par la richesse nutritive du milieu. L’incubation en milieu LB liquide favorise l’expression de la plupart des systèmes d’adhésion par rapport à un milieu LB liquide sans agitation ou un milieu LB solide. En somme, l’expression des systèmes d’adhésion de S. Typhi a été observée et est influencée par des conditions environnementales. Dans un second volet, nous avons tent de surexprimer les différents systèmes d’adhésion chez une souche d’E. coli ou de S. Typhi afimbriaire. Avec cette approche, nous avons été en mesure de démontrer que l’opéron tcf encode pour un fimbria fonctionnel que l’on a pu observer en microscopie électronique. L’expression de tcf chez une souche afimbriaire d’E. coli et S. Typhi a également diminué leur capacité d’adhésion à des cellules épithéliales intestinales humaines lors d’essais in vitro. Nos observations démontrent que l’expression des systèmes d’adhésion retrouvés chez S. Typhi est influencée par les conditions enviroi9onnementales. Au moins un de ces systèmes est fonctionnel. Ceci suggère une contribution des systèmes d’adhésion retrouvés chez S. Typhi lors de l’interaction de ce pathogène avec l’humain.

An approximation method using approximate approximations

The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Branching random motions, nonlinear hyperbolic systems and traveling waves

A branching random motion on a line, with abrupt changes of direction, is studied. The branching mechanism, being independient of random motion, and intensities of reverses are defined by a particle's current direction. A soluton of a certain hyperbolic system of coupled non-linear equations (Kolmogorov type backward equation) have a so-called McKean representation via such processes. Commonly this system possesses traveling-wave solutions. The convergence of solutions with Heaviside terminal data to the travelling waves is discussed.This Paper realizes the McKean programme for the Kolmogorov-Petrovskii-Piskunov equation in this case. The Feynman-Kac formula plays a key role.

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

Parallel Monte Carlo approach for integration of the rendering equation

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images. In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method. This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

Role of NleH, a Type III Secreted Effector from Attaching and Effacing Pathogens, in Colonization of the Bovine, Ovine, and Murine Gut

The human pathogen enterohemorrhagic Escherichia coli (EHEC) O157:H7 colonizes human and animal gut via formation of attaching and effacing lesions. EHEC strains use a type III secretion system to translocate a battery of effector proteins into the mammalian host cell, which subvert diverse signal transduction pathways implicated in actin dynamics, phagocytosis, and innate immunity. The genomes of sequenced EHEC O157: H7 strains contain two copies of the effector protein gene nleH, which share 49% sequence similarity with the gene for the Shigella effector OspG, recently implicated in inhibition of migration of the transcriptional regulator NF-kappa B to the nucleus. In this study we investigated the role of NleH during EHEC O157: H7 infection of calves and lambs. We found that while EHEC Delta nleH colonized the bovine gut more efficiently than the wild-type strain, in lambs the wild-type strain exhibited a competitive advantage over the mutant during mixed infection. Using the mouse pathogen Citrobacter rodentium, which shares many virulence factors with EHEC O157: H7, including NleH, we observed that the wild-type strain exhibited a competitive advantage over the mutant during mixed infection. We found no measurable differences in T-cell infiltration or hyperplasia in colons of mice inoculated with the wild-type or the nleH mutant strain. Using NF-kappa B reporter mice carrying a transgene containing a luciferase reporter driven by three NF-kappa B response elements, we found that NleH causes an increase in NF-kappa B activity in the colonic mucosa. Consistent with this, we found that the nleH mutant triggered a significantly lower tumor necrosis factor alpha response than the wild-type strain.