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All papers have been typed and translated trying to reproduce as much as possible their original aspect. We only used common fonts and the same style for the titles, the abstracts and the bibliography. [ABSTRACT FROM AUTHOR]

This set of notes was used to complement my short course on the convergence to equilibrium for the Boltzmann equation, given at Institut Henri Poincaré in November\2-December 2001, as part of the Hydrodynamic limits program organized by Stefano Olla and François Golse. The informal style is in accordance with the fact that this is neither a reference book nor a research paper. The reader can use my review paper, A review of mathematical topics in collisional kinetic theory, as a reference source to dissipate any ambiguity with respect to notation for instance. Apart from minor corrections here and there, the main changes with respect to the original version of the notes were the addition of a final section to present some more recent developments and open directions, and the change of the sign convention for the entropy, to agree with physical tradition. Irene Mazzella is warmly thanked for kindly typesetting a preliminary version of this manuscript. [ABSTRACT FROM AUTHOR]

In this chapter, we review and discuss precursory and alternative theories. We start in the first section with Lindenbaum et al. Their papers contain a theory of shape detection whose setting is essentially the same as the one developed in this book. The Bienenstock et al. compositional model discussed in Sections 15.2 and 15.5 is an ambitious theory attempting to build directly a grammar of visual primitives. A nice illustration of these compositional approaches is the work of Zhu et al. described in Section 15.2. Section 15.3 discusses the link among meaningful events, hypothesis testing, and Signal Detection Theory. It also shows that the Number of False Alarms (NFA) can be put in a classical statistical framework where multiple testing is involved. In Section 15.4 the Arias-Castro et al. geometric detection theory is addressed. This theory is very close in spirit to the tools in this book and are actually partly inspired from it. It gives complementary information on asymptotic geometric detection thresholds and hints on how to speed up detection algorithm. Section 15.5 discusses the Bayesian theory according to which the probability of the image interpretation given the observation must be maximized. An extension of this theory, the Minimum Description Length, is also invoked in the compositional model. In both cases, a probability is maximized. In contrast, meaningful events were obtained by minimizing an a-contrario probability. This point is discussed and the complementarity of both approaches are indicated. [ABSTRACT FROM AUTHOR]

Multiple Recursive Generators (MRGs) have become the most popular random number generators recently. They compute the next value iteratively from the previous k values using a k-th order recurrence equation which, in turn, corresponds to a k-th degree primitive polynomial under a prime modulus p. In general, when k and p are large, checking if a k-th degree polynomial is primitive under a prime modulus p is known to be a hard problem. A common approach is to check the conditions given in Alanen and Knuth [1964] and Knuth [1998]. However, as mentioned in Deng [2004], this approach has two obvious problems: (a) it requires the complete factorization of pk - 1, which can be difficult; (b) it does not provide any early exit strategy for non-primitive polynomials. To avoid (a), one can consider a prime order k and prime modulus p such that (pk - 1)/(p - 1) is also a prime number as considered in L'Ecuyer [1999] and Deng [2004]. To avoid (b), one can use a more efficient iterative irreducibility test proposed in Deng [2004]. In this paper, we survey several leading probabilistic and deterministic methods for the problems of primality testing and irreducibility testing. To test primality of a large number, it is known that probabilistic methods are much faster than deterministic methods. On the other hand, a probabilistic algorithm in fact has a very tiny probability of, say, 10-200 to commit a false positive error in the test result. Moreover, even when such an unlikely event had happened, for a speci.c choice of k and p, it can be argued that such an error has a negligible e.ect on the successful search of a primitive polynomial. We perform a computer search for large-order DX generators proposed in Deng and Xu [2003] and present many such generators in the paper for ready implementation. An extensive empirical study shows that these large-order DX generators have passed the stringent Crush battery of the TestU01 package. [ABSTRACT FROM AUTHOR]

It is well known that there are "nonclassical" situations where, in contrast to Lax's and Glimm's results, the Cauchy problem for a system of conservation laws does not possess a weak L∞-solution except for some particular initial data. To solve the Cauchy problem in this "nonclassical" situation, it is necessary to introduce new singular solutions called δ-shocks and singular shocks. The components of these solutions contain delta functions [ASh05], [B94], [DSh03]- [LW02], [S02]- [Sh04], [TZZ94]. The exact structure of such type solutions is given below in (2), (7) and Definition 1. The theory of δ-shocks and singular shocks has been intensively developed in the last 10 years. In particular, in numerous papers δ-shock type solutions of "zero-pressure gas dynamics" have been studied. Moreover, in the recent papers [PSh06], [Sh06] the theory of δ′-shocks was established, and a concept of δ(n)-shocks was introduced, n = 2, 3,. … They are new type singular solutions such that their components contain delta functions and their derivatives. [ABSTRACT FROM AUTHOR]

Many equations arising in continuum fluid dynamics do not, or at least are not known to, possess smooth solutions for general data. Therefore it is necessary to identify a larger class of "weak solutions" in order to develop a rigorous mathematical theory. Following the seminal paper of Leray [10] we introduce a class of weak solutions to the Navier-Stokes-Fourier system based on the concept of generalized derivatives (distributions). The main objective of the present paper is to illustrate the strength of the abstract theory reviewing several theoretical results on the asymptotic behavior of the weak solutions that are in good agreement with practical experiments as well as their numerical analysis.The main topics to be discussed include the following: [ABSTRACT FROM AUTHOR]

Starting from the modelling requirements of the early design phase of the product development, the paper will show a possible strategy to overcome some limitations of current CAS/CAD systems. In fact, the styling stage involves both technical knowledge and fuzzy and dynamic aspects, which have to be taken into account for a proper management. The paper focuses on high-level modelling tools developed to deform surfaces with semantic (aesthetic) constraints, i.e. the crucial design elements for the stylist. Furthermore, the communication among the other actors of the design process is consequently facilitated. [ABSTRACT FROM AUTHOR]

In finance, the strong convergence properties of discretisations of stochastic differential equations (SDEs) are very important for the hedging and valuation of exotic options. In this paper we show how the use of the Milstein scheme can improve the convergence of the multilevel Monte Carlo method, so that the computational cost to achieve an accuracy of O(e) is reduced to O(ϵ−2) for a Lipschitz payoff. The Milstein scheme gives first order strong convergence for all one-dimensional systems (one Wiener process). However, for processes with two or more Wiener processes, such as correlated portfolios and stochastic volatility models, there is no exact solution for the iterated integrals of second order (Lévy area) and the Milstein scheme neglecting the Lévy area gives the same order of convergence as the Euler-Maruyama scheme. The purpose of this paper is to show that if certain conditions are satisfied, we can avoid the calculation of the Lévy area and obtain first convergence order by applying an orthogonal transformation. We demonstrate when the conditions of the two-dimensional problem permit this and give an exact solution for the orthogonal transformation. We present examples of pricing exotic options to demonstrate that the use of both the orthogonal Milstein scheme and the multilevel Monte Carlo give a substantial reduction in the computation cost. [ABSTRACT FROM AUTHOR]

This paper is meant to be the continuation of the previous work [1] where a coupled ODE/PDE method for the simulation of semiconductor devices was introduced. From a strictly mathematical viewpoint, analytical results on coupled PDE/ODE systems (as arising in integrated circuit simulation) can be found in [2]. In particular, in the present paper, we investigate numerically new algorithms of Domain Decomposition type for the simulation of circuits containing distributed devices (Sect. 2) as well as semiconductors in which some part is modeled with lumped parameters (Sect. 3). The results presented here have been investigated in the seminal work [3], while a more extended analysis is ongoing [4]. [ABSTRACT FROM AUTHOR]

This paper is concerned with the generation of C0 semigroup associated with the Oseen equation with rotating effect and its Lp-Lq decay estimate. The theorems presented in this paper give us one of the key steps in order to show a globally in time existence of solutions to the Navier-Stokes equations describing the motion of viscous incompressible fluid flow past a rotating rigid body. [ABSTRACT FROM AUTHOR]

In this paper we explain the notion of stochastic backward differential equations and its relationship with classical (backward) parabolic differential equations of second order. The paper contains a mixture of stochastic processes like Markov processes and martingale theory and semi-linear partial differential equations of parabolic type. Some emphasis is put on the fact that the whole theory generalizes Feynman-Kac formulas. A new method of proof of the existence of solutions is given. All the existence arguments are based on rather precise quantitative estimates. [ABSTRACT FROM AUTHOR]

For centuries celestial mechanics has been an exceptional rich source of problems and results in mathematics. To some extent this is still the case. Today one can discern, rather artificially, three problem fields. The first one is the study of classical problems like perturbed Kepler motion, orbits in the threebody problem, the theory of asteroids and comets, etc. The second one is a small but relatively important field in which the astrophysicists are interested; we are referring to systems with evolution like for instance changes caused by tidal effects or by exchange of mass. The third field is what one could call ‘mathematical celestial mechanics', a subject which is part of the theory of dynamical systems. The distinction between the fields is artificial. There is some interplay between the fields and hopefully, this will increase in the future. An interesting example of a study combining the first and the third field is the paper by Brjuno [41]. A typical example of an important mathematical paper which has found already some use in classical celestial mechanics is Moser's study on the geometrical interpretation of the Kepler problem [193]. Surveys of mathematical aspects of celestial mechanics have been given in [194] and [3]. [ABSTRACT FROM AUTHOR]

In this paper, new natural element approximations are proposed, in order to address issues associated with incompressibility as well as to increase the accuracy in the Natural Element Method (NEM). The NEM exhibits attractive features such as interpolant shape functions or auto-adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well known inf-sup, or LBB condition. The first proposed approach constructs richer NEM approximation schemes by means of bubbles associated with the topological entities of the underlying Delaunay tessellation, allowing to pass the LBB and to remove pressure oscillations in the incompressible limit. Despite of its simplicity, this approach does not construct approximation with higher order consistency. The second part of the paper deals with a discussion on the construction of second-order accurate NEM approximations. For this purpose, two techniques are investigated : (a) the enrichment in the MLS framework of the bubbles with higher-order polynomials and (b) the use of a new Hermite-NEM formulation. [ABSTRACT FROM AUTHOR]

In this paper we present the application of the meshfree method of finite spheres to the solution of thin and thick plates composed of isotropic as well as functionally graded materials. For the solution of such problems it is observed that using Gaussian and adaptive quadrature schemes are computationally inefficient. In this paper a new technique, presented in [26, 21], in which the integration points and weights are generated using genetic algorithms and stored in a lookup table using normalized coordinates as part of an offline preprocessing step, is shown to provide significant reduction of computational time without sacrificing accuracy. [ABSTRACT FROM AUTHOR]

In this short paper, we present a formalism which specifies the notions of consistency and stability of finite element methods for the numerical approximation of nonlinear partial differential equations of elliptic and parabolic type. This formalism can be found in [4], [7], [10], and allows to establish a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods. In concrete cases, the Cléement's interpolation technique [6] is very useful in order to establish local a posteriori error estimates. This paper uses some ideas of [10] and its main goal is to show in a very simple setting, the mathematical arguments which lead to the stability and convergence of Galerkin methods. The bibliography concerning this subject is very large and the references of this paper are no exhaustive character. In order to obtain a large bibliography on the a posteriori error estimates, we report the lecturer to Verfürth's book and its bibliography [12]. [ABSTRACT FROM AUTHOR]

Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n−1+δ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper. [ABSTRACT FROM AUTHOR]

It has been found recently that an increase in phase space dimension by including simulated auxiliary random variables in the number of phase coordinates can be effective for the construction of weight modifications. In this paper the effectiveness of "value" and partial "value" modelling is considered. These types of modelling are related to the construction of simulated distribution for some auxiliary random variable by multiplying the initial density by the "value" function which is usually corresponds to the solution of adjoint integral equation of the second kind. It is proved that the weight estimator variance in case of the partial value modelling is finite. On the basis of this fact a new criterion based on the use of majorant adjoint equation was proposed for finiteness of the weight estimator variance. Using this criterion the classical "exponential transformation" method is studied for the free path simulation in one and three dimensional modifications. [ABSTRACT FROM AUTHOR]

The (weighted) dyadic diaphony is a measure for the irregularity of distribution modulo one of a sequence. Recently it has been shown that the (weighted) dyadic diaphony can be interpreted as the worst-case error for QMC integration in a certain Hilbert space of functions. In this paper we give upper bounds on the weighted dyadic diaphony of digital (t, s)-sequences over ℤ2. [ABSTRACT FROM AUTHOR]

We study approximation of functions belonging to Sobolev spaces Wrp (Q) by randomized algorithms based on function values. Here 1 ≤ p ≤ ∞, Q = [0, 1]d, and r, d ∈ N. The error is measured in Lq(Q), with 1 ≤ q < ∞, and we assume r/d > 1/p − 1/q, guaranteeing that Wrp(Q) is embedded into Lq(Q). The optimal order of convergence for the case that Wrp(Q) is embedded even into C(Q) is wellknown. It is n−r/d+max(1/p−1/q,0) (n the number of function evaluations). This rate is already reached by deterministic algorithms, and randomization gives no speedup. In this paper we are concerned with the case that Wrp(Q) is not embedded into C(Q) (but, of course, still into Lq(Q)). For this situation approximation based on function values was not studied before. We prove that for randomized algorithms the above rate also holds, while for deterministic algorithms no rate whatsoever is possible. Thus, in the case of low smoothness, Monte Carlo approximation algorithms reach a considerable speedup over deterministic ones (up to n−1+ε for any ε > 0). We also give some applications to integration of functions and to approximation of solutions of elliptic PDE. [ABSTRACT FROM AUTHOR]

We study approximation of continuous linear functionals Id defined over reproducing kernel weighted Hilbert spaces of d-variate functions. Let n(ε, Id) denote the minimal number of function values needed to solve the problem to within ε. There are many papers studying polynomial tractability for which n(ε, Id) is to be bounded by a polynomial in ε−1 and d. We study generalized tractability for which we want to guarantee that either n(ε, Id) is not exponentially dependent on ε−1 and d, which is called weak tractability, or is bounded by a power of T(ε−1, d) for (ε−1, d) ∈ Ω ⊆ [1,∞) × N, which is called (T,Ω)-tractability. Here, the tractability function T is non-increasing in both arguments and does not depend exponentially on ε−1 and d. We present necessary conditions on generalized tractability for arbitrary continuous linear functionals Id defined on weighted Hilbert spaces whose kernel has a decomposable component, and sufficient conditions on generalized tractability for multivariate integration for general reproducing kernel Hilbert spaces. For some weighted Sobolev spaces these necessary and sufficient conditions coincide. They are expressed in terms of necessary and sufficient conditions on the weights of the underlying spaces. [ABSTRACT FROM AUTHOR]

In this paper we show that the Milstein scheme can be used to improve the convergence of the multilevel Monte Carlo method for scalar stochastic differential equations. Numerical results for Asian, lookback, barrier and digital options demonstrate that the computational cost to achieve a root-mean-square error of ε is reduced to O(ε-2). This is achieved through a careful construction of the multilevel estimator which computes the difference in expected payoff when using different numbers of timesteps. [ABSTRACT FROM AUTHOR]

In analogy to a recent paper by Kuo, Sloan, and Woźniakowski, which studied lattice rule algorithms for approximation in weighted Korobov spaces, we consider the approximation problem in a weighted Hilbert space of Walsh series. Our approximation uses a truncated Walsh series with Walsh coefficients approximated by numerical integration using digital nets. We show that digital nets (or more precisely, polynomial lattices) tailored specially for the approximation problem lead to better error bounds. The error bounds can be independent of the dimension s, or depend only polynomially on s, under certain conditions on the weights defining the function space. [ABSTRACT FROM AUTHOR]

DX-k, proposed by Deng and Xu [2003], is a special class of multiple Recursive Generators (MRGs) where all nonzero coefficients of the k-th order recurrence are equal. In particular, a DX-k generator requires only up to four nonzero coefficients in its recurrence equation, hence is very efficient in computation. However, a random number generator with few nonzero coefficients has a drawback that, when the k-dimensional state vector is close to the zero vector, the subsequent numbers generated may stay within a neighborhood of zero for quite many of them before they can break away from this near-zero land, a property apparently not desirable in the sense of randomness. Consequently, two generated sequences using the same DX generator with nearly identical initial state vectors may not depart from each other quickly enough. To avoid the above potential problem, we consider MRGs with very few zero coefficients. To make such generators efficient and portable, we propose selecting the same nonzero value for all coefficients (with at most one exception) in the recurrence equation. With this feature, the proposed generators can be implemented efficiently via a higher-order recurrence of few zero coefficients. Note that the new class of generators is an opposite of the DX generators in terms of the number of nonzero coefficients. Several such generators with the maximum period have been found via computer search and presented in the paper for ready implementation. [ABSTRACT FROM AUTHOR]

Some computer graphics applications, such as architectural design, generate visually realistic images of computer models. This is accomplished by either explicitly or implicitly solving the light transport equations. Accurate solutions involve high-dimensional equations, and Monte Carlo (MC) techniques are used with an emphasis on importance sampling rather than stratification. For many applications, approximate solutions are adequate, and the dimensionality of the problem can be reduced. In these cases, the distribution of samples is important, and quasi-Monte Carlo (QMC) methods are often used. It is still unknown what sampling schemes are best for these lower dimensional graphics problems, or what "best" even means in this case. This paper reviews the work in MC and QMC computer graphics, and poses some open problems in the field. [ABSTRACT FROM AUTHOR]

In this paper I present some new approaches to the random field simulation, and show in four different examples how this simulation technique works. The first example deals with a transport in turbulent flows, where the Lagrangian trajectories are described by a stochastic differential equation whose drift term involves the Eulerian velocity as a random field with a given spectral tensor. Studies of the second example concern with the flows in porous medium governed by the Darcy equation with random hydraulic conductivity. Elasticity system of elliptic Lamé equations with random loads is considered in the third example. Finally, in the fourth example we solve a nonlinear Smoluchowski equation which is used to model the process of crystal growth. [ABSTRACT FROM AUTHOR]

In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert -valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options. [ABSTRACT FROM AUTHOR]

Nets and (t, s)-sequences are standard sources of quasirandom points for quasi-Monte Carlo methods. Connections between nets and error-correcting codes have been noticed for a long time, and these links have become even more pronounced with the development of the duality theory for digital nets. In this paper, we further explore these fascinating connections. We present also a recent construction of digital (t, s)-sequences using global function fields and new general constructions of nets and (t, s)-sequences. [ABSTRACT FROM AUTHOR]

In a recent paper, Herty and Rascle (SIAM J. Appl. Math. 38:595-616, 2006) introduced some coupling conditions to model road junctions with the Aw-Rascle traffic flow model (Aw and Rascle, SIAM J. Appl. Math. 60:916-944, 2000). In this chapter, we study an extension of the work in Herty and Rascle (SIAM J. Appl. Math. 38:595-616, 2006). Here, in contrast to Herty and Rascle (SIAM J. Appl. Math. 38:595-616, 2006), we propose a general formulation for suitable coupling conditions at an intersection, without imposing any fixed mixture principle. The solution conserves mass and "pseudomomentum" and additionally maximizes the total (mass) flux at an intersection. This problem has been completely solved in the cases of 2 → 1 (merge) and 1 → 2 (diverge) junctions. Furthermore these two simple cases of merging and diverging junctions can be combined to model more complex junctions, like roundabouts. [ABSTRACT FROM AUTHOR]

We present second-order accurate central finite volume methods adapted here to three-dimensional problems in ideal magnetohydrodynamics. These methods alternate between two staggered grids, thus leading to Riemann solver-free algorithms with relatively favorable computing times. The original grid considered in this paper is Cartesian, while the dual grid features either Cartesian or diamond-shaped oblique dual cells. The div.B = 0 constraint on the magnetic field is enforced with a suitable adaptation of the constrained transport method to our central schemes. Numerical experiments show the feasibility of these methods and our results are in good agreement with existing results in the literature. [ABSTRACT FROM AUTHOR]

This paper deals with a coupled system of non-linear elliptic differential equations arising in electrodeposition modelling process. We show the existence and uniqueness of the solution. A numerical algorithm to compute an approximation of the weak solution is described. We introduce a domain decomposition method to take in account the anisotropy of the solution. We show the domain decomposition method convergence. A numerical example is presented and commented. [ABSTRACT FROM AUTHOR]

State estimation is necessary in diagnosing anomalies inWater Demand Systems (WDS). In this paper we present a neural network performing such a task. State estimation is performed by using optimization, which tries to reconcile all the available information. Quantification of the uncertainty of the input data (telemetry measures and demand predictions) can be achieved by means of robust estate estimation. Using a mathematical model of the network, fuzzy estimated states for anomalous states of the network can be obtained. They are used to train a neural network capable of assessing WDS anomalies associated with particular sets of measurements. [ABSTRACT FROM AUTHOR]

Thin-film flows occur in a variety of physical contexts including, for example, industry, biology and nature, and have been the subject of considerable theoretical research. (See, for example, the review by Oron, Davis and Bankoff [4].) In particular, there are several practically important situations in which an external airflow has a significant effect on the behaviour of a film of fluid, and consequently there has been considerable theoretical and numerical work done to try to understand better the various flows that can occur. (See, for example, the studies by King and Tuck [2] and Villegas-Díaz, Power and Riley [6].) The flow of a rivulet on a planar substrate subject to a shear stress at its free surface has been investigated by several authors, notably Myers, Liang and Wetton [3], Saber and El-Genk [5], and Wilson and Duffy [9]. All of these works concern a non-perfectly wetting fluid; the flow of a rivulet of a perfectly wetting fluid in the absence of a shear stress at its free surface has been treated by Alekseenko, Geshev and Kuibin [1], and by Wilson and Duffy [7,8]. In the present short paper we use the lubrication approximation to obtain a complete description of the steady unidirectional flow of a thin rivulet of a perfectly wetting fluid on an inclined substrate subject to a prescribed uniform longitudinal shear stress at its free surface. [ABSTRACT FROM AUTHOR]

The evaporation of liquid droplets is of fundamental importance to industry, with a vast number of applications including ink-jet printing, spray cooling and DNA mapping, and has been the subject of considerable theoretical and experimental research in recent years. Significant recent papers include those by Deegan [1], Deegan et al. [2], Hu and Larson [3], Poulard et al. [4], Sultan et al. [5], and Shahidzadeh-Bonn et al. [6]. [ABSTRACT FROM AUTHOR]

This paper introduces the principles of creating the web-tool on differential equations. It can be used to support European Master Program for Mathematics in Industry. Such a Program is working already on the leading partner universities of ECMI and now the use of e-study as an innovational step is being discussed. [ABSTRACT FROM AUTHOR]

In this paper we will show an industrial application of a new framework, called LaTEX2WeB, which translates LaTEX material into an interactive Web-based document. The more important characteristic of LaTEX2WeB is the possibility of integrating, in the Web-based document, external programs produced in every languages. We exploited LaTEX2WeB to create an interactive Web-based manual, which illustrates a new software for the multiobjective optimization applied to the parameter extraction in circuit design. Thanks to LaTEX2WeB it was possible to simulate the algorithms written in C, C++, and FORTRAN, used in the multiobjective optimization software. [ABSTRACT FROM AUTHOR]

In this paper we consider the recovery of ellipsoidal 3D shapes with piecewise constant coefficients in Diffuse Optical Tomography (DOT). We use an adjoint scheme for calculating gradients for the shape parameters defining the unknown ellipsoids, and a Newton-type optimisation process for the minimization of a least squares data misfit functional. A boundary integral formulation is used for the forward modelling. An advantage of the proposed method is the implicit regularisation effect arising from the reduced dimensionality of the inverse problem. Results of a numerical experiment in 3D are shown which demonstrate the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate model order reduction of a nonlinear academic model of a diode chain. Two reduction methods, which are suitable for nonlinear differential algebraic equation systems are used, the trajectory piecewise linear approach and the proper orthogonal decomposition with missing point estimation. [ABSTRACT FROM AUTHOR]

The general question which will make the object of this paper is the homogenization of some nonlinear problems arising in the modelling of thermal diffusion in a two-component composite. We shall consider, at the microscale, a periodic structure formed by two materials with different thermal properties. We shall deal with two situations: in the first one, we assume that we have some nonlinear sources acting in both components and that at the interface between our two materials the temperature and the flux are continuous, while in the second problem we shall address here, we assume that the flux is still continuous, but depends in a nonlinear way on the jump of the temperature field. In both cases, since the characteristic sizes of these two components are small compared with the macroscopic length-scale of the flow domain, we can apply an homogenization procedure. [ABSTRACT FROM AUTHOR]

The correct simulation of industrial plants firing pulverized fuels (pf: coal, biomass, etc.) by means of commercial CFD codes relies on a number of submodels for the various processes, including, e.g. heat transfer (radiation, conduction through deposits, etc.) and particle combustion. The latter is of major importance in the design of the combustion chamber and the selection of the mills or, conversely, regarding the feasibility of burning a new fuel in an existing boiler. In the last decade, the introduction of new, internationally traded coals and alternative fuels into the power market has motivated renewed interest in the experimental and theoretical characterization of the combustion of these fuels. Regarding experimentation, it is generally accepted that the ‘reactivity' of a fuel can not be determined in desktop analytical instruments; instead, drop tube furnaces or entrained flow reactors (EFR) must be used in order to reproduce the high temperature, high heating rate conditions found in a real pf combustion chamber [1]. Several alternative experimental procedures have been developed in the past and are still used (see, e.g. [2, 3]). On the other hand, two general approaches are used in the literature to model pulverized coal/biomass char combustion: one intends to characterize the evolution of the pores inside the burning particle, and considers both internal and external diffusion, whereas the kinetics for the basic homogeneous and heterogeneous reactions are taken from low temperature analysis or fundamental knowledge of the chemistry involved (e.g. [4]); the other one, followed here, makes use of an apparent kinetics based on the outer particle surface, and includes external diffusion [5]. In the latter case, two parameters governing an Arrhenius-like kinetics are the main unknowns to be determined from the experiments performed in an EFR. The aim of this paper is to discuss some aspects of the mathematical procedure for the determination of those parameters. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to contribute to the mathematical modelling of the combustion of coal particles in pulverised coal furnaces, and also to propose an algorithm for its numerical solution. The mathematical model includes two coupled phases: the solid phase, for the coal particles, where a Lagrangian description is used and an Eulerian description for the gas phase, where the effects of the combustion of coal particles are homogenised. [ABSTRACT FROM AUTHOR]

This paper is concerned with a fascinating phenomenon in boundary layer transition, namely, three-dimensional disturbances undergo rapid amplification despite that they have smaller linear growth rates than two-dimensional ones. Physical mechanisms are sought by considering two types of nonlinear interactions between oblique and planar instability modes. The first is the well-known subharmonic resonance. The relevant mathematical theory and its main predictions are briefly summarised. This mechanism, however, operates only among a very restrictive set of modes, and hence is unable to explain the broadband nature of the amplifying disturbances observed in experiments. The second mechanism involves the interaction between a planar and a pair of oblique Tollmien—Schlichting (T—S) waves which are phase-locked in that they travel with (nearly) the same phase speed. It is a more general type of interaction than subharmonic resonance since no further restriction is imposed on the frequencies. Yet similar to subharmonic resonance, this interaction also leads to super-exponential growth of the oblique modes, while the planar mode remains to follow linear stability theory. The dominant planar mode therefore plays the role of a catalyst, the implications of which for the eN-method and for transition control are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with mathematical modelling and numerical simulation of induction heating furnaces for axisymmetric geometries. The mathematical model presented consists in a coupled thermo-magneto-hydrodynamic problem with phase change. We propose a finite element method and an iterative algorithm to solve the equations. Some numerical results for an industrial furnace used for silicon purification are shown. [ABSTRACT FROM AUTHOR]

The objective of this paper is to provide an analytic theory for pricing of Asian options of European type. We present a partial differential equation describing the fair price process of an Asian option. This appears as $$ \left( {\partial _t - A - x \cdot \nabla _y } \right)u = 0 $$ and the associated payoff function as the end value. Here the operator A is the d-dimensional Black-Scholes operator, and B = x·∇y represents the path dependence in terms of the price averaging in Asian options. The main result will be to prove, that a solution of this partial differential equation exists, is unique, and depends continuously on the data in appropriate function spaces, i.e., that the problem is well posed. On our way we are going to employ semigroup methods, in particular the Lumer-Phillips theorem. [ABSTRACT FROM AUTHOR]

The paper is concerned with the general theory of nonlinear agedependent population dynamics. We present (a) a principle of linearized stability and (b) a result on regularity of solutions to the general nonlinear model. [ABSTRACT FROM AUTHOR]

This paper is concerned with time-dependent relatively continuous perturbations of analytic semigroups and applications to convective reaction-diffusion systems. A general class of time-dependent semilinear evolution equations of the form ut = (A + B(t))u(t), t ∈ (s, τ); u(s) = v ∈ D(s) is introduced in a general Banach space X. Here A is the generator of an analytic semigroup in X and B(t) is a possibly nonlinear operator from a subset of the domain of a fractional power (−A)α into X and D(t) = D(B(t)) ⊂ D((−A)α). This type of semilinear evolution equations admit only local and mild solutions in general. In order to restrict the growth of mild solutions and formulate a Lipschitz conditions in a local sense for B(t), a lower semicontinuous functional ϕ: D((−A)α) → [0,+∞] is introduced and the growth condition of u(·) is formulated in terms of the nonnegative function ϕ(u(·)) and the nonlinear operator B(t) is assumed to be Lipschitz continuous on Dρ(t) ≡ {v ∈ D(t): ϕ(v) ≤ ρ for ρ > 0. The main objective is to establish a generation theorem for a nonlinear evolution operator which provides mild solutions to the semilinear evolution equation under the assumption that a consistent discrete scheme exists under a growth condition with respect to ϕ as well as closedness condition for the noncylindrical domain ∪({t}×Dρ(t)). Moreover, a characterization theorem for the existence of such evolution operator is established in terms of the existence of ϕ-bounded discrete schemes. Our generation theorem can be applied to a variety of semilinear convective reaction-diffusion systems. We here make an attempt to apply our result to a mathematical model which describes a complex physiological phenomena of bone remodeling. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the generation of analytic semigroup for a family of degenerate elliptic operators (with unbounded coefficients) which includes well-known operators arising in mathematical finance. The generation property is proved by assuming some compensation conditions among the coefficients and applying a suitable modification of the techniques developed in [16]. Using the results proved in [11] concerning the generation in the space L2(ℝd), we prove the generation results in Lp(ℝd) for p ∈ [1,+∞]. These results have several consequences in connection with the financial applications [3, 11]. [ABSTRACT FROM AUTHOR]

The mathematical theory of viscous multipolar fluids, based on the general ideas of Green and Rivlin [8], was proposed by Nečas and Šilhavý [17] (see also Nečas et al. [15], [16] for relevant existence theory) in order to develop a general framework for studying viscous fluids and to present a suitable alternative to the boundary layer theory (see Bellout et al. [1]). The theory is compatible with the basic principles of thermodynamics as well as with the principle of material frame indifference. The present paper is concerned with the mathematical description of the motion of one or several rigid bodies immersed in a viscous multipolar fluid. The principal and very natural idea behind the analysis presented below is the fact that the dissipation of mechanical energy, being much stronger than for classical newtonian fluids, yields better estimates on the gradient of the velocity field, in particular, the streamlines are well defined, which seems crucial for this class of problems partially formulated in terms of the Lagrangian coordinate system. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove a priori error estimates for the semi-discrete solution of the dual mixed method for the heat diffusion equation in a polygonal domain. Due to the geometric singularities of the domain, the solution is not regular in the context of classical Sobolev spaces. Instead, one must use weighted Sobolev spaces. In order to recapture the optimal order of convergence, the meshes are refined in an appropriate fashion near the reentrant corners of the domain. [ABSTRACT FROM AUTHOR]

In this paper we show that the curve shorting flow with contact angle and triple junction in a mirror symmetric configuration is locally well posed in suitable Hölder spaces. [ABSTRACT FROM AUTHOR]

The integral equation that plays a key role in AeroElasticity is known as the Possio Integral Equation, named after its discoverer. From its inception in 1938, this equation was formulated in the Fourier Transform domain using divergent integrals, until 2002 when a more precise formulation valid in a right half-plane was given. In this paper we express it in the time-domain, which requires the language of Functional Analysis, Lp spaces, 1 < p < 2, and Semigroup Theory. A key role is played by the Finite Hilbert Transform and the Tricomi-Sohngen airfoil equation, which may actually be considered a special case. [ABSTRACT FROM AUTHOR]