ALBERT

Beschreibung: This article evaluates 186 papers, published between 1978 and 2013 in 16 representative scientific journals, related to maintenance and reliability problems that were tackled from a multi-criteria (MC) perspective. An overview and insights are presented. This study may be useful to researchers and others concerned with maintenance and reliability who seek not only to understand the potential of MC and multi-objective models but also to develop and apply an MC decision model to help solve a real problem in these areas. There are some discussions on some principles for the application of MC in maintenance and reliability and some guidance on how to choose a suitable MC method is given, based on previous applications.

Beschreibung: A credit-linked note (CLN) is a type of credit derivative, constructed with a bond and an embedded credit default swap, which allows the issuer to transfer a specific credit risk to credit investors. In this paper, we model CLNs with and without counterparty risk in the reduced-form framework. For CLNs with counterparty risk, we consider two different scenarios, i.e. the issuer of CLNs and reference assets have either positive correlation or negative correlation. Assuming the interest rate follows the Cox–Ingersoll–Ross (CIR) model (Cox et al ., 1985) and the default events mainly depend on the interest rate, we model the two different correlations. Explicit formulas for value functions are obtained through a partial differential equation approach. In addition, counterparty valuation adjustment and the dependence on related parameters are also investigated.

Beschreibung: The class of elastic bodies, that is bodies incapable of dissipation in whatever motion that they undergo, has been significantly enlarged recently (see Rajagopal 2003, On implicit constitutive theories. Appl. Math. , 48 , 279–319; Rajagopal 2007, The elasticity of elasticity. Z. Angew. Math. Phys. 58 , 309–317; Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids. Proc. R. Soc. Lond. A , 463 , 357–367). The new classes include fully implicit constitutive relations for the stress and the deformation gradient, and the interesting sub-class wherein the Cauchy–Green tensor or the linearized strain tensor bears a non-linear relationship to the stress. While a fully thermodynamic treatment of such elastic bodies, when defined through implicit constitutive relations between the Piola stress and the Green–St. Venant strain, within a 3D framework has been carried out (see Rajagopal, K. R. & Srinivasa, A. R. 2007, On the response of non-dissipative solids, Proc. R. Soc. Lond. A , 463 , 357–367), other possible implicit relationships between other stress and kinematic measures have not been investigated. This paper is devoted to the determination of the consequences of thermodynamics on the new class of elastic bodies, when they are expressed through implicit relations for different stress and stretch/strain measures.

Beschreibung: This work investigates the existence and non-existence of travelling wave solutions for Kolmogorov-type delayed lattice reaction–diffusion systems. Employing the cross iterative technique coupled with the explicit construction of upper and lower solutions in the theory of quasimonotone dynamical systems, we can find two threshold speeds $c^* $ and $c_* $ with $c^* \geq c_* 〉 0.$ If the wave speed is greater than $c^* ,$ then we establish the existence of travelling wave solutions connecting two different equilibria. On the other hand, if the wave speed is smaller than $c_* ,$ we further prove the non-existence result of travelling wave solutions. Finally, several ecological examples including one-species, two-species and three-species models with various functional responses and time delays are presented to illustrate the analytical results.

Beschreibung: In this paper, sufficient criteria for global asymptotic stability of a general stochastic non-autonomous Lotka–Volterra system with infinite delays are established. Then these criteria are applied to a competitive system, a predator–prey system and a cooperative system to demonstrate their applicability and effectiveness. Some recent results are extended and improved.

Beschreibung: In the paper, the problem of existence of almost periodic solutions to non-linear impulsive systems of Caputo fractional order is addressed. Sufficient conditions for stability of almost periodic solutions are also established. The investigations are carried out by using of a fractional comparison principle and the Lyapunov's function method. We apply our results to an impulsive cellular neural network model of fractional order.

Beschreibung: In this paper we propose a limited memory method, called the Broyden rank reduction (BRR) method, to efficiently compute fixed points of high-dimensional non-linear maps. The method is based on the singular value decomposition and has good convergence properties. We discuss some basic properties of the BRR method and prove convergence for linear systems. The method is developed for a concrete problem in engineering. In chemical engineering, periodically forced processes in packed bed reactors are generally simulated using pseudo-homogeneous one-dimensional models on a coarse grid discretization. In these simulations, one typically uses Broyden's method because of its simplicity and the fact that appropriate initial conditions can be chosen in a natural way. A disadvantage of Broyden's method is the memory usage to store the Broyden matrix. The BRR method resolves this issue and allows us to consider more complicated and accurate models. We show that the BRR method developed to simulate periodically forced processes is in certain situations a good option to compute fixed points of high-dimensional non-linear maps and illustrate this using examples from integral equations and parabolic partial differential equations. To benchmark our method, we conclude the paper with a comparison of the BRR method with the generally preferred Newton–GMRES( m ) method.

Beschreibung: In this paper, the relationship between representation formulas for unique regularized solutions of inverse source, as well as backward problems with final overdetermination for evolution equations and singular value decomposition (SVD) of corresponding input–output operators, is analysed. For each considered inverse problem this representation formula is derived via the solution of appropriate adjoint problems and the parameter of regularization. The Lagrange multiplier method is then used to show that each regularized inverse problem is equivalent to a coupled system of two (direct and adjoint) parabolic (or hyperbolic) problems, and the Lagrange multiplier is the weak solution of the corresponding adjoint problem. In the constant coefficient linear parabolic and hyperbolic equations cases, it is proved that the coupled problem is equivalent to the normal equation corresponding to each inverse/backward problem. As a result, an equivalence of the representation formula for unique regularized solution and the SVD of the input–output operator is obtained. This approach, in particular, allows one to construct regularizing filters for a considered class of inverse problems. Some numerical examples are presented to illustrate the usefulness and effectiveness of the proposed approach.

Beschreibung: This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier–Stokes equations, complexified in time. Specifically, for periodic boundary conditions on [0, L ] 2 , and a force g D( A (α–1)/2 ), we show that there is a fixed strip about the real-time axis on which a uniform bound $|A^{\alpha}u| 〈 m_{\alpha}\nu\kappa_{0}^{\alpha}$ holds for each $\alpha \in {\mathbb {N}}$ . Here is viscosity, $\kappa_{0}=2\pi/L$ , and m α is explicitly given in terms of g and α. We show that if any element in A is in $\mathcal{D}(A^{\alpha})$ , then all of A is in $\mathcal{D}(A^{\alpha})$ , and likewise with $\mathcal{D}(A^{\alpha})$ replaced by $C^{\infty}(\Omega)$ . We demonstrate the universality of this ‘all for one, one for all’ law is also shown to hold for the union of a hierarchical set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a non-zero force by showing that if this is so, the force must be in a particular function class.

Beschreibung: An exact solution is obtained for the steady plane magnetohydrodynamics oblique stagnation-point flow of a homogeneous, incompressible, electrically conducting micropolar fluid over a rigid uncharged dielectric at rest. The space is permeated by a non-uniform external magnetic field and by a suitable uniform external electric field. The total magnetic field in the fluid is parallel to the velocity at infinity. The problem is reduced to a non-linear ordinary differential boundary value problem by using similarity transformations. The effects of the magnetic field on the velocity and on the microrotation profiles are presented graphically and discussed.

Beschreibung: In this paper, we analyse the connection between approximate non-linear self-adjointness of perturbed partial differential equations (PDEs) and non-linear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate non-linear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation laws of the corresponding unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximately non-linearly self-adjoint and construct its general approximate conservation law formula. The specific examples demonstrate that approximate non-linear self-adjointness can be used to construct new approximate conservation laws of perturbed PDEs.

Beschreibung: Practical physical applications of mathematical nature are frequently met in engineering technology and involve the low-Reynolds number flow of micropolar conducting fluids under the effect of magnetic fields. Here, we consider the three-dimensional (3D) creeping motion (Stokes flow), in steady state, of a non-conductive colloidal suspension of ferromagnetic material embedded within an electrically conductive, viscous and incompressible, carrier liquid. In such cases, the ferromagnetic particles behave as rigid magnetic dipoles and react in the presence of an externally applied magnetic field, which is of general form and arbitrarily orientated in the 3D space. Therein, an induced magnetic field of minor significance is created, while the effective viscosity of the fluid is increasing and an additional magnetic pressure is appeared. The consistency of the governing set of partial differential equations with the principles of both ferrohydrodynamics and magnetohydrodynamics is established by taking into account magnetization and electrical conductivity of the fluid, respectively. Our main intension is to use the potential representation theory to improve previous models and construct a new complete and unique integro-differential representation of the magnetic Stokes flow of conducting liquids, valid for any non-axisymmetric geometry, which provides the velocity and total pressure fields in a closed form and in terms of easy-to-find potentials, via a semi-analytical formalism. To demonstrate the usefulness of our analytical approach, we assume a degenerate case of the aforementioned method to simulate the creeping flow of a micropolar fluid with conductive properties inside a circular duct.

Beschreibung: Written to coincide with the anniversary of the launch of our journal, this survey reviews some of the highlights from the first fifty years of the IMA Journal of Applied Mathematics and its predecessor the Journal of the IMA. Rather than an exhaustive survey, we have chosen some of our personal favourite articles from the last fifty years. For each, we have given only the briefest of summaries—readers can download each of these papers for themselves, as they have each been put on open access to coincide with this Issue. We also attempt to put the work in context. More importantly, these papers were chosen as they have had, and continue to have, a considerable influence. For each, we try to spell out the impact of the work, and by implication, the impact of our journal.

Beschreibung: Social media data tend to cluster around events and themes. Local newsworthy events, sports team victories or defeats, abnormal weather patterns and globally trending topics all influence the content of online discussion. The automated discovery of these underlying themes from corpora of text is of interest to numerous academic fields as well as to law enforcement organizations and commercial users. One useful class of tools to deal with such problems are topic models, which attempt to recover latent groups of word associations from the text. However, it is clear that these topics may also exhibit patterns in both time and space. The recovery of such patterns complements the analysis of the text itself and in many cases provides additional context. In this work we describe two methods for mining interesting spatio-temporal dynamics and relations among topics, one that compares the topic distributions as histograms in space and time and another that models topics over time as temporal or spatio-temporal Hawkes process with exponential trigger functions. Both methods may be used to discover topics with abnormal distributions in space and time. The second method also allows for self-exciting topics and can recover intertopic relationships (excitation or inhibition) in both time and space. We apply these methods to a geo-tagged Twitter dataset and provide analysis and discussion of the results.

Beschreibung: In 1952, Lighthill introduced the simplest possible model of a swimming micro-organism of finite size, intended as a model of a single-celled protozoan covered in beating cilia. The model consisted of a sphere, on the surface of which material points undergo small-amplitude oscillations. In 1971, Lighthill’s student, John Blake, completed the calculations and applied the theory to various species of ciliate. Subsequently we, and many others, have used the even simpler (though less realistic) model of a steady squirmer, a sphere whose surface moves tangentially with time-independent velocity. In this article we survey: -low-Reynolds-number locomotion, nutrient uptake and optimisation of individual squirmers; -hydrodynamic interactions between pairs of steady squirmers and their influence on clustering, self-diffusion and rheology in suspensions of squirmers, including the effect of being bottom-heavy; -measurements and modelling of metachronal waves in Volvox , the only truly sphericalmulti-celled organism, culminating in predictions of the mean swimming speed and angular velocity of free-swimming Volvox . The predictions are compared with experiment.

Beschreibung: New mathematics has often been inspired by new insights into the natural world. Here we describe some ongoing and possible future interactions among the massive data sets being collected in neuroscience, methods for their analysis and mathematical models of the underlying, still largely uncharted neural substrates that generate these data. We start by recalling events that occurred in turbulence modelling when substantial space-time velocity field measurements and numerical simulations allowed a new perspective on the governing equations of fluid mechanics. While no analogous global mathematical model of neural processes exists, we argue that big data may enable validation or at least rejection of models at cellular to brain area scales and may illuminate connections among models. We give examples of such models and survey some relatively new experimental technologies, including optogenetics and functional imaging, that can report neural activity in live animals performing complex tasks. The search for analytical techniques for these data is already yielding new mathematics, and we believe their multi-scale nature may help relate well-established models, such as the Hodgkin–Huxley equations for single neurons, to more abstract models of neural circuits, brain areas and larger networks within the brain. In brief, we envisage a closer liaison, if not a marriage, between neuroscience and mathematics.

Beschreibung: The properties of spatially localized structures in systems with a conservation law are reviewed and related to the properties of fronts in such systems. The theory is illustrated using the conserved Swift–Hohenberg equation, and the insights gained used to shed light on the process of crystallization from a melt and on two problems arising in hydrodynamics: convection in a rotating layer and convection in an imposed magnetic field. Spatially localized structures in systems with time-periodic forcing are also considered. Several open problems are highlighted.

Beschreibung: Thixotropic yield stress fluids have a yield stress and viscosity which change slowly over time as the fluid undergoes structural changes. This article reviews recent work which shows how such behaviour can arise as a limit of viscoelastic flow when a relaxation time becomes large. The large relaxation time introduces a small parameter which can be used as a basis for singular perturbation studies. A simple prototype for a viscoelastic model will be used to illustrate the analysis. We review the behaviour in startup of shear flow, oscillatory shear flow and elongational flow and identify potential directions for future research.

Beschreibung: The 120-year-old so-called Painlevé paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with a single contact point. Various practical consequences of the theory are revisited, including models for robotic manipulators and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including Poisson or energetic impact laws. The general problem in 2D with a single point of contact is discussed and cases of inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularization are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully 3D analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painlevé paradox is highlighted and open problems are discussed.

Beschreibung: This article surveys the important role, in a variety of applied mathematical contexts, played by the so-called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th century investigators, its theoretical significance for applications has only been realized in the last decade or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It is shown here that, in terms of it, many well-known results pertaining only to the simply connected case (no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing those well-known results within a more general framework of much broader applicability. Given the wide-ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we introduce both a new theoretical formulation for its computation, as well as two distinct numerical methods to implement the construction. The combination of these theoretical and computational developments renders the S–K prime function a powerful new tool in applied mathematics.

Beschreibung: This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.

Beschreibung: Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Beschreibung: In this paper, the rigorous linking of exact stochastic models to mean-field approximations is studied. Using a continuous-time Markov chain, we start from the exact formulation of a simple epidemic model on a certain class of networks, including completely connected and regular random graphs, and rigorously derive the well-known mean-field approximation that is usually justified based on biological hypotheses. We propose a unifying framework that incorporates and discusses the details of two existing proofs and we put forward a new ordinary differential equation (ODE)-based proof. The more well-known proof is based on a first-order partial differential equation approximation, while the other, more technical one, uses Martingale and Semigroup theory. We present the main steps of both proofs to investigate their applicability in different modelling contexts and to make these ideas more accessible to a broader group of applied researchers. The main result of the paper is a new ODE-based proof that may serve as a building block to prove similar convergence results for more complex networks. The new proof is based on deriving a countable system of ODEs for the moments of a distribution of interest and proving a perturbation theorem for this infinite system.

Beschreibung: In this paper, we extend certain key results from the classical theory of isotropic elasticity to the generalized theory of elasticity for decagonal quasicrystaline composites. These results include: (i) the dependence of the solution on the number of elastic constants, (ii) Green's functions for bimaterials consisting of two bonded half-planes, (iii) Green's functions for a circular elastic inclusion, (iv) the oscillatory singular stress field in the vicinity of an interface crack tip and (v) the inverse problem corresponding to the design of harmonic shapes.

Beschreibung: The note develops an approximate approach to mixed boundary value problems in linear elasticity starting from an explicit asymptotic model for the Rayleigh surface wave. It is demonstrated that the original vector mixed problem may be reduced to a scalar problem for the Laplace equation. As an illustration, the steady-state motion of a rigid stamp is analysed. Comparison of asymptotic and exact results is presented.

Beschreibung: In this paper, we investigate the advantages of the 2D Navier–Stokes–Voight (NSV) turbulence model for use in algorithms and explore its limits in the context of image inpainting. We begin by giving a brief review of the work of Bertalmio et al. in 2001 when an elegant analogy between the image intensity function for the image inpainting problem and the stream function in 2D incompressible fluid was established. An approximate solution to the inpainting problem was then obtained by numerically approximating the steady-state solution of the 2D Navier–Stokes vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the 3D NSV model of viscoelastic fluid was suggested by Cao et al. as an inviscid regularization to the 3D Navier–Stokes equations (NSEs). We give some background on the NSV model, describe why it is a good candidate sub-grid scale turbulence model and then we propose this model as an alternative partial differential equation for image inpainting. We describe an implementation of the inpainting procedure using the NSV model and then present numerical results comparing the images obtained when using the NSE versus the NSV model. Our results show that the NSV model allows for a larger time step to converge to the steady-state solution, yielding a more efficient numerical process when automating the inpainting process.We compare the quality of the resulting images using a subjective measure (human evaluation) and an objected measure (by calculating the peak signal-to-noise ratio). We also present some new theoretical results based on energy methods comparing the sufficient conditions for numerical stability for the two model equations. These theoretical and numerical studies shed some light on what can be expected from this category of approach when automating the inpainting problem.

Beschreibung: On the basis of the general non-linear theory of a hyperelastic material with initial stress, initially without consideration of the origin of the initial stress, we determine explicit expressions for the stress-dependent tensor of incremental elastic moduli. In considering three special cases of initial stress within the general framework, namely hydrostatic stress, uniaxial stress and planar shear stress, we then elucidate in general form the dependence of various elastic moduli on the initial stress. In each case, the effect of initial stress on the wave speed of homogeneous plane waves is studied and it is shown how various special theories from the earlier literature fit within the general framework. We then consider the situation in which the initial stress is a pre-stress associated with a finite deformation and, in particular, we discuss the specialization to the second-order theory of elasticity and highlight connections between several classical approaches to the topic, again with special reference to the influence of higher-order terms on the speed of homogeneous plane waves. Some discrepancies arising in the earlier literature are noted.

Beschreibung: In this paper, we introduce a macroscopic model for road traffic accidents along highway sections. We discuss the motivation and the derivation of such a model, and we present its mathematical properties. The results are presented by means of examples where a section of a crowded one-way highway contains in the middle a cluster of drivers whose dynamics are prone to road traffic accidents. We discuss the coupling conditions and present some existence results of weak solutions to the associated Riemann Problems. Furthermore, we illustrate some features of the proposed model through some numerical simulations.

Beschreibung: Transverse vibrations of Euler–Bernoulli beams coupled with van der Waals elastic forces are considered for multi-walled carbon nanotubes. Eigenfunctions are obtained for general boundary conditions of interest in fields such as nano-electronics and atomic force microscopy. It is employed a matrix basis generated by a Green matrix function of initial value. Uniform modes are characterized in closed form. Double-walled carbon nanotubes are discussed in detail for a cantilever beam.

Beschreibung: In this paper, we prove that the Cauchy problem for a non-linear wave equation (1) has a unique global generalized solutions in and a unique global classical solution in . We also prove that the Cauchy problem for the equation (1) admits a unique global generalized solution in C 2 ([0, ); W m , p ( R )   L ( R ))( m  ≥ 0 is an integer,1 ≤  p  ≤ ) and a unique global classical solution in .

Beschreibung: Generalized models provide a framework for the study of evolution equations without specifying all functional forms in the defining vector field. The generalized formulation of problems has been shown to facilitate the analytical investigation of local dynamics and has been used successfully to answer applied questions. Yet their potential to facilitate analytical computations has not been realized in the mathematical literature. In the present paper, we introduce the method of generalized modelling in mathematical terms, supporting the key steps of the procedure by rigorous proofs. Further, we point out open questions that are in the scope of present mathematical research and if answered could greatly increase the predictive power of generalized models.

Beschreibung: This paper investigates a time consistent multiperiod mean–variance (MV) portfolio selection problem under the Markov regime-switching framework. We use a vector auto-regression model to forecast the values of market factors and then predict the returns of risky assets by using a regime-dependent linear multifactor model. By introducing the notion of separable expected conditional mapping, we construct a time consistent multiperiod MV model with regime switching. Under the self-financing constraint, we derive an analytical optimal investment policy satisfying time consistency and the corresponding MV efficient frontier by using dynamic programming. Empirical results are provided to illustrate the reasonability and practicality of the proposed new model and the derived explicit investment strategy. Especially, we find that the investor invests more in risky assets under a bull market than that under a consolidation market or a bear market; the MV efficient frontier under a bull market is much superior to those determined under a consolidation market and a bear market.

Beschreibung: ABC classifications can be constructed based on a wide range of approaches (varying from formal multi-criteria optimization models to more subjective approaches like the analytic hierarchy processes). Several multi-criteria inventory classification (MCIC) models in particular have recently been proposed in the academic literature. However, even in the presence of the same criteria and the same input data, these models may offer considerably different ABC classifications. That is, for example, an item that is classified based on a certain model in the most important class (class A for example) could be classified by another model in the least important class (class C for example). Such discrepancies have motivated the work described in this paper the objective of which is to tackle the inconsistent operation of existing models for ABC classifications. Based on two well-known MCIC models proposed in the literature, we first develop a new hybrid model which succeeds in reducing the conflict in the resulting classifications. We then propose some new procedures that may take as an input differing ABC classifications to reach as an output a consensus between them. The efficiency of such procedures is evaluated through a number of computational experiments performed on a dataset commonly used in the inventory literature. Using such a dataset allows us to contrast our results with those reported by other researchers.

Beschreibung: An M/M/1 retrial queue is studied for local area network applications in which the server will take delayed vacations upon completion of a service. An announced price charged by the server is imposed on the customers who decide to enter the system for service. These customers have to make decisions upon arrival to balk or to enter the system for service based on their expected payoff and available information about the status of the server. On the other hand, the server faces the revenue maximization issue once the customers' Nash equilibrium strategies are obtained for any announced price. We further study how the social planner, whose goal is to maximize the overall welfare of the customers and the server, determines the price to charge a joining customer. Finally, the sensitivity of the solution of revenue optimization and the socially optimal price as well as the balking behaviour of customers with respect to some main indicators of the system are explored via numerical experiments.

Beschreibung: Amateur baseball contests, particularly those involving young players, often feature a run limit, which establishes a maximum number of runs that may be scored in a half-inning of play. Although conventional wisdom can be used for designing the batting order under standard baseball rules, the composition of the optimal lineup under run limit is still unclear, and the best batting order could differ significantly from the traditional ordering. In this paper, we consider the impact of run limits on the design of the optimal batting order, by developing a Markov chain model for calculating the expected number of runs scored (ENRS) under 3- and 5-run limits. We apply this model to both a high-quality and a lower-quality team, using the data from the 2011 Little League World Series, and illustrate the effect of run limit on the ENRS and the optimal batting order.

Beschreibung: The close–open vehicle routing problem (VRP) with time windows is a realistic variant of the classical VRP with customers that must be served in a specific time interval and where some routes are open and others are closed. A typical situation in logistics and transport activities is when many companies opt for collaboration and recruitment of other specialist companies to carry them out. The company has its own fleet of vehicles and hires the services of other companies that provide complementary resources for specific service needs or business strategy. The aim of this paper is to test the variable neighbourhood search algorithm for solving this problem. In this paper, we present some results using different strategies to obtain the neighbourhoods, applying real problem instances and some Solomon instances.

Beschreibung: We discuss phase resetting behaviour when non-small, near instantaneous, perturbations are applied to a dynamical system containing a strong attractor that is equivalent to a winding map on an $n$ -dimensional torus. Almost all of the literature to date has focused on the application of phase transition curves derivable for systems with attracting limit cycles, when $n = 1$ , and the consequent possibilities for entrainment of the system subject to such periodic largish perturbations. In higher dimensions, the familiar tongues in the amplitude versus period plane, describing solutions entrained with periodic fast perturbations, have subtle structure (since only locally two-dimensional subsets of the $n$ -dimensional torus may admit such solutions) whilst the topological class of the $n$ -dimensional phase transition mapping is represented by a matrix of winding numbers. In turn this governs the existence of asymptotic solutions in the limits of small perturbations. This paper is an attempt to survey the entrainment behaviour for the full set of alternative (near rigid) phase transition mappings definable on the 2-torus. We also discuss higher dimensional effects and we suggest how this might be important for non-linear neural circuits including delays that routinely exhibit such attractors and may drive one another in cascades of periodic behaviour.

Beschreibung: This article considers the theoretical modelling of a novel electrostatic transducer in which the backplate consists of many spherical resonators. Three analytical models are considered, each of which produce impedance profiles of the device, in addition to transmission voltage responses and reception force responses, all of which closely agree. Design parameters are then varied to investigate their influence on the resonant frequencies and other model outputs.

Beschreibung: A class of backward doubly stochastic differential equations (BDSDEs), which has a more general form of the forward Ito integrals (compared with the ones which have been studied until now), is observed. The existence of the solution for this class of BDSDEs with continuous coefficients is given, a comparison theorem for this class of BDSDE is proved and the existence of minimal and maximal solution is derived. At the end, Kneser-type theorem for a larger class of BDSDEs is obtained. It is shown that for the elements of this class of equations, there is either unique or there are uncountable solutions.

Beschreibung: In this paper, three dynamic linear models of the main span of suspension bridges are studied. The models describe vertical and torsional oscillations of the deck under the action of lateral wind. The problems correspond to the situations when both the midspan cable bands are loosened, both the midspan bands are fixed, and one midspan band is fixed and the other is loosened. Three corresponding eigenvalue and eigenvector problems are formulated and analysed. The continuous dependence of eigenvalues and eigenvectors on data are proved. The analysis of the three eigenvalue and eigenvector problems against flutter is carried out, which reveals possible reasons of the collapse of the original Tacoma suspension bridge.

Beschreibung: We present a complete description on the spectrum and eigenfunctions of the following two-point boundary value problem (0.1) \begin{equation} \begin{cases} (p(x)f')'-(q(x)- \lambda r(x))f=0,\quad 0 〈 x 〈 L \\ f'(0)=(\alpha_{1} \lambda +\alpha_{2})f(0)\\ f'(L)=(\beta_{1}\lambda -\beta_{2})f(L) \end{cases} \end{equation} where $\lambda $ and $\alpha _{i},\ \beta _{i}$ are spectral and physical parameters, respectively, and $p(x),\ q(x),\ r(x)$ are positive and regular functions in $[0,L]$ . Our survey is focused mainly on the case $\alpha _{1} 〉 0$ and $\beta _{1} 〈 0$ , where neither self-adjoint operator theorems on Hilbert spaces nor Sturm's comparison results can be used directly. We describe the spectrum and the oscillatory results of the eigenfunctions from a geometrical approach, using a function related to the Prüfer angle. The proofs of the asymptotic results of the eigenvalues and separation theorem of the eigenfunctions are developed through classical second-order differential equation tools. Finally, the results on the spectrum of ( 0.1 ) are used to study the linear stability of three-layer Hele–Shaw flows, which is a simple model for the fingering phenomenon on the flooding oil recovery process.

Beschreibung: As known from the theory of centre manifolds, it is important to approximate invariant manifolds when discussing high order terms of the reduced systems in the investigation of bifurcations. In this paper, we give a method to approximate invariant manifolds which are obtained under pseudo-hyperbolicity. This method enables us to avoid the trouble of increasing dimension of the so-called approximate inertial manifold for smaller thickness of attractive neighbourhood. Unlike centre manifolds, exponential expansion and contraction may exist on such manifolds. Consequently, the technique of the contraction principle cannot be used directly. We not only overcome the difficulty so as to approximate such $C^{1}$ manifolds at a possibly higher order but also estimate, for non-local bifurcations, the lower bound of their locality radius such that the approximation is valid. Our approximation is demonstrated by examples of a wave equation and an ordinary differential equation.

Beschreibung: The Riemann problem for the pressureless Euler system with the Coulomb-like friction term is solved constructively, whose solutions involve delta shock wave and vacuum state. The position, propagation speed and strength of the delta shock wave are calculated explicitly from the generalized Rankine–Hugoniot conditions of the delta shock wave which are derived in detail, such that the influence of the Coulomb-like friction term on the Riemann solutions can be seen clearly. Furthermore, the generalized Riemann problem with measure-valued initial data is also considered and four kinds of different structures are displayed for the Riemann solutions. It is shown that the Riemann solutions converge to the corresponding ones of the pressureless Euler system when the friction term vanishes. In addition, the balance relations associated with the mass and momentum transportation are established for the delta shock wave solution.

Beschreibung: This paper is devoted to the attraction–repulsion chemotaxis system with a logistic source: \[ \begin{cases} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\mu\nabla\cdot(u\nabla w)+\mathcal{R}(u),\quad x\in\Omega,\ t 〉 0,\\ \varrho v_t=\Delta v-\alpha_1 v+\beta_1 u,\quad x\in\Omega,\ t 〉 0,\\ \varrho w_t=\Delta w-\alpha_2 w+\beta_2 u,\quad x\in\Omega,\ t 〉 0, \end{cases} \] where $\Omega \subset \mathbb {R}^N(N\geqslant 1)$ is a bounded domain with smooth boundary and $\mathcal {R}(s)\leqslant a-bs^\tau $ . For the case $\varrho =0$ , we show that when the repulsion prevails over the attraction in the sense that $\mu \beta _2-\chi \beta _1 〉 0$ , there exist global bounded classical solutions for any logistic damping $\tau \geqslant 1$ . When the attraction dominates the repulsion in the sense that $\mu \beta _2-\chi \beta _1 〈 0$ , the classical solutions are still global and bounded provided that the logistic damping is strong. For the case $\varrho 〉 0$ , we will investigate the similar problem for $N=1$ and $N=2$ . We will also study the regularity of stationary solutions.

Beschreibung: This paper presents a new heuristic for fast approximation of VaR (Value-at-Risk) and CVaR (conditional Value-at-Risk) for financial portfolios, where the net worth of a portfolio is a non-linear function of possibly non-Gaussian risk factors. The proposed method is based on mapping non-normal marginal distributions into normal distributions via a probability conserving transformation and then using a quadratic, i.e. Delta–Gamma, approximation for the portfolio value. The method is very general and can deal with a wide range of marginal distributions of risk factors, including non-parametric distributions. Its computational load is comparable with the Delta–Gamma–Normal method based on Fourier inversion. However, unlike the Delta–Gamma–Normal method, the proposed heuristic preserves the tail behaviour of the individual risk factors, which may be seen as a significant advantage. We demonstrate the utility of the new method with comprehensive numerical experiments on simulated as well as real financial data.

Beschreibung: This paper focuses on an optimal management problem for a general insurance company which contains an insurer and a reinsurer. The general company aims to maximize the expected exponential utility of the weighted sum of the insurer's and the reinsurer's terminal wealth. In our model, the basic claim process is assumed to follow a Brownian motion with drift. The insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset, respectively. The prices of risky assets are described by the constant elasticity of variance (CEV) models. In addition, the insurer can purchase proportional reinsurance from the reinsurer. By solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation, we derive the optimal reinsurance and investment strategies for the insurer and the reinsurer, respectively. Finally, numerical simulations are presented to show the effects of model parameters on the optimal reinsurance and investment strategies.

Beschreibung: This paper deals with the interaction between polluting firms, tax inspectors and politicians in a corrupted context. We construct a theoretical game model with incomplete information to discuss the effects of such interaction on environmental policy. In this respect, we believe that the State may pursue environmental protection by employing two alternative strategies: on the one hand, the State can, through greater incentive for the tax inspector, increase the monitoring level that reduces the evasion and thus increase tax revenues ( incentive channel ); on the other hand, the State can, through greater environmental expenses, increase the compliance of the polluting firm which means lower evasion and, thus greater tax revenues ( compliance channel ). Clearly, more environmental expenses mean, ceteris paribus , less public resources for the tax inspector's incentive, and vice versa. In this context, we demonstrate that, for a country with a high (low) level of incentives, the incentive (compliance) channel is more efficient than the compliance (incentive) channel.

Beschreibung: This paper presents a new approach for proving that the presence of a bounded defect in a uniform open waveguide cannot produce trapped modes, contrary to the case of a closed waveguide. The originality of the proof lies in the fact that it relies on a modal decomposition. It shows in particular that the absence of trapped modes results from a strong connection between the various modal components of the field. The case of the 3D scalar wave equation is considered.

Beschreibung: This paper is concerned with travelling wave solutions of periodic monostable differential equations on a two-dimensional lattice. The existence, stability and uniqueness of supercritical travelling wave solutions are obtained by comparison principle.

Beschreibung: This paper considers a stochastic delay Lotka–Volterra model whose coefficients are dependent on expectations. By applying the M -matrix technique, the existence and uniqueness of the global positive solution and its exponential extinction are studied. An example is provided to illustrate the effectiveness of our results.

Beschreibung: We consider the selection and stability of periodic wave trains that form behind invasion fronts in predator–prey reaction–diffusion models where the prey compete in a non-local manner. We modify the selection criterion developed in earlier work, also applying it to the non-local system, and study how the properties of selected wave trains vary with the standard deviation of the non-local prey competition kernel. We find that the wavelength of selected wave trains decreases with the standard deviation of the non-local kernel and also that unstable wave trains are selected for a larger parameter range, suggesting that spatiotemporal chaos may be more common in highly non-local systems.

Beschreibung: We investigate the different wave-modes generated by a wave-maker in compressible flow. In addition to the propagating and evanescent waves found in the incompressible case, new radially propagating acoustic-gravity modes appear. We discuss the asymptotic behaviour of these waves, and give an example for a simple line wave-maker configuration.

Beschreibung: This paper concerns the derivation and preliminary analysis of a model for judder in devices such as brakes and clutches. The model is based on the competition between wear and thermal expansion which, in the multiple-pin configuration proposed by Barber (1969, Thermoelastic instabilities in the sliding of conforming solids. Proc. Roy. Soc. (London) , A312 , 381–394), leads to a novel, non-local dynamic contact problem.

Beschreibung: An averaged model of a coupled-inductor boost converter using the piecewise complementarity model of the converter under sliding motions is obtained. The model takes into account the idealized voltage–current characteristic of passive switches (diodes) present in the converter. Because of its lower complexity, the averaged model is more suitable for control design purposes when compared with the linear complementarity systems (LCS) model of the converter. The dynamic performance of the LCS model and the averaged models of the converter are validated through computer simulations using Matlab.

Beschreibung: A spherically symmetric isolated point defect in a 3D uniaxial nematic liquid crystal sample is often called a radial hedgehog . We use topological methods to describe local configurations of uniaxial and biaxial states into which a hedgehog naturally deforms under small perturbations: these include the biaxial torus and split core configurations studied in the literature using analytical and numerical methods. The topological results here take no account of the governing physical laws but provide a library of options from which the physics must make a choice.

Beschreibung: In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.

Beschreibung: The border collision normal form is a family of continuous 2D piecewise smooth maps describing dynamics close to a critical parameter at which a fixed point intersects the switching surface. It is well known that if the fixed point is stable on one side of the bifurcation point, then after the bifurcation the system may have stable periodic orbits and/or chaotic attractors with a quasi-1D structure (robust chaos). We show that it is also possible to have a robust transition from a stable fixed point to an attractor with topological dimension two, i.e. the highest dimension possible in the phase space.

Beschreibung: The inventory routing problem (IRP) deals with the study of the inter-relationship between two important activities in the supply chain: transportation of commodities and inventory management. In this paper, we study a variant of the multi-product IRP and we propose a general variable neighbourhood search metaheuristic for solving it. We present several neighbourhood structures based on the movement of quantities between routes and periods. Computational experiments show the efficiency of the proposed metaheuristic when compared with the state-of-the-art based on variable neighbourhood search.

Beschreibung: Several variants of variable neighbourhood search (VNS) for solving unconstrained and constrained continuous optimization problems have been proposed in the literature. In this paper, we suggest two new variants, one of which uses the recent modified Nelder–Mead (MNM) direct search method as a local search and the other an extension of the MNM method obtained by increasing the size of the simplex each time the search cannot be continued. For these new and some previous VNS variants, extensive computational experiments are performed on standard and large non-differentiable test instances. Some interesting observations regarding comparison of some VNS variants with NM based local search are made.

Beschreibung: In this paper we report the results of an extensive computational study on a variety of rich vehicle routing problems in which two fundamentally different paradigms or strategies of applying (the same set of) different neighbourhoods are assessed with respect to effectivity (solution quality) and efficiency (speed). We especially compare the structured approach known from variable neighbourhood search where neighbourhoods are applied in some fixed sequential order with an approach where neighbourhoods are applied concurrently. Since all approaches are implemented using the same basic modules, differences can be attributed to this strategic difference. We can show that, first, the application of large neighbourhoods is essential for quality, and, second, that the concurrent strategy outperforms the sequential strategy.

Beschreibung: Variable neighbourhood search (VNS) and all its variants have been successfully proved in hard combinatorial optimization problems. However, there are only few works concerning parallel VNS algorithms, compared with the amount of works devoted to sequential VNS design. In this paper, we propose different parallel designs for the VNS schema. We illustrate the performance of these general strategies by parallelizing a new VNS variant called variable formulation search (VFS). Specifically, we propose six different variants which differ in the VNS stages to be parallelized as well as in the communication mechanisms among processes. We group these variants into three different templates. The first one is oriented to parallelize the whole VNS method. The second one parallelizes the shake and the local search procedures. Finally, the third one explores in parallel the set of predefined neighbourhoods. We test the resulting designs on the cutwidth minimization problem (CMP). Experimental results show that the parallel implementation of the VFS outperforms previous methods in the state of the art for the CMP. This fact is also confirmed by non-parametric statistical tests.

Beschreibung: In this paper, we propose a new heuristic for the covering design problem based on a large neighbourhood search (LNS) metaheuristic that can be seen as a special case of a variable neighbourhood search. As the initial solution, we use a well-known greedy heuristic as well as a new tie-braking rule within greedy algorithms for choosing blocks in the covering. Some theoretical aspects of the greedy heuristic are discussed. The proposed LNS-based heuristic called level reduction can be applied to any covering design and different pre-defined orders, such as lex, colex, etc. With our simple approach, we establish 21 new best known upper bounds on the covering number.

Beschreibung: The dual market phenomenon, in which a product's market is segmented into early and main markets, has attracted much attention in the field of innovation diffusion. It is known that main market adopters, also called the majority, begin to enter the market after a certain period, based on their levels of price sensitivity, which differ from those of early market adopters. This paper proposes a new dual market model that explicitly considers the delayed entry of the majority. We empirically estimate the delayed entry time (DET) for various consumer electronic products, comparing them with two other related phenomena of the early stages of diffusion: take-off and saddle. We also examine whether the DET of a new product can be predicted before it is reached. The result shows that the DET is predictable, using the coefficients of external and internal influence of the early market, also using those of the single market estimated from limited early data. The findings are managerially helpful because being aware of DET at an early stage enables managers to make timely decisions for serving the mass market.

Beschreibung: After demonstrating four basic properties a multi-period risk measure should satisfy, a class of generalized convex multi-period risk measures is defined, which improves and extends existing coherent and convex multi-period risk measures. With respect to this kind of generalized convex multi-period risk measure, the equivalence among different notions of time consistency of risk measure is established. Furthermore, by introducing a fair property that any sophisticated multi-period risk measure should satisfy, we show that if the multi-period risk measure is time consistent, then optimal decisions derived under this risk measure also satisfy the time consistency of optimal policy. Finally, we present the concrete structure of the time consistent generalized convex multi-period risk measure, and illustrate through several examples how to construct time consistent generalized convex multi-period risk measures from existing single-period risk measures.

Beschreibung: We consider a finite-horizon differential game with a legal producer and a counterfeiter as players. The planning horizon is divided into two periods. In the first period, the legal producer is alone in the market, while in the second both players compete. Each player controls his own retail price and advertising budget. We characterize the equilibrium pricing and advertising strategies of the two players, and assess the impact of the counterfeiter's entry date on these strategies. Further, by contrasting the results of the scenarios with and without counterfeiting, we determine the conditions under which the presence of the illegal producer is beneficial to consumers and even to the legal firm.

Beschreibung: Organizations need to better design, manage and improve their supply chains as these become global and more complex. To do this, they need to learn from other organizations and sectors, preempt problems before they occur, and understand the future challenges they may face. Although over 40,000 articles and books have been published on supply chain management since the term was coined in 1982, a clear understanding of the emerging trends, current knowledge gaps and potential areas for future development is only now emerging. Our bibliometric analysis of the existing literature suggests we still need to better understand how to manage security, insourcing, sustainability, competition, risk and disruption, and human behaviour within supply chains. Equally, there is still a lack of research within healthcare, disaster and humanitarian supply chains, as well as within small and medium enterprises. supply chain management; research; surveys; bibliometrics.

Beschreibung: This paper extends the Malmquist productivity index by considering an important factor called the ‘balance factor’. Although factors such as efficiency, scale and technology have been considered before, here the effect of imposed strategies on decision-making units (DMUs) is considered. For this purpose, we develop a new model to calculate the balance of DMUs and take into account the effect of this factor in the Malmquist index in addition to three previous factors. The objective of defining the balance factor is to compute to what extent DMUs are aligned with strategies defined by policy-makers. An empirical case study illustrates the need for such a new index and compares the traditional and extended Malmquist index.

Beschreibung: We consider a passive investor who divides his capital between two assets: a risk-free money market instrument and an index fund, or an exchange-traded fund, tracking a broad market index. We model the evolution of the market index by a lognormal diffusion. The agent faces both fixed and proportional transaction costs and solvency constraints. The objective is to maximize the expected utility from the portfolio liquidation at a fixed horizon, but if the portfolio reaches a pre-set target value, then the position in the risky asset is liquidated. The model is formulated as a parabolic impulse control problem and we characterize the value function as the unique constrained viscosity solution of the associated quasi-variational inequality. We show the existence of an impulse policy which is arbitrarily close to the optimal one by reducing the model to a sequence of iterated optimal stopping problems. The value function and the quasi-optimal policy are computed numerically by an iterative finite element discretization technique. We present extended numerical results in the case of a constant relative risk aversion utility function, showing the non-stationary shape of the optimal strategy and how it varies with respect to the model parameters. The numerical experiments reveal that, even with small transaction costs and distant horizons, the optimal strategy is essentially a buy-and-hold trading strategy where the agent recalibrates his portfolio very few times.

Beschreibung: Maritime security assessment is moving towards a proactive risk-based regime. This opens the way for security analysts and managers to explore and exploit flexible and advanced risk modelling and decision-making approaches in maritime transport. In this article, following a review of maritime security risk assessment, a generic quantitative security assessment methodology is developed. Novel mathematical models for security risk analysis and management are outlined and integrated to demonstrate their use in the developed framework. Such approaches may be used to facilitate security risk modelling and decision making in situations where conventional quantitative risk analysis techniques cannot be appropriately applied. Finally, recommendations on further exploitation of advances in risk and uncertainty modelling technology are suggested with respect to maritime security risk quantification and management.

Beschreibung: In this paper, a bivariate semi-Markov reward chain (BVSMC) model is presented. Equations for the higher order moments of the reward process are presented for the first time and applied to the problem of modelling the credit spread evolution of an obligor by considering the dynamic of its own credit rating and that of a dependent obligor called the counterpart. The paper shows how to compute the expected value of the accumulated credit spread (expressed in basis points) that the obligor should expect to pay in addition to the risk free interest rate. Higher order moments of the accumulated credit spread process convey important financial information in terms of variance, skewness and kurtosis of the total basis points the obligor should with pay in a given time horizon. The paper contributes to the literature by extending on previous results on the semi-Markov reward chains, BVSMC and credit spread modelling by providing unifying approach to these problems. The model and the validity of the results are illustrated through a numerical example.

Beschreibung: Combining wave field decomposition, invariant imbedding and general phase theory can reformulate the ill-posed elliptic wave propagation problems into the exact, well-posed, one-way problems. As a main focus in the direct and inverse analysis of this approach, the singularity structure (turning points, focal points, cusps, kinks and poles) of the scattering and Dirichlet-to-Neumann operator symbols are presented and analysed in the transversely homogeneous limit. A specific 2D example is provided with numerical results.

Beschreibung: We mainly study the Cauchy problem of modified Camassa–Holm and Degasperis–Procesi equations. First, we establish the local well-posedness for the equation in Besov space. Secondly, we derive the conservation laws and a precise blow-up scenario. Moreover, we prove the existence of blow-up solutions and obtain its blow-up rate, provided the initial data satisfy certain conditions. Finally, we present the persistence properties of the equation.

Beschreibung: Subdiffusion equations with distributed-order fractional derivatives describe some important physical phenomena. In this paper, we consider the time distributed-order and Riesz space fractional diffusions on bounded domains with Dirichlet boundary conditions. Here, the time derivative is defined as the distributed-order fractional derivative in the Caputo sense, and the space derivative is defined as the Riesz fractional derivative. First, we discretize the integral term in the time distributed-order and Riesz space fractional diffusions using numerical approximation. Then the given equation can be written as a multi-term time–space fractional diffusion. Secondly, we propose an implicit difference method for the multi-term time–space fractional diffusion. Thirdly, using mathematical induction, we prove the implicit difference method is unconditionally stable and convergent. Also, the solvability for our method is discussed. Finally, two numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Beschreibung: We rigorously derive the leading-order terms in asymptotic expansions for the scattered electric and magnetic fields in the presence of a small object at distances that are large compared with its size. Our expansions hold for fixed wavenumber when the scatterer is a (lossy) homogeneous dielectric object with constant material parameters or a perfect conductor. We also derive the corresponding leading-order terms in expansions for the fields for a low-frequency problem when the scatterer is a non-lossy homogeneous dielectric object with constant material parameters or a perfect conductor. In each case, we express our results in terms of polarization tensors.

Beschreibung: The permafrost methane emission problem is the focus of attention on different climate models. Here, we present a mathematical model for permafrost lake methane emission and its influence on the climate system. We model this process using the theory of non-linear phase transitions. Further, we find that a climate catastrophe possibility depends on a value of feedback connecting the methane concentration in the atmosphere and temperature, and on the tundra permafrost methane pool. We note that the permafrost lake model that we developed for the methane emission positive feedback loop problem is a conceptual climate model.

Beschreibung: This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact dispersion property is adopted as the governing equation for unidirectional wave packet evolution. Following the idea from Zakharov's seminal paper (Zakharov, V. E. (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. , 9 , 190–194), the equation is transformed from the spatial–temporal domain to the wavenumber–temporal domain. The solution of the transformed equation is sought using the perturbation theory, for which the ansatz is expressed in the form of a regular expansion in the increasing order of a small parameter. After implementing the naïve perturbation method, due to non-linear mode generation and particular combinations of wavenumbers, the third-order solution contains spurious secular growth terms that appear as a consequence of resonant interaction and non-linear mode generation. These spurious secular growth terms can be prevented by implementing the method of strained parameters for which non-linear dispersion relation terms are produced for particular combination of wavenumbers.

Beschreibung: The evolution equations for a small spherical low-Reynolds-number swimmer actuated by means of an imposed treadmilling action on its surface are derived by means of an asymptotic analysis. The analysis rests on the assumption that the swimmer radius is small compared with its distance from the wall. It generalizes, to the more realistic 3D case, recent work on 2D treadmilling swimmers near a no-slip wall. The study is motivated by a recent work on the dynamics of Volvox algae near solid surfaces.

Beschreibung: We propose an approach to extend the isogeometric analysis (IGA) method to solve material interface problems. The development is carried out through incorporating the advantages of the extended finite element method into the standard IGA approach for solving problems with discontinuities. By applying both the XIGA and IGA methods to solve Poisson's equation problem containing weak discontinuities, we demonstrate that the XIGA achieves the optimal convergence rate, whereas the IGA only converges suboptimally. The proposed method is then successfully applied to solve bimaterial and curved material interface problems.

Beschreibung: In this work, we consider epitaxial growth of thin crystalline films. Thereby, we propose a new diffuse-interface approximation of a semi-continuous model resolving atomic distances in the growth direction but being coarse-grained in the lateral directions. Mathematically, this leads to a free boundary problem proposed by Burton, Cabrera and Frank for steps separating terraces of different atomic heights. The evolution of the steps is coupled to a diffusion equation for the adatom (adsorbed atom) concentration fulfilling Robin-type boundary conditions at the steps. Our approach allows to incorporate an Ehrlich–Schwoebel barrier as well as diffusion along step edges into a diffuse-interface model. This model results in a Cahn–Hilliard equation with a degenerate mobility coupled to diffusion equations on the terraces with a diffuse-interface description of the boundary conditions at the steps. We provide a justification by matched asymptotic expansions formally showing the convergence of the diffuse-interface model towards the sharp-interface model as the interface width shrinks to zero. The results of the asymptotic analysis are numerically reproduced by a finite element discretization.

Beschreibung: In this paper, we study the quasi-neutral limit of a 3D viscous capillary model for plasmas and semiconductors. Guided by the convergence-stability principle previously formulated by the second author for singular limit problems, we show that when the Debye length is sufficiently small, the initial value problem of the model has a unique smooth solution in the time interval where the corresponding incompressible Navier–Stokes equation has a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Debye length goes to zero. Moreover, we obtain the convergence of smooth solutions for the model towards those for the incompressible Navier–Stokes equation in the quasi-neutral limit, with a sharp convergence rate.

Beschreibung: In this paper, we study competition and coordination issues in a duopoly distribution channel where two boundedly rational agents compete on their retail price and warranty policy. The paper analyses the dynamics of competition for three different possible cases where retailers compete: (i) exclusively on price, (ii) exclusively on warranty duration and (iii) both on price and warranty duration. We show that price and warranty competition are dynamically stable in nature under certain condition(s). In such cases, each competition model converges to an equilibrium. However the speed of the parameter adjustment determines how quickly the dynamic game will reach equilibrium and the paper analyses the stability of such an equilibrium. The model is illustrated through a numerical study and the results show that though coordination enhances system profit, it may affect the consumers with higher product price and lower warranty duration. Hence, in some cases, competition serves the firms better than coordination, particularly in terms of ‘ social welfare ’.

Beschreibung: Based on empirical analysis of the Capesize and Panamax indices, we propose different continuous-time stochastic processes to model their dynamics. The models go beyond the standard geometric Brownian motion, and incorporate observed effects like heavy-tailed returns, stochastic volatility and memory. In particular, we suggest stochastic dynamics based on exponential Lévy processes with normal inverse Gaussian distributed logarithmic returns. The Barndorff-Nielsen and Shephard stochastic volatility model is shown to capture time-varying volatility in the data. Finally, continuous-time autoregressive processes provide a class of models sufficiently rich to incorporate short-term persistence of freight rates. Our proposed dynamical models are fitted to market data. Finally, a numerical Value-at-Risk exercise is provided that illustrate the significance of using more sophisticated models than geometric Brownian motion.

Beschreibung: This paper investigates the stabilization problem of neural networks with unbounded continuously distributed delays via impulsive control. By establishing an impulsive infinite delay differential inequality from the impulsive control point of view, some sufficient conditions ensuring the existence and global exponential stabilization of the unique equilibrium point are derived. Our result shows that impulses can be used to globally and exponentially stabilize the neural networks with unbounded continuously distributed delays if they are not originally stable or exponentially stable. Finally, two examples with their simulations are given to show the effectiveness of the proposed method.

Beschreibung: We formulate a mathematical model to study the in vivo keratin organization in terms of repartition of the keratin material between three different structural states: soluble proteins, filament precursors and filamentous state. The model describes a cycle of assembly and disassembly of keratin material. A generic nucleation function and structural stability of filament precursors are assumed. It is established that the three structural states always coexist and that the repartition of keratin material never exhibits any cyclic behaviour. Under some conditions, it is shown that the choice of the nucleation function does not affect the qualitative behaviour of the system. However, it might change the stable repartition of the keratin material in the cell.

Beschreibung: We present an extension of the numerical method of Loeper and Rapetti (2005, Numerical solution of the Monge–Ampère equation by a Newton's algorithm. C.R. Acad. Sci. Paris , I , 319–324) for the Monge–Ampère equation to non-uniform target densities and adopt it to solve the optimal transport problem with quadratic cost. The method employs a damped Newton algorithm to solve the Monge–Ampère equation. We show that the algorithm converges for sufficiently large damping coefficients, for the case where the source and target densities are sufficiently smooth, periodic and bounded away from zero. At each Newton iteration, we solve a non-constant coefficient linear partial differential equation. To improve the efficiency of the procedure, we use an analytically preconditioned fast Fourier transform method coupled with GMRES (Strain, J. (1994) Fast spectrally-accurate solution of variable-coefficients elliptic problems. Proc. Amer. Math. Sci. , 122 , 843–850) to solve this equation, as opposed to a more straightforward approach based on a second-order finite-difference discretization combined with biconjugate gradient used in the original LOEPER and RAPETTI paper. Finally, we present some numerical experiments in image processing to demonstrate the efficiency of the method.

Beschreibung: Herein we consider Rayleigh waves propagating in a given direction along the traction-free boundary of an anisotropic and prestressed elastic half-space, the effective incremental elasticity tensor and mass density of which are smooth functions of depth from the traction-free surface. We derive a high-frequency asymptotic formula which, for a large wave number, expresses the phase velocity of the Rayleigh waves in question in terms of the values of the effective incremental elasticity tensor, the mass density, and their first and higher-order normal derivatives at the traction-free boundary. We develop a procedure which, through an asymptotic expansion of the surface impedance matrix, can deliver an expression for each term of the asymptotic expansion of the Rayleigh-wave velocity. Our asymptotic formula thus gives a characterization of the frequency dependence of Rayleigh waves as caused by the vertical inhomogeneity of the medium, thereby providing a mathematical foundation for work on non-destructive evaluation of the depth dependence of near-surface stress and/or material parameters through their effect on the dispersion of Rayleigh waves.

Beschreibung: The compressible flow between a slider and a moving rough surface is examined asymptotically and numerically in the limit of very small gap height. The roughness is assumed to consist of a well-defined periodic pattern of grooves whose amplitude and wavelength are of the order of the gap height. A two-scale homogenization analysis is employed to determine a non-linear elliptic partial differential equation governing the leading order pressure in the gap on the scale of the slider. The equation involves coefficient functions that are determined numerically by averaging Stokes flows on the scale of the patterned roughness. Numerical results are obtained for the cases of transverse and longitudinal roughness for both no-slip and slip-flow boundary conditions, and for the limiting case of large bearing number. Comments and a brief analysis are given on the reduction of the governing equation for pressure in the conventional limit of long wavelength of the surface roughness.

Beschreibung: Inspired by the paper (Gong, W., Li, R., Yan, N. N. & Zhao, W. B. (2008) An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math. , 26 , 1–13), we consider an optimal finite element error estimate for an inverse problem in multispectral bioluminescence tomography. Different from achromatic or monochromatic measurements, hyperspectral or multispectral data can reduce the ill-posedness of the inverse problem and yield improved depth reconstruction. Compared with Gong et al. (2008, An improved error analysis for finite element approximation of bioluminescence tomography. J. Comput. Math. , 26 , 1–13), which just improved the error order of piecewise constant light source function, error order in this paper is optimal and all error estimates here are valid for a general smooth domain rather than a polyhedral/polygonal one. Moreover, under a boundedness assumption for the admissible source set, the constants in our error estimates do not depend on the regularization parameter, and therefore are bounded.

Beschreibung: In this paper, we prove that the parameter-to-solution map associated to the inverse problem of determining the diffusion coefficient in a parabolic partial differential equation satisfies the local tangential cone condition . In particular, we show stability and convergence of the regularized solutions by means of Landweber iteration. Our result has an immediate application to the local volatility calibration problem of the Black–Scholes model for European call options. We present a numerical validation based on simulated data to this calibration problem and discuss the results. We also prove convergence and stability of Kaczmarz-type strategies of the local volatility calibration problem by transforming the problem into a system of non-linear ill-posed equations.

Beschreibung: This paper presents a new class of eighth-order canonical polynomials in R n . Based on the sequential canonical dual transformation, all extrema are obtained. The method can be used to solve the associated seventh-order non-linear algebraic equations. Optimality conditions are provided to find both global minimizers and local extrema. Applications are illustrated by examples.

Beschreibung: An improved Weiszfeld algorithm, termed the fortified Weiszfeld algorithm, for the solution of the single facility Weber problem with Euclidean distances is proposed. The new approach is based on a parabolic approximation of the objective function and testing demand points for optimality. Computational experiments show the superiority of the fortified approach to the original Weiszfeld algorithm.

Beschreibung: This paper presents a semi-parametric method for bootstrapping non-stochastic estimates of Malmquist indices and the second-stage regressions with a coherent data-generating process (DGP) for panel data. It emphasizes the importance of using censoring techniques instead of truncation in the DGP. To keep the panel structure, the fixed effects censored least square (CLS) method is preferred in the algorithm. Two smoothing processes in the DGP using fixed effects CLSs and cross-sectional censored least absolute deviations are proposed and compared. This semi-parametric bootstrap process can be used to delete artificial correlation amongst the estimated efficiencies in panel data. Although the exposition gives only output-oriented indices, the method can easily be extended to the input-oriented model. Finally, an empirical case study is carried out on Irish dairy farms.

Beschreibung: Conventional data envelopment analysis models have been introduced for technologies only with non-negative inputs and outputs. But in real-world applications, we have some outputs and/or inputs which can take negative values. Especially, in some cases both positive and negative data appear simultaneously in output/input data. In this paper, we review and compare the most recent approaches introduced in the literature. Then, with recourse to some numerical examples, we show that present approaches may fail under two main categories in performance analysis: (a) in introducing non-negative efficient targets in the Pareto sense and (b) in introducing a complete efficiency score for the units. Subsequently, we propose a two-phase slack-based measure model which tackles both problems in an aggregated model. An empirical application in banking is used to illustrate the approach.

Beschreibung: Operating theatre scheduling is a critical task that directly impacts the efficient delivery of surgical care. In this context, we propose a comprehensive stochastic programming modelling framework which handles the inherent uncertainty characterizing the arrival of emergency patients and the duration of surgery. In particular, three recourse strategies are presented with the aim of modelling different reactive scheduling policies actually adopted by hospital managers. In order to solve realistic-sized instances in a reasonable amount of time, we develop tailored heuristic solution strategies that exploit the problem structure. Computational results obtained on a set of randomly generated problems show the effective impact of the stochastic programming approach and the efficiency of the proposed heuristics.

Beschreibung: We consider the transmission eigenvalue problem for an impenetrable obstacle with Dirichlet boundary condition surrounded by a thin layer of non-absorbing inhomogeneous material. We derive a rigorous asymptotic expansion for the first transmission eigenvalue with respect to the thickness of the thin layer. Our convergence analysis is based on a Max–Min principle and an iterative approach which involves estimates on the corresponding eigenfunctions. We provide explicit expressions for the terms in the asymptotic expansion up to order 3.