We introduce a game theoretical model of stealing interactions. We model the situation asan extensive form game when one individual may attempt to steal a valuable item fromanother who may in turn defend it. The population is not homogeneous, but rather eachindividual has a different Resource Holding Potential (RHP). We assume that RHP not onlyinfluences the outcome of the potential aggressive contest (the individual with the largerRHP is more likely to win), but that it also influences how an individual values aparticular resource. We investigate several valuation scenarios and study the prevalenceof aggressive behaviour. We conclude that the relationship between RHP and resource valueis crucial, where some cases lead to fights predominantly between pairs of strongindividuals, and some between pairs of weak individuals. Other cases lead to no fightswith one individual conceding, and the order of strategy selection is crucial, where theindividual which picks its strategy first often has an advantage.

The aim of this paper is to introduce and study multilinear pseudo-differential operatorson Zn and Tn =(Rn/2πZn) then-torus.More precisely, we give sufficient conditions and sometimes necessary conditions forLp-boundedness of theseclasses of operators. L2-boundedness results for multilinearpseudo-differential operators on Zn and Tn withL2-symbols are stated. The proofs of theseresults are based on elementary estimates on the multilinear Rihaczek transforms forfunctions in L2(Zn)respectively L2(Tn)which are also introduced.

We study the weak continuity of multilinear operators on the m-fold product of Lebesguespaces Lpj(Zn),j =1,...,m and thelink with the continuity of multilinear pseudo-differential operators on Zn.

Necessary and sufficient conditions for multilinear pseudo-differential operators onZn or Tn to be aHilbert-Schmidt operators are also given. We give a necessary condition for a multilinearpseudo-differential operators on Zn to be compact. A sufficientcondition for compactness is also given.

Cancer treatment using the antiangiogenic agents targets the evolution of the tumor vasculature. The aim is to significantly reduce supplies of oxygen and nutrients, and thus starve the tumor and induce its regression. In the paper we consider well established family of tumor angiogenesis models together with their recently proposed modification, that increases accuracy in the case of treatment using VEGF antibodies. We consider the optimal control problem of minimizing the tumor volume when the maximal admissible drug dose (the total amount of used drug) and the final level of vascularization are also taken into account. We investigate the solution of that problem for a fixed therapy duration. We show that the optimal strategy consists of the drug-free, full-dose and singular (with intermediate values of the control variable) intervals. Moreover, no bang-bang switch of the control is possible, that is the change from the no-dose to full-dose protocol (or in opposite direction) occurs on the interval with the singular control. For two particular models, proposed by Hahnfeldt et al. and Ergun et al., we provide additional theorems about the optimal control structure. We investigate the optimal controls numerically using the customized software written in MATLAB®, which we make freely available for download. Utilized numerical scheme is based on the composition of the well known gradient and shooting methods.

A feature often observed in epidemiological networks is significant heterogeneity indegree. A popular modelling approach to this has been to consider large populations withhighly heterogeneous discrete contact rates. This paper defines an individual-levelnon-Markovian stochastic process that converges on standard ODE models of such populationsin the appropriate asymptotic limit. A generalised Sellke construction is derived for thismodel, and this is then used to consider final outcomes in the case where heterogeneityfollows a truncated Zipf distribution.

The majority of species are under predatory risk in their natural habitat and targeted bypredators as part of the food web. During the evolution of ecosystems, manifold mechanismshave emerged to avoid predation. So called secondary defences, which are used after apredator has initiated prey-catching behaviour, commonly involve the expression of toxinsor deterrent substances which are not observable by the predator. Hence, the possession ofsuch secondary defence in many prey species comes with a specific signal of that defence(aposematism). This paper builds on the ideas of existing models of such signallingbehaviour, using a model of co-evolution and generalisation of aversive information andintroduces a new methodology of numerical analysis for finite populations. This newmethodology significantly improves the accessibility of previous models.

In finite populations, investigating the co-evolution of defence and signalling requiresan understanding of natural selection as well as an assessment of the effects of drift asan additional force acting on stability. The new methodology is able to reproduce thepredicted solutions of preceding models and finds additional solutions involving negativecorrelation between signal strength and the extent of secondary defence. In addition,genetic drift extends the range of stable aposematic solutions through the introduction ofa new pseudo-stability and gives new insights into the diversification of aposematicdisplays.

We prove that localization operators associated to ridgelet transforms withLp symbols are boundedlinear operators on L2(Rn).Operators closely related to these localization operators are shown to be in the traceclass and a trace formula for them is given.

Properties of blood cells and their interaction determine their distribution in flow. Itis observed experimentally that erythrocytes migrate to the flow axis, platelets to thevessel wall, and leucocytes roll along the vessel wall. In this work, a three-dimensionalmodel based on Dissipative Particle Dynamics method and a new hybrid (discrete-continuous)model for blood cells is used to study the interaction of erythrocytes with platelets andleucocytes in flow. Erythrocytes are modelled as elastic highly deformable membranes,while platelets and leucocytes as elastic membranes with their shape close to a sphere.Separation of erythrocytes and platelets in flow is shown for different values ofhematocrit. Erythrocyte and platelet distributions are in a good qualitative agreementwith the existing experimental results. Migration of leucocyte to the vessel wall and itsrolling along the wall is observed.

We present a fully automatic approach to recover boundary conditions and locations of thevessel wall, given a crude initial guess and some velocity cross-sections, which can becorrupted by noise. This paper contributes to the body of work regarding patient-specificnumerical simulations of blood flow, where the computational domain and boundaryconditions have an implicit uncertainty and error, that derives from acquiring andprocessing clinical data in the form of medical images. The tools described in this paperfit well in the current approach of performing patient-specific simulations, where areasonable segmentation of the medical images is used to form the computational domain,and boundary conditions are obtained as velocity cross-sections from phase-contrastmagnetic resonance imaging. The only additional requirement in the proposed methods is toobtain additional velocity cross-section measurements throughout the domain. The toolsdeveloped around optimal control theory, would then minimize a user defined cost functionto fit the observations, while solving the incompressible Navier-Stokes equations.Examples include two-dimensional idealized geometries and an anatomically realisticsaccular geometry description.

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.

We present sufficient conditions of local controllability for a class of models of treatment response to combined anticancer therapies which include delays in control strategies. The combined therapy is understood as combination of direct anticancer strategy e.g. chemotherapy and indirect modality (in this case antiangiogenic therapy). Controllability of the models in the form of semilinear second order dynamic systems with delays in control enables to answer the questions of realizability of different objectives of multimodal therapy in the presence of PK/PD effects. We compare results for the models without delays and conditions for relative local controllability of models with delays.

In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1 / 2and H1/200and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.

We give a necessary and sufficient condition for pseudo-differential operators with SGsymbols to be compact from L2(ℝn)into L2(ℝn).

We consider Schrödinger operators Hα given by equation(1.1) below. We study the asymptotic behavior of the spectral density E(Hα,λ)for λ → 0 andthe L1 →L∞ dispersive estimates associated to theevolution operator e−itHα.In particular we prove that for positive values of α, the spectral densityE(Hα,λ)tends to zero as λ →0 with higher speed compared to the spectral density of Schrödingeroperators with a short-range potential V. We then show how the long time behavior ofe−itHαdepends on α.More precisely we show that the decay rate of e−itHαfor t → ∞ canbe made arbitrarily large provided we choose α large enough and consider a suitable operatornorm.

We review mathematical results about the qualitative structure of chemotherapy protocols that were obtained with the methods of optimal control. As increasingly more complex features are incorporated into the mathematical model—progressing from models for homogeneous, chemotherapeutically sensitive tumor cell populations to models for heterogeneous agglomerations of subpopulations of various sensitivities to models that include tumor immune-system interactions—the structures of optimal controls change from bang-bang solutions (which correspond to maximum dose rate chemotherapy with restperiods) to solutions that favor singular controls (representing reduced dose rates). Medically, this corresponds to a transition from standard MTD (maximum tolerated dose) type protocols to chemo-switch strategies towards metronomic dosing.

We formulate an immuno-epidemiological model of coupled “within-host” model of ODEs and“between-host” model of ODE and PDE, using the Human Immunodeficiency Virus (HIV) forillustration. Existence and uniqueness of solution to the “between-host” model isestablished, and an explicit expression for the basic reproduction number of the“between-host” model derived. Stability of disease-free and endemic equilibria isinvestigated. An optimal control problem with drug-treatment control on the within-hostsystem is formulated and analyzed; these results are novel for optimal control of ODEslinked with such first order PDEs. Numerical simulations based on the forward-backwardsweep method are obtained.

This paper investigates the output controllability problem of temporal Boolean networkswith inputs (control nodes) and outputs (controlled nodes). A temporal Boolean network isa logical dynamic system describing cellular networks with time delays. Using semi-tensorproduct of matrices, the temporal Boolean networks can be converted into discrete timelinear dynamic systems. Some necessary and sufficient conditions on the outputcontrollability via two kinds of inputs are obtained by providingcorresponding reachable sets. Two examples are given to illustrate the obtainedresults.

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction of a new variable, we show that if the limit system is exponentially stable,then this evolutionary system is uniformly − with respect to the calibration parameter −exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

Galerkin reduced-order models for the semi-discrete wave equation, that preserve thesecond-order structure, are studied. Error bounds for the full state variables are derivedin the continuous setting (when the whole trajectory is known) and in the discrete settingwhen the Newmark average-acceleration scheme is used on the second-order semi-discreteequation. When the approximating subspace is constructed using the proper orthogonaldecomposition, the error estimates are proportional to the sums of the neglected singularvalues. Numerical experiments illustrate the theoretical results.

We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phaseflow model, that is able to cope with arbitrarily small values of the statistical phasefractions. The solver relies on a relaxation approximation of the model for which theRiemann problem is exactly solved for subsonic relative speeds. In an original manner, theRiemann solutions to the linearly degenerate relaxation system are allowed to dissipatethe total energy in the vanishing phase regimes, thereby enforcing the robustness andstability of the method in the limits of small phase fractions. The scheme is proved tosatisfy a discrete entropy inequality and to preserve positive values of the statisticalfractions and densities. The numerical simulations show a much higher precision and a morereduced computational cost (for comparable accuracy) than standard numerical schemes usedin the nuclear industry. Finally, two test-cases assess the good behavior of the schemewhen approximating vanishing phase solutions.

Inspired by the growing use of non linear discretization techniques for the lineardiffusion equation in industrial codes, we construct and analyze various explicit nonlinear finite volume schemes for the heat equation in dimension one. These schemes areinspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I348 (2010) 691–695]. They preserve the maximum principle and admita finite volume formulation. We provide a original functional setting for the analysis ofconvergence of such methods. In particular we show that the fourth discrete derivative isbounded in quadratic norm. Finally we construct, analyze and test a new explicit nonlinear maximum preserving scheme with third order convergence: it is optimal on numericaltests.