In this work we consider a stabilized Lagrange multiplier method in order toapproximate the Coulomb frictional contact model in linear elastostatics. Theparticularity of the method is that no discrete inf-sup condition is needed. Westudy the existence and the uniqueness of solution of the discrete problem.

We are interested in optimizing the co-administration of two drugs for some acute myeloidleukemias (AML), and we are looking for in vitro protocols as a first step. This issue canbe formulated as an optimal control problem. The dynamics of leukemic cell populations inculture is given by age-structured partial differential equations, which can be reduced toa system of delay differential equations, and where the controls represent the action ofthe drugs. The objective function relies on eigenelements of the uncontrolled model and ongeneral relative entropy, with the idea to maximize the efficiency of the protocols. Theconstraints take into account the toxicity of the drugs. We present in this paper themodeling aspects, as well as theoretical and numerical results on the optimal controlproblem that we get.

In this paper we study the complex dynamics of predator-prey systems with nonmonotonicfunctional response and harvesting. When the harvesting is constant-yield for prey, it isshown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopfbifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. Theexistence of two limit cycles and a homoclinic loop is established by numericalsimulations. When the harvesting is seasonal for both species, sufficient conditions forthe existence of an asymptotically stable periodic solution and bifurcation of a stableperiodic orbit into a stable invariant torus of the model are given. Numerical simulationsare carried out to demonstrate the existence of bifurcation of a stable periodic orbitinto an invariant torus and transition from invariant tori to periodic solutions,respectively, as the amplitude of seasonal harvesting increases.

We analyze the problem of shear-induced electrokinetic lift on a particle freely suspended near a solid wall, subject to a homogeneous (simple) shear. To this end, we apply the large-Péclet-number generic scheme recently developed by Yariv et al. (J. Fluid Mech., Vol. 685, 2011, p. 306). For a force- and torque-free particle, the driving flow comprises three components, respectively describing (i) a particle translating parallel to the wall; (ii) a particle rotating with an angular velocity vector normal to the plane of shear; and (iiii) a stationary particle in a shear flow. Symmetry arguments reveal that the electro-viscous lift, normal to the wall, is contributed by Maxwell stresses accompanying the induced electric field, while electro-viscous drag and torque corrections, parallel to the wall, are contributed by the Newtonian stresses accompanying the induced flow. We focus upon the near-contact limit, where all electro-viscous contributions are dominated by the intense electric field in the narrow gap between the particle and the wall. This field is determined by the gap-region pressure distributions associated with the translational and rotational components of the driving Stokes flow, with the shear-component contribution directly affecting only higher-order terms. Owing to the similarity of the corresponding pressure distributions, the induced electric field for equal particle–wall zeta potentials is proportional to the sum of translation and rotation speeds. The electro-viscous loads result in induced particle velocities, normal and tangential to the wall, inversely proportional to the second power of particle–wall separation.

We review recent results on stability of traveling waves in partly parabolicreaction-diffusion systems with stable or marginally stable equilibria. We explain howattention to what are apparently mathematical technicalities has led to theorems thatallow one to convert spectral calculations, which are used in the sciences and engineeringto study stability of a wave, into detailed, theoretically-based information about thebehavior of perturbations of the wave.

Building on earlier work, we have given in [29] aproof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model ofsuperconductivity and shown that the triangular lattice gives the lowest energy perlattice cell. After [29] was published, we realizedthat it proves a stronger result than was stated there. This result is recorded in thepresent paper. The proofs remain the same as in [29], apart from some streamlining.

We extrapolate from the exact master equations of epidemic dynamics on fully connectedgraphs to non-fully connected by keeping the size of the state space N + 1, whereN is thenumber of nodes in the graph. This gives rise to a system of approximate ODEs (ordinarydifferential equations) where the challenge is to compute/approximate analytically thetransmission rates. We show that this is possible for graphs with arbitrary degreedistributions built according to the configuration model. Numerical tests confirm that:(a) the agreement of the approximate ODEs system with simulation is excellent and (b) thatthe approach remains valid for clustered graphs with the analytical calculations of thetransmission rates still pending. The marked reduction in state space gives good results,and where the transmission rates can be analytically approximated, the model provides astrong alternative approximate model that agrees well with simulation. Given that thetransmission rates encompass information both about the dynamics and graph properties, thespecific shape of the curve, defined by the transmission rate versus the number ofinfected nodes, can provide a new and different measure of network structure, and themodel could serve as a link between inferring network structure from prevalence orincidence data.

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have beenstudied extensively. We introduce in this paper multi-dimensional Stockwell transformsthat include multi-dimensional Gabor transforms as special cases. Continuous inversionformulas for multi-dimensional Stockwell transforms are proved.

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.

A possible control strategy against the spread of an infectious disease is the treatmentwith antimicrobials that are given prophylactically to those that had contact with an infective person.The treatment continues until recovery or until it becomes obvious that there was no infectionin the first place. The model considers susceptible, treated uninfected exposed, treated infected,(untreated) infectious, and recovered individuals. The overly optimistic assumptions are made thattreated uninfected individuals are not susceptible and treated infected individuals are not infectious.Since treatment lengths are considered that have an arbitrary distribution, the model systemconsists of ordinary differential and integral equations. We study the impact of the treatment lengthdistribution on the large-time behavior of the model solutions, namely whether the solutions convergeto an equilibrium or whether they are driven into undamped oscillations.

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.

Let {pn}∞n=0 be a sequence of orthogonal polynomials. We brieflyreview properties of pn that have been usedto derive upper and lower bounds for the largest and smallest zero ofpn. Bounds for theextreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained usingdifferent approaches are numerically compared and new bounds for extreme zeros ofq-Laguerre and little q-Jacobi polynomials are proved.

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.

A decision analytical model is presented and analysed to assess the effectiveness andcost-effectiveness of routine vaccination against varicella and herpes-zoster, orshingles. These diseases have as common aetiological agent the varicella-zoster virus(VZV). Zoster can more likely occur in aged people with declining cell-mediated immunity.The general concern is that universal varicella vaccination might lead to more cases ofzoster: with more vaccinated children exposure of the general population to varicellainfectives become smaller and thus a larger proportion of older people will have weakerimmunity to VZV, leading to more cases of reactivation of zoster. Our compartment modelshows that only two possible equilibria exist, one without varicella and the other onewhere varicella and zoster both thrive. Threshold quantities to distinguish these casesare derived. Cost estimates on a possible herd vaccination program are discussedindicating a possible tradeoff choice.

We review recent stability and separation results in volume comparison problems and usethem to prove several hyperplane inequalities for intersection and projection bodies.

Pharmacokinetics is an excellent way to introduce biomathematical modeling at the sophomore level. Students have the opportunity to develop a mathematical model of a biological phenomenon to which they all can relate. Exploring pharmacokinetics takes students through the necessary stages of mathematical modeling: determining the goals of the model, deciphering between the biological aspects to include in the model, defining the assumptions of the model, and finally, building, analyzing, using, and refining the model to answer questions and test hypotheses. Readily accessible data allows students to use the model to test hypotheses that are meaningful to them on an individual level. Students make interdisciplinary connections between this model and their previous personal, mathematical, and other classroom experiences. By beginning with a simple model involving the half-life of a drug, students take advantage of their mathematical abilities to explore the biology. They can then use the new knowledge gained from analyzing the simple model to create more complicated models, thus gaining mathematical and modeling maturity through improving the biological accuracy of the model. Through this experiences, students actually get to do applied mathematics, and they take ownership of the model.

Wilkins' method has been successfully used since early 60s for numericalsimulation of high velocity contact elastic-plastic flows. The present workproposes some effective modifications of this method including moresophisticated material model including the Baushinger effect; modification ofthe algorithm based on correction of the initial configuration of a solid; acontact algorithm based on the idea of a mild contact.

Neutropenia is a significant dose-limiting toxicity of cancerchemotherapy, especially in dose-intensified regimens. It iswidely treated by injections of Granulocyte Colony-StimulatingFactor (G-CSF). However, optimal schedules of G-CSF administrationare still not determined. In order to aid in identifying moreefficacious and less neutropenic treatment protocols, we studied adetailed physiologically-based computer model of granulopoiesis,as affected by different treatment schedules of doxorubicin and/orGranulocyte Colony-Stimulating Factor (G-CSF). We validated themodel as evident from accurate prediction of clinical data onhuman granulopoiesis in healthy individuals and indoxorubicin-treated cancer patients, with or without G-CSFsupport. Based on our model, we suggest new G-CSF administrationregimens. These regimens include reduced G-CSF doses, optimallytimed post-chemotherapy. Application of these regimens can lead tominimization of G-CSF side effects, as well as more cost-effectiveand less myelotoxic protocols. Currently clinical trials are beingdesigned in order to test these new treatment regimens.

We study optimal sustainable policies in a benchmark logistic world (where both population and technological progress follow logistic laws of motion) subject to a pollution ceiling. The main policy in the hands of the benevolent planner is pollution abatement, ultimately leading to the control of a dirtiness index as in the early literature of the limits to growth literature. Besides inclusion of demographic dynamics, we also hypothesize that population size affects negatively the natural regeneration or assimilation rate, as a side product of human activities (like increasing pollution, deforestation,...). We first characterize optimal sustainable policies. Under certain conditions, the planner goes to the pollution ceiling value and stays on, involving a more stringent environmental policy and a sacrifice in terms of consumption per capita. Second, we study how the sustainable problem is altered when we depart from the logistic world by considering exponential technical progress (keeping population growth logistic). It’s shown that, as expected, introducing such an asymmetry widens the margins of optimal policies as sustainable environmental policies are clearly less stringent under exponential technical progress. Third, we connect our model to the data, using in particular UN population projections.

Over the past decade or so, there have been a large number of modelling approaches aimedat elucidating the most important mechanisms affecting the formation of new capillariesfrom parent blood vessels — a process known as angiogenesis. Most studies have focussedupon the way in which capillary sprouts are initiated and migrate in response todiffusible chemical stimuli supplied by hypoxic stromal cells and leukocytes in thecontexts of solid tumour growth and wound healing. However, relatively few studies haveexamined the important role played by blood perfusion during angiogenesis and fewer stillhave explored the ways in which a dynamically evolving vascular bed architecture canaffect the distribution of flow within it. From the perspective of solid tumour growthand, perhaps more importantly, its treatment (e.g. chemotherapy), it would clearly be ofsome benefit to understand this coupling between vascular structure and perfusion morefully. This paper focuses on the implications of such a coupling upon chemotherapeutic,anti-angiogenic, and anti-vascular treatments.

In an extension to previous work by the authors, the issue of pericyte recruitment duringvessel maturation is considered in order to study the effects of different anti-vascularand anti-angiogenic therapies from a more rigorous modelling standpoint. Pericytes are aprime target for new vascular disrupting agents (VDAs) currently in clinical trials.However, different compounds attack different components of the vascular network and theimplications of targeting only certain elements of the capillary bed are not immediatelyclear. In light of these uncertainties, the effects of anti-angiogenic and anti-vasculardrugs are re-examined by using an extended model that includes an interdependency betweenvessel remodelling potential and local pericyte density. Two- and three-dimensionalsimulation results are presented and suggest that it may be possible to identify aVDA-specific “plasticity window” (a time period corresponding to low pericyte density),within which a given VDA would be most effective.