We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

In this work we describe an efficient model for the simulation of atwo-phase flow made of a gas and a granular solid. The starting point is the two-velocitytwo-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented bya relaxation source term in orderto take into account the pressure equilibrium between the two phases andthe granular stress in the solid phase. We show that the relaxationprocess can be made thermodynamically coherent with an adequate choice of the granular stress. We then propose a numerical scheme based on a splitting approach. Each step of the time marchingalgorithm is made of two stages. In the first stage, the homogeneous convection equations are solved by a standard finite volume Rusanov scheme. In the second stage, the volume fractionis updated in order to take into account the equilibrium source term.The whole procedure is entropy dissipative.For simplified pressure laws (stiffened gas laws) we are able to prove that the approximated volumefraction stays within its natural bounds.

We study here the water waves problem for uneven bottoms in a highly nonlinear regime wherethe small amplitude assumption of the Korteweg-de Vries (KdV) equation is enforced. It is knownthat, for such regimes, a generalization of the KdV equation (somehow linked to the Camassa-Holm equation) can be derived and justified [Constantin and Lannes, Arch. Ration. Mech. Anal. 192 (2009) 165–186] when the bottom isflat. We generalize here this resultwith a new class of equations taking into account variable bottom topographies. Of course, many variable depth KdV equations existing in the literature are recovered as particular cases.Various regimes for the topography regimes are investigated and we prove consistency of these models, as well as a full justificationfor some of them. We also study the problem of wave breaking for our newvariable depth and highly nonlinear generalizations of the KdV equations.

Epidemiologic data suggest that schools and daycare facilities likely play a major rolein the dissemination of influenza. Pathogen transmission within such small, inhomogenouslymixed populations is difficult to model using traditional approaches. We developedsimulation based mathematical tool to investigate the effects of social contact networkson pathogen dissemination in a setting analogous to a daycare center or grade school. Herewe show that interventions that decrease mixing within child care facilities, includinglimiting the size of social clusters, reducing the contact frequency between socialclusters, and eliminating large gatherings, could diminish pathogen dissemination.Moreover, these measures may amplify the effectiveness of vaccination or antiviralprophylaxis, even if the vaccine is not uniformly effective or antiviral compliance isincomplete. Similar considerations should apply to other small, imperfectly mixedpopulations, such as offices and schools.

For an SI type endemic model with one host and two parasite strains, we study thestability of the endemic coexistence equilibrium, where the host and both parasite strainsare present. Our model, which is a system of three ordinary differential equations,assumes complete cross-protection between the parasite strains and reduced fertility andincreased mortality of infected hosts. It also assumes that one parasite strain isexclusively vertically transmitted and cannot persists just by itself. We give severalsufficient conditions for the equilibrium to be locally asymptotically stable. One of themis that the horizontal transmission is of density-dependent (mass-action) type. If thehorizontal transmission is of frequency-dependent (standard) type, we show that, undercertain conditions, the equilibrium can be unstable and undamped oscillations can occur.We support and extend our analytical results by numerical simulations and bytwo-dimensional plots of stability regions for various pairs of parameters.

In this work we study the optimal control problem for a class of nonlinear time-delaysystems via paratingent equation with delayed argument. We use an equivalence theorembetween solutions of differential inclusions with time-delay and solutions of paratingentequations with delayed argument. We study the problem of optimal control which minimizes acertain cost function. To show the existence of optimal control, we use the maintopological properties of the set solutions of paratingent equation with delayedargument

Approximate aggregation techniques consist of introducing certain approximations thatallow one to reduce a complex system involving many coupled variables obtaining a simplerʽʽaggregated systemʼʼ governed by a few variables. Moreover, they give results that allowone to extract information about the complex original system in terms of the behavior ofthe reduced one. Often, the feature that allows one to carry out such a reduction is thepresence of different time scales in the system under consideration. In this work we dealwith aggregation techniques in stochastic discrete time models and their application tothe study of multiregional models, i.e., of models for an age structured populationdistributed amongst different spatial patches and in which migration between the patchesis usually fast with respect to the demography (reproduction-survival) in each patch.Stochasticity in population models can be of two kinds: environmental and demographic. Wereview the formulation and the main properties of the dynamics of the different models forpopulations evolving in discrete time and subjected to the effects of environmental anddemographic stochasticity. Then we present different stochastic multiregional models withtwo time scales in which migration is fast with respect to demography and we review themain relationships between the dynamics of the original complex system and the aggregatedsimpler one. Finally, and within the context of models with environmental stochasticity inwhich the environmental variation is Markovian, we make use these techniques to analyzequalitatively the behavior of two multiregional models in which the original complexsystem is intractable. In particular we study conditions under which the population goesextinct or grows exponentially.

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation oftruncated vectorial series, for path following problems [2]. In this paper, we present anddiscuss some techniques to define local parameterization [4, 6, 7] in the ANM. We givesome numerical comparisons of pseudo arc-length parameterization and localparameterization on non-linear elastic shells problems

We analyze a stochastic neuronal network model which corresponds to an all-to-all networkof discretized integrate-and-fire neurons where the synapses are failure-prone. Thisnetwork exhibits different phases of behavior corresponding to synchrony and asynchrony,and we show that this is due to the limiting mean-field system possessing multipleattractors. We also show that this mean-field limit exhibits a first-order phasetransition as a function of the connection strength — as the synapses are made morereliable, there is a sudden onset of synchronous behavior. A detailed understanding of thedynamics involves both a characterization of the size of the giant component in a certainrandom graph process, and control of the pathwise dynamics of the system by obtainingexponential bounds for the probabilities of events far from the mean.

We study the coexistence of multiple periodic solutions for an analogue of theintegrate-and-fire neuron model of two-neuron recurrent inhibitory loops with delayedfeedback, which incorporates the firing process and absolute refractory period. Uponreceiving an excitatory signal from the excitatory neuron, the inhibitory neuron emits aspike with a pattern-related delay, in addition to the synaptic delay. We present atheoretical framework to view the inhibitory signal from the inhibitory neuron as aself-feedback of the excitatory neuron with this additional delay. Our analysis shows thatthe inhibitory feedbacks with firing and the absolute refractory period can generate fourbasic types of oscillations, and the complicated interaction among these basicoscillations leads to a large class of periodic patterns and the occurrence ofmultistability in the recurrent inhibitory loop. We also introduce the average time ofconvergence to a periodic pattern to determine which periodic patterns have the potentialto be used for neural information transmission and cognition processing in the nervoussystem.

We study controllability for a nonhomogeneous string and ring under an axial stretchingtension that varies with time. We consider the boundary control for a string anddistributed control for a ring. For a string, we are looking for a controlf(t) ∈ L 2(0,T) that drives the state solution to rest. We show that for a ring, two forcesare required to achieve controllability. The controllability problem is reduced to amoment problem for the control. We describe the set of initial data which may be driven torest by the control. The proof is based on an auxiliary basis property result.

In topology optimization problems, we are often forced to deal with large-scale numericalproblems, so that the domain decomposition method occurs naturally. Consider a typicaltopology optimization problem, the minimum compliance problem of a linear isotropicelastic continuum structure, in which the constraints are the partial differentialequations of linear elasticity. We subdivide the partial differential equations into twosubproblems posed on non-overlapping sub-domains. In this paper, we consider the resultingproblem as multilevel one and show that it can be written as one level problem

Early studies of the novel swine-origin 2009 influenza A (H1N1) epidemic indicateclinical attack rates in children much higher than in adults. Non-medical interventionssuch as school closings are constrained by their large socio-economic costs. Here wedevelop a mathematical model to ascertain the roles of pre-symptomatic influenzatransmission as well as symptoms surveillance of children to assess the utility of schoolclosures. Our model analysis indicates that school closings are advisable whenpre-symptomatic transmission is significant or when removal of symptomatic children isinefficient. Our objective is to provide a rational basis for school closings decisionsdependent on virulence characteristics and local surveillance implementation, applicableto the current epidemic and future epidemics.

We have carried out a polysystem analysis of the population dynamics by using a varietyof hypotheses and their respective models based on different system interpretations of thephenomenon under investigation. Each of the models supplements standard dynamic equationsfor explaining the effects observed. A qualitative model-based analysis is made of theage-specific male mortality for a Siberian industrial city. The study revealed thetendencies for background mortality to increase with age and over time, whichcharacterizes in an integral manner the influence of socio-ecological factors on thedecline in population viability. It is shown that these tendencies are similar fordifferent years and for different population age groups.

In this paper, a new hybrid simulated annealing algorithm for constrained globaloptimization is proposed. We have developed a stochastic algorithm called ASAPSPSA thatuses Adaptive Simulated Annealing algorithm (ASA). ASA is a series of modifications to thebasic simulated annealing algorithm (SA) that gives the region containing the globalsolution of an objective function. In addition, Simultaneous Perturbation StochasticApproximation (SPSA) method, for solving unconstrained optimization problems, is used torefine the solution. We also propose Penalty SPSA (PSPSA) for solving constrainedoptimization problems. The constraints are handled using exterior point penalty functions.The combination of both techniques ASA and PSPSA provides a powerful hybrid optimizationmethod. The proposed method has a good balance between exploration and exploitation withvery fast computation speed, its performance as a viable large scale optimization methodis demonstrated by testing it on a number of benchmark functions with 2 - 500 dimensions.In addition, applicability of the algorithm on structural design was tested and successfulresults were obtained

Propagation of polymerization fronts with liquid monomer and liquid polymer is consideredand the influence of vibrations on critical conditions of convective instability isstudied. The model includes the heat equation, the equation for the concentration and theNavier-Stokes equations considered under the Boussinesq approximation. Linear stabilityanalysis of the problem is fulfilled, and the convective instability boundary is founddepending on the amplitude and on the frequency of vibrations