We consider the problem of state and parameter estimation for a class of nonlinearoscillators defined as a system of coupled nonlinear ordinary differential equations.Observable variables are limited to a few components of state vector and an input signal.This class of systems describes a set of canonic models governing the dynamics of evokedpotential in neural membranes, including Hodgkin-Huxley, Hindmarsh-Rose, FitzHugh-Nagumo,and Morris-Lecar models. We consider the problem of state and parameter reconstruction forthese models within the classical framework of observer design. This framework offerscomputationally-efficient solutions to the problem of state and parameter reconstructionof a system of nonlinear differential equations, provided that these equations are in theso-called adaptive observer canonic form. We show that despite typical neural oscillatorsbeing locally observable they are not in the adaptive canonic observer form. Furthermore,we show that no parameter-independent diffeomorphism exists such that the originalequations of these models can be transformed into the adaptive canonic observer form. Wedemonstrate, however, that for the class of Hindmarsh-Rose and FitzHugh-Nagumo models,parameter-dependent coordinate transformations can be used to render these systems intothe adaptive observer canonical form. This allows reconstruction, at least partially andup to a (bi)linear transformation, of unknown state and parameter values with exponentialrate of convergence. In order to avoid the problem of only partial reconstruction and atthe same time to be able to deal with more general nonlinear models in which the unknownparameters enter the system nonlinearly, we present a new method for state and parameterreconstruction for these systems. The method combines advantages of standardLyapunov-based design with more flexible design and analysis techniques based on thenotions of positive invariance and small-gain theorems. We show that this flexibilityallows to overcome ill-conditioning and non-uniqueness issues arising in this problem.Effectiveness of our method is illustrated with simple numerical examples.

Plant growth depends essentially on nutrients coming from the roots and metabolitesproduced by the plant. Appearance of new branches is determined by concentrations ofcertain plant hormones. The most important of them are Auxin and Cytokinin. Auxin isproduced in the growing, Cytokinin in either roots or in growing parts. Many dynamicalmodels of this phenomena have been studied in [1]. In [5], the authors deal with onebranch model. In this work, we focus our interest on a multiple branch model. We deal withthe transport equation in domains of different sizes. A variational reduction type method[3] based on asymptotic partial decomposition introduced in [2] (see also [4]) is used. Inthis work we consider the transport equation in decomposed domain with a general righthand side

In this paper we study the error rate of RNA synthesis in the look-ahead model for therandom walk of RNA polymerase along DNA during transcription. The model’s centralassumption is the existence of a window of activity in whichribonucleoside triphosphates (rNTPs) bind reversibly to the template DNA strand beforebeing hydrolyzed and linked covalently to the nascent RNA chain. An unknown, butimportant, integer parameter of this model is the window size w. Here, weuse mathematical analysis and computer simulation to study the rate at whichtranscriptional errors occur as a function of w. We find dramaticreduction in the error rate of transcription as w increases, especiallyfor small values of w. The error reduction method provided by look-aheadoccurs before hydrolysis and covalent linkage of rNTP to the nascent RNAchain, and is therefore distinct from error correction mechanisms that have previouslybeen considered.

In this paper we present a new hybrid method, called SASP method. We propose thehybridization of two methods, the simulated annealing (SA), which belong to the class ofglobal optimization based on the principles of thermodynamics, and the descent method werewe estimate the gradient using the simultaneous perturbation. This hybrid method givesbetter results. We use the Normal Boundary Intersection approach (NBI) based on the SASPmethod to solve a portfolio optimization problem. Such problem is a multi-objectiveoptimization problem, in order to solve this problem we use three statistical quantities:the expected value, the variance and the Conditional Value-at-Risk (CVaR). The purpose ofthis work is to find the efficient boundary of the considered multi-objective problemusing the NBI method based on the SASP method

Motivated by structured parasite populations in aquaculture we consider a class ofsize-structured population models, where individuals may be recruited into the populationwith distributed states at birth. The mathematical model which describes the evolution ofsuch a population is a first-order nonlinear partial integro-differential equation ofhyperbolic type. First, we use positive perturbation arguments and utilise results fromthe spectral theory of semigroups to establish conditions for the existence of a positiveequilibrium solution of our model. Then, we formulate conditions that guarantee that thelinearised system is governed by a positive quasicontraction semigroup on the biologicallyrelevant state space. We also show that the governing linear semigroup is eventuallycompact, hence growth properties of the semigroup are determined by the spectrum of itsgenerator. In the case of a separable fertility function, we deduce a characteristicequation, and investigate the stability of equilibrium solutions in the general case usingpositive perturbation arguments.

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

The interplay between intrinsic and network dynamics has been the focus of manyinvestigations. Here we use a combination of theoretical and numerical approaches to studythe effects of delayed global feedback on the information transmission properties ofneural networks. Specifically, we compare networks of neurons that display intrinsicinterspike interval correlations (nonrenewal) to networks that do not (renewal). We findthat excitatory and inhibitory delays can tune information transmission by single neuronsbut not by the entire network. Most surprisingly, addition of a delay can change thedependence of the information on the coupling strength for renewal neurons and not fornonrenewal neurons. Our results show that intrinsic ISI correlations can have nontrivialinteractions with network-induced phenomena.

In this paper, we consider one-dimensional wave equation with real-valued square-summablepotential. We establish the long-time asymptotics of solutions by, first, studying thestationary problem and, second, using the spectral representation for the evolutionequation. In particular, we prove that part of the wave travels ballistically ifq ∈ L 2(ℝ+) and this result issharp.

In this paper, we investigate the complex dynamics of a spatial plankton-fish system withHolling type III functional responses. We have carried out the analytical study for bothone and two dimensional system in details and found out a condition for diffusiveinstability of a locally stable equilibrium. Furthermore, we present a theoreticalanalysis of processes of pattern formation that involves organism distribution and theirinteraction of spatially distributed population with local diffusion. The results ofnumerical simulations reveal that, on increasing the value of the fish predation rates,the sequences spots → spot-stripe mixtures → stripes → hole-stripe mixtures holes → wavepattern is observed. Our study shows that the spatially extended model system has not onlymore complex dynamic patterns in the space, but also has spiral waves.

In vitro transmesothelial migration assays of ovarian cancer cells, isolated oraggregated in multicellular spheroids, are reproduced deducing suitable Cellular PottsModels (CPM). We show that the simulations are in good agreement with the experimentalevidence and that the overall process is regulated by the activity of matrixmetalloproteinases (MMPs) and by the interplay of the adhesive properties of the cellswith the extracellular matrix and between cells, both of the same type and of differenttypes. In particular, the process depends on the ability of the cell to induce theloosening of cadherin-mediated junctions. Coherently with experiments, it is found thatsingle cell invasion is more conservative with a crucial role played by MMPs. A similarimportant role is played in cell spheroid invasion, which in comparison is moredisruptive. It achieves monofocal or multifocal characteristics according to the relativeadhesion affinity among cells or between them and the mesothelial layer.

The aim of this study was to describe and analyze the regulation and spatio-temporaldynamics of melanocyte migration in vitro and its coupling to celldivision and interaction with the matrix. The melanocyte lineage is particularlyinteresting because it is involved in both embryonic development andoncogenesis/metastasis (melanoma). Biological experiments were performed on two melanocytecell lines established from wild-type and β-catenin-transgenic mice(bcat*). The multi-functional β-catenin molecule is known to be able toregulate the transcription of various genes involved in cell proliferation and migration,particularly in the melanocyte lineage. We also investigated fibronectin, anextra-cellular matrix protein that binds integrins, thereby providing adhesion points forcells and encouraging migration. As the migration of individual cells were followed,automated methods were required for processing the large amount of data generated by thetime-lapse video-microscopy. A model-based approach for automated cell tracking wasevaluated on a sample by comparison with manual tracking. This method was found reliablein studying overall cell behaviour. Its application to all the observed sequences providedinsight into the factors affecting melanocyte migration in vitro:melanocytes of mutated form of β-catenin showed higher division rates andno contact inhibition of growth was induced by the resulting increase in cell density.However, cell density limited the amplitude of cell displacements, although their motilitywas less affected. The high fibronectin concentration bound to substratum promoted cellmigration and motility, the effect being more intense for wild-type cells than for cellswith β-catenin over-expression. During the division process, cellmigration speed increased rapidly then decreased slowly. Analyses of such data is expectedto lead both to biological answers and to a framework for a better modeling processes inthe future.

2D shallow water equations with depth-averaged k−εmodel is considered. A meshless method based on multiquadric radial basis functions isdescribed. This methods is based on the collocation formulation and does not require thegeneration of a grid and any integral evaluation. The application of this method to a flowin horizontal channel, taken as an experimental device, is presented. The results ofcomputations are compared with experimental data and are found to be satisfactory

We present an iterative method based on an infinite dimensional adaptation of thesuccessive overrelaxation (SOR) algorithm for solving the 2-D neutron transport equation.In a wide range of application, the neutron transport operator admits a Self-Adjoint andm-Accretive Splitting (SAS). This splitting leads to an ADI-like iterative method whichconverges unconditionally and is equivalent to a fixed point problem where the operator isa 2 by 2 matrix of operators. An infinite dimensional adaptation of a SOR algorithm isthen applied to solve the matrix operator equation. Theoretical and numerical results ofconvergence are given

Real-world medical decisions rarely involve binary Ðsole condition present or absent-patterns of patient pathophysiology. Similarly, provider interventions are rarely unitaryin nature: the clinician often undertakes multiple interventions simultaneously.Conventional approaches towards complex physiologic derangements and their associatedmanagement focus on the frequencies of joint appearances, treating the individualderangements of physiology or elements of intervention as conceptually isolated. Thisframework is ill suited to capture either the integrated patterns of derangement displayedby a particular patient or the integrated patterns of provider intervention. Here weillustrate the application of a different approach-that of symbolic dynamics-in which theintegrated pattern of each patients derangement, and the associated provider response, arecaptured by defining words based on the elements of the pattern offailure. We will use as an example provider practices in the context of mechanicalventilation- a common, potentially harmful, and complex life support technology. We alsodelineate other domains in which symbolic dynamics approaches might aid in quantitatingpractice patterns, assessing quality of care, and identifying best practices.

Tuberculosis (TB) is the leading cause of death among individuals infected with thehepatitis B virus (HBV). The study of the joint dynamics of HBV and TB present formidablemathematical challenges due to the fact that the models of transmission are quitedistinct. We formulate and analyze a deterministic mathematical model which incorporatesof the co-dynamics of hepatitis B and tuberculosis. Two sub-models, namely: HBV-only andTB-only sub-models are considered first of all. Unlike the HBV-only sub-model, which has aglobally-asymptotically stable disease-free equilibrium whenever the associatedreproduction number is less than unity, the TB-only sub-model undergoes the phenomenon ofbackward bifurcation, where a stable disease-free equilibrium co-exists with a stableendemic equilibrium, for a certain range of the associated reproduction number less thanunity. Thus, for TB, the classical requirement of having the associated reproductionnumber to be less than unity, although necessary, is not sufficient for its elimination.It is also shown, that the full HBV-TB co-infection model undergoes a backward bifurcationphenomenon. Through simulations, we mainly find that i) the two diseases will co-existwhenever their partial reproductive numbers exceed unity; (ii) the increased progressionrate due to exogenous reinfection from latent to active TB in co-infected individuals mayplay a significant role in the rising prevalence of TB; and (iii) the increasedprogression rates from acute stage to chronic stage of HBV infection have increased theprevalence levels of HBV and TB prevalences.

The Bilevel Knapsack Problem (BKP) is a hierarchical optimization problem in which thefeasible set is determined by the set of optimal solutions of parametric Knapsack Problem.In this paper, we propose two stages exact method for solving the BKP. In the first stage,a dynamic programming algorithm is used to compute the set of reactions of the follower.The second stage consists in solving an integer program reformulation of BKP. We show thatthe integer program reformulation is equivalent to the BKP. Numerical results show theefficiency of our method compared with those obtained by the algorithm of Moore andBard

In the present paper, a general multiobjective optimization problem is stated as a Nashgame. In the nonrestrictive case of two objectives, we address the problem of thesplitting of the design variable between the two players. The so-called territorysplitting problem is solved by means of an allocative approach. We propose two algorithmsin order to find fair allocation tables

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systemsdepending rationally on the spectral parameter is treated and its applications tononlinear equations are given. Explicit solutions of direct and inverse problems forDirac-type systems, including systems with singularities, and for the system auxiliary tothe N-wave equation are reviewed. New results on explicit construction ofthe wave functions for radial Dirac equation are obtained.