We investigate several equivalent notions of the Jost solution associated with a unitaryCMV matrix and provide a necessary and sufficient conditions for the Jost solution toconsist of entire functions of finite growth order in terms of super exponential decay ofVerblunsky coefficients. We also establish several one-to-one correspondences between CMVmatrices with super-exponentially decaying Verblunsky coefficients and spectral dataassociated with the first component of the Jost solution.

We present and characterize a multi-host epidemic model of Rift Valley fever (RVF) virusin East Africa with geographic spread on a network, rule-based mitigation measures, andmosquito infection and population dynamics. Susceptible populations are depleted bydisease and vaccination and are replenished with the birth of new animals. We observe thatthe severity of the epidemics is strongly correlated with the duration of the rainy seasonand that even severe epidemics are abruptly terminated when the rain stops. Becausenaturally acquired herd immunity is established, total mortality across 25 years isrelatively insensitive to many mitigation approaches. Strong reductions in cattlemortality are expected, however, with sufficient reduction in population densities ofeither vectors or susceptible (ie. unvaccinated) hosts. A better understanding of RVFepidemiology would result from serology surveys to quantify the importance of herdimmunity in epidemic control, and sequencing of virus from representative animals toquantify the realative importance of transportation and local reservoirs in nucleatingyearly epidemics. Our results suggest that an effective multi-layered mitigation strategywould include vector control, movement control, and vaccination of young animals yearly,even in the absence of expected rainfall.

We first investigate linear size-structured population models with spacial diffusion. Existence of a unique mild solution is established. Then we consider a harvesting problem for linear size-structured models with diffusion and show the existence of an optimal harvesting effort to maximize the total price or total harvest.

We show that simple diffusion processes are weak limits of piecewise continuous processesconstructed within a totally deterministic framework. The proofs are based on thecontinuous mapping theorem and the functional central limit theorem.

We consider a family of discrete Jacobi operators on the one-dimensional integer latticewith Laplacian and potential terms modulated by a primitive invertible two-lettersubstitution. We investigate the spectrum and the spectral type, the fractal structure andfractal dimensions of the spectrum, exact dimensionality of the integrated density ofstates, and the gap structure. We present a review of previous results, some applications,and open problems. Our investigation is based largely on the dynamics of trace maps. Thiswork is an extension of similar results on Schrödinger operators, although some of theresults that we obtain differ qualitatively and quantitatively from those for theSchrödinger operators. The nontrivialities of this extension lie in the dynamics of theassociated trace map as one attempts to extend the trace map formalism from theSchrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are,in a sense, a test item, as many other models can be attacked via the same techniques, andwe present an extensive discussion on this.

In this paper we consider a new kind of Mumford–Shah functionalE(u, Ω) for mapsu : ℝm → ℝnwith m ≥ n. The most important novelty is that theenergy features a singular set Su ofcodimension greater than one, defined through the theory of distributional jacobians.After recalling the basic definitions and some well established results, we prove anapproximation property for the energy E(u, Ω) viaΓ −convergence, in the same spirit of the work by Ambrosio andTortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math.43 (1990) 999–1036].

In this work we propose a method for analysis of postsurgical haemodynamics after femoralartery treatment of occlusive vascular disease. Patient specific reconstruction algorithmof 1D core network based on MRI data is proposed as a tool for such analysis. Along withpresurgical ultrasound data fitting it provides effective personalizing predictive methodthat is validated with clinical observations.

Consider the Schrödinger operator −∇2 + q with a smooth compactly supportedpotential q,q =q(x),x ∈R3.

Let A(β,α,k)be the corresponding scattering amplitude, k2 be the energy, α ∈S2 be the incident direction,β ∈S2 be the direction of scattered wave,S2 be the unit sphere in R3. Assume thatk =k0> 0 is fixed, andα =α0 is fixed. Then the scattering data areA(β) =A(β,α0,k0)= Aq(β)is a function on S2. The following inverse scatteringproblem is studied: IP: Given an arbitrary f ∈L2(S2)and an arbitrary small number ϵ> 0, can one find q ∈ C0∞(D) , where D ∈R3 is an arbitrary fixed domain, suchthat ||Aq(β) −f(β)||L2(S2)<ϵ? A positive answer to this question is given. A method for constructing such aq isproposed. There are infinitely many such q, not necessarily real-valued.

The liner parabolic equation \hbox{$\frac{\pp y}{\pp t}-\frac12\,\D y+F\cdot\na y={\vec{1}}_{\calo_0}u$}$\frac{\mathit{\partial y}}{\mathit{\partial t}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\mathit{\Delta y}\mathrm{+}\mathit{F}\mathrm{·}\mathrm{\nabla }\mathit{y}\mathrm{=}{{1}}_{{\mathrm{𝒪}}_{\mathrm{0}}}\mathit{u}$ with Neumann boundary condition on a convex open domain 𝒪 ⊂ ℝdwith smooth boundary is exactly null controllable on each finite interval if 𝒪0is an open subset of 𝒪which contains a suitable neighbourhood of the recession cone of \hbox{$\ov\calo$}$\overline{\mathrm{𝒪}}$. Here, F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

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.

The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease ina population of constant size is considered. In order to control the spread of infection,we propose the model with four bounded controls which describe vaccination of newborns,vaccination of the susceptible, treatment of infected, and indirect strategies aimed at areduction of the incidence rate (e. g. quarantine). The optimal control problem ofminimizing the total number of the infected individuals on a given time interval is statedand solved. The optimal solutions are obtained with the use of the Pontryagin MaximumPrinciple and investigated analytically. Numerical results are presented to illustrate theoptimal solutions.

A traceless variant of the Johnson-Segalman viscoelastic model is presented and appliedto blood flow simulations. The viscoelastic extra stress tensor is decomposed into itstraceless (deviatoric) and spherical parts, leading to a reformulation of the classicalJohnson-Segalman model. The equivalence of the two models is established comparing modelpredictions for simple test cases. The new model is validated using several 2D benchmarkproblems, designed to reproduce difficulties that arise in the simulation of blood flow inblood vessels or medical devices. The structure and behaviour of the new model arediscussed and the future use of the new model in envisioned, both on the theoretical andnumerical perspectives.

In this article, we consider a swimmer (i.e. a self-deformable body)immersed in a fluid, the flow of which is governed by the stationary Stokes equations.This model is relevant for studying the locomotion of microorganisms or micro robots forwhich the inertia effects can be neglected. Our first main contribution is to prove thatany such microswimmer has the ability to track, by performing a sequence of shape changes,any given trajectory in the fluid. We show that, in addition, this can be done by means ofarbitrarily small body deformations that can be superimposed to any preassigned sequenceof macro shape changes. Our second contribution is to prove that, when no macrodeformations are prescribed, tracking is generically possible by means of shape changesobtained as a suitable combination of only four elementary deformations. Eventually, stillconsidering finite dimensional deformations, we state results about the existence ofoptimal swimming strategies on short time intervals, for a wide class of costfunctionals.

Efficiency of mixing, resulting from the reflection of an internal wave field imposed onthe oscillatory background flow with a three-dimensional bottom topography, isinvestigated using a linear approximation. The radiating wave field is associated with thespectrum of the linear model, which consists of those mode numbers n and slope valuesα, forwhich the solution represents the internal waves of frequencies ω =nω0 radiating upwrad ofthe topography, where ω0 is the fundamental frequency at whichinternal waves are generated at the topography. The effects of the bottom topography andthe earth’s rotation on the spectrum is analyzed analytically and numerically in thevicinity of the critical slope

αn,θc = arcsin (n 2ω02-f 2 / N 2-f 2) 1/2

which is a slope with the same angle to the horizontal as the internal wavecharacteristic. In this notation, θ is latitude, f is the Coriolis parameterand N is thebuoyancy frequency, which is assumed to be a constant, which corresponds to the uniformstratification.

Nothing is more fundamental to life than the ability to reproduce and duplicate theinformation cells store in their genomes. The mechanism of duplication of DNA has beenconserved from prokaryotes to eukaryotes. The aim of the study was to quantify whichevolutionary forces could produce the pattern of genome replication architecture observedin present-day organisms. This was achieved using an evolutionary simulation, combiningrandom genome sequence shuffling, mutation, selection and the mathematical modeling of DNAreplication. We have found parameter values which explained evolutionary pressures of DNAreplication in E.coli, P.calidifontis and S.cerevisae. Surprisingly, the results of the evolutionary simulation suggeststhat for a fixed cost per replication origin it is more advantageous for genomes to reducethe number of replication origins under increasing uncertainty in origin activationtiming.

Recent advances in high-resolution fluorescence microscopy have enabled the systematicstudy of morphological changes in large populations of cells induced by chemical andgenetic perturbations, facilitating the discovery of signaling pathways underlyingdiseases and the development of new pharmacological treatments. In these studies, though,due to the complexity of the data, quantification and analysis of morphological featuresare for the vast majority handled manually, slowing significantly data processing andlimiting often the information gained to a descriptive level. Thus, there is an urgentneed for developing highly efficient automated analysis and processing tools forfluorescent images. In this paper, we present the application of a method based on theshearlet representation for confocal image analysis of neurons. The shearletrepresentation is a newly emerged method designed to combine multiscale data analysis withsuperior directional sensitivity, making this approach particularly effective for therepresentation of objects defined over a wide range of scales and with highly anisotropicfeatures. Here, we apply the shearlet representation to problems of soma detection ofneurons in culture and extraction of geometrical features of neuronal processes in braintissue, and propose it as a new framework for large-scale fluorescent image analysis ofbiomedical data.

This work is devoted to analyze a numerical scheme for the approximation of the linear heat equation’s controls. It is known that, due to the regularizing effect, the efficient computation of the null controls for parabolic type equations is a difficult problem. A possible cure for the bad numerical behavior of the approximating controls consists of adding a singular perturbation depending on a small parameter ε which transforms the heat equation into a wave equation. A space discretization of step h leads us to a system of ordinary differential equations. The aim of this paper is to show that there exists a sequence of exact controls of the corresponding perturbed semi-discrete systems which converges to a control of the original heat equation when both h (the mesh size) and ε (the perturbation parameter) tend to zero.