The problem of transformation of quasimonochromatic wavetrains of surface gravity waveswith narrow spatial-temporal spectra on the bottom shelf is considered in the linearapproximation. By means of numerical modeling, the transmission and reflectioncoefficients are determined as functions of the depth ratio and wave number (frequency) ofan incident wave. The approximation formulae are proposed for the coefficients of wavetransformation. The characteristic features of these formulae are analyzed. It is shownthat the numerical results agree quite satisfactorily with the proposed approximationformulae.

We perform a sensitivity analysis for a thus far unstudied mathematical model for theformation, growth and lysis of clots in vitro. The sensitivity analysis procedure uses anensemble standard deviation for species concentrations, and is equivalent to a variancedecomposition procedure also available in the literature. Our analysis shows that fibrinproduction is most sensitive to the rate constant governing activation of prothrombin tothrombin. Further, the time-averaged sum of all species’ concentrations is most sensitiveto the rate constants governing the inactivation of VIIIa (intrinsic as well as by APC).We therefore conclude that the rate constants for VIIIa inactivation affect the model thegreatest: this conclusion must be experimentally verified to determine if such is indeedthe case for hemostasis.

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.

Thrombus formation in flowing blood is a complex time- and space-dependent process ofcell adhesion and fibrin gel formation controlled by huge intricate networks ofbiochemical reactions. This combination of complex biochemistry, non-Newtonianhydrodynamics, and transport processes makes thrombosis difficult to understand. That iswhy numerous attempts to use mathematical modeling for this purpose were undertaken duringthe last decade. In particular, recent years witnessed something of a transition from the“systems biology” to the “systems pharmacology/systems medicine” stage: computationalmodeling is being increasingly applied to practical problems such as drug development,investigation of particular events underlying disease, analysis of the mechanism(s) ofdrug’s action, determining an optimal dosing protocols, etc. Here we review recentadvances and challenges in our understanding of thrombus formation.

The paper investigates an age-structured infinite-horizon optimal control model ofharvesting a biological resource, interpreted as fish. Time and age are considered ascontinuum variables. The main result shows that in case of selective fishing, where onlyfish of prescribed sizes is harvested, it may be advantageous in the log run to implementa periodic fishing effort, rather than constant (the latter suggested by single-fishmodels that disregard the age-heterogeneity). Thus taking into account the age-structureof the fish may qualitatively change the theoretically optimal fishing mode. This resultis obtained by developing a technique for reliable numerical verification of second ordernecessary optimality conditions for the considered problem. This technique could be usefulfor other optimal control problems of periodic age-structured systems.

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.