We discuss the issues of implementation of a higher order discontinuous Galerkin (DG)scheme for aerodynamics computations. In recent years a DG method has intensively beenstudied at Central Aerohydrodynamic Institute (TsAGI) where a computational code has beendesigned for numerical solution of the 3-D Euler and Navier-Stokes equations. Ourdiscussion is mainly based on the results of the DG study conducted in TsAGI incollaboration with the NUMECA International. The capacity of a DG scheme to tacklechallenging computational problems is demonstrated and its potential advantages over FVschemes widely used in modern computational aerodynamics are highlighted.

Let L be a nonsymmetric second order uniformly elliptic operator with generalWentzellboundary conditions. We show that a suitable version of L generates a quasicontractive semigroupon an Lp space that incorporates both the underlying domain and its boundary. This extends theearlier work of the authors on the symmetric case.

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.

We determine the steady-state structures that result from liquid-liquid demixing in afree surface film of binary liquid on a solid substrate. The considered model correspondsto the static limit of the diffuse interface theory describing the phase separationprocess for a binary liquid (model-H), when supplemented by boundary conditions at thefree surface and taking the influence of the solid substrate into account. The resultingvariational problem is numerically solved employing a Finite Element Method on an adaptivegrid. The developed numerical scheme allows us to obtain the coupled steady-state filmthickness profile and the concentration profile inside the film. As an example wedetermine steady state profiles for a reflection-symmetric two-dimensional droplet forvarious surface tensions of the film and various preferential attraction strength of onecomponent to the substrate. We discuss the relation of the results of the present diffuseinterface theory to the sharp interface limit and determine the effective interfacetension of the diffuse interface by several means.

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

Infectious diseases may have multiple infectious stages with very different epidemiological attributes,including infectivity and disease progression. These stages are oftenassumed to have exponentially distributed durations in epidemiological models. However,models that use the exponential distribution assumption (EDA) may generate biased andeven misleading results in some cases. This discrepancy is particularly damaging if the modelsare employed to assist policy-makers in disease control and interventions. This paperstudies a mathematical model that includes multiple infectious stages and general distributionsfor the stage durations (with the exponential distribution as a special case). Formulasfor the control reproductive number, Rc, and the basic reproductive number, R0, are derived,which can be conveniently applied to models in which specific stage distributions areassumed. It is also shown that the disease dynamics are determined by the reproductivenumbers.

Propagation of chemical waves at very low temperatures, observedexperimentally [V.V. Barelko et al., Advances in Chem. Phys. 74 (1988), 339-384.] at velocities of order  10 cm/s, is due toa very non- standard physical mechanism. The energy liberated bythe chemical reaction induces destruction of the material, therebyfacilitating the reaction, a process very different from standardcombustion. In this work we present recent experimental results and develop anew mathematical model which takes into account mechanical and chemicalprocesses. We carry out numerical simulations and suggest a simplified model inorder to obtain an explicit expression for the wave speed. This modelling allowsus to give a possible explanation of the existence of two modes of propagationobserved experimentally and of the dependence of the speed of propagation on theradiation dose.

The goal of this paper is to apply the Continuous Hopfield Networks (CHN) to thePlacement of Electronic Circuit Problem (PECP). This assignment problem has been expressedas Quadratic Knapsack Problem (QKP). To solve the PECP via the CHN, we choose an energyfunction which ensures an appropriate balance between minimization of the cost functionand simultaneous satisfaction of the PECP constraints. In addition, the parameters of thisfunction must avoid some bad local minima. Finally, some computational experiments solvingthe PECP are included

The flow trough the Strait of Gibraltar could be analyzed as a problem oftwo-layerhydraulic exchange between the Atlantic ocean and the Mediterranean sea. Theshallow waterequations in both layers coupled together are an important tool to simulate thisphenomenon. Inthis paper we perform an upwind schemes for hyperbolic equations based on theRoe approximateRiemann solver, to study the resulting model. The main goal assigned was topredict the locationof the interface between the two layers. Therefore the computational resultsobtained are comparedto previous results and experiments.

Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

We consider the number of trophic levels in a food chain given by the equilibrium state for a simple mathematical model with ordinary differential equations which govern the temporal variation of the energy reserve in each trophic level. When a new trophic level invades over the top of the chain, the chain could lengthen by one trophic level. We can derive the condition that such lengthening could occur, and prove that the possibly longest chain is globally stable. In some specific cases, we find that the possibly longest chain is such that the lower trophic level has a greater energy reserve than the higher has, so that the distribution of energy reserves can be regarded to have a pyramid shape, whereas, if any of its trophic levels is removed, the pyramid shape cannot be maintained. Further, we find the condition that arbitrary long chain can be established. In such unbounded case, we prove that any chain could not have the pyramid shape of energy reserve distribution.

In this paper, we show finite time blow-up of solutions of the p−waveequation in ℝN, with critical Sobolev exponent. Our workextends a result by Galaktionov and Pohozaev [4]

The self-organization of porous nanostructures in anodic metal oxideis considered. A mathematical model which incorporates the chemicalreactions at the metal-oxide and oxide-electrolyte interfaces andelastic stress caused by the electrostrictive effects isdeveloped. It is shown through linear stability analysis, that ashort-wave instability exists in certain parameter regimes which canlead to the formation of hexagonally ordered pores observed inanodized aluminum oxide.

During the last two decades, molecular genetic studies and the completion of thesequencing of the Arabidopsis thaliana genome have increased knowledge ofhormonal regulation in plants. These signal transduction pathways act in concert throughgene regulatory and signalling networks whose main components have begun to be elucidated.Our understanding of the resulting cellular processes is hindered by the complex, andsometimes counter-intuitive, dynamics of the networks, which may be interconnected throughfeedback controls and cross-regulation. Mathematical modelling provides a valuable tool toinvestigate such dynamics and to perform in silico experiments that may not be easilycarried out in a laboratory. In this article, we firstly review general methods formodelling gene and signalling networks and their application in plants. We then describespecific models of hormonal perception and cross-talk in plants. This mathematicalanalysis of sub-cellular molecular mechanisms paves the way for more comprehensivemodelling studies of hormonal transport and signalling in a multi-scale setting.

Existence and stability of periodic solutions are studied for a system of delaydifferential equations with two delays, with periodic coefficients. It models theevolution of hematopoietic stem cells and mature neutrophil cells in chronic myelogenousleukemia under a periodic treatment that acts only on mature cells. Existence of a guidingfunction leads to the proof of the existence of a strictly positive periodic solution by atheorem of Krasnoselskii. The stability of this solution is analysed.

Dirac structures are used as the underlying structure to mathematically formalizeport-Hamiltonian systems. This note approaches the Dirac structures forinfinite-dimensional systems using the theory of linear relations on Hilbert spaces.First, a kernel representation for a Dirac structure is proposed. The one-to-onecorrespondence between Dirac structures and unitary operators is revisited. Further, theproposed kernel representation and a scattering representation are constructively related.Several illustrative examples are also presented in the paper.

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.

This paper is concerned with the methods of mathematical modeling of thesocio-economicdevelopment of regions. The objective of this study is to analyze and assess thesocio-economicpotential of the territory having regard to its individual socio-economic andeconomic-geographicalcharacteristics. The territory's individual characteristics, in turn, aretreated as invariantconditions of its development. The study territory includes regions and federaldistricts of theRussian Federation.
An assessment of the regions' development potential is made from the perspectiveof analyzingtheir current socio-economic situation for the time interval 1999-2002 takenfrom official statisticalhandbooks. To accomplish this, a special-purpose mathematical model is developedin the formof a hierarchical fractal indicative function relating the generalized indicatorcharacterizing theregions' development potential, or the “indicated” (that which is indicatedversus indicator), to theparticular socio-economic characteristics of the territory, or the developmentindicators of the localeconomy, via the model coefficients which is termed regulators.
As a result, the economic-mathematical model is generated in the form of anindicative functioncharacterizing the current socio-economic situation of the regions and federaldistricts of Russiaon different levels of economic organization (from local and regional tonational). It reflects appropriatelyand dramatically the specific character of the region's socio-economic potentialby theexample of the connection of investments with the volumes of production. Themodel permitsthe position of the regions and districts to be identified in the system ofinter-regional and nationalsocio-economic links and economic-geographical conditions of development. Theresultsderived from modeling the socio-economic development potential of Russia'sregions have beenused to construct the control surface of the country's economy characterizingthe changes in the development potential of the federal districts toward itsdecrease.

This paper is devoted to the study of global existence of periodic solutions of a delayedtumor-immune competition model. Also some numerical simulations are given to illustratethe theoretical results

Deformable cell model is developed to study pattern formation and to simulate planttissue growth. Each cell represents a polygon with a number of vertices connected bysprings. Some cells in the tissue can grow and divide, other cells are differentiated anddo not grow or divide but remain deformable. The model is used to investigate formation ofself-similar structures which reproduce the same cell organization during their growth. Innumerical experiments we observed that self-similar solutions can exist for a ratherprecise choice of plant structure and mechanical properties of cell walls. We test themodel for simulation of apical meristems functioning which represent self-similar cellstructures in plants. At the next stage of modelling, auxin distribution is introduced bymeans of diffusion and polar transport mechanisms. The existence of steady auxindistribution in a growing root is investigated. Single as well as multiple auxin maximahave been observed in model solutions.