Investigation of the Structure of the Set of Continuous Solutions of Systems of Linear Difference Equations with Continuous Argument

We obtain conditions for the existence of continuous and N-periodic solutions, where N is a positive integer number, for systems of linear difference equations with continuous argument and investigate the structure of the set of these solutions. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 351–359, July–September, 2005.     

Dynamics of a chain system of rigid bodies with gravity-friction seismic dampers: fixed supports

A design model for a chain system of N elastically linked rigid bodies with a spheroidal gravity-friction damper is proposed. The Lagrange–Painlevé equations of the first kind are used to construct nonlinear dynamical models of a mechanical system undergoing translational vibrations about the equilibrium position. The conditions under which the system moves in one plane are established. The double nonstationary phase–frequency resonance of a system with N = 2 is analyze. After the numerical integration of the systems of differential equations, the phase–frequency surfaces are plotted and examined for several combinations of system parameters under two-frequency loading     

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

The Bogoliubov–Dirac–Fock (BDF) model is the mean-field approximation of no-photon quantum electrodynamics. The present paper is devoted to the study of the minimization of the BDF energy functional under a charge constraint. An associated minimizer, if it exists, will usually represent the ground state of a system of N electrons interacting with the Dirac sea, in an external electrostatic field generated by one or several fixed nuclei. We prove that such a minimizer exists when a binding (HVZ-type) condition holds. We also derive, study and interpret the equation satisfied by such a minimizer. Finally, we provide two regimes in which the binding condition is fulfilled, obtaining the existence of a minimizer in these cases. The first is the weak coupling regime for which the coupling constant α is small whereas αZ and the particle number N are fixed. The second is the non-relativistic regime in which the speed of light tends to infinity (or equivalently α tends to zero) and Z, N are fixed. We also prove that the electronic solution converges in the non-relativistic limit towards a Hartree–Fock ground state.     

Viscoelastic properties of ultra-high viscosity alginates

Ultra-high viscosity alginates were extracted from the brown seaweeds Lessonia nigrescens (UHV_N, containing 61% mannuronate (M) and 2% guluronate (G)) and Lessonia trabeculata (UHV_T, containing 22% M and 78% G). The viscoelastic behavior of the aqueous solutions of these alginates was determined in shear flow in terms of the shear stress σ ₂₁, the first normal stress difference N ₁, and the shear viscosity η in isotonic NaCl solutions (0.154 mol/L) at T = 298 K in dependence of the shear rate [(g)\dot]\dot{\gamma} for solutions of varying concentrations and molar masses (3–10 × 10⁵ g/mol, homologous series was prepared by ultrasonic degradation). Data obtained in small-amplitude oscillatory shear (SAOS) experiments obey the Cox–Merz rule. For comparison, a commercial alginate with intermediate chemical composition was additionally characterized. Particulate substances which are omnipresent in most alginates influenced the determination of the material functions at low shear rates. We have calculated structure–property relationships for the prediction of the viscosity yield, e.g., η–M _w–c–[(g)\dot]\dot{\gamma} for the Newtonian and non-Newtonian region. For the highest molar masses and concentrations, the elasticity yield in terms of N ₁ could be determined. In addition, the extensional flow behavior of the alginates was measured using capillary breakup extensional rheometry. The results demonstrate that even samples with the same average molar mass but different molar mass distributions can be differentiated in contrast to shear flow or SAOS experiments.     

Lyapunov and Sacker–Sell Spectral Intervals

In this work, we show that for linear upper triangular systems of differential equations, we can use the diagonal entries to obtain the Sacker and Sell, or Exponential Dichotomy, and also –under some restrictions– the Lyapunov spectral intervals. Since any bounded and continuous coefficient matrix function can be smoothly transformed to an upper triangular matrix function, our results imply that these spectral intervals may be found from scalar homogeneous problems. In line with our previous work [Dieci and Van Vleck (2003), SIAM J. Numer. Anal. 40, 516–542], we emphasize the role of integral separation. Relationships between different spectra are shown, and examples are used to illustrate the results and define types of coefficient matrix functions that lead to continuous Sacker–Sell spectrum and/or continuous Lyapunov spectrum.      

A comparison of three different methods for measuring both normal stress differences of viscoelastic liquids in torsional rheometers

A novel pressure sensor plate (normal stress sensor (NSS) from RheoSense, Inc.) was adapted to an Advanced Rheometrics Expansion System rheometer in order to measure the radial pressure profile for a standard viscoelastic fluid, a poly(isobutylene) solution, during cone–plate and parallel-plate shearing flows at room temperature. We observed in our previous experimental work that use of the NSS in cone-and-plate shearing flow is suitable for determining the first and second normal stress differences N ₁ and N ₂ of various complex fluids. This is true, in part, because the uniformity of the shear rate at small cone angles ensures the existence of a simple linear relationship between the pressure [i.e., the vertical diagonal component of the total stress tensor (Π₂₂)] and the logarithm of the radial position r (Christiansen and coworkers, Magda et al.). However, both normal stress differences can also be calculated from the radial pressure distribution measured in parallel-plate torsional flows. This approach has rarely been attempted, perhaps because of the additional complication that the shear rate value increases linearly with radial position. In this work, three different methods are used to investigate N ₁ and N ₂ as a function of shear rate in steady shear flow. These methods are: (1) pressure distribution cone–plate (PDCP) method, (2) pressure distribution parallel-plate (PDPP) method, and (3) total force cone–plate parallel-plate (TFCPPP) method. Good agreement was obtained between N ₁ and N ₂ values obtained from the PDCP and PDPP methods. However, the measured N ₁ values were 10–15% below the certified values for the standard poly(isobutylene) solution at higher shear rates. The TFCPPP method yielded N ₁ values that were in better agreement with the certified values but gave positive N ₂ values at most shear rates, in striking disagreement with published results for the standard poly(isobutylene) solution.

Revisiting the Drainage Relative Permeability Measurement by Centrifuge Method Using a Forward–backward Modeling Scheme

Measurement of drainage relative permeability by the centrifuge method was first introduced by Hagoort (SPE J. 29(3):139–150, 1980). It has been shown that capillary end effects can cause error in the measurement of relative permeability if a minimum rotational speed is not honoured. To determine the minimum rotational speed that makes the capillary end effect negligible, ω _min, we propose that the value of capillary-gravity number, N _cg, should be of the order of 10⁻² or smaller. This conclusion is based on the use a Forward–backward scheme consisting of a forward numerical simulator developed for centrifuge experiments and applying Hagoort’s method as a backward model. The article presents the use of this Forward–backward scheme as a powerful tool for error analysis such as determining the impact of capillary end effects. By using this loop, we first determine ω _min for specific core and fluid properties. Later, we generalize the ω _min calculations by using the definition of N _cg as a “rule of thumb” for designing relative permeability experiments by centrifuge method. We also demonstrate another use of this loop for controlling the quality of the experimental data.     

Asymptotic approximation for the solution of a boundary-value problem with varying type of boundary conditions in a thick two-level junction

We consider a mixed boundary-value problem for a Poisson equation in a plane two-level junction Ω_ε that is the union of a domain Ω₀ and a large number 3N of thin rods with thickness of order . The thin rods are divided into two levels depending on their length. In addition, the thin rods from each level are ε-periodically alternated. The homogeneous Dirichlet conditions and inhomogeneous Neumann conditions are given on the sides of the thin rods from the first level and the second level, respectively. Using the method of matched asymptotic expansions and special junction-layer solutions, we construct an asymptotic approximation for the solution and prove the corresponding estimates in the Sobolev space H ¹(Ω_ε) as ε → 0 (N → +∞). Published in Neliniini Kolyvannya, Vol. 9, No. 3, pp. 336–355, July–September, 2006.     

Sufficient Conditions for the Existence of Bounded Solutions of Systems of Nonlinear Difference Equations

We obtain sufficient conditions for systems of nonlinear difference equations x(n + 1) = A(x(n))x(n) + f(n), n ∈ ℤ, where A(x) is a matrix function continuous on ℝ ^m , to have solutions in the space of bilateral number sequences. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 165–173, April–June, 2005.     

On Bifurcations of Equilibria of Intrinsically Curved, Electrically Charged, Rod-like Structures that Model DNA Molecules in Solution

DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair, which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general 6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations occur at a single value of c.      

Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model

Linear viscoelastic (LVE) measurements as well as non-linear elongation measurements have been performed on stoichiometrically imbalanced polymeric networks to gain insight into the structural influence on the rheological response (Jensen et al., Rheol Acta 49(1):1–13, 2010). In particular, we seek knowledge about the effect of dangling ends and soluble structures. To interpret our recent experimental results, we exploit a molecular model that can predict LVE data and non-linear stress–strain data. The slip-link model has proven to be a robust tool for both LVE and non-linear stress–strain predictions for linear chains (Khaliullin and Schieber, Phys Rev Lett 100(18):188302–188304, 2008, Macromolecules 42(19):7504–7517, 2009; Schieber, J Chem Phys 118(11):5162–5166, 2003), and it is thus used to analyze the experimental results. Initially, we consider a stoichiometrically balanced network, i.e., all strands in the ensemble are attached to the network in both ends. Next we add dangling strands to the network representing the stoichiometric imbalance, or imperfections during curing. By considering monodisperse network strands without dangling ends, we find that the relative low-frequency plateau, G₀/G_N⁰G_0/G_N^0, decreases linearly with the average number of entanglements. The decrease from G_N⁰G_N^0 to G ₀ is a result of monomer fluctuations between entanglements, which is similar to “longitudinal modes” in tube theory. It is found that the slope of G′ is dependent on the fraction of network strands and the structural distribution of the network. The power-law behavior of G ^″ is not yet captured quantitatively by the model, but our results suggest that it is a result of polydisperse dangling and soluble structures.     

Unique Solutions to Hartree–Fock Equations for Closed Shell Atoms

In this paper we study the problem of uniqueness of solutions to the Hartree and Hartree–Fock equations of atoms. We show, for example, that the Hartree–Fock ground state of a closed shell atom is unique provided the atomic number Z is sufficiently large compared to the number N of electrons. More specifically, a two-electron atom with atomic number Z\geqq 35{Z\geqq 35} has a unique Hartree–Fock ground state given by two orbitals with opposite spins and identical spatial wave functions. This statement is wrong for some Z > 1, which exhibits a phase segregation.     

Generation and registration of disturbances in a gas flow. 1. Formation of arrays of tubular microheaters and microsensors

A new method is proposed for creating “smart” surfaces for suppressing turbulence and retaining a laminar supersonic flow. Methods of formation of super-fast-response sensors and actuators for such surfaces are developed. Such sensors and actuators are structurally designed as microtubes made of SiO ₂ /Si ₃ N ₄ /Au and InGaAs/GaAs/Au heterofilms and suspended above a substrate; the wall thickness of these tubes is in the nanometer range; the tubes are connected to electrical contacts. Models of distributed arrays of tubular microsensors and microactuators are fabricated in a single technological process, which involves the well-established planar technology and the technology of rolling of stressed heterofilms. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 145–151, March–April, 2009.     

Optimal Gradient Estimates and Asymptotic Behaviour for the Vlasov–Poisson System with Small Initial Data

The Vlasov–Poisson system describes interacting systems of collisionless particles. For solutions with small initial data in three dimensions it is known that the spatial density of particles decays as t ⁻³ at late times. In this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay, so that in N dimensions for N \geqq 3{N \geqq 3} the derivative of the density of order k decays as t ^−N-k . An asymptotic formula for the solution at late times is also obtained.     

Homogenization of a Boundary-Value Problem with Varying type of Boundary Conditions in a Thick Two-Level Junction

We consider a mixed boundary-value problem for the Poisson equation in a plane two-level junction Ω _ɛ that is the union of a domain Ω₀ and a large number 2N of thin rods with thickness of order ɛ = (N ⁻¹). Depending on their lengths, the thin rods are divided into two levels. In addition, the rods from each level are ɛ-periodically alternated. Inhomogeneous Neumann boundary conditions are given on the vertical sides of the thin rods of the first level, and homogeneous Dirichlet boundary conditions are given on the vertical sides of the rods of the second level. We investigate the asymptotic behavior of a solution of this problem as ɛ → 0 and prove a convergence theorem and the convergence of the energy integral. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 2, pp. 241–257, April–June, 2005.     

Fokker–Planck Equations for a Free Energy Functional or Markov Process on a Graph

The classical Fokker–Planck equation is a linear parabolic equation which describes the time evolution of the probability distribution of a stochastic process defined on a Euclidean space. Corresponding to a stochastic process, there often exists a free energy functional which is defined on the space of probability distributions and is a linear combination of a potential and an entropy. In recent years, it has been shown that the Fokker–Planck equation is the gradient flow of the free energy functional defined on the Riemannian manifold of probability distributions whose inner product is generated by a 2-Wasserstein distance. In this paper, we consider analogous matters for a free energy functional or Markov process defined on a graph with a finite number of vertices and edges. If N ≧ 2 is the number of vertices of the graph, we show that the corresponding Fokker–Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. However, in contrast to stochastic processes defined on Euclidean spaces, the situation is more subtle for discrete spaces. We have different choices for inner products on the space of probability distributions resulting in different Fokker–Planck equations for the same process. It is shown that there is a strong connection but there are also substantial discrepancies between the systems of ordinary differential equations and the classical Fokker–Planck equation on Euclidean spaces. Furthermore, both systems of ordinary differential equations are gradient flows for the same free energy functional defined on the Riemannian manifolds of probability distributions with different metrics. Some examples are also discussed.     

Asymptotics of the trajectory of an interval that contains the preimage of a periodic point

We consider dynamical systems defined by continuous maps of an interval I of the real axis into itself. We prove that if an interval J in I contains the preimage of a periodic point of period p of a map f ∈ C ⁰(I, I), then the sequence of intervals f ^2pn (J), n= 0, 1, 2,…, is convergent. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 130–133, January–March, 2009.     

In situ monitoring the sol–gel transition for polyacrylamide gel

The effects of heating rate and shear stress on the sol–gel transition for a polyacrylamide gel during in situ preparation were investigated by dynamic rheology. The gelation evolution was also studied through monitoring of the static gelation process by means of temperature, pH, and conductivity variation. The gels were prepared by polymerization of acrylamide crosslinked with N,N′-methylenebisacrylamide using a redox initiation based on potassium persulfate/ascorbic acid. In situ gelation process was studied using oscillatory deformation tests at constant frequency of 0.1 Hz for different heating rate (from 0.5 to 5 °C/min) and shear stress (in the range of 0.1 to 10 Pa). The paper was presented at the third annual rheology conference, AERC 2006, Crete, Greece, 27–29 April 2006.     

Vortex ring instability and collapse in a stably stratified fluid

Exploratory measurements of the effect of a stable continuous vertical stratification on horizontally propagating vortex rings show that the rings are subject to a stratification induced instability and subsequent collapse, which forms a well mixed intrusion. For initial ring Froude numbersF ₀=U ₀/Nd ₀=1.0–2.0, instability and collapse occurs when the ring Froude number has decreased to a value in the range 0.6–1.0.     

A simplified model system for <Emphasis Type="Italic">Toxoplasma gondii</Emphasis> spread within a heterogeneous environment

This study is dedicated to building and analyzing the spatial spread of Toxoplasma gondii through a heterogeneous predator–prey system. The spatial domain is made of N patches hosting various population species, some of them being prey, others being predators. Predators offer strong heterogeneities with respect to local sustainable resources yielding variable growth rates, from exponential decay to logistic regulation. T. gondii life cycle goes through several stages, starting in the environment where oocysts are released from cat feces, reaching prey within which asexual reproduction yields cysts and then predators wherein sexual reproduction takes place. The resulting model system is complex to handle. We consider some relevant toy models with three patches, two resident predator species and Lotka–Volterra functional responses to predation. We provide the existence and local stability of a persistent stationary state for the underlying predator–prey model systems. The reproduction number R ₀ is computed in the quasistationary case; it simplifies when slow–fast dynamics are considered. Numerical experiments illustrate our analysis.