Teaching for Abstraction: A Model

This article outlines a theoretical model for teaching elementary mathematical concepts that we have developed over the past 10 years. We begin with general ideas about the abstraction process and differentiate between abstract-general and abstract-apart concepts. A 4-phase model of teaching, called Teaching for Abstraction, is then proposed that is explicitly designed to promote abstract-general learning. Studies investigating the model with 4 different topics (angles, decimals, percentages, and ratios) are reported, and the 4 phases are further elucidated. The article concludes with a discussion of the effectiveness of the model and its applicability to other mathematical concepts.     

On the Turbulence Modelling of 3D-Atmospheric Boundary Layer Flow

The paper presents a mathematical and numerical investigation of the 3D–flow in the atmospheric boundary layer (ABL) over complex relief. The two–equation k - ε model is applied to account for the turbulence. The flow is also supposed to be viscous, incompressible and stationary. The boundary conditions are realized through the wall-functions. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thin elastic and periodic plates

This paper is devoted to the study of a three dimensional model of elastic periodic plate when the thickness e of the plate and the size ω of the periods are small. In the three studied limits (e → 0 then ω → 0), (ω → 0 then e → 0) and lately (e and ω → 0 together) the three dimensional equation of elasticity are approached by the two dimensional general equations of a linear anisotropic plate, the stretching and bending being coupled. This study points out the importance of the ratio of the two small parameters, indeed the moduli occuring in the two dimensional equations are different in the three limits. In each case a convergence proof is carried out.     

Global existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity

We prove the global‐in‐time existence and uniqueness of weak solutions in critical spaces for a mathematical model in superfluidity, with initial data ψ₀,A₀ ∈ L³,u₀ ∈ L^{3 ∕ 2},u₀ ≥ 0 in three dimension and in two dimension. Copyright © 2014 John Wiley & Sons, Ltd.     

Delamination in visco-elastic materials with thermal effects

This contribution reports on a model describing the evolution of brittle delamination between two visco-elastic bodies Ω₊ and Ω₋, bonded along a prescribed contact surface Γ_C, over a fixed time interval (0, T); a detailed discussion and a rigorous mathematical analysis can be found in [1]. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

PDE Models for Chemotactic Movements: Parabolic,Hyperbolic and Kinetic

Modeling the movement of cells (bacteria, amoeba) is a long standing subject and partial differential equations have been used several times. The most classical and successful system was proposed by Patlak and Keller & Segel and is formed of parabolic or elliptic equations coupled through a drift term. This model exhibits a very deep mathematical structure because smooth solutions exist for small initial norm (in the appropriate space) and blow-up for large norms. This reflects experiments on bacteria like Escherichia coli or amoeba like Dictyostelium discoïdeum exhibiting pointwise concentrations.For human endothelial cells, several experiments show the formation of networks that can be interpreted as the initiation of angiogenesis. To recover such patterns a hydrodynamical model seems better adapted.The two systems can be unified by a kinetic approach that was proposed for Escherichia coli, based on more precise experiments showing a movement by jump and tumble. This nonlinear kinetic model is interesting by itself and the existence theory is not complete. It is also interesting from a scaling point of view; in a diffusion limit one recovers the Keller-Segel model and in a hydrodynamical limit one recovers the model proposed for human endothelial cells.We also mention the mathematical interest of analyzing another degenerate parabolic system (exhibiting different properties) proposed to describe the angiogenesis phenomena i.e. the formation of capillary blood vessels.     

Multicriteria movement synchronization scheduling problems and algorithms

This article deals with multicriteria optimization models and algorithms of movement scheduling for many objects to synchronize their movement (2CMSS problem). The model consists of two parts: (1) node–disjoint path planning visiting specified nodes for K objects with a given vector of intermediate nodes for each one (NDSP problem); (2) movement synchronization in some intermediate nodes (MS problem). For synchronous movement, two categories of criteria are defined: time of movement and ‘distance’ of K-moved objects from the movement pattern. We defined the problem as a discrete-continuous, non-linear, two-criteria mathematical programming problem. We proposed to use a two-stage algorithm to solve the 2CMSS problem (as lexicographic solution): At first we have to find the vector of node–disjoint shortest paths for K objects visiting intermediate nodes to set optimal paths under the assumption that we use maximal possible velocities on each arc belonging to a path for each object (solution of the NDSP problem), and next we try to decrease the values of velocities to optimize the second criterion (synchronization, solution of the MS problem). Experimental analyses of effectiveness and complexity of the algorithms are presented.     

A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation

A side-by-side single sex age-structured population dynamic model is presented in this paper. The model consists of two coupled von Foerster-McKendrick-type quasi-linear partial differential equations, two initial conditions, and two boundary conditions. The state variables of the model are male and female population densities. The solutions of these partial differential equations provide explicit time and age dependence of the variables. The initial conditions define the male and female population densities at the initial time, while the boundary conditions compute the male and female births at zero-age by using fertility rates. The assumptions of the nontime-dependence of the death and fertility rates and a specific factorization of the migratory balances allow us to obtain exact solutions for male and female population densities. In addition, the hypotheses about the mathematical structure of the input variables are formulated, and the exact solution of the model is obtained. Next, the model is applied to the case study of Spain for the time period 1996–2004. Model validation demonstrates that this approach is a powerful prediction tool. Code and data are available upon request.     

Global sharp interface limit of the Hele–Shaw–Cahn–Hilliard system

In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two‐phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by ? , and the mobility coefficient (the diffusional Peclet number) is ? ^α . We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as ? →0 in the case of 0≤α  < 1. Copyright © 2016 John Wiley & Sons, Ltd.     

A Reduced-Load Equivalence for Generalised Processor Sharing Networks with Long-Tailed Input Flows

We consider networks where traffic is served according to the Generalised Processor Sharing (GPS) principle. GPS-based scheduling algorithms are considered important for providing differentiated quality of service in integrated-services networks. We are interested in the workload of a particular flow i at the bottleneck node on its path. Flow i is assumed to have long-tailed traffic characteristics. We distinguish between two traffic scenarios, (i) flow i generates instantaneous traffic bursts and (ii) flow i generates traffic according to an on/off process. In addition, we consider two configurations of feed-forward networks. First we focus on the situation where other flows join the path of flow i. Then we extend the model by adding flows which can branch off at any node, with cross traffic as a special case. We prove that under certain conditions the tail behaviour of the workload distribution of flow i is equivalent to that in a two-node tandem network where flow i is served in isolation at constant rates. These rates only depend on the traffic characteristics of the other flows through their average rates. This means that the results do not rely on any specific assumptions regarding the traffic processes of the other flows. In particular, flow i is not affected by excessive activity of flows with heavier-tailed traffic characteristics. This confirms that GPS has the potential to protect individual flows against extreme behaviour of other flows, while obtaining substantial multiplexing gains.     

Identification of Cl(Ca) channel distributions in olfactory cilia

Identification of detailed features of neuronal systems is an important challenge in the biosciences today. Transduction of an odor into an electrical signal occurs in the membranes of the cilia. The Cl(Ca) channels that reside in the ciliary membrane are activated by calcium, allow a depolarizing efflux of Cl⁻ and are thought to amplify the electrical signal to the brain. In this paper, a mathematical model consisting of partial differential equations is developed to study two different experiments; one involving the interaction of the cyclic-nucleotide-gated (CNG) and Cl(Ca) channels and the other, the diffusion of Ca²⁺ into cilia. This work builds on an earlier study (Mathematical modeling of the Cl(Ca) ion channels in frog olfactory cilia. Ph.D. Thesis, University of Cincinnati, Cincinnati, OH, 2006; Math. Comput. Modelling 2006; 43 :945–956; Biophys. J. 2006; 91 :179–188), which suggested that the CNG channels are clustered at about 0.28 of the length of a cilium from its open end. Closed-form solutions are derived after certain reductions in the model are made. These special solutions provide estimates of the channel distributions. Scientific computation is also used. This preliminary study suggests that the Cl(Ca) ion channels are also clustered at about one-third of the length of a cilium from its open end. Copyright © 2008 John Wiley & Sons, Ltd.     

Chance and necessity in sociological theory

This paper proposes a mathematical model of financial markets as networks. The model examines the effect of network structure on market behavior (price volatility and trading volume). In the model, investors are arrayed in various network configurations through which they gather information to make trading decisions. The basic network considered is a chain graph with two parameters, number of investors (n) and the length of time in which information is transmitted (k). Closed‐form expressions for price volatility and expected trading volume are provided. The model is generalized to more complex networks, focusing on the hub‐and‐spoke network. The network configurations analyzed do not represent the real (and unknown) communication network among investors, but predictions from the model are consistent with price and volume patterns observed in sociological and economic research on financial markets. The main result is that network structure alone influences price volatility and expected trading volume even though investors are homogeneous and the information introduced into the system is unbiased and random. This result suggests that the structure of the real communication network among investors may influence market behavior.     

Education campaign and family understanding affect stability and qualitative behavior of an online game addiction model for children and youth in Thailand

Online game addiction has become a large problem worldwide, and it could give negative impacts on children in many ways such as physical health, learning, emotion, and behavior. Online game addiction is mostly found in children and youth at the age of 15 to 24 years. Therefore, in this paper, we have developed a mathematical model of online game addiction to explore the effects of education campaign and family understanding on online game addiction in Thailand. Analysis of this model reveals two main equilibria, addiction‐free (AFE) and addiction‐present (APE) ones. Results show that the AFE is locally asymptotically stable when the value of basic reproduction number (R₀) is less than the unity; otherwise, there is a unique endemic equilibrium point, and it is locally stable when satisfies the Routh‐Hurwitz criterion. Further, the conditions of AFE to be globally stable are demonstrated. Finally, the results of sensitivity analysis and numerical simulation show that the effectiveness of both education campaign and family understanding is an important factor in reducing the value of R₀ and holds great promise for lowering the number of children and youth who addict to online game in Thailand.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Man-made plutonium in freshwater and marine environments

Significant concentrations of plutonium isotopes are found in lake and coastal marine sediments accumulating at rates of 1–5 mm/yr. The activity levels of^{239, 240}Pu range between 0·7–3·3 dpm/g for post 1960 sediments. The measured and the calculated integrated activities of²³⁹Pu in the lake sediments are in agreement. This suggests that plutonium is chemically reactive in freshwater environments and it is removed to sediments from the water column shortly after its injection. A delay of about three years has been observed between the peak of plutonium fallout and its incorporation in the varved sediment from Santa Barbara basin. Based on this observation the mean size of particles transporting plutonium to the basin sediments is estimated to be about three microns. The usefulness of plutonium isotopes as a tracer nuclide for estimating sedimentation rates for the last two decades have been evaluated.     

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ₀ of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ₀. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.     

On the convergence of general regularization and smoothing schemes for mathematical programs with complementarity constraints

We extend the convergence analysis of a smoothing method [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg.] to a general class of smoothing functions and show that a weak second-order necessary optimality condition holds at the limit point of a sequence of stationary points found by the smoothing method. We also show that convergence and stability results in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] hold for a relaxation problem suggested by Scholtes [S. Scholtes (2003). Private communications.] using a class of smoothing functions. In addition, the relationship between two technical, yet critical, concepts in [M. Fukushima and J.-S. Pang (2000). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: M. Théra and R. Tichatschke (Eds.), Ill-posed Variational Problems and Regularization Techniques, pp. 99–110. Springer, Berlin/Heidelberg; S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] for the convergence analysis of the smoothing and regularization methods is discussed and a counter-example is provided to show that the stability result in [S. Scholtes (2001). Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim., 11, 918–936.] cannot be extended to a weaker regularization.     

A simpler GMRES

The generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar₀ instead of r₀, we obtain simpler Gram–Schmidt and Householder implementations of the GMRES method that do not require upper Hessenberg factorization. The Gram–Schmidt implementation also maintains the residual vector at each iteration, which allows cheaper restarts of GMRES(m) and may otherwise be useful.