The design of an on-demand transport system with origin and destination matrices is studied. Two objectives are considered; one is to minimize the sum (over all users) of travel time per unit time and the other is to minimize the sum of the total cost of track built. Firstly, we show that even the minimum total cost problem without consideration of the travel time is not a simple minimum spanning tree problem. Then, for the bi-objective problem, we transform one objective to a constraint condition. With this transformation, we transform the bi-objective problem to a single-objective problem and give the optimal solution by the branch and bound method. Finally, we propose a fast approximate algorithm and analyze its complexity.

This paper describes the modelling of the toner behaviour in the development nip of the Océ Direct Imaging print process. The dynamic motion of and mechanical interactions between toner particles are explicitly modelled. The mechanical interactions are due to collisions, friction, adhesion, and electromagnetic forces. The discrete element method (DEM) is used as the simulation tool for a quantitative description of the system. The interaction rules are determined for the toner particles and the surfaces of the development rollers. The model is validated with print quality results. It is shown that it is possible to achieve quantitative agreement between DEM simulations and experimental print quality results.

In this paper we study some aspects of the integrability problem for polynomial vector fields $\dot{x}=P(x,y)$, $\skew1\dot{y}=Q(x,y)$. We analyze the possible existence of first integrals of the form $I(x,y)=e^{ h_1(x) \prod_{k=1}^r (y-a_k(x))/ \prod_{j=1}^s(y-f_j(x))} h_2(x)$$\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, where $g_i(x)$ and $f_j(x)$ are unknown particular solutions of $dy/dx=Q(x,y)/P(x,y)$, $\alpha_i \in \mathbb{C}$ are unknown constants, and $a_k(x)$, $h_1(x)$ and $h_2(x)$ are unknown functions. We give an algorithmic method to determine if the polynomial vector field has a first integral of the form above described. In the case when some of the particular solutions remain arbitrary and the other ones are explicitly determined or are functionally related to the arbitrary particular solutions, we will obtain a generalized nonlinear superposition principle, see [6]. In the case when all the particular solutions $g_i(x)$ and $f_j(x)$ are determined, they are algebraic functions and our algorithm gives an alternative method for determining such type of solutions.

This paper studies many fundamental aspects of a newly proposed multi-class traffic flow model. For the first time it presents a complete discussion on the hyperbolicity of the system; and based upon this, the admissible waves of the Riemann problem are deeply investigated. Many important conclusions are made in discussions, and the related physical meanings are interpreted. For the confirmation of main conclusions, numerical examples are given at the end.

The notion of polarization tensor is employed for the derivation of the leading-order boundary perturbations in the steady-state voltage potentials that are due to the presence of conductivity inclusions of small diameter. Recently, Capdeboscq and Vogelius obtained optimal bounds of Hashin-Shtrikman type for the trace of the polarization tensor, showing that every pair satisfying these optimal bounds arises as the eigenvalues of a polarization tensor associated with a coated ellipse. In this paper, we give numerical evidence of the fact that the set of possible polarization tensor eigenvalue pairs can also be obtained using simply connected domains. Our numerical computations are based on a boundary integral method.

The scalar initial value problem \[ u_t = \rho Du + f(u), \] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $\Omega \in \mathbb{R}^n$ and time $t$, and $u(\cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=\Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=\Delta$). We develop a technique for proving that indeed convergence does hold for small $\rho$ and show by constructing a counter-example that this result does not hold in general for all $\rho$.

We study the propagation of a crack in critical equilibrium for a brittle material in a Mode III field. The energy variations for small virtual extensions of the crack are handled in a novel way: the amount of energy released is written as a functional over a family of univalent functions on the upper half plane. Classical techniques developed in connection to the Bieberbach Conjecture are used to quantify the energy-shape relationship. By means of a special family of trial paths generated by the so-called Löwner equation we impose a stability condition on the field which derives in a local crack propagation criterion. We called this the anti-symmetry principle, being closely related to the well known symmetry principle for the in-plane fields.