Properties of the aggregated quasi-hydrodynamic system of equations for a homogeneous gas mixture with a common regularizing velocity

We study a quasi-hydrodynamic system of equations for a homogeneous (with common velocity and temperature) multicomponent gas mixture in the absence of chemical reactions, with a regularizing velocity common for the components. We derive the entropy balance equation with a non-negative entropy production taking into account the diffusion fluxes of the mixture components. In the absence of diffusion fluxes, a system of equations linearized at a constant solution is constructed by a new technique, In the absence of diffusion fluxes, a system of equations linearized on a constant solution is constructed by a new technique. It is reduced to a symmetric form, the L^2-dissipativity of its solutions is proved, and a degeneration (with respect to the densities of the mixture components) of the parabolicity property for the original system is established. Actually, the system has the composite type. The obtained properties strictly reflect its physical correctness and dissipative nature of the quasi-hydrodynamic regularization.

Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establish a priori estimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

On L2-dissipativity of a linearized difference scheme on staggered meshes with a quasi-hydrodynamic regularization for 1D barotropic gas dynamics equations

We study an explicit two-level finite difference scheme on staggered meshes, with a quasi-hydrodynamic regularization, for 1D barotropic gas dynamics equations. We derive necessary conditions and sufficient conditions close to each other for L²-dissipativity of solutions to the Cauchy problem for its linearization on a constant solution, for any background Mach number M. We apply the spectral approach and analyze matrix inequalities containing symbols of symmetric matrices of convective and regularizing terms. We consider the cases where either the artificial viscosity coefficient or the physical viscosity one is used. A comparison with the spectral von Neumann condition is also given for M=0.

Global Bifurcation of Stationary Solutions for a Volume-Filling Chemotaxis Model with Logistic Growth

In this paper, we show how the global bifurcation theory for nonlinear Fredholm operators (Theorem 4.3 of [Shi & Wang, 2009]) and for compact operators (Theorem 1.3 of [Rabinowitz, 1971]) can be used in the study of the nonconstant stationary solutions for a volume-filling chemotaxis model with logistic growth under Neumann boundary conditions. Our results show that infinitely many local branches of nonconstant solutions bifurcate from the positive constant solution [Formula: see text] at [Formula: see text]. Moreover, for each [Formula: see text], we prove that each [Formula: see text] can be extended into a global curve, and the projection of the bifurcation curve [Formula: see text] onto the [Formula: see text]-axis contains [Formula: see text].

Learner-generated graphic representations for word problems: an intervention and evaluation study in grade 3

The use of written or graphic representations is essential in mathematics. Graphic representations are mainly used and researched as instruments for problem solving. There is a gap in research for interventions that use learner-generated graphic representations as documents for reflection processes for promoting the development of children’s graphic representation competences. This is the focus of the study presented here. The study examines to what extent such an intervention has an effect on how the children take into account a mathematical structure in their self-generated graphic representations, how they ensure a mathematical matching with the word problem, and what degree of abstraction they choose. Additionally, the effect on the solution rates is investigated. The results show that children in the intervention group more frequently pay attention to a mathematically appropriate structure, compared with children in the control groups. This result is statistically significant. At the same time, children keep the degree of abstraction relatively constant. Solution rates improve continuously, but the difference is not significant.

Global and exponential attractor of the repulsive Keller–Segel model with logarithmic sensitivity

We consider a Keller–Segel model that describes the cellular chemotactic movement away from repulsive chemical subject to logarithmic sensitivity function over a confined region in ${{\mathbb{R}}^n},\,n \le 2$ . This sensitivity function describes the empirically tested Weber–Fecher’s law of living organism’s perception of a physical stimulus. We prove that, regardless of chemotaxis strength and initial data, this repulsive system is globally well-posed and the constant solution is the global and exponential in time attractor. Our results confirm the ‘folklore’ that chemorepulsion inhibits the formation of non-trivial steady states within the logarithmic chemotaxis model, hence preventing cellular aggregation therein.

Inhomogeneous Partition Regularity

We say that the system of equations $Ax = b$, where $A$ is an integer matrix and $b$ is a (non-zero) integer vector, is partition regular if whenever the integers are finitely coloured there is a monochromatic vector $x$ with $Ax = b.$ Rado proved that the system $Ax = b$ is partition regular if and only if it has a constant solution. Byszewski and Krawczyk asked if this remains true when the integers are replaced by a general (commutative) ring $R$. Our aim in this note is to answer this question in the affirmative. The main ingredient is a new 'direct' proof of Rado’s result.

Bifurcation Behaviors of Steady-State Solution to a Discrete General Brusselator Model

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear u in the standard Brusselator model to the nonlinear fu. Assume that f∈C0,∞,0,∞ is a strictly increasing function, and f′f−1a∈0,∞. Taking b as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point b0j,f−1a,ab/f−1a2. Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that fs/s2 is nonincreasing in 0,∞. In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem.

Some models with repulsion effect on superinfecting viruses by infected cells

In this paper, we study some models with repulsion effect on superinfecting viruses by infected cells [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are the density of uninfected cells, infected cells and viruses at time [Formula: see text] at location [Formula: see text], respectively. The functions [Formula: see text] and [Formula: see text] are assumed to be positive, continuous and bounded. [Formula: see text] denotes the production rate of uninfected cells. The infection rate is [Formula: see text] and the function [Formula: see text] is the production rate of free viruses. And [Formula: see text] is the rate of transfer from uninfected cells to infected cells. The positive constants [Formula: see text] and [Formula: see text] denote the death rate of uninfected cells, infected cells and viruses, respectively. The stability of the infection-free equilibrium solution and infection equilibrium solution is discussed. It is shown that if the basic reproduction number [Formula: see text] then the chemotaxis has no effect, that is, the infection-free constant solution is stable. For the system with chemotactic sensitivity [Formula: see text], if [Formula: see text], then the infection constant solution will be unstable under some conditions.

On the weakly degenerate Allen-Cahn equation

In this paper we consider a one-dimensional Allen-Cahn equation with degeneracy in the interior of the domain and Neumann boundary conditions. We allow the diffusivity coefficient vanish at some point of the space domain and we are addressed on the existence of stable non-constant solution.