The sign-coherence property of $c$-vectors plays an important role in the structure theory of cluster algebras. However, all known proofs depend on either the invariant theory of geometry or the representation theory. In this paper we give an elementary proof for the sign-coherence property of $c$-vectors for cluster algebras of rank $2$. As an application, we prove that a cluster is determined by its associated subset of positive $c$-vectors.

In this paper, we study the asymptotic dynamics of a single-species model with resource-dependent dispersal in one dimension.

To overcome the analytical difficulties brought by the resource-dependent dispersal, we use the idea of changing variables to transform the model into a uniform dispersal one. Then the existence and uniqueness of positive stationary solution to the model can be verified by the squeezing argument, where the solution plays a crucial role in later analyses. Moreover, the asymptotic behavior of solutions to the model is obtained by the upper-lower solutions method. The result indicates that the solutions of the model converge to the corresponding positive stationary solution locally uniformly in one dimension as time goes to infinity.

In this paper, we construct a new sixth order iterative method for solving nonlinear equations. The local convergence and order of convergence of the new iterative method is demonstrated. In order to check the validity of the new iterative method, we

employ several chemical engineering applications and academic test problems. Numerical results show the good numerical performance of the new iterative method. Moreover, the dynamical study of the new method also supports the theoretical results.

In this paper, we first give the general forms of skew commuting maps and skew anti-commuting maps by the Peirce decomposition on a unital ring with a nontrivial idempotent, respectively, and then, as applications, we obtain the concrete characterizations of all nonadditive skew (anti-)commuting maps on some operator algebras.

This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation. Applying the maximum principle and the moving frame method and based on the location of the maximum point, the boundary gradient estimation of the solutions to the equation is obtained.

As a generalization of the two-term conjugate gradient method (CGM), the spectral CGM is one of the effective methods for solving unconstrained optimization. In this paper, we enhance the JJSL conjugate parameter, initially proposed by Jiang et al. (Computational and Applied Mathematics, 2021, 40: 174), through the utilization of a convex combination technique.

And this improvement allows for an adaptive search direction by integrating a newly constructed spectral gradient-type restart strategy. Then, we develop a new spectral CGM by employing an inexact line search to determine the step size. With the application of the weak Wolfe line search, we establish the sufficient descent property of the proposed search direction. Moreover, under general assumptions, including the employment of the strong Wolfe line search for step size calculation, we demonstrate the global convergence of our new algorithm. Finally, the given unconstrained optimization test results show that the new algorithm is effective.

In this paper we introduce the notions of mean dimension and metric mean dimension for non-autonomous iterated function systems

(NAIFSs for short) on countably infinite alphabets which can be regarded as generalizations of the mean dimension and the Lindenstrauss metric mean dimension for non-autonomous iterated function systems. We also show the relationship between the mean topological dimension and the metric mean dimension.