The random walk is one of the most basic dynamic properties of complex networks, which has gradually become a research hotspot in recent years due to its many applications in actual networks. An important characteristic of the random walk is the mean time to absorption, which plays an extremely important role in the study of topology, dynamics and practical application of complex networks. Analyzing the mean time to absorption on the regular iterative self-similar network models is an important way to explore the influence of self-similarity on the properties of random walks on the network. The existing literatures have proved that even local self-similar structures can greatly affect the properties of random walks on the global network, but they have failed to prove whether these effects are related to the scale of these self-similar structures. In this article, we construct and study a class of Horizontal Partitioned Sierpinski Gasket network model based on the classic Sierpinski gasket network, which is composed of local self-similar structures, and the scale of these structures will be controlled by the partition coefficient $k.$ Then, the analytical expressions and approximate expressions of the mean time to absorption on the network model are obtained, which prove that the size of the self-similar structure in the network will directly restrict the influence of the self-similar structure on the properties of random walks on the network. Finally, we also analyzed the mean time to absorption of different absorption nodes on the network to find the location of the node with the highest absorption efficiency.

In this note, we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials. We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an $L^2$ initial datum for positive time. So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.

Noting that the particles communicate with each other within a finite distance, in the paper, we investigate the hierarchical model with free will and cut-off function (linear or nonlinear). As our observation, the cut-off size $r$ is a sensitive coefficient to achieve the flocking behavior. Besides we will use energy function to achieve some sufficient conditions to achieve a flock. The free-will represents the tendency of the particle itself to move. It will either facilitate or prevent cluster generation. Some numerical simulations will validate our conclusions.

We study existence and uniqueness results for the Yamabe problem on non-compact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In this context, our result requires only a partial $C^2$ decay of the metric, namely the full decay of the metric in $C^1$ and the decay of the scalar curvature. In particular, no decay of the Ricci curvature is assumed. In our second result we establish that a local volume ratio condition, when combined with negativity of the scalar curvature at infinity, is sufficient for existence of a solution. Our volume ratio condition appears tight. This paper is based on the DPhil thesis of the first author.

The aim of this paper is to establish the boundedness of bilinear pseudo-differential operator $T_σ$ and its commutator $[b_1, b_2, T_σ]$ generated by $T_σ$ and $b_1, b_2∈ {\rm BMO}(\mathbb{R}^n)$ on generalized fractional weighted Morrey spaces $L^{p,η,\varphi} (ω).$ Under assumption that a weight satisfies a certain condition, the authors prove that $T_σ$ is bounded from products of spaces $L^{p_1,η_1,\varphi}(ω_1)×L^{p_2,η_2,\varphi}(ω_2)$ into spaces $L^{p,η,\varphi} (\vec{ω}),$ where $\vec{ω}= (ω_1, ω_2) ∈ A_{\vec{P}},$ $\vec{P} = (p_1, p_2),$ $η = η_1 + η_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ with $p_1, p_2 ∈ (1, ∞).$ Furthermore, the authors show that the $[b_1, b_2, T_σ]$ is bounded from products of generalized fractional Morrey spaces $L^{p_1 ,η_1 ,\varphi} (\mathbb{R}^n)×L^{p_2,η_2,\varphi} (\mathbb{R}^n)$ into $L^{p,η,\varphi}(\mathbb{R}^n).$ As corollaries, the boundedness of the $T_σ$ and $[b_1, b_2, T_σ]$ on generalized weighted Morrey spaces $L^{p,\varphi} (ω)$ and on generalized Morrey spaces $L^{p,\varphi}(\mathbb{R}^n)$ is also obtained.