Let $H \xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min \{ \Delta(H): H \xrightarrow{s} G \textrm{ and } \chi(H) = 2 \}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with fixed $m$. We also evaluate $br_{\Delta}(G;s)$ for paths and other trees.

Let $T$ be a tree, a vertex of degree one is a leaf of $T$ and a vertex of degree at least three is a branch vertex of $T$. The set of leaves of $T$ is denoted by $L(T)$ and the set of branch vertices of $T$ is denoted by $B(T)$. For two distinct vertices $u$, $v$ of $T$, let $P_T[u,v]$ denote the unique path in $T$ connecting $u$ and $v$. Let $T$ be a tree with $B(T) \neq \emptyset$, for each vertex $x \in L(T)$, set $y_x \in B(T)$ such that $(V(P_T[x,y_x]) \setminus \{y_x\}) \cap B(T) = \emptyset$. We delete $V(P_T[x,y_x]) \setminus \{y_x\}$ from $T$ for all $x \in L(T)$. The resulting graph is a subtree of $T$ and is denoted by $\operatorname{R\_Stem}(T)$. It is called the reducible stem of $T$. A leaf of $\operatorname{R\_Stem}(T)$ is called a peripheral branch vertex of $T$. In this paper, we give some sharp sufficient conditions on the independence number and the degree sum for a graph $G$ to have a spanning tree with few peripheral branch vertices.

Given a field $k$, a finite-dimensional $k$-algebra $A$, and a finite acyclic quiver $Q$, let $AQ$ be the path algebra of $Q$ over $A$. Then the category of representations of $Q$ over $A$ is equivalent to the category of $AQ$-modules. The main result of this paper explicitly describes the strongly Gorenstein-projective $AQ$-modules via the separated monic representations with a local strongly Gorenstein-property. As an application, a necessary and sufficient condition is given on when each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective. As a direct result, for an integer $t \geq 2$, let $A = k[x]/\langle x^t \rangle$, each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective if and only if $A = k[x]/\langle x^2 \rangle$.

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincaré polynomial of the compactified space.

We introduce the notion of generalized MR log canonical surfaces and establish the minimal model theory for generalized MR log canonical surfaces in full generality.

The purpose of this paper is to work out the theoretical aspects of the vibration problem of rubber wiper blade on convex windshield. Over the past 20 years, some $2$-dimensional spring-mass models were presented in engineering science to simulate the vibration of rubber wiper blade on windshield. In this paper, we will consider the elasticity perspective on this $3$-dimensional vibration problem. Our theoretical analysis suggests that there should exist two classes of vibration frequencies corresponding to “$\ast$-exact deformations (Class I)” and “$\ast$-closed deformations (Class II)”. We prove mathematical theorems on the characterization of deformations of Class I. We also explain how elementary deformations of Class II can be constructed. We then deduce two mathematical formulas, for the vibration problem of rubber wiper blade on convex windshield, from our theoretical analysis. Our theoretical predictions are in almost perfect agreement with experimental data. One of the crucial steps of our analysis is a decomposition theorem motivated by the de Rham Cohomology and the Hodge Theory.

Using the method of moving planes, we establish the radial symmetry of positive solutions to the fractional system \[ \begin{cases} (-\Delta)^s_p u = f(u,v), \\ (-\Delta)^t_q v = g(u,v) \end{cases} \] in the entire Euclidean space $\mathbb{R}^n$ and in the unit ball, where $0 \lt s,t \lt 1$ and $p,q \geq 2$. In particular, our result can be applied to the nonlinearities $f(u,v) \equiv u^a v^b$ and $g(u,v) \equiv u^c v^d$, where $a,d \in \mathbb{R}$ and $b,c \gt 0$.

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces \[ V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \in \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}} \in \big( \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d) \big)^r \bigg\} \] of mixed Lebesgue spaces $L^{p,q}(\mathbf{R} \times \mathbf{R}^{d})$, $1 \leq p,q \leq \infty$, where $\Phi_r = (\varphi_1, \varphi_2, \ldots, \varphi_r)^T$ with $\varphi_i \in L^{p,q}(\mathbf{R} \times \mathbf{R}^d)$ and $c = (c_1,c_2,\ldots,c_r)^T$ with $c_i \in \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d)$, $i = 1,2,\ldots,r$, under the assumption that the family $\{ \varphi_{i}(x-k_{1},y-k_{2}): (k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}, 1 \leq i \leq r \}$ constitutes a $(p,q)$-frame of $V_{p,q}(\Phi_r)$. First, iterative approximation projection algorithms for two kinds of average sampling functionals are established. Then, we estimate the convergence rates of the corresponding algorithms.

In this paper, a diffusive tumour-immune model is presented. By comparing the effect of Neumann boundary conditions and Dirichlet boundary conditions on the stability of trivial equilibrium, we derive that the former can provide more mechanisms for spatial pattern formation of the model. By taking the diffusion rate of tumour cells as a parameter, we first give the local and global steady-state bifurcations emitting from the positive equilibrium of the model. Then the stability of the bifurcation solution is discussed by computing the second derivative of an appropriate function, which is different from the general case. Furthermore, numerical simulations provide an indication of the wealth of patterns that the system can exhibit. In particular, periodic oscillation and spot-like patterns can be observed in one-dimensional and two-dimensional simulations, respectively. All results obtained reveal the mechanism of interaction between tumour cells and immune system, which have profound significance for the development of tumour immunotherapy.

In this paper, we provide an effective method to compute the topological entropies of $G$-subshifts of finite type ($G$-SFTs) with $G = F_{d}$ and $S_{d}$, the free group and free semigroup with $d$ generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of $S_{2}$-SFTs, namely the types $\mathbf{E}$, $\mathbf{D}$, $\mathbf{C}$ and $\mathbf{O}$, are introduced, and we could compute their entropies explicitly. This enables us to give the complete characterization of $S_{2}$-SFTs on two symbols. That is, the set of entropies of $S_{2}$-SFTs on two symbols is equal to $\mathbf{E} \cup \mathbf{D} \cup \mathbf{C} \cup \mathbf{O}$. The methods developed in $S_{d}$-SFTs will also apply to the study of the entropy theory of $F_{d}$-SFTs. The entropy formulae of $S_{d}$-, $F_{d}$-golden mean shifts and $k$-colored chessboards are also presented herein.

The influence of the geometry of the domain on the behavior of generalized solutions of Dirichlet problems for elliptic partial differential equations has been an important subject for over a century. We investigate the boundary behavior of variational solutions $f$ of Dirichlet problems for prescribed mean curvature equations in a domain $\Omega \subset \mathbb{R}^{2}$ near a point $\mathcal{O} \in \partial \Omega$ under different assumptions about the curvature of $\partial \Omega$ on each side of $\mathcal{O}$. We prove that the radial limits at $\mathcal{O}$ of $f$ exist under different assumptions about the Dirichlet boundary data $\phi$, depending on the curvature properties of $\partial \Omega$ near $\mathcal{O}$.

This paper is concerned with the biholomorphism of two equidimensional Hartogs type domains over irreducible bounded symmetric domains $H_{\Omega}(p)$ (see (1.1)) which is a Hua construction of Cartan–Hartogs domain. We develop a new simple methods to give an sufficient and necessary condition for the two Hua domains to be biholomorphic equivalent by using the function $\mathcal{L}_{\Omega}(z,\omega)$ (see (1.2)). Furthermore as an application, we can also give an equivalent description for the automorphism of Hua domain.

In the paper a second order differential system with a periodic boundary data is considered. Using some variational methods sufficient conditions for the continuous dependence of trajectories on controls are proved. The obtained results are applied then to the optimal control problem governed by the above system and the cost functional of a Bolza-type. In the end of the paper, an example of periodic optimal control problem demonstrating the applicability of the results is presented.