In 2011, Caro et al. introduced the monochromatic connection of graphs. An edge-coloring of a connected graph $G$ is called a monochromatically connecting (MC-coloring, for short) if there is a monochromatic path joining any two vertices. The monochromatic connection number $\operatorname{mc}(G)$ of a graph $G$ is the maximum integer $k$ such that there is a $k$-edge-coloring, which is an MC-coloring of $G$. Clearly, a monochromatic spanning tree can monochromatically connect any two vertices. So for a graph $G$ of order $n$ and size $m$, $\operatorname{mc}(G) \geq m-n+2$. Caro et al. proved that both triangle-free graphs and graphs of diameter at least three meet the lower bound.

In this paper, we consider the monochromatic connectivity of graphs containing triangles which meet the lower bound too. Also, in order to study the graphs of diameter two, we present the formula for the monochromatic connectivity of join graphs. This will be helpful to solve the problem for graphs of diameter two.

In this paper, it is proved that with at most $O(N^{17/42+\varepsilon})$ exceptions, all even positive integer $n$, $n \in [N/2,N]$, can be represented in the form $p_{1}^{2} + p_{2}^{2} + p_{3}^{3} + p_{4}^{3} + p_{5}^{4} + p_{6}^{4}$, where $p_{1}$, $p_{2}$, $p_{3}$, $p_{4}$, $p_{5}$, $p_{6}$ are prime numbers. This improves a recent result $O(N^{13/16+\varepsilon})$ due to Zhang and Li [13].

Let $R$ be a commutative Noetherian ring with non-zero identity, $n$ a non-negative integer, $\mathfrak{a}$ an ideal of $R$ with $\dim(R/\mathfrak{a}) \leq n+1$, and $X$ an arbitrary $R$-module. In this paper, we prove the following results:

(i) If $X$ is an $\mathfrak{a}$-torsion $R$-module such that $\operatorname{Hom}_{R}(R/\mathfrak{a},X)$ and $\operatorname{Ext}_{R}^{1}(R/\mathfrak{a},X)$ are $\operatorname{FD}_{\lt n}$ $R$-modules, then $X$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-module;

(ii) The category of $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-modules is an Abelian category;

(iii) $\operatorname{H}^{i}_{\mathfrak{a}}(X)$ is an $(\operatorname{FD}_{\lt n},\mathfrak{a})$-cofinite $R$-module and $\{ \mathfrak{p} \in \operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X)) : \dim(R/\mathfrak{p}) \geq n \}$ is a finite set for all $i$ when $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt n}$ $R$-module for all $i$.

We observe that, among other things, $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(X))$ is a finite set for all $i$ whenever $R$ is a semi-local ring with $\dim(R/\mathfrak{a}) \leq 2$ and $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is an $\operatorname{FD}_{\lt 1}$ $R$-module for all $i$.

We discuss a generalization of the Bogomolov-Miyaoka-Yau inequality to Deligne-Mumford surfaces of general type.

We consider the existence of singular limit solutions for a nonlinear elliptic system of Liouville type with Navier boundary conditions. We use the nonlinear domain decomposition method and a Pohozaev type identity.

This paper is devoted to the well-posedness and long-time behavior of a stochastic Kirchhoff type suspension bridge equation with strong damping. The existence of the random attractor for a Kirchhoff type suspension bridge equation with white noise is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.

This paper is concerned with the global stability of non-monotone traveling waves for a discrete diffusion equation with a monostable convolution type nonlinearity. It has been proved by Yang and Zhang (Sci. China Math. 61 (2018), 1789--1806) that all noncritical traveling waves (waves with speeds $c \gt c_*$, $c_*$ is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. In this paper, we further prove that all traveling waves with large speed are globally stable, when the initial perturbations around the waves in a weighted Sobolev space can be arbitrarily large. The approaches adopted are the nonlinear Halanay's inequality, the technical weighted energy method and Fourier's transform.

Let $G$ be a locally compact group with left Haar measure. We study the closed convex left invariant subsets of $L^{\Phi}(G)$ and characterize affine mappings from the space of nonnegative functions in $L^{1}(G)$ of norm $1$ into $L^{\Phi}(G)$ spaces. We apply the results to the study of the multipliers of $L^{\Phi}(G)$. We also investigate the homological properties of $L^{\Phi}(G)$ as a Banach left $L^{1}(G)$-module such as projectivity, injectivity and flatness.

In this paper, we apply the modified potential well method and the logarithmic Sobolev inequality to study the fourth-order parabolic equation with $p$-Laplacian and logarithmic nonlinearity. Some results are obtained under the different initial data conditions. More precisely, we give the global existence of weak solution by combining the classical Galerkin's method with the modified potential well method, decay estimates, and blow-up in finite time when the initial energy is subcritical and critical, respectively. In addition, sufficient conditions for the global existence and blow-up of the weak solution are also provided for supercritical initial energy. These results extend and improve many results in the literature.

This paper deals with an inverse source problem for the multi-term time-fractional diffusion equation with a diffusion parameter by using final overdetermination. On the basis of analytic Fredholm theory, a generic well-posedness of the inverse source problem in some suitable function space is proved.

In this paper, we aim to find efficient solutions of a multi-objective optimization problem over a linear matrix inequality (LMI in short), in which the objective functions are SOS-convex polynomials. We do this by using two scalarization approaches, that is, the $\epsilon$-constraint method and the hybrid method. More precisely, we first transform the considered multi-objective optimization problem into their scalar forms by the $\epsilon$-constraint method and the hybrid method, respectively. Then, strong duality results, between each formulated scalar problem and its associated semidefinite programming dual problem, are given, respectively. Moreover, for each proposed scalar problem, we show that its optimal solution can be found by solving an associated single semidefinite programming problem, under a suitable regularity condition. As a consequence, we prove that finding efficient solutions to the considered problem can be done by employing any of the two scalarization approaches. Besides, we illustrate our methods through some nontrivial numerical examples.