An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n$; (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r$; (3) the minimum order of a $k$-regular almost self-centered graph; (4) the maximum size of an almost peripheral graph of order $n$; (5) possible maximum degrees of an almost peripheral graph of order $n$ and (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ with maximum degree $n-4$ which is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.

In this paper, we consider a depth inequality of Auslander which holds for finitely generated Tor-rigid modules over commutative Noetherian local rings. We raise the question of whether such a depth inequality can be extended for $n$-Tor-rigid modules, and obtain an affirmative answer for $2$-Tor-rigid modules that are generically free. Furthermore, in the appendix, we use Dao's eta function and determine new classes of Tor-rigid modules over hypersurfaces that are quotient of unramified regular local rings.

In the paper, we study an inverse problem of recovering a time-dependent convection coefficient from the measured data at an interior/boundary point in a one-dimensional nonlinear subdiffusion model with non-homogeneous boundary conditions. Due to the nonlinearity and non-homogeneous boundary conditions of the system, such an inverse problem is novel and important. We first investigate the unique existence and some regularities of the solution to forward problem by using the transposition method and the fixed point theorem. Then a conditional stability of the inverse problem is obtained based on the regularity of solution for the direct problem and some generalized Gronwall's inequalities. Finally, we transform the inverse problem into a variational problem. The existence and convergence of the regularization solution for the variational problem are proved and we use a modified Levenberg–Marquardt method to find an approximate convection coefficient function. The efficiency and accuracy of the algorithm are illustrated with two numerical examples.

RK-RK and REK-RK methods are the two latest while very effective randomized iteration solvers for factorized linear system $UV \beta = y$ by interlacing randomized Kaczmarz (RK) and randomized extended Kaczmarz (REK) updates. This paper considers two latest randomized iterative methods for solving large-scale linear systems and linear least-squares problems—greedy randomized Kaczmarz (GRK) method and greedy randomized Gauss–Seidel (GRGS) method. By introducing a relaxation parameter $\omega$ into the iterates of GRK and GRGS, we construct relaxed GRK and GRGS methods, respectively. In addition, by interlacing their updates, we propose relaxed GRK-GRK and GRGS-GRK methods to solve consistent and inconsistent factorized linear systems, respectively. We prove the exponential convergence of these two interlaced methods and show that relaxed GRK-GRK and GRGS-GRK can be more efficient than RK-RK and REK-RK, respectively, if the relaxation parameters are chosen appropriately.

The uniform stabilization of a semilinear wave equation with variable coefficients and nonlinear boundary conditions is considered. The uniform decay rate is established by the Riemannian geometry method.

We study the local-in-time well-posedness of Vlasov–Poisson equation in Besov space for the large initial data. To accomplish it, we establish commutator estimates in Besov space which are quite useful in dealing with the electronic term $\nabla_{x} \phi$. Also, the $L^p$-$L^q$ type estimates for the electronic term $\nabla_{x} \phi$ are established, which are not only useful in the estimate for Poisson equation, but also play a fundamentally important role in commutator estimates involving the electronic term $\nabla_{x} \phi$.

In this paper, we investigate the $L^{2}$ boundedness of Fourier integral operator $T_{\phi,a}$ with rough symbol $a \in L^{\infty} S^{m}_{\rho}$ and rough phase $\phi \in L^{\infty} \Phi^{2}$ which satisfies $\big| \{ x: |\nabla_{\xi} \phi(x,\xi) - y| \leq r \} \big| \leq C(r^{n-1}+r^{n})$ for any $\xi,y \in \mathbb{R}^{n}$ and $r \gt 0$. We obtain that $T_{\phi,a}$ is bounded on $L^2$ if $m \lt \rho(n-1)/2 - n/2$ when $0 \leq \rho \leq 1/2$ or $m \lt -(n+1)/4$ when $1/2 \leq \rho \leq 1$. When $\rho = 0$ or $n = 1$, the condition of $m$ is sharp. Moreover, the maximal wave operator is a special class of $T_{\phi,a}$ which is studied in this paper. Thus, our main theorem substantially extends and improves some known results about the maximal wave operator.

We consider how distribution of eigenvalues depends on boundary conditions of a discrete Laplacian operator on lattices. We study the Laplacian with boundary conditions given by a linear combination of Dirichlet and Neumann conditions. In particular, we derive a secular equation and investigate the Laplacian operator's eigenvalues with different boundary conditions, including the interlacing property, the first eigenvalue gaps, and the monotonicity property.

In this paper, we concentrate on the generalized multiple-set split feasibility problems in Hilbert spaces and propose a new iterative method for this problem. One of the most important of this method is using dynamic step-sizes, in which the information of the previous step is the only requirement to compute the next approximation. The strong convergence result of the suggested algorithm is proven theoretically under some feasible assumptions. When considering the main results in some special cases, we also obtain some applications regarding the solution of the multiple-set split feasibility problem, the split feasibility problem with multiple output sets, and the split feasibility problem as well as the linear optimal control problem. Some numerical experiments on infinite-dimensional spaces and applications in optimal control problems are conducted to demonstrate the advantages and computational efficiency of the proposed algorithms over some existing results.