Let $X$ be a complex nonsingular projective surface and let $L$ be an ample line bundle on $X$. We study multi-point Seshadri constants of $L$ at singular points of certain connected arrangements of curves on $X$. We pose some questions about such Seshadri constants and prove results in many cases, including star arrangements of curves. We also study the configurational Seshadri constants for curve arrangements on surfaces and compare them with the usual Seshadri constants. We give several examples illustrating the properties that we study.

In this paper, we report on the $\tau$-tilting finiteness of some classes of finite-dimensional algebras over an algebraically closed field, including symmetric algebras of polynomial growth, $0$-Hecke algebras and $0$-Schur algebras. Consequently, we find that derived equivalence preserves the $\tau$-tilting finiteness over symmetric algebras of polynomial growth, and self-injective cellular algebras of polynomial growth are $\tau$-tilting finite. Furthermore, the representation-finiteness and $\tau$-tilting finiteness over $0$-Hecke algebras and $0$-Schur algebras (with few exceptions) coincide.

In this article, we investigate a high-order numerical method for the variable-order (VO) subdiffusion equation with the Caputo–Hadamard derivative. The temporal variable and the spatial variable are discretized by the finite difference method and the local discontinuous Galerkin method, respectively. Furthermore, for all variable-order $\alpha(t) \in (0,1)$, the stability and the optimal error estimates are proved for the presented scheme. Finally, several numerical tests are given to demonstrate optimal rates of convergence and show the efficiency of the method.

In this paper, it is considered an inverse space-dependent source problem of time-space fractional diffusion equation from the noisy final data in a bounded domain. Such a problem is mildly ill-posed. A new regularization method called the exponential Tikhonov method with a parameter $\gamma$ is utilized to solve the problem, and its convergence rates are analyzed under an a-priori and an a-posteriori regularization parameter choice rule. A novel result indicates that the optimal convergence rate can be obtained and it is independent of the regularity information of the unknown source term when $\gamma$ is less than or equal to zero. However, when $\gamma$ is greater than zero, the optimal convergence rate depends on the value of $\gamma$ related to the regularity of the unknown source but it does not have convergence saturation limit and can theoretically approach 1, which is superior to Tikhonov's regularization framework using the usual Sobolev space norm as a penalty term in a minimized functional. Numerical examples show that the proposed regularization method is effective and stable, and both parameter choice rules work well.

In this work, a biharmonic elliptic system is investigated, which involves a Rellich-type potential and a homogeneous term of a critical Rellich–Sobolev degree. By using the variational methods and analytic techniques, we obtain multiplicity result of nontrivial solutions for the system.

Consider nonautonomous evolution equation $\dot{u} = A(t) u(t) + Bg(u)(t)$ in which the family of operators $(A(t))_{t \geq 0}$ generates the evolution family $(\mathcal{U}(t,s))_{t \geq s \geq 0}$ of $(X,Y,\varphi)$-type, i.e., $\|\mathcal{U}(t,0)x\|_{Y} \leq \varphi(t) \|x\|_{X}$, $t > 0$, for certain couple of Banach spaces $(X,Y)$ and real-valued, positive function $\varphi$ satisfying $\lim_{t \to \infty} \varphi(t) = 0$. Inspired by Serrin's technique, we develop a unified approach toward the problems on the existence of periodic solutions to above equation. As illustrations of our abstract results, we give applications to the existence and uniqueness of periodic solutions to Oseen–Navier–Stokes and damped wave equations, as well as the existence of local stable manifolds nearby the periodic solution to the damped wave equations.

The aim of this paper is to give existence and uniqueness results for solutions to the Cauchy problem of semilinear heat equations on stratified Lie groups $\mathbb{G}$ with the small initial data belonging to the critical Sobolev space. We consider a power type nonlinearity that behaves like $|u|^{\alpha}$ or $|u|^{\alpha-1}u$ ($\alpha > 1$).

In this paper we consider the relationship between the number of separated maximum points of an entire curve and the Baernstein's $T^{*}$-function. The results of Edrei, Goldberg, Krytov, Ostrovskii, Teichmüller are generalized. We also give example showing that the obtained estimate is sharp.

In this paper, we propose a penalty dual-primal balanced-based augmented Lagrangian method for solving linearly constrained convex minimization problems. Convergence and convergence rate of the penalty dual-primal balanced-based augmented Lagrangian method are established by the tool of variational inequality. Further, we generalize the penalty dual-primal balanced-based augmented Lagrangian method to solve linearly constrained multi-block separable convex minimization problems with full splitting technique and partial splitting technique. Numerical results on the basic pursuit problem and the Lasso model are presented to illustrate the efficiency of the proposed methods.

In this research article, a concept of $\varepsilon$-quasi subdifferential for locally Lipschitz functions is proposed. Calculuses of scalar product rule and sum rule for $\varepsilon$-quasi subdifferentials are investigated. A notion of $\varepsilon$-quasi normal set is introduced and its properties are presented. Based on the obtained results, optimality conditions for $\varepsilon$-quasi solutions in Karush–Kuhn–Tucker type of some classes of nonsmooth optimization problems are established. Several illustrative examples are also given.