Acta Mathematica Sinica-English Series最新文献

First we investigate relative n-regionally proximal tuples. Let π: (X, G) → (Y, G) be a Bronstein extension between minimal systems. It turns out that if (x₁,…, x_n) is a minimal point and (x_i, x_i+1) is relative regionally proximal for 1 ≤ i ≤ n − 1, then (x₁,…, x_n) is relative n-regionally proximal. We consider the relative versions of sensitivity, including relative n-sensitivity and relative block ℱ_t-n-sensitivity, where ℱ_t is the family of thick sets. We show that π is relatively n-sensitive if and only if the relative n-regionally proximal relation contains a point whose coordinates are distinct, and the structure of π which is relatively n-sensitive but not relatively n + 1-sensitive is determined. We also characterize relatively block ℱ_t-n-sensitive via relative regionally proximal tuples.

We prove the observability inequalities at two time points for the Schrödinger equation in a uniform magnetic field in dimensions 2 and 3. The proofs mainly rely on Nazarov’s uncertainty principle. In particular, the observability inequality in three dimensions can also be derived from the approach used to establish the Amerin–Berthier uncertainty principle.

In this paper, we study the p-Laplacian Choquard equation

on a finite lattice graph Nⁿ with n ∈ ℕ₊, where p > 1, q > 1 and 0 ≤ α ≤ n are some constants, V(x) is a positive function on Nⁿ. Using the Nehari method, we prove that if 1 < p < q < +∞, then the above equation admits a ground state solution. Previously, the p-Laplacian Choquard equation on finite lattice graph has not been studied, and our result contains the critical cases α = 0 and α = n, which further improves the study of Choquard equations on lattice graphs.

In this paper, we explore the convergence and convergence rate results for a new methodology termed the half-proximal symmetric splitting method (HPSSM). This method is designed to address linearly constrained two-block non-convex separable optimization problem. It integrates a half-proximal term within its first subproblem to cancel out complicated terms in applications where the subproblem is not easy to solve or lacks a simple closed-form solution. To further enhance adaptability in selecting relaxation factor thresholds during the two Lagrange multiplier update steps, we strategically incorporate a relaxation factor as a disturbance parameter within the iterative process of the second subproblem. Building on several foundational assumptions, we establish the subsequential convergence, global convergence, and iteration complexity of HPSSM. Assuming the presence of the Kurdyka-Łojasiewicz inequality of Łojasiewicz-type within the augmented Lagrangian function (ALF), we derive the convergence rates for both the ALF sequence and the iterative sequence. To substantiate the effectiveness of HPSSM, sufficient numerical experiments are conducted. Moreover, expanding upon the two-block iterative scheme, we present the theoretical results for the symmetric splitting method when applied to a three-block case.

Consider a branching process {Z_n}_n≥0 with immigration in varying environments. For a ∈ {0, 1, 2, …}, let C(a) = {n ≥ 0: Z_n = a} be the collection of times at which the population size of the process attains level a. We give a criterion to determine whether the set C(a) is finite or not. For the critical Galton–Watson process, based on a moment method, we show that ({{| {C(a) cap [1,n]} |} over {log ;n to S}}) in distribution, where S is an exponentially distributed random variable with P(S > t) = e^−t, t > 0.

Let A be an Artin algebra and M be a presilting radical complex. We show that M is silting provided its some left part or some right part is silting.

We consider β-Jacobi ensembles with parameters p₁, p₂ ≥ n. We prove that the empirical measure of the rescaled β-Jacobi ensembles converges weakly to a modified Wachter law via the spectral measure method. We also provide the central limit theorem and the large deviation for the corresponding rescaled spectral measure.

In this paper, we formulate two new families of fourth-order explicit exponential Runge–Kutta (ERK) methods with four stages for solving first-order differential systems y′(t)+ My(t) = f(y(t)). The order conditions of these ERK methods are derived by comparing the Taylor series of the exact solution, which are exactly identical to the order conditions of explicit Runge–Kutta methods, and these ERK methods reduce to classical Runge–Kutta methods once M → 0. Moreover, we analyze the stability properties and the convergence of these new methods. Several numerical examples are implemented to illustrate the accuracy and efficiency of these ERK methods by comparison with standard exponential integrators.

Perfect and complete Lie conformal algebras will be discussed in this paper. We give the characterizations of complete Lie conformal algebras. We demonstrate that every perfectly complete Lie conformal algebra can be uniquely decomposed to a direct sum of indecomposable perfectly complete ideals. And we show the existence of a sympathetic decomposition in every perfect Lie conformal algebra. Finally, we study a class of ideals of Lie conformal algebras such that the quotients are perfectly complete Lie conformal algebras.

In this paper, we study the rigidity of k(≥ 1)-extremal submanifolds in a sphere and prove various pinching theorems under different curvature conditions, including sectional and Ricci curvatures in pointwise and integral sense.