Acta Mathematica Sinica-English Series最新文献

In this paper we will be concerned with the problem

where V is a potential continuous and f: ℝ → ℝ is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method in order to show existence of nonnegative solutions and existence of nodal solutions. Our results complement the classical result of “Solutions for quasilinear Schrdinger equations via the Nehari method” due to Jia–Quan Liu, Ya–Qi Wang and Zhi-Qiang Wang in the sense that in this article we are considering nonlinearity of the exponential type.

Let {X_n}_n≥0 be a p-type (p ≥ 2) supercritical branching process with immigration and mean matrix M. Suppose that M is positively regular and ρ is the maximal eigenvalue of M with the corresponding left and right eigenvectors v and u. Let ρ > 1 and (Y_{n}=rho^{-n}left[{bf u}cdot{X}_{n}-{{{rho}^{n+1}-1} over {rho}-1}left({boldsymbol u} cdot {boldsymbol lambda}right)right]), where the vector λ denotes the mean immigration rate. In this paper, we will show that Y_n is a martingale and converges to an r.v. Y as n → ∞. We study the rates of convergence to 0 as n → ∞ of

for any ε > 0, i = 1,…,p, 1 = (1,…,1) and l ∈ ℝ^p, the p-dimensional Euclidean space. It is shown that under certain moment conditions, the first two decay geometrically, while conditionally on the event Y ≥ α (α > 0) supergeometrically. The decay rate of the last probability is always supergeometric under a finite moment generating function assumption.

We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.

In this paper, we introduce the spectral Einstein functional for perturbations of Dirac operators on manifolds with boundary. Furthermore, we provide the proof of the Dabrowski–Sitarz–Zalecki type theorems associated with the spectral Einstein functionals for perturbations of Dirac operators, particularly in the cases of on 4-dimensional manifolds with boundary.

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.