Acta Mathematica Sinica-English Series最新文献

In this paper, we study asymptotic properties of the approximated maximum likelihood estimator (MLE) for the drift coefficient in an Ornstein-Uhlenbeck process with discrete observations. By the change of measure method and asymptotic analysis technique, we establish an exponential nonuniform Berry-Esseen bound of the approximated MLE. Then, the Cramér-type moderate deviation can be obtained. As applications, the global and local powers for the hypothesis test are shown to approach one at exponential rates. Simulation experiments are conducted to confirm the theoretical results.

In this paper, we study the local well-posedness of classical solutions to the ideal Hall–MHD equations whose magnetic field is supposed to be azimuthal in the L²-based Sobolev spaces. By introducing a good unknown coupling with the original unknowns, we overcome difficulties arising from the lack of magnetic resistance, and establish a self-closed H^m with (3 ≤ m ∈ ℕ) local energy estimate of the system. Here, a key cancellation related to θ derivatives is discovered. In order to apply this cancellation, part of the high-order energy estimates is performed in the cylindrical coordinate system, even though our solution is not assumed to be axially symmetric. During the proof, high-order derivative tensors of unknowns in the cylindrical coordinates system are carefully calculated, which would be useful in further researches on related topics.

We provide three types of invariance pressure for uncertain control systems, namely, invariance pressure, strong invariance pressure, and invariance feedback pressure. The first two respectively extend the corresponding pressures for deterministic control systems proposed by Colonius, Cossich, and Santana (2018) and by Nie, Wang, and Huang (2022); and the third generalizes invariance feedback entropy of uncertain control systems presented by Tomar, Rungger, and Zamani (2020), by adding potentials on the control range. Then we prove that (1) an explicit formula for invariance pressure of a controlled invariant set with respect to a potential by the logarithm of the spectral radius of the admissible weighted matrix determined by this potential under some suitable conditions; (2) an explicit formula for pressure of invariant quasi-partitions by maximum mean weight over all irreducible periodic sequences; (3) the invariance feedback pressure of a controlled invariant set is equal to the pressure of an atom partition under some technical assumptions; (4) lower and upper bounds for pressure of invariant quasi-partitions w.r.t. a potential by the logarithm of the spectral radius of the weighted adjacency matrix determined by this potential; (5) a variational principle for strong invariance pressure.

In this paper, we develop some new variational and analytic techniques to study the multiplicity and concentration of positive solutions for a planar Schrödinger-Poisson system involving competing weight potentials and the nonlinearity K(x)∣u∣^p−2u (2 < p < 4) in ℝ². By Nehari manifold and Ljusternik-Schnirelmann category, we relate the number of positive solutions to the category of the global minima set of a suitable ground energy function. Our results improve and extend the ones in [Du, Weth, Nonlinearity, 30, 3492–3515 (2017)] and [Chen, Tang, J. Differ. Equ., 268, 945–976 (2020)]. In particular, we do not need the assumption K(x) ≡ 1 and the C¹ smoothness of V(x). Furthermore, we do not use the axially symmetric condition of the potential in our second main result. Moreover, we shall show that there is a great difference in our results between N = 2 and N ≥ 3.

We construct a new class of subspace lattices ({cal L}) on an infinite-dimensional Hilbert space ({cal K}). We show that the bounded cohomology groups (H^{n}({rm Alg} , {cal L},,{cal B}({cal K}))) of the corresponding lattice algebras Alg ({cal L}) with coefficients in ({cal B}({cal K})) are trivial for all n ≥ 1, and every derivation ϕ from Alg ({cal L}) into Alg ({cal L}) is an inner derivation under some conditions. We also prove that every Lie derivation δ from Alg ({cal L}) into ({cal B}({cal K})) is standard.

Polya–Carlson theorem asserts that if a power series with integer coefficients and convergence radius 1 can be extended holomorphically out of the unit disc, it must represent a rational function. In this note, we give a generalization of this result to multivariate case and give an application to rationality theorem about D-finite power series.

We construct an explicit example of a smooth isotopy {ξ_t}_t∈[0,1] of volume- and orientation-preserving diffeomorphisms on [0, 1]ⁿ (n ≥ 3) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of (M, g), a “topologically complicated” Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with {ξ_t} at t = 0 and t = 1 but of finite total kinetic energy.

DeBiasio and Krueger showed the following result: For all 0 ≤ δ ≤ 1 and ϵ > 0, there exists n₀ such that if G is a balanced bipartite graph on 2n ≥ 2n₀ vertices with δ(G) = δn, then in every 2-coloring of G, there exists a monochromatic cycle of order at least (f(δ) − ϵ)n, where

Zhang and Peng (2023) extended the above result to off-diagonal cases when ({delta} > {3 over 4}). In this paper, we relax the condition ({delta} > {3 over 4}) to ({delta} > {2 over 3}). We show the following result: For every η > 0, there exists a positive integer N₀ such that for every integer N > N₀ the following holds. Let ({2 over 3} < {delta} leq {3 over 4}), and let ({alpha_1} geq {{deltaalpha}_{2} over {3delta - 2}} > 0) such that α₁ + α₂ = 1. Let G[X, Y] be a balanced bipartite graph on 2N vertices with δ(G) = (δ + 3η)N. Then for each red-blue-edge-coloring of G, either there exist red even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3η²)α₁N}, or there exist blue even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3δ²)α₂N}. There are constructions of colorings showing that the length of a longest monochromatic cycle is asymptotically tight and the condition ({alpha_1} geq {{deltaalpha}_{2} over {3delta - 2}}) cannot be removed.

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl–Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

We study the L^p boundedness of singular integral operators along surfaces of revolution on product spaces. The L^p boundedness for these operators are obtained under very weak conditions on kernels. Our results are new and they improve several previously known results. Furthermore, they are natural extensions of many known results on singular integrals in the one-parameter setting and they subsume many other corresponding results on the product space setting.