Acta Mathematica Sinica-English Series最新文献

Let m, n be two positive integers, ({mathbb k}) be an algebraically closed field with ({rm char}({mathbb k}) , nmid , mn). Radford constructed an mn²-dimensional Hopf algebra R_mn(q) such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double D(R_mn(q)) of Radford Hopf algebra R_mn(q) has ribbon elements if and only if n is odd. Moreover, if m is even and n is odd, then D(R_mn(q)) has two ribbon elements, if both m and n are odd, then D(R_mn(q)) has only one ribbon element. Moreover, we compute explicitly all ribbon elements of D(R_mn(q)).

Let {X, X_n; n ≥ 1} be a sequence of i.i.d. non-degenerate real-valued random variables with ({mathbb E}{X}^{2} < infty). Let (S_{n}=sumnolimits_{i=1}^{n} X_{i}), n ≥ 1. Let g(·): [0, ∞) → [0, ∞) be a nondecreasing regularly varying function with index ρ ≥ 0 and (limnolimits_{{trightarrowinfty}} g(t)=infty). Let (mu = {mathbb E}X) and ({sigma^{2}}={mathbb E}{(X-mu)}^{2}). In this paper, on the scale g(log n), we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form (log , {mathbb P}(S_{n} - {n}mu > x{sqrt {ng(log n)})}), (log , {mathbb P}(S_{n} - {n}mu < -x{sqrt {ng(log n)})}), and (log , {mathbb P}(vert S_{n} - {n}mu vert > x{sqrt {ng(log n)})}) for all x > 0. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of X.

In this paper, we investigate Hermitian weighted composition operators on the Hardy space ({H}^{2}({{mathbb D}^{2}})) over the bidisk ({{mathbb D}^{2}}). Concretely, we characterize Hermitian weighted composition operators C_ψ,φ on ({H}^{2}({{mathbb D}^{2}})) into two classes. To our surprise, we find that φ₁ and φ₂ are depending only on one variable in each class, where φ = (φ₁, φ₂). Moreover, spectra and spectral decompositions of Hermitian weighted composition operators are described. In addition, semigroups of weighted composition operators over the bidisk are studied. Our results extend those of Cowen and Ko [Trans. Amer. Math. Soc., 362, 5771–5801 (2010)].

We study inhomogeneous projective oscillator representations of Lie superalgebras of Q-type on supersymmetric polynomial algebras. These representations are infinite-dimensional. We prove that they are completely reducible. Moreover, these modules are explicitly decomposed as direct sums of two irreducible submodules.

In this paper, we study asymptotic behavior of small perturbation for path-distribution dependent stochastic differential equations driven simultaneously by a fractional Brownian motion with Hurst parameter (H in ({1 over 2}, 1)) and a standard Brownian motion. We establish large and moderate deviation principles by utilising the weak convergence approach.

In this article, we study the limit behavior of solutions to an energy-critical complex Ginzburg–Landau equation. Via energy method, we establish a rigorous theory of the zero-dispersion limit from energy-critical complex Ginzburg–Landau equation to energy-critical nonlinear heat equation in dimensions three and four for both the defocusing and focusing cases. Furthermore, we derive the inviscid limit of energy-critical complex Ginzburg–Landau equation from energy-critical nonlinear Schrödinger equation in dimension four for the focusing case.

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.