Acta Mathematica Sinica-English Series最新文献

By giving a series of estimates of multiplication operators, investigating Cauchy transforms and using L²-bounded Calderón–Zygmund operators, we provide the corona theorem with countably many functions for multiplier algebras of weighted Dirichlet spaces ({cal D}_{K}). Our result thereby gives an extension of a corona theorem with infinitely many functions from the Dirichlet space ({cal D}) to a weighted Dirichlet space ({cal D}_{K}) for a general weight function K.

In this paper, we consider a delayed Ait-Sahalia type model driven by Poisson jumps. The analytical properties including uniqueness, positiveness and boundedness of the true solution to this model are investigated. Since the coefficients are sup-linear, a tamed Euler–Maruyama (EM) method is constructed, and it is shown that the tamed EM solution converges strongly to the true solution.

Matrix-valued data have found extensive applications in various fields, such as modern biomedical imaging, chemometrics, and economics. In this paper, we address the problem of generalized trace regression involving matrix-valued covariates. To estimate the unknown parameters, we propose a penalty that combines the MCP nuclear norm and two-dimensional spline lasso. This penalty accounts for the potential low-rank and row/column order structures in the matrix-valued covariates. We establish the general theory and explicit statistical convergence rate of the resulting estimator. Through simulations, we demonstrate the advantages of our proposed method compared to other competing methods. Finally, we apply our approach to analyze the brain-image datasets related to Alzheimer’s disease, identifying several efficient regions that illustrate the mechanism of Alzheimer.

In this paper, we introduce a communication-efficient distributed estimation method tailored for massive datasets exhibiting skewness. The data are stored across multiple machines. We construct a surrogate likelihood which only need to transfer subgradient from local machines to approximate higher-order derivatives of the global likelihood. An enhanced EM algorithm is developed for computations. The proposed method not only addresses the non-normality of data by utilizing first-order gradient information in each transmission, ensuring low communication overhead, but also ensures privacy protection. Simulation studies illustrate the superior performance of the proposed methods.

In this paper, we study the vector fields X with a global Poincaré cross-section on a 2n + 1-dimensional presymplectic manifold (M, ῶ) under certain conditions. We use (M, ῶ) to construct a 2n + 2k dimensional symplectic manifold (({tilde M}, Lambda)), on which the vector field X can be extended to a Hamiltonian vector field ({tilde X}) with a smooth Hamiltonian (H:{tilde M}rightarrow R). We also consider vector fields X with a first integral F and a Jacobi multiplier J on an n-dimensional manifold (M, Ω). On a level set Σ of F, we get an n − 1-volume form ω_n on Σ and prove that X is a volume-preserving vector field with respect to ω_n. Specifically, when X is a 3 dimensional devergence-free vector field, the results have been discussed by Lerman in 2019.

In this paper, entropy and pressure are investigated for a random dynamical system φ over ℤ^k-actions on a compact metric space. The pressure P(φ, f) of φ with respect to a random continuous function f and the measure-theoretic entropy h_μ(φ) for a φ-invariant measure μ are defined. A variational principle for pressure P(φ, f) is established, which states that P(φ, f) is the supremum of the sum of h_μ(φ) and the integral of f taken over all invariant measures μ. We also obtain some basic properties for equilibrium states.

In this article, we study the hyperbolic dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. By hyperbolic dynamics, we focus on the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows on the set of rank 1 non-wandering set Ω₁ under the conditions of bounded asymptote and uniform visibility. As an application, we further discuss on some generic properties of the set of invariant probability measures.

Federated learning (FL) has becoming a prevailing paradigm which enables small-scale devices to collaboratively learn a shared model efficiently and trains a machine learning model without exchanging data. However, though the original data never leave the local machines in federated learning, possible privacy leakage still exists. To make strong privacy guarantee, in this paper, we incorporate the notion of differential privacy (DP) to study the federated averaging (FedAvg) algorithm. In particular, by adding calibrated gaussian noise, we propose a set of differentially private federated averaging algorithms (DP-FedAvg) under the full and partial participation schemes. We provide tight analysis of the privacy bound by using advanced composition and privacy amplification techniques. We also analyze the convergence bound of DP-FedAvg without having the assumptions: (i) the data are the independent identically distribution (IID), and (ii) all the devices are active. It turns out that the convergence rate is consistent with the one without DP guarantee. The effectiveness of our algorithms is demonstrated by synthetic and real datasets.

Let G be a complex, connected reductive Lie group which is the complexification of a compact Lie group K. Let M be a ℚ-Fano G-compactification. In this paper, we first prove a uniqueness result of K × K-invariant (singular) Kähler–Einstein metrics on M. Then we show the existence of (singular) Kähler–Einstein metric implies properness of the reduced Ding functional. This gives a refinement of the properness conjecture on group compactifications.

This paper discusses variable selection for interval-censored failure time data, a general type of failure time data that commonly arise in many areas such as clinical trials and follow-up studies. Although some methods have been developed in the literature for the problem, most of the existing procedures apply only to specific models. In this paper, we consider the data arising from a general class of partly linear additive generalized odds rate models and propose a penalized variable selection approach through maximizing a derived penalized likelihood function. In the method, the Bernsetin polynomials are employed to approximate both the unknown baseline hazard functions and the nonlinear covariate effects functions, and for the implementation of the method, a coordinate descent algorithm is developed. Also the asymptotic properties of the proposed estimators, including the oracle property, are established. An extensive simulation study is conducted to assess the finite-sample performance of the proposed estimators and indicates that it works well in practice. Finally, the proposed method is applied to a set of real data on Alzheimer’s disease.