Acta Mathematica Sinica-English Series最新文献

Given a finite tensor category ({cal C}), an exact indecomposable ({cal C})-module category ({cal M}), and a tensor subcategory ({cal D} subseteq {cal C}_{cal M}^{ast}), we describe a way to produce exact commutative algebras in the center (Z({cal C})), measuring this inclusion. The construction of such algebras is done in an analogous way as presented by Shimizu [20], but using instead the relative (co)end, a categorical tool developed in [1] in the realm of representations of tensor categories. We provide some explicit computations.

Let V ⋃_S W be a Heegaard splitting of M with distance n ≥ 2 and F a boundary component of ∂_V. A simple closed curve J in F is called distance degenerating if the distance of M_J = V_J ⋃_S W is less than n, where M_J is obtained by attaching a 2-handle to M along J. In this paper, by considering the distance between J and the image of the essential disks of W under the projection map, we obtain a sufficient condition for the diameter of the set of distance degenerating curves in F to be bounded in C(F). Moreover, for F = ∂M, an upper bound of the diameter of the set of the boundary reducible curves in F is given under some circumstance.

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton–Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton–Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.

In this paper, we give a slight improvement of El Soufi–Ilias–Ros’s upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class. We also optimize Montiel–Ros’s argument to obtain a better upper bound of the conformal area for certain rectangular tori.

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.