Acta Mathematica Sinica-English Series最新文献

(mathfrak{L}_{nu}) operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of (mathfrak{L}_{nu}^{2}) operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of (mathfrak{L}_{nu}^{2}) operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the (mathcal{L}_{II}) operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate.

On Hom-Lie algebras and superalgebras, we introduce the notions of biderivations and linear commuting maps, and compute them for some typical Hom-Lie algebras and superalgebras, including the q-deformed W(2,2) algebra, the q-deformed Witt algebra and superalgebra.

After reviewing Grunsky operator and Faber operator acting on Dirichlet space, we discuss the boundedness of Faber operator on BMOA, a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space. In particular, we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space. Meanwhile, we obtain several results on quasiconformal mappings, BMO-Teichmüller space and chord-arc curves as well. As by-products, this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.

By comprehensive utilizing of the geometry structure of 2D Burgers equation and the stochastic noise, we find the decay properties of the solution to the stochastic 2D Burgers equation with Dirichlet boundary conditions. Consequently, the expected ergodicity for this turbulence model is established.

In this article, we study deformations of conjugate self-dual Galois representations. The study is twofold. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.

Let G be a symplectic or orthogonal group defined over a finite field with odd characteristic and let D ≤ G be a Sylow 2-subgroup. In this paper, we classify the essential 2-subgroups and determine the essential 2-rank of the Frobenius category ℱ_D(G). Together with the results of An-Dietrich and Cao–An–Zeng, this completes the work of essential subgroups and essential ranks of classical groups.

Let {X_υ: υ ∈ ℤ^d} be i.i.d. random variables. Let (S(pi) = sumnolimits_{upsilonin pi} {{X_upsilon}}) be the weight of a self-avoiding lattice path π. Let

We are interested in the asymptotics of M_n as n → ∞. This model is closely related to the first passage percolation when the weights {X_υ: υ ∈ ℤ^d} are non-positive and it is closely related to the last passage percolation when the weights {X_υ, υ ∈ ℤ^d} are non-negative. For general weights, this model could be viewed as an interpolation between first passage models and last passage models. Besides, this model is also closely related to a variant of the position of right-most particles of branching random walks. Under the two assumptions that (exists alpha > 0,,E{(X_0^ + )^d}{({log ^ + }X_0^ + )^{d + alpha }} < + ,infty) and that (E[X_0^ - ] < + ,infty), we prove that there exists a finite real number M such that M_n/n converges to a deterministic constant M in L¹ as n tends to infinity. And under the stronger assumptions that (exists alpha > 0,,,E{(X_0^ + )^d}{({log ^ + },X_0^ + )^{d + alpha }} < , + ,infty) and that (E[{(X_0^ - )^4}] < , + ,infty), we prove that M_n/n converges to the same constant M almost surely as n tends to infinity.

In this correspondence, we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements. Most of the results pertain to random elements which are M-dependent. We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces. One of the main contributions of the paper is to simplify and improve a recent result of Li, Presnell, and Rosalsky [Journal of Mathematical Inequalities, 16, 117–126 (2022)]. A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest. The sharpness of the results is illustrated by four examples.

In this paper, we study compressed data separation (CDS) problem, i.e., sparse data separation from a few linear random measurements. We propose the nonconvex ℓ_q-split analysis with ℓ_∞-constraint and 0 < q ≤ 1. We call the algorithm ℓ_q-split-analysis Dantzig selector (ℓ_q-split-analysis DS). We show that the two distinct subcomponents that are approximately sparse in terms of two different dictionaries could be stably approximated via the ℓ_q-split-analysis DS, provided that the measurement matrix satisfies either a classical D-RIP (Restricted Isometry Property with respect to Dictionaries and ℓ₂ norm) or a relatively new (D, q)-RIP (RIP with respect to Dictionaries and ℓ_q-quasi norm) condition and the two different dictionaries satisfy a mutual coherence condition between them. For the Gaussian random measurements, the measurement number needed for the (D, q)-RIP condition is far less than those needed for the D-RIP condition and the (D, 1)-RIP condition when q is small enough.

The penalized variable selection methods are often used to select the relevant covariates and estimate the unknown regression coefficients simultaneously, but these existing methods may fail to be consistent for the setting with highly correlated covariates. In this paper, the semi-standard partial covariance (SPAC) method with Lasso penalty is proposed to study the generalized linear model with highly correlated covariates, and the consistencies of the estimation and variable selection are shown in high-dimensional settings under some regularity conditions. Some simulation studies and an analysis of colon tumor dataset are carried out to show that the proposed method performs better in addressing highly correlated problem than the traditional penalized variable selection methods.