Acta Mathematica Sinica-English Series最新文献

Let (mathbb{B}) be a unit ball in ℝ², (W_{0}^{1,2}(mathbb{B})) be the standard Sobolev space. For any ϵ > 0, de Figueiredo, do Ó, dos Santons, Yang and Zhu proved the existence of extremals of a Trudinger-Moser inequality in the unit ball. Precisely,

can be attained by some radially symmetric function (u_{epsilon}in W_{0}^{1,2}(mathbb{B})) with (int_{mathbb{B}}vertnabla u_{epsilon}vert^{2}dx=1). In this note, we concern the compactness of the function family {u_ϵ}_ϵ>0 and prove that up to a subsequence u_ϵ converges to some function u₀ in (C^{1}(overline{mathbb{B}})) as ϵ → 0. Furthermore, u₀ is an extremal function of the supremum

Let us explain the result in geometry. Denote (omega_{0}=dx_{1}^{2}+dx_{2}^{2}) be the standard Euclidean metric. Define a conical metric (omega_{epsilon}=vert xvert^{2epsilon}omega_{0}) for (xinmathbb{B}). Then the extremal family {u_ϵ}_ϵ>0 of the following Trudinger-Moser functionals

under the constraint (uin W_{0}^{1,2}(mathbb{B})) and (int_{mathbb{B}}vertnabla_{omega_{epsilon}u}vert^{2}dv_{omega_{epsilon}}leq 1) is compact as ϵ → 0.

Let L/F be a finite Galois extension of number fields of degree n and let p be a prime which does not divide n. We shall study the p^j-rank of (K_{2i}(mathcal{O}_{L})) via its Galois module structure following the approaches of Iwasawa and Komatsu–Nakano. Along the way, we generalize previous observations of Browkin, Wu and Zhou on K₂-groups to higher even K-groups. We also give examples to illustrate our results. Finally, we apply our discussion to refine a result of Kitajima pertaining to the p-rank of even K-groups in the cyclotomic ℤ_l-extension, where l ≠ p.

This paper focuses on a question raised by Holm and Jørgensen, who asked if the induced cotorsion pairs (Φ(X), Φ(X)^⊥) and (^⊥Ψ(Y), Ψ(Y)) in Rep(Q, A)—the category of all A-valued representations of a quiver Q—are complete whenever (X, Y) is a complete cotorsion pair in an abelian category A satisfying some mild conditions. We give an affirmative answer if the quiver Q is rooted. As an application, we show under certain mild conditions that if a subcategory L, which is not necessarily closed under direct summands, of A is special precovering (resp., preenveloping), then Φ(L)(resp., Ψ(L)) is special precovering (resp., preenveloping) in Rep(Q, A).

Let f be any arithmetic function and define ({S_{f}}(x):=sumnolimits_{{n le x}}f([x/n])). If the function f is small, namely, f(n) ≪ n^ε, then the error term E_f(x) in the asymptotic formula of S_f(x) has the form O(x^1/2+ε). In this paper, we shall study the mean square of E_f(x) and establish some new results of E_f(x) for some special functions.

Using methods from complex analysis in one variable, we define an integral operator that solves ({bar partial}) with supnorm estimates on product domains in ℂⁿ.

In this paper, we studied the metric mean dimension in Feldman–Katok (FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subsets. And we established two variational principles.

A model space is a subspace of the Hardy space which is invariant under the backward shift, and a truncated Toeplitz operator is the compression of a Toeplitz operator on some model space. In this paper we prove a necessary and sufficient condition for the commutator of two truncated Toeplitz operators on a model space to be compact.

The distributional properties of a multi-dimensional continuous-state branching process are determined by its cumulant semigroup, which is defined by the backward differential equation. We provide a proof of the assertion of Rhyzhov and Skorokhod (Theory Probab. Appl., 1970) on the uniqueness of the solutions to the equation, which is based on a characterization of the process as the pathwise unique solution to a system of stochastic equations.

In this paper, we consider the reducibility of three-dimensional skew symmetric systems. We obtain a reducibility result if the base frequency is high-dimensional weak Liouvillean and the parameter is sufficiently small. The proof is based on a modified KAM theory for 3-dimensional skew symmetric systems.

By generalizing a criterion of Mufa Chen for Markov jump processes, we establish the necessary and sufficient conditions for the extinction, explosion and coming down from infinity of a continuous-state nonlinear Neveu’s branching process.