Acta Mathematica Sinica-English Series最新文献

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.

In this paper, the entropy of discrete Heisenberg group actions is considered. Let α be a discrete Heisenberg group action on a compact metric space X. Two types of entropies, (tilde{h}(alpha)) and h(α) are introduced, in which (tilde{h}(alpha)) is defined in Ruelle’s way and h(α) is defined via the natural extension of α. It is shown that when X is the torus and α is induced by integer matrices then (tilde{h}(alpha)) is zero and h(α) can be expressed via the eigenvalues of the matrices.

Using operator algebras, we extend the theory of quantum computation on a graph to a theory of computation on an arbitrary topological space. Quantum computation is usually implemented on finite discrete sets, and the purpose of this study is to extend this to theories on arbitrary sets. The conventional theory of quantum computers can be viewed as a simplified algebraic geometry theory in which the action of SU(2) is defined on each point of a discrete set. In this study, we extend this in general as a theory of quantum fibrations in which the action of the von Neumann algebra is defined on an arbitrary topological space. The quantum channel is then naturally extended as a net of von Neumann algebras. This allows for a more mathematically rigorous discussion of general theories, including physics and chemistry, which are defined on sets that are not necessarily discrete, from the perspective of quantum computer science.