Acta Mathematica Sinica-English Series最新文献

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.

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.

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.

How to analyze flocking behaviors of a multi-agent system with local interaction functions is a challenging problem in theory. Motsch and Tadmor in 2011 also stressed the significance to assume that the interaction function is rapidly decaying or cut-off at a finite distance (cf. Motsch and Tadmor in J. Stat. Phys. 2011). In this paper, we study the flocking behavior of a Cucker–Smale type model with compactly supported interaction functions. Using properties of a connected stochastic matrix, together with an elaborate analysis on perturbations of a linearized system, we obtain a sufficient condition imposed only on model parameters and initial data to guarantee flocking. Moreover, it is shown that the system achieves flocking at an exponential rate.

Let π: (X, T) → (Y, S) be a factor map between two topological dynamical systems, and (cal{F}) a Furstenberg family of ℤ. We introduce the notion of relative broken (cal{F})-sensitivity. Let (cal{F}_{s}) (resp. (cal{F}_{text{pubd}},cal{F}_{text{inf}})) be the families consisting of all syndetic subsets (resp. positive upper Banach density subsets, infinite subsets). We show that for a factor map π: (X, T) → (Y, S) between transitive systems, π is relatively broken (cal{F})-sensitive for (cal{F}=cal{F}_{s}) or (cal{F}_{text{pubd}}) if and only if there exists a relative sensitive pair which is an (cal{F})-recurrent point of (R_π, T⁽²⁾); is relatively broken (cal{F}_{text{inf}})-sensitive if and only if there exists a relative sensitive pair which is not asymptotic. For a factor map π: (X, T) → (Y, S) between minimal systems, we get the structure of relative broken (cal{F})-sensitivity by the factor map to its maximal equicontinuous factor.

In this paper, we derive the optimal Cauchy–Schwarz inequalities on a class of Hilbert and Krein modules over a Clifford algebra, which heavily depend on the Clifford algebraic structure. The obtained inequalities further lead to very general uncertainty inequalities on these modules. Some new phenomena arise, due to the non-commutative nature, the Clifford-valued inner products and the Krein geometry. Taking into account applications, special attention is given to the Dirac operator and the Howe dual pair (text{Pin}(m)timesmathfrak{osp}(1vert2)). Moreover, it is surprisingly to find that the recent highly non-trivial uncertainty relation for triple observables is indeed a direct consequence of our Cauchy–Schwarz inequality. This new observation leads to refined uncertainty relations in terms of the Wigner–Yanase–Dyson skew information for mixed states and other generalizations. These show that the obtained uncertainty inequalities on Clifford modules can be considered as new uncertainty relations for multiple observables.