Acta Mathematica Sinica-English Series最新文献

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.

Let Mⁿ be an embedded closed submanifold of ℝ^k+1 or a smooth bounded domain in ℝⁿ, where n ≥ 3. We show that the local smooth solution to the heat flow of self-induced harmonic map will blow up at a finite time, provided that the initial map u₀ is in a suitable nontrivial homotopy class with energy small enough.

Let (cal{A}) be a unital C*-algebra and (cal{B}) a unital C*-algebra with a faithful trace τ. Let n be a positive integer. We give the definition of weakly approximate diagonalization (with respect to τ) of a unital homomorphism (phi:cal{A}rightarrow M_{n}(cal{B})). We give an equivalent characterization of McDuff II₁ factors. We show that, if (cal{A}) is a unital nuclear C*-algebra and (cal{B}) is a type II₁ factor with faithful trace τ, then every unital *-homomorphism (phi:cal{A}rightarrow M_{n}(cal{B})) is weakly approximately diagonalizable. If (cal{B}) is a unital simple infinite dimensional separable nuclear C*-algebra, then any finitely many elements in (M_{n}(cal{B})) can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same.

(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.

Consider a class of quasi-periodically forced logistic maps

where ω is an irrational frequency and α(θ) is a specific bimodal function. We prove that under weak Liouvillean condition on frequency, the strange non-chaotic attractor occurs with negative Lyapunov exponent. This extends the result in [Bjerklov, CMP, 2009].

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.