In this paper, the authors consider the non-relativistic limit of solutions to the defocusing cubic nonlinear Klein-Gordon equations. Inspired by the work of Lei and Wu (2023), which considered the real-valued case, the authors show that the solutions to the complex Klein-Gordon equation can be described by using a system of two coupled nonlinear Schrödinger equations as the speed of light tends to infinity both in two and three dimensions. On the one hand, if the initial data belong to a lower-regularity Sobolev space H^α, α ∈ [1, 4], the error is proportionate to the α order of the reciprocal of the speed of light, with a constant that grows linearly in time. On the other hand, if initial data are smoother (at least in H⁴), the error is proportionate to the square of the reciprocal of the speed of light, with a constant that grows linearly in time. The error of two different regularity requirements holds globally over time.

Let H be a complex Hilbert space. Denote by \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}(H)$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}_{s}(H)$$\end{document} the algebra of all bounded linear operators on H and the Jordan algebra of all self-adjoint operators in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}(H)$$\end{document}, respectively. In this paper, the authors first give some elementary properties about higher dimensional numerical range of generalized (Jordan) product of operators in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}(H)$$\end{document}. Based on these results, all surjective maps on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}(H)$$\end{document} (respectively, \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}_{s}(H)$$\end{document}) that preserve higher dimensional numerical range of generalized (Jordan) product of operators are completely characterized. Particularly, they give complete characterizations of all surjective maps preserving higher dimensional numerical range of operator products, Jordan semi-triple products and Jordan products on \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}(H)$$\end{document} and \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{B}}_{s}(H)$$\end{document}, respectively.

In this paper, high-order layered complex networks are proposed. The synchronization is discussed in detail. The relations of synchronization, individual coupling matrices and the intrinsic function of the uncoupled system are given. As special cases, the synchronization of monolayer networks and multiplex networks discussed in the literature can be obtained.

In this paper, the authors prove a Schwarz lemma at the boundary for holomorphic mappings between non-equidimensional unit polydiscs and consider an inequality for holomorphic self-mappings of the unit polydisc.

This paper is concerned with logarithmic Laplacian equations and systems with gradient terms in bounded convex domains. Under some assumptions on the nonlinearities, the authors prove the strict monotonicity and symmetry of positive solutions. In particular, the authors establish one new estimate of the logarithmic Laplacian operator that plays an important role in applying the direct method of moving planes. This paper seems to be the first one to deal with the monotonicity and symmetry of positive solutions to logarithmic Laplacian equations and systems with gradient terms.

The main aim of this paper is to extend the Bayesian approach to finding quadratic optimal control to a wider range of stochastic linear systems. These systems involve an unknown parameter in the drift term, which is observed through a noisy linear channel. The author demonstrates the effectiveness of the Bayesian strategy by comparing the cost it incurs with that of an optimal control that possesses complete knowledge of the parameter. The findings reveal that the Bayesian strategy minimizes the worst-case multiplicative regret. Furthermore, the author provides a proof that the corresponding adaptive scheme is optimal when the unknown system parameter belongs to an infinite space. This result further validates the effectiveness of the Bayesian strategy in handling systems with unknown parameters. In summary, this paper contributes to the generalization of the Bayesian strategy for the optimal control in stochastic linear systems with unknown parameters. It also establishes a theoretical basis for its optimality.

Quasitoric orbifolds and locally 1-standard T-manifolds are the topological generalizations of simplicial projective toric varieties and good contact toric manifolds respectively. In this paper, the authors show that there is a resolution of singularities of a quasitoric orbifold. Then, they give an explicit construction of a (2n + 1)-dimensional smooth orientable manifold with Tⁿ⁺¹-action whose boundary is a given locally 1-standard T-manifold. Moreover, they conclude that good contact toric manifolds and generalized lens spaces are equivariantly cobordant to zero.

By using Shen’s technique of mixed product, Xu and Zhao constructed an inhomogeneous representation on the tensor space of any finite-dimensional irreducible o(n)-module with the polynomial space, and found a sufficient condition for this representation to be irreducible. In this paper, the author gives a sufficient and necessary condition for the irreducibility of this representation. Note that the Laplace operator and partial differential operators play important roles in the proof.

Given a positive integer n ≥ 2, let D(n) denote the smallest positive integer m such that a³ + a (a = 1, ⋯, n) are pairwise distinct modulo m². A conjecture of Sun states that D(n) = 3^k, where 3^k is the least power of 3 no less than