15 April 2022, Volume 38 Issue 4

    

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  Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces with Generalized Smoothness and Their Applications

  Zi Wei LI, Da Chun YANG, Wen YUAN

  Acta Mathematica Sinica. 2022, 38(4): 623-661. https://doi.org/10.1007/s10114-022-1086-6

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  In this article, the authors first establish the pointwise characterizations of Besov and Triebel-Lizorkin spaces with generalized smoothness on $\mathbb{R}^n$ via the Hajłlasz gradient sequences, which serve as a way to extend these spaces to more general metric measure spaces. Moreover, on metric spaces with doubling measures, the authors further prove that the Besov and the Triebel-Lizorkin spaces with generalized smoothness defined via Hajłlasz gradient sequences coincide with those defined via hyperbolic fillings. As an application, some trace theorems of these spaces on Ahlfors regular spaces are established.
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  Symmetrical Independence Tests for Two Random Vectors with Arbitrary Dimensional Graphs

  Jia Min LIU, Gao Rong LI, Jian Qiang ZHANG, Wang Li XU

  Acta Mathematica Sinica. 2022, 38(4): 662-682. https://doi.org/10.1007/s10114-022-0045-6

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  Test of independence between random vectors $X$ and $Y$ is an essential task in statistical inference. One type of testing methods is based on the minimal spanning tree of variables $X$ and $Y$. The main idea is to generate the minimal spanning tree for one random vector $X$, and for each edges in minimal spanning tree, the corresponding rank number can be calculated based on another random vector $Y$. The resulting test statistics are constructed by these rank numbers. However, the existed statistics are not symmetrical tests about the random vectors $X$ and $Y$ such that the power performance from minimal spanning tree of $X$ is not the same as that from minimal spanning tree of $Y$. In addition, the conclusion from minimal spanning tree of $X$ might conflict with that from minimal spanning tree of $Y$. In order to solve these problems, we propose several symmetrical independence tests for $X$ and $Y$. The exact distributions of test statistics are investigated when the sample size is small. Also, we study the asymptotic properties of the statistics. A permutation method is introduced for getting critical values of the statistics. Compared with the existing methods, our proposed methods are more efficient demonstrated by numerical analysis.
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  Projection-based High-dimensional Sign Test

  Hui CHEN, Chang Liang ZOU, Run Ze LI

  Acta Mathematica Sinica. 2022, 38(4): 683-708. https://doi.org/10.1007/s10114-022-0435-9

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  This article is concerned with the high-dimensional location testing problem. For highdimensional settings, traditional multivariate-sign-based tests perform poorly or become infeasible since their Type I error rates are far away from nominal levels. Several modifications have been proposed to address this challenging issue and shown to perform well. However, most of modified sign-based tests abandon all the correlation information, and this results in power loss in certain cases. We propose a projection weighted sign test to utilize the correlation information. Under mild conditions, we derive the optimal direction and weights with which the proposed projection test possesses asymptotically and locally best power under alternatives. Benefiting from using the sample-splitting idea for estimating the optimal direction, the proposed test is able to retain type-I error rates pretty well with asymptotic distributions, while it can be also highly competitive in terms of robustness. Its advantage relative to existing methods is demonstrated in numerical simulations and a real data example.
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  On Quantitative Sphere Equivalence and Extrapolation Phenomenon

  Jun Xi CHEN, Qing Jin CHENG

  Acta Mathematica Sinica. 2022, 38(4): 709-717. https://doi.org/10.1007/s10114-022-1346-5

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  Let $X,Y$ be Banach spaces. We prove that if their unit spheres $S(X)$ and $S(Y)$ are $(\alpha,\beta)$-Hölder homeomorphic for some $\alpha,\beta\in (0,1]$. Then the unit spheres of $L_p(\Omega,\mu, X)$ and $L_q(\Omega,\mu, Y)$ are ($\min\{\frac{p}{q}, \alpha\}, \min\{\frac{q}{p}, \beta\}$)-Hölder homeomorphic for every measure space $(\Omega,\mu)$ and for every $1\leq p,q<\infty$. In particular, the result yields an interesting application involving extrapolation phenomenon.
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  On the Unitriangular Groups over Rational Numbers Field

  Rui GAO, Jun LIAO, He Guo LIU, Xing Zhong XU

  Acta Mathematica Sinica. 2022, 38(4): 718-734. https://doi.org/10.1007/s10114-022-0485-z

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  Let $U(n,\mathbb{Q})$ be the group of all $n \times n$ (upper) unitriangular matrices over rational numbers field $\mathbb{Q}$. Let $S$ be a subset of $U(n,\mathbb{Q})$. In this paper, we prove that $S$ is a subgroup of $U(n,\mathbb{Q})$ if and only if the $(i,j)$-th entry $S_{ij}$ satisfies some condition (see Theorem 3.5). Furthermore, we compute the upper central series and the lower central series for $S$, and obtain the condition that the upper central series and the lower central series of $S$ coincide.
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  Multiplicity of Homoclinic Solutions for Second Order Hamiltonian Systems with Local Conditions at the Origin

  Peng LIU, Fei GUO

  Acta Mathematica Sinica. 2022, 38(4): 735-744. https://doi.org/10.1007/s10114-022-0626-4

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  In this paper, the multiplicity of homoclinic solutions for second order non-autonomous Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla_{u} W (t,u(t))={\textbf{0}}$ is obtained via a new Symmetric Mountain Pass Lemma established by Kajikiya, where $L \in C(\mathbb{R}, \mathbb{R}^{N\times N})$ is symmetric but non-periodic, $W\in C^{1}(\mathbb{R}\times \mathbb{R}^{N}, \mathbb{R})$ is locally even in $u$ and only satisfies some growth conditions near $u={\textbf{0}}$, which improves some previous results.
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  Rapid Time-Decay Phenomenon of the Incompressible Navier-Stokes Flow in Exterior Domains

  Qing Hua ZHANG, Yue Ping ZHU

  Acta Mathematica Sinica. 2022, 38(4): 745-760. https://doi.org/10.1007/s10114-022-1116-4

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  This paper focuses on the rapid time-decay phenomenon of the 3D incompressible Navier-Stokes flow in exterior domains. By using the representation of the flow in exterior domains, together with the estimates of the Gaussian kernel, the tensor kernel, and the Stokes semigroup, we prove that under the assumption $\int_{0}^{\infty}\int_{\partial\Omega}T[u,p](y,t)\cdot\nu dS_{y}dt=0$ for the body pressure tensor $T[u,p]$, if $u_{0}\in L^{1}(\Omega)\cap L_{\sigma}^{3}(\Omega)\cap W^{\frac{2}{5},\frac{5}{4}}(\Omega)$ with $\|u_{0}\|_{3}\leq\eta$ for some sufficiently small number $\eta>0$, then rapid time-decay phenomenon of the Navier-Stokes flow appears. If additionally $|x|^{\alpha}u_{0}\in L^{r_{0}}(\Omega)$ for some $0<\alpha<1$ and $1<r_{0}<(1-\frac{\alpha}{3})^{-1}$ or $\alpha=1$ and $r_{0}=1$, then the flow exhibits higher decay rates as $t\rightarrow\infty$.
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  Topological r-entropy and Measure Theoretic r-entropy of Flows

  Yong JI, Yun Ping WANG

  Acta Mathematica Sinica. 2022, 38(4): 761-769. https://doi.org/10.1007/s10114-022-0573-0

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  The topological r-entropy and measure theoretic r-entropy of a flow are studied. For a flow (X, φ), it is shown that topological (measure theoretic) r-entropy is equal to the topological (measure theoretic) entropy of the time one map φ₁ as r decreases to zero. The Brin-Katok's entropy formula for r-entropy is also established.
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  Strict Neighbor-Distinguishing Total Index of Graphs

  Jing GU, Wei Fan WANG, Yi Qiao WANG

  Acta Mathematica Sinica. 2022, 38(4): 770-776. https://doi.org/10.1007/s10114-022-0593-9

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  A total-coloring of a graph $G$ is strict neighbor-distinguishing if for any two adjacent vertices $u$ and $v$, the set of colors used on $u$ and its incident edges and the set of colors used on $v$ and its incident edges are not included with each other. The strict neighbor-distinguishing total index $\chi''_{\rm snd}(G)$ of $G$ is the minimum number of colors in a strict neighbor-distinguishing total-coloring of $G$. In this paper, we prove that every simple graph $G$ with $\Delta(G)\ge 3$ satisfies $\chi''_{\rm snd}(G)\le 2\Delta(G)$.
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  Non-asymptotic Error Bound for Optimal Prediction of Function-on-Function Regression by RKHS Approach

  Hong Zhi TONG, Ling Fang HU, Michael NG

  Acta Mathematica Sinica. 2022, 38(4): 777-796. https://doi.org/10.1007/s10114-021-9346-4

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  In this paper, we study and analyze the regularized least squares for function-on-function regression model. In our model, both the predictors (input data) and responses (output data) are multivariate functions (with $d$ variables and $\tilde{d}$ variables respectively), and the model coefficient lies in a reproducing kernel Hilbert space (RKHS). We show under mild condition on the reproducing kernel and input data statistics that the convergence rate of excess prediction risk by the regularized least squares is minimax optimal. Numerical examples based on medical image analysis and atmospheric point spread function estimation are considered and tested, and the results demonstrate that the performance of the proposed model is comparable with that of other testing methods.