Acta mathematica scientia,Series B

NOTES ON THE LOG-BRUNN-MINKOWSKI INEQUALITY^*

Yunlong YANG, Nan JIANG, Deyan ZHANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2333-2346. DOI: 10.1007/s10473-023-0601-x

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Böröczky-Lutwak-Yang-Zhang proved the log-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane in a way that is stronger than for the classical Brunn-Minkowski inequality. In this paper, we investigate the relative positive center set of planar convex bodies. As an application of the relative positive center, we prove the log-Minkowski inequality and the log-Brunn-Minkowski inequality.

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ENTIRE SOLUTIONS OF LOTKA-VOLTERRA COMPETITION SYSTEMS WITH NONLOCAL DISPERSAL^*

Yuxia HAO, Wantong LI, Jiabing WANG, Wenbing XU

Acta mathematica scientia,Series B. 2023, 43 (6): 2347-2376. DOI: 10.1007/s10473-023-0602-9

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This paper is mainly concerned with entire solutions of the following two-species Lotka-Volterra competition system with nonlocal (convolution) dispersals:Here

$a\neq 1$ ,

$b\neq1$ ,

$d$ , and

$r$ are positive constants. By studying the eigenvalue problem of (0.1) linearized at

$(\phi_c(\xi), 0)$ , we construct a pair of super- and sub-solutions for (0.1), and then establish the existence of entire solutions originating from

$(\phi_c(\xi), 0)$ as

$t\rightarrow -\infty$ , where

$\phi_c$ denotes the traveling wave solution of the nonlocal Fisher-KPP equation

$u_t=k*u-u+u\left(1-u\right)$ . Moreover, we give a detailed description on the long-time behavior of such entire solutions as

$t\rightarrow \infty$ . Compared to the known works on the Lotka-Volterra competition system with classical diffusions, this paper overcomes many difficulties due to the appearance of nonlocal dispersal operators.

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ISOMETRY AND PHASE-ISOMETRY OF NON-ARCHIMEDEAN NORMED SPACES^*

Ruidong WANG, Wenting YAO

Acta mathematica scientia,Series B. 2023, 43 (6): 2377-2386. DOI: 10.1007/s10473-023-0603-8

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In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry

$f:S_{r}(X)\rightarrow S_{r}(X)$ , where

$X$ is a finite-dimensional non-Archimedean normed space and

$S_{r}(X)$ is a sphere with radius

$r\in \|X\|$ , is surjective if and only if

$\mathbb{K}$ is spherically complete and

$k$ is finite. Moreover, we prove that if

$X$ and

$Y$ are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with

$|2|=1$ , any phase-isometry

$f:X\rightarrow Y$ is phase equivalent to an isometric operator.

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THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS^*

Yu DONG, Yaofang HUANG, Li LI, Qing LU

Acta mathematica scientia,Series B. 2023, 43 (6): 2387-2397. DOI: 10.1007/s10473-023-0604-7

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In this paper, we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations. To be more precise, under some conditions on the swirling component of vorticity, we can conclude that the weak solutions are regular.

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A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES^*

Shanli YE, Guanghao FENG

Acta mathematica scientia,Series B. 2023, 43 (6): 2398-2412. DOI: 10.1007/s10473-023-0605-6

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Let

$\mu$ be a positive Borel measure on the interval

$[0,1)$ . The Hankel matrix

$\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries

$\mu_{n,k}=\mu_{n+k}$ , where

$\mu_{n}=\int_{[0,1)}t^n{\rm d}\mu(t)$ , induces formally the operator as

$\mathcal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\Big(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\Big)(n+1)z^n , z\in \mathbb{D},$ where

$f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ is an analytic function in

$\mathbb{D}$ .We characterize the positive Borel measures on

$[0,1)$ such that

$\mathcal{DH}_\mu(f)(z)= \int_{[0,1)} \frac{f(t)}{{(1-tz)^2}} {\rm d}\mu(t)$ for all

$f$ in the Hardy spaces

$H^p(0<p<\infty)$ , and among these we describe those for which

$\mathcal{DH}_\mu$ is a bounded (resp., compact) operator from

$H^p(0<p <\infty)$ into

$H^q(q > p$ and

$q\geq 1$ ). We also study the analogous problem in the Hardy spaces

$H^p(1\leq p\leq 2)$ .

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CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM^*

Huiyang XU, Cece LI

Acta mathematica scientia,Series B. 2023, 43 (6): 2413-2429. DOI: 10.1007/s10473-023-0606-5

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In this paper, we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric. As a main result, we classify these hypersurfaces as not being of a flat affine metric. In particular, 2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

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RELATIVE ENTROPY DIMENSION FOR COUNTABLE AMENABLE GROUP ACTIONS^*

Zubiao XIAO, Zhengyu YIN

Acta mathematica scientia,Series B. 2023, 43 (6): 2430-2448. DOI: 10.1007/s10473-023-0607-4

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We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence

$\{F_n\}_{n=0}^{+\infty}$ , we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.

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A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY^*

Zhiguo LIU

Acta mathematica scientia,Series B. 2023, 43 (6): 2449-2470. DOI: 10.1007/s10473-023-0608-3

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Using Hartogs' fundamental theorem for analytic functions in several complex variables and

$q$ -partial differential equations, we establish a multiple

$q$ -exponential differential formula for analytic functions in several variables. With this identity, we give new proofs of a variety of important classical formulas including Bailey's

$_6\psi_6$ series summation formula and the Atakishiyev integral. A new transformation formula for a double

$q$ -series with several interesting special cases is given. A new transformation formula for a

$_3\psi_3$ series is proved.

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CHARACTEIZATIONS OF WOVEN g-FRAMES AND WEAVING g-FRAMES IN HILBERT SPACES AND C^*-MODULES^*

Amir KHOSRAVI, Mohammad Reza FARMANI

Acta mathematica scientia,Series B. 2023, 43 (6): 2471-2482. DOI: 10.1007/s10473-023-0609-2

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In this paper, using Parseval frames we generalize Sun's results to g-frames in Hilbert

$C^*$ -modules. Moreover, for g-frames in Hilbert spaces, we present some characterizations in terms of a family of frames, not only for orthonormal bases. Also, we have a note about a comment and a relation in the proof of Proposition 5.3 in [D. Li et al., On weaving g-frames for Hilbert spaces, Complex Analysis and Operator Theory, 2020]. Finally, we have some results for g-Riesz bases, woven and P-woven g-frames.

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ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS^*

Jia LIU, Saisai SHI, Yuan ZHANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2483-2492. DOI: 10.1007/s10473-023-0610-9

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In this paper, we consider the graph of the product of continuous functions in terms of Hausdorff and packing dimensions. More precisely, we show that, given a real number

$1\leq\beta\leq2$ , any real-valued continuous function in C([0,1]) can be decomposed into a product of two real-valued continuous functions, each having a graph of Hausdorff dimension

$\beta$ . In addition, a product decomposition result for the packing dimension is obtained. This work answers affirmatively two questions raised by Verma and Priyadarshi [14].

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THEORETICAL RESULTS ON THE EXISTENCE, REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS: APPLICATIONS TO 3D PRIMITIVE EQUATIONS^*

Renhai WANG, Boling GUO, Daiwen HUANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2493-2518. DOI: 10.1007/s10473-023-0611-8

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Several new concepts of enhanced pullback attractors for nonautonomous dynamical systems are introduced here by uniformly enhancing the compactness and attraction of the usual pullback attractors over an infinite forward time-interval under strong and weak topologies. Then we provide some theoretical results for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which can be applied to a large class of PDEs. The existence of these enhanced attractors is harder to obtain than the backward case [33], since it is difficult to uniformly control the long-time pullback behavior of the systems over the forward time-interval. As applications of our theoretical results, we consider the famous 3D primitive equations modelling the large-scale ocean and atmosphere dynamics, and prove the existence, regularity and asymptotic stability of the enhanced pullback attractors in

$V\times V$ and

$H^2\times H^2$ for the time-dependent forces which satisfy some weak conditions.

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TRANSPORTATION COST-INFORMATION INEQUALITY FOR A STOCHASTIC HEAT EQUATION DRIVEN BY FRACTIONAL-COLORED NOISE^*

Ruinan LI, Xinyu WANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2519-2532. DOI: 10.1007/s10473-023-0612-7

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In this paper, we prove Talagrand's

$T_2$ transportation cost-information inequality for the law of stochastic heat equation driven by Gaussian noise, which is fractional for a time variable with the Hurst index

$H\in\left(\frac12,\,1\right)$ , and is correlated for the spatial variable. The Girsanov theorem for fractional-colored Gaussian noise plays an important role in the proof.

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ZERO DISSIPATION LIMIT TO A RAREFACTION WAVE WITH A VACUUM FOR A COMPRESSIBLE, HEAT CONDUCTING REACTING MIXTURE^*

Shengchuang CHANG, Ran DUAN

Acta mathematica scientia,Series B. 2023, 43 (6): 2533-2552. DOI: 10.1007/s10473-023-0613-6

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In this paper, we study the zero dissipation limit with a vacuum for the reacting mixture Navier-Stokes equations. For proper smooth initial data that the initial density tends to zero as the relevant physical coefficients tend to zero, we demonstrate that the solution tends to a rarefaction wave connected to a vacuum on the left side coupled with a zero mass fraction of reactant. What is more, the uniform convergence rate is obtained.

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A CLASS OF INVERSE QUOTIENT CURVATURE FLOW IN THE ADS-SCHWARZSCHILD MANIFOLD^*

Zhengchao JI

Acta mathematica scientia,Series B. 2023, 43 (6): 2553-2572. DOI: 10.1007/s10473-023-0614-5

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In this paper, we study the asymptotic behavior of a class of inverse quotient curvature flow in the anti-de Sitter-Schwarzschild manifold. We prove that under suitable convex conditions for the initial hypersurface, one can get the long-time existence for the inverse curvature flow. Moreover, we also get that the principal curvatures of the evolving hypersurface converge to

$1$ when

$t\rightarrow+\infty$ .

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JONES TYPE C^*-BASIC CONSTRUCTION IN NON-EQUILIBRIUM HOPF SPIN MODELS^*

Xiaomin WEI, Lining JIANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2573-2588. DOI: 10.1007/s10473-023-0615-4

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Let H be a finite dimensional Hopf

${C}^*$ -algebra, and let K be a Hopf *-subalgebra of H. Considering that the field algebra

$\mathscr{F}_{K}$ of a non-equilibrium Hopf spin model carries a

$D(H,K)$ -invariant subalgebra

$\mathscr{A}_{K}$ , this paper shows that the

${C}^*$ -basic construction for the inclusion

$\mathscr{A}_{K} \subseteq \mathscr{F}_{K}$ {can be expressed as} the crossed product

${C}^*$ -algebra

$\mathscr{F}_{K} \rtimes D(H,K)$ . Here,

$D(H,K)$ is a bicrossed product of the opposite dual

$\widehat{H^{op}}$ and

$K$ . Furthermore, the natural action of

$\widehat{D(H,K)}$ on

$D(H,K)$ gives rise to the iterated crossed product

$\mathscr{F}_{K} \rtimes D(H,K) \rtimes \widehat{D(H,K)}$ , which coincides with the

${C}^*$ -basic construction for the inclusion

$\mathscr{F}_{K} \subseteq \mathscr{F}_{K} \rtimes D(H,K)$ . In the end, the Jones type tower of field algebra

$\mathscr{F}_{K}$ is obtained, and the new field algebra emerges exactly as the iterated crossed product.

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ON A SUPER POLYHARMONIC PROPERTY OF A HIGHER-ORDER FRACTIONAL LAPLACIAN^*

Meiqing XU

Acta mathematica scientia,Series B. 2023, 43 (6): 2589-2596. DOI: 10.1007/s10473-023-0616-3

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Let

$0<\alpha<2$ ,

$p\geq 1$ ,

$m\in\mathbb{N}_+$ . Consider the positive solution

$u$ of the PDE

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (-\Delta)^{\frac{\alpha}{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0.1)$
In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition

$u\in\mathcal{L}_\alpha$ , (0.1) possesses a super polyharmonic property

$(-\Delta)^{k+\frac{\alpha}{2}}u\geq 0$ for

$k=0,1,\cdots ,m-1$ . In this paper, we show another kind of super polyharmonic property

$(-\Delta)^k u> 0$ for

$k=1,\cdots ,m-1$ , under the conditions

$(-\Delta)^mu\in\mathcal{L}_\alpha$ and

$(-\Delta)^m u\geq 0$ . Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation

$u(x)=\int_{\mathbb{R}^n}\frac{u^p(y)}{|x-y|^{n-2m-\alpha}}{\rm d}y$ . One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.

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THE ANALYTIC SMOOTHING EFFECT OF LINEAR LANDAU EQUATION WITH SOFT POTENTIALS^*

Haoguang LI, Chaojiang XU

Acta mathematica scientia,Series B. 2023, 43 (6): 2597-2614. DOI: 10.1007/s10473-023-0617-2

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In this work, we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in

$L^{2}(\mathbb{R}^3)$ enjoys an analytic regularization effect, and that the evolution of the analytic radius is the same as the heat equations.

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THE EXISTENCE AND MULTIPLICITY OF k-CONVEX SOLUTIONS FOR A COUPLED k-HESSIAN SYSTEM^*

Chenghua GAO, Xingyue HE, Jingjing WANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2615-2628. DOI: 10.1007/s10473-023-0618-1

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In this paper, we focus on the following coupled system of

$k$ -Hessian equations:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{equation*} \left\{\begin{aligned}&S_k(\lambda(D^2u))=f_1(|x|,-v)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&S_k(\lambda(D^2v))=f_2(|x|,-u)\ \ \ \ \ \ \ \ {\rm in}\ B,\\&u=v=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm on}\ \partial B.\end{aligned}\right.\end{equation*}$
Here B is a unit ball with center 0 and

$f_i (i=1,2)$ are continuous and nonnegative functions. By introducing some new growth conditions on the nonlinearities

$f_1$ and

$f_2$ , which are more flexible than the existing conditions for the k-Hessian systems (equations), several new existence and multiplicity results for k-convex solutions for this kind of problem are obtained.

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AN INFORMATIC APPROACH TO A LONG MEMORY STATIONARY PROCESS^*

Yiming DING, Liang WU, Xuyan XIANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2629-2648. DOI: 10.1007/s10473-023-0619-0

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Long memory is an important phenomenon that arises sometimes in the analysis of time series or spatial data. Most of the definitions concerning the long memory of a stationary process are based on the second-order properties of the process. The mutual information between the past and future

$I_{p-f}$ of a stationary process represents the information stored in the history of the process which can be used to predict the future. We suggest that a stationary process can be referred to as long memory if its

$I_{p-f}$ is infinite. For a stationary process with finite block entropy,

$I_{p-f}$ is equal to the excess entropy, which is the summation of redundancies that relate the convergence rate of the conditional (differential) entropy to the entropy rate. Since the definitions of the

$I_{p-f}$ and the excess entropy of a stationary process require a very weak moment condition on the distribution of the process, it can be applied to processes whose distributions are without a bounded second moment. A significant property of

$I_{p-f}$ is that it is invariant under one-to-one transformation; this enables us to know the

$I_{p-f}$ of a stationary process from other processes. For a stationary Gaussian process, the long memory in the sense of mutual information is more strict than that in the sense of covariance. We demonstrate that the

$I_{p-f}$ of fractional Gaussian noise is infinite if and only if the Hurst parameter is

$H \in (1/2, 1)$ .

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THE EXISTENCE OF GROUND STATE NORMALIZED SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS^*

Yu MAO, Xingping WU, Chunlei TANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2649-2661. DOI: 10.1007/s10473-023-0620-7

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In this paper, we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in

$H^{1}(\mathbb{R}^{2})$ . When the nonlinearity satisfies some general 3-superlinear conditions, we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in [L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. (1997)].

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BIFURCATION CONTROL FOR A FRACTIONAL-ORDER DELAYED SEIR RUMOR SPREADING MODEL WITH INCOMMENSURATE ORDERS^*

Maolin YE, Haijun JIANG

Acta mathematica scientia,Series B. 2023, 43 (6): 2662-2682. DOI: 10.1007/s10473-023-0621-6

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A fractional-order delayed SEIR rumor spreading model with a nonlinear incidence function is established in this paper, and a novel strategy to control the bifurcation of this model is proposed. First, Hopf bifurcation is investigated by considering time delay as bifurcation parameter for the system without a feedback controller. Then, a state feedback controller is designed to control the occurrence of bifurcation in advance or to delay it by changing the parameters of the controller. Finally, in order to verify the theoretical results, some numerical simulations are given.

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