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25 February 2021, Volume 41 Issue 1 Previous Issue    Next Issue

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CONTINUOUS DEPENDENCE ON DATA UNDER THE LIPSCHITZ METRIC FOR THE ROTATION-CAMASSA-HOLM EQUATION Xinyu TU, Chunlai MU, Shuyan QIU Acta mathematica scientia,Series B. 2021, 41 (1):  1-18.  DOI: 10.1007/s10473-021-0101-9 Abstract ( 67 )   RICH HTML PDF   Save In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions. References | Related Articles | Metrics

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WEAK SOLUTION TO THE INCOMPRESSIBLE VISCOUS FLUID AND A THERMOELASTIC PLATE INTERACTION PROBLEM IN 3D Srđan TRIFUNOVIĆ, Yaguang WANG Acta mathematica scientia,Series B. 2021, 41 (1):  19-38.  DOI: 10.1007/s10473-021-0102-8 Abstract ( 37 )   RICH HTML PDF   Save In this paper we deal with a nonlinear interaction problem between an incompressible viscous fluid and a nonlinear thermoelastic plate. The nonlinearity in the plate equation corresponds to nonlinear elastic force in various physically relevant semilinear and quasilinear plate models. We prove the existence of a weak solution for this problem by constructing a hybrid approximation scheme that, via operator splitting, decouples the system into two sub-problems, one piece-wise stationary for the fluid and one time-continuous and in a finite basis for the structure. To prove the convergence of the approximate quasilinear elastic force, we develop a compensated compactness method that relies on the maximal monotonicity property of this nonlinear function. References | Related Articles | Metrics

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ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS Junqiang ZHANG, Dachun YANG Acta mathematica scientia,Series B. 2021, 41 (1):  39-66.  DOI: 10.1007/s10473-021-0103-7 Abstract ( 41 )   RICH HTML PDF   Save Let $L:=-\Delta+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq3$, where $V$ is a non-negative potential satisfying $\Delta^{-1}(V)\in L^\infty(\mathbb{R}^n)$. Let $w$ be an $L$-harmonic function, determined by $V$, satisfying that there exists a positive constant $\delta$ such that, for any $x\in\mathbb{R}^n$, $0<\delta\leq w(x)\leq 1$. Assume that $p(\cdot):\ \mathbb{R}^n\to (0,\,1]$ is a variable exponent satisfying the globally $\log$-Hölder continuous condition. In this article, the authors show that the mappings $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto wf\in H^{p(\cdot)}(\mathbb{R}^n)$ and $H_L^{p(\cdot)}(\mathbb{R}^n)\ni f\mapsto (-\Delta)^{1/2}L^{-1/2}(f)\in H^{p(\cdot)}(\mathbb{R}^n)$ are isomorphisms between the variable Hardy spaces $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated with $L$, and the variable Hardy spaces $H^{p(\cdot)}(\mathbb{R}^n)$. References | Related Articles | Metrics

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HITTING PROBABILITIES OF WEIGHTED POISSON PROCESSES WITH DIFFERENT INTENSITIES AND THEIR SUBORDINATIONS Heng ZUO, Zhaohui SHEN, Guanglin RANG Acta mathematica scientia,Series B. 2021, 41 (1):  67-84.  DOI: 10.1007/s10473-021-0104-6 Abstract ( 29 )   RICH HTML PDF   Save In this article, we study the hitting probabilities of weighted Poisson processes and their subordinated versions with different intensities. Furthermore, we simulate and analyze the asymptotic properties of the hitting probabilities in different weights and give an example in the case of subordination. References | Related Articles | Metrics

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INHERITANCE OF DIVISIBILITY FORMS A LARGE SUBALGEBRA Qingzhai FAN, Xiaochun FANG, Xia ZHAO Acta mathematica scientia,Series B. 2021, 41 (1):  85-96.  DOI: 10.1007/s10473-021-0105-5 Abstract ( 21 )   RICH HTML PDF   Save Let $A$ be an infinite dimensional stably finite unital simple separable ${\rm C^*}$-algebra. Let $B\subset A$ be a stably (centrally) large subalgebra in $A$ such that $B$ is $m$-almost divisible ($m$-almost divisible, weakly $(m,n)$-divisible). Then $A$ is $2(m+1)$-almost divisible (weakly $m$-almost divisible, secondly weakly $(m,n)$-divisible). References | Related Articles | Metrics

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UNDERSTANDING SCHUBERT'S BOOK (I) Banghe LI Acta mathematica scientia,Series B. 2021, 41 (1):  97-113.  DOI: 10.1007/s10473-021-0106-4 Abstract ( 108 )   RICH HTML PDF   Save Hilbert Problem 15 required an understanding of Schubert’s book [1], both its methods and its results. In this paper, following his idea, we prove that the formulas in §6, §7, §10, about the incidence of points, lines and planes, are all correct. As an application, we prove formulas 8 and 9 in §12, which are frequently used in his book. References | Related Articles | Metrics

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SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW Jianwei DONG, Manwai YUEN Acta mathematica scientia,Series B. 2021, 41 (1):  114-126.  DOI: 10.1007/s10473-021-0107-3 Abstract ( 26 )   RICH HTML PDF   Save In this article, we are concerned with analytical solutions for a model of inviscid liquid-gas two-phase flow. On the basis of Yuen's works [25, 27-29] on self-similar solutions for compressible Euler equations, we present some special self-similar solutions for a model of inviscid liquid-gas two-phase flow in radial symmetry with and without rotation, and in elliptic symmetry without rotation. Some blowup phenomena and the global existence of the solutions obtained are classified. References | Related Articles | Metrics

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THE AVERAGE ABUNDANCE FUNCTION WITH MUTATION OF THE MULTI-PLAYER SNOWDRIFT EVOLUTIONARY GAME MODEL Ke XIA, Xianjia WANG Acta mathematica scientia,Series B. 2021, 41 (1):  127-163.  DOI: 10.1007/s10473-021-0108-2 Abstract ( 38 )   RICH HTML PDF   Save This article explores the characteristics of the average abundance function with mutation on the basis of the multi-player snowdrift evolutionary game model by analytical analysis and numerical simulation. The specific field of this research concerns the approximate expressions of the average abundance function with mutation on the basis of different levels of selection intensity and an analysis of the results of numerical simulation on the basis of the intuitive expression of the average abundance function. In addition, the biological background of this research lies in research on the effects of mutation, which is regarded as a biological concept and a disturbance to game behavior on the average abundance function. The mutation will make the evolutionary result get closer to the neutral drift state. It can be deduced that this affection is not only related to mutation, but also related to selection intensity and the gap between payoff and aspiration level. The main research findings contain four aspects. First, we have deduced the concrete expression of the expected payoff function. The asymptotic property and change trend of the expected payoff function has been basically obtained. In addition, the intuitive expression of the average abundance function with mutation has been obtained by taking the detailed balance condition as the point of penetration. It can be deduced that the effect of mutation is to make the average abundance function get close to 1/2. In addition, this affection is related to selection intensity and the gap. Secondly, the first-order Taylor expansion of the average abundance function has been deduced for when selection intensity is sufficiently small. The expression of the average abundance function with mutation can be simplified from a composite function to a linear function because of this Taylor expansion. This finding will play a significant role when analyzing the results of the numerical simulation. Thirdly, we have obtained the approximate expressions of the average abundance function corresponding to small and large selection intensity. The significance of the above approximate analysis lies in that we have grasped the basic characteristics of the effect of mutation. The effect is slight and can be neglected when mutation is very small. In addition, the effect begins to increase when mutation rises, and this effect will become more remarkable with the increase of selection intensity. Fourthly, we have explored the influences of parameters on the average abundance function with mutation through numerical simulation. In addition, the corresponding results have been explained on the basis of the expected payoff function. It can be deduced that the influences of parameters on the average abundance function with mutation will be slim when selection intensity is small. Moreover, the corresponding explanation is related to the first-order Taylor expansion. Furthermore,the influences will become notable when selection intensity is large. References | Related Articles | Metrics

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GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES Muhammad Zainul ABIDIN, Jiecheng CHEN Acta mathematica scientia,Series B. 2021, 41 (1):  164-176.  DOI: 10.1007/s10473-021-0109-1 Abstract ( 35 )   RICH HTML PDF   Save In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space $\mathcal{F\dot{N}}_{p(\cdot),h(\cdot),q}^{s(\cdot)}(\mathbb{R}^3)$ with $s(\cdot) = 4-2\alpha-\frac{3}{p(\cdot)} $. We prove global well-posedness result with small initial data belonging to $\mathcal{F\dot{N}}_{p(\cdot),h(\cdot),q}^{4-2\alpha-\frac{3}{p(\cdot)} }(\mathbb{R}^3)$. The result of this paper extends some recent work. References | Related Articles | Metrics

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A GENERALIZED RESULT ON THE POLYNOMIAL-LIKE ITERATIVE EQUATION Pingping ZHANG, Yingying ZENG Acta mathematica scientia,Series B. 2021, 41 (1):  177-186.  DOI: 10.1007/s10473-021-0110-8 Abstract ( 25 )   RICH HTML PDF   Save Most results on the polynomial-like iterative equation are given under the condition that the given function is monotone, while a work by L. Liu and X. Gong gets non-monotone PM solutions with height 1 when the given function is of the same case. Removing the condition on height for the given function, we first give a method to assert the nonexistence of $C^0$ solutions, then present equivalent conditions for the existence of PM solutions with finite height. Finally, as an application of the equivalent conditions, we construct the PM solutions in the case that the given function has one fort. References | Related Articles | Metrics

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A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES Yunhua WENG, Xuesong LI, Nanjing HUANG Acta mathematica scientia,Series B. 2021, 41 (1):  187-206.  DOI: 10.1007/s10473-021-0111-7 Abstract ( 42 )   RICH HTML PDF   Save This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space. Utilizing the KKM theorem, a result concerned with the upper semicontinuity and measurability of the solution set of a hemi-variational inequality is established. By using a fixed point theorem for a condensing set-valued map, the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions. Finally, an example is presented to illustrate our main results. References | Related Articles | Metrics

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WEIGHTED LASSO ESTIMATES FOR SPARSE LOGISTIC REGRESSION:NON-ASYMPTOTIC PROPERTIES WITH MEASUREMENT ERRORS Huamei HUANG, Yujing GAO, Huiming ZHANG, Bo LI Acta mathematica scientia,Series B. 2021, 41 (1):  207-230.  DOI: 10.1007/s10473-021-0112-6 Abstract ( 25 )   RICH HTML PDF   Save For high-dimensional models with a focus on classification performance, the $\ell_{1}$-penalized logistic regression is becoming important and popular. However, the Lasso estimates could be problematic when penalties of different coefficients are all the same and not related to the data. We propose two types of weighted Lasso estimates, depending upon covariates determined by the McDiarmid inequality. Given sample size $n$ and a dimension of covariates $p$, the finite sample behavior of our proposed method with a diverging number of predictors is illustrated by non-asymptotic oracle inequalities such as the $\ell_{1}$-estimation error and the squared prediction error of the unknown parameters. We compare the performance of our method with that of former weighted estimates on simulated data, then apply it to do real data analysis. References | Related Articles | Metrics

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THE LOCAL WELL-POSEDNESS OF A CHEMOTAXIS-SHALLOW WATER SYSTEM WITH VACUUM Jishan FAN, Fucai LI, Gen NAKAMURA Acta mathematica scientia,Series B. 2021, 41 (1):  231-240.  DOI: 10.1007/s10473-021-0113-5 Abstract ( 34 )   RICH HTML Save In this paper we prove the local well-posedness of strong solutions to a chemotaxis-shallow water system with initial vacuum in a bounded domain $\Omega\subset\mathbb{R}^2$ without the standard compatibility condition for the initial data. This improves some results obtained in [J. Differential Equations 261(2016), 6758-6789]. References | Related Articles | Metrics

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ON THE MIXED RADIAL-ANGULAR INTEGRABILITY OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS Ronghui LIU, Feng LIU, Huoxiong WU Acta mathematica scientia,Series B. 2021, 41 (1):  241-256.  DOI: 10.1007/s10473-021-0114-4 Abstract ( 22 )   RICH HTML PDF   Save This paper studies the mixed radial-angular integrability of parametric Marcinki-ewicz integrals along "polynomial curves". Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness, the authors prove that such operators are bounded on the mixed radial-angular spaces. Meanwhile, corresponding vector-valued versions are also obtained. References | Related Articles | Metrics

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ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS Hua CHEN, Nian LIU Acta mathematica scientia,Series B. 2021, 41 (1):  257-282.  DOI: 10.1007/s10473-021-0115-3 Abstract ( 26 )   RICH HTML PDF   Save In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted $p$-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincar$\acute{e}$ inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family. References | Related Articles | Metrics

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DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE Peide LIU Acta mathematica scientia,Series B. 2021, 41 (1):  283-296.  DOI: 10.1007/s10473-021-0116-2 Abstract ( 41 )   RICH HTML PDF   Save In this paper we deal with the martingales in variable Lebesgue space over a probability space. We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space. The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob's maximal operators in martingale Lebesgue space with a variable exponent. In particular, we present two kinds of weak-type Doob's maximal inequalities and some necessary and sufficient conditions for strong-type Doob's maximal inequalities. Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1. References | Related Articles | Metrics

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PRECISE VALUES OF THE BLOCH CONSTANTS OF CERTAIN LOG-p-HARMONIC MAPPINGS Mingsheng LIU, Lifang LUO Acta mathematica scientia,Series B. 2021, 41 (1):  297-310.  DOI: 10.1007/s10473-021-0117-1 Abstract ( 20 )   RICH HTML PDF   Save The aim of this article is twofold. One aim is to establish the precise forms of Landau-Bloch type theorems for certain polyharmonic mappings in the unit disk by applying a geometric method. The other is to obtain the precise values of Bloch constants for certain log-p-harmonic mappings. These results improve upon the corresponding results given in Bai et al. (Complex Anal. Oper. Theory, 13(2): 321-340, 2019). References | Related Articles | Metrics

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ON THE CAUCHY PROBLEM FOR AW-RASCLE SYSTEM WITH LINEAR DAMPING Juan C. JUAJIBIOY Acta mathematica scientia,Series B. 2021, 41 (1):  311-318.  DOI: 10.1007/s10473-021-0118-0 Abstract ( 19 )   RICH HTML PDF   Save The existence of global BV solutions for the Aw-Rascle system with linear damping is considered. In order to get approximate solutions we consider the system in Lagrangian coordinates, then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of [1] we get a sequence of approximate solutions. Finally we show the convergence of this approximate sequence to the weak entropic solution. References | Related Articles | Metrics

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THE EXISTENCE AND STABILITY OF STATIONARY SOLUTIONS OF THE INFLOW PROBLEM FOR FULL COMPRESSIBLE NAVIER-STOKES-POISSON SYSTEM Hakho HONG Acta mathematica scientia,Series B. 2021, 41 (1):  319-336.  DOI: 10.1007/s10473-021-0119-z Abstract ( 22 )   RICH HTML PDF   Save In this paper, we consider an inflow problem for the non-isentropic Navier-Stokes-Poisson system in a half line (0,∞). For the general gas including ideal polytropic gas, we first give some results for the existence of the stationary solution with the aid of center manifold theory on a 4×4 system of autonomous ordinary differential equations. We also show the time asymptotic stability of the stationary solutions with small strength under smallness assumptions on the initial perturbations by using an elementary energy method. References | Related Articles | Metrics

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GLEASON'S PROBLEM ON FOCK-SOBOLEV SPACES Jineng DAI, Jingyun ZHOU Acta mathematica scientia,Series B. 2021, 41 (1):  337-348.  DOI: 10.1007/s10473-021-0120-6 Abstract ( 24 )   RICH HTML PDF   Save In this article, we solve completely Gleason's problem on Fock-Sobolev spaces $F^{p,m}$ for any non-negative integer $m$ and $0 < p\leq\infty$. References | Related Articles | Metrics