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25 June 2021, Volume 41 Issue 3 Previous Issue    Next Issue

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SEQUENCES OF POWERS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE Yong CHEN, Kei Ji IZUCHI, Kou Hei IZUCHI, Young Joo LEE Acta mathematica scientia,Series B. 2021, 41 (3):  657-669.  DOI: 10.1007/s10473-021-0301-3 Abstract ( 143 )   RICH HTML PDF   Save We consider Toeplitz operators $T_u$ with symbol $u$ on the Bergman space of the unit ball, and then study the convergences and summability for the sequences of powers of Toeplitz operators. We first charactreize analytic symbols $\varphi$ for which the sequence $T^{*k}_\varphi f$ or $T^{k}_\varphi f$ converges to 0 or $\infty$ as $k\to\infty$ in norm for every nonzero Bergman function $f$. Also, we characterize analytic symbols $\varphi$ for which the norm of such a sequence is summable or not summable. We also study the corresponding problems on an infinite direct sum of Bergman spaces as a generalization of our result. References | Related Articles | Metrics

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A REMARK ON GENERAL COMPLEX (α,β) METRICS Hongchuan XIA, Chunping ZHONG Acta mathematica scientia,Series B. 2021, 41 (3):  670-678.  DOI: 10.1007/s10473-021-0302-2 Abstract ( 113 )   RICH HTML PDF   Save In this paper, we give a characterization for the general complex (α,β) metrics to be strongly convex. As an application, we show that the well-known complex Randers metrics are strongly convex complex Finsler metrics, whereas the complex Kropina metrics are only strongly pseudoconvex. References | Related Articles | Metrics

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GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS Yabo REN, Boling GUO, Shu WANG Acta mathematica scientia,Series B. 2021, 41 (3):  679-702.  DOI: 10.1007/s10473-021-0303-1 Abstract ( 108 )   RICH HTML PDF   Save Global in time weak solutions to the $\alpha$-model regularization for the three dimensional Euler-Poisson equations are considered in this paper. We prove the existence of global weak solutions to $\alpha$-model regularization for the three dimension compressible Euler-Poisson equations by using the Fadeo-Galerkin method and the compactness arguments on the condition that the adiabatic constant satisfies $\gamma>\frac{4}{3}$. References | Related Articles | Metrics

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MULTIPLE SOLUTIONS FOR THE SCHRÖDINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY Yongsheng JIANG, Na WEI, Yonghong WU Acta mathematica scientia,Series B. 2021, 41 (3):  703-711.  DOI: 10.1007/s10473-021-0304-0 Abstract ( 89 )   RICH HTML PDF   Save We are concerned with the nonlinear Schrödinger-Poisson equation $$\begin{equation} \tag{P} \left\{\begin{array}{ll} -\Delta u +(V(x) -\lambda)u+\phi (x) u =f(u), \\ -\Delta\phi = u^2,\ \lim\limits_{|x|\rightarrow +\infty}\phi(x)=0, \ \ \ x\in \mathbb{R}^3, \end{array}\right. \end{equation}$$ where $\lambda$ is a parameter, $V(x)$ is an unbounded potential and $f(u)$ is a general nonlinearity. We prove the existence of a ground state solution and multiple solutions to problem (P). References | Related Articles | Metrics

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HOMOCLINIC SOLUTIONS OF NONLINEAR LAPLACIAN DIFFERENCE EQUATIONS WITHOUT AMBROSETTI-RABINOWITZ CONDITION Antonella NASTASI, Stepan TERSIAN, Calogero VETRO Acta mathematica scientia,Series B. 2021, 41 (3):  712-718.  DOI: 10.1007/s10473-021-0305-z Abstract ( 86 )   RICH HTML PDF   Save The aim of this paper is to establish the existence of at least two non-zero homoclinic solutions for a nonlinear Laplacian difference equation without using Ambrosetti-Rabinowitz type-conditions. The main tools are mountain pass theorem and Palais-Smale compactness condition involving suitable functionals. References | Related Articles | Metrics

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SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS Yueying YANG, Weimao QIAN, Hongwei ZHANG, Yuming CHU Acta mathematica scientia,Series B. 2021, 41 (3):  719-728.  DOI: 10.1007/s10473-021-0306-y Abstract ( 93 )   RICH HTML PDF   Save In the article, we prove that the double inequalities $$\begin{array}{l}{G^p}[{{\rm{\lambda }}_1}a + (1 - {{\rm{\lambda }}_1})b,{{\rm{\lambda }}_1}b + (1 - {{\rm{\lambda }}_1})a]{A^{1 - p}}(a,b) < [A(a,b),G(a,b)]\\ < {G^p}[{\mu _1}a + (1 - {\mu _1})b,{\mu _1}b + (1 - {\mu _1})a]{A^{1 - p}}(a,b),\\{C^s}[{\rm{\lambda }}2a + (1 - {{\rm{\lambda }}_2})b,{{\rm{\lambda }}_2}b + (1 - {{\rm{\lambda }}_2})a]{A^{1 - p}}(a,b) < [A(a,b),Q(a,b)]\\ < {C^s}[\mu 2a + (1 - \mu 2)b,\mu 2b + (1 - {\mu _2})a]{A^{1 - p}}(a,b)\end{array}$$ hold for all a, b > 0 with $a\neq b$ if and only if $\lambda_{1}\leq 1/2-\sqrt{1-(2/\pi)^{2/p}}/2$, $\mu_{1}\geq 1/2-\sqrt{2p}/(4p)$, $\lambda_{2}\leq1/2+\sqrt{2^{3/(2s)}(\mathcal{E}(\sqrt{2}/2)/\pi)^{1/s}-1}/2$ and $\mu_{2}\geq 1/2+\sqrt{s}/(4s)$ if $\lambda_{1}, \mu_{1}\in (0, 1/2)$, $\lambda_{2}, \mu_{2}\in (1/2, 1)$, $p\geq 1$ and $s\geq 1/2$, where $G(a, b)=\sqrt{ab}$, $A(a,b)=(a+b)/2$, $T(a,b)=2\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}{\rm d}t/\pi$, $Q(a,b)=\sqrt{\left(a^{2}+b^{2}\right)/2}$, $C(a, b)=(a^{2}+b^{2})/(a+b)$ and $\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}t}{\rm d}t$. References | Related Articles | Metrics

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THE PROXIMAL RELATION, REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS Yuan LIAN, Xiaojun HUANG, Zhiqiang LI Acta mathematica scientia,Series B. 2021, 41 (3):  729-752.  DOI: 10.1007/s10473-021-0307-x Abstract ( 58 )   RICH HTML PDF   Save In this paper, we study the proximal relation, regionally proximal relation and Banach proximal relation of a topological dynamical system for amenable group actions. A useful tool is the support of a topological dynamical system which is used to study the structure of the Banach proximal relation, and we prove that above three relations all coincide on a Banach mean equicontinuous system generated by an amenable group action. References | Related Articles | Metrics

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THE VON NEUMANN PARADOX FOR THE EULER EQUATIONS Li WANG Acta mathematica scientia,Series B. 2021, 41 (3):  753-763.  DOI: 10.1007/s10473-021-0308-9 Abstract ( 55 )   RICH HTML PDF   Save The reflection of a weak shock wave is considered using a shock polar. We present a sufficient condition under which the von Neumann paradox appears for the Euler equations. In an attempt to resolve the von Neumann paradox for the Euler equations, two new types of reflection configuration, one called the von Neumann reflection (vNR) and the other called the Guderley reflection (GR), are observed in numerical calculations. Finally, we obtain that GR is a reasonable configuration and vNR is an unreasonable configuration to resolve the von Neumann paradox. References | Related Articles | Metrics

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NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL MINHAJUL, T RAJA SEKHAR Acta mathematica scientia,Series B. 2021, 41 (3):  764-780.  DOI: 10.1007/s10473-021-0309-8 Abstract ( 64 )   RICH HTML PDF   Save In this article, we study the exhaustive analysis of nonlinear wave interactions for a 2× 2 homogeneous system of quasilinear hyperbolic partial differential equations (PDEs) governing the macroscopic production. We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations. Furthermore, we study the interaction between simple waves in detail through exact solution of general initial value problem. Finally, we discuss the all possible interaction of elementary waves using the solution of Riemann problem. References | Related Articles | Metrics

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STABILITY ANALYSIS OF CAUSAL INTEGRAL EVOLUTION IMPULSIVE SYSTEMS ON TIME SCALES Jiafa XU, Bakhtawar PERVAIZ, Akbar ZADA, Syed Omar SHAH Acta mathematica scientia,Series B. 2021, 41 (3):  781-800.  DOI: 10.1007/s10473-021-0310-2 Abstract ( 58 )   RICH HTML PDF   Save In this article, we present the existence, uniqueness, Ulam-Hyers stability and Ulam-Hyers-Rassias stability of semilinear nonautonomous integral causal evolution impulsive integro-delay dynamic systems on time scales, with the help of a fixed point approach. We use Grönwall's inequality on time scales, an abstract Gröwall's lemma and a Picard operator as basic tools to develop our main results. To overcome some difficulties, we make a variety of assumptions. At the end an example is given to demonstrate the validity of our main theoretical results. References | Related Articles | Metrics

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HIGH-ORDER NUMERICAL METHOD FOR SOLVING A SPACE DISTRIBUTED-ORDER TIME-FRACTIONAL DIFFUSION EQUATION Jing LI, Yingying YANG, Yingjun JIANG, Libo FENG, Boling GUO Acta mathematica scientia,Series B. 2021, 41 (3):  801-826.  DOI: 10.1007/s10473-021-0311-1 Abstract ( 72 )   RICH HTML PDF   Save This article proposes a high-order numerical method for a space distributed-order time-fractional diffusion equation. First, we use the mid-point quadrature rule to transform the space distributed-order term into multi-term fractional derivatives. Second, based on the piecewise-quadratic polynomials, we construct the nodal basis functions, and then discretize the multi-term fractional equation by the finite volume method. For the time-fractional derivative, the finite difference method is used. Finally, the iterative scheme is proved to be unconditionally stable and convergent with the accuracy $O(\sigma^2+\tau^{2-\beta}+h^3)$, where $\tau$ and $h$ are the time step size and the space step size, respectively. A numerical example is presented to verify the effectiveness of the proposed method. References | Related Articles | Metrics

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MARTINGALE REPRESENTATION AND LOGARITHMIC-SOBOLEV INEQUALITY FOR THE FRACTIONAL ORNSTEIN-UHLENBECK MEASURE Xiaoxia SUN, Feng GUO Acta mathematica scientia,Series B. 2021, 41 (3):  827-842.  DOI: 10.1007/s10473-021-0312-0 Abstract ( 49 )   RICH HTML PDF   Save In this paper, we consider the measure determined by a fractional Ornstein-Uhlenbeck process. For such a measure, we establish an explicit form of the martingale representation theorem and consequently obtain an explicit form of the Logarithmic-Sobolev inequality. To this end, we also present the integration by parts formula for such a measure, which is obtained via its pull back formula and the Bismut method. References | Related Articles | Metrics

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ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS Jinchuan HOU, Wenli WANG Acta mathematica scientia,Series B. 2021, 41 (3):  843-874.  DOI: 10.1007/s10473-021-0313-z Abstract ( 63 )   RICH HTML PDF   Save For $n\geq 3$, we construct a class $\{W_{n,\pi_1,\pi_2}\}$ of $n^2\times n^2$ hermitian matrices by the permutation pairs and show that, for a pair $\{\pi_1,\pi_2\}$ of permutations on $(1,2,\ldots,n)$, $W_{n,\pi_1,\pi_2}$ is an entanglement witness of the $n\otimes n$ system if $\{\pi_1,\pi_2\}$ has the property (C). Recall that a pair $\{\pi_1,\pi_2\}$ of permutations of $(1,2,\ldots,n)$ has the property (C) if, for each $i$, one can obtain a permutation of $(1,\ldots,i-1,i+1,\ldots,n)$ from $(\pi_1(1),\ldots,\pi_1(i-1),\pi_1(i+1),\ldots,\pi_1(n))$ and $(\pi_2(1),\ldots,\pi_2(i-1),\pi_2(i+1),\ldots,\pi_2(n))$. We further prove that $W_{n,\pi_1,\pi_2}$ is not comparable with $W_{n,\pi}$, which is the entanglement witness constructed from a single permutation $\pi$; $W_{n,\pi_1,\pi_2}$ is decomposable if $\pi_1\pi_2={\rm id}$ or $\pi_1^2=\pi_2^2={\rm id}$. For the low dimensional cases $n\in\{3,4\}$, we give a sufficient and necessary condition on $\pi_1,\pi_2$ for $W_{n,\pi_1,\pi_2}$ to be an entanglement witness. We also show that, for $n\in\{3,4\}$, $W_{n,\pi_1,\pi_2}$ is decomposable if and only if $\pi_1\pi_2={\rm id}$ or $\pi_1^2=\pi_2^2={\rm id}$; $W_{3,\pi_1,\pi_2}$ is optimal if and only if $(\pi_1,\pi_2)=(\pi,\pi^2)$, where $\pi=(2,3,1)$. As applications, some entanglement criteria for states and some decomposability criteria for positive maps are established. References | Related Articles | Metrics

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SPREADING SPEED IN THE FISHER-KPP EQUATION WITH NONLOCAL DELAY Ge TIAN, Haoyu WANG, Zhicheng WANG Acta mathematica scientia,Series B. 2021, 41 (3):  875-886.  DOI: 10.1007/s10473-021-0314-y Abstract ( 74 )   RICH HTML PDF   Save This paper is concerned with the Fisher-KPP equation with diffusion and nonlocal delay. Firstly, we establish the global existence and uniform boundedness of solutions to the Cauchy problem. Then, we establish the spreading speed for the solutions with compactly supported initial data. Finally, we investigate the long time behavior of solutions by numerical simulations. References | Related Articles | Metrics

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NEW NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON Sp(n) Shaoxiang ZHANG, Huibin CHEN, Shaoqiang DENG Acta mathematica scientia,Series B. 2021, 41 (3):  887-898.  DOI: 10.1007/s10473-021-0315-x Abstract ( 71 )   RICH HTML PDF   Save In this paper, we consider a class of left invariant Riemannian metrics on Sp(n), which is invariant under the adjoint action of the subgroup Sp(n-3)×Sp(1)×Sp(1)×Sp(1). Based on the related formulae in the literature, we show that the existence of Einstein metrics is equivalent to the existence of solutions of some homogeneous Einstein equations. Then we use a technique of the Gröbner basis to get a sufficient condition for the existence, and show that this method will lead to new non-naturally reductive metrics. References | Related Articles | Metrics

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THE BLOCH SPACE ON THE UNIT BALL OF A HILBERT SPACE: MAXIMALITY AND MULTIPLIERS Pablo GALINDO, Mikael LINDSTRÖM Acta mathematica scientia,Series B. 2021, 41 (3):  899-906.  DOI: 10.1007/s10473-021-0316-9 Abstract ( 61 )   RICH HTML PDF   Save We prove that, as in the finite dimensional case, the space of Bloch functions on the unit ball of a Hilbert space contains, under very mild conditions, any semi-Banach space of analytic functions invariant under automorphisms. The multipliers for such Bloch space are characterized and some of their spectral properties are described. References | Related Articles | Metrics

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THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF *-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE Xiaomin WEI, Lining JIANG, Qiaoling XIN Acta mathematica scientia,Series B. 2021, 41 (3):  907-924.  DOI: 10.1007/s10473-021-0317-8 Abstract ( 90 )   RICH HTML PDF   Save Denote a finite dimensional Hopf $C^*$-algebra by $H$, and a Hopf $*$-subalgebra of $H$ by $H_{1}$. In this paper, we study the construction of the field algebra in Hopf spin models determined by $H_{1}$ together with its symmetry. More precisely, we consider the quantum double $D(H,H_{1})$ as the bicrossed product of the opposite dual $\widehat{H^{op}}$ of $H$ and $H_{1}$ with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between $H_{1}$ and $\widehat{H}$ we define the observable algebra $\mathcal{A}_{H_{1}}$. Then using a comodule action of $D(H,H_{1})$ on $\mathcal{A}_{H_{1}}$, we obtain the field algebra $\mathcal{F}_{H_{1}}$, which is the crossed product $\mathcal{A}_{H_{1}} \rtimes \widehat{D(H,H_{1})}$, and show that the observable algebra $\mathcal{A}_{H_{1}}$ is exactly a $D(H,H_{1})$-invariant subalgebra of $\mathcal{F}_{H_{1}}$. Furthermore, we prove that there exists a duality between $D(H,H_{1})$ and $\mathcal{A}_{H_{1}}$, implemented by a $*$-homomorphism of $D(H,H_{1})$. References | Related Articles | Metrics

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UNIQUENESS OF THE INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION Jianli XIANG, Guozheng YAN Acta mathematica scientia,Series B. 2021, 41 (3):  925-940.  DOI: 10.1007/s10473-021-0318-7 Abstract ( 56 )   RICH HTML PDF   Save This paper considers the inverse acoustic wave scattering by a bounded penetrable obstacle with a conductive boundary condition. We will show that the penetrable scatterer can be uniquely determined by its far-field pattern of the scattered field for all incident plane waves at a fixed wave number. In the first part of this paper, adequate preparations for the main uniqueness result are made. We establish the mixed reciprocity relation between the far-field pattern corresponding to point sources and the scattered field corresponding to plane waves. Then the well-posedness of a modified interior transmission problem is deeply investigated by the variational method. Finally, the a priori estimates of solutions to the general transmission problem with boundary data in $L^{p}(\partial\Omega)$ ($1 References | Related Articles | Metrics

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EXISTENCE AND UNIQUENESS OF THE GLOBAL L1 SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS Tingting CHEN, Aifang QU, Zhen WANG Acta mathematica scientia,Series B. 2021, 41 (3):  941-958.  DOI: 10.1007/s10473-021-0319-6 Abstract ( 61 )   RICH HTML PDF   Save In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space $L_{\rm loc}^1$. The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties. References | Related Articles | Metrics

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ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY Deguang ZHONG, Fanning MENG, Wenjun YUAN Acta mathematica scientia,Series B. 2021, 41 (3):  959-967.  DOI: 10.1007/s10473-021-0320-0 Abstract ( 86 )   RICH HTML PDF   Save Let $f$ be a twice continuously differentiable self-mapping of a unit disk satisfying Poisson differential inequality $|\Delta f(z)|\leq B\cdot|D f(z)|^{2}$ for some $B>0$ and $f(0)=0.$ In this note, we show that $f$ does not always satisfy the Schwarz-Pick type inequality $$\frac{1-|z|^{2}}{1-|f(z)|^{2}}\leq C(B),$$ where $C(B)$ is a constant depending only on $B.$ Moreover, a more general Schwarz-Pick type inequality for mapping that satisfies general Poisson differential inequality is established under certain conditions. References | Related Articles | Metrics

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BILINEAR SPECTRAL MULTIPLIERS ON HEISENBERG GROUPS Naiqi SONG, Heping LIU, Jiman ZHAO Acta mathematica scientia,Series B. 2021, 41 (3):  968-990.  DOI: 10.1007/s10473-021-0321-z Abstract ( 69 )   RICH HTML PDF   Save As we know, thus far, there has appeared no definition of bilinear spectral multipliers on Heisenberg groups. In this article, we present one reasonable definition of bilinear spectral multipliers on Heisenberg groups and investigate its boundedness. We find some restrained conditions to separately ensure its boundedness from $\mathcal{C}_{0}(\mathbb{H}^{n})\times L^{2}(\mathbb{H}^{n})$ to $L^{2}(\mathbb{H}^{n})$, from $L^{2}(\mathbb{H}^{n}) \times \mathcal{C}_{0}(\mathbb{H}^{n})$ to $L^{2}(\mathbb{H}^{n})$, and from $L^{p}\times L^{q}$ to $L^{r}$ with $2 < p,q < \infty, 2\leq r \leq \infty$. References | Related Articles | Metrics

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DYNAMICS ANALYSIS OF A DELAYED HIV INFECTION MODEL WITH CTL IMMUNE RESPONSE AND ANTIBODY IMMUNE RESPONSE Junxian YANG, Leihong WANG Acta mathematica scientia,Series B. 2021, 41 (3):  991-1016.  DOI: 10.1007/s10473-021-0322-y Abstract ( 95 )   RICH HTML PDF   Save In this paper, dynamics analysis of a delayed HIV infection model with CTL immune response and antibody immune response is investigated. The model involves the concentrations of uninfected cells, infected cells, free virus, CTL response cells, and antibody antibody response cells. There are three delays in the model: the intracellular delay, virus replication delay and the antibody delay. The basic reproductive number of viral infection, the antibody immune reproductive number, the CTL immune reproductive number, the CTL immune competitive reproductive number and the antibody immune competitive reproductive number are derived. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions for the stability of each equilibrium is established. The results show that the intracellular delay and virus replication delay do not impact upon the stability of each equilibrium, but when the antibody delay is positive, Hopf bifurcation at the antibody response and the interior equilibrium will exist by using the antibody delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results. References | Related Articles | Metrics

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ANALYSIS OF THE GENOMIC DISTANCE BETWEEN BAT CORONAVIRUS RATG13 AND SARS-COV-2 REVEALS MULTIPLE ORIGINS OF COVID-19 Shaojun PEI, Stephen S. -T. YAU Acta mathematica scientia,Series B. 2021, 41 (3):  1017-1022.  DOI: 10.1007/s10473-021-0323-x Abstract ( 247 )   RICH HTML PDF   Save The severe acute respiratory syndrome COVID-19 was discovered on December 31, 2019 in China. Subsequently, many COVID-19 cases were reported in many other countries. However, some positive COVID-19 samples had been reported earlier than those officially accepted by health authorities in other countries, such as France and Italy. Thus, it is of great importance to determine the place where SARS-CoV-2 was first transmitted to human. To this end, we analyze genomes of SARS-CoV-2 using k-mer natural vector method and compare the similarities of global SARS-CoV-2 genomes by a new natural metric. Because it is commonly accepted that SARS-CoV-2 is originated from bat coronavirus RaTG13, we only need to determine which SARS-CoV-2 genome sequence has the closest distance to bat coronavirus RaTG13 under our natural metric. From our analysis, SARS-CoV-2 most likely has already existed in other countries such as France, India, Netherland, England and United States before the outbreak at Wuhan, China. References | Related Articles | Metrics