当期目录

2021年, 第41卷, 第3期　刊出日期：2021-06-25 上一期    下一期

本期栏目： 论文　

全选:

合并摘要

导出引用 EndNote Reference Manager ProCite BibTeX RefWorks

隐藏/显示图片

论文

Select

SEQUENCES OF POWERS OF TOEPLITZ OPERATORS ON THE BERGMAN SPACE 陈泳, Kei Ji IZUCHI, Kou Hei IZUCHI, Young Joo LEE 数学物理学报(英文版). 2021 (3):  657-669.  DOI: 10.1007/s10473-021-0301-3 摘要 ( 123 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

A REMARK ON GENERAL COMPLEX (α,β) METRICS 夏红川, 钟春平 数学物理学报(英文版). 2021 (3):  670-678.  DOI: 10.1007/s10473-021-0302-2 摘要 ( 77 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

GLOBAL WEAK SOLUTIONS TO THE α-MODEL REGULARIZATION FOR 3D COMPRESSIBLE EULER-POISSON EQUATIONS 任亚伯, 郭柏灵, 王术 数学物理学报(英文版). 2021 (3):  679-702.  DOI: 10.1007/s10473-021-0303-1 摘要 ( 84 )   RICH HTML PDF   收藏 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 $\gamma>\frac{4}{3}$. 参考文献 | 相关文章 | 计量指标

Select

MULTIPLE SOLUTIONS FOR THE SCHRÖDINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY 蒋永生, 魏娜, 吴永洪 数学物理学报(英文版). 2021 (3):  703-711.  DOI: 10.1007/s10473-021-0304-0 摘要 ( 45 )   RICH HTML PDF   收藏 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). 参考文献 | 相关文章 | 计量指标

Select

HOMOCLINIC SOLUTIONS OF NONLINEAR LAPLACIAN DIFFERENCE EQUATIONS WITHOUT AMBROSETTI-RABINOWITZ CONDITION Antonella NASTASI, Stepan TERSIAN, Calogero VETRO 数学物理学报(英文版). 2021 (3):  712-718.  DOI: 10.1007/s10473-021-0305-z 摘要 ( 47 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

SHARP BOUNDS FOR TOADER-TYPE MEANS IN TERMS OF TWO-PARAMETER MEANS 杨月英, 钱伟茂, 张宏伟, 褚玉明 数学物理学报(英文版). 2021 (3):  719-728.  DOI: 10.1007/s10473-021-0306-y 摘要 ( 52 )   RICH HTML PDF   收藏 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 $\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}t}{\rm d}t$. 参考文献 | 相关文章 | 计量指标

Select

THE PROXIMAL RELATION, REGIONALLY PROXIMAL RELATION AND BANACH PROXIMAL RELATION FOR AMENABLE GROUP ACTIONS 连媛, 黄小军, 李智强 数学物理学报(英文版). 2021 (3):  729-752.  DOI: 10.1007/s10473-021-0307-x 摘要 ( 41 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

THE VON NEUMANN PARADOX FOR THE EULER EQUATIONS 王丽 数学物理学报(英文版). 2021 (3):  753-763.  DOI: 10.1007/s10473-021-0308-9 摘要 ( 38 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL MINHAJUL, T RAJA SEKHAR 数学物理学报(英文版). 2021 (3):  764-780.  DOI: 10.1007/s10473-021-0309-8 摘要 ( 35 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

STABILITY ANALYSIS OF CAUSAL INTEGRAL EVOLUTION IMPULSIVE SYSTEMS ON TIME SCALES 徐家发, Bakhtawar PERVAIZ, Akbar ZADA, Syed Omar SHAH 数学物理学报(英文版). 2021 (3):  781-800.  DOI: 10.1007/s10473-021-0310-2 摘要 ( 36 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

HIGH-ORDER NUMERICAL METHOD FOR SOLVING A SPACE DISTRIBUTED-ORDER TIME-FRACTIONAL DIFFUSION EQUATION 李景, 杨莹莹, 姜英军, 封利波, 郭柏灵 数学物理学报(英文版). 2021 (3):  801-826.  DOI: 10.1007/s10473-021-0311-1 摘要 ( 37 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

MARTINGALE REPRESENTATION AND LOGARITHMIC-SOBOLEV INEQUALITY FOR THE FRACTIONAL ORNSTEIN-UHLENBECK MEASURE 孙晓霞, 郭峰 数学物理学报(英文版). 2021 (3):  827-842.  DOI: 10.1007/s10473-021-0312-0 摘要 ( 31 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

ENTANGLEMENT WITNESSES CONSTRUCTED BY PERMUTATION PAIRS 侯晋川, 王文丽 数学物理学报(英文版). 2021 (3):  843-874.  DOI: 10.1007/s10473-021-0313-z 摘要 ( 34 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

SPREADING SPEED IN THE FISHER-KPP EQUATION WITH NONLOCAL DELAY 田歌, 王浩雨, 王智诚 数学物理学报(英文版). 2021 (3):  875-886.  DOI: 10.1007/s10473-021-0314-y 摘要 ( 49 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

NEW NON-NATURALLY REDUCTIVE EINSTEIN METRICS ON Sp(n) 张绍祥, 陈慧斌, 邓少强 数学物理学报(英文版). 2021 (3):  887-898.  DOI: 10.1007/s10473-021-0315-x 摘要 ( 42 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

THE BLOCH SPACE ON THE UNIT BALL OF A HILBERT SPACE: MAXIMALITY AND MULTIPLIERS Pablo GALINDO, Mikael LINDSTRÖM 数学物理学报(英文版). 2021 (3):  899-906.  DOI: 10.1007/s10473-021-0316-9 摘要 ( 33 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

THE FIELD ALGEBRA IN HOPF SPIN MODELS DETERMINED BY A HOPF *-SUBALGEBRA AND ITS SYMMETRIC STRUCTURE 魏晓敏, 蒋立宁, 辛巧玲 数学物理学报(英文版). 2021 (3):  907-924.  DOI: 10.1007/s10473-021-0317-8 摘要 ( 48 )   RICH HTML PDF   收藏 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 $D(H,H_{1})$. 参考文献 | 相关文章 | 计量指标

Select

UNIQUENESS OF THE INVERSE TRANSMISSION SCATTERING WITH A CONDUCTIVE BOUNDARY CONDITION 向建立, 严国政 数学物理学报(英文版). 2021 (3):  925-940.  DOI: 10.1007/s10473-021-0318-7 摘要 ( 30 )   RICH HTML PDF   收藏 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 参考文献 | 相关文章 | 计量指标

Select

EXISTENCE AND UNIQUENESS OF THE GLOBAL L1 SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS 陈停停, 屈爱芳, 王振 数学物理学报(英文版). 2021 (3):  941-958.  DOI: 10.1007/s10473-021-0319-6 摘要 ( 39 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

ON SCHWARZ-PICK TYPE INEQUALITY FOR MAPPINGS SATISFYING POISSON DIFFERENTIAL INEQUALITY 钟德光, 孟凡宁, 袁文俊 数学物理学报(英文版). 2021 (3):  959-967.  DOI: 10.1007/s10473-021-0320-0 摘要 ( 35 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

BILINEAR SPECTRAL MULTIPLIERS ON HEISENBERG GROUPS 宋乃琪, 刘和平, 赵纪满 数学物理学报(英文版). 2021 (3):  968-990.  DOI: 10.1007/s10473-021-0321-z 摘要 ( 39 )   RICH HTML PDF   收藏 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 $2 < p,q < \infty, 2\leq r \leq \infty$. 参考文献 | 相关文章 | 计量指标

Select

DYNAMICS ANALYSIS OF A DELAYED HIV INFECTION MODEL WITH CTL IMMUNE RESPONSE AND ANTIBODY IMMUNE RESPONSE 杨俊仙, 王雷宏 数学物理学报(英文版). 2021 (3):  991-1016.  DOI: 10.1007/s10473-021-0322-y 摘要 ( 75 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标

Select

ANALYSIS OF THE GENOMIC DISTANCE BETWEEN BAT CORONAVIRUS RATG13 AND SARS-COV-2 REVEALS MULTIPLE ORIGINS OF COVID-19 裴少君, 丘成栋 数学物理学报(英文版). 2021 (3):  1017-1022.  DOI: 10.1007/s10473-021-0323-x 摘要 ( 222 )   RICH HTML PDF   收藏 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. 参考文献 | 相关文章 | 计量指标