Table of Content

25 October 2023, Volume 43 Issue 5 Previous Issue    Next Issue

For Selected:

View Abstracts

Download Citations EndNote Reference Manager ProCite BibTeX RefWorks

Toggle Thumbnails

Select

OBSTACLE PROBLEMS ON $RCD(K, N)$ METRIC MEASURE SPACES* Sitan Lin Acta mathematica scientia,Series B. 2023, 43 (5):  1925-1944.  DOI: 10.1007/s10473-023-0501-0 Abstract ( 75 )   RICH HTML Save In this paper, we solve the obstacle problems on metric measure spaces with generalized Ricci lower bounds. We show the existence and Lipschitz continuity of the solutions, and then we establish some regularities of the free boundaries. References | Related Articles | Metrics

Select

HYDROSTATIC LIMIT OF THE NAVIER-STOKES-ALPHA MODEL* Léo Glangetas, Van-Sang Ngo, El Mehdi Said Acta mathematica scientia,Series B. 2023, 43 (5):  1945-1980.  DOI: 10.1007/s10473-023-0502-z Abstract ( 56 )   RICH HTML Save In this paper we study the hydrostatic limit of the Navier-Stokes-alpha model in a very thin strip domain. We derive some Prandtl-type limit equations for this model and we prove the global well-posedness of the limit system for small initial conditions in an appropriate analytic function space. References | Related Articles | Metrics

Select

THE FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES* Yanyan han, Huoxiong wu Acta mathematica scientia,Series B. 2023, 43 (5):  1981-1996.  DOI: 10.1007/s10473-023-0503-y Abstract ( 44 )   RICH HTML Save This paper is devoted to studying the behaviors of the fractional type Marcinkiewicz integrals $\mu_{\Omega, \beta}$ and the commutators $\mu_{\Omega, \beta}^b$ generated by $\mu_{\Omega, \beta}$ with $b\in L_{\rm loc}(\mathbb{R}^n)$ on weighted Hardy spaces. Under the assumption of that the homogeneous kernel $\Omega$ satisfies certain regularities, the authors obtain the boundedness of $\mu_{\Omega, \beta}$ from the weighted Hardy spaces $H^p_{\omega^p}(\mathbb{R}^n)$ to the weighted Lebesgue spaces $L^q_{\omega^q}(\mathbb{R}^n)$ for $n/(n+\beta)\le p\le 1$ with $1/q=1/p-\beta/n$, as well as the same $(H^p_{\omega^p}, L^q_{\omega^q})$-boudedness of $\mu_{\Omega, \beta}^b$ when $b$ belongs to $\mathcal{BMO}_{\omega^p, p}(\mathbb{R}^n)$, which is a non-trivial subspace of ${\rm BMO}(\mathbb{R}^n)$. References | Related Articles | Metrics

Select

THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS* Qinghui LIU, Zhiyi Tang Acta mathematica scientia,Series B. 2023, 43 (5):  1997-2004.  DOI: 10.1007/s10473-023-0504-x Abstract ( 26 )   RICH HTML Save Let $\tau$ be a generalized Thue-Morse substitution on a two-letter alphabet $\{a, b\}$:$\tau(a)=a^mb^m$, $\tau(b)=b^ma^ms$ for the integer $m\ge 2$. Let $\xi$ be a sequence in $\{a, b\}^{\mathbb{Z}}$ that is generated by $\tau$. We study the one-dimensional Schrödinger operator $H_{m, \lambda}$ on $l^2(\mathbb{Z})$ with a potential given by $$v(n)=\lambda V_{\xi}(n), $$ where $\lambda>0$ is the coupling and $V_\xi(n)=1$ ($V_\xi(n)=-1$) if $\xi(n)=a$ ($\xi(n)=b$). Let $\Lambda_2=2$, and for $m>2$, let $\Lambda_m=m$ if $m\equiv0\mod 4$; let $\Lambda_m=m-3$ if $m\equiv1\mod 4$; let $\Lambda_m=m-2$ if $m\equiv2\mod 4$; let $\Lambda_m=m-1$ if $m\equiv3\mod 4$. We show that the Hausdorff dimension of the spectrum $\sigma(H_{m, \lambda})$ satisfies that $$\dim_H \sigma(H_{m, \lambda})> \frac{\log \Lambda_m}{\log 64m+4}.$$ It is interesting to see that $\dim_H \sigma(H_{m, \lambda})$ tends to $1$ as $m$ tends to infinity. References | Related Articles | Metrics

Select

INVERSE CONDUCTIVE MEDIUM SCATTERING WITH UNKNOWN BURIED OBJECTS* Fenglong Qu, Ruixue Jia, Yanli Cui Acta mathematica scientia,Series B. 2023, 43 (5):  2005-2025.  DOI: 10.1007/s10473-023-0505-9 Abstract ( 28 )   RICH HTML Save This paper is concerned with inverse acoustic scattering in an inhomogeneous medium with a conductive boundary condition and the unknown buried impenetrable objects inside. Using a variational approach, we establish the well-posedness of the direct problem. For the inverse problem, we shall numerically reconstruct the inhomogeneous medium from the far-field data for different kinds of cases. For the case when a Dirichlet boundary condition is imposed on the buried object, the classical factorization method proposed in [2] is justified as valid for reconstructing the inhomogeneous medium from the far-field data. For the case when a Neumann boundary condition is imposed on the buried object, the classical factorization method of [1] cannot be applied directly, since the middle operator of the factorization of the far-field operator is only compact. In this case, we develop a modified factorization method to locate the inhomogeneous medium with a conductive boundary condition and the unknown buried objects. Some numerical experiments are provided to demonstrate the practicability of the inversion algorithms developed. References | Related Articles | Metrics

Select

A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS* Yirang Yuan, Changfeng Li, Tongjun Sun, Qing Yang Acta mathematica scientia,Series B. 2023, 43 (5):  2026-2042.  DOI: 10.1007/s10473-023-0506-8 Abstract ( 28 )   RICH HTML Save In this paper a mixed finite element-characteristic mixed finite element method is discussed to simulate an incompressible miscible Darcy-Forchheimer problem. The flow equation is solved by a mixed finite element and the approximation accuracy of Darch-Forchheimer velocity is improved one order. The concentration equation is solved by the method of mixed finite element, where the convection is discretized along the characteristic direction and the diffusion is discretized by the zero-order mixed finite element method. The characteristics can confirm strong stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. In actual computations the characteristics adopts a large time step without any loss of accuracy. The scalar unknowns and its adjoint vector function are obtained simultaneously and the law of mass conservation holds in every element by the zero-order mixed finite element discretization of diffusion flux. In order to derive the optimal $3/2$-order error estimate in $L^2$ norm, a post-processing technique is included in the approximation to the scalar unknowns. Numerical experiments are illustrated finally to validate theoretical analysis and efficiency. This method can be used to solve such an important problem. References | Related Articles | Metrics

Select

GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA* Yaxian Ma, Xianwen Zhang Acta mathematica scientia,Series B. 2023, 43 (5):  2043-2060.  DOI: 10.1007/s10473-023-0507-7 Abstract ( 21 )   RICH HTML Save We investigate the global classical solutions of the non-relativistic Vlasov-Darwin system with generalized variables (VDG) in three dimensions. We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials. Then, we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system (VP) at the asymptotic rate of $\frac{1}{c^2}$ as the speed of light $c$ tends to infinity for all time. Moreover, we obtain rigorously an asymptotic estimate of the difference between the two systems. References | Related Articles | Metrics

Select

TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS* Khalid BOUABID, Rachid ECHARGHAOUI, Mohssine EL MANSOUR Acta mathematica scientia,Series B. 2023, 43 (5):  2061-2074.  DOI: 10.1007/s10473-023-0508-6 Abstract ( 17 )   RICH HTML Save In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents $ \left\{\begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u +\frac{ \vert u \vert ^{2^{*}(s)-2}u}{ \vert x \vert ^{s}}+ a(x) \vert u \vert ^{q-2} u & \;{\rm in} \; \Omega, \\ u=0 & \;{\rm on} \; \partial \Omega, \end{array}\right.$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with $0\in \partial \Omega$ and all the principle curvatures of $ \partial \Omega$ at 0 are negative, $a \in \mathcal{C}^{1}(\bar{\Omega}, \mathbb{R^{\ast}}^{+}), $ $ \mu> 0, $ $0<s<2, $ $1<q<2$ and $N > 2\frac{q+1}{q -1}.$ By $2^{*}:=\frac{2 N}{N-2}$ and $2^{*}(s):=\frac{2 (N-s)}{N-2}$ we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively. References | Related Articles | Metrics

Select

THE FEKETE-SZEGÖ INEQUALITY AND SUCCESSIVE COEFFICIENTS DIFFERENCE FOR A SUBCLASS OF CLOSE-TO-STARLIKE MAPPINGS IN COMPLEX BANACH SPACES* Qinghua XU, Weikang FANG, Weiheng FENG, Taishun LIU Acta mathematica scientia,Series B. 2023, 43 (5):  2075-2088.  DOI: 10.1007/s10473-023-0509-5 Abstract ( 27 )   RICH HTML Save Let $\mathcal{C}$ be the familiar class of normalized close-to-convex functions in the unit disk. In [17], Koepf demonstrated that, as to a function $f(\xi)=\xi+\sum\limits_{m=2}^\infty a_m\xi^m$ in the class $\mathcal{C}$, $$ \max\limits_{f\in \mathcal{C}}|a_3-\lambda a_2^2|\leq \left\{\begin{array}{ll} 3-4\lambda, \quad & \lambda\in[0, \frac{1}{3}], \\[3mm] \frac{1}{3}+\frac{4}{9\lambda}, \quad & \lambda\in[\frac{1}{3}, \frac{2}{3}], \\[3mm] 1, \quad & \lambda\in[\frac{2}{3}, 1]. \end{array}\right.$$ By applying this inequality, it can be proven that $||a_3|-|a_2||\leq 1$ for close-to-convex functions. Now we generalized the above conclusions to a subclass of close-to-starlike mappings defined on the unit ball of a complex Banach space. References | Related Articles | Metrics

Select

GLOBAL WEAK SOLUTIONS TO A THREE-DIMENSIONAL QUANTUM KINETIC-FLUID MODEL* Fucai Li, Yue Li, Baoyan Sun Acta mathematica scientia,Series B. 2023, 43 (5):  2089-2107.  DOI: 10.1007/s10473-023-0510-z Abstract ( 20 )   RICH HTML Save In this paper, we study a quantum kinetic-fluid model in a three-dimensional torus. This model is a coupling of the Vlasov-Fokker-Planck equation and the compressible quantum Navier-Stokes equations with degenerate viscosity. We establish a global weak solution to this model for arbitrarily large initial data when the pressure takes the form $ p(\rho)=\rho^\gamma+p_c(\rho)$, where $\gamma>1$ is the adiabatic coefficient and $p_c(\rho)$ satisfies \begin{equation*} p_c(\rho)=\left\{ \begin{array}{ll}-c\rho^{-4k}\; \;\;\;&{\rm{if}}\;\;\rho\leq 1, \\ \rho^{\gamma}\;\;\;\;&{\rm{if}}\;\;\rho>1 \end{array} \right. \end{equation*} for $k\geq 4$ and some constant $c>0$. References | Related Articles | Metrics

Select

DYNAMICS OF THE STOCHASTIC $g$-NAVIER-STOKES EQUATIONS DRIVEN BY NONLINEAR NOISE* Tao Yan, Lu Zhang, Aihong Zou, Ji Shu Acta mathematica scientia,Series B. 2023, 43 (5):  2108-2120.  DOI: 10.1007/s10473-023-0511-y Abstract ( 19 )   RICH HTML Save This paper deals with the asymptotic behavior of solutions of the stochastic $g$-Navier-Stokes equation driven by nonlinear noise. The existence and uniqueness of weak pullback mean random attractors for the equation in Bochner space is proven for when the diffusion terms are Lipschitz nonlinear functions. Furthermore, we also establish the existence of invariant measures for the equation. References | Related Articles | Metrics

Select

COMPLEX SYMMETRIC $C_0$-SEMIGROUPS ON THE WEIGHTED HARDY SPACES $H_\gamma(\mathbb{D})$* Xiaohe Hu Acta mathematica scientia,Series B. 2023, 43 (5):  2121-2132.  DOI: 10.1007/s10473-023-0512-x Abstract ( 15 )   RICH HTML Save In this paper, we study the complex symmetric $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on the weighted Hardy spaces $H_\gamma$ of the unit disk $\mathbb{D}$. It is well-known that there are only two classes of weighted composition conjugations $\mathcal{A}_{u, v}$ on $H_\gamma(\mathbb{D})$: either $\mathcal{C}_1$ or $\mathcal{C}_2$. We completely characterize $\mathcal{C}_1$-symmetric ($\mathcal{C}_2$-symmetric) $C_0$-semigroups of weighted composition operators $W_{\psi, \varphi}$ on $H_\gamma(\mathbb{D})$.} References | Related Articles | Metrics

Select

THE ASYMPTOTIC STABILITY OF PHASE SEPARATION STATES FOR COMPRESSIBLE IMMISCIBLE TWO-PHASE FLOW IN 3D* Yazhou CHEN, Hakho HONG, Xiaoding SHI Acta mathematica scientia,Series B. 2023, 43 (5):  2133-2158.  DOI: 10.1007/s10473-023-0513-9 Abstract ( 16 )   RICH HTML Save This paper is concerned with a diffuse interface model called Navier-Stokes/Cahn-Hilliard system. This model is usually used to describe the motion of immiscible two-phase flows with a diffusion interface. For the periodic boundary value problem of this system in torus $\mathbb{T}^3$, we prove that there exists a global unique strong solution near the phase separation state, which means that no vacuum, shock wave, mass concentration, interface collision or rupture will be developed in finite time. Furthermore, we establish the large time behavior of the global strong solution of this system. In particular, we find that the phase field decays algebraically to the phase separation state. References | Related Articles | Metrics

Select

THE UNIFORM CONVERGENCE OF A DG METHOD FOR A SINGULARLY PERTURBED VOLTERRA INTEGRO-DIFFERENTIAL EQUATION* Xia Tao, Ziqing Xie Acta mathematica scientia,Series B. 2023, 43 (5):  2159-2178.  DOI: 10.1007/s10473-023-0514-8 Abstract ( 19 )   RICH HTML Save The purpose of this work is to implement a discontinuous Galerkin (DG) method with a one-sided flux for a singularly perturbed Volterra integro-differential equation (VIDE) with a smooth kernel. First, the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided. Then the existence and uniqueness of the DG solution are proven. Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established. Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants, the DG method achieves the uniform convergence in the $L^2$ norm with respect to the singular perturbation parameter $\epsilon$ when the space of polynomials with degree $p$ is used. A numerical experiment validates the theoretical results. Furthermore, an ultra-convergence order $2p+1$ at the nodes for the one-sided flux, uniform with respect to the singular perturbation parameter $\epsilon$, is observed numerically. References | Related Articles | Metrics

Select

NONLINEAR STABILITY OF RAREFACTION WAVES TO THE COMPRESSIBLE NAVIER-STOKES EQUATIONS FOR A REACTING MIXTURE WITH ZERO HEAT CONDUCTIVITY* Lishuang Peng, Yong Li Acta mathematica scientia,Series B. 2023, 43 (5):  2179-2203.  DOI: 10.1007/s10473-023-0515-7 Abstract ( 17 )   RICH HTML Save In this paper, we study the time-asymptotically nonlinear stability of rarefaction waves for the Cauchy problem of the compressible Navier-Stokes equations for a reacting mixture with zero heat conductivity in one dimension. If the corresponding Riemann problem for the compressible Euler system admits the solutions consisting of rarefaction waves only, it is shown that its Cauchy problem has a unique global solution which tends time-asymptotically towards the rarefaction waves, while the initial perturbation and the strength of rarefaction waves are suitably small. References | Related Articles | Metrics

Select

MEAN APPROXIMATION BY DILATATIONS IN BERGMAN SPACES ON THE UPPER HALF-PLANE* Ali Abkar Acta mathematica scientia,Series B. 2023, 43 (5):  2204-2214.  DOI: 10.1007/s10473-023-0516-6 Abstract ( 17 )   RICH HTML Save We study sufficient conditions on radial and non-radial weight functions on the upper half-plane that guarantee norm approximation of functions in weighted Bergman, weighted Dirichlet, and weighted Besov spaces on the upper half-plane by dilatations and eventually by analytic polynomials. References | Related Articles | Metrics

Select

ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION WITH LARGE INITIAL DATA* Mingjuan Chen, Shuai Zhang Acta mathematica scientia,Series B. 2023, 43 (5):  2215-2233.  DOI: 10.1007/s10473-023-0517-5 Abstract ( 13 )   RICH HTML Save We consider the fourth-order nonlinear Schrödinger equation (4NLS) \begin{align*} ({\rm i}\partial_t+\varepsilon\Delta+\Delta^2)u=c_1u^m+c_2(\partial u)u^{m-1}+c_3(\partial u)^2u^{m-2}, \end{align*} and establish the conditional almost sure global well-posedness for random initial data in $H^s(\mathbb{R}^d)$ for $s\in (s_c-1/2, \ s_c]$, when $d\geq3$ and $m\geq5$, where $s_c:=d/2-2/(m-1)$ is the scaling critical regularity of 4NLS with the second order derivative nonlinearities. Our proof relies on the nonlinear estimates in a new $M$-norm and the stability theory in the probabilistic setting. Similar supercritical global well-posedness results also hold for $d=2, \ m\geq4$ and $ d\geq3, \ 3\leq m<5$. References | Related Articles | Metrics

Select

GENERAL COUPLED MEAN-FIELD REFLECTED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS* Junsong LI, Chao MI, Chuanzhi XING, Dehao ZHAO Acta mathematica scientia,Series B. 2023, 43 (5):  2234-2262.  DOI: 10.1007/s10473-023-0518-4 Abstract ( 16 )   RICH HTML Save In this paper we consider general coupled mean-field reflected forward-backward stochastic differential equations (FBSDEs), whose coefficients not only depend on the solution but also on the law of the solution. The first part of the paper is devoted to the existence and the uniqueness of solutions for such general mean-field reflected backward stochastic differential equations (BSDEs) under Lipschitz conditions, and for the one-dimensional case a comparison theorem is studied. With the help of this comparison result, we prove the existence of the solution for our mean-field reflected forward-backward stochastic differential equation under continuity assumptions. It should be mentioned that, under appropriate assumptions, we prove the uniqueness of this solution as well as that of a comparison theorem for mean-field reflected FBSDEs in a non-trivial manner. References | Related Articles | Metrics

Select

THE GLOBAL LIPSCHITZ SOLUTION FOR A PEELING MODEL* Qianfeng Li, Yongqian Zhang Acta mathematica scientia,Series B. 2023, 43 (5):  2263-2278.  DOI: 10.1007/s10473-023-0519-3 Abstract ( 22 )   RICH HTML Save This paper focusses on a peeling phenomenon governed by a nonlinear wave equation with a free boundary. Under the hypotheses that the total variation of the intial data and the boundary data are small, the global existence of a weak solution to the nonlinear problem (1.1)-(1.3) is proven by a modified Glimm scheme. The regularity of the peeling front is established, and the asymptotic behaviour of the obtained solution and the peeling front at infinity is also studied. References | Related Articles | Metrics

Select

POSITIVE CLASSICAL SOLUTIONS OF DIRICHLET PROBLEM FOR THE STEADY RELATIVISTIC HEAT EQUATION* Tianjie YANG, Guangwei YUAN Acta mathematica scientia,Series B. 2023, 43 (5):  2279-2290.  DOI: 10.1007/s10473-023-0520-x Abstract ( 14 )   RICH HTML Save In this paper, for a bounded $C^2$ domain, we prove the existence and uniqueness of positive classical solutions to the Dirichlet problem for the steady relativistic heat equation with a class of restricted positive $C^2$ boundary data. We have a non-existence result, which is the justification for taking into account the restricted boundary data. There is a smooth positive boundary datum that precludes the existence of the positive classical solution. References | Related Articles | Metrics

Select

SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH CRITICAL GROWTH* Yinbin Deng, Wei Shuai, Xiaolong Yang Acta mathematica scientia,Series B. 2023, 43 (5):  2291-2308.  DOI: 10.1007/s10473-023-0521-9 Abstract ( 19 )   RICH HTML Save In this paper, we study the following Schrödinger-Poisson system with critical growth: \begin{equation*} \begin{cases}-\Delta u+V(x)u+ \phi(x)u =f(u)+|u|^4u, \ & x\in\mathbb{R}^3, \\ -\Delta \phi=u^2, \ & x\in\mathbb{R}^3. \end{cases} \end{equation*} We establish the existence of a positive ground state solution and a least energy sign-changing solution, providing that the nonlinearity $f$ is super-cubic, subcritical and that the potential $V(x)$ has a potential well. References | Related Articles | Metrics

Select

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS* Yali PAN Acta mathematica scientia,Series B. 2023, 43 (5):  2309-2319.  DOI: 10.1007/s10473-023-0522-8 Abstract ( 23 )   RICH HTML Save Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds. References | Related Articles | Metrics

Select

RIGHT MEAN FOR THE $\alpha$-z BURES-WASSERSTEIN QUANTUM DIVERGENCE* Miran Jeong, Jinmi Hwangm, Sejong Kim Acta mathematica scientia,Series B. 2023, 43 (5):  2320-2332.  DOI: 10.1007/s10473-023-0523-7 Abstract ( 30 )   RICH HTML Save The optimization problem to minimize the weighted sum of $\alpha$-z Bures-Wasserstein quantum divergences to given positive definite Hermitian matrices has been solved. We call the unique minimizer the $\alpha$-z weighted right mean, which provides a new non-commutative version of generalized mean (Hölder mean). We investigate its fundamental properties, and give many interesting operator inequalities with the matrix power mean including the Cartan mean. Moreover, we verify the trace inequality with the Wasserstein mean and provide bounds for the Hadamard product of two right means. References | Related Articles | Metrics