#### Table of Content

25 February 2022, Volume 42 Issue 1
Articles
 UNDERSTANDING SCHUBERT'S BOOK (II) Banghe LI Acta mathematica scientia,Series B. 2022, 42 (1):  1-48.  DOI: 10.1007/s10473-022-0101-4 In this paper, we give rigorous justification of the ideas put forward in §20, Chapter 4 of Schubert's book; a section that deals with the enumeration of conics in space. In that section, Schubert introduced two degenerate conditions about conics, i.e., the double line and the two intersection lines. Using these two degenerate conditions, he obtained all relations regarding the following three conditions:conics whose planes pass through a given point, conics intersecting with a given line, and conics which are tangent to a given plane. We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert's idea.
 MOMENTS AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION IN RANDOM ENVIRONMENTS Chunmao HUANG, Chen WANG, Xiaoqiang WANG Acta mathematica scientia,Series B. 2022, 42 (1):  49-72.  DOI: 10.1007/s10473-022-0102-3 Let (Zn) be a branching process with immigration in a random environment ξ, where ξ is an independent and identically distributed sequence of random variables. We show asymptotic properties for all the moments of Zn and describe the decay rates of the n-step transition probabilities. As applications, a large deviation principle for the sequence log Zn is established, and related large deviations are also studied.
 FURTHER EXTENSIONS OF SOME TRUNCATED HECKE TYPE IDENTITIES Helen W. J. ZHANG Acta mathematica scientia,Series B. 2022, 42 (1):  73-90.  DOI: 10.1007/s10473-022-0103-2 The main purpose of this paper is to generalize the study of the Hecke-Rogers type series, which are the extensions of truncated theorems obtained by Andrews, Merca, Wang and Yee. Our proofs rely heavily on the theory of Bailey pairs.
 ASYMPTOTIC GROWTH BOUNDS FOR THE VLASOV-POISSON SYSTEM WITH RADIATION DAMPING Yaxian MA, Xianwen ZHANG Acta mathematica scientia,Series B. 2022, 42 (1):  91-104.  DOI: 10.1007/s10473-022-0104-1 We consider asymptotic behaviors of the Vlasov-Poisson system with radiation damping in three space dimensions. For any smooth solution with compact support, we prove a sub-linear growth estimate of its velocity support. As a consequence, we derive some new estimates of the charge densities and the electrostatic field in this situation.
 CONVERGENCE RESULTS FOR NON-OVERLAP SCHWARZ WAVEFORM RELAXATION ALGORITHM WITH CHANGING TRANSMISSION CONDITIONS Minh-Phuong TRAN, Thanh-Nhan NGUYEN, Phuoc-Toan HUYNH, Nhu-Binh LY, Minh-Dang NGUYEN, Quoc-Anh HO Acta mathematica scientia,Series B. 2022, 42 (1):  105-126.  DOI: 10.1007/s10473-022-0105-0 In this paper, we establish a new algorithm to the non-overlapping Schwarz domain decomposition methods with changing transmission conditions for solving one dimensional advection reaction diffusion problem. More precisely, we first describe the new algorithm and prove the convergence results under several natural assumptions on the sequences of parameters which determine the transmission conditions. Then we give a simple method to estimate the new value of parameters in each iteration. The interesting advantage of our method is that one may update the better parameters in each iteration to save the computational cost for optimizing the parameters after many steps. Finally some numerical experiments are performed to show the behavior of the convergence rate for the new method.
 RIEMANN-HILBERT PROBLEMS AND SOLITON SOLUTIONS OF NONLOCAL REVERSE-TIME NLS HIERARCHIES Wenxiu MA Acta mathematica scientia,Series B. 2022, 42 (1):  127-140.  DOI: 10.1007/s10473-022-0106-z The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrödinger (NLS) hierarchies associated with higher-order matrix spectral problems. The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations. A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix, corresponding to the reflectionless inverse scattering transforms, is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.
 THEORETICAL AND NUMERICAL STUDY OF THE BLOW UP IN A NONLINEAR VISCOELASTIC PROBLEM WITH VARIABLE-EXPONENT AND ARBITRARY POSITIVE ENERGY Ala A. TALAHMEH, Salim A. MESSAOUDI, Mohamed ALAHYANE Acta mathematica scientia,Series B. 2022, 42 (1):  141-154.  DOI: 10.1007/s10473-022-0107-y In this paper, we consider the following nonlinear viscoelastic wave equation with variable exponents:$u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(x,\tau){\rm d}\tau +\mu u_t=|u|^{p(x)-2}u,$ where $\mu$ is a nonnegative constant and the exponent of nonlinearity $p(\cdot)$ and $g$ are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function $g$, we prove a finite-time blow-up result. We also give some numerical applications to illustrate our theoretical results.
 EXISTENCE OF PERIODIC SOLUTIONS TO AN ISOTHERMAL RELATIVISTIC EULER SYSTEM Fei WU, Zejun WANG Acta mathematica scientia,Series B. 2022, 42 (1):  155-171.  DOI: 10.1007/s10473-022-0108-x In this paper, we study the global existence of periodic solutions to an isothermal relativistic Euler system in BV space. First, we analyze some properties of the shock and rarefaction wave curves in the Riemann invariant plane. Based on these properties, we construct the approximate solutions of the isothermal relativistic Euler system with periodic initial data by using a Glimm scheme, and prove that there exists an entropy solution $V(x,t)$ which belongs to $L^{\infty}\cap {\rm BV}_{\rm loc}(\mathbb{R}\times\mathbb{R}_+)$.
 THE VALUE DISTRIBUTION OF GAUSS MAPS OF IMMERSED HARMONIC SURFACES WITH RAMIFICATION Zhixue LIU, Yezhou LI, Xingdi CHEN Acta mathematica scientia,Series B. 2022, 42 (1):  172-186.  DOI: 10.1007/s10473-022-0109-9 Motivated by the result of Chen-Liu-Ru[1], we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in $\Bbb{R}^n$ with ramification, which can be seen as a generalization of the results in the case of the minimal surfaces. In addition, we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.
 PERIODIC AND ALMOST PERIODIC SOLUTIONS FOR A NON-AUTONOMOUS RESPIRATORY DISEASE MODEL WITH A LAG EFFECT Lei SHI, Longxing QI, Sulan ZHAI Acta mathematica scientia,Series B. 2022, 42 (1):  187-211.  DOI: 10.1007/s10473-022-0110-3 This paper studies a kind of non-autonomous respiratory disease model with a lag effect. First of all, the permanence and extinction of the system are discussed by using the comparison principle and some differential inequality techniques. Second, it assumes that all coefficients of the system are periodic. The existence of positive periodic solutions of the system is proven, based on the continuation theorem in coincidence with the degree theory of Mawhin and Gaines. In the meantime, the global attractivity of positive periodic solutions of the system is obtained by constructing an appropriate Lyapunov functional and using the Razumikin theorem. In addition, the existence and uniform asymptotic stability of almost periodic solutions of the system are analyzed by assuming that all parameters in the model are almost periodic in time. Finally, the theoretical derivation is verified by a numerical simulation.
 GENERALIZED CESÀRO OPERATORS ON DIRICHLET-TYPE SPACES Jianjun JIN, Shuan TANG Acta mathematica scientia,Series B. 2022, 42 (1):  212-220.  DOI: 10.1007/s10473-022-0111-2 In this note, we introduce and study a new kind of generalized Cesàro operator, $\mathcal{C}_{\mu}$, induced by a positive Borel measure $\mu$ on $[0, 1)$ between Dirichlet-type spaces. We characterize the measures $\mu$ for which $\mathcal{C}_{\mu}$ is bounded (compact) from one Dirichlet-type space, $\mathcal{D}_{\alpha}$, into another one, $\mathcal{D}_{\beta}$.
 A STRONG CONVERGENCE THEOREM FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES Mehdi MOHAMMADI, G. Zamani ESKANDANI Acta mathematica scientia,Series B. 2022, 42 (1):  221-232.  DOI: 10.1007/s10473-022-0112-1 In this paper, we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces. We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.
 THE GLOBAL EXISTENCE OF STRONG SOLUTIONS TO THE 3D COMPRESSIBLE ISOTHERMAL NAVIER-STOKES EQUATIONS Haibo YU Acta mathematica scientia,Series B. 2022, 42 (1):  233-246.  DOI: 10.1007/s10473-022-0113-0 This paper concerns the global existence of strong solutions to the 3D compressible isothermal Navier-Stokes equations with a vacuum at infinity. Based on the special structure of the Zlotnik inequality, the time uniform upper bounds for density are established through some time-dependant a priori estimates under the assumption that the total mass is suitably small.
 A GENERALIZED PENALTY METHOD FOR DIFFERENTIAL VARIATIONAL-HEMIVARIATIONAL INEQUALITIES Liang LU, Lijie LI, Mircea SOFONEA Acta mathematica scientia,Series B. 2022, 42 (1):  247-264.  DOI: 10.1007/s10473-022-0114-z We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in[1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.
 GEVREY CLASS REGULARITY FOR THE GLOBAL ATTRACTOR OF A TWO-DIMENSIONAL NON-NEWTONIAN FLUID Caidi ZHAO, Zehan LIN, T. Tachim MEDJO Acta mathematica scientia,Series B. 2022, 42 (1):  265-282.  DOI: 10.1007/s10473-022-0115-y This article investigates Gevrey class regularity for the global attractor of an incompressible non-Newtonian fluid in a two-dimensional domain with a periodic boundary condition. This Gevrey class regularity reveals that the solutions lying in the global attractor are analytic in time with values in a Gevrey class of analytic functions in space.
 OPTIMAL CONTROL OF A POPULATION DYNAMICS MODEL WITH HYSTERESIS Bin CHEN, Sergey A. TIMOSHIN Acta mathematica scientia,Series B. 2022, 42 (1):  283-298.  DOI: 10.1007/s10473-022-0116-x This paper addresses a nonlinear partial differential control system arising in population dynamics. The system consist of three diffusion equations describing the evolutions of three biological species:prey, predator, and food for the prey or vegetation. The equation for the food density incorporates a hysteresis operator of generalized stop type accounting for underlying hysteresis effects occurring in the dynamical process. We study the problem of minimization of a given integral cost functional over solutions of the above system. The set-valued mapping defining the control constraint is state-dependent and its values are nonconvex as is the cost integrand as a function of the control variable. Some relaxation-type results for the minimization problem are obtained and the existence of a nearly optimal solution is established.
 ANISOTROPIC (p,q)-EQUATIONS WITH COMPETITION PHENOMENA Zhenhai LIU, Nikolaos S. PAPAGEORGIOU Acta mathematica scientia,Series B. 2022, 42 (1):  299-322.  DOI: 10.1007/s10473-022-0117-9 We consider a nonlinear Robin problem driven by the anisotropic (p, q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear (concave) term and of a superlinear (convex) term. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies. We also prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.
 THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS Lidan WANG Acta mathematica scientia,Series B. 2022, 42 (1):  323-348.  DOI: 10.1007/s10473-022-0118-8 This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains. After establishing the weak maximum principles, the global boundary Hölder estimates and the boundary Harnack inequalities of the equations, we show that all solutions bounded from below are linear combinations of two special solutions (exponential growth at one end and exponential decay at the other) with a bounded solution to the degenerate equations.
 TOEPLITZ OPERATORS FROM HARDY SPACES TO WEIGHTED BERGMAN SPACES IN THE UNIT BALL OF Cn Ru PENG, Yaqing FAN Acta mathematica scientia,Series B. 2022, 42 (1):  349-363.  DOI: 10.1007/s10473-022-0119-7 We study Toeplitz operators from Hardy spaces to weighted Bergman spaces in the unit ball of $C^{n}$. Toeplitz operators are closely related to many classical mappings, such as composition operators, the Volterra type integration operators and Carleson embeddings. We characterize the boundedness and compactness of Toeplitz operators from Hardy spaces $H^{p}$ to weighted Bergman spaces $A_{\alpha}^{q}$ for the different values of $p$ and $q$ in the unit ball.
 DYNAMICAL BEHAVIOR OF AN INNOVATION DIFFUSION MODEL WITH INTRA-SPECIFIC COMPETITION BETWEEN COMPETING ADOPTERS Rakesh KUMAR, Anuj Kumar SHARMA, Govind Prasad SAHU Acta mathematica scientia,Series B. 2022, 42 (1):  364-386.  DOI: 10.1007/s10473-022-0120-1 In this paper, we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation (product) in a particular region. The model exhibits two equilibria, namely, the adopter-free and an interior equilibrium. The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number (BIN) RA. It is investigated that the adopter free steady-state is stable if RA < 1. By considering τ (the adoption experience of the adopters) as the bifurcation parameter, we have been able to obtain the critical value of τ responsible for the periodic solutions due to Hopf bifurcation. The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem. Exhaustive numerical simulations in the support of analytical results have been presented.
 A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY Weishan ZHENG, Yanping CHEN Acta mathematica scientia,Series B. 2022, 42 (1):  387-402.  DOI: 10.1007/s10473-022-0121-0 In this paper, a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay, which contains a weakly singular kernel. We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1, 1], so that the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high order accuracy for the approximation, the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules. In the end, we provide a rigorous error analysis for the proposed method. The spectral rate of convergence for the proposed method is established in both the L∞-norm and the weighted L2-norm.
 HYBRID REGULARIZED CONE-BEAM RECONSTRUCTION FOR AXIALLY SYMMETRIC OBJECT TOMOGRAPHY Xinge LI, Suhua WEI, Haibo XU, Chong CHEN Acta mathematica scientia,Series B. 2022, 42 (1):  403-419.  DOI: 10.1007/s10473-022-0122-z In this paper, we consider 3D tomographic reconstruction for axially symmetric objects from a single radiograph formed by cone-beam X-rays. All contemporary density reconstruction methods in high-energy X-ray radiography are based on the assumption that the cone beam can be treated as fan beams located at parallel planes perpendicular to the symmetric axis, so that the density of the whole object can be recovered layer by layer. Considering the relationship between different layers, we undertake the cone-beam global reconstruction to solve the ambiguity effect at the material interfaces of the reconstruction results. In view of the anisotropy of classical discrete total variations, a new discretization of total variation which yields sharp edges and has better isotropy is introduced in our reconstruction model. Furthermore, considering that the object density consists of continually changing parts and jumps, a high-order regularization term is introduced. The final hybrid regularization model is solved using the alternating proximal gradient method, which was recently applied in image processing. Density reconstruction results are presented for simulated radiographs, which shows that the proposed method has led to an improvement in terms of the preservation of edge location.
 UNBOUNDED COMPLEX SYMMETRIC TOEPLITZ OPERATORS Kaikai HAN, Maofa WANG, Qi WU Acta mathematica scientia,Series B. 2022, 42 (1):  420-428.  DOI: 10.1007/s10473-022-0123-y In this paper, we study unbounded complex symmetric Toeplitz operators on the Hardy space $H^{2}(\mathbb{D})$ and the Fock space $\mathscr{F}^{2}$. The technique used to investigate the complex symmetry of unbounded Toeplitz operators is different from that used to investigate the complex symmetry of bounded Toeplitz operators.
 OPERATOR NORM AND LOWER BOUND OF FOUR-DIMENSIONAL GENERALIZED HAUSDORFF MATRICES Gholamreza TALEBI Acta mathematica scientia,Series B. 2022, 42 (1):  429-436.  DOI: 10.1007/s10473-022-0124-x The problem addressed is the exact determination of the operator norm and lower bound of four-dimensional generalized Hausdorff matrices on the double sequence spaces $\mathcal{L}_p$. A Hardy type formulae is found as an exact value for their operator norm and a Copson type formulae is established as a lower estimate for their lower bound. Further, exact values are found for the operator norm and lower bound of the transpose of generalized Hausdorff matrices.