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    MAXIMAL $L^1$-REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES
    Myong-Hwan RI, Reinhard FARWIG
    Acta mathematica scientia,Series B    2022, 42 (4): 1261-1272.   DOI: 10.1007/s10473-022-0401-8
    Abstract87)      PDF       Save
    In this paper, we prove that the generator of any bounded analytic semigroup in $(\theta,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.
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    SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES
    Xiaosong LIU
    Acta mathematica scientia,Series B    2022, 42 (2): 454-466.   DOI: 10.1007/s10473-022-0202-0
    Abstract69)      PDF       Save
    In this paper, we first establish the sharp growth theorem and the distortion theorem of the Frechét derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cn with some restricted conditions. We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of Cn with an arbitrary norm and the unit polydisk in Cn under certain restricted assumptions. Finally we obtain the sharp Goluzin type distortion theorem for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cn with some additional conditions. The results derived all reduce to the corresponding classical results in one complex variable, and include some known results from the prior literature.
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    RELAXED INERTIAL METHODS FOR SOLVING SPLIT VARIATIONAL INEQUALITY PROBLEMS WITHOUT PRODUCT SPACE FORMULATION
    Grace Nnennaya OGWO, Chinedu IZUCHUKWU, Oluwatosin Temitope MEWOMO
    Acta mathematica scientia,Series B    2022, 42 (5): 1701-1733.   DOI: 10.1007/s10473-022-0501-5
    Abstract66)            Save
    Many methods have been proposed in the literature for solving the split variational inequality problem. Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space, or that the underlying operators are co-coercive. However, it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem. On the other hand, the co-coercive assumption of the underlying operators would preclude the potential applications of these methods. To avoid these setbacks, we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation, and for which the underlying operators are freed from the restrictive co-coercive assumption. The methods proposed, involve projections onto half-spaces only, and originate from an explicit discretization of a dynamical system, which combines both the inertial and relaxation techniques in order to achieve high convergence speed. Moreover, the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces. Furthermore, numerical implementations and comparisons are given to support our theoretical findings.
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    UNDERSTANDING SCHUBERT'S BOOK (III)
    Banghe LI
    Acta mathematica scientia,Series B    2022, 42 (2): 437-453.   DOI: 10.1007/s10473-022-0201-1
    Abstract64)      PDF       Save
    In §13 of Schubert's famous book on enumerative geometry, he provided a few formulas called coincidence formulas, which deal with coincidence points where a pair of points coincide. These formulas play an important role in his method. As an application, Schubert utilized these formulas to give a second method for calculating the number of planar curves in a one dimensional system that are tangent to a given planar curve. In this paper, we give proofs for these formulas and justify his application to planar curves in the language of modern algebraic geometry. We also prove that curves that are tangent to a given planar curve is actually a condition in the space of planar curves and other relevant issues.
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    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
    Abstract58)      PDF       Save
    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.
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    STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS
    Li-Xin ZHANG
    Acta mathematica scientia,Series B    2022, 42 (2): 467-490.   DOI: 10.1007/s10473-022-0203-z
    Abstract52)      PDF       Save
    Limit theorems for non-additive probabilities or non-linear expectations are challenging issues which have attracted a lot of interest recently. The purpose of this paper is to study the strong law of large numbers and the law of the iterated logarithm for a sequence of random variables in a sub-linear expectation space under a concept of extended independence which is much weaker and easier to verify than the independence proposed by Peng[20]. We introduce a concept of extended negative dependence which is an extension of the kind of weak independence and the extended negative independence relative to classical probability that has appeared in the recent literature. Powerful tools such as moment inequality and Kolmogorov's exponential inequality are established for these kinds of extended negatively independent random variables, and these tools improve a lot upon those of Chen, Chen and Ng[1]. The strong law of large numbers and the law of iterated logarithm are also obtained by applying these inequalities.
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    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
    Abstract48)      PDF       Save
    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.
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    UNDERSTANDING SCHUBERT'S BOOK (II)
    Banghe LI
    Acta mathematica scientia,Series B    2022, 42 (1): 1-48.   DOI: 10.1007/s10473-022-0101-4
    Abstract47)      PDF       Save
    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.
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    GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION
    Xueting Jin, Yuelong Xiao, Huan Yu
    Acta mathematica scientia,Series B    2022, 42 (4): 1293-1309.   DOI: 10.1007/s10473-022-0403-6
    Abstract42)      PDF       Save
    In this paper, we prove the global well-posedness of the 2D Boussinesq equations with three kinds of partial dissipation; among these the initial data $(u_0,\theta_0)$ is required such that its own and the derivative of one of its directions $(x,y)$ are assumed to be $L^2(\mathbb R^2)$. Our results only need the lower regularity of the initial data, which ensures the uniqueness of the solutions.
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    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
    Abstract42)      PDF       Save
    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.
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    A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM
    Jin DENG, Benniao LI
    Acta mathematica scientia,Series B    2022, 42 (5): 1743-1764.   DOI: 10.1007/s10473-022-0503-3
    Abstract36)            Save
    In this paper, we consider the Chern-Simons-Schrödinger system \begin{equation*}\left\{\begin{array}{lll} - \Delta u+\left[e^{2}|\mathbf{A}|^{2}+\left(V(x)+2e A_{0}\right)+2\left(1+\frac{\kappa q}{2 }\right) N\right] u+ q |u|^{p-2}u=0, \\ -\Delta N+\kappa^{2} q^{2} N+q\left(1+\frac{\kappa q}{2}\right) u^{2}=0, \\ \kappa\left(\partial_{1} A_{2}-\partial_{2} A_{1}\right)= - e u^{2}, \, \, \partial_{1} A_{1}+\partial_{2} A_{2}=0, \\ \kappa \partial_{1} A_{0}= e^{2} A_{2} u^{2}, \, \, \kappa \partial_{2} A_{0}= - e^{2} A_{1} u^{2}, \, \, \end{array} \right.{\rm (P)} \end{equation*} where $u \in H^{1}(\mathbb{R}^{2})$, $p \in (2, 4)$, $A_{\alpha}: \mathbb{R}^{2} \rightarrow \mathbb{R}$ are the components of the gauge potential $(\alpha=0, 1, 2)$, $N: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a neutral scalar field, $V(x)$ is a potential function, the parameters $ \kappa, q>0$ represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and $ e>0$ is the coupling constant. In this paper, the truncation function is used to deal with a neutral scalar field and a gauge field in the Chern-Simons-Schrödinger problem. The ground state solution of the problem (P) is obtained by using the variational method.
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    ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE
    Tiehong ZHAO, Miaokun WANG, Yuming CHU
    Acta mathematica scientia,Series B    2022, 42 (2): 491-501.   DOI: 10.1007/s10473-022-0204-y
    Abstract36)      PDF       Save
    In this paper, we present new bounds for the perimeter of an ellipse in terms of harmonic, geometric, arithmetic and quadratic means; these new bounds represent improvements upon some previously known results.
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    WEIGHTED NORM INEQUALITIES FOR COMMUTATORS OF THE KATO SQUARE ROOT OF SECOND ORDER ELLIPTIC OPERATORS ON $\mathbb R^n$
    Yanping CHEN, Yong DING, Kai ZHU
    Acta mathematica scientia,Series B    2022, 42 (4): 1310-1332.   DOI: 10.1007/s10473-022-0404-5
    Abstract33)      PDF       Save
    Let $L=-\mathrm{div}(A\nabla)$ be a second order divergence form elliptic operator with bounded measurable coefficients in ${\Bbb R}^n$. We establish weighted $L^p$ norm inequalities for commutators generated by $\sqrt{L}$ and Lipschitz functions, where the range of $p$ is different from $(1,\infty)$, and we isolate the right class of weights introduced by Auscher and Martell. In this work, we use good-$\lambda$ inequality with two parameters through the weighted boundedness of Riesz transforms $\nabla L^{-1/2}$. Our result recovers, in some sense, a previous result of Hofmann.
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    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
    Abstract30)      PDF       Save
    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.
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    $\mathcal{O}(t^{-\beta})$-SYNCHRONIZATION AND ASYMPTOTIC SYNCHRONIZATION OF DELAYED FRACTIONAL ORDER NEURAL NETWORKS
    Anbalagan PRATAP, Ramachandran RAJA, Jinde CAO, Chuangxia HUANG, Chuangxia HUANG, Ovidiu BAGDASAR
    Acta mathematica scientia,Series B    2022, 42 (4): 1273-1292.   DOI: 10.1007/s10473-022-0402-7
    Abstract30)      PDF       Save
    This article explores the $\mathcal{O}(t^{-\beta})$ synchronization and asymptotic synchronization for fractional order BAM neural networks (FBAMNNs) with discrete delays, distributed delays and non-identical perturbations. By designing a state feedback control law and a new kind of fractional order Lyapunov functional, a new set of algebraic sufficient conditions are derived to guarantee the $\mathcal{O}(t^{-\beta})$ Synchronization and asymptotic synchronization of the considered FBAMNNs model; this can easily be evaluated without using a MATLAB LMI control toolbox. Finally, two numerical examples, along with the simulation results, illustrate the correctness and viability of the exhibited synchronization results.
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    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
    Abstract29)      PDF       Save
    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.
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    GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY
    Mei WANG, Zilai LI, Zhenhua GUO
    Acta mathematica scientia,Series B    2022, 42 (2): 521-539.   DOI: 10.1007/s10473-022-0207-8
    Abstract27)      PDF       Save
    In this paper, we consider the 3D compressible isentropic Navier-Stokes equations when the shear viscosity μ is a positive constant and the bulk viscosity is λ(ρ) = ρβ with β > 2, which is a situation that was first introduced by Vaigant and Kazhikhov in [1]. The global axisymmetric classical solution with arbitrarily large initial data in a periodic domain Ω = {(r, z)|r = √x2 + y2, (x, y, z) ∈ R3, rI ⊂ (0, +∞), −∞ < z < +∞} is obtained. Here the initial density keeps a non-vacuum state p > 0 when z → ±∞. Our results also show that the solution will not develop the vacuum state in any finite time, provided that the initial density is uniformly away from the vacuum.
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    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
    Abstract25)      PDF       Save
    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.
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    TIME ANALYTICITY FOR THE HEAT EQUATION ON GRADIENT SHRINKING RICCI SOLITONS
    Jiayong WU
    Acta mathematica scientia,Series B    2022, 42 (4): 1690-1700.   DOI: 10.1007/s10473-022-0424-1
    Abstract25)      PDF       Save
    On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.
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    THE MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF-CHOQUARD EQUATION WITH MAGNETIC FIELDS
    Li WANG, Kun CHENG, Jixiu WANG
    Acta mathematica scientia,Series B    2022, 42 (4): 1453-1484.   DOI: 10.1007/s10473-022-0411-6
    Abstract24)      PDF       Save
    In this paper, we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields: \begin{equation*} (a\varepsilon^{2s}+b\varepsilon^{4s-3}[u]^2_{\varepsilon,A/\varepsilon}) (-\Delta)_{A/\varepsilon}^{s} u+V(x)u = \varepsilon^{-\alpha}(I_\alpha*F(|u|^2))f(|u|^2)u\ \ \text{in }\ \mathbb{R}^3. \end{equation*} Here $\varepsilon > 0$ is a small parameter, $a,b > 0$ are constants, $s \in (0% \frac{3} {4} ,1), (-\Delta)_{A}^{s}$ is the fractional magnetic Laplacian, $A: \mathbb{R}^3 \to \mathbb{R}^3$ is a smooth magnetic potential, $I_{\alpha}=\frac{\Gamma(\frac{3-\alpha}{2})}{2^{\alpha}\pi^{\frac{3}{2}}\Gamma(\frac{\alpha}{2})}\cdot\frac{1}{|x|^{\alpha} }$ is the Riesz potential, the potential $V$ is a positive continuous function having a local minimum, and $f: \mathbb{R} \to \mathbb{R}$ is a $C^1$ subcritical nonlinearity. Under some proper assumptions regarding $V$ and $f, $ we show the multiplicity and concentration of positive solutions with the topology of the set $M:= \{x \in \mathbb{R}^3 : V (x) = \inf V \}$ by applying the penalization method and Ljusternik-Schnirelmann theory for the above equation.
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