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    THE UNIQUENESS OF THE Lp MINKOWSKI PROBLEM FOR q-TORSIONAL RIGIDITY
    Guangling SUN, Lu XU, Ping ZHANG
    Acta mathematica scientia,Series B    2021, 41 (5): 1405-1416.   DOI: 10.1007/s10473-021-0501-x
    Abstract252)      PDF       Save
    In this paper, we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case. Meanwhile, the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.
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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
    Abstract219)      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.
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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
    Abstract119)      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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    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
    Abstract116)      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.
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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
    Abstract111)      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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    THE EXISTENCE OF GLOBAL SOLUTIONS FOR THE FULL NAVIER-STOKES-KORTEWEG SYSTEM OF VAN DER WAALS GAS
    Hakho Hong
    Acta mathematica scientia,Series B    2023, 43 (2): 469-491.   DOI: 10.1007/s10473-023-0201-9
    Abstract96)      PDF       Save
    The aim of this work is to prove the existence for the global solution of a non-isothermal or non-isentropic model of capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), in the case of van der Waals gas. Under the small initial perturbation, the proof of the global existence is based on an elementary energy method using the continuation argument of local solution. Moreover, the uniqueness of global solutions and large time behavior of the density are given. It is one of the main difficulties that the pressure $p$ is not the increasing function of the density $\rho$.
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    CONSTANT DISTANCE BOUNDARIES OF THE $t$-QUASICIRCLE AND THE KOCH SNOWFLAKE CURVE*
    Xin Wei, Zhi-Ying Wen
    Acta mathematica scientia,Series B    2023, 43 (3): 981-993.   DOI: 10.1007/s10473-023-0301-6
    Abstract93)      PDF       Save
    Let $\Gamma$ be a Jordan curve in the complex plane and let $\Gamma_\lambda$ be the constant distance boundary of $\Gamma$. Vellis and Wu \cite{VW} introduced the notion of a $(\zeta,r_0)$-chordal property which guarantees that, when $\lambda$ is not too large, $\Gamma_\lambda$ is a Jordan curve when $\zeta=1/2$ and $\Gamma_\lambda$ is a quasicircle when $0<\zeta<1/2$. We introduce the $(\zeta,r_0,t)$-chordal property, which generalizes the $(\zeta,r_0)$-chordal property, and we show that under the condition that $\Gamma$ is $(\zeta,r_0,\sqrt t)$-chordal with $0<\zeta < r_0^{1-\sqrt t}/2$, there exists $\varepsilon>0$ such that $\Gamma_\lambda$ is a $t$-quasicircle once $\Gamma_\lambda$ is a Jordan curve when $0<\zeta<\varepsilon$. In the last part of this paper, we provide an example: $\Gamma$ is a kind of Koch snowflake curve which does not have the $(\zeta,r_0)$-chordal property for any $0<\zeta\le 1/2$, however $\Gamma_\lambda$ is a Jordan curve when $\zeta$ is small enough. Meanwhile, $\Gamma$ has the $(\zeta,r_0,\sqrt t)$-chordal property with $0<\zeta < r_0^{1- \sqrt t}/2$ for any $t\in (0,1/4)$. As a corollary of our main theorem, $\Gamma_\lambda$ is a $t$-quasicircle for all $0<t<1/4$ when $\zeta$ is small enough. This means that our $(\zeta,r_0,t)$-chordal property is more general and applicable to more complicated curves.
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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
    Abstract90)            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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    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
    Abstract89)      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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    ANALYTIC PHASE RETRIEVAL BASED ON INTENSITY MEASUREMENTS
    Wei QU, Tao QIAN, Guantie DENG, Youfa LI, Chunxu ZHOU
    Acta mathematica scientia,Series B    2021, 41 (6): 2123-2135.   DOI: 10.1007/s10473-021-0619-x
    Abstract88)      PDF       Save
    This paper concerns the reconstruction of a function $f$ in the Hardy space of the unit disc $\mathbb{D}$ by using a sample value $f(a)$ and certain $n$-intensity measurements $|\langle f,E_{a_1\cdots a_n}\rangle|,$ where $a_1,\cdots,a_n\in \mathbb{D},$ and $E_{a_1\cdots a_n}$ is the $n$-th term of the Gram-Schmidt orthogonalization of the Szegökernels $k_{a_1},\cdots,k_{a_n},$ or their multiple forms. Three schemes are presented. The first two schemes each directly obtain all the function values $f(z).$ In the first one we use Nevanlinna's inner and outer function factorization which merely requires the $1$-intensity measurements equivalent to know the modulus $|f(z)|.$ In the second scheme we do not use deep complex analysis, but require some $2$- and $3$-intensity measurements. The third scheme, as an application of AFD, gives sparse representation of $f(z)$ converging quickly in the energy sense, depending on consecutively selected maximal $n$-intensity measurements $|\langle f,E_{a_1\cdots a_n}\rangle|.$
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    CONSTRUCTION OF IMPROVED BRANCHING LATIN HYPERCUBE DESIGNS
    Hao CHEN, Jinyu YANG, Min-Qian LIU
    Acta mathematica scientia,Series B    2021, 41 (4): 1023-1033.   DOI: 10.1007/s10473-021-0401-0
    Abstract84)      PDF       Save
    In this paper, we propose a new method, called the level-collapsing method, to construct branching Latin hypercube designs (BLHDs). The obtained design has a sliced structure in the third part, that is, the part for the shared factors, which is desirable for the qualitative branching factors. The construction method is easy to implement, and (near) orthogonality can be achieved in the obtained BLHDs. A simulation example is provided to illustrate the effectiveness of the new designs.
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    REVISITING A NON-DEGENERACY PROPERTY FOR EXTREMAL MAPPINGS
    Xiaojun HUANG
    Acta mathematica scientia,Series B    2021, 41 (6): 1829-1838.   DOI: 10.1007/s10473-021-0602-6
    Abstract83)      PDF       Save
    We extend an earlier result obtained by the author in[7].
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    A SUPERLINEARLY CONVERGENT SPLITTING FEASIBLE SEQUENTIAL QUADRATIC OPTIMIZATION METHOD FOR TWO-BLOCK LARGE-SCALE SMOOTH OPTIMIZATION*
    Jinbao Jian, Chen Zhang, Pengjie Liu
    Acta mathematica scientia,Series B    2023, 43 (1): 1-24.   DOI: 10.1007/s10473-023-0101-z
    Abstract82)      PDF       Save
    This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints. Based on the ideas of splitting and sequential quadratic optimization (SQO), a new feasible descent method for the discussed problem is proposed. First, we consider the problem of quadratic optimal (QO) approximation associated with the current feasible iteration point, and we split the QO into two small-scale QOs which can be solved in parallel. Second, a feasible descent direction for the problem is obtained and a new SQO-type method is proposed, namely, splitting feasible SQO (SF-SQO) method. Moreover, under suitable conditions, we analyse the global convergence, strong convergence and rate of superlinear convergence of the SF-SQO method. Finally, preliminary numerical experiments regarding the economic dispatch of a power system are carried out, and these show that the SF-SQO method is promising.
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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
    Abstract81)      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 $\gamma>\frac{4}{3}$.
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    PREFACE
    Shaoji FENG, Caiheng OUYANG, Quanhua XU, Lixin YAN, Xiangyu ZHOU
    Acta mathematica scientia,Series B    2021, 41 (6): 1827-1828.   DOI: 10.1007/s10473-021-0601-7
    Abstract80)            Save
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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
    Abstract75)      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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    GLOBAL STRONG SOLUTION AND EXPONENTIAL DECAY OF 3D NONHOMOGENEOUS ASYMMETRIC FLUID EQUATIONS WITH VACUUM
    Guochun WU, Xin ZHONG
    Acta mathematica scientia,Series B    2021, 41 (5): 1428-1444.   DOI: 10.1007/s10473-021-0503-8
    Abstract75)      PDF       Save
    We prove the global existence and exponential decay of strong solutions to the three-dimensional nonhomogeneous asymmetric fluid equations with nonnegative density provided that the initial total energy is suitably small. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time-weighted techniques.
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    RIGIDITY RESULTS FOR SELF-SHRINKING SURFACES IN $\mathbb{R}^4$
    Xuyong JIANG, Hejun SUN, Peibiao ZHAO
    Acta mathematica scientia,Series B    2021, 41 (5): 1417-1427.   DOI: 10.1007/s10473-021-0502-9
    Abstract74)      PDF       Save
    In this paper, we give some rigidity results for complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ under some assumptions regarding their Gauss images. More precisely, we prove that this has to be a plane, provided that the images of either Gauss map projection lies in an open hemisphere or $\mathbb{S}^2(1/\sqrt{2})\backslash \bar{\mathbb{S}}^1_+(1/\sqrt{2})$. We also give the classification of complete self-shrinking surfaces properly immersed in $\mathbb{R}^4$ provided that the images of Gauss map projection lies in some closed hemispheres. As an application of the above results, we give a new proof for the result of Zhou. Moreover, we establish a Bernstein-type theorem.
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    THE BEREZIN TRANSFORM AND ITS APPLICATIONS
    Kehe ZHU
    Acta mathematica scientia,Series B    2021, 41 (6): 1839-1858.   DOI: 10.1007/s10473-021-0603-5
    Abstract73)      PDF       Save
    We give a survey on the Berezin transform and its applications in operator theory. The focus is on the Bergman space of the unit disk and the Fock space of the complex plane. The Berezin transform is most effective and most successful in the study of Hankel and Toepltiz operators.
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    MARTINGALE INEQUALITIES UNDER G-EXPECTATION AND THEIR APPLICATIONS
    Hanwu LI
    Acta mathematica scientia,Series B    2021, 41 (2): 349-360.   DOI: 10.1007/s10473-021-0201-6
    Abstract73)      PDF       Save
    In this paper, we study the martingale inequalities under $G$-expectation and their applications. To this end, we introduce a new kind of random time, called $G$-stopping time, and then investigate the properties of a $G$-martingale (supermartingale) such as the optional sampling theorem and upcrossing inequalities. With the help of these properties, we can show the martingale convergence property under $G$-expectation.
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