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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
    Abstract228)      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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    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
    Abstract72)      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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    Ground state solution to the Chern-Simons-Schrodinger systems
    Jin Deng, Benniao Li
    Acta mathematica scientia,Series B   
    Accepted: 24 April 2022
    Online available: 28 May 2022

    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
    Abstract69)      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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    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
    Abstract60)      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
    Abstract59)      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
    Abstract58)      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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    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
    Abstract58)            Save
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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
    Abstract56)      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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    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
    Abstract53)      PDF       Save
    We extend an earlier result obtained by the author in[7].
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    CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS
    Shukai CHEN, Zenghu LI
    Acta mathematica scientia,Series B    2021, 41 (5): 1445-1473.   DOI: 10.1007/s10473-021-0504-7
    Abstract50)      PDF       Save
    A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.
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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
    Abstract49)      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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    LIMIT CYCLE BIFURCATIONS OF A PLANAR NEAR-INTEGRABLE SYSTEM WITH TWO SMALL PARAMETERS
    Feng LIANG, Maoan HAN, Chaoyuan JIANG
    Acta mathematica scientia,Series B    2021, 41 (4): 1034-1056.   DOI: 10.1007/s10473-021-0402-z
    Abstract45)      PDF       Save
    In this paper we consider a class of polynomial planar system with two small parameters, $\varepsilon$ and $\lambda$, satisfying $0<\varepsilon\ll\ lambda\ll1$. The corresponding first order Melnikov function $M_1$ with respect to $\varepsilon$ depends on $\lambda$ so that it has an expansion of the form $M_1(h,\lambda)=\sum\limits_{k=0}^\infty M_{1k}(h)\lambda^k.$ Assume that $M_{1k'}(h)$ is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of $M_{1k'}(h)$, we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for $0<\varepsilon\ll\lambda\ll1$, when $k'=0$ or $1$. In addition, for each $k\in \mathbb{N}$, an upper bound of the maximal number of zeros of $M_{1k}(h)$, taking into account their multiplicities, is presented.
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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
    Abstract45)      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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    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
    Abstract42)      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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    SOME OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS WITH A DELAY ARGUMENT ON TIME SCALES
    Xin WU
    Acta mathematica scientia,Series B    2021, 41 (5): 1474-1492.   DOI: 10.1007/s10473-021-0505-6
    Abstract41)      PDF       Save
    In this paper, we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form \begin{align*}[r_n\varphi(\cdots r_2(r_1x^{\Delta})^{\Delta}\cdots)^{\Delta}]^{\Delta}(t)+h(t)f(x(\tau(t)))=0 \end{align*} on an arbitrary time scale $\mathbb{T}$ with $\sup\mathbb{T}=\infty$, where $n\geq 2$, $\varphi(u)=|u|^{\gamma}$sgn$(u)$ for $\gamma>0$, $r_i(1\leq i\leq n)$ are positive rd-continuous functions and $h\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},(0,\infty))$. The function $\tau\in {\mathrm{C}_{\mathrm{rd}}}(\mathbb{T},\mathbb{T})$ satisfies $\tau(t)\leq t$ and $\lim\limits_{t\rightarrow\infty}\tau(t)=\infty$ and $f\in {\mathrm{C}}(\mathbb{R},\mathbb{R})$. By using a generalized Riccati transformation, we give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. The obtained results are new for the corresponding higher order differential equations and difference equations. In the end, some applications and examples are provided to illustrate the importance of the main results.
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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
    Abstract39)      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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    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
    Abstract32)      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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    SLOW MANIFOLD AND PARAMETER ESTIMATION FOR A NONLOCAL FAST-SLOW DYNAMICAL SYSTEM WITH BROWNIAN MOTION
    Hina ZULFIQAR, Ziying HE, Meihua YANG, Jinqiao DUAN
    Acta mathematica scientia,Series B    2021, 41 (4): 1057-1080.   DOI: 10.1007/s10473-021-0403-y
    Abstract31)      PDF       Save
    We establish a slow manifold for a fast-slow dynamical system with anomalous diffusion, where both fast and slow components are influenced by white noise. Furthermore, we verify the exponential tracking property for the established random slow manifold, which leads to a lower dimensional reduced system. Alongside this we consider a parameter estimation method for a nonlocal fast-slow stochastic dynamical system, where only the slow component is observable. In terms of quantifying parameters in stochastic evolutionary systems, the provided method offers the advantage of dimension reduction.
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    COARSE ISOMETRIES BETWEEN FINITE DIMENSIONAL BANACH SPACES
    Yuqi SUN, Wen ZHANG
    Acta mathematica scientia,Series B    2021, 41 (5): 1493-1502.   DOI: 10.1007/s10473-021-0506-5
    Abstract29)      PDF       Save
    Assume that $X$ and $Y$ are real Banach spaces with the same finite dimension. In this paper we show that if a standard coarse isometry $f:X\rightarrow Y$ satisfies an integral convergence condition or weak stability on a basis, then there exists a surjective linear isometry $U:X\rightarrow Y$ such that $\|f(x)-Ux\|=o(\|x\|)$ as $\|x\|\rightarrow\infty$. This is a generalization about the result of Lindenstrauss and Szankowski on the same finite dimensional Banach spaces without the assumption of surjectivity. As a consequence, we also obtain a stability result for $\varepsilon$-isometries which was established by Dilworth.
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