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25 December 2021, Volume 41 Issue 6 Previous Issue    Next Issue

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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 Abstract ( 114 )   RICH HTML Save Related Articles | Metrics

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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 Abstract ( 107 )   RICH HTML PDF   Save We extend an earlier result obtained by the author in[7]. References | Related Articles | Metrics

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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 Abstract ( 99 )   RICH HTML 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. References | Related Articles | Metrics

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QUANTIZATION COMMUTES WITH REDUCTION, A SURVEY Xiaonan MA Acta mathematica scientia,Series B. 2021, 41 (6):  1859-1872.  DOI: 10.1007/s10473-021-0604-4 Abstract ( 84 )   RICH HTML PDF   Save We review the themes relating to the proposition that "quantization commutes with reduction" ([Q, R]=0), from symplectic manifolds to Cauchy-Riemann manifolds. References | Related Articles | Metrics

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CONFORMAL RESTRICTION MEASURES ON LOOPS SURROUNDING AN INTERIOR POINT Yong HAN, Yuefei WANG Acta mathematica scientia,Series B. 2021, 41 (6):  1873-1886.  DOI: 10.1007/s10473-021-0605-3 Abstract ( 63 )   RICH HTML PDF   Save A conformal restriction measure is a probability measure which is used to describe the law of a random connected subset in a simply connected domain that satisfies a certain conformal restriction property. Usually there are three kinds of conformal restriction measures:one (called the chordal restriction measure) has two given boundary points of the random set, the second (called the radial restriction measure) has one boundary point and one interior point in the random set, and the third (called the tri-chordal restriction measure) has three boundary points in the random set. In this article, we will define a new probability measure such that the random set associated to it contains one given interior point and does not intersect with the boundary. Furthermore, we will show that this measure can be characterized by one parameter; we will also construct this one-parameter family of measures in two ways and obtain several properties. References | Related Articles | Metrics

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NORMAL CRITERIA FOR A FAMILY OF HOLOMORPHIC CURVES Daochun SUN, Yingying HUO, Fujie CHAI Acta mathematica scientia,Series B. 2021, 41 (6):  1887-1895.  DOI: 10.1007/s10473-021-0606-2 Abstract ( 93 )   RICH HTML PDF   Save In this paper, we extend the concept of holomorphic curves sharing hyperplanes and introduce definitions of restricted hyperplanes and partial shared hypersurfaces. Then, we prove several normal criteria of the family of holomorphic curves and holomorphic mappings that concern restricted hyperplanes and partial shared hypersurfaces. These results generalize the Montel-type normal criterion of holomorphic curves. References | Related Articles | Metrics

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MULTIFRACTAL ANALYSIS OF THE CONVERGENCE EXPONENT IN CONTINUED FRACTIONS Lulu FANG, Jihua MA, Kunkun SONG, Min WU Acta mathematica scientia,Series B. 2021, 41 (6):  1896-1910.  DOI: 10.1007/s10473-021-0607-1 Abstract ( 53 )   RICH HTML PDF   Save Let $x \in (0,1)$ be a real number with continued fraction expansion $[a_1(x),a_2(x),$ $a_3(x),\cdots]$. This paper is concerned with the multifractal spectrum of the convergence exponent of $\{a_n(x)\}_{n \geq 1}$ defined by $$\tau(x):=\inf\bigg\{s \geq 0:\sum_{n \geq 1} a^{-s}_n(x)<\infty\bigg\}.$$ References | Related Articles | Metrics

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THE GROWTH OF DIFFERENCE EQUATIONS AND DIFFERENTIAL EQUATIONS Zongxuan CHEN, Ranran ZHANG, Shuangting LAN, Chuangxin CHEN Acta mathematica scientia,Series B. 2021, 41 (6):  1911-1920.  DOI: 10.1007/s10473-021-0608-0 Abstract ( 80 )   RICH HTML PDF   Save In this paper, we mainly apply a new, asymptotic method to investigate the growth of meromorphic solutions of linear higher order difference equations and differential equations. We delete the condition (1.6) of Theorems E and F, yet obtain the same results for Theorems E and F. We also weaken the condition (1.4) of Theorems C and D. References | Related Articles | Metrics

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DERIVED SEQUENCES AND THE FACTOR SPECTRUM OF THE PERIOD-DOUBLING SEQUENCE Yuke HUANG, Zhiying WEN Acta mathematica scientia,Series B. 2021, 41 (6):  1921-1937.  DOI: 10.1007/s10473-021-0609-z Abstract ( 58 )   RICH HTML PDF   Save Factor properties and their structures are important themes in combinatorics on words. Let $\mathbb{D}$ be the infinite one-sided sequence over the alphabet $\{a,b\}$ generated by the period-doubling substitution $\sigma(a)=ab$ and $\sigma(b)=aa$. In this paper, we determine the derived sequence $D_w$($\mathbb{D}$) for any factor ω $\prec$ $\mathbb{D}$, and study some factor spectra using the structures of derived sequences. We also prove the reflexivity property of derived sequences. References | Related Articles | Metrics

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LAPLACE TRANSFORMS FOR ANALYTIC FUNCTIONS IN TUBULAR DOMAINS Guantie DENG, Qian FU, Hui CAO Acta mathematica scientia,Series B. 2021, 41 (6):  1938-1948.  DOI: 10.1007/s10473-021-0610-6 Abstract ( 88 )   RICH HTML PDF   Save Assume that $0< p<\infty$ and that $B$ is a connected nonempty open set in $\mathbb{R}^n$, and that $A^{p}(B)$ is the vector space of all holomorphic functions $F$ in the tubular domains $\mathbb{R}^n+{\rm i}B$ such that for any compact set $K \subset B,$ $$\|y\mapsto \|x\mapsto F(x+{\rm i}y)\|_{L^p(\mathbb{R}^n)}\|_{L(K)}<\infty,$$ so $A^p(B)$ is a Fréchet space with the Heine-Borel property, its topology is induced by a complete invariant metric, is not locally bounded, and hence is not normal. Furthermore, if $1\leq p\leq 2$, then the element $F$ of $A^{p}(B)$ can be written as a Laplace transform of some function $f\in L(\mathbb{R}^n)$. References | Related Articles | Metrics

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FINITE TYPE CONDITIONS ON REAL HYPERSURFACES WITH ONE DEGENERATE EIGENVALUE Wei CHEN, Yingxiang CHEN, Wanke YIN Acta mathematica scientia,Series B. 2021, 41 (6):  1949-1958.  DOI: 10.1007/s10473-021-0611-5 Abstract ( 53 )   RICH HTML PDF   Save Let $M$ be a smooth pseudoconvex hypersurface in $\mathbb{C}^{n+1}$ whose Levi form has at most one degenerate eigenvalue. For any tangent vector field $L$ of type $(1,0)$, we prove the equality of the commutator type and the Levi form type associated to $L$. We also show that the regular contact type, the commutator type and the Levi form type of the real hypersurface are the same. References | Related Articles | Metrics

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GEOMETRY ON THE WASSERSTEIN SPACE OVER A COMPACT RIEMANNIAN MANIFOLD Hao DING, Shizan FANG Acta mathematica scientia,Series B. 2021, 41 (6):  1959-1984.  DOI: 10.1007/s10473-021-0612-4 Abstract ( 53 )   RICH HTML PDF   Save We revisit the intrinsic differential geometry of the Wasserstein space over a Riemannian manifold, due to a series of papers by Otto, Otto-Villani, Lott, Ambrosio-Gigli-Savaré, etc. References | Related Articles | Metrics

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ON F(p,q,s) SPACES Ruhan ZHAO Acta mathematica scientia,Series B. 2021, 41 (6):  1985-2020.  DOI: 10.1007/s10473-021-0613-3 Abstract ( 43 )   RICH HTML PDF   Save The family of spaces F(p,q,s) was introduced by the author in 1996. Since then, there has been great development in the theory of these spaces, due to the fact that these spaces include many classical function spaces, and have connections with many other areas of mathematics. In this survey we present some basic properties and recent results on F(p,q,s) spaces. References | Related Articles | Metrics

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L2 EXTENSIONS WITH SINGULAR METRICS ON KÄHLER MANIFOLDS Xiangyu ZHOU, Langfeng ZHU Acta mathematica scientia,Series B. 2021, 41 (6):  2021-2038.  DOI: 10.1007/s10473-021-0614-2 Abstract ( 57 )   RICH HTML PDF   Save In this paper, we give a survey of our recent results on extension theorems on Kähler manifolds for holomorphic sections or cohomology classes of (pluri)canonical line bundles twisted with holomorphic line bundles equipped with singular metrics, and also discuss their applications and the ideas contained in the proofs. References | Related Articles | Metrics

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$Q_K$ SPACES: A BRIEF AND SELECTIVE SURVEY Guanlong BAO, Hasi WULAN Acta mathematica scientia,Series B. 2021, 41 (6):  2039-2054.  DOI: 10.1007/s10473-021-0615-1 Abstract ( 75 )   RICH HTML PDF   Save This article traces several prominent trends in the development of Möbius invariant function spaces $Q_K$ with emphasis on theoretic aspects. References | Related Articles | Metrics

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SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS Gilles PISIER Acta mathematica scientia,Series B. 2021, 41 (6):  2055-2085.  DOI: 10.1007/s10473-021-0616-0 Abstract ( 64 )   RICH HTML PDF   Save We show that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in the following sense: there is a factorization of the identity of $M$ $$Id_M=vu:M{\buildrel u\over\longrightarrow} B(H) {\buildrel v\over\longrightarrow} M$$ with $u$ normal, unital, positive and $v$ completely contractive. As a corollary, if $M$ has a separable predual, $M$ is isomorphic (as a Banach space) to $B(\ell_2)$. For instance this applies (rather surprisingly) to the von Neumann algebra of any free group. Nevertheless, since $B(H)$ fails the approximation property (due to Szankowski) there are $M$'s (namely $B(H)^{**}$ and certain finite examples defined using ultraproducts) that are not seemingly injective. Moreover, for $M$ to be seemingly injective it suffices to have the above factorization of $Id_M$ through $B(H)$ with $u,v$ positive (and $u$ still normal). References | Related Articles | Metrics

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ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)∞ Changgui ZHANG Acta mathematica scientia,Series B. 2021, 41 (6):  2086-2106.  DOI: 10.1007/s10473-021-0617-z Abstract ( 58 )   RICH HTML PDF   Save As in our previous work[14], a function is said to be of theta-type when its asymptotic behavior near any root of unity is similar to what happened for Jacobi theta functions. It is shown that only four Euler infinite products have this property. That this is the case is obtained by investigating the analyticity obstacle of a Laplace-type integral of the exponential generating function of Bernoulli numbers. References | Related Articles | Metrics

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OLD AND RECENT RESULTS IN THE ANALYTIC THEORY OF DIRICHLET SERIES: A SURVEY Hervé QUEFFÉLEC Acta mathematica scientia,Series B. 2021, 41 (6):  2107-2122.  DOI: 10.1007/s10473-021-0618-y Abstract ( 42 )   RICH HTML PDF   Save This survey is dedicated to the memory of Professor Jiarong Yu, who recently passed away. It is concerned by a topic of which he was fond, an interest shared by myself:the analytic theory of Dirichlet series. References | Related Articles | Metrics

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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 Abstract ( 111 )   RICH HTML 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 $|\langle f,E_{a_1\cdots a_n}\rangle|.$ References | Related Articles | Metrics

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SOME QUESTIONS REGARDING VERIFICATION OF CARLESON MEASURES Fangwen DENG, Caiheng OUYANG, Guantie DENG Acta mathematica scientia,Series B. 2021, 41 (6):  2136-2148.  DOI: 10.1007/s10473-021-0620-4 Abstract ( 96 )   RICH HTML PDF   Save In this paper, we give a survey of some recent progress in terms of verifying Carleson measures; this includes the difference between two definitions of a Carleson measure, the Bergman tree condition, the T1 condition for Besov-Sobolev spaces on a complex ball, vector-valued Carleson measures, Carleson measures in strongly pseudoconvex domains and reverse Carleson measures. References | Related Articles | Metrics

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HANDEL'S FIXED POINT THEOREM: A MORSE THEORETICAL POINT OF VIEW Patrice LE CALVEZ Acta mathematica scientia,Series B. 2021, 41 (6):  2149-2172.  DOI: 10.1007/s10473-021-0621-3 Abstract ( 48 )   RICH HTML PDF   Save Michael Handel has proved in[10] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that turned out to be an efficient tool in the study of the dynamics of surface homeomorphisms. The present article fits into a series of articles by the author[13] and by Juliana Xavier[21, 22], where proofs were given, related to the classical Brouwer Theory, instead of the Homotopical Brouwer Theory used in the original article. Like in[13, 21] and[22], we will use "free brick decompositions" but will present a more conceptual Morse theoretical argument. It is based on a new preliminary lemma, that gives a nice "condition at infinity" for our problem. References | Related Articles | Metrics

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NOTES ON REAL INTERPOLATION OF OPERATOR Lp-SPACES Marius JUNGE, Quanhua XU Acta mathematica scientia,Series B. 2021, 41 (6):  2173-2182.  DOI: 10.1007/s10473-021-0622-2 Abstract ( 59 )   RICH HTML PDF   Save Let $\mathcal{M}$ be a semifinite von Neumann algebra. We equip the associated noncommutative $L_p$-spaces with their natural operator space structure introduced by Pisier via complex interpolation. On the other hand, for $1 < p < \infty$ let $$L_{p,p}(\mathcal{M})=\big(L_{\infty}(\mathcal{M}),\,L_{1}(\mathcal{M})\big)_{\frac1p,\,p}$$ be equipped with the operator space structure via real interpolation as defined by the second named author (J. Funct. Anal. 139 (1996), 500——539). We show that $L_{p,p}(\mathcal{M})=L_{p}(\mathcal{M})$ completely isomorphically if and only if $\mathcal{M}$ is finite dimensional. This solves in the negative the three problems left open in the quoted work of the second author. \\ We also show that for $1 < p < \infty$ and $1\le q\le\infty$ with $p\neq q$ $$\big(L_{\infty}(\mathcal{M};\ell_q),\,L_{1}(\mathcal{M};\ell_q)\big)_{\frac1p,\,p}=L_p(\mathcal{M}; \ell_q)$$ with equivalent norms, i.e., at the Banach space level if and only if $\mathcal{M}$ is isomorphic, as a Banach space, to a commutative von Neumann algebra. \\ Our third result concerns the following inequality: $$\Big\|\big(\sum_ix_i^q\big)^{\frac1q}\Big\|_{L_p(\mathcal{M})}\le \Big\|\big(\sum_ix_i^r\big)^{\frac1r}\Big\|_{L_p(\mathcal{M})}$$ for any finite sequence $(x_i)\subset L_p^+(\mathcal{M})$, where $0 < r < q < \infty$ and $0 < p\le\infty$. If $\mathcal{M}$ is not isomorphic, as a Banach space, to a commutative von Meumann algebra, then this inequality holds if and only if $p\ge r$. References | Related Articles | Metrics

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ENTROPICAL OPTIMAL TRANSPORT, SCHRÖDINGER'S SYSTEM AND ALGORITHMS Liming WU Acta mathematica scientia,Series B. 2021, 41 (6):  2183-2197.  DOI: 10.1007/s10473-021-0623-1 Abstract ( 46 )   RICH HTML PDF   Save In this exposition paper we present the optimal transport problem of MongeAmpère-Kantorovitch (MAK in short) and its approximative entropical regularization. Contrary to the MAK optimal transport problem, the solution of the entropical optimal transport problem is always unique, and is characterized by the Schrödinger system. The relationship between the Schrödinger system, the associated Bernstein process and the optimal transport was developed by Léonard[32, 33] (and by Mikami[39] earlier via an h-process). We present Sinkhorn's algorithm for solving the Schrödinger system and the recent results on its convergence rate. We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence, whose rate might be independent of the regularization constant. This exposition is motivated by recent applications of optimal transport to different domains such as machine learning, image processing, econometrics, astrophysics etc.. References | Related Articles | Metrics

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PENALIZED LEAST SQUARE IN SPARSE SETTING WITH CONVEX PENALTY AND NON GAUSSIAN ERRORS Doualeh ABDILLAHI-ALI, Nourddine AZZAOUI, Arnaud GUILLIN, Guillaume LE MAILLOUX, Tomoko MATSUI Acta mathematica scientia,Series B. 2021, 41 (6):  2198-2216.  DOI: 10.1007/s10473-021-0624-0 Abstract ( 53 )   RICH HTML PDF   Save This paper consider the penalized least squares estimators with convex penalties or regularization norms. We provide sparsity oracle inequalities for the prediction error for a general convex penalty and for the particular cases of Lasso and Group Lasso estimators in a regression setting. The main contribution is that our oracle inequalities are established for the more general case where the observations noise is issued from probability measures that satisfy a weak spectral gap (or Poincaré) inequality instead of Gaussian distributions. We illustrate our results on a heavy tailed example and a sub Gaussian one; we especially give the explicit bounds of the oracle inequalities for these two special examples. References | Related Articles | Metrics