Loading...

Table of Content

    25 December 2021, Volume 41 Issue 6 Previous Issue   
    Articles
    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
    Related Articles | Metrics
    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 ( 30 )   RICH HTML PDF   Save
    We extend an earlier result obtained by the author in[7].
    References | Related Articles | Metrics
    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 ( 23 )   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
    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 ( 16 )   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
    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 ( 11 )   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
    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 ( 11 )   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
    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
    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
    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 ( 16 )   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
    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
    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
    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
    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
    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 ( 10 )   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
    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
    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
    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
    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
    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
    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
    $Q_K$ 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
    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
    SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS
    Gilles PISIER
    Acta mathematica scientia,Series B. 2021, 41 (6):  2055-2085.  DOI: 10.1007/s10473-021-0616-0
    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
    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
    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
    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
    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
    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 ( 61 )   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
    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
    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
    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
    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
    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 ( 11 )   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
    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
    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
    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
    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