数学物理学报(B辑)
主办 中国科学院精密测量科学与技术创新研究院
编辑  《数学物理学报》编辑委员会
   地址: 湖北武汉武昌小洪山西30号
       中科院精密测量科学与技术创新研究院
   电话: 027-87199206(中、英文版)
       027-87199087(英文版)
   E-mail: actams@wipm.ac.cn
   Http://actams.wipm.ac.cn
ISSN 1003-3998
CN  42-1226/O
2022年, 第42卷, 第2期 刊出日期:2022-04-25 上一期   
本期栏目: 论文 
论文
UNDERSTANDING SCHUBERT'S BOOK (III)
李绑河
数学物理学报(英文版). 2022 (2):  437-453.  DOI: 10.1007/s10473-022-0201-1
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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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SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES
刘小松
数学物理学报(英文版). 2022 (2):  454-466.  DOI: 10.1007/s10473-022-0202-0
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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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STRONG LIMIT THEOREMS FOR EXTENDED INDEPENDENT RANDOM VARIABLES AND EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES UNDER SUB-LINEAR EXPECTATIONS
张立新
数学物理学报(英文版). 2022 (2):  467-490.  DOI: 10.1007/s10473-022-0203-z
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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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ON THE BOUNDS OF THE PERIMETER OF AN ELLIPSE
赵铁洪, 王淼坤, 褚玉明
数学物理学报(英文版). 2022 (2):  491-501.  DOI: 10.1007/s10473-022-0204-y
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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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A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY
雷春雨, 廖家锋, 储昌木, 索洪敏
数学物理学报(英文版). 2022 (2):  502-510.  DOI: 10.1007/s10473-022-0205-x
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We consider the logarithmic elliptic equation with singular nonlinearity \begin{equation*} \begin{cases} \Delta u+u\log u^2 +\frac{\lambda}{u^\gamma}=0, &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ ($N\geq3$) is a bounded domain with a smooth boundary, $0<\gamma<1$ and $\lambda$ is a positive constant. By using a variational method and the critical point theory for a nonsmooth functional, we obtain the existence of two positive solutions. This result generalizes and improves upon recent results in the literature.
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COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITELY MANY GAMMA FUNCTIONS
祁锋
数学物理学报(英文版). 2022 (2):  511-520.  DOI: 10.1007/s10473-022-0206-9
In this paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finitely many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews the complete monotonicity of several linear combinations of finitely many digamma or trigamma functions.
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GLOBAL SOLUTIONS TO A 3D AXISYMMETRIC COMPRESSIBLE NAVIER-STOKES SYSTEM WITH DENSITY-DEPENDENT VISCOSITY
王梅, 李自来, 郭真华
数学物理学报(英文版). 2022 (2):  521-539.  DOI: 10.1007/s10473-022-0207-8
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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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AN AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL DELAY EQUATIONS DRIVEN BY TIME-CHANGED LÉVY NOISE
申广君, 徐文涛, 吴奖伦
数学物理学报(英文版). 2022 (2):  540-550.  DOI: 10.1007/s10473-022-0208-7
In this paper, we aim to derive an averaging principle for stochastic differential equations driven by time-changed Lévy noise with variable delays. Under certain assumptions, we show that the solutions of stochastic differential equations with time-changed Lévy noise can be approximated by solutions of the associated averaged stochastic differential equations in mean square convergence and in convergence in probability, respectively. The convergence order is also estimated in terms of noise intensity. Finally, an example with numerical simulation is given to illustrate the theoretical result.
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THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN
王文清, 毛安民
数学物理学报(英文版). 2022 (2):  551-560.  DOI: 10.1007/s10473-022-0209-6
We investigate the bi-harmonic problem $$\left\{\begin{array}{ll} \Delta^{2}u - \alpha\nabla \cdot (f(\nabla u)) - \beta\Delta_{p}u = g(x,u) &\hbox{in}\ \ \Omega,\\[2mm] \frac{\partial u}{\partial n}=0, \frac{\partial(\Delta u)}{\partial n}=0 &\hbox{on}\ \ \partial\Omega,\end{array} \right. $$ where $\Delta^{2}u = \Delta(\Delta u), \Delta_{p}u =\div\left(|\nabla u|^{p-2}\nabla u\right)$ with $p > 2.$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N \geq 1.$ By using a special function space with the constraint $\int_{\Omega}u {\rm d}x = 0$, under suitable assumptions on $f$ and $g(x,u)$, we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.
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ARBITRARILY SMALL NODAL SOLUTIONS FOR PARAMETRIC ROBIN (p,q)-EQUATIONS PLUS AN INDEFINITE POTENTIAL
Salvatore LEONARDI, Nikolaos S. PAPAGEORGIOU
数学物理学报(英文版). 2022 (2):  561-574.  DOI: 10.1007/s10473-022-0210-0
We consider a nonlinear Robin problem driven by the $(p,q)$-Laplacian plus an indefinite potential term and with a parametric reaction term. Under minimal conditions on the reaction function, which concern only its behavior near zero, we show that, for all $\lambda >0$ small, the problem has a nodal solution $y_{\lambda} \in C^1(\bar{Ω})$ and we have $y_{\lambda} \rightarrow 0$ in $C^1(\bar{Ω})$ as $\lambda \rightarrow 0^+$.
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SUP-ADDITIVE METRIC PRESSURE OF DIFFEOMORPHISMS
张文达, 李智强
数学物理学报(英文版). 2022 (2):  575-587.  DOI: 10.1007/s10473-022-0211-z
In this paper, we study the pressure of C1-smooth partially hyperbolic diffeomorphisms with sup-additive potentials. We give several definitions of the so called unstable (measure theoretic) pressure in terms of Bowen's picture and the capacity picture. We show that all such unstable metric pressures of a given ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure.
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GLOBAL STABILITY OF LARGE SOLUTIONS TO THE 3D MAGNETIC BÉNARD PROBLEM
秦绪龙, 邱华, 姚正安
数学物理学报(英文版). 2022 (2):  588-610.  DOI: 10.1007/s10473-022-0212-y
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In this paper, we consider the 3D magnetic Bénard problem. More precisely, we prove that the large solutions are stable under certain conditions. And we obtain the equivalent condition with respect to this stability condition. Finally, we also establish the stability of 2D magnetic Bénard problem under 3D perturbations.
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THE SUBORDINATION PRINCIPLE AND ITS APPLICATION TO THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR
王建飞, 张晓飞
数学物理学报(英文版). 2022 (2):  611-622.  DOI: 10.1007/s10473-022-0213-x
This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties. First, we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination. As applications, we obtain that if $\beta\in[0,1],\gamma\in[0,\frac{1}{r}]$ and $\beta+\gamma\leq1$, then the Roper-Suffridge extension operator $$ \Phi_{\beta,\,\gamma}(f)(z)=\left(f(z_{1}), \left(\frac{f(z_1)}{z_1}\right)^{\beta}(f'(z_{1}))^{\gamma}w\right),\,\,z\in \Omega_{p,r} $$ preserves an almost starlike mapping of complex order $\lambda$ on $\Omega_{p,r}=\{z=(z_1,w)\in \mathbb C\times X :|z_1|^{p}+\|w\|_X^{r}<1\}$, where $1\leq p\leq 2$, $r\geq 1$ and $X$ is a complex Banach space. Second, by applying the principle of subordination, we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order $\lambda$. Finally, we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator. This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator, and enables us to generalize many known results from $p=2$ to $1\leq p\leq 2$.
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ORLICZ-LORENTZ SEQUENCE SPACES EQUIPPED WITH THE ORLICZ NORM
崔云安, Paweƚ FORALEWSKI, Joanna KOŃCZAK
数学物理学报(英文版). 2022 (2):  623-652.  DOI: 10.1007/s10473-022-0214-9
In this article, we consider Orlicz-Lorentz sequence spaces equipped with the Orlicz norm $\left(\lambda _{\varphi,\omega}, \Vert\cdot\Vert_{\varphi, \omega}^{O}\right)$ generated by any Orlicz function and any non-increasing weight sequence. As far as we know, research on such a general case is conducted for the first time. After showing that the Orlicz norm is equal to the Amemiya norm in general and giving some important properties of this norm, we study the problem of existence of order isomorphically isometric copies of $l^{\infty}$ in the space $\left(\lambda _{\varphi,\omega}, \Vert\cdot\Vert_{\varphi, \omega}^{O}\right)$ and we find criteria for order continuity and monotonicity properties of this space. We also find criteria for monotonicity properties of $n$-dimensional subspaces $\lambda _{\varphi,\omega}^{n}$ ($n\geq 2$) and the subspace $\left(\lambda _{\varphi,\omega}\right)_{a}$ of order continuous elements of $\lambda _{\varphi,\omega}$. Finally, as an immediate consequence of the criteria considered in this article, the properties of Orlicz sequence spaces equipped with the Orlicz norm are deduced.
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HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS
王军, 陈振龙
数学物理学报(英文版). 2022 (2):  653-670.  DOI: 10.1007/s10473-022-0215-8
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Let $X^H=\{X^H(s), s\in \mathbb{R}^{N_1}\} $ and $X^K=\{X^K(t), t\in \mathbb{R}^{N_2}\} $ be two independent time-space anisotropic random fields with indices $H \in (0,1)^{N_1}$ and $K \in (0,1)^{N_2}$, which may not possess Gaussianity, and which take values in $\mathbb{R}^d$ with a space metric $\tau$. Under certain general conditions with density functions defined on a bounded interval, we study problems regarding the hitting probabilities of time-space anisotropic random fields and the existence of intersections of the sample paths of random fields $X^H$ and $X^K$. More generally, for any Borel set $F \subset \mathbb{R}^d$, the conditions required for $F$ to contain intersection points of $X^H$ and $X^K$ are established. As an application, we give an example of an anisotropic non-Gaussian random field to show that these results are applicable to the solutions of non-linear systems of stochastic fractional heat equations. }
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ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES
王艳青, 魏巍, 吴刚, 叶嵎林
数学物理学报(英文版). 2022 (2):  671-689.  DOI: 10.1007/s10473-022-0216-7
In this paper, we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant $\varepsilon$ such that the solution $(\rho,u,\theta)$ to the full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds:
(1) $\rho \in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $u\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and \begin{equation}\label{L1}\| u\|_{L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=1,\ \ q > 3;\end{equation} (2) $\lambda < 3\mu,$ $\rho \in L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{3}))$, $\theta\in L^{p,\infty}(0,T;L^{q,\infty}(\mathbb{R}^{3}))$ and \begin{equation}\label{L12}\|\theta\|_{L^{p,\infty}(0,T; L^{q,\infty}(\mathbb{R}^{3}))}\leq \varepsilon, ~~\text{with}~~ {2/p}+ {3/q}=2,\ \ q > 3/2.\end{equation} To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time $T^{\ast}$:
(1) assuming that the pair $(p,\overrightarrow{q})$ satisfies $ {2}/{p }+{1}/{q_{1} }+{1}/{q_{2} }+{1}/{q_{3} }=1$ $(1 < q_{i} < \infty)$ and (1.17), then \begin{equation}\label{AL1}\limsup_{t\rightarrow T^*}( \|\rho \|_{L^{\infty}(0,t;L^{\infty}(\mathbb{R}^{3}))}+ \| u \|_{L^{p }(0,t; L_{1}^{ q_{1} }L_{2}^{ q_{2} } L_{3}^{q_{3} }(\mathbb{R}^{3}) )} )= \infty; \end{equation} (2) letting the pair $(p,\overrightarrow{q})$ satisfy ${2}/{p }+{1}/{q_{1} }+{1}/{q_{2} }+{1}/{q_{3} }=2$ $(1 < q_{i} < \infty)$ and (1.17), then \begin{equation}\label{AL2}\limsup_{t\rightarrow T^*}( \|\rho \|_{L^{\infty}(0,t;L^{\infty}(\mathbb{R}^{3}))}+ \| \theta \|_{L^{p }(0,t; L_{1}^{ q_{1} }L_{2}^{ q_{2} } L_{3}^{q_{3} }(\mathbb{R}^{3}) )} )= \infty, (\lambda < 3\mu). \end{equation} Third, without the condition on $\rho$ in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.
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MIXED LIPSCHITZ SPACES AND THEIR APPLICATIONS
何少勇, 陈杰诚
数学物理学报(英文版). 2022 (2):  690-714.  DOI: 10.1007/s10473-022-0217-6
The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory. As an application, we derive a boundedness criterion for singular integral operators in a mixed Journé class on mixed Lipschitz spaces. Key elements of the paper are the development of the Littlewood-Paley theory for a special mixed Besov spaces, and a density argument for the mixed Lipschitz spaces in the weak sense.
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TRAVELING WAVES IN A SIRH MODEL WITH SPATIO-TEMPORAL DELAY AND NONLOCAL DISPERSAL
杨璐, 杨赟瑞, 宋雪
数学物理学报(英文版). 2022 (2):  715-736.  DOI: 10.1007/s10473-022-0218-5
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This paper deals mainly with the existence and asymptotic behavior of traveling waves in a SIRH model with spatio-temporal delay and nonlocal dispersal based on Schauder's fixed-point theorem and analysis techniques, which generalize the results of nonlocal SIRH models without relapse and delay. In particular, the difficulty of obtaining the asymptotic behavior of traveling waves for the appearance of spatio-temporal delay is overcome by the use of integral techniques and analysis techniques. Finally, the more general nonexistence result of traveling waves is also included.
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IMPULSIVE EXPONENTIAL SYNCHRONIZATION OF FRACTIONAL-ORDER COMPLEX DYNAMICAL NETWORKS WITH DERIVATIVE COUPLINGS VIA FEEDBACK CONTROL BASED ON DISCRETE TIME STATE OBSERVATIONS
李瑞鸿, 武怀勤, 曹进德
数学物理学报(英文版). 2022 (2):  737-754.  DOI: 10.1007/s10473-022-0219-4
This article aims to address the global exponential synchronization problem for fractional-order complex dynamical networks (FCDNs) with derivative couplings and impulse effects via designing an appropriate feedback control based on discrete time state observations. In contrast to the existing works on integer-order derivative couplings, fractional derivative couplings are introduced into FCDNs. First, a useful lemma with respect to the relationship between the discrete time observations term and a continuous term is developed. Second, by utilizing an inequality technique and auxiliary functions, the rigorous global exponential synchronization analysis is given and synchronization criterions are achieved in terms of linear matrix inequalities (LMIs). Finally, two examples are provided to illustrate the correctness of the obtained results.
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ON (α, β)-METRICS OF CONSTANT FLAG CURVATURE
陈光祖, 程新跃
数学物理学报(英文版). 2022 (2):  755-768.  DOI: 10.1007/s10473-022-0220-y
In this paper, we study the (α, β)-metrics of constant flag curvature. We characterize almost regular (α, β)-metrics of constant flag curvature under the condition that β is a homothetic 1-form with respect to α. Furthermore, we prove that if a regular (α, β)-metric is of constant flag curvature and β is a Killing 1-form with constant length, then it must be a Riemannian metric or locally Minkowskian.
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A NOTE ON MEASURE-THEORETIC EQUICONTINUITY AND RIGIDITY
罗炽逸, 赵云
数学物理学报(英文版). 2022 (2):  769-773.  DOI: 10.1007/s10473-022-0221-x
Given a topological dynamical system $(X,T)$ and a $T$-invariant measure $\mu$, let $\mathcal{B}$ denote the Borel $\sigma$-algebra on $X$. This paper proves that $(X,\mathcal{B},\mu,T)$ is rigid if and only if $(X,T)$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$, and a function $f\in L^2(\mu)$ is rigid if and only if $f$ is $\mu$-$A$-equicontinuous in the mean for some subsequence $A$ of $\mathbb{N}$. In particular, this gives a positive answer to Question 4.11 in [1].}
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COMPLEX INTERPOLATION OF Lp(C, H) SPACES WITH RESPECT TO CULLEN-REGULAR
贺凯莉, 何建勋, 娄珍珍
数学物理学报(英文版). 2022 (2):  774-788.  DOI: 10.1007/s10473-022-0222-9
The aim of this paper is to prove a new version of the Riesz-Thorin interpolation theorem on Lp(C, H). In the sense of Cullen-regular, we show Hadamard's three-lines theorem by means of the Maximum modulus principle on a symmetric slice domain. In addition, two applications of the Riesz-Thorin theorem are presented. Finally, we investigate two kinds of Calderón's complex interpolation methods in Lp(C, H).
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MAPS PRESERVING THE NORM OF THE POSITIVE SUM IN Lp SPACES
郝晶晶, 董云柏, 李磊
数学物理学报(英文版). 2022 (2):  789-794.  DOI: 10.1007/s10473-022-0223-8
{For $1 < p < \infty$, let $S(L_p)_+$ be the set of positive elements in $L_p$ with norm one. Assume that $V_0: S(L_p(\Omega_1))_{+}\to S(L_p(\Omega_2))_{+}$ is a surjective norm-additive map; that is, \[\|V_0(x)+V_0(y)\|=\|x+y\|,\quad\forall\,x, y\in S(L_p(\Omega_1 ))_{+}.\] In this paper, we show that $V_0$ can be extended to an isometry from $L_p(\Omega_1)$ onto $L_p(\Omega_2)$.
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STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES
Nguyen Xuan LINH, Duong Viet THONG, Prasit CHOLAMJIAK, Pham Anh TUAN, Luong Van LONG
数学物理学报(英文版). 2022 (2):  795-812.  DOI: 10.1007/s10473-022-0224-7
In this work, we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space, and present a projection-type approximation method for solving this problem. Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping. A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions. Finally, we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.
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α-LIMIT SETS AND LYAPUNOV FUNCTION FOR MAPS WITH ONE TOPOLOGICAL ATTRACTOR
丁义明, 孙运
数学物理学报(英文版). 2022 (2):  813-824.  DOI: 10.1007/s10473-022-0225-6
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We consider the topological behaviors of continuous maps with one topological attractor on compact metric space X. This kind of map is a generalization of maps such as topologically expansive Lorenz map, unimodal map without homtervals and so on. Under the finiteness and basin conditions, we provide a leveled A-R pair decomposition for such maps, and characterize α-limit set of each point. Based on weak Morse decomposition of X, we construct a bounded Lyapunov function V (x), which gives a clear description of orbit behavior of each point in X except a meager set.
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