中国科学院数学与系统科学研究院期刊网

15 March 2022, Volume 65 Issue 2
    

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  • Decay of Solution for the MGT Equations with Degenerate Memory Arising in High Frequency Ultrasound
    Wen Jun LIU, Zhi Yu TU, Dan Hua WANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 205-220. https://doi.org/10.12386/A2022sxxb0016
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    This paper is concerned with a third order in time dissipative Moore-Gibson-Thompson equation with infinite degenerate memory
    $\tau u_{ttt}+\alpha(x)u_{tt}-c^{2}\Delta u - b\Delta u_{t}+\displaystyle\int_{0}^{\infty}g(s)\text{div}[a(x)\nabla u(t-s)]{d}s= 0$,
    in which the function $a(x)\geq0 $ and $\alpha(x)\geq0$ can be degenerate but satisfy $a(x)+\alpha(x)\geq \delta >0$. This equation arises as a linearization of a model for wave propagation in viscous thermally relaxing fluids. By using the Faedo-Galerkin approximations together with some energy estimates, we prove that the above system is well-posedness. Besides, under appropriate assumptions, we establish the exponential or general decay results of energy via constructing suitable Lyapunov functionals.
  • Results of Certain Types of Nonlinear Delay Differential Equation
    Man Li LIU, Pei Chu HU, Zhi LI, Qiong Yan WANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 221-234. https://doi.org/10.12386/A2022sxxb0017
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    We show that if the following delay differential equation
    $ \left[w(z+1)w(z)-1\right]\left[w(z)w(z-1)-1\right]+a(z)\dfrac{w'(z)}{w(z)}=\dfrac{\sum_{i=0}^pa_i(z)w^i}{\sum_{j=0}^qb_j(z)w^j}$
    with rational coefficients $a(z), a_i(z), b_j(z)$, admits a transcendental meromorphic solution $w$ of finite many poles with hyper-order less than one, then it reduces into a more simple delay differential equation, which improves some known theorems obtained most recently by Liu and Song. Moreover, we also study the delay differential equations of Tumura-Clunie type and obtain some quantitative properties of transcendental meromorphic solutions.
  • The Measure and Dimension of the First Return Exceptional Sets of Finite Type
    Chun WEI, Man Li LOU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 235-242. https://doi.org/10.12386/A2022sxxb0018
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    Let $(\Sigma_M,\sigma)$ be the shift of finite type where $M=(m_{ij})$ is a $b\times b$ matrix with $m_{ij}\in\{0,1\}$. In this paper, we consider the first return rate of the system $(\Sigma_M,\sigma)$. Let $\tau_k(x)$ be the first return time of $x\in \Sigma_M$ to the $k$-th cylinder containing $x$. Denote
    $E_{\alpha,\beta}=\Big\{x\in\Sigma_M: \liminf_{k\rightarrow\infty}\frac{\log \tau_k(x)}{k}=\alpha,\, \limsup_{k\rightarrow\infty}\frac{\log \tau_k(x)}{k}=\beta\Big\}.$
    We prove that if $M$ is irreducible, then $E_{\alpha,\beta}$ has full Hausdorff dimension for any $0\leq\alpha\leq\beta\leq+\infty$ and has Markov measure either 0 or 1.
  • Exact Continuous Capped-L1 Relaxation for Group Sparse Optimization Problems
    Ding Tao PENG, Qi TANG, Xian ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 243-262. https://doi.org/10.12386/A2022sxxb0019
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    We consider the constrained group sparse regression problem, where the loss function is convex. In order to obtain the exact continuous relaxation of the problem, we use the group Capped-L1 regularization to relax the group sparse regularization. First, we introduce three types of stationary points for the relaxed problem, that is, D(irectional)-stationary points, C(ritical)-stationary points, and L(ifted)-stationary points. We provide some equivalent descriptions for the three types of stationary points, based on which, we investigate the relationship among the three types of stationary points and obtain some properties of them. Furthermore, we analyze the necessary and sufficient optimality conditions for the original group sparse problem and the relaxed problem. At last, we investigate the relationship of the solutions between the original group sparse problem and the relaxed problem, not only from the point of view of global minimizers but also from the point of view of local minimizers. The results reveal the equivalence of the original problem and the relaxed problem.
  • Potential Bifurcation Theorem for a Nonhomogeneous Operator Equation and Its Application
    Guo Wei DAI, Ru Yun MA
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 263-274. https://doi.org/10.12386/A2022sxxb0020
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    The bifurcation phenomenon of the operator equation $\lambda (f_1'(x)+f_2'(x))=g_1'(x)+g_2'(x)$ is studied in this paper. Suppose $f_2'\equiv0$, $f_1$ and $g_1$ are $a$-homogeneous, and some other suitable conditions hold, Fučík et al. obtained that each normalized LS-eigenvalue of $\lambda f_1'(x)= g_1'(x)$ is a bifurcation point of the operator equation above. This paper studies the inhomogeneous case of $f_1+f_2$. We establish the same results as theirs when $f_1$, $f_2$, $g_1$ and $g_2$ satisfy some suitable conditions. A Lyusternik-Shnirel'man theorem is obtained as a preliminary result. And for the application of our abstract theorems, the bifurcation phenomenon from arbitrary LS-eigenvalues is studied for a nonlocal elliptic problem.
  • Gaps Between Zeros of Automorphic L-functions for GL(2)
    Heng Cai TANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 275-286. https://doi.org/10.12386/A2022sxxb0021
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    Let f(z) be a holomorphic Hecke eigenform of even weight k for the full modular group. L(s, f) is the automorphic L-function associated of f. By the smooth shifted second moment of L(s, f), it is proved that there exist infinitely many consecutive zeros of L(s, f) on the critical line whose gaps are greater than 1.88 times of the averaging spacing.
  • Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations
    Yong Xiang LI, Li Juan ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 287-300. https://doi.org/10.12386/A2022sxxb0022
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    This paper is concerned with the existence of periodic solutions for the fully second order ordinary differential equation
    $ -u''(t)=f(t,\,u(t),\,u'(t)),\ \ t\in\mathbb{R} $
    where the nonlinearity $f:\mathbb{R}^{3}\to\mathbb{R}$ is continuous and $f(t, x, y)$ is $2\pi$-periodic in $t$. Some existence results of odd $2\pi$-periodic solutions are obtained under that $f$ satisfies some precise inequality conditions. These inequality conditions allow that $f(t, x, y)$ may be superlinear or sublinear growth on $(x, y)$ as $|(x, y)|\to 0$ and $|(x, y)|\to \infty$.
  • Crossed Product of Finite von Neumann Algebras with Property Γ
    Wen Hua QIAN, Jun Hao SHEN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 301-308. https://doi.org/10.12386/A2022sxxb0023
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    Let $\mathscr{M}$ be a separable type II$_1$ von Neumann algebra. We prove that, if $\mathscr{M}$ has Property $\Gamma$, $G$ is a countable amenable group and $\alpha$ is a trace preserving, properly outer action of $G$ on $\mathscr M$, then the crossed product $\mathscr{M} \rtimes_{\alpha} G$ is a type II$_1$ von Neumann algebra with Property $\Gamma$.
  • Endpoint Estimates for Commutators of Pseudodifferential Operators
    Yu Long DENG, Shun Chao LONG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 309-316. https://doi.org/10.12386/A2022sxxb0024
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    In this paper, we establish two endpoint estimates for the commutator, $[b,T]$, of a class of pseudodifferential operators $T$ with symbols in Hörmander class $S^{m}_{\rho,\delta}(\mathbb R^{n})$. The first one is that the commutator $[b,T]$ is bounded from Hardy space $H^{1}(\mathbb R^{n})$ into weak $L^{1}(\mathbb R^{n})$ space. The second one is an estimate in the Hardy type spaces associated with $b$, where $b\in {\rm BMO}(\mathbb R^{n})$.
  • Order Boundedness of Weighted Composition Operators Between Two Classes of Function Spaces
    Qing Ze LIN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 317-324. https://doi.org/10.12386/A2022sxxb0025
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    In this paper, we first investigate the correspondence between order boundedness and Hilbert-Schmidt of weighted composition operators $W_{\phi,\varphi}(f):=\phi f\circ\varphi$. Then, by resorting to the estimates of the norms of point evaluation functionals $\delta_z$ and derivative point evaluation functionals $\delta'_z$ on weighted Dirichlet spaces $D_{\beta}^q (0<q<\infty,\, -1<\beta<\infty)$ and derivative Hardy spaces $S^p (0<p<\infty)$, the order boundedness of weighted composition operators $W_{\phi,\varphi}$ between weighted Dirichlet spaces $D_{\beta}^q$ and derivative Hardy spaces $S^p$ are completely characterized.
  • Adjacent Vertex Distinguishing Edge Coloring of Planar Graphs with Girth at Least 5
    Xiao Xiu ZHANG, Dan Jun HUANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 325-334. https://doi.org/10.12386/A2022sxxb0026
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    An adjacent vertex distinguishing edge-colorings of a graph $G$ is a proper edge coloring of $G$ such that any pair of adjacent vertices have distinct sets of colors. The minimum number of color required for an adjacent vertex distinguishing edge-coloring of $G$ is denoted by $\chi_{a}'(G)$. In this paper, we prove that if $G$ is a planar graph with girth at least 5 and without isolated edges, then $\chi_a'(G)\leq$ max$\{8,\Delta(G)+1\}$.
  • Biharmonic Hypersurfaces in $\mathbb E_s^5$
    Yu FU, Zhong Hua HOU, Dan YANG, Xin ZHAN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 335-352. https://doi.org/10.12386/A2022sxxb0027
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    We study the geometry and classification problems of biharmonic hypersurfaces in a pseudo-Euclidean 5-space $\mathbb{E}^5_s \ (s=1,2,3,4)$. We prove that if $M^4_{r}$ is a nondegenerate hypersurface in $\mathbb{E}^5_s$ with diagonal shape operator, then $M^4_{r}$ is minimal. Furthermore, based on the results due to Turgay et al. we show that any Lorentz biharmonic hypersurfaces in $\mathbb{E}^5_1$ is minimal. This result gives supporting answers to the biharmonic conjecture for hypersurfaces in 5-dimensional pseudo-Euclidean space.
  • m-th Residue Codes with Length the Product of Two Odd Primes over Finite Fields
    Yuan Bo LIU, Qun Ying LIAO
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 353-370. https://doi.org/10.12386/A2022sxxb0028
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    Let q be a power of the prime, m ≥ 2 be an integer and p1, p2 be two distinct odd primes with gcd(q, p1p2) = 1 and m | gcd(p1 - 1, p2 - 1). Based on the idea of m-th residues, the present paper gives two constructions for the m-th residue code with length n = p1p2 over finite fields. For each construction, a necessary and sufficient condition for the q-ary m-th residue code and the corresponding counting formula are given. Furthermore, a criterion for that these codes are self-orthogonal or complementary dual is obtained, respectively. In some cases, a lower bound of the minimal distance for these codes is obtained.
  • On Transcendental Meromorphic Solutions of a Certain Type of Nonlinear Complex Differential Equations
    Zheng LI, Jun Fan CHEN
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 371-386. https://doi.org/10.12386/A2022sxxb0029
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    We study transcendental meromorphic solutions of a certain type of nonlinear complex differential equations $f^{4}+a(z)ff^{(k)}=p_{1}(z){\rm e}^{\alpha_{1}(z)}+p_{2}(z){\rm e}^{\alpha_{2}(z)}$, where $a$, $p_{1}$, $p_{2}$ are non-zero rational functions and $\alpha_{1}$, $\alpha_{2}$ are nonconstant polynomials. Further, we can derive the conditions concerning the terms $\alpha_{1}$, $\alpha_{2}$, $p_{1}$ and $p_{2}$ that are necessary for the existence and the form of a transcendental meromorphic solution of the equation above. In addition, we analyze the existence of meromorphic solutions of another type of nonlinear complex differential equations $f^{3}+a(z)f'=p_{1}(z){\rm e}^{\nu(z)}+p_{2}(z){\rm e}^{-\nu(z)}$, where $a$, $p_{1}$, $p_{2}$ are non-zero rational functions and $\nu$ is a nonconstant polynomial. These results extend some known results obtained most recently.
  • Hermite Ring Conjecture on Valuation Rings
    Jin Wang LIU, Dong Mei LI, Tao WU
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 387-392. https://doi.org/10.12386/A2022sxxb0030
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    This paper mainly studies the Hermite ring conjecture on valuation rings. According to the properties of the univariate polynomial ring $V[x]$ on the valuation ring $V$, we investigate and obtain a series of equivalent properties for the unimodular row vector $(a_1(x),a_2(x),\ldots,a_n(x))$ on $V[x]$. And then we prove that the Hermite ring conjecture on the valuation ring holds, that is, for an arbitrary valuation ring $V$, $V[x]$ is a Hermite ring.
  • On Moment Estimates for Solutions of Mixed SDEs under Non-Lipschitz Condition
    Xiao Xia SUN, Xuan Ming NI, Jun Yu ZHANG
    Acta Mathematica Sinica, Chinese Series. 2022, 65(2): 393-404. https://doi.org/10.12386/A2022sxxb0031
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    We consider stochastic differential equations (SDEs) driven by mixed fractional Brownian motions under non-Lipschitz conditions. The mixed fractional Brownian motion is a linear combination of Brownian motion and fractional Brownian moiton. We give the p-th moment estimates and the continuity for solutions of considered SDEs by divergence-type Itô formula and Malliavin calculus for mixed fractional Brownianmotion.
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