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Smooth invariant manifolds and foliations for the differential equations with piecewise constant argument 具有分段常数参数的微分方程的光滑不变流形和叶状
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-22 DOI: 10.1016/j.bulsci.2025.103579
Weijie Lu , Donal O'Regan , Yonghui Xia
{"title":"Smooth invariant manifolds and foliations for the differential equations with piecewise constant argument","authors":"Weijie Lu ,&nbsp;Donal O'Regan ,&nbsp;Yonghui Xia","doi":"10.1016/j.bulsci.2025.103579","DOIUrl":"10.1016/j.bulsci.2025.103579","url":null,"abstract":"<div><div>In this work, we establish the theory of smooth invariant manifolds and smooth invariant foliations for the differential equations with piecewise constant argument of a generalized type (DEPCAGs). Suppose that the linear DEPCAGs admits a <em>α</em>-exponential dichotomy, we obtain the existence of Lipschitz stable (unstable) invariant manifolds and Lipschitz stable (unstable) invariant foliations, which are based on the Lyapunov-Perron integrals with piecewise constant argument and other non-trivial techniques (such as, dichotomy inequalities with piecewise constant argument). Furthermore, we formulate and prove the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-smoothness of these manifolds and foliations for DEPCAGs by means of the fiber contraction theorem.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103579"},"PeriodicalIF":1.3,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A new method based on the semi-tensor product of matrices for solving communicative quaternion matrix equation ∑i=1kAiXBi=C and its application 基于矩阵半张量积求解交流四元数矩阵方程∑i=1kAiXBi=C的新方法及其应用
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-21 DOI: 10.1016/j.bulsci.2025.103576
Mingcui Zhang, Ying Li, Jianhua Sun, Xueling Fan, Anli Wei
{"title":"A new method based on the semi-tensor product of matrices for solving communicative quaternion matrix equation ∑i=1kAiXBi=C and its application","authors":"Mingcui Zhang,&nbsp;Ying Li,&nbsp;Jianhua Sun,&nbsp;Xueling Fan,&nbsp;Anli Wei","doi":"10.1016/j.bulsci.2025.103576","DOIUrl":"10.1016/j.bulsci.2025.103576","url":null,"abstract":"<div><div>This paper studies the least squares problem of the commutative quaternion matrix equation <span><span>(1.1)</span></span>, finds its minimal norm least squares (anti-)Hermitian solution. In the process of completing this work, we generalize the semi-tensor product of real matrices to the commutative quaternion matrices, then use it to extend the vector operators to the commutative quaternion matrix and propose the <em>L</em>-representation, which transforms the intricate commutative quaternion matrix equation into a solvable system of real linear equations, we also use <em>GH</em>-representation to reduce the complexity of the operation and greatly save the operation time. This can be illustrated by numerical examples in the paper. In addition, we take a special kind of commutative quaternion: reduced biquaternion as an example, and compare our method with another method in reference <span><span>[33]</span></span> to prove the effectiveness of our method. Finally, we apply the method used in this paper to symmetric color image restoration.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103576"},"PeriodicalIF":1.3,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Characterizations of complex symmetric Toeplitz operators 复对称Toeplitz算子的刻画
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-21 DOI: 10.1016/j.bulsci.2025.103578
Sudip Ranjan Bhuia, Deepak Pradhan, Jaydeb Sarkar
{"title":"Characterizations of complex symmetric Toeplitz operators","authors":"Sudip Ranjan Bhuia,&nbsp;Deepak Pradhan,&nbsp;Jaydeb Sarkar","doi":"10.1016/j.bulsci.2025.103578","DOIUrl":"10.1016/j.bulsci.2025.103578","url":null,"abstract":"<div><div>We characterize Toeplitz operators that are complex symmetric. This follows as a by-product of characterizations of conjugations on Hilbert spaces. Notably, we prove that every conjugation admits a canonical factorization. As a consequence, we prove that a Toeplitz operator is complex symmetric if and only if the Toeplitz operator is <em>S</em>-Toeplitz for some unilateral shift <em>S</em> and the transpose of the Toeplitz operator matrix is equal to the matrix of the Toeplitz operator corresponding to the basis of the unilateral shift <em>S</em>. Also, we characterize complex symmetric Toeplitz operators on the Hardy space over the open unit polydisc. Our results are related to a question raised by K. Guo and S. Zhu <span><span>[9]</span></span>.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103578"},"PeriodicalIF":1.3,"publicationDate":"2025-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bohr inequalities for the class of unimodular bounded functions on shifted disks 移盘上一类非模有界函数的玻尔不等式
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-20 DOI: 10.1016/j.bulsci.2025.103577
Molla Basir Ahamed , Vasudevarao Allu , Himadri Halder
{"title":"Bohr inequalities for the class of unimodular bounded functions on shifted disks","authors":"Molla Basir Ahamed ,&nbsp;Vasudevarao Allu ,&nbsp;Himadri Halder","doi":"10.1016/j.bulsci.2025.103577","DOIUrl":"10.1016/j.bulsci.2025.103577","url":null,"abstract":"<div><div>Let <span><math><mi>H</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> denote the class of analytic functions in the unit disk <span><math><mi>D</mi><mo>:</mo><mo>=</mo><mo>{</mo><mi>z</mi><mo>∈</mo><mi>C</mi><mo>:</mo><mo>|</mo><mi>z</mi><mo>|</mo><mo>&lt;</mo><mn>1</mn><mo>}</mo></math></span>. The classical Bohr's inequality <span><span>[21]</span></span> states that if <span><math><mi>f</mi><mo>∈</mo><mi>H</mi><mo>(</mo><mi>D</mi><mo>)</mo></math></span> is given by <span><math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that <span><math><mo>|</mo><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>|</mo><mo>&lt;</mo><mn>1</mn></math></span> for <span><math><mi>z</mi><mo>∈</mo><mi>D</mi></math></span>, then<span><span><span><math><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mo>∞</mo></mrow></munderover><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>≤</mo><mn>1</mn><mspace></mspace><mspace></mspace><mtext>for</mtext><mspace></mspace><mspace></mspace><mi>r</mi><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></span></span></span> and the constant 1/3 cannot be improved. The constant 1/3 is known as Bohr radius. In this paper, we study Bohr phenomenon for classes of analytic as well as harmonic mappings on shifted disks. We prove several sharp results on improved Bohr radius for the classes of analytic functions as well as for the class of harmonic mappings on certain shifted disks.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103577"},"PeriodicalIF":1.3,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Infinitesimal prolongation of the Ψ-Hilfer derivative Ψ-Hilfer导数的无穷小延伸
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-10 DOI: 10.1016/j.bulsci.2024.103574
F.S. Costa , J.C.A. Soares , J.V.C. Sousa , G.S.F. Frederico , G.L. Araujo
{"title":"Infinitesimal prolongation of the Ψ-Hilfer derivative","authors":"F.S. Costa ,&nbsp;J.C.A. Soares ,&nbsp;J.V.C. Sousa ,&nbsp;G.S.F. Frederico ,&nbsp;G.L. Araujo","doi":"10.1016/j.bulsci.2024.103574","DOIUrl":"10.1016/j.bulsci.2024.103574","url":null,"abstract":"<div><div>In this paper, a study on the infinitesimal prolongation of Ψ-Hilfer fractional derivative is performed. The properties of Lie group transformation are presented, which are applied in the fractional Ψ-Thomas-Fermi equation. Numerical simulations are presented to the model.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103574"},"PeriodicalIF":1.3,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Subdirect sums of partially doubly strictly diagonally matrices 部分双严格对角矩阵的次直和
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-08 DOI: 10.1016/j.bulsci.2024.103571
Fude Zhang, Lanlan Liu, Deshu Sun
{"title":"Subdirect sums of partially doubly strictly diagonally matrices","authors":"Fude Zhang,&nbsp;Lanlan Liu,&nbsp;Deshu Sun","doi":"10.1016/j.bulsci.2024.103571","DOIUrl":"10.1016/j.bulsci.2024.103571","url":null,"abstract":"<div><div>The class of partially doubly strictly diagonally (for shortly, <span><math><mi>P</mi><mi>D</mi><mi>S</mi><mi>D</mi><mi>D</mi></math></span>) matrices is a new subclass of nonsingular <em>H</em>-matrices. In this paper, we focus on the subdirect sum of <span><math><mi>P</mi><mi>D</mi><mi>S</mi><mi>D</mi><mi>D</mi></math></span> matrices, and some sufficient conditions guaranteeing that the subdirect sum of <span><math><mi>P</mi><mi>D</mi><mi>S</mi><mi>D</mi><mi>D</mi></math></span> matrices is in the class of <span><math><mi>P</mi><mi>D</mi><mi>S</mi><mi>D</mi><mi>D</mi></math></span> matrices are provided. Moreover, we also present some numerical examples to illustrate the corresponding results.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103571"},"PeriodicalIF":1.3,"publicationDate":"2025-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On directional preservation of orthogonality and its application to isometries 正交性的方向保持及其在等距图中的应用
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-07 DOI: 10.1016/j.bulsci.2025.103575
Jayanta Manna , Kalidas Mandal , Kallol Paul , Debmalya Sain
{"title":"On directional preservation of orthogonality and its application to isometries","authors":"Jayanta Manna ,&nbsp;Kalidas Mandal ,&nbsp;Kallol Paul ,&nbsp;Debmalya Sain","doi":"10.1016/j.bulsci.2025.103575","DOIUrl":"10.1016/j.bulsci.2025.103575","url":null,"abstract":"<div><div>We study the local preservation of Birkhoff-James orthogonality by linear operators between normed linear spaces, at a point and in a particular direction. We obtain a complete characterization of the same, which allows us to present refinements of the local preservation of orthogonality explored earlier. We also study the directional preservation of orthogonality with respect to certain special subspaces of the domain space, and apply the results towards identifying the isometries on a polyhedral normed linear space. In particular, we obtain refinements of the Blanco-Koldobsky-Turnšek Theorem for polyhedral normed linear spaces, including <span><math><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mi>n</mi></mrow></msubsup><mo>,</mo><msubsup><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103575"},"PeriodicalIF":1.3,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Higher Gaussian maps for special classes of curves 特殊曲线类的高高斯映射
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-03 DOI: 10.1016/j.bulsci.2024.103573
Dario Faro, Paola Frediani, Antonio Lacopo
{"title":"Higher Gaussian maps for special classes of curves","authors":"Dario Faro,&nbsp;Paola Frediani,&nbsp;Antonio Lacopo","doi":"10.1016/j.bulsci.2024.103573","DOIUrl":"10.1016/j.bulsci.2024.103573","url":null,"abstract":"<div><div>In this paper we study higher Gaussian (or Wahl) maps for the canonical bundle of certain smooth projective curves. More precisely, we determine the rank of higher Gaussian maps of the canonical bundle for plane curves, for curves contained in certain linear systems in a surface given by a product of two curves and for curves contained in a sufficiently ample line bundle on an Enriques surface.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103573"},"PeriodicalIF":1.3,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Congruences corresponding to hypergeometric identities II. Generalized hypergeometric transformations 对应于超几何恒等式的同余2。广义超几何变换
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2025-01-03 DOI: 10.1016/j.bulsci.2024.103570
Guo-Shuai Mao , Hao Pan
{"title":"Congruences corresponding to hypergeometric identities II. Generalized hypergeometric transformations","authors":"Guo-Shuai Mao ,&nbsp;Hao Pan","doi":"10.1016/j.bulsci.2024.103570","DOIUrl":"10.1016/j.bulsci.2024.103570","url":null,"abstract":"<div><div>We consider the congruences modulo <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> for the truncated generalized hypergeometric functions <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mi>s</mi></mrow><none></none><mprescripts></mprescripts><mrow><mi>r</mi></mrow><none></none></mmultiscripts><msub><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><mrow></mrow></mtd></mtr></mtable></mrow><mrow><mtable><mtr><mtd><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mrow><mtable><mtr><mtd><mo>…</mo></mtd></mtr><mtr><mtd><mo>…</mo></mtd></mtr></mtable></mrow><mrow><mtable><mtr><mtd><msub><mrow><mi>α</mi></mrow><mrow><mi>r</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>β</mi></mrow><mrow><mi>s</mi></mrow></msub></mtd></mtr></mtable></mrow><mo>|</mo><mspace></mspace><mn>1</mn><mo>]</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In particular, we give the <span><math><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> analogue of Whipple's <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>4</mn></mrow><none></none></mmultiscripts></math></span> and <span><math><mmultiscripts><mrow><mi>F</mi></mrow><mrow><mn>6</mn></mrow><none></none><mprescripts></mprescripts><mrow><mn>7</mn></mrow><none></none></mmultiscripts></math></span> transformations.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103570"},"PeriodicalIF":1.3,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Shubin calculi for actions of graded Lie groups 分级李群作用的舒宾演算
IF 1.3 3区 数学
Bulletin des Sciences Mathematiques Pub Date : 2024-12-30 DOI: 10.1016/j.bulsci.2024.103572
Eske Ewert, Philipp Schmitt
{"title":"Shubin calculi for actions of graded Lie groups","authors":"Eske Ewert,&nbsp;Philipp Schmitt","doi":"10.1016/j.bulsci.2024.103572","DOIUrl":"10.1016/j.bulsci.2024.103572","url":null,"abstract":"<div><div>In this article, we develop a calculus of Shubin type pseudodifferential operators on certain non-compact spaces, using a groupoid approach similar to the one of van Erp and Yuncken. More concretely, we consider actions of graded Lie groups on graded vector spaces and study pseudodifferential operators that generalize fundamental vector fields and multiplication by polynomials. Our two main examples of elliptic operators in this calculus are Rockland operators with a potential and the generalizations of the harmonic oscillator to the Heisenberg group due to Rottensteiner–Ruzhansky.</div><div>Deforming the action of the graded group, we define a tangent groupoid which connects pseudodifferential operators to their principal (co)symbols. We show that our operators form a calculus that is asymptotically complete. Elliptic elements in the calculus have parametrices, are hypoelliptic, and can be characterized in terms of a Rockland condition. Moreover, we study the mapping properties as well as the spectra of our operators on Sobolev spaces and compare our calculus to the Shubin calculus on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and its anisotropic generalizations.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"199 ","pages":"Article 103572"},"PeriodicalIF":1.3,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143173503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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