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Global planar dynamics with a star node and contracting nonlinearity 带有星形节点和收缩非线性的全局平面动力学
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-07 DOI: 10.1007/s40687-024-00427-0
Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau
{"title":"Global planar dynamics with a star node and contracting nonlinearity","authors":"Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau","doi":"10.1007/s40687-024-00427-0","DOIUrl":"https://doi.org/10.1007/s40687-024-00427-0","url":null,"abstract":"<p>This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part is a contracting homogeneous polynomial. The contracting nonlinearity provides the existence of an invariant circle and allows us to obtain a classification through a complete invariant for the dynamics, extending previous work by other authors that was mainly concerned with the existence and number of limit cycles. The general results are also applied to some classes of examples: definite nonlinearities, <span>({textbf {Z}}_2oplus {textbf {Z}}_2)</span> symmetric systems and nonlinearities of degree 3, for which we provide complete sets of phase-portraits.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071668","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
Milnor fibration theorem for differentiable maps 可变映射的米尔诺纤维定理
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-05 DOI: 10.1007/s40687-024-00431-4
José Luis Cisneros-Molina, Aurélio Menegon
{"title":"Milnor fibration theorem for differentiable maps","authors":"José Luis Cisneros-Molina, Aurélio Menegon","doi":"10.1007/s40687-024-00431-4","DOIUrl":"https://doi.org/10.1007/s40687-024-00431-4","url":null,"abstract":"<p>In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps <span>(f:({mathbb {R}}^n,0) rightarrow ({mathbb {R}}^k,0))</span>, where <span>(nge kge 2)</span>, with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class <span>(C^{ell })</span>, <span>(ell ge 2)</span>, with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class <span>(C^{ell })</span> between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"38 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045961","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
De Rham-Witt KZ equations 德拉姆-维特 KZ方程
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-03-04 DOI: 10.1007/s40687-024-00425-2
{"title":"De Rham-Witt KZ equations","authors":"","doi":"10.1007/s40687-024-00425-2","DOIUrl":"https://doi.org/10.1007/s40687-024-00425-2","url":null,"abstract":"<h3>Abstract</h3> <p>In this paper we propose a de Rham-Witt version of the derived KZ equations and their hypergeometric realizations.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"10 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140033002","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
Duality and geometry of horocyclic evolutes in hyperbolic plane 双曲面中角环演化的对偶性和几何学
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-29 DOI: 10.1007/s40687-024-00434-1
Liang Chen, Shyuichi Izumiya, Masatomo Takahashi
{"title":"Duality and geometry of horocyclic evolutes in hyperbolic plane","authors":"Liang Chen, Shyuichi Izumiya, Masatomo Takahashi","doi":"10.1007/s40687-024-00434-1","DOIUrl":"https://doi.org/10.1007/s40687-024-00434-1","url":null,"abstract":"<p>We investigate geometric properties of a special kind of evolutes, so-called horocyclic evolutes, of smooth curves in hyperbolic plane from the viewpoint of duality. To do that, we first review the basic notions of (spacelike) frontals in hyperbolic plane, which developed by the first and the third authors by using basic Legendrian duality theorem developed by the second author. Moreover, two kinds of horocyclic evolutes are defined and the relationship between these two different evolutes are studied. As results, they are Legendrian dual to each other.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"50 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011582","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
Image of iterated polynomial maps of the real plane 实平面多项式迭代映射的图像
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-28 DOI: 10.1007/s40687-024-00433-2
Tat Thang Nguyen
{"title":"Image of iterated polynomial maps of the real plane","authors":"Tat Thang Nguyen","doi":"10.1007/s40687-024-00433-2","DOIUrl":"https://doi.org/10.1007/s40687-024-00433-2","url":null,"abstract":"<p>Let <span>(F: {mathbb {R}}^2rightarrow {mathbb {R}}^2)</span> be a polynomial mapping. We consider the image of the compositions <span>(F^k)</span> of <i>F</i>. We prove that under some condition then the image of the iterated map <span>(F^k)</span> is stable when <i>k</i> is large.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"20 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009354","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
Zariski invariant for quasi-ordinary hypersurfaces 准平凡超曲面的扎里斯基不变式
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-28 DOI: 10.1007/s40687-024-00430-5
R. A. Barbosa, M. E. Hernandes
{"title":"Zariski invariant for quasi-ordinary hypersurfaces","authors":"R. A. Barbosa, M. E. Hernandes","doi":"10.1007/s40687-024-00430-5","DOIUrl":"https://doi.org/10.1007/s40687-024-00430-5","url":null,"abstract":"<p>We introduced an <span>(tilde{mathcal {A}})</span>-invariant for quasi-ordinary parameterizations, and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"18 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009630","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 the topology of complex projective hypersurfaces 论复杂投影超曲面的拓扑学
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-27 DOI: 10.1007/s40687-024-00435-0
Laurenţiu G. Maxim
{"title":"On the topology of complex projective hypersurfaces","authors":"Laurenţiu G. Maxim","doi":"10.1007/s40687-024-00435-0","DOIUrl":"https://doi.org/10.1007/s40687-024-00435-0","url":null,"abstract":"<p>This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"263 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009353","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
$$omega $$ -Symplectic algebra and Hamiltonian vector fields $$omega $$ -Symplectic 代数和哈密顿向量场
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-26 DOI: 10.1007/s40687-024-00423-4
Patrícia H. Baptistelli, Maria Elenice R. Hernandes, Eralcilene Moreira Terezio
{"title":"$$omega $$ -Symplectic algebra and Hamiltonian vector fields","authors":"Patrícia H. Baptistelli, Maria Elenice R. Hernandes, Eralcilene Moreira Terezio","doi":"10.1007/s40687-024-00423-4","DOIUrl":"https://doi.org/10.1007/s40687-024-00423-4","url":null,"abstract":"<p>The purpose of this paper is to present an algebraic theoretical basis for the study of <span>(omega )</span>-Hamiltonian vector fields defined on a symplectic vector space <span>((V,omega ))</span> with respect to coordinates that are not necessarily symplectic. We introduce the concepts of <span>(omega )</span>-symplectic and <span>(omega )</span>-semisymplectic groups, and describe some of their properties that may not coincide with the classical context. We show that the Lie algebra of such groups is a useful tool in the recognition and construction of <span>(omega )</span>-Hamiltonian vector fields.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"38 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140009505","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
Reconstruction of a hypersurface singularity from its moduli algebra 从模代数重构超曲面奇点
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-26 DOI: 10.1007/s40687-024-00432-3
João Hélder Olmedo Rodrigues
{"title":"Reconstruction of a hypersurface singularity from its moduli algebra","authors":"João Hélder Olmedo Rodrigues","doi":"10.1007/s40687-024-00432-3","DOIUrl":"https://doi.org/10.1007/s40687-024-00432-3","url":null,"abstract":"<p>In this paper we present a constructive method to characterize ideals of the local ring <span>({mathscr {O}}_{{mathbb {C}}^n,0})</span> of germs of holomorphic functions at <span>(0in {mathbb {C}}^n)</span> which arise as the moduli ideal <span>(langle f,{mathfrak {m}}, j(f)rangle )</span>, for some <span>(fin {mathfrak {m}}subset {mathscr {O}}_{{mathbb {C}}^n,0})</span>. A consequence of our characterization is an effective solution to a problem dating back to the 1980s, called the Reconstruction Problem of the hypersurface singularity from its moduli algebra. Our results work regardless of whether the hypersurface singularity is isolated or not.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"57 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969402","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
The variance and correlations of the divisor function in $${mathbb {F}}_q [T]$$ , and Hankel matrices $${mathbb {F}}_q [T]$$ 中除数函数的方差和相关性，以及汉克尔矩阵
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-17 DOI: 10.1007/s40687-023-00418-7
Michael Yiasemides
{"title":"The variance and correlations of the divisor function in $${mathbb {F}}_q [T]$$ , and Hankel matrices","authors":"Michael Yiasemides","doi":"10.1007/s40687-023-00418-7","DOIUrl":"https://doi.org/10.1007/s40687-023-00418-7","url":null,"abstract":"<p>We prove an exact formula for the variance of the divisor function over short intervals in <span>({mathcal {A}}:= {mathbb {F}}_q [T])</span>, where <i>q</i> is a prime power; and for correlations of the form <span>(d(A) d(A+B))</span>, where we average both <i>A</i> and <i>B</i> over certain intervals in <span>({mathcal {A}})</span>. We also obtain an exact formula for correlations of the form <span>(d(KQ+N) d (N))</span>, where <i>Q</i> is prime and <i>K</i> and <i>N</i> are averaged over certain intervals with <span>({{,textrm{deg},}}N le {{,textrm{deg},}}Q -1 le {{,textrm{deg},}}K)</span>; and we demonstrate that <span>(d(KQ+N))</span> and <i>d</i>(<i>N</i>) are uncorrelated. We generalize our results to <span>(sigma _z)</span> defined by <span>(sigma _z (A):= sum _{E mid A} |A |^z)</span> for all monics <span>(A in {mathcal {A}})</span>. Our approach is to use the orthogonality relations of additive characters on <span>({mathbb {F}}_q)</span> to translate the problems to ones involving the ranks of Hankel matrices over <span>({mathbb {F}}_q)</span>. We prove several results regarding the rank and kernel structure of these matrices, thus demonstrating their number-theoretic properties. We also discuss extending our method to other divisor sums, such as those involving <span>(d_k)</span>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":"6 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139903881","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
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