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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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
On a new absolute version of Siegel’s lemma 关于西格尔定理的新绝对版本
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-15 DOI: 10.1007/s40687-024-00422-5
{"title":"On a new absolute version of Siegel’s lemma","authors":"","doi":"10.1007/s40687-024-00422-5","DOIUrl":"https://doi.org/10.1007/s40687-024-00422-5","url":null,"abstract":"<h3>Abstract</h3> <p>We establish a new version of Siegel’s lemma over a number field <em>k</em>, providing a bound on the maximum of heights of basis vectors of a subspace of <span> <span>(k^N)</span> </span>, <span> <span>(N ge 2)</span> </span>. In addition to the small-height property, the basis vectors we obtain satisfy certain sparsity condition. Further, we produce a nontrivial bound on the heights of all the possible subspaces generated by subcollections of these basis vectors. Our bounds are absolute in the sense that they do not depend on the field of definition. The main novelty of our method is that it uses only linear algebra and does not rely on the geometry of numbers or the Dirichlet box principle employed in the previous works on this subject.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750966","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 Ribet’s lemma for GL $$_2$$ modulo prime powers 关于 GL $_$2$$ 素幂模的里贝特定理
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-14 DOI: 10.1007/s40687-023-00419-6
{"title":"On Ribet’s lemma for GL $$_2$$ modulo prime powers","authors":"","doi":"10.1007/s40687-023-00419-6","DOIUrl":"https://doi.org/10.1007/s40687-023-00419-6","url":null,"abstract":"<h3>Abstract</h3> <p>Let <span> <span>(rho :Grightarrow {{,textrm{GL},}}_2(K))</span> </span> be a continuous representation of a compact group <em>G</em> over a complete discretely valued field <em>K</em> with ring of integers <span> <span>(mathcal {O})</span> </span> and uniformiser <span> <span>(pi )</span> </span>. We prove that <span> <span>({{,textrm{tr},}}rho )</span> </span> is reducible modulo <span> <span>(pi ^n)</span> </span> if and only if <span> <span>(rho )</span> </span> is reducible modulo <span> <span>(pi ^n)</span> </span>. More precisely, there exist characters <span> <span>(chi _1,chi _2 :Grightarrow (mathcal {O}/pi ^nmathcal {O})^times )</span> </span> such that <span> <span>(det (t - rho (g))equiv (t-chi _1(g))(t-chi _2(g))pmod {pi ^n})</span> </span> for all <span> <span>(gin G)</span> </span>, if and only if there exists a <em>G</em>-stable lattice <span> <span>(Lambda subseteq K^2)</span> </span> such that <span> <span>(Lambda /pi ^nLambda )</span> </span> contains a <em>G</em>-invariant, free, rank one <span> <span>(mathcal {O}/pi ^nmathcal {O})</span> </span>-submodule. Our result applies in the case that <span> <span>(rho )</span> </span> is not residually multiplicity-free, in which case it answers a question of Bellaïche and Chenevier (J Algebra 410:501–525, 2014, pp. 524). As an application, we prove an optimal version of Ribet’s lemma, which gives a condition for the existence of a <em>G</em>-stable lattice <span> <span>(Lambda )</span> </span> that realises a non-split extension of <span> <span>(chi _2)</span> </span> by <span> <span>(chi _1)</span> </span>.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139750963","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
Extensions of MacMahon’s sums of divisors 麦克马洪除数和的扩展
IF 1.2 3区 数学
Research in the Mathematical Sciences Pub Date : 2024-02-05 DOI: 10.1007/s40687-024-00421-6
Tewodros Amdeberhan, George E. Andrews, Roberto Tauraso
{"title":"Extensions of MacMahon’s sums of divisors","authors":"Tewodros Amdeberhan, George E. Andrews, Roberto Tauraso","doi":"10.1007/s40687-024-00421-6","DOIUrl":"https://doi.org/10.1007/s40687-024-00421-6","url":null,"abstract":"<p>In 1920, P. A. MacMahon generalized the (classical) notion of divisor sums by relating it to the theory of partitions of integers. In this paper, we extend the idea of MacMahon. In doing so, we reveal a wealth of divisibility theorems and unexpected combinatorial identities. Our initial approach is quite different from MacMahon and involves <i>rational</i> function approximation to MacMahon-type generating functions. One such example involves multiple <i>q</i>-harmonic sums </p><span>$$begin{aligned} sum _{k=1}^nfrac{(-1)^{k-1}genfrac[]{0.0pt}{}{n}{k}_{q}(1+q^k)q^{left( {begin{array}{c}k 2end{array}}right) +tk}}{[k]_q^{2t}genfrac[]{0.0pt}{}{n+k}{k}_{q}} =sum _{1le k_1le cdots le k_{2t}le n}frac{q^{n+k_1+k_3cdots +k_{2t-1}}+q^{k_2+k_4+cdots +k_{2t}}}{[n+k_1]_q[k_2]_qcdots [k_{2t}]_q}. end{aligned}$$</span>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139690107","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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