首页 > 最新文献

Arnold Mathematical Journal最新文献

英文 中文
Red Sizes of Quivers 红色大小的箭袋
Q3 Mathematics Pub Date : 2023-03-06 DOI: 10.1007/s40598-023-00226-5
Eric Bucher, John Machacek

In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a red number. In this paper, we generalized this notion to what we call unrestricted red numbers which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.

在这篇文章中,我们将扩展与簇代数相关的四元组的最大绿化序列和红化序列的概念。这些序列的存在已在与福明和泽列文斯基的簇代数有关的各种应用中得到研究。艾哈迈德和李考虑了一种数字度量方法,即红色数字,来衡量一个震颤体有多接近于接纳一个最大绿色序列。在本文中,我们将这一概念推广到了所谓的无限制红数,它与红化序列有关。除了建立这个更一般的框架,我们还完全确定了所有有限突变类型的震颤体的红数和非限制红数。此外,我们还给出了关于一般红数和非限制红数可能值的猜想。
{"title":"Red Sizes of Quivers","authors":"Eric Bucher,&nbsp;John Machacek","doi":"10.1007/s40598-023-00226-5","DOIUrl":"10.1007/s40598-023-00226-5","url":null,"abstract":"<div><p>In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated with cluster algebras. The existence of these sequences has been studied for a variety of applications related to Fomin and Zelevinsky’s cluster algebras. Ahmad and Li considered a numerical measure of how close a quiver is to admitting a maximal green sequence called a <i>red number</i>. In this paper, we generalized this notion to what we call <i>unrestricted red numbers</i> which are related to reddening sequences. In addition to establishing this more general framework, we completely determine the red numbers and unrestricted red numbers for all finite mutation type of quivers. Furthermore, we give conjectures on the possible values of red numbers and unrestricted red numbers in general.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46327127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sandpile Solitons in Higher Dimensions 高维中的沙堆孤子
Q3 Mathematics Pub Date : 2023-01-30 DOI: 10.1007/s40598-023-00224-7
Nikita Kalinin

Let (pin {mathbb {Z}}^n) be a primitive vector and (Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min (pcdot z, 0)). The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to (Psi ) “at infinity”. We apply this result to sandpile models on ({mathbb {Z}}^n). We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function

$$begin{aligned} Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min _{pin Acap {mathbb {Z}}^n}(pcdot z+c_p), c_pin {mathbb {Z}}end{aligned}$$

there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with (Psi ) “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

设(pin{mathbb{Z}}^n。外壳理论允许我们证明在所有整数值超调和函数中存在一个点态极小函数,该函数等于“在无穷大”的(Psi)。我们将这一结果应用于({mathbb{Z}}^n)上的沙堆模型。我们证明了沙堆模型中所谓孤子的存在,这是S.Caracciolo、G.Paoletti和a.Sportiello在二维环境中发现的,并由作者和M.Shkolnikov在以前的论文中进行了研究。我们证明,与二维情况类似,当我们应用沙堆波算子时,使用我们的外壳程序定义的沙堆状态是不变的(这就是为什么我们称它们为孤子)。对于除顶点外没有格点的每个格多面体A,我们证明了一个类似的结果。也就是说,对于每一个函数$$begin{aligned}Psi:{mathbb{Z}}^nrightarrow{math bb{Z}}}、ZrightarrowmainAcap{mattbb{Z}}^n}(pcdot Z+cp)、c_pin{mathbb}}end{align}$$,在所有与(Psi)“无穷大”重合的整数值超调和函数中都存在一个逐点极小函数。后一个函数的拉普拉斯算子对应于我们在对应于A边缘的孤立子相交时观察到的情况(见图1)。
{"title":"Sandpile Solitons in Higher Dimensions","authors":"Nikita Kalinin","doi":"10.1007/s40598-023-00224-7","DOIUrl":"10.1007/s40598-023-00224-7","url":null,"abstract":"<div><p>Let <span>(pin {mathbb {Z}}^n)</span> be a primitive vector and <span>(Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min (pcdot z, 0))</span>. The theory of <i>husking</i> allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to <span>(Psi )</span> “at infinity”. We apply this result to sandpile models on <span>({mathbb {Z}}^n)</span>. We prove existence of so-called <i>solitons</i> in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope <i>A</i> without lattice points except its vertices. Namely, for each function </p><div><div><span>$$begin{aligned} Psi :{mathbb {Z}}^nrightarrow {mathbb {Z}}, zrightarrow min _{pin Acap {mathbb {Z}}^n}(pcdot z+c_p), c_pin {mathbb {Z}}end{aligned}$$</span></div></div><p>there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with <span>(Psi )</span> “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of <i>A</i>, intersect (see Fig. 1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41268245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Solvability of Some Systems of Integro-differential Equations in Population Dynamics Depending on the Natality and Mortality Rates 基于出生率和死亡率的人口动力学中某些积分微分方程组的可解性
Q3 Mathematics Pub Date : 2023-01-27 DOI: 10.1007/s40598-023-00225-6
Vitali Vougalter, Vitaly Volpert

We establish the existence of stationary solutions for certain systems of reaction–diffusion-type equations in the corresponding (H^{2}) spaces. Our method relies on the fixed point theorem when the elliptic problem contains second-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates involved in the equations of the systems.

我们在相应的 (H^{2}) 空间中建立了某些反应扩散型方程系统的静态解的存在性。当椭圆问题包含具有或不具有弗雷德霍姆性质的二阶微分算子时,我们的方法依赖于定点定理,而弗雷德霍姆性质可能取决于系统方程中涉及的出生率和死亡率之间的竞争结果。
{"title":"Solvability of Some Systems of Integro-differential Equations in Population Dynamics Depending on the Natality and Mortality Rates","authors":"Vitali Vougalter,&nbsp;Vitaly Volpert","doi":"10.1007/s40598-023-00225-6","DOIUrl":"10.1007/s40598-023-00225-6","url":null,"abstract":"<div><p>We establish the existence of stationary solutions for certain systems of reaction–diffusion-type equations in the corresponding <span>(H^{2})</span> spaces. Our method relies on the fixed point theorem when the elliptic problem contains second-order differential operators with and without the Fredholm property, which may depend on the outcome of the competition between the natality and the mortality rates involved in the equations of the systems.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44453917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Classification of Schubert Galois Groups in (textit{Gr},(4,9)) Schubert-Galois群在(textit{Gr},(4,9))中的分类
Q3 Mathematics Pub Date : 2023-01-17 DOI: 10.1007/s40598-022-00221-2
Abraham Martín del Campo, Frank Sottile, Robert Lee Williams

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.

我们将9维空间中4平面的Grassmann中的Schubert问题按它们的Galois群进行分类。在这个格拉斯曼的31806个基本舒伯特问题中,只有149个问题的伽罗瓦群不包含交替群。我们确定了这149个群中的伽罗瓦群——每个群都是一个无印置换群。根据它们的几何形状,这149个分为两个家族。这项研究提出了一种可能的舒伯特问题的分类,其伽罗瓦群不是完全对称群,这是舒伯特微积分反伽罗瓦问题的第一步。
{"title":"Classification of Schubert Galois Groups in (textit{Gr},(4,9))","authors":"Abraham Martín del Campo,&nbsp;Frank Sottile,&nbsp;Robert Lee Williams","doi":"10.1007/s40598-022-00221-2","DOIUrl":"10.1007/s40598-022-00221-2","url":null,"abstract":"<div><p>We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, there are only 149 whose Galois group does not contain the alternating group. We identify the Galois groups of these 149—each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and is a first step toward the inverse Galois problem for Schubert calculus.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00221-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50490033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Classification of Schubert Galois Groups in $$textit{Gr},(4,9)$$ Gr ( 4 , 9 ) $$textit{Gr},(4,9)$$Gr(4,9)中Schubert-Galois群的分类
Q3 Mathematics Pub Date : 2023-01-17 DOI: 10.1007/s40598-022-00221-2
Abraham Martín del Campo, F. Sottile, Robert Williams
{"title":"Classification of Schubert Galois Groups in \u0000 \u0000 \u0000 \u0000 $$textit{Gr},(4,9)$$\u0000 \u0000 \u0000 Gr\u0000 \u0000 (\u0000 4\u0000 ,\u0000 9\u0000 )\u0000 \u0000 ","authors":"Abraham Martín del Campo, F. Sottile, Robert Williams","doi":"10.1007/s40598-022-00221-2","DOIUrl":"https://doi.org/10.1007/s40598-022-00221-2","url":null,"abstract":"","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45286327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum 自相似群与全纯动力学:重整化、可积性和谱
Q3 Mathematics Pub Date : 2023-01-05 DOI: 10.1007/s40598-022-00223-0
N.-B. Dang, R. Grigorchuk, M. Lyubich

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a spectral current). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.

本文从动力学和代数几何的角度探讨了几个自相似群(Grigorchuk、Lamplighter和Hanoi群)在Schreier图上的拉普拉斯谱测度。对于这些图,经典的舒尔重整化变换作用于适当的谱参数作为两个变量的有理映射。我们表明,所讨论的谱可以被解释为切片的渐近分布,通过在相应的有理映射下的某些代数曲线的迭代回调线(导致我们得到谱电流的概念)。接着给出了谱离散性的动力学判据。对于原子光谱,给出了极限光谱测度的有限尺度近似的精确收敛速率。对于所考虑的三组,相应的有理映射恰好是在一个变量的多项式上编织的。我们揭示了这种可积现象的代数几何性质。
{"title":"Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum","authors":"N.-B. Dang,&nbsp;R. Grigorchuk,&nbsp;M. Lyubich","doi":"10.1007/s40598-022-00223-0","DOIUrl":"10.1007/s40598-022-00223-0","url":null,"abstract":"<div><p>In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two variables. We show that the spectra in question can be interpreted as asymptotic distributions of slices by a line of iterated pullbacks of certain algebraic curves under the corresponding rational maps (leading us to a notion of a <i>spectral current</i>). We follow up with a dynamical criterion for discreteness of the spectrum. In case of atomic spectrum, the precise rate of convergence of finite-scale approximands to the limiting spectral measure is given. For the three groups under consideration, the corresponding rational maps happen to be fibered over polynomials in one variable. We reveal the algebro-geometric nature of this integrability phenomenon.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41470153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Planar Sections of a Surface Close to an Umbilic 靠近脐带的平面剖面
Q3 Mathematics Pub Date : 2022-12-20 DOI: 10.1007/s40598-022-00222-1
Peter Giblin, Aleksandr Pukhlikov

We study the intersection between a smooth algebraic surface with an umbilic point and a plane parallel and close to the tangent plane at the umbilic. The problem has its origin in the study of isophote (equal illumination) curves in a 2-dimensional image. In particular, we study the circles which have exceptional tangency to this intersection curve: ordinary tangency at one point and osculating at another; ordinary tangency at three points; and 4-point tangency at a vertex. The centres of circles having ordinary tangency at two points trace out a curve whose closure is the symmetry set of the intersection curve, and the exceptional circles above give respectively cusps, triple crossings and endpoints of this set. We analyse the curves traced out by the contact points and centres of the exceptional circles as the plane approaches the tangent plane at the umbilic. We also briefly discuss the global structure of the symmetry set by means of a typical example.

研究了带脐点的光滑代数曲面与在脐点处平行且接近切平面的平面的交。这个问题起源于对二维图像中等光照曲线的研究。特别地,我们研究了与这条相交曲线有特殊相切的圆:一点相切,另一点相切;三点的常切线;和一个顶点的四点相切。两点常相切的圆心画出一条曲线,该曲线的闭包是交点曲线的对称集,上述例外圆分别给出了交点曲线的顶点、三交点和端点。我们分析了由接触点和异常圆的中心跟踪的曲线,当平面接近切平面在脐带。并通过一个典型的例子简要地讨论了对称集的整体结构。
{"title":"Planar Sections of a Surface Close to an Umbilic","authors":"Peter Giblin,&nbsp;Aleksandr Pukhlikov","doi":"10.1007/s40598-022-00222-1","DOIUrl":"10.1007/s40598-022-00222-1","url":null,"abstract":"<div><p>We study the intersection between a smooth algebraic surface with an umbilic point and a plane parallel and close to the tangent plane at the umbilic. The problem has its origin in the study of isophote (equal illumination) curves in a 2-dimensional image. In particular, we study the circles which have exceptional tangency to this intersection curve: ordinary tangency at one point and osculating at another; ordinary tangency at three points; and 4-point tangency at a vertex. The centres of circles having ordinary tangency at two points trace out a curve whose closure is the <i>symmetry set</i> of the intersection curve, and the exceptional circles above give respectively cusps, triple crossings and endpoints of this set. We analyse the curves traced out by the contact points and centres of the exceptional circles as the plane approaches the tangent plane at the umbilic. We also briefly discuss the global structure of the symmetry set by means of a typical example.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00222-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42119963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Generalized Problem Associated to the Kummer–Vandiver Conjecture 与Kummer-Vandiver猜想有关的一个广义问题
Q3 Mathematics Pub Date : 2022-11-07 DOI: 10.1007/s40598-022-00220-3
Hiroki Sumida-Takahashi

To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let p be an odd prime number and (zeta _p) a primitive p-th root of unity. Using new programs, we compute the Iwasawa invariants of ({textbf{Q}}(sqrt{d},zeta _p)) in the range (|d|<200) and (200<p <1{,}000{,}000). From our data, the actual numbers of exceptional cases seem to be near the expected numbers for (p<1{,}000{,}000). Moreover, we find a few rare exceptional cases for (|d|<10) and (p>1{,}000{,}000). We give two partial reasons why it is difficult to find exceptional cases for (d=1) including counter-examples to the Kummer–Vandiver conjecture.

为了讨论Kummer–Vandiver猜想的有效性,我们考虑了与该猜想相关的一个广义问题。设p是奇数素数,(zeta_p)是单位的原始p根。使用新程序,我们计算了({textbf{Q}})(sqrt{d},zeta _p))在(|d|<;200)和(200<;p<;1{,}000{,}000)范围内的Iwasawa不变量。根据我们的数据,异常情况的实际数量似乎接近(p<;1{,}000{,}000 )的预期数量。此外,我们发现了(|d|<;10)和(p>;1{,}000{,}000)的一些罕见的例外情况。我们给出了为什么很难找到(d=1)的例外情况的两个部分原因,包括Kummer–Vandiver猜想的反例。
{"title":"A Generalized Problem Associated to the Kummer–Vandiver Conjecture","authors":"Hiroki Sumida-Takahashi","doi":"10.1007/s40598-022-00220-3","DOIUrl":"10.1007/s40598-022-00220-3","url":null,"abstract":"<div><p>To discuss the validity of the Kummer–Vandiver conjecture, we consider a generalized problem associated to the conjecture. Let <i>p</i> be an odd prime number and <span>(zeta _p)</span> a primitive <i>p</i>-th root of unity. Using new programs, we compute the Iwasawa invariants of <span>({textbf{Q}}(sqrt{d},zeta _p))</span> in the range <span>(|d|&lt;200)</span> and <span>(200&lt;p &lt;1{,}000{,}000)</span>. From our data, the actual numbers of exceptional cases seem to be near the expected numbers for <span>(p&lt;1{,}000{,}000)</span>. Moreover, we find a few rare exceptional cases for <span>(|d|&lt;10)</span> and <span>(p&gt;1{,}000{,}000)</span>. We give two partial reasons why it is difficult to find exceptional cases for <span>(d=1)</span> including counter-examples to the Kummer–Vandiver conjecture.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42514584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Polyhedral Homotopy Algorithm for Real Zeros 实零点的多面体同胚算法
Q3 Mathematics Pub Date : 2022-10-27 DOI: 10.1007/s40598-022-00219-w
Alperen A. Ergür, Timo de Wolff

We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry.

我们设计了一种基于Viro的拼接方法的同源延拓算法,用于寻找稀疏多项式系统的实零。该算法针对具有满足某些凹性条件的系数的多项式系统,它跟踪最优解路径数,并且它完全在实数上运算。在更专业的术语中,我们设计了一种算法,该算法可以正确地计算并找到多项式系统的实零,这些多项式系统位于底层A判别变形虫的补码的无界分量中。我们详细阐述了Viro的拼接工作方法、a-判别变形虫补的凸几何和计算实代数几何之间的联系。
{"title":"A Polyhedral Homotopy Algorithm for Real Zeros","authors":"Alperen A. Ergür,&nbsp;Timo de Wolff","doi":"10.1007/s40598-022-00219-w","DOIUrl":"10.1007/s40598-022-00219-w","url":null,"abstract":"<div><p>We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46281127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Jordan Types of Triangular Matrices over a Finite Field 有限域上三角矩阵的Jordan型
Q3 Mathematics Pub Date : 2022-10-13 DOI: 10.1007/s40598-022-00214-1
Dmitry Fuchs, Alexandre Kirillov Sr.

Let (lambda ) be a partition of an integer n and ({mathbb F}_q) be a finite field of order q. Let (P_lambda (q)) be the number of strictly upper triangular (ntimes n) matrices of the Jordan type (lambda ). It is known that the polynomial (P_lambda ) has a tendency to be divisible by high powers of q and (Q=q-1), and we put (P_lambda (q)=q^{d(lambda )}Q^{e(lambda )}R_lambda (q)), where (R_lambda (0)ne 0) and (R_lambda (1)ne 0). In this article, we study the polynomials (P_lambda (q)) and (R_lambda (q)). Our main results: an explicit formula for (d(lambda )) (an explicit formula for (e(lambda )) is known, see Proposition 3.3 below), a recursive formula for (R_lambda (q)) (a similar formula for (P_lambda (q)) is known, see Proposition 3.1 below), the stabilization of (R_lambda ) with respect to extending (lambda ) by adding strings of 1’s, and an explicit formula for the limit series (R_{lambda 1^infty }). Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.

设(lambda)是整数n的一个分区,并且({mathbb F}_q)是q阶有限域。设(P_lambda(q))是Jordan类型的严格上三角(ntimes n)矩阵的个数。已知多项式(P_lambda )具有可被q和(q=q-1)的高幂整除的趋势,我们将(P_ lambda(q)=q^{d(lambda)}q^{e(lamba)}R_ lambda。本文研究了多项式(P_λ(q))和(R_λ(q))。我们的主要结果:(d(lambda))的显式公式((e(lamba))是已知的显式表达式,见下面的命题3.3),(R_lambda(q))((P_lamba(q))的类似公式是已知的,见下面命题3.1),,以及极限级数(R_。我们的研究是由轨道方法在有限域上幂零代数群表示理论中的投影应用所推动的。
{"title":"Jordan Types of Triangular Matrices over a Finite Field","authors":"Dmitry Fuchs,&nbsp;Alexandre Kirillov Sr.","doi":"10.1007/s40598-022-00214-1","DOIUrl":"10.1007/s40598-022-00214-1","url":null,"abstract":"<div><p>Let <span>(lambda )</span> be a partition of an integer <i>n</i> and <span>({mathbb F}_q)</span> be a finite field of order <i>q</i>. Let <span>(P_lambda (q))</span> be the number of strictly upper triangular <span>(ntimes n)</span> matrices of the Jordan type <span>(lambda )</span>. It is known that the polynomial <span>(P_lambda )</span> has a tendency to be divisible by high powers of <i>q</i> and <span>(Q=q-1)</span>, and we put <span>(P_lambda (q)=q^{d(lambda )}Q^{e(lambda )}R_lambda (q))</span>, where <span>(R_lambda (0)ne 0)</span> and <span>(R_lambda (1)ne 0)</span>. In this article, we study the polynomials <span>(P_lambda (q))</span> and <span>(R_lambda (q))</span>. Our main results: an explicit formula for <span>(d(lambda ))</span> (an explicit formula for <span>(e(lambda ))</span> is known, see Proposition 3.3 below), a recursive formula for <span>(R_lambda (q))</span> (a similar formula for <span>(P_lambda (q))</span> is known, see Proposition 3.1 below), the stabilization of <span>(R_lambda )</span> with respect to extending <span>(lambda )</span> by adding strings of 1’s, and an explicit formula for the limit series <span>(R_{lambda 1^infty })</span>. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00214-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41782923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
期刊
Arnold Mathematical Journal
全部 Acc. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1