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Remarks on Joachimsthal Integral and Poritsky Property 关于Joachimshal积分和Poritsky性质的注记
Q3 Mathematics Pub Date : 2021-06-01 DOI: 10.1007/s40598-021-00180-0
Maxim Arnold, Serge Tabachnikov

The billiard in an ellipse has a conserved quantity, the Joachimsthal integral. We show that the existence of such an integral characterizes conics. We extend this result to the spherical and hyperbolic geometries and to higher dimensions. We connect the existence of Joachimsthal integral with the Poritsky property, a property of billiard curves, called so after H. Poritsky whose important paper Poritsky (Ann Math 51:446–470, 1950) was one of the early studies of the billiard problem.

椭圆中的台球有一个守恒量,Joachimshal积分。我们证明了这种积分的存在性是二次曲线的特征。我们将这一结果推广到球面和双曲几何以及更高维度。我们将Joachimshal积分的存在性与台球曲线的性质Poritsky联系起来,这是以H.Poritsky的重要论文Poritsky(Ann Math 51:446–4701950)命名的,Poritsky是台球问题的早期研究之一。
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引用次数: 2
Simplicity of Spectra for Bethe Subalgebras in $${mathrm {Y}}({mathfrak {gl}}_2)$$ Y ( gl 2 $${mathrm{Y}}({mathfrak{gl}}_2)$$Y(gl2)中Bethe子代数的谱的简单性
Q3 Mathematics Pub Date : 2021-06-01 DOI: 10.1007/s40598-020-00171-7
I. Mashanova-Golikova
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引用次数: 3
Inflation of Poorly Conditioned Zeros of Systems of Analytic Functions 解析函数系统的弱条件零的膨胀
Q3 Mathematics Pub Date : 2021-05-27 DOI: 10.1007/s40598-021-00177-9
Michael Burr, Anton Leykin

Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.

给定一个解析函数系统和一个近似零,我们引入通货膨胀将这个系统转化为一个具有正则二次零的系统。这导致了一种用于隔离给定系统的零簇的方法。
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引用次数: 3
The Yomdin–Gromov Algebraic Lemma Revisited 对Yomdin–Gromov代数引理的再认识
Q3 Mathematics Pub Date : 2021-05-03 DOI: 10.1007/s40598-021-00176-w
Gal Binyamini, Dmitry Novikov

In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for (C^infty ) maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.

1987年,Yomdin证明了一个关于半代数集光滑参数化的引理,作为他对(C^infty)映射的Shub熵猜想解的一部分。格罗莫夫进一步完善了这一说法,产生了现在所知的约姆丁-格罗莫夫代数引理。文献中出现了一些基于格罗莫夫素描的完整证明,但由于一些技术问题,这些证明比格罗莫夫最初的陈述要复杂得多。在这篇注释中,我们给出了一个紧跟格罗莫夫最初陈述的证明。我们证明了一个更强的陈述,其中参数化映射保证是细胞的。事实证明,这种额外的限制,以及o-极小结构中可微函数的一些初等引理,使得归纳可以在没有技术困难的情况下进行。
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引用次数: 4
On the Density of Dispersing Billiard Systems with Singular Periodic Orbits 具有奇异周期轨道的分散Billiard系统的密度
Q3 Mathematics Pub Date : 2021-04-21 DOI: 10.1007/s40598-020-00173-5
Otto Vaughn Osterman

Dynamical billiards, or the behavior of a particle traveling in a planar region D undergoing elastic collisions with the boundary has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of particular interest are the dispersing billiards, where D consists of a union of finitely many open convex regions. These billiard flows are known to be ergodic and to possess the K-property. However, Turaev and Rom-Kedar proved that for dispersing systems permitting singular periodic orbits, there exists a family of smooth Hamiltonian flows with regions of stability near such orbits, converging to the billiard flow. They conjecture that systems possessing such singular periodic orbits are dense in the space of all dispersing billiard systems and remark that if this conjecture is true then every dispersing billiard system is arbitrarily close to a non-ergodic smooth Hamiltonian flow with regions of stability [6]. We present a partial solution to this conjecture by showing that any system with a near-singular periodic orbit satisfying certain conditions can be perturbed to a system that permits a singular periodic orbit. We comment on the assumptions of our theorem that must be removed to prove the conjecture of Turaev and Rom-Kedar.

动力学台球,或在平面区域D中运动的粒子与边界发生弹性碰撞的行为,已经被广泛研究,并被用于模拟许多物理现象,如玻尔兹曼气体。特别令人感兴趣的是分散台球,其中D由有限多个开凸区域的并集组成。已知这些台球流是遍历的,并且具有K性质。然而,Turaev和Rom-Kedar证明,对于允许奇异周期轨道的分散系统,存在一组光滑的哈密顿流,其稳定区域靠近这些轨道,收敛到台球流。他们推测,具有这种奇异周期轨道的系统在所有分散台球系统的空间中是稠密的,并指出,如果这个猜想成立,那么每个分散台球系统都任意接近于具有稳定区域的非遍历光滑哈密顿流[6]。我们通过证明任何具有满足某些条件的近奇异周期轨道的系统都可以被摄动到允许奇异周期轨道存在的系统,给出了这一猜想的部分解。我们评论了我们的定理的假设,必须删除这些假设才能证明图拉耶夫和罗姆·凯达尔的猜想。
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引用次数: 0
Interpolation of Weighted Extremal Functions 加权极值函数的插值
Q3 Mathematics Pub Date : 2021-03-16 DOI: 10.1007/s40598-021-00175-x
Alexander Rashkovskii

An approach to interpolation of compact subsets of ({{mathbb {C}}}^n), including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.

在[8]中,通过给定集合的相对极值函数之间的亚调和测地线,提出了一种对({mathbb{C}})的紧子集进行插值的方法,包括插值集容量的Brunn–Minkowski型不等式。在这里,我们证明了通过加权相对极值函数之间的测地线可以实现更好的控制。特别地,我们建立了比Brunn–Minkowski不等式给出的容量更强的容量的凸性性质。
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引用次数: 1
Fifty New Invariants of N-Periodics in the Elliptic Billiard 椭圆台球n周期的50个新不变量
Q3 Mathematics Pub Date : 2021-02-18 DOI: 10.1007/s40598-021-00174-y
Dan Reznik, Ronaldo Garcia, Jair Koiller

We introduce 50+ new invariants manifested by the dynamic geometry of N-periodics in the Elliptic Billiard, detected with an experimental/interactive toolbox. These involve sums, products and ratios of distances, areas, angles, etc. Though curious in their manifestation, said invariants do all depend upon the two fundamental conserved quantities in the Elliptic Billiard: perimeter and Joachimsthal’s constant. Several proofs have already been contributed (references are provided); these have mainly relied on algebraic geometry. We very much welcome new proofs and contributions.

我们引入了50多个新的不变量,这些不变量由椭圆台球中N周期的动态几何所表现,并用实验/交互式工具箱进行了检测。这些不变量涉及距离、面积、角度等的和、乘积和比率。尽管它们的表现形式很奇怪,但所说的不变量都取决于椭圆台球中的两个基本守恒量:周长和约阿希姆塔尔常数。已经提供了一些证据(提供了参考资料);这些主要依赖于代数几何。我们非常欢迎新的证明和贡献。
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引用次数: 27
Near Parabolic Renormalization for Unicritical Holomorphic Maps 单临界全纯映射的近抛物重整化
Q3 Mathematics Pub Date : 2021-02-10 DOI: 10.1007/s40598-020-00172-6
Arnaud Chéritat

Inou and Shishikura provided a class of maps that is invariant by near-parabolic renormalization, and that has proved extremely useful in the study of the dynamics of quadratic polynomials. We provide here another construction, using more general arguments. This will allow to extend the range of applications to unicritical polynomials of all degrees.

Inou和Shishikura提供了一类通过近抛物重整化不变的映射,该映射在研究二次多项式的动力学中被证明是非常有用的。我们在这里提供了另一种构造,使用了更一般的论点。这将允许将应用范围扩展到所有阶的单临界多项式。
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引用次数: 5
Billiard Trajectories in Regular Polygons and Geodesics on Regular Polyhedra 规则多边形上的台球轨迹与规则多面体上的大地测量学
Q3 Mathematics Pub Date : 2021-01-07 DOI: 10.1007/s40598-020-00170-8
Dmitry Fuchs

This article is devoted to the geometry of billiard trajectories in a regular polygon and geodesics on the surface of a regular polyhedron. Main results are formulated as conjectures based on ample computer experimentation.

本文研究了正多边形台球轨迹的几何和正多面体表面的测地线。主要结果被表述为基于大量计算机实验的推测。
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引用次数: 0
Accessible Boundary Points in the Shift Locus of a Family of Meromorphic Functions with Two Finite Asymptotic Values 具有两个有限渐近值的亚纯函数族位移轨迹上的可达边界点
Q3 Mathematics Pub Date : 2021-01-07 DOI: 10.1007/s40598-020-00169-1
Tao Chen, Yunping Jiang, L. Keen
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引用次数: 0
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