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Catastrophe in Elastic Tensegrity Frameworks 弹性张量框架中的突变
Q3 Mathematics Pub Date : 2022-02-18 DOI: 10.1007/s40598-021-00193-9
Alexander Heaton, Sascha Timme

We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.

我们讨论了由刚性杆和弹性索制成的弹性张拉整体框架,这取决于许多参数。对于任何固定的参数值,通过最小化受代数约束的能量函数来确定框架的稳定平衡位置。随着参数的平稳变化,稳定的平衡可能会消失。这种平衡的损失被称为灾难,因为尽管参数变化很小,框架仍会经历大规模的形状变化。使用非线性代数,我们刻画了参数空间的半代数子集,突变集,它检测来自约束优化问题的参数化族的局部极值的合并,从而检测可能的突变。数值非线性代数的工具允许可靠和有效地计算所有稳定的平衡位置以及突变集本身。
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引用次数: 3
The ({{mathbb {F}}}_p)-Selberg Integral ({mathbb{F}})_p)-Selberg积分
Q3 Mathematics Pub Date : 2022-01-24 DOI: 10.1007/s40598-021-00191-x
Richárd Rimányi, Alexander Varchenko

We prove an ({{mathbb {F}}}_p)-Selberg integral formula, in which the ({{mathbb {F}}}_p)-Selberg integral is an element of the finite field ({{mathbb {F}}}_p) with odd prime number p of elements. The formula is motivated by the analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo p.

我们证明了一个({mathbb{F}}}_p)-Selberg积分公式,其中({{math bb{F}}}_p)-Selberg积分是元素素数为p的有限域({mathbb{F}}}_p)的一个元素。该公式是由KZ方程的多维超几何解和同一方程的多项式解模p的相似性推动的。
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引用次数: 0
The Equivalence Problem in Analytic Dynamics for 1-Resonance 1-共振解析动力学中的等价问题
Q3 Mathematics Pub Date : 2022-01-20 DOI: 10.1007/s40598-021-00196-6
Christiane Rousseau

When are two germs of analytic systems conjugate or orbitally equivalent under an analytic change of coordinates in a neighborhood of a singular point? The present paper, of a survey nature, presents a research program around this question. A way to answer is to use normal forms. However, there are large classes of dynamical systems for which the change of coordinates to a normal form diverges. In this paper, we discuss the case of singularities for which the normalizing transformation is k-summable, thus allowing to provide moduli spaces. We explain the common geometric features of these singularities, and show that the study of their unfoldings allows understanding both the singularities themselves, and the geometric obstructions to convergence of the normalizing transformations. We also present some moduli spaces for generic k-parameter families unfolding such singularities.

在奇异点邻域中坐标的解析变化下,解析系统的两个芽何时共轭或轨道等价?本文具有调查性质,提出了一个围绕这个问题的研究方案。答案的一种方法是使用正规形式。然而,有很大一类动力系统的坐标向正规形式的变化是发散的。在本文中,我们讨论了奇异性的情况,对于该奇异性,归一化变换是k和的,从而允许提供模空间。我们解释了这些奇点的共同几何特征,并表明研究它们的展开可以理解奇点本身,以及归一化变换收敛的几何障碍。我们还给出了展开此类奇点的一般k参数族的一些模空间。
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引用次数: 0
Open Problems on Billiards and Geometric Optics 台球与几何光学的开放问题
Q3 Mathematics Pub Date : 2022-01-17 DOI: 10.1007/s40598-022-00198-y
Misha Bialy, Corentin Fierobe, Alexey Glutsyuk, Mark Levi, Alexander Plakhov, Serge Tabachnikov

This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021.

这是由2021年10月4日至8日在CIRM举行的“微分几何、台球和几何光学”研讨会的一些参与者撰写的问题集。
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引用次数: 0
Partial Duality of Hypermaps 超映射的部分对偶性
Q3 Mathematics Pub Date : 2022-01-03 DOI: 10.1007/s40598-021-00194-8
S. Chmutov, F. Vignes-Tourneret

We introduce partial duality of hypermaps, which include the classical Euler–Poincaré duality as a particular case. Combinatorially, hypermaps may be described in one of three ways: as three involutions on the set of flags (bi-rotation system or (tau )-model), or as three permutations on the set of half-edges (rotation system or (sigma )-model in orientable case), or as edge 3-coloured graphs. We express partial duality in each of these models. We give a formula for the genus change under partial duality.

我们引入了超映射的偏对偶,其中包括作为特例的经典欧拉-庞加莱对偶。组合起来,超映射可以用三种方式之一来描述:作为标志集上的三个对合(双旋转系统或(tau)-模型),或者作为半边集上的两个排列(旋转系统或可定向情况下的( sigma)-模型),或者边缘3-色图。我们在每一个模型中都表达了部分对偶性。给出了偏对偶条件下亏格变化的一个公式。
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引用次数: 3
A Symplectic Dynamics Proof of the Degree–Genus Formula 度-亏格公式的辛动力学证明
Q3 Mathematics Pub Date : 2021-12-20 DOI: 10.1007/s40598-021-00195-7
Peter Albers, Hansjörg Geiges, Kai Zehmisch

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

我们对三球面上标准接触形式的Reeb流的截面的全局表面进行了分类(定义了Hopf纤维化),边界由流正向定向。作为一个应用,我们利用辛场论中全纯建筑物语言启发的初等退化过程,证明了复射影曲线的度亏格公式。
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引用次数: 5
Equivalence of Neighborhoods of Embedded Compact Complex Manifolds and Higher Codimension Foliations 嵌入紧复流形的邻域等价与高余维叶
Q3 Mathematics Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00192-w
Xianghong Gong, Laurent Stolovitch

We consider an embedded n-dimensional compact complex manifold in (n+d) dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert’s formal principle program. We will give conditions ensuring that a neighborhood of (C_n) in (M_{n+d}) is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold’s result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in (M_{n+d}) having (C_n) as a compact leaf, extending Ueda’s theory to the high codimension case. Both problems appear as a kind of linearization problems involving small divisors condition arising from solutions to their cohomological equations.

我们在(n+d)维复流形中考虑一个嵌入的n维紧致复流形。作为Grauert形式原理程序的一部分,我们对邻域的全纯分类感兴趣。我们将给出确保(M_{n+d})中(C_n)的邻域是其正规丛的零部分的邻域的双全纯的条件。这扩展了Arnold关于曲面中复杂环面邻域的结果。我们还证明了(C_n)为紧叶的(M_{n+d})中全纯叶理的存在性,将Ueda的理论推广到高余维情形。这两个问题都表现为一类线性化问题,涉及由其上同调方程的解引起的小因子条件。
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引用次数: 4
On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field 完全非阿基米德域上Schneider的连分数映射
Q3 Mathematics Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00190-y
A. Haddley, R. Nair
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引用次数: 0
On Schneider’s Continued Fraction Map on a Complete Non-Archimedean Field 关于完全非阿基米德域上Schneider的连分式映射
Q3 Mathematics Pub Date : 2021-10-15 DOI: 10.1007/s40598-021-00190-y
A. Haddley, R. Nair

Let ({mathcal {M}}) denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote (k^{times }), and a uniformizer we denote (pi ). In this paper, we consider the map (T_{v}: {mathcal {M}} rightarrow {mathcal {M}}) defined by

$$begin{aligned} T_v(x) = frac{pi ^{v(x)}}{x} - b(x), end{aligned}$$

where b(x) denotes the equivalence class to which (frac{pi ^{v(x)}}{x}) belongs in (k^{times }). We show that (T_v) preserves Haar measure (mu ) on the compact abelian topological group ({mathcal {M}}). Let ({mathcal {B}}) denote the Haar (sigma )-algebra on ({mathcal {M}}). We show the natural extension of the dynamical system (({mathcal {M}}, {mathcal {B}}, mu , T_v)) is Bernoulli and has entropy (frac{#( k)}{#( k^{times })}log (#( k))). The first of these two properties is used to study the average behaviour of the convergents arising from (T_v). Here for a finite set A its cardinality has been denoted by (# (A)). In the case (K = {mathbb {Q}}_p), i.e. the field of p-adic numbers, the map (T_v) reduces to the well-studied continued fraction map due to Schneider.

设({mathcal{M}})表示具有余类域K的非阿基米德域K的整数环的最大理想,我们表示其可逆元素(K^{times}),并且我们表示一致化器(pi)。在本文中,我们考虑由$$beagin{aligned}T_v(x)=frac{pi^{v(x。我们证明了紧阿贝尔拓扑群({mathcal{M}})上的(T_v)保持Haar测度(mu)。设({mathcal{B}})表示({{math cal{M})上的Haar( sigma)-代数。我们证明了动力系统({mathcal{M}},{math cal{B},mu,T_v))的自然扩展是伯努利的,并且具有熵(frac{#(k)}{#(k^{times})}log(#(k)))。这两个性质中的第一个用于研究由(T_v)引起的收敛的平均行为。这里,对于有限集a,其基数用(#(a))表示。在情形(K={mathbb{Q}}_p),即p-adic数的域中,映射(T_v)由于Schneider而简化为研究得很好的连分式映射。
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引用次数: 2
Varieties in Cages: A Little Zoo of Algebraic Geometry 笼中的变种:代数几何的小动物园
Q3 Mathematics Pub Date : 2021-09-30 DOI: 10.1007/s40598-021-00189-5
Gabriel Katz

A (d^{{n}})-cage (mathsf K) is the union of n groups of hyperplanes in (mathbb P^n), each group containing d members. The hyperplanes from the distinct groups are in general position, thus producing (d^n) points where hyperplanes from all groups intersect. These points are called the nodes of (mathsf K). We study the combinatorics of nodes that impose independent conditions on the varieties (X subset mathbb P^n) containing them. We prove that if X, given by homogeneous polynomials of degrees (le d), contains the points from such a special set (mathsf A) of nodes, then it contains all the nodes of (mathsf K). Such a variety X is very special: in particular, X is a complete intersection.

一个(d^{{n}})-笼(mathsf K)是(math bb P^n )中n组超平面的并集,每组包含d个成员。来自不同群的超平面处于一般位置,从而产生来自所有群的超平相交的(d^n)点。这些点被称为(mathsf K)的节点。我们研究了对包含它们的变种(Xsubetmathbb P^n)施加独立条件的节点的组合学。我们证明了如果由次齐次多项式(le d)给出的X包含来自这样一个特殊节点集(mathsf a)的点,那么它包含(math fK)的所有节点。这样一个变种X是非常特殊的:特别是,X是一个完全的交集。
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Arnold Mathematical Journal
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