Examples and Counterexamples最新文献

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On the effect of different samplings to the solution of parametric PDE eigenvalue problems

Examples and Counterexamples

Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100170

Daniele Boffi , Abdul Halim , Gopal Priyadarshi

The use of sparse sampling is a consolidated technique for the reduced order modeling of parametric PDEs. In this note we investigate the choice of sampling points within the framework of reduced order techniques for the approximation of eigenvalue problems originating from parametric PDEs. We use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerkin orthogonal technique to get the reduced problem. We present some numerical results and observe that, as in the case of the source problem, also for eigenvalue problems the use of sparse sampling is a good idea and that, when the number of sampling points is assigned, sparse sampling provides better results than uniform sampling.

In the spirit of the journal, we present our results in the form of examples and counterexamples.

引用次数: 0

An example for application of Lax–Milgram’s theorem and Riesz–Schauder’s theorem

Examples and Counterexamples

Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100169

Tujin Kim

In this note reviewing Lax–Migram’s theorem, we show an example of its application to prove the existence of a solution to an equation in complex Hilbert space arising in the field of electromagnetic heating.

引用次数: 0

Corrigendum to “Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis” [Examples and Counterexamples 4 (2023) 100128]

Examples and Counterexamples

Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100149

Subrata Paul , Animesh Mahata , Supriya Mukherjee , Prakash Chandra Mali , Banamali Roy

引用次数: 0

New Andrews–Curtis trivializations for Miller–Schupp group presentations 米勒-舒普群列报的新安德鲁斯-柯蒂斯三段论

Examples and Counterexamples

Pub Date : 2024-11-06 DOI: 10.1016/j.exco.2024.100168

Alexei Lisitsa

We present recent developments in the applications of automated theorem proving in the investigation of the Andrews–Curtis conjecture. We demonstrate previously unknown trivializations of group presentations from a parametric family

M S_{n} (w_{*})

of trivial group presentations for

n = 3, 4, 5, 6, 7, 8

(subset of well-known Miller–Schupp family). Based on the human analysis of these trivializations we formulate two conjectures on the structure of simplifications for the infinite family

M S_{n} (w_{*})

n \geq 3

我们介绍了在研究安德鲁斯-柯蒂斯猜想过程中应用自动定理证明的最新进展。我们从 n=3,4,5,6,7,8 的琐碎群呈现的参数族 MSn(w∗)（众所周知的米勒-舒普族的子集）中展示了之前未知的群呈现的琐碎化。基于对这些琐碎化的人为分析，我们提出了两个关于无穷族 MSn(w∗)（n≥3）简化结构的猜想。

引用次数: 0

Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey–Dodd–Gibbon equation 时间分数考德里-多德-吉本方程的对称分析、精确解和守恒定律

Examples and Counterexamples

Pub Date : 2024-10-28 DOI: 10.1016/j.exco.2024.100166

Jicheng Yu , Yuqiang Feng

In this paper, Lie symmetry analysis method is applied to time fractional Caudrey–Dodd–Gibbon equation. We obtain a symmetric group spanned by two generators for the governing equation. The obtained group generators are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative, thereby getting one trivial solution and one convergent power series solution for the reduced equations. Then we present the dynamic behavior of the obtained analytical solutions graphically. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

本文将李对称分析方法应用于时间分数 Caudrey-Dodd-Gibbon 方程。我们为支配方程获得了一个由两个生成器跨越的对称群。利用所得到的群生成器，将所研究的分式偏微分方程还原为一些具有黎曼-刘维尔分式导数或埃尔德利-科贝尔分式导数的分式常微分方程，从而得到还原方程的一个微分解和一个收敛幂级数解。然后，我们用图形展示了所获解析解的动态行为。此外，我们还提出了新的守恒定理和诺特算子广义，以构建所研究方程的守恒定律。

引用次数: 0

On the influence of non-linearity within two-phase poro-elasticity: Numerical examples and counterexamples 关于两相孔弹性中非线性的影响：数值示例和反例

Examples and Counterexamples

Pub Date : 2024-10-28 DOI: 10.1016/j.exco.2024.100167

Maximilian Brodbeck , Franziska S. Egli , Marlon Suditsch , Seyed Morteza Seyedpour , Tim Ricken

Porous materials can be described either by Biot’s consolidation theory or the Theory of Porous Media (TPM). Depending on the loading regime, permeability or compressibility of the solid matrix, either small or finite deformations occur. Numerical solution procedures for the case of finite deformation are prone to instabilities and computationally costly. Simplified models assuming small deformations increase stability of the solution process. Within this work, limitations of two simplified models in comparison with the fully non-linear TPM are studied. Therefore a Mandel-like problem is considered. Differences arise especially for rapid consolidation processes and for elongations larger than 3%. It can be further shown, that the simplified models have an inherent mass error.

多孔材料可以用毕奥固结理论或多孔介质理论（TPM）来描述。根据加载制度、固体基质的渗透性或可压缩性，会发生小变形或有限变形。有限变形情况下的数值求解程序容易产生不稳定性，且计算成本高昂。假设小变形的简化模型可提高求解过程的稳定性。在这项工作中，研究了两种简化模型与完全非线性 TPM 相比的局限性。因此，我们考虑了类似曼德尔的问题。尤其是在快速固结过程和伸长率大于 3% 的情况下，两者之间会出现差异。可以进一步证明，简化模型存在固有的质量误差。

引用次数: 0

On the elastic instability of a simple truss structure subjected to a tensile dead load 关于简单桁架结构在拉伸自重作用下的弹性失稳问题

Examples and Counterexamples

Pub Date : 2024-10-23 DOI: 10.1016/j.exco.2024.100165

Alfredo Canelas, Ana I. Abreu

There are a significant number of examples of structures that fail due to elastic instability of elements subjected to compressive forces. There are also examples of elastic instability caused by tensile forces, but they are less well known. This paper presents an interesting example of the latter type, whose main feature is its counterintuitive post-critical behavior. In fact, the applied dead load does negative work in the movement from the critical to the post-critical equilibrium configuration. The example admits a complete analytical solution, which makes it ideal for teaching use and as a benchmark problem for testing computational codes.

由于受压元素弹性失稳而导致结构失效的例子不胜枚举。拉力导致弹性失稳的例子也有，但不太为人所知。本文介绍了后一种类型的一个有趣实例，其主要特点是临界后的反直觉行为。事实上，在从临界平衡构型向临界后平衡构型移动的过程中，施加的死载荷会产生负功。该示例提供了完整的分析解决方案，因此非常适合教学使用，也可作为测试计算代码的基准问题。

引用次数: 0

Small examples of mosaics of combinatorial designs 组合设计马赛克小实例

Examples and Counterexamples

Pub Date : 2024-10-16 DOI: 10.1016/j.exco.2024.100163

Vedran Krčadinac

We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-

(13, 3, 1)

, 2-

(13, 4, 2)

, and 2-

(13, 6, 5)

. Furthermore, we give examples of mosaics of 2-

(9, 3, 2)

designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order 3 that cannot be obtained by tiling groups with difference sets.

我们首次给出了三个组合设计的马赛克的例子，这三个组合设计的参数分别为 2-(13,3,1)、2-(13,4,2) 和 2-(13,6,5)。此外，我们还举例说明了不可解的 2-(9,3,2) 设计的马赛克，从而回答了 M. Wiese 和 H. Boche 提出的问题。最后，我们给出了一个阶数为 3 的投影面马赛克的例子，该马赛克无法通过差集平铺组获得。

引用次数: 0

Whether the singularities are removable in some metrics 奇点在某些度量中是否可移动

Examples and Counterexamples

Pub Date : 2024-10-15 DOI: 10.1016/j.exco.2024.100164

Ting-Han Pei

For a long time, the singularities in certain metrics have been problematic and must be removed to avoid unreasonable results in spacetime. Therefore, some new metrics were proposed to eliminate these singularities through coordinate transformations, but they seem not to be workable. In this paper, we re-examine the mathematical structures of the Schwarzschild metric, Reissner-Nordström metric, and Kerr metric. We find that after some transformations, the timelike Eddington-Finkelstein coordinate and the Kruskal-Szekeres coordinates do not delete the singularity problem in the original Schwarzschild metric. It is also true for the tortoise coordinates that it does not solve the singularities at two event horizons in the Kerr metric. After some discussions on those coordinate transformations, a counterexample is given where the singularities are not eliminated.

长期以来，某些度量中的奇点一直是个问题，必须消除这些奇点才能避免在时空中出现不合理的结果。因此，人们提出了一些新的度量，通过坐标变换来消除这些奇点，但似乎并不可行。在本文中，我们重新研究了施瓦兹柴尔德度量、赖斯纳-诺德斯特伦度量和克尔度量的数学结构。我们发现，经过一些变换后，时间性的爱丁顿-芬克尔斯坦坐标和克鲁斯卡尔-塞克斯克坐标并不能消除原来的施瓦兹柴尔德公度量中的奇点问题。同样，龟坐标也不能解决克尔公度量中两个事件视界处的奇点问题。在对这些坐标变换进行了一些讨论之后，给出了一个没有消除奇点的反例。

引用次数: 0

Counting graphs induced by Gauss diagrams and families of mutant alternating knots 高斯图诱导的计数图和突变交替结族

Examples and Counterexamples

Pub Date : 2024-10-11 DOI: 10.1016/j.exco.2024.100162

Alexei Lisitsa , Alexei Vernitski

The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these graphs of small sizes, and we published the number of these graphs as sequence A343358 in the OEIS. The aim of this article is to clarify several subtle theoretical points concerning A343358. Most importantly, we explain why our numbers, produced using graph-theoretical constructions, reflect the number of so-called mutant knots.

被称为高斯图或高斯词的构造是结理论中最古老的构造之一，在结和具有自交的封闭曲线中都得到了广泛的研究。当我们研究高斯图所诱导的图形时，我们产生了这些图形的所有小尺寸示例，并在 OEIS 中以序列 A343358 公布了这些图形的编号。本文旨在澄清有关 A343358 的几个微妙的理论观点。最重要的是，我们解释了为什么我们利用图论构造得出的数字反映了所谓突变结的数量。

引用次数: 0

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