Pub Date : 2024-12-24DOI: 10.1016/j.exco.2024.100173
Vilmar Steffen , Carlos Catusso Della Pasqua , Maiquiel Schmidt de Oliveira , Edson Antonio da Silva
Hybrid methods are widely used in many areas of applied mathematics. One of the simplest and most common problems in this field is root finding, for which various methods exist. Some of the most efficient approaches combine two or more techniques into hybrid methods. Among these are the Dekker and Brent methods, for which we propose a modification to ensure that the search interval is halved in each iteration. We apply this modification to two examples: a transcendental equation and a cubic equation of state. The results demonstrate that the proposed modifications guarantee at least interval halving and offer a slight improvement in the efficiency of the root-finding process.
{"title":"Halving interval guaranteed for Dekker and Brent root finding methods","authors":"Vilmar Steffen , Carlos Catusso Della Pasqua , Maiquiel Schmidt de Oliveira , Edson Antonio da Silva","doi":"10.1016/j.exco.2024.100173","DOIUrl":"10.1016/j.exco.2024.100173","url":null,"abstract":"<div><div>Hybrid methods are widely used in many areas of applied mathematics. One of the simplest and most common problems in this field is root finding, for which various methods exist. Some of the most efficient approaches combine two or more techniques into hybrid methods. Among these are the Dekker and Brent methods, for which we propose a modification to ensure that the search interval is halved in each iteration. We apply this modification to two examples: a transcendental equation and a cubic equation of state. The results demonstrate that the proposed modifications guarantee at least interval halving and offer a slight improvement in the efficiency of the root-finding process.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100173"},"PeriodicalIF":0.0,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291592","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}
Pub Date : 2024-12-18DOI: 10.1016/j.exco.2024.100171
Eliane Kummer, Stephan Simonis
Recently, the construction of finite difference schemes from lattice Boltzmann schemes has been rigorously analyzed [Bellotti et al. (2022), Numer. Math. 152, pp. 1–40]. It is thus known that any lattice Boltzmann scheme can be expressed in terms of a corresponding multi-step finite difference scheme on the conserved variables. In the present work, we provide counterexamples for the conjecture that any multi-step finite difference scheme has a unique lattice Boltzmann formulation. Based on that, we indicate the existence of equivalence classes for discretized relaxation systems.
最近，格子Boltzmann格式的有限差分格式的构造得到了严格的分析[Bellotti et al. (2022), number。数学。152，第1-40页]。由此可知，任何晶格玻尔兹曼格式都可以用守恒变量上相应的多步有限差分格式来表示。在本工作中，我们提供了关于任何多步有限差分格式具有唯一晶格玻尔兹曼公式的猜想的反例。在此基础上，给出了离散松弛系统的等价类的存在性。
{"title":"Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods","authors":"Eliane Kummer, Stephan Simonis","doi":"10.1016/j.exco.2024.100171","DOIUrl":"10.1016/j.exco.2024.100171","url":null,"abstract":"<div><div>Recently, the construction of finite difference schemes from lattice Boltzmann schemes has been rigorously analyzed [Bellotti et al. (2022), Numer. Math. 152, pp. 1–40]. It is thus known that any lattice Boltzmann scheme can be expressed in terms of a corresponding multi-step finite difference scheme on the conserved variables. In the present work, we provide counterexamples for the conjecture that any multi-step finite difference scheme has a unique lattice Boltzmann formulation. Based on that, we indicate the existence of equivalence classes for discretized relaxation systems.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100171"},"PeriodicalIF":0.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163828","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}
Pub Date : 2024-12-01DOI: 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.
{"title":"On the effect of different samplings to the solution of parametric PDE eigenvalue problems","authors":"Daniele Boffi , Abdul Halim , Gopal Priyadarshi","doi":"10.1016/j.exco.2024.100170","DOIUrl":"10.1016/j.exco.2024.100170","url":null,"abstract":"<div><div>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.</div><div>In the spirit of the journal, we present our results in the form of examples and counterexamples.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100170"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160098","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}
Pub Date : 2024-12-01DOI: 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.
{"title":"An example for application of Lax–Milgram’s theorem and Riesz–Schauder’s theorem","authors":"Tujin Kim","doi":"10.1016/j.exco.2024.100169","DOIUrl":"10.1016/j.exco.2024.100169","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100169"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160099","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}
Pub Date : 2024-12-01DOI: 10.1016/j.exco.2024.100149
Subrata Paul , Animesh Mahata , Supriya Mukherjee , Prakash Chandra Mali , Banamali Roy
{"title":"Corrigendum to “Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis” [Examples and Counterexamples 4 (2023) 100128]","authors":"Subrata Paul , Animesh Mahata , Supriya Mukherjee , Prakash Chandra Mali , Banamali Roy","doi":"10.1016/j.exco.2024.100149","DOIUrl":"10.1016/j.exco.2024.100149","url":null,"abstract":"","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100149"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127956","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}
Pub Date : 2024-11-06DOI: 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 of trivial group presentations for (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 , .
{"title":"New Andrews–Curtis trivializations for Miller–Schupp group presentations","authors":"Alexei Lisitsa","doi":"10.1016/j.exco.2024.100168","DOIUrl":"10.1016/j.exco.2024.100168","url":null,"abstract":"<div><div>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 <span><math><mrow><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>)</mo></mrow></mrow></math></span> of trivial group presentations for <span><math><mrow><mi>n</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn></mrow></math></span> (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 <span><math><mrow><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100168"},"PeriodicalIF":0.0,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662768","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}
Pub Date : 2024-10-28DOI: 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.
{"title":"Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey–Dodd–Gibbon equation","authors":"Jicheng Yu , Yuqiang Feng","doi":"10.1016/j.exco.2024.100166","DOIUrl":"10.1016/j.exco.2024.100166","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100166"},"PeriodicalIF":0.0,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554185","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}
Pub Date : 2024-10-28DOI: 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.
{"title":"On the influence of non-linearity within two-phase poro-elasticity: Numerical examples and counterexamples","authors":"Maximilian Brodbeck , Franziska S. Egli , Marlon Suditsch , Seyed Morteza Seyedpour , Tim Ricken","doi":"10.1016/j.exco.2024.100167","DOIUrl":"10.1016/j.exco.2024.100167","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100167"},"PeriodicalIF":0.0,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571811","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}
Pub Date : 2024-10-23DOI: 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.
{"title":"On the elastic instability of a simple truss structure subjected to a tensile dead load","authors":"Alfredo Canelas, Ana I. Abreu","doi":"10.1016/j.exco.2024.100165","DOIUrl":"10.1016/j.exco.2024.100165","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100165"},"PeriodicalIF":0.0,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528944","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}
Pub Date : 2024-10-16DOI: 10.1016/j.exco.2024.100163
Vedran Krčadinac
We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-, 2-, and 2-. Furthermore, we give examples of mosaics of 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。
{"title":"Small examples of mosaics of combinatorial designs","authors":"Vedran Krčadinac","doi":"10.1016/j.exco.2024.100163","DOIUrl":"10.1016/j.exco.2024.100163","url":null,"abstract":"<div><div>We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>, and 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></math></span>. Furthermore, we give examples of mosaics of 2-<span><math><mrow><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span> 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.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100163"},"PeriodicalIF":0.0,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528855","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}
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