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Asymptotic behavior of the empirical checkerboard copula process for binary data: An educational presentation 二进制数据的经验棋盘联结过程的渐近行为：一个教育演示
Examples and Counterexamples Pub Date : 2025-01-31 DOI: 10.1016/j.exco.2025.100176
Christian Genest, Johanna G. Nešlehová
{"title":"Asymptotic behavior of the empirical checkerboard copula process for binary data: An educational presentation","authors":"Christian Genest,&nbsp;Johanna G. Nešlehová","doi":"10.1016/j.exco.2025.100176","DOIUrl":"10.1016/j.exco.2025.100176","url":null,"abstract":"<div><div>The empirical multilinear or checkerboard copula process is a promising tool for statistical inference in copula models for data with ties (Genest et al., 2019a). The large-sample behavior of this process was determined in Genest et al. (2014, 2017) under very broad conditions. The purpose of this note is to provide a detailed description of this asymptotic result and to derive an expression for the limit of the process in the simplest possible case in which the data form a random sample of pairs of Bernoulli random variables. Although one would never actually fit a copula model to a 2 × 2 contingency table, this case is particularly well suited for explicit calculations and didactic explanations of the intricacies of the limiting behavior of this process and make it clear why the conditions in Genest et al. (2014, 2017) are needed and cannot be simplified.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100176"},"PeriodicalIF":0.0,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143240690","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
The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses PDE反问题的Hadamard-PINN：具有远初始猜测的收敛性
Examples and Counterexamples Pub Date : 2025-01-28 DOI: 10.1016/j.exco.2025.100175
Yohan Chandrasukmana, Helena Margaretha, Kie Van Ivanky Saputra
{"title":"The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses","authors":"Yohan Chandrasukmana,&nbsp;Helena Margaretha,&nbsp;Kie Van Ivanky Saputra","doi":"10.1016/j.exco.2025.100175","DOIUrl":"10.1016/j.exco.2025.100175","url":null,"abstract":"<div><div>This paper presents the Hadamard-Physics-Informed Neural Network (H-PINN) for solving inverse problems in partial differential equations (PDEs), specifically the heat equation and the Korteweg–de Vries (KdV) equation. H-PINN addresses challenges in convergence and accuracy when initial parameter guesses are far from their actual values. The training process is divided into two phases: data fitting and parameter optimization. This phased approach is based on Hadamard’s conditions for well-posed problems, which emphasize that the uniqueness of a solution relies on the specified initial and boundary conditions. The model is trained using the Adam optimizer, along with a combined learning rate scheduler. To ensure reliability and consistency, we repeated each numerical experiment five times across three different initial guesses. Results showed significant improvements in parameter accuracy compared to the standard PINN, highlighting H-PINN’s effectiveness in scenarios with substantial deviations in initial guesses.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100175"},"PeriodicalIF":0.0,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163826","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 note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs 关于离散莫尔斯理论技术在非连通图复合体上应用的注解
Examples and Counterexamples Pub Date : 2025-01-15 DOI: 10.1016/j.exco.2025.100174
Anupam Mondal , Pritam Chandra Pramanik
{"title":"A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs","authors":"Anupam Mondal ,&nbsp;Pritam Chandra Pramanik","doi":"10.1016/j.exco.2025.100174","DOIUrl":"10.1016/j.exco.2025.100174","url":null,"abstract":"<div><div>Robin Forman’s highly influential 2002 paper <em>A User’s Guide to Discrete Morse Theory</em> presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract simplicial complex of disconnected graphs of order <span><math><mi>n</mi></math></span> (which was previously done by Victor Vassiliev using classical topological methods) using discrete Morse theoretic techniques, which are purely combinatorial in nature. The techniques involve the construction (and verification) of a discrete gradient vector field on the complex. However, the verification part relies on a claim that does not seem to hold. In this note, we provide a couple of counterexamples against this specific claim. We also provide an alternative proof of the bigger claim that the constructed discrete vector field is indeed a gradient vector field. Our proof technique relies on a key observation which is not specific to the problem at hand, and thus is applicable while verifying a constructed discrete vector field is a gradient one in general.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100174"},"PeriodicalIF":0.0,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163827","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
Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods 由有限差分方法导出的晶格Boltzmann格式的非唯一性
Examples and Counterexamples Pub Date : 2024-12-18 DOI: 10.1016/j.exco.2024.100171
Eliane Kummer, Stephan Simonis
{"title":"Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods","authors":"Eliane Kummer,&nbsp;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}
引用次数: 0
On the effect of different samplings to the solution of parametric PDE eigenvalue problems 不同采样对参数偏微分方程特征值问题解的影响
Examples and Counterexamples Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100170
Daniele Boffi , Abdul Halim , Gopal Priyadarshi
{"title":"On the effect of different samplings to the solution of parametric PDE eigenvalue problems","authors":"Daniele Boffi ,&nbsp;Abdul Halim ,&nbsp;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}
引用次数: 0
An example for application of Lax–Milgram’s theorem and Riesz–Schauder’s theorem 一个应用拉克斯-米尔格拉姆定理和里兹-肖德定理的例子
Examples and Counterexamples Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100169
Tujin Kim
{"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}
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] “具有多时滞的分数阶SEIR流行病模型的动力学行为及其稳定性分析”的勘误表[例与反例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
{"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 ,&nbsp;Animesh Mahata ,&nbsp;Supriya Mukherjee ,&nbsp;Prakash Chandra Mali ,&nbsp;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}
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
{"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}
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
{"title":"Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey–Dodd–Gibbon equation","authors":"Jicheng Yu ,&nbsp;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}
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
{"title":"On the influence of non-linearity within two-phase poro-elasticity: Numerical examples and counterexamples","authors":"Maximilian Brodbeck ,&nbsp;Franziska S. Egli ,&nbsp;Marlon Suditsch ,&nbsp;Seyed Morteza Seyedpour ,&nbsp;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}
引用次数: 0
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