Examples and Counterexamples最新文献

{"title":"Weak and very weak solutions of the Laplace equation and the Stokes system with prescribed regularity","authors":"Thomas Apel ,&nbsp;Katharina Lorenz ,&nbsp;Serge Nicaise","doi":"10.1016/j.exco.2025.100198","DOIUrl":"10.1016/j.exco.2025.100198","url":null,"abstract":"<div><div>To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for elliptic boundary value problems.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100198"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145094828","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}

{"title":"A catch-all discrete distribution: Short description and main properties","authors":"Nuno Fachada ,&nbsp;João P. Matos-Carvalho ,&nbsp;Carlos M. Fernandes","doi":"10.1016/j.exco.2025.100196","DOIUrl":"10.1016/j.exco.2025.100196","url":null,"abstract":"<div><div>Consider a population of size <span><math><mi>N</mi></math></span> where each element has an independent probability <span><math><mi>p</mi></math></span> of being a success. By sampling this population without replacement, how many elements need to be drawn to find all successes? This paper describes this discrete distribution, derives its main properties, and validates the results through simulation.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100196"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144887018","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}

{"title":"Automated theorem proving reveals a lengthy Andrews–Curtis trivialization for a Miller-Schupp trivial group presentation","authors":"Alexei Lisitsa","doi":"10.1016/j.exco.2025.100201","DOIUrl":"10.1016/j.exco.2025.100201","url":null,"abstract":"<div><div>We present an independently discovered Andrews–Curtis trivialization of the balanced trivial group presentation <span><math><mrow><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mn>9</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo>〈</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>∣</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>b</mi></mrow><mrow><mn>9</mn></mrow></msup><mi>a</mi><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup><mo>,</mo><mspace></mspace><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>b</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>a</mi><mi>b</mi><msup><mrow><mi>a</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>〉</mo></mrow></mrow></math></span> obtained through automated theorem proving. At the time of discovery, the AC-status of this case was unsettled; it has since been resolved by other methods. The resulting simplification sequence, consisting of 8,634 elementary moves, remains the longest AC-simplification sequence found by any computational method to date (as of January 2025). The sequence was generated by the <span>Prover9</span> theorem prover, which required approximately 74 days and over 56 GB of memory to compute.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100201"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145218954","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}

{"title":"Cauchy’s stress theorem and its influence on the first elasticity tensor","authors":"Stefan Buchen","doi":"10.1016/j.exco.2025.100190","DOIUrl":"10.1016/j.exco.2025.100190","url":null,"abstract":"<div><div>The well-known Clausius–Planck inequality for solids is derived from the second law of thermodynamics and provides the important relation between stress and strain tensors according to the Coleman–Noll procedure. For the detailed derivation of this inequality, mass continuity equation, the equation of motion and the first law of thermodynamics in local form are required. The transfer of integral to local formulation can be calculated using Cauchy’s stress theorem and the divergence theorem. In the literature, however, there are two definitions for both, the Cauchy’s stress theorem and the divergence theorem for second-order tensors. The aim of this article is to show the resulting differences in the first principles for polar solids up to the first elasticity tensor for hyperelastic materials.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100190"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828740","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}

{"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":"2025-06-01","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}

{"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":"2025-06-01","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}

{"title":"Distance equienergetic graphs of diameter 4","authors":"B.J. Manjunatha ,&nbsp;B.R. Rakshith ,&nbsp;R.G. Veeresha","doi":"10.1016/j.exco.2025.100184","DOIUrl":"10.1016/j.exco.2025.100184","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> be graphs with pairwise disjoint vertex sets. The graph <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> is obtained from the graphs <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∘</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> (the corona product) and <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> by joining each vertices of <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> in <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∘</mo><mspace></mspace><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span> with every vertices in <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. Two connected graphs are called distance equienergetic graphs if their distance energies are the same. Several methods for constructing distance equienergetic graphs have been presented in the literature, most constructed distance equienergetic graphs have diameters of 2 or 3. So the problem of constructing distance equienergetic graphs of diameter greater than 3 would be interesting. Another interesting problem posed by Indulal (2020) is to construct a pair of graphs which are both adjacency equienergetic and distance equienergetic. Motivated by these two problems, in this paper, we obtain the distance spectrum of <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></mrow></mrow></math></span> when all these graphs are regular. As an application, we give a method to obtain distance equienergetic graphs of diameter 4. Also we construct a pair of graphs on <span><math><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices (<span><math><mrow><mi>n</mi><mo>≥</mo><mn>6</mn></mrow></math></span>) which are both adjacency equienergetic and distance equienergetic graphs.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100184"},"PeriodicalIF":0.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739188","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}

{"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":"2025-06-01","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}

{"title":"Jensen-type generalized quadratic fractional integral inequalities","authors":"McSylvester Ejighikeme Omaba","doi":"10.1016/j.exco.2025.100189","DOIUrl":"10.1016/j.exco.2025.100189","url":null,"abstract":"<div><div>This work introduces a generalized quadratic (double) fractional integral operator and derives a two-dimensional extension of Jensen’s inequality for functions integrable on rectangular domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. To illustrate and validate our findings, some examples are provided. Many established results in the literature are derived as special cases of our results.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100189"},"PeriodicalIF":0.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144203229","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}

{"title":"The bound on the locating-chromatic number for a generalized Petersen graphs P(N,2)","authors":"Redha Sakri ,&nbsp;Boualem Slimi","doi":"10.1016/j.exco.2025.100183","DOIUrl":"10.1016/j.exco.2025.100183","url":null,"abstract":"<div><div>The concept of the locating-chromatic number for graphs was introduced by Chartrand et al. (2002). Recently, Sakri and Abbas (2024), presented the locating-chromatic number of generalized Petersen graphs <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> when <span><math><mrow><mi>n</mi><mo>≤</mo><mn>12</mn></mrow></math></span>. In this paper, We determine a lower and upper bound for the locating chromatic number of generalized Petersen graphs <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> when <span><math><mi>n</mi></math></span> even and <span><math><mrow><mi>n</mi><mo>≥</mo><mn>14</mn></mrow></math></span>.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100183"},"PeriodicalIF":0.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143748667","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}