Pub Date : 2025-10-10DOI: 10.1016/j.exco.2025.100203
Daniel B. Rubin
This note gives a counterexample to a conjecture of Richard Gill that was stated as open problem related to the statistical analysis of Bell experiments in quantum mechanics, and which can formulated as an elementary probability problem.
{"title":"On a conjecture of Gill on Bell experiments","authors":"Daniel B. Rubin","doi":"10.1016/j.exco.2025.100203","DOIUrl":"10.1016/j.exco.2025.100203","url":null,"abstract":"<div><div>This note gives a counterexample to a conjecture of Richard Gill that was stated as open problem related to the statistical analysis of Bell experiments in quantum mechanics, and which can formulated as an elementary probability problem.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100203"},"PeriodicalIF":0.0,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320239","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 : 2025-09-29DOI: 10.1016/j.exco.2025.100202
Mohamed M. Awad , A.A. Elsadany , Mohammed K. Elboree
In this study, we employ a symbolic computation approach to construct various rogue wave solutions of the (3+1)-dimensional nonlinear wave equation modeling wave propagation in a liquid medium containing gas bubbles. Through the application of the Hirota bilinear method, we systematically derive first-order, second-order, and third-order rogue wave solutions. By selecting appropriate parameter values, we generate graphical illustrations that reveal the structural features and interaction dynamics of these rogue waves. A variety of soliton solutions were obtained from the analysis. Upon plotting these solutions, several rogue wave solutions emerged and periodic waveforms. These results are visually represented through two-dimensional, three-dimensional, and contour plots to highlight the different features and behaviors of each solution. The analysis enhances our understanding of rogue wave behavior within the context of the modeled physical system.
{"title":"Multiple rogue waves solutions for a (3 + 1)-dimensional nonlinear wave in liquid with gas bubbles via Hirota bilinear equation method","authors":"Mohamed M. Awad , A.A. Elsadany , Mohammed K. Elboree","doi":"10.1016/j.exco.2025.100202","DOIUrl":"10.1016/j.exco.2025.100202","url":null,"abstract":"<div><div>In this study, we employ a symbolic computation approach to construct various rogue wave solutions of the (3+1)-dimensional nonlinear wave equation modeling wave propagation in a liquid medium containing gas bubbles. Through the application of the Hirota bilinear method, we systematically derive first-order, second-order, and third-order rogue wave solutions. By selecting appropriate parameter values, we generate graphical illustrations that reveal the structural features and interaction dynamics of these rogue waves. A variety of soliton solutions were obtained from the analysis. Upon plotting these solutions, several rogue wave solutions emerged and periodic waveforms. These results are visually represented through two-dimensional, three-dimensional, and contour plots to highlight the different features and behaviors of each solution. The analysis enhances our understanding of rogue wave behavior within the context of the modeled physical system.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100202"},"PeriodicalIF":0.0,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219497","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 : 2025-09-22DOI: 10.1016/j.exco.2025.100200
Ahmed Al Fares , Gizem Karaali
Quasigroups are algebraic structures in which divisibility is always defined. This paper illustrates some similarities and differences between quasigroup theory and group theory, by singling out a special family of quasigroups which seem to be most “grouplike”.
{"title":"On a “grouplike” family of quasigroups","authors":"Ahmed Al Fares , Gizem Karaali","doi":"10.1016/j.exco.2025.100200","DOIUrl":"10.1016/j.exco.2025.100200","url":null,"abstract":"<div><div>Quasigroups are algebraic structures in which divisibility is always defined. This paper illustrates some similarities and differences between quasigroup theory and group theory, by singling out a special family of quasigroups which seem to be most “grouplike”.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100200"},"PeriodicalIF":0.0,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157221","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 : 2025-09-22DOI: 10.1016/j.exco.2025.100199
Horst J. Klepp
Equations for checking the approximate solutions of nonlinear differential equations are given. These solutions should predict the behavior of the systems for which these differential equations are the mathematical models. It is general consensus, regardless of which integration method is used to determine approximate functions, it must be checked whether these functions satisfy control equations with the desired accuracy in order to be considered approximate solutions. Here, the control equations are the differential equations of motion and the balance equation for the variation of the kinetic energy and the mechanical work of the forces. The errors in these control equations, residues respectively deviations, must be less than desired limits. As example, the sinking motion of a mass point in a liquid medium with the drag force assumed to be proportional to the square of the velocity is considered. Therefore, the differential equation of motion is nonlinear. The exact solution is determined and compared with the approximate function established with Euler’s polygon method. It turns out that for integration steps larger than steps for which bifurcation occurs the residues and the deviations are too large and such functions are not approximate solutions and cannot be used for making predictions about the behavior of the system. The energetic interpretation of the result is given.
{"title":"Verification of approximate solutions for nonlinear differential equations","authors":"Horst J. Klepp","doi":"10.1016/j.exco.2025.100199","DOIUrl":"10.1016/j.exco.2025.100199","url":null,"abstract":"<div><div>Equations for checking the approximate solutions of nonlinear differential equations are given. These solutions should predict the behavior of the systems for which these differential equations are the mathematical models. It is general consensus, regardless of which integration method is used to determine approximate functions, it must be checked whether these functions satisfy control equations with the desired accuracy in order to be considered approximate solutions. Here, the control equations are the differential equations of motion and the balance equation for the variation of the kinetic energy and the mechanical work of the forces. The errors in these control equations, residues respectively deviations, must be less than desired limits. As example, the sinking motion of a mass point in a liquid medium with the drag force assumed to be proportional to the square of the velocity is considered. Therefore, the differential equation of motion is nonlinear. The exact solution is determined and compared with the approximate function established with Euler’s polygon method. It turns out that for integration steps larger than steps for which bifurcation occurs the residues and the deviations are too large and such functions are not approximate solutions and cannot be used for making predictions about the behavior of the system. The energetic interpretation of the result is given.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100199"},"PeriodicalIF":0.0,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145121181","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 : 2025-09-22DOI: 10.1016/j.exco.2025.100201
Alexei Lisitsa
We present an independently discovered Andrews–Curtis trivialization of the balanced trivial group presentation 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 Prover9 theorem prover, which required approximately 74 days and over 56 GB of memory to compute.
{"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":"2025-09-22","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}
Pub Date : 2025-09-15DOI: 10.1016/j.exco.2025.100198
Thomas Apel , Katharina Lorenz , Serge Nicaise
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.
{"title":"Weak and very weak solutions of the Laplace equation and the Stokes system with prescribed regularity","authors":"Thomas Apel , Katharina Lorenz , 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":"2025-09-15","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}
Pub Date : 2025-09-13DOI: 10.1016/j.exco.2025.100197
Márcia Lemos-Silva, Delfim F.M. Torres
We propose a new dynamic logistic equation that, in contrast with the one available in the literature, preserves the non-negativity of its solutions, which is an essential property for biological meaning. We derive the exact solution of the proposed model and analyze its behavior on arbitrary time scales. Finally, we illustrate our results with concrete examples. Our key novelty is: the model’s ability to maintain positivity across general time scales and its dynamic consistency with the classical logistic equation.
{"title":"Logistic equation on time scales","authors":"Márcia Lemos-Silva, Delfim F.M. Torres","doi":"10.1016/j.exco.2025.100197","DOIUrl":"10.1016/j.exco.2025.100197","url":null,"abstract":"<div><div>We propose a new dynamic logistic equation that, in contrast with the one available in the literature, preserves the non-negativity of its solutions, which is an essential property for biological meaning. We derive the exact solution of the proposed model and analyze its behavior on arbitrary time scales. Finally, we illustrate our results with concrete examples. Our key novelty is: the model’s ability to maintain positivity across general time scales and its dynamic consistency with the classical logistic equation.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100197"},"PeriodicalIF":0.0,"publicationDate":"2025-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145060624","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 : 2025-08-20DOI: 10.1016/j.exco.2025.100196
Nuno Fachada , João P. Matos-Carvalho , Carlos M. Fernandes
Consider a population of size where each element has an independent probability 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.
{"title":"A catch-all discrete distribution: Short description and main properties","authors":"Nuno Fachada , João P. Matos-Carvalho , 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":"2025-08-20","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}
Pub Date : 2025-08-06DOI: 10.1016/j.exco.2025.100195
Lennin Mallma Ramirez , Nelson Maculan , Adilson Elias Xavier , Vinicius Layter Xavier
In this paper, we study the hyperbolic augmented Lagrangian function, which is based on the hyperbolic penalty function. We ensure the existence of a local saddle point for the hyperbolic augmented Lagrangian function in nonconvex constrained optimization problems by considering the classical second-order sufficiency conditions. Finally, we present some computational illustrations of the theory proposed in this work.
{"title":"A local saddle point for the hyperbolic augmented Lagrangian function","authors":"Lennin Mallma Ramirez , Nelson Maculan , Adilson Elias Xavier , Vinicius Layter Xavier","doi":"10.1016/j.exco.2025.100195","DOIUrl":"10.1016/j.exco.2025.100195","url":null,"abstract":"<div><div>In this paper, we study the hyperbolic augmented Lagrangian function, which is based on the hyperbolic penalty function. We ensure the existence of a local saddle point for the hyperbolic augmented Lagrangian function in nonconvex constrained optimization problems by considering the classical second-order sufficiency conditions. Finally, we present some computational illustrations of the theory proposed in this work.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100195"},"PeriodicalIF":0.0,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827609","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 : 2025-07-17DOI: 10.1016/j.exco.2025.100193
João Dias, Bruno Dinis
In analogy with flasque sheaves, we introduce the notion of flasque meadow as a common meadow where the transition maps are all surjective. We study some properties of flasque meadows and illustrate them with several examples and counterexamples.
{"title":"Flasque meadows","authors":"João Dias, Bruno Dinis","doi":"10.1016/j.exco.2025.100193","DOIUrl":"10.1016/j.exco.2025.100193","url":null,"abstract":"<div><div>In analogy with flasque sheaves, we introduce the notion of flasque meadow as a common meadow where the transition maps are all surjective. We study some properties of flasque meadows and illustrate them with several examples and counterexamples.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100193"},"PeriodicalIF":0.0,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679750","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}
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