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Absolutely fractional differentiable functions: Results, examples and counterexamples 绝对分数可微函数：结果，例子和反例
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-11-06 DOI: 10.1016/j.exco.2025.100207
Sudhir Kumar, Jyotindra C. Prajapati
{"title":"Absolutely fractional differentiable functions: Results, examples and counterexamples","authors":"Sudhir Kumar,&nbsp;Jyotindra C. Prajapati","doi":"10.1016/j.exco.2025.100207","DOIUrl":"10.1016/j.exco.2025.100207","url":null,"abstract":"<div><div>Authors have defined absolutely fractional differentiable functions on a closed bounded interval <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span> of <span><math><mi>R</mi></math></span>. Some significant results and equivalent conditions for the function to be absolutely fractional differentiable on a closed bounded interval <span><math><mrow><mo>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>]</mo></mrow></math></span> have been discussed. Several examples of absolutely fractional differentiable functions have been constructed. Counterexamples have also been constructed for some results.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100207"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465966","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}
引用次数: 0
A family of polynomials that defy general convergence in three methods 用三种方法都不能普遍收敛的多项式族
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-06-14 DOI: 10.1016/j.exco.2025.100191
Jared Collins , Grant Hunter
{"title":"A family of polynomials that defy general convergence in three methods","authors":"Jared Collins ,&nbsp;Grant Hunter","doi":"10.1016/j.exco.2025.100191","DOIUrl":"10.1016/j.exco.2025.100191","url":null,"abstract":"<div><div>We consider the root-finding algorithms of Newton’s method, Halley’s method and Schröder’s method. These methods are known to have polynomials for which the method produces extraneous attracting cycles, i.e. the method is not generally convergent for these polynomials. We produce a family of polynomials that have an extraneous attracting cycle in all three methods. By doing so we show that mixing methods is not beneficial to overcoming the issues of general convergence. We also address a more specific result for the degree three case.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100191"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307989","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}
引用次数: 0
Verification of approximate solutions for nonlinear differential equations 非线性微分方程近似解的验证
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-09-22 DOI: 10.1016/j.exco.2025.100199
Horst J. Klepp
{"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":"2025-12-01","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}
引用次数: 0
Existence and uniqueness of solutions for nonlocal elliptic problems with Neumann boundary conditions and some examples 具有Neumann边界条件的非局部椭圆型问题解的存在唯一性及一些例子
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-11-07 DOI: 10.1016/j.exco.2025.100208
Baoqiang Yan , Donal O’Regan
{"title":"Existence and uniqueness of solutions for nonlocal elliptic problems with Neumann boundary conditions and some examples","authors":"Baoqiang Yan ,&nbsp;Donal O’Regan","doi":"10.1016/j.exco.2025.100208","DOIUrl":"10.1016/j.exco.2025.100208","url":null,"abstract":"<div><div>In this paper, we consider a nonlocal elliptic equation under Neumann boundary conditions with parameter <span><math><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>. We show that the number of positive solutions of our problem is equivalent to the number of solutions of an algebraic equation and we also consider the existence and uniqueness of positive solutions of our problem. Some examples are listed to illustrate our theorems.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100208"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145570922","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}
引用次数: 0
An example of Disease-induced fractal–fractional dynamic system for potato spots 马铃薯斑疹病致分形-分形动态系统的一个实例
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-10-10 DOI: 10.1016/j.exco.2025.100204
Shaher Momani , Rabha W. Ibrahim
{"title":"An example of Disease-induced fractal–fractional dynamic system for potato spots","authors":"Shaher Momani ,&nbsp;Rabha W. Ibrahim","doi":"10.1016/j.exco.2025.100204","DOIUrl":"10.1016/j.exco.2025.100204","url":null,"abstract":"<div><div>The biological significance of fractal–fractional calculus resides to analyzing and describing complex biological systems (like the dynamic system of potato spots). These systems frequently have erratic patterns or processes with memory and long-term dependencies. We seek an example to enhance models of dynamic system of potato spots.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100204"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145320238","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}
引用次数: 0
On a conjecture of Gill on Bell experiments 吉尔对贝尔实验的一个猜想
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-10-10 DOI: 10.1016/j.exco.2025.100203
Daniel B. Rubin
{"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":"2025-12-01","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}
引用次数: 0
Wildest SL2-tilings 最疯狂的SL2-tilings
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-11-04 DOI: 10.1016/j.exco.2025.100206
Andrei Zabolotskii
{"title":"Wildest SL2-tilings","authors":"Andrei Zabolotskii","doi":"10.1016/j.exco.2025.100206","DOIUrl":"10.1016/j.exco.2025.100206","url":null,"abstract":"<div><div>Tame <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings are related to Farey graph and friezes; much less is known about wild (not tame) <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings. In this note, we demonstrate <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings that are maximally wild: we prove that the maximum wild density of an integer <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tiling is <span><math><mfrac><mrow><mn>2</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> and present <span><math><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tilings over <span><math><mrow><mi>Z</mi><mo>/</mo><mi>N</mi><mi>Z</mi></mrow></math></span> with wild density 1.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100206"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465967","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}
引用次数: 0
The third axiom of filling properties 填充性质的第三个公理
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-06-30 DOI: 10.1016/j.exco.2025.100192
Ridwan Pandiya
{"title":"The third axiom of filling properties","authors":"Ridwan Pandiya","doi":"10.1016/j.exco.2025.100192","DOIUrl":"10.1016/j.exco.2025.100192","url":null,"abstract":"<div><div>Since its first appearance in 1983, the filled function method, which was initiated to solve global optimization problems, has developed very rapidly. From the results conducted by many scholars, the ideal filled function has at least two properties: parameter-free and continuously differentiable. Several researchers have attempted to provide filled functions with such properties that meet the three axioms (filling properties) required by the filled function definition. The third axiom specifically states that the filled function has a minimum point in the region of attraction. This paper examines the fact that the currently available continuously differentiable parameter-free filled functions do not fulfil the third axiom of the filling properties by providing several counterexamples.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100192"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144514374","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}
引用次数: 0
Flasque meadows Flasque草地
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-07-17 DOI: 10.1016/j.exco.2025.100193
João Dias, Bruno Dinis
{"title":"Flasque meadows","authors":"João Dias,&nbsp;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":"2025-12-01","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}
引用次数: 0
Uniqueness of the smallest chameleon in the class of monoids 一元类中最小变色龙的唯一性
Examples and Counterexamples Pub Date : 2025-12-01 Epub Date: 2025-11-27 DOI: 10.1016/j.exco.2025.100209
Nicholas Johnston, Edmond W.H. Lee, Vehbi E. Paksoy
{"title":"Uniqueness of the smallest chameleon in the class of monoids","authors":"Nicholas Johnston,&nbsp;Edmond W.H. Lee,&nbsp;Vehbi E. Paksoy","doi":"10.1016/j.exco.2025.100209","DOIUrl":"10.1016/j.exco.2025.100209","url":null,"abstract":"<div><div>A semigroup is a <em>chameleon</em> if it is the reduct of both a finitely based involution semigroup and a non-finitely based involution semigroup. Recently, a certain monoid <span><math><mi>K</mi></math></span> of order six was shown to be a smallest chameleon in the class of monoids. The main objective of the present article is to show that although there exist 2237 monoids of order six up to isomorphism, examples of chameleons different from <span><math><mi>K</mi></math></span> do not exist. Consequently, up to isomorphism, <span><math><mi>K</mi></math></span> is the unique smallest example of chameleon in the class of monoids.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100209"},"PeriodicalIF":0.0,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145683694","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}
引用次数: 0
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