Pub Date : 2025-07-13DOI: 10.1016/j.exco.2025.100194
Zhekai Shen
In this paper, we address the problem of finding functions with predetermined Lipschitz continuous points. More precisely, given , we are interested in the existence of function which is Lipschitz continuous exactly on . Our result is related to Liouville numbers.
{"title":"Lipschitz continuous points of functions on an interval","authors":"Zhekai Shen","doi":"10.1016/j.exco.2025.100194","DOIUrl":"10.1016/j.exco.2025.100194","url":null,"abstract":"<div><div>In this paper, we address the problem of finding functions with predetermined Lipschitz continuous points. More precisely, given <span><math><mrow><mi>A</mi><mo>⊆</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, we are interested in the existence of function <span><math><mrow><mi>f</mi><mo>:</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>→</mo><mi>R</mi></mrow></math></span> which is Lipschitz continuous exactly on <span><math><mi>A</mi></math></span>. Our result is related to Liouville numbers.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"8 ","pages":"Article 100194"},"PeriodicalIF":0.0,"publicationDate":"2025-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144631377","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 : 2025-07-11DOI: 10.1016/j.exco.2025.100190
Stefan Buchen
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.
{"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":"2025-07-11","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}
Pub Date : 2025-06-30DOI: 10.1016/j.exco.2025.100192
Ridwan Pandiya
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.
{"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":"2025-06-30","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}
Pub Date : 2025-06-14DOI: 10.1016/j.exco.2025.100191
Jared Collins , Grant Hunter
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.
{"title":"A family of polynomials that defy general convergence in three methods","authors":"Jared Collins , 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":"2025-06-14","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}
Pub Date : 2025-06-01DOI: 10.1016/j.exco.2025.100189
McSylvester Ejighikeme Omaba
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 . To illustrate and validate our findings, some examples are provided. Many established results in the literature are derived as special cases of our results.
{"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}
Pub Date : 2025-05-14DOI: 10.1016/j.exco.2025.100187
William L. Blair
We show there exist solutions to higher-order Vekua equations that, along with their -derivatives, have finite Hardy space norm, but their -derivatives do not.
我们证明了高阶Vekua方程的解及其z导数具有有限Hardy空间范数，但它们的z导数没有。
{"title":"Meta-analytic functions with derivative not in a Hardy space","authors":"William L. Blair","doi":"10.1016/j.exco.2025.100187","DOIUrl":"10.1016/j.exco.2025.100187","url":null,"abstract":"<div><div>We show there exist solutions to higher-order Vekua equations that, along with their <span><math><mover><mrow><mi>z</mi></mrow><mrow><mo>̄</mo></mrow></mover></math></span>-derivatives, have finite Hardy space norm, but their <span><math><mi>z</mi></math></span>-derivatives do not.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100187"},"PeriodicalIF":0.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143948723","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 : 2025-05-14DOI: 10.1016/j.exco.2025.100188
Zoran Stanić
An extended mixed graph contains edges declared to be either positive, or negative, or oriented from one end to the other. They unify concepts of mixed graphs, signed graphs, oriented graphs and ordinary graphs. We demonstrate a method that establishes an upper bound for the largest singular value of the corresponding adjacency matrix in a bit general setting allowing the existence of loops at vertices. An upper bound for the largest singular value of the Laplacian matrix is extracted. The obtained bounds remain valid for the largest modulus of the corresponding eigenvalues, and cover the aforementioned particular structures.
{"title":"An upper bound for the largest singular value of extended mixed graphs","authors":"Zoran Stanić","doi":"10.1016/j.exco.2025.100188","DOIUrl":"10.1016/j.exco.2025.100188","url":null,"abstract":"<div><div>An extended mixed graph contains edges declared to be either positive, or negative, or oriented from one end to the other. They unify concepts of mixed graphs, signed graphs, oriented graphs and ordinary graphs. We demonstrate a method that establishes an upper bound for the largest singular value of the corresponding adjacency matrix in a bit general setting allowing the existence of loops at vertices. An upper bound for the largest singular value of the Laplacian matrix is extracted. The obtained bounds remain valid for the largest modulus of the corresponding eigenvalues, and cover the aforementioned particular structures.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100188"},"PeriodicalIF":0.0,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143948724","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 : 2025-04-28DOI: 10.1016/j.exco.2025.100185
Monika Swami , M.R. Jadeja
In this article, we introduce the -Tseng’s extragradient method, inspired by the extragradient method defined by Korpelevich, for solving -variational inequality problems in Hilbert space. We also address a fixed point problem in a Hilbert space endowed with a graph using the proposed -Tseng’s extragradient method. In the context of Hilbert space, we establish weak and strong convergence theorems for the algorithm. Additionally, we provide numerical examples to support our findings.
{"title":"G-Tseng’s extragradient method for approximating G-variational inequality problem in Hilbert space endowed with graph","authors":"Monika Swami , M.R. Jadeja","doi":"10.1016/j.exco.2025.100185","DOIUrl":"10.1016/j.exco.2025.100185","url":null,"abstract":"<div><div>In this article, we introduce the <span><math><mi>G</mi></math></span>-Tseng’s extragradient method, inspired by the extragradient method defined by Korpelevich, for solving <span><math><mi>G</mi></math></span>-variational inequality problems in Hilbert space. We also address a fixed point problem in a Hilbert space endowed with a graph using the proposed <span><math><mi>G</mi></math></span>-Tseng’s extragradient method. In the context of Hilbert space, we establish weak and strong convergence theorems for the algorithm. Additionally, we provide numerical examples to support our findings.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100185"},"PeriodicalIF":0.0,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143895012","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 : 2025-04-25DOI: 10.1016/j.exco.2025.100186
M. Chara , R. Toledano
In Chara and Toledano (2015) general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this paper the genus of a family of wild towers of function fields will be considered together with a result with less restrictive sufficient conditions for a wild tower to have infinite genus.
{"title":"A family of asymptotically bad wild towers of function fields","authors":"M. Chara , R. Toledano","doi":"10.1016/j.exco.2025.100186","DOIUrl":"10.1016/j.exco.2025.100186","url":null,"abstract":"<div><div>In Chara and Toledano (2015) general conditions were given to prove the infiniteness of the genus of certain towers of function fields over a perfect field. It was shown that many examples where particular cases of those general results. In this paper the genus of a family of wild towers of function fields will be considered together with a result with less restrictive sufficient conditions for a wild tower to have infinite genus.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100186"},"PeriodicalIF":0.0,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143874282","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 : 2025-03-28DOI: 10.1016/j.exco.2025.100184
B.J. Manjunatha , B.R. Rakshith , R.G. Veeresha
Let , and be graphs with pairwise disjoint vertex sets. The graph is obtained from the graphs (the corona product) and by joining each vertices of in with every vertices in . 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 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 vertices () which are both adjacency equienergetic and distance equienergetic graphs.
{"title":"Distance equienergetic graphs of diameter 4","authors":"B.J. Manjunatha , B.R. Rakshith , 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":"2025-03-28","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}
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