Examples and Counterexamples最新文献

{"title":"G-Tseng’s extragradient method for approximating G-variational inequality problem in Hilbert space endowed with graph","authors":"Monika Swami ,&nbsp;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}

{"title":"A family of asymptotically bad wild towers of function fields","authors":"M. Chara ,&nbsp;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}

{"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":"2025-03-28","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}

{"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":"2025-03-27","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}

{"title":"On the uniqueness of collections of pennies and marbles","authors":"Sean Dewar ,&nbsp;Georg Grasegger ,&nbsp;Kaie Kubjas ,&nbsp;Fatemeh Mohammadi ,&nbsp;Anthony Nixon","doi":"10.1016/j.exco.2025.100181","DOIUrl":"10.1016/j.exco.2025.100181","url":null,"abstract":"<div><div>In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit <span><math><mi>d</mi></math></span>-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100181"},"PeriodicalIF":0.0,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453961","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":"Exceptional surgeries on hyperbolic knots arising from the 3-chain link","authors":"Alberto Cavicchioli,&nbsp;Fulvia Spaggiari","doi":"10.1016/j.exco.2025.100182","DOIUrl":"10.1016/j.exco.2025.100182","url":null,"abstract":"<div><div>We study some closed connected orientable 3–manifolds obtained by Dehn surgery along the oriented components of the 3-chain link. For such manifolds, we describe exceptional surgeries related to some results from Audoux et al. (2018) and Martelli and Petronio (2006). Then we construct a related family of hyperbolic knots in the 3-sphere, which admit two consecutive Seifert fibered surgeries and two toroidal fillings at distance 3. Such additional examples are not mentioned in the quoted papers.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100182"},"PeriodicalIF":0.0,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143437914","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":"Automation of image processing through ML algorithms of GRASS GIS using embedded Scikit-Learn library of Python","authors":"Polina Lemenkova","doi":"10.1016/j.exco.2025.100180","DOIUrl":"10.1016/j.exco.2025.100180","url":null,"abstract":"<div><div>Image processing using Machine Learning (ML) and Artificial Neural Network (ANN) methods was investigated by employing the algorithms of Geographic Resources Analysis Support System (GRASS) Geographic Information System GIS with embedded Scikit-Learn library of Python language. The data are obtained from the United States Geological Survey (USGS) and include the Landsat 8 Operational Land Imager/Thermal Infrared Sensor (OLI/TIRS) multispectral satellite images. The images were collectedon 2013 and 2023 to evaluate land cover categories in each of the year. The study area covers the region of Nile Delta and the Faiyum Oasis, Egypt. A series of modules for raster image processing was applied using scripting language of GRASS GIS to process the remote sensing data. The satellite images were classified into raster maps presenting the land cover types. These include ‘i.cluster’ and ‘i.maxlik’ for non-supervised classification used as training dataset of random pixel seeds, ‘r.random’, ‘r.learn.train’, ‘r.learn.predict’ and ‘r.category’ for ML part of image processing. The consequences of various ML parameters on the cartographic outputs are analysed, such as speed and accuracy, randomness of nodes, analytical determination of the output weights, and dependence distribution of pixels for each algorithm. Supervised learning models of GRASS GIS were tested and compared including the Gaussian Naive Bayes (GaussianNB), Multi-layer Perceptron classifier (MLPClassifier), Support Vector Machines (SVM) Classifier, and Random Forest Classifier (RF). Though each algorithms was developed to serve different objectives of ML applications in RS data processing, their technical implementation and practical purposes present valuable approaches to cartographic data processing and image analysis. The results shown that the most time-consuming algorithms was noted as SVM classification, while the fastest results were achieved by the GaussianNB approach to image processing and the best results are achieved by RF Classifier.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100180"},"PeriodicalIF":0.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143163825","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":"Counterexamples for your calculus course","authors":"Jürgen Appell ,&nbsp;Simon Reinwand","doi":"10.1016/j.exco.2025.100177","DOIUrl":"10.1016/j.exco.2025.100177","url":null,"abstract":"<div><div>We present 2 theorems and 20 counterexamples illustrating the surprising behaviour of functions between metric spaces.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100177"},"PeriodicalIF":0.0,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143240880","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":"Solving change of basis from Bernstein to Chebyshev polynomials","authors":"D.A. Wolfram","doi":"10.1016/j.exco.2025.100178","DOIUrl":"10.1016/j.exco.2025.100178","url":null,"abstract":"<div><div>We provide two closed-form solutions to the change of basis from Bernstein polynomials to shifted Chebyshev polynomials of the fourth kind and show them to be equivalent by applying Zeilberger’s algorithm. The first solution uses orthogonality properties of the Chebyshev polynomials. The second is “modular” which enables separately verified sub-problems to be composed and re-used in other basis transformations. These results have applications in change of basis of orthogonal, and non-orthogonal polynomials.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100178"},"PeriodicalIF":0.0,"publicationDate":"2025-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349023","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":"Hölder’s inequality for shifted quantum integral operator","authors":"Andrea Aglić Aljinović,&nbsp;Lana Horvat Dmitrović,&nbsp;Ana Žgaljić Keko","doi":"10.1016/j.exco.2025.100179","DOIUrl":"10.1016/j.exco.2025.100179","url":null,"abstract":"<div><div>We show by two counterexamples that Hölder’s inequality for shifted quantum integral operator does not hold in general and we prove the case in which it is valid.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100179"},"PeriodicalIF":0.0,"publicationDate":"2025-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143240692","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}