Pub Date : 2025-03-27DOI: 10.1016/j.exco.2025.100183
Redha Sakri , Boualem Slimi
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 when . In this paper, We determine a lower and upper bound for the locating chromatic number of generalized Petersen graphs when even and .
图的定位色数概念是由Chartrand et al.（2002）提出的。最近，Sakri和Abbas（2024）给出了n≤12时广义Petersen图P（n,k）的定位色数。本文确定了广义Petersen图P（n,2）在n为偶数且n≥14时的色数定位的下界和上界。
{"title":"The bound on the locating-chromatic number for a generalized Petersen graphs P(N,2)","authors":"Redha Sakri , 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}
Pub Date : 2025-02-14DOI: 10.1016/j.exco.2025.100181
Sean Dewar , Georg Grasegger , Kaie Kubjas , Fatemeh Mohammadi , Anthony Nixon
In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit -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.
{"title":"On the uniqueness of collections of pennies and marbles","authors":"Sean Dewar , Georg Grasegger , Kaie Kubjas , Fatemeh Mohammadi , 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}
Pub Date : 2025-02-14DOI: 10.1016/j.exco.2025.100182
Alberto Cavicchioli, Fulvia Spaggiari
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.
{"title":"Exceptional surgeries on hyperbolic knots arising from the 3-chain link","authors":"Alberto Cavicchioli, 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}
Pub Date : 2025-02-03DOI: 10.1016/j.exco.2025.100180
Polina Lemenkova
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.
{"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}
Pub Date : 2025-02-03DOI: 10.1016/j.exco.2025.100177
Jürgen Appell , Simon Reinwand
We present 2 theorems and 20 counterexamples illustrating the surprising behaviour of functions between metric spaces.
我们给出了2个定理和20个反例来说明函数在度量空间之间的惊人行为。
{"title":"Counterexamples for your calculus course","authors":"Jürgen Appell , 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}
Pub Date : 2025-02-02DOI: 10.1016/j.exco.2025.100178
D.A. Wolfram
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.
{"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}
Pub Date : 2025-02-02DOI: 10.1016/j.exco.2025.100179
Andrea Aglić Aljinović, Lana Horvat Dmitrović, Ana Žgaljić Keko
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.
通过两个反例证明了平移量子积分算子Hölder的不等式一般不成立，并证明了其成立的情况。
{"title":"Hölder’s inequality for shifted quantum integral operator","authors":"Andrea Aglić Aljinović, Lana Horvat Dmitrović, 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}
Pub Date : 2025-01-31DOI: 10.1016/j.exco.2025.100176
Christian Genest, Johanna G. Nešlehová
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.
经验多元线性或棋盘联结过程是一种很有前途的工具，用于对具有联系的数据进行联结模型的统计推断（Genest等人，2019a）。Genest et al.（2014, 2017）在非常广泛的条件下确定了该过程的大样本行为。本文的目的是详细描述这一渐近结果，并在数据构成伯努利随机变量对的随机样本的最简单情况下推导出该过程的极限表达式。尽管人们永远不会真正将联结模型拟合到2 × 2列联表中，但这种情况特别适合于对该过程的限制行为的复杂性进行明确的计算和说明性解释，并明确为什么需要Genest等人（2014,2017）中的条件并且不能简化。
{"title":"Asymptotic behavior of the empirical checkerboard copula process for binary data: An educational presentation","authors":"Christian Genest, 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":"2025-01-31","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}
Pub Date : 2025-01-28DOI: 10.1016/j.exco.2025.100175
Yohan Chandrasukmana, Helena Margaretha, Kie Van Ivanky Saputra
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.
{"title":"The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses","authors":"Yohan Chandrasukmana, Helena Margaretha, 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":"2025-01-28","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}
Pub Date : 2025-01-15DOI: 10.1016/j.exco.2025.100174
Anupam Mondal , Pritam Chandra Pramanik
Robin Forman’s highly influential 2002 paper A User’s Guide to Discrete Morse Theory 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 (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.
Robin Forman在2002年发表的极具影响力的论文《离散莫尔斯理论的用户指南》以一种非常可读的方式概述了该主题。作为概念证明，作者利用纯粹组合性质的离散莫尔斯理论技术，确定了n阶断连图的抽象简单复合体的拓扑（同伦类型）（先前由Victor Vassiliev用经典拓扑方法完成）。该技术涉及在复合体上构建（和验证）离散梯度向量场。然而，核查部分依赖于一项似乎站不住脚的主张。在本文中，我们提供了几个反例来反驳这一特定的说法。我们还提供了另一种证明，证明所构造的离散向量场确实是一个梯度向量场。我们的证明技术依赖于一个关键的观察，而不是特定于手头的问题，因此适用于验证构造的离散向量场通常是梯度场。
{"title":"A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs","authors":"Anupam Mondal , 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":"2025-01-15","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}
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