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The locating chromatic number of generalized Petersen graphs with small order 小阶广义彼得森图的定位色度数
Examples and Counterexamples Pub Date : 2024-06-01 Epub Date: 2024-02-17 DOI: 10.1016/j.exco.2024.100141
Redha Sakri , Moncef Abbas
{"title":"The locating chromatic number of generalized Petersen graphs with small order","authors":"Redha Sakri ,&nbsp;Moncef Abbas","doi":"10.1016/j.exco.2024.100141","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100141","url":null,"abstract":"<div><p>It was conjectured by Asmiati (2018) that the generalized Petersen graph <span><math><mrow><mi>P</mi><mfenced><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></mfenced></mrow></math></span> has a locating chromatic number 4 if and only if <span><math><mrow><mo>(</mo><mi>n</mi><mspace></mspace><mi>o</mi><mi>d</mi><mi>d</mi><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></math></span> or <span><math><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>4</mn><mspace></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mspace></mspace><mi>k</mi><mo>=</mo><mn>2</mn><mo>)</mo></mrow></math></span>. In this paper, we give a negative answer to the conjecture posed by Asmiati. As a consequence, we are able to exhibit many counterexamples to the recent conjecture proposed, by proving that if <span><math><mrow><mo>(</mo><mn>5</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>12</mn><mo>)</mo></mrow></math></span> and <span><math><mrow><mo>(</mo><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌋</mo></mrow><mo>)</mo></mrow></math></span> and <span><math><mrow><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow><mo>≠</mo><mrow><mo>(</mo><mn>12</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>L</mi></mrow></msub></mrow></msub><mfenced><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mrow></mfenced><mo>=</mo><mn>4</mn></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100141"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000077/pdfft?md5=0c1ce0bbc9c76ab3ef2eb212405914a8&pid=1-s2.0-S2666657X24000077-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898722","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}
引用次数: 0
On some topological indices of zero divisor graphs of direct product of three finite fields 论三个有限域直积零除子图的一些拓扑指数
Examples and Counterexamples Pub Date : 2024-06-01 Epub Date: 2023-12-08 DOI: 10.1016/j.exco.2023.100129
Subhash Mallinath Gaded, Nithya Sai Narayana
{"title":"On some topological indices of zero divisor graphs of direct product of three finite fields","authors":"Subhash Mallinath Gaded,&nbsp;Nithya Sai Narayana","doi":"10.1016/j.exco.2023.100129","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100129","url":null,"abstract":"<div><p>In this paper, we determine some degree-based topological indices, such as the Sombor index, the first and second Zagreb indices, the forgotten topological index, the Narumi–Katayama index, the first and second multiplicative Zagreb indices, the atom-bond connectivity index, and eccentricity-based topological indices such as total eccentricity, the first and second Zagreb eccentricity indices, and the eccentricity connectivity index of the zero divisor graph with vertex set non-zero zero divisors of the reduced ring of the direct product of three finite fields.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100129"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000319/pdfft?md5=8d93cb32e7a18cf629d5d68fa7ae9cda&pid=1-s2.0-S2666657X23000319-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138557836","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}
引用次数: 0
‘Incomplete’ Pál type interpolation problems on zeros of polynomials with complex coefficients 关于复系数多项式零点的 "不完全 "帕尔型插值问题
Examples and Counterexamples Pub Date : 2024-06-01 Epub Date: 2024-01-05 DOI: 10.1016/j.exco.2023.100132
Poornima Tiwari , A.K. Pathak
{"title":"‘Incomplete’ Pál type interpolation problems on zeros of polynomials with complex coefficients","authors":"Poornima Tiwari ,&nbsp;A.K. Pathak","doi":"10.1016/j.exco.2023.100132","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100132","url":null,"abstract":"<div><p>We termed the Pál type interpolation problem as PTIP. Here we studied the regularity of <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-PTIP and <span><math><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-PTIP, where we omitted a real or complex node from a set of zeros of polynomials with complex coefficients.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100132"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000344/pdfft?md5=31254183438f8f7872d17f7a05aaaad5&pid=1-s2.0-S2666657X23000344-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139107245","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}
引用次数: 0
Solving recurrences for Legendre–Bernstein basis transformations 求解legende - bernstein基变换的递归式
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-07-25 DOI: 10.1016/j.exco.2023.100117
D.A. Wolfram
{"title":"Solving recurrences for Legendre–Bernstein basis transformations","authors":"D.A. Wolfram","doi":"10.1016/j.exco.2023.100117","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100117","url":null,"abstract":"<div><p>The change of basis matrix <span><math><mi>M</mi></math></span> from shifted Legendre to Bernstein polynomials and <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> have applications in computer graphics. Algorithms use their properties to find the matrix elements efficiently. We give new functions for the elements of <span><math><mi>M</mi></math></span> as a summation, and a complete hypergeometric function. We find that Gosper’s algorithm does not produce closed-form expressions for the elements of either <span><math><mi>M</mi></math></span> or <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. Zeilberger’s algorithm produces four second-order recurrences for the elements of the matrices that enable them to be computed in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> time, and for the derivation of closed-form functions by row and column for the elements. Two row recurrences are special cases of those found by Woźny (2013) who used a different method. We show that the recurrences for rows of <span><math><mi>M</mi></math></span> and columns of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> are equivalent. The recurrences for columns of <span><math><mi>M</mi></math></span> and rows of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> generate functions that are the Lagrange interpolation polynomials of their elements. These polynomials are equal to hypergeometric functions, which are solutions of the recurrences.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100117"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882576","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: 1
A family of c-cyclic graphs with a Θ(|V|2log|V|) Kirchhoff index 具有Θ(|V|2log|V|) Kirchhoff指数的c-循环图族
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-10-07 DOI: 10.1016/j.exco.2023.100124
José Luis Palacios
{"title":"A family of c-cyclic graphs with a Θ(|V|2log|V|) Kirchhoff index","authors":"José Luis Palacios","doi":"10.1016/j.exco.2023.100124","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100124","url":null,"abstract":"<div><p>By means of a recurrence, we provide a family of <span><math><mi>c</mi></math></span>-cyclic graphs, <span><math><mrow><mi>c</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, whose Kirchhoff index is <span><math><mrow><mi>Θ</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>log</mo><mrow><mo>|</mo><mi>V</mi><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100124"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49883265","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
Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis 多时滞分数阶SEIR流行病模型的动力学行为及其稳定性分析
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-11-18 DOI: 10.1016/j.exco.2023.100128
Subrata Paul , Animesh Mahata , Supriya Mukherjee , Prakash Chandra Mali , Banamali Roy
{"title":"Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis","authors":"Subrata Paul ,&nbsp;Animesh Mahata ,&nbsp;Supriya Mukherjee ,&nbsp;Prakash Chandra Mali ,&nbsp;Banamali Roy","doi":"10.1016/j.exco.2023.100128","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100128","url":null,"abstract":"<div><p>With multiple time delays, we investigated a Caputo fractional order dynamical system involving susceptible, exposed, infected, and recovered individuals. Positivity and boundedness are also theoretically demonstrated using Laplace transform and Mittag-Leffler function. The stability of the disease-free and epidemic equilibrium points has been studied for both delayed and non-delayed model. For generating numerical solutions to the model system, we used the Adam-Bashforth-Moulton predictor-corrector technique. With the help of MATLAB (2018a), we were able to conduct graphical demonstrations and numerical simulations. The system displays Hopf bifurcation and the solutions are no longer periodic beyond a certain threshold value of the time delay parameters.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100128"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000307/pdfft?md5=edab84a58f8ff0eb993a5fb57ba503c9&pid=1-s2.0-S2666657X23000307-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138480350","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}
引用次数: 0
Diagonalization of the cross-product matrix 叉乘矩阵的对角化
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-08-02 DOI: 10.1016/j.exco.2023.100118
Oskar Maria Baksalary , Götz Trenkler
{"title":"Diagonalization of the cross-product matrix","authors":"Oskar Maria Baksalary ,&nbsp;Götz Trenkler","doi":"10.1016/j.exco.2023.100118","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100118","url":null,"abstract":"<div><p>The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100118"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882668","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
Non-existence of perturbed solutions under a second-order sufficient condition 二阶充分条件下摄动解的不存在性
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-09-06 DOI: 10.1016/j.exco.2023.100122
Gerd Wachsmuth
{"title":"Non-existence of perturbed solutions under a second-order sufficient condition","authors":"Gerd Wachsmuth","doi":"10.1016/j.exco.2023.100122","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100122","url":null,"abstract":"<div><p>We present an optimization problem in infinite dimensions which satisfies the usual second-order sufficient condition but for which perturbed problems fail to possess solutions.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100122"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49883264","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
Counterexamples for Noise Models of Stochastic Gradients 随机梯度噪声模型的反例
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-08-27 DOI: 10.1016/j.exco.2023.100123
Vivak Patel
{"title":"Counterexamples for Noise Models of Stochastic Gradients","authors":"Vivak Patel","doi":"10.1016/j.exco.2023.100123","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100123","url":null,"abstract":"<div><p>Stochastic Gradient Descent (SGD) is a widely used, foundational algorithm in data science and machine learning. As a result, analyses of SGD abound making use of a variety of assumptions, especially on the noise behavior of the stochastic gradients. While recent works have achieved a high-degree of generality on assumptions about the noise behavior of the stochastic gradients, it is unclear that such generality is necessary. In this work, we construct a simple example that shows that less general assumptions will be violated, while the most general assumptions will hold.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100123"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882666","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
Identification of brain tumors based on digitized parameters from magnetic resonance imaging results 基于磁共振成像结果数字化参数的脑肿瘤识别
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-11-19 DOI: 10.1016/j.exco.2023.100125
A.M. Al-Ansi , M. Almadi , V. Ryabtsev , T. Utkina
{"title":"Identification of brain tumors based on digitized parameters from magnetic resonance imaging results","authors":"A.M. Al-Ansi ,&nbsp;M. Almadi ,&nbsp;V. Ryabtsev ,&nbsp;T. Utkina","doi":"10.1016/j.exco.2023.100125","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100125","url":null,"abstract":"<div><p>A methodology is proposed for identifying brain tumors by dividing the database into four parts. The results obtained from the study of sample specimens for each type of brain tumor showed a high degree of similarity in recognition. This methodology can be applied in healthcare facilities to improve the accuracy of disease diagnosis.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100125"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000277/pdfft?md5=f3100a0e2172cdec05b25d019b3236c5&pid=1-s2.0-S2666657X23000277-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138395174","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}
引用次数: 0
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