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On two Kuznetsov’s conjectures 关于库兹涅佐夫的两个猜想
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-11-15 DOI: 10.1016/j.exco.2023.100127
Florian Oschmann
{"title":"On two Kuznetsov’s conjectures","authors":"Florian Oschmann","doi":"10.1016/j.exco.2023.100127","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100127","url":null,"abstract":"<div><p>We provide a proof and a counterexample to two conjectures made by N. Kuznetsov.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100127"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000290/pdfft?md5=3d37c77be6fcc33ff02da1f947a414bc&pid=1-s2.0-S2666657X23000290-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136572155","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 the fractional Allee logistic equation in the Caputo sense 卡普托意义上的分数阶Allee logistic方程
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-08-11 DOI: 10.1016/j.exco.2023.100121
I. Area , Juan J. Nieto
{"title":"On the fractional Allee logistic equation in the Caputo sense","authors":"I. Area ,&nbsp;Juan J. Nieto","doi":"10.1016/j.exco.2023.100121","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100121","url":null,"abstract":"<div><p>In the framework of population models, logistic growth and fractional logistic growth has been analyzed. In some situations the so-called Allee effect gives more accurate approximation. In this work, fractional Allee differential equation in the Caputo sense is considered. The solution is obtained by considering formal power series. Numerical computations are presented to compare the truncating series with the classical Allee differential equation.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100121"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882667","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
New example of strongly regular graph with parameters (81,30,9,12) and a simple group A5 as the automorphism group 参数为(81,30,9,12)的强正则图的新例子，自同构群为一个简单群A5
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-07-28 DOI: 10.1016/j.exco.2023.100119
Dean Crnković, Andrea Švob
{"title":"New example of strongly regular graph with parameters (81,30,9,12) and a simple group A5 as the automorphism group","authors":"Dean Crnković,&nbsp;Andrea Švob","doi":"10.1016/j.exco.2023.100119","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100119","url":null,"abstract":"<div><p>A new strongly regular graph with parameters <span><math><mrow><mo>(</mo><mn>81</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></math></span> is found as a graph invariant under certain subgroup of the full automorphism group of the previously known strongly regular graph discovered in 1981 by J. H. van Lint and A. Schrijver.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100119"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882574","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
Infinitesimal phase response functions can be misleading 无穷小的相位响应函数可能会产生误导
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-08-07 DOI: 10.1016/j.exco.2023.100120
Christoph Börgers
{"title":"Infinitesimal phase response functions can be misleading","authors":"Christoph Börgers","doi":"10.1016/j.exco.2023.100120","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100120","url":null,"abstract":"<div><p>Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase shifts as a function of the phase at which the input is received. When the pulse is weak, it is customary to linearize around zero pulse strength. The result is called the <em>infinitesimal</em> phase response function. These ideas have been used extensively in theoretical biology, and also in some areas of engineering. I give examples showing that the infinitesimal phase response function may predict that two oscillators, as they exchange pulses back and fourth, will converge to synchrony, yet this is false when the exact phase response function is used, for all positive interaction strengths. For short, the analogue of the Hartman–Grobman theorem that one might expect to hold at first sight is invalid. I give a condition under which the prediction derived using the infinitesimal phase response function does hold for the exact phase response function when interactions are sufficiently weak but of positive strength. However, I argue that this condition may often fail to hold.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100120"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882573","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
Exact solutions of a variable coefficient KdV equation: Power law in time-coefficients 变系数KdV方程的精确解:时间系数的幂律
Examples and Counterexamples Pub Date : 2023-12-01 Epub Date: 2023-11-21 DOI: 10.1016/j.exco.2023.100126
Motlatsi Molati
{"title":"Exact solutions of a variable coefficient KdV equation: Power law in time-coefficients","authors":"Motlatsi Molati","doi":"10.1016/j.exco.2023.100126","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100126","url":null,"abstract":"<div><p>The Lie symmetry analysis of a power law in-time coefficients Korteweg–de Vries (KdV) equation is performed with the aim of specifying the model parameters (powers of <span><math><mi>t</mi></math></span>). That is, the symmetries of the resulting subclasses of the underlying equation are obtained. Further, symmetry reductions and some exact solutions are obtained.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100126"},"PeriodicalIF":0.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000289/pdfft?md5=e7b499b9a817b8e0282ff728e9b6db5f&pid=1-s2.0-S2666657X23000289-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138395175","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
A short note on Wedderburn decomposition of a group algebra 关于群代数的Wedderburn分解的一个注记
Examples and Counterexamples Pub Date : 2023-11-01 Epub Date: 2023-04-04 DOI: 10.1016/j.exco.2023.100105
Gaurav Mittal , R.K. Sharma
{"title":"A short note on Wedderburn decomposition of a group algebra","authors":"Gaurav Mittal ,&nbsp;R.K. Sharma","doi":"10.1016/j.exco.2023.100105","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100105","url":null,"abstract":"<div><p>In this paper, we extend the result of Mittal and Sharma (Bull. Korean Math. Soc. 2022) on Wedderburn decomposition (WD) of a finite semisimple group algebra. It is known that, under certain conditions, WD of a finite semisimple group algebra <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mi>G</mi></mrow></math></span> can be computed from WD of its subalgebra <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>/</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mi>H</mi></math></span> is a normal subgroup of <span><math><mi>G</mi></math></span> of prime order and <span><math><mrow><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span> for some prime <span><math><mi>p</mi></math></span> and positive integer <span><math><mi>k</mi></math></span>. We extend this result to any normal subgroup <span><math><mi>H</mi></math></span> of <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100105"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203800","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
General solutions and applications of the coupled Drinfel’d–Sokolov–Wilson equation 耦合Drinfel'd-Sokolov-Wilson方程的通解及其应用
Examples and Counterexamples Pub Date : 2023-11-01 Epub Date: 2023-04-04 DOI: 10.1016/j.exco.2023.100108
Shreya Mitra , A. Ghose-Choudhury , Sudip Garai
{"title":"General solutions and applications of the coupled Drinfel’d–Sokolov–Wilson equation","authors":"Shreya Mitra ,&nbsp;A. Ghose-Choudhury ,&nbsp;Sudip Garai","doi":"10.1016/j.exco.2023.100108","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100108","url":null,"abstract":"<div><p>We report a new batch of wave solutions for the coupled Drinfel’d–Sokolov–Wilson equation which represents a coupled system of nonlinear partial differential equations (NLPDEs). Firstly by making a travelling wave ansatz, we decouple the system and obtain a second-order ordinary differential equation (ODE). Thereafter we perform phase space and bifurcation analysis of that second-order ODE and proceed to construct the general solution for the envelope of the wave packet. The solutions are expressed in terms of the Jacobi elliptic sine function from which one can obtain solitary wave (particular) solutions by imposing appropriate conditions on the roots of certain quartic polynomials as discussed thereafter.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100108"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203802","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
Algebraic constructions of group divisible designs 群可分设计的代数构造
Examples and Counterexamples Pub Date : 2023-11-01 Epub Date: 2022-12-07 DOI: 10.1016/j.exco.2022.100094
Shyam Saurabh , Kishore Sinha
{"title":"Algebraic constructions of group divisible designs","authors":"Shyam Saurabh ,&nbsp;Kishore Sinha","doi":"10.1016/j.exco.2022.100094","DOIUrl":"https://doi.org/10.1016/j.exco.2022.100094","url":null,"abstract":"<div><p>Some series of Group divisible designs using generalized Bhaskar Rao designs over Dihedral, Symmetric and Alternating groups are obtained.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100094"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203720","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
Families of finite sets in which no set is covered by the union of the others 有限集族，其中没有集被其他集的并集覆盖
Examples and Counterexamples Pub Date : 2023-11-01 Epub Date: 2022-12-10 DOI: 10.1016/j.exco.2022.100095
Guillermo Alesandroni
{"title":"Families of finite sets in which no set is covered by the union of the others","authors":"Guillermo Alesandroni","doi":"10.1016/j.exco.2022.100095","DOIUrl":"https://doi.org/10.1016/j.exco.2022.100095","url":null,"abstract":"<div><p>Let <span><math><mi>ℱ</mi></math></span> be a finite nonempty family of finite nonempty sets. We prove the following: (1) <span><math><mi>ℱ</mi></math></span> satisfies the condition of the title if and only if for every pair of distinct subfamilies <span><math><mrow><mo>{</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>}</mo></mrow></math></span>, <span><math><mrow><mo>{</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>}</mo></mrow></math></span> of <span><math><mi>ℱ</mi></math></span>, <span><math><mrow><munderover><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></munderover><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><munderover><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></munderover><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></math></span>. (2) If <span><math><mi>ℱ</mi></math></span> satisfies the condition of the title, then the number of subsets of <span><math><mrow><munder><mrow><mo>⋃</mo></mrow><mrow><mi>A</mi><mo>∈</mo><mi>ℱ</mi></mrow></munder><mi>A</mi></mrow></math></span> containing at least one set of <span><math><mi>ℱ</mi></math></span> is odd. We give two applications of these results, one to number theory and one to commutative algebra.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100095"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203716","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: 4
Minimal counterexamples to Hendrickson’s conjecture on globally rigid graphs Hendrickson关于全局刚性图猜想的极小反例
Examples and Counterexamples Pub Date : 2023-11-01 Epub Date: 2023-04-06 DOI: 10.1016/j.exco.2023.100106
Georg Grasegger
{"title":"Minimal counterexamples to Hendrickson’s conjecture on globally rigid graphs","authors":"Georg Grasegger","doi":"10.1016/j.exco.2023.100106","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100106","url":null,"abstract":"<div><p>In this paper we consider the class of graphs which are redundantly <span><math><mi>d</mi></math></span>-rigid and <span><math><mrow><mo>(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-connected but not globally <span><math><mi>d</mi></math></span>-rigid, where <span><math><mi>d</mi></math></span> is the dimension. This class arises from counterexamples to a conjecture by Bruce Hendrickson. It seems that there are relatively few graphs in this class for a given number of vertices. Using computations we show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn><mo>,</mo><mn>5</mn></mrow></msub></math></span> is indeed the smallest counterexample to the conjecture.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100106"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203803","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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