Examples and Counterexamples最新文献

{"title":"An arithmetic term for the factorial function","authors":"Mihai Prunescu ,&nbsp;Lorenzo Sauras-Altuzarra","doi":"10.1016/j.exco.2024.100136","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100136","url":null,"abstract":"<div><p>As proved by Marchenkov and Mazzanti, every Kalmar function can be represented by arithmetic terms. We display one of such terms to represent the factorial function, and as a consequence, we get an example of an arithmetic term which represents a function whose image is the set of primes.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100136"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000028/pdfft?md5=14034c2031c53802d6653cf6837b9961&pid=1-s2.0-S2666657X24000028-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139674621","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":"Hexagonal finite differences for the two-dimensional variable coefficient Poisson equation","authors":"R. Itza Balam ,&nbsp;M. Uh Zapata ,&nbsp;U. Iturrarán-Viveros","doi":"10.1016/j.exco.2024.100144","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100144","url":null,"abstract":"<div><p>For many years, finite differences in hexagonal grids have been developed to solve elliptic problems such as the Poisson and Helmholtz equations. However, these schemes are limited to constant coefficients, which reduces their usefulness in many applications. The main challenge is accurately approximating the diffusive term. This paper presents examples of both successful and unsuccessful attempts to obtain accurate finite differences based on a hexagonal stencil with equilateral triangles to approximate two-dimensional Poisson equations. Local truncation error analysis reveals that a second-order scheme can be achieved if the derivative of the diffusive coefficient is included. Finally, we provide numerical examples to verify the accuracy of the proposed methods.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100144"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000107/pdfft?md5=bd79b118d40b9d2dc1de56be1a5d51b9&pid=1-s2.0-S2666657X24000107-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140645431","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":"Assemblies as semigroups","authors":"Ulderico Dardano ,&nbsp;Bruno Dinis ,&nbsp;Giuseppina Terzo","doi":"10.1016/j.exco.2024.100143","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100143","url":null,"abstract":"<div><p>In this paper we give an algebraic characterization of assemblies in terms of bands of groups. We also consider substructures and homomorphisms of assemblies. We give many examples and counterexamples.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100143"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000090/pdfft?md5=cb668f90949bc416ea00c880ec4aa3e0&pid=1-s2.0-S2666657X24000090-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140180647","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":"Trifferent codes with small lengths","authors":"Sascha Kurz","doi":"10.1016/j.exco.2024.100139","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100139","url":null,"abstract":"<div><p>A code <span><math><mrow><mi>C</mi><mo>⊆</mo><msup><mrow><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> of length <span><math><mi>n</mi></math></span> is called trifferent if for any three distinct elements of <span><math><mi>C</mi></math></span> there exists a coordinate in which they all differ. By <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> we denote the maximum cardinality of trifferent codes with length <span><math><mi>n</mi></math></span>. The values <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow><mo>=</mo><mn>10</mn></mrow></math></span> and <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow><mo>=</mo><mn>13</mn></mrow></math></span> were recently determined (Fiore et al., 2022). Here we determine <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mo>=</mo><mn>16</mn></mrow></math></span>, <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow><mo>=</mo><mn>20</mn></mrow></math></span>, and <span><math><mrow><mi>T</mi><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow><mo>=</mo><mn>27</mn></mrow></math></span>. For the latter case <span><math><mrow><mi>n</mi><mo>=</mo><mn>9</mn></mrow></math></span> there also exist linear codes attaining the maximum possible cardinality 27.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100139"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000053/pdfft?md5=6d4ca67bb2a4151b63492ee97290bf7c&pid=1-s2.0-S2666657X24000053-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139699378","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":"Ree groups as automorphism groups of block designs","authors":"Ashraf Daneshkhah","doi":"10.1016/j.exco.2024.100137","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100137","url":null,"abstract":"<div><p>A recent classification of flag-transitive 2-designs with parameters <span><math><mrow><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow></math></span> whose replication number <span><math><mi>r</mi></math></span> is coprime to <span><math><mi>λ</mi></math></span> gives rise to eight possible infinite families of 2-designs, some of which are with new parameters. In this note, we give explicit constructions for two of these families of 2-designs, and show that for a given positive integer <span><math><mrow><mi>q</mi><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⩾</mo><mn>27</mn></mrow></math></span>, there exist 2-designs with parameters <span><math><mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></mrow></math></span>, for <span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></math></span>, admitting the Ree group <span><math><mrow><msup><mrow></mrow><mrow><mn>2</mn></mrow></msup><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow></mrow></math></span> as their automorphism groups.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100137"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X2400003X/pdfft?md5=874ac10905c9399343d40e6310933a30&pid=1-s2.0-S2666657X2400003X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139674623","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":"Infinite families of trees with equal spectral radius","authors":"Francesco Belardo,&nbsp;Maurizio Brunetti","doi":"10.1016/j.exco.2024.100138","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100138","url":null,"abstract":"<div><p>In this note we show that for each positive integer <span><math><mrow><mi>a</mi><mo>⩾</mo><mn>2</mn></mrow></math></span> there exist infinitely many trees whose spectral radius is equal to <span><math><msqrt><mrow><mn>2</mn><mi>a</mi></mrow></msqrt></math></span>. Such trees are obtained by replacing the central edge of the double star <span><math><mrow><mi>S</mi><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mn>2</mn><mi>a</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> with suitable bidegreed caterpillars.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100138"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000041/pdfft?md5=163e05dcfa0673ec0b2a9629bf2ab099&pid=1-s2.0-S2666657X24000041-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139674622","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":"Existence conditions for 2-periodic solutions to a non-homogeneous differential equations with piecewise constant argument","authors":"Mukhiddin I. Muminov ,&nbsp;Tirkash A. Radjabov","doi":"10.1016/j.exco.2024.100145","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100145","url":null,"abstract":"<div><p>This paper provides a method of finding 2-periodical solutions for the first-order non-homogeneous differential equations with piecewise constant arguments. All existence conditions are described for 2-periodical solutions and obtained explicit formula for these solutions. An example for the problem that has infinitely many solutions is constructed.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100145"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000119/pdfft?md5=fe96a491c51b2a83cd2df8741cd75203&pid=1-s2.0-S2666657X24000119-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140807543","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":"A closer look at some new lower bounds on the minimum singular value of a matrix","authors":"Avleen Kaur ,&nbsp;S.H. Lui","doi":"10.1016/j.exco.2024.100142","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100142","url":null,"abstract":"<div><p>There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, <span><math><mrow><mi>P</mi><mo>+</mo><mi>Q</mi></mrow></math></span>, in relation to the eigenvalues of <span><math><mi>P</mi></math></span> and <span><math><mi>Q</mi></math></span>. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mo>min</mo></mrow></msub><mrow><mo>(</mo><mi>P</mi><mo>+</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span>, under the conditions that matrices <span><math><mi>P</mi></math></span> and <span><math><mi>Q</mi></math></span> are symmetric positive semi-definite and their sum <span><math><mrow><mi>P</mi><mo>+</mo><mi>Q</mi></mrow></math></span> is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices <span><math><mi>P</mi></math></span> and <span><math><mi>Q</mi></math></span>, which are denoted by <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span>, respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. One significant aspect of the two novel lower bounds on <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mo>min</mo></mrow></msub><mrow><mo>(</mo><mi>P</mi><mo>+</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> is the distinction of the case where <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>P</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span> have no principal angles between 0 and <span><math><mfrac><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span>, which is employed to formulate the lower bounds for <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mo>min</mo></mrow></msub><mrow><mo>(</mo><mi>P</mi><mo>+</mo><mi>Q</mi><mo>)</mo></mrow></mrow></math></span>. At last, an example that showcases the potential for improving these two lower bounds is presented.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100142"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000089/pdfft?md5=0c2ae22f7c329a636b6ee13795d2840d&pid=1-s2.0-S2666657X24000089-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139985367","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":"Effect of local fractional derivatives on Riemann curvature tensor","authors":"Muhittin Evren Aydin","doi":"10.1016/j.exco.2023.100134","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100134","url":null,"abstract":"<div><p>In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated <span><math><mrow><mi>M</mi><mo>−</mo></mrow></math></span> and <span><math><mrow><mi>V</mi><mo>−</mo></mrow></math></span>fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100134"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X23000368/pdfft?md5=cdf1658da9063967d38270f28537f406&pid=1-s2.0-S2666657X23000368-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139100860","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":"Corrigendum to “On the conjecture of Sombor energy of a graph” [Examples and Counterexamples 3 (2023) 100115]","authors":"Harishchandra S. Ramane ,&nbsp;Deepa V. Kitturmath","doi":"10.1016/j.exco.2024.100135","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100135","url":null,"abstract":"<div><p>In this corrigendum, we correct some errors in the proof of Theorem 2.1 in “On the conjecture of Sombor energy of a graph” [Examples and Counterexamples 3 (2023) 100115].</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100135"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000016/pdfft?md5=9eaf8b07dc7392eca6de0037bd6665cc&pid=1-s2.0-S2666657X24000016-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139653566","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}