Examples and Counterexamples最新文献

{"title":"Small examples of mosaics of combinatorial designs","authors":"Vedran Krčadinac","doi":"10.1016/j.exco.2024.100163","DOIUrl":"10.1016/j.exco.2024.100163","url":null,"abstract":"<div><div>We give the first example of a mosaic of three combinatorial designs with distinct parameters 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>, and 2-<span><math><mrow><mo>(</mo><mn>13</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow></math></span>. Furthermore, we give examples of mosaics of 2-<span><math><mrow><mo>(</mo><mn>9</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span> designs that are not resolvable, thereby answering a question posed by M. Wiese and H. Boche. Finally, we give an example of a mosaic of projective planes of order 3 that cannot be obtained by tiling groups with difference sets.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100163"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142528855","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":"New Andrews–Curtis trivializations for Miller–Schupp group presentations","authors":"Alexei Lisitsa","doi":"10.1016/j.exco.2024.100168","DOIUrl":"10.1016/j.exco.2024.100168","url":null,"abstract":"<div><div>We present recent developments in the applications of automated theorem proving in the investigation of the Andrews–Curtis conjecture. We demonstrate previously unknown trivializations of group presentations from a parametric family <span><math><mrow><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>)</mo></mrow></mrow></math></span> of trivial group presentations for <span><math><mrow><mi>n</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>8</mn></mrow></math></span> (subset of well-known Miller–Schupp family). Based on the human analysis of these trivializations we formulate two conjectures on the structure of simplifications for the infinite family <span><math><mrow><mi>M</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>w</mi></mrow><mrow><mo>∗</mo></mrow></msub><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100168"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662768","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":"An example for application of Lax–Milgram’s theorem and Riesz–Schauder’s theorem","authors":"Tujin Kim","doi":"10.1016/j.exco.2024.100169","DOIUrl":"10.1016/j.exco.2024.100169","url":null,"abstract":"<div><div>In this note reviewing Lax–Migram’s theorem, we show an example of its application to prove the existence of a solution to an equation in complex Hilbert space arising in the field of electromagnetic heating.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100169"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160099","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 effect of different samplings to the solution of parametric PDE eigenvalue problems","authors":"Daniele Boffi ,&nbsp;Abdul Halim ,&nbsp;Gopal Priyadarshi","doi":"10.1016/j.exco.2024.100170","DOIUrl":"10.1016/j.exco.2024.100170","url":null,"abstract":"<div><div>The use of sparse sampling is a consolidated technique for the reduced order modeling of parametric PDEs. In this note we investigate the choice of sampling points within the framework of reduced order techniques for the approximation of eigenvalue problems originating from parametric PDEs. We use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerkin orthogonal technique to get the reduced problem. We present some numerical results and observe that, as in the case of the source problem, also for eigenvalue problems the use of sparse sampling is a good idea and that, when the number of sampling points is assigned, sparse sampling provides better results than uniform sampling.</div><div>In the spirit of the journal, we present our results in the form of examples and counterexamples.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100170"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160098","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 influence of non-linearity within two-phase poro-elasticity: Numerical examples and counterexamples","authors":"Maximilian Brodbeck ,&nbsp;Franziska S. Egli ,&nbsp;Marlon Suditsch ,&nbsp;Seyed Morteza Seyedpour ,&nbsp;Tim Ricken","doi":"10.1016/j.exco.2024.100167","DOIUrl":"10.1016/j.exco.2024.100167","url":null,"abstract":"<div><div>Porous materials can be described either by Biot’s consolidation theory or the Theory of Porous Media (TPM). Depending on the loading regime, permeability or compressibility of the solid matrix, either small or finite deformations occur. Numerical solution procedures for the case of finite deformation are prone to instabilities and computationally costly. Simplified models assuming small deformations increase stability of the solution process. Within this work, limitations of two simplified models in comparison with the fully non-linear TPM are studied. Therefore a Mandel-like problem is considered. Differences arise especially for rapid consolidation processes and for elongations larger than 3%. It can be further shown, that the simplified models have an inherent mass error.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100167"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142571811","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":"Exact solution of a mathematical model for human muscular motion","authors":"Motlatsi Molati","doi":"10.1016/j.exco.2024.100160","DOIUrl":"10.1016/j.exco.2024.100160","url":null,"abstract":"<div><div>An ordinary differential equation (ODE) which models human muscular movement is considered. A functional form of the model parameter is specified through the Lie symmetry approach, yielding a different expression from the one derived in the previous study (Kosugi et al., 2019). The Lie point symmetries corresponding to the model parameter are employed for derivation of exact solution.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100160"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416532","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":"An advection–diffusion equation with a generalized advection term: Well-posedness analysis and examples","authors":"Tetyana Malysheva ,&nbsp;Luther W. White","doi":"10.1016/j.exco.2024.100159","DOIUrl":"10.1016/j.exco.2024.100159","url":null,"abstract":"<div><div>We consider a Cauchy–Dirichlet problem for a semilinear advection–diffusion equation with a generalized advection term. Specific examples include an incision–diffusion landscape evolution model and a viscous Hamilton–Jacobi equation with an absorbing gradient term. We establish existence and uniqueness of a weak solution and its continuous dependence on initial data and a source term.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100159"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416533","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 “Dynamical behavior of fractional order SEIR epidemic model with multiple time delays and its stability analysis” [Examples and Counterexamples 4 (2023) 100128]","authors":"Subrata Paul ,&nbsp;Animesh Mahata ,&nbsp;Supriya Mukherjee ,&nbsp;Prakash Chandra Mali ,&nbsp;Banamali Roy","doi":"10.1016/j.exco.2024.100149","DOIUrl":"10.1016/j.exco.2024.100149","url":null,"abstract":"","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100149"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143127956","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":"Titchmarsh’s-type theorem for two-sided quaternion Fourier transform and sharp Hausdorff–Young inequality for quaternion linear canonical transform","authors":"Mawardi Bahri","doi":"10.1016/j.exco.2024.100154","DOIUrl":"10.1016/j.exco.2024.100154","url":null,"abstract":"<div><p>In this work, we first introduce the two-sided quaternion Fourier transform and demonstrate its essential properties. We generalize Titchmarsh’s-type theorem in the framework of the two-sided quaternion Fourier transform. Based on the interaction between the quaternion Fourier transform and quaternion linear canonical transform we explore sharp Hausdorff–Young inequality for the quaternion linear canonical transform. The obtained result can be considered as a generalized version of sharp Hausdorff–Young inequality for the two-dimensional quaternion Fourier transformation in the literature.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100154"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X2400020X/pdfft?md5=c0ae250d9b341ddca5ab6361f0c8f3be&pid=1-s2.0-S2666657X2400020X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141951684","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 Some families of Path-related graphs with their edge metric dimension","authors":"Lianglin Li,&nbsp;Shu Bao,&nbsp;Hassan Raza","doi":"10.1016/j.exco.2024.100152","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100152","url":null,"abstract":"<div><p>Locating the origin of diffusion in complex networks is an interesting but challenging task. It is crucial for anticipating and constraining the epidemic risks. Source localization has been considered under many feasible models. In this paper, we study the localization problem in some path-related graphs and study the edge metric dimension. A subset <span><math><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>⊆</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msub></mrow></math></span> is known as an edge metric generator for <span><math><mi>G</mi></math></span> if, for any two distinct edges <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mi>E</mi></mrow></math></span>, there exists a vertex <span><math><mrow><mi>a</mi><mo>⊆</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>E</mi></mrow></msub></mrow></math></span> such that <span><math><mrow><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>a</mi><mo>)</mo></mrow><mo>≠</mo><mi>d</mi><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>a</mi><mo>)</mo></mrow></mrow></math></span>. An edge metric generator that contains the minimum number of vertices is termed an edge metric basis for <span><math><mi>G</mi></math></span>, and the number of vertices in such a basis is called the edge metric dimension, denoted by <span><math><mrow><mi>d</mi><mi>i</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>e</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. An edge metric generator with the fewest vertices is called an edge metric basis for <span><math><mi>G</mi></math></span>. The number of vertices in such a basis is the edge metric dimension, represented as <span><math><mrow><mi>d</mi><mi>i</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>e</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. In this paper, the edge metric dimension of some path-related graphs is computed, namely, the middle graph of path <span><math><mrow><mi>M</mi><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and the splitting graph of path <span><math><mrow><mi>S</mi><mrow><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100152"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000181/pdfft?md5=c2a7ee3861dc78370607917771d98ef6&pid=1-s2.0-S2666657X24000181-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141605065","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}