Examples and Counterexamples最新文献

{"title":"On an inequality related to the volume of a parallelepiped","authors":"Oskar Maria Baksalary ,&nbsp;Götz Trenkler","doi":"10.1016/j.exco.2024.100155","DOIUrl":"10.1016/j.exco.2024.100155","url":null,"abstract":"<div><p>The problem of establishing an upper bound for the volume of a parallelepiped is considered by utilizing an original approach involving a skew-symmetric matrix of order four (along with its Moore–Penrose inverse). It is shown that the commonly known inequality characterizing the bound can be virtually sharpened. Similarly, a sharpening is established with respect to the Cauchy–Schwarz inequality. General properties of the Moore–Penrose inverse of a skew-symmetric matrix are discussed as well.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100155"},"PeriodicalIF":0.0,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000211/pdfft?md5=133540ffa4027abba6ae5558b006daa1&pid=1-s2.0-S2666657X24000211-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141992966","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":"2024-08-05","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":"A dispersive dynamical system that is chain transitive","authors":"Eduardo C. Viscovini,&nbsp;Josiney A. Souza","doi":"10.1016/j.exco.2024.100153","DOIUrl":"10.1016/j.exco.2024.100153","url":null,"abstract":"<div><p>This paper compares the notions of Auslander generalized recurrence and chain recurrence of dynamical systems. Generalized recurrent points are chain recurrent, however, chain recurrent points may not be generalized recurrent of any order. Because of the absence of recursiveness, the dispersive systems have no generalized recurrent point. Examples of systems with generalized recurrent points and systems with no generalized recurrent points are presented. The main example shows a dispersive system that is chain transitive.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100153"},"PeriodicalIF":0.0,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000193/pdfft?md5=b5a91c98f35a5ef489f7935a208858b4&pid=1-s2.0-S2666657X24000193-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141622845","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":"2024-07-13","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}

{"title":"Quadrature based innovative techniques concerning nonlinear equations having unknown multiplicity","authors":"Farooq Ahmed Shah ,&nbsp;Muhammad Waseem","doi":"10.1016/j.exco.2024.100150","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100150","url":null,"abstract":"<div><p>Solution of nonlinear equations is one of the most frequently encountered issue in engineering and applied sciences. Most of the intricateed engineering problems are modeled in the frame work of nonlinear equation <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>.</mo></mrow></math></span> The significance of iterative algorithms executed by computers in resolving such functions is of paramount importance and undeniable in contemporary times. If we study the simple roots and the roots having multiplicity greater of any nonlinear equations we come to the point that finding the roots of nonlinear equations having multiplicity greater than one is not trivialvia classical iterative methods. Instability or slow convergence rate is faced by these methods, and also sometimes these methods diverge. In this article, we give some innovative and robust iterative techniques for obtaining the approximate solution of nonlinear equations having multiplicity <span><math><mrow><mi>m</mi><mo>&gt;</mo><mn>1</mn></mrow></math></span>. Quadrature formulas are implemented to obtain iterative techniques for finding roots of nonlinear equations having unknown multiplicity. The derived methods are the variants of modified Newton method with high order of convergence and better accuracy. The convergence criteria of the new techniques are studied by using Taylor series method. Some examples are tested for the sack of implementations of these techniques. Numerical and graphical comparison shows the performance and efficiency of these new techniques.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100150"},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000168/pdfft?md5=1401656be5f763d7bc65918548e7c152&pid=1-s2.0-S2666657X24000168-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594856","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":"Estimation of time delay functions for design of traffic systems","authors":"N. Hossam,&nbsp;U. Gazder","doi":"10.1016/j.exco.2024.100151","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100151","url":null,"abstract":"<div><p>Main aim of this research was to apply multiple approaches for the development of time delay functions on three highways in Bahrain, namely; Dry Dock Highway, Arad Highway and Zallaq Highway. Four equations were obtained from previous studies and two equations were, additionally, tailored for each of the three highways. The results were used to obtain two parameters that aid in design, optimum flow rate and level of service.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100151"},"PeriodicalIF":0.0,"publicationDate":"2024-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X2400017X/pdfft?md5=19e6cddc4e57c5ac2232cf8fa66a6629&pid=1-s2.0-S2666657X2400017X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594855","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":"Studies in fractal–fractional operators with examples","authors":"Rabha W. Ibrahim","doi":"10.1016/j.exco.2024.100148","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100148","url":null,"abstract":"<div><p>By using the generalization of the gamma function (<span><math><mi>p</mi></math></span>-gamma function: <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></math></span>), we introduce a generalization of the fractal–fractional calculus which is called <span><math><mi>p</mi></math></span>-fractal fractional calculus. We extend the proposed operators into the symmetric complex domain, specifically the open unit disk. Normalization for each operator is formulated. This allows us to explore the most important geometric properties. Examples are illustrated including the basic power functions.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100148"},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000144/pdfft?md5=eb86f085d4d25f908eda02f5243db74c&pid=1-s2.0-S2666657X24000144-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485923","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 extension of quadrant dependence","authors":"João Lita da Silva","doi":"10.1016/j.exco.2024.100146","DOIUrl":"https://doi.org/10.1016/j.exco.2024.100146","url":null,"abstract":"<div><p>In this short note, it is propounded an extension for quadrant dependence, and shown that some of the original proprieties of this popular concept remain valid, while others are necessarily generalized. A second Borel–Cantelli lemma due to Petrov (Statist. Probab. Lett. 58: 283–286, 2002) is revisited for events enjoying this new dependence notion and demonstrated by means of simpler arguments.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"5 ","pages":"Article 100146"},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000120/pdfft?md5=7ab76f0ec02449bb41d1ed97e3dbd4c2&pid=1-s2.0-S2666657X24000120-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242913","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":"2024-04-29","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":"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":"2024-04-26","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}