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An upper bound for the largest singular value of extended mixed graphs 扩展混合图的最大奇异值的上界
Examples and Counterexamples Pub Date : 2025-06-01 Epub Date: 2025-05-14 DOI: 10.1016/j.exco.2025.100188
Zoran Stanić
{"title":"An upper bound for the largest singular value of extended mixed graphs","authors":"Zoran Stanić","doi":"10.1016/j.exco.2025.100188","DOIUrl":"10.1016/j.exco.2025.100188","url":null,"abstract":"<div><div>An extended mixed graph contains edges declared to be either positive, or negative, or oriented from one end to the other. They unify concepts of mixed graphs, signed graphs, oriented graphs and ordinary graphs. We demonstrate a method that establishes an upper bound for the largest singular value of the corresponding adjacency matrix in a bit general setting allowing the existence of loops at vertices. An upper bound for the largest singular value of the Laplacian matrix is extracted. The obtained bounds remain valid for the largest modulus of the corresponding eigenvalues, and cover the aforementioned particular structures.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100188"},"PeriodicalIF":0.0,"publicationDate":"2025-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143948724","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
Halving interval guaranteed for Dekker and Brent root finding methods Dekker和Brent寻根方法保证的减半间隔
Examples and Counterexamples Pub Date : 2024-12-24 DOI: 10.1016/j.exco.2024.100173
Vilmar Steffen , Carlos Catusso Della Pasqua , Maiquiel Schmidt de Oliveira , Edson Antonio da Silva
{"title":"Halving interval guaranteed for Dekker and Brent root finding methods","authors":"Vilmar Steffen ,&nbsp;Carlos Catusso Della Pasqua ,&nbsp;Maiquiel Schmidt de Oliveira ,&nbsp;Edson Antonio da Silva","doi":"10.1016/j.exco.2024.100173","DOIUrl":"10.1016/j.exco.2024.100173","url":null,"abstract":"<div><div>Hybrid methods are widely used in many areas of applied mathematics. One of the simplest and most common problems in this field is root finding, for which various methods exist. Some of the most efficient approaches combine two or more techniques into hybrid methods. Among these are the Dekker and Brent methods, for which we propose a modification to ensure that the search interval is halved in each iteration. We apply this modification to two examples: a transcendental equation and a cubic equation of state. The results demonstrate that the proposed modifications guarantee at least interval halving and offer a slight improvement in the efficiency of the root-finding process.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"7 ","pages":"Article 100173"},"PeriodicalIF":0.0,"publicationDate":"2024-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291592","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
Comparison of different elastic strain definitions for largely deformed SEI of chemo-mechanically coupled silicon battery particles 化学机械耦合硅电池颗粒大变形 SEI 不同弹性应变定义的比较
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-09-13 DOI: 10.1016/j.exco.2024.100157
R. Schoof , G.F. Castelli , W. Dörfler
{"title":"Comparison of different elastic strain definitions for largely deformed SEI of chemo-mechanically coupled silicon battery particles","authors":"R. Schoof ,&nbsp;G.F. Castelli ,&nbsp;W. Dörfler","doi":"10.1016/j.exco.2024.100157","DOIUrl":"10.1016/j.exco.2024.100157","url":null,"abstract":"<div><p>Amorphous silicon is a highly promising anode material for next-generation lithium-ion batteries. Large volume changes of the silicon particle have a critical effect on the surrounding solid-electrolyte interphase (SEI) due to repeated fracture and healing during cycling. Based on a thermodynamically consistent chemo-elasto-plastic continuum model we investigate the stress development inside the particle and the SEI. Using the example of a particle with SEI, we apply a higher order finite element method together with a variable-step, variable-order time integration scheme on a nonlinear system of partial differential equations. Starting from a single silicon particle setting, the surrounding SEI is added in a first step with the typically used elastic Green–St-Venant (GSV) strain definition for a purely elastic deformation. For this type of deformation, the definition of the elastic strain is crucial to get reasonable simulation results. In case of the elastic GSV strain, the simulation aborts. We overcome the simulation failure by using the definition of the logarithmic Hencky strain. However, the particle remains unaffected by the elastic strain definitions in the particle domain. Compared to GSV, plastic deformation with the Hencky strain is straightforward to take into account. For the plastic SEI deformation, a rate-independent and a rate-dependent plastic deformation are newly introduced and numerically compared for three half cycles for the example of a radial symmetric particle.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100157"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000235/pdfft?md5=eb47ebf0227fdbfbf64434e6a7ff02b1&pid=1-s2.0-S2666657X24000235-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142228535","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 an inequality related to the volume of a parallelepiped 关于平行四边形体积的不等式
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-08-16 DOI: 10.1016/j.exco.2024.100155
Oskar Maria Baksalary , Götz Trenkler
{"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":"2024-12-01","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}
引用次数: 0
Counting graphs induced by Gauss diagrams and families of mutant alternating knots 高斯图诱导的计数图和突变交替结族
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-10-11 DOI: 10.1016/j.exco.2024.100162
Alexei Lisitsa , Alexei Vernitski
{"title":"Counting graphs induced by Gauss diagrams and families of mutant alternating knots","authors":"Alexei Lisitsa ,&nbsp;Alexei Vernitski","doi":"10.1016/j.exco.2024.100162","DOIUrl":"10.1016/j.exco.2024.100162","url":null,"abstract":"<div><div>The construction known as Gauss diagrams or Gauss words is one of the oldest in knot theory and has been studied extensively both in the context of knots and in the context of closed curves with self-intersections. When we studied graphs induced by Gauss diagrams, we produced all examples of these graphs of small sizes, and we published the number of these graphs as sequence A343358 in the OEIS. The aim of this article is to clarify several subtle theoretical points concerning A343358. Most importantly, we explain why our numbers, produced using graph-theoretical constructions, reflect the number of so-called mutant knots.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100162"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445039","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
Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey–Dodd–Gibbon equation 时间分数考德里-多德-吉本方程的对称分析、精确解和守恒定律
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-10-28 DOI: 10.1016/j.exco.2024.100166
Jicheng Yu , Yuqiang Feng
{"title":"Symmetry analysis, exact solutions and conservation laws of time fractional Caudrey–Dodd–Gibbon equation","authors":"Jicheng Yu ,&nbsp;Yuqiang Feng","doi":"10.1016/j.exco.2024.100166","DOIUrl":"10.1016/j.exco.2024.100166","url":null,"abstract":"<div><div>In this paper, Lie symmetry analysis method is applied to time fractional Caudrey–Dodd–Gibbon equation. We obtain a symmetric group spanned by two generators for the governing equation. The obtained group generators are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative, thereby getting one trivial solution and one convergent power series solution for the reduced equations. Then we present the dynamic behavior of the obtained analytical solutions graphically. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100166"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554185","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 Sombor index and graph energy of some chemically important graphs 论一些重要化学图的松博指数和图能
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-09-27 DOI: 10.1016/j.exco.2024.100158
Md Selim Reja, Sk. Md. Abu Nayeem
{"title":"On Sombor index and graph energy of some chemically important graphs","authors":"Md Selim Reja,&nbsp;Sk. Md. Abu Nayeem","doi":"10.1016/j.exco.2024.100158","DOIUrl":"10.1016/j.exco.2024.100158","url":null,"abstract":"<div><div>Sombor index of a graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is provided by the expression <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mi>v</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></msub><msqrt><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msqrt></mrow></math></span>, where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> is the degree of the vertex <span><math><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The energy of a graph is the quantity given by the total of the absolute values of its adjacency matrix’s eigenvalues. In this article, we improve the relation between the Sombor index and graph energy and derive the relation between them for unicyclic, bicyclic and tricyclic graphs, trees, triangular chain, square cactus chain and hexagonal cactus chain graphs. At last, we find the bounds of graph energy for zigzag and linear hexagonal chains.</div></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100158"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142356888","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
Estimation of time delay functions for design of traffic systems 为设计交通系统估算时延函数
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-06-30 DOI: 10.1016/j.exco.2024.100151
N. Hossam, U. Gazder
{"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":"2024-12-01","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}
引用次数: 0
Optimal boundary control in a micromixer 微搅拌器中的最佳边界控制
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-09-19 DOI: 10.1016/j.exco.2024.100156
Manuel Wegmann , Julius Jeßberger , Gudrun Thäter , Mathias J. Krause
{"title":"Optimal boundary control in a micromixer","authors":"Manuel Wegmann ,&nbsp;Julius Jeßberger ,&nbsp;Gudrun Thäter ,&nbsp;Mathias J. Krause","doi":"10.1016/j.exco.2024.100156","DOIUrl":"10.1016/j.exco.2024.100156","url":null,"abstract":"<div><p>This work studies mathematical foundations for optimal boundary control of mixers which mix two fluid phases. Existence of optima is proved and the influence of common objective functionals is discussed.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"6 ","pages":"Article 100156"},"PeriodicalIF":0.0,"publicationDate":"2024-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2666657X24000223/pdfft?md5=f6459a02db3942e0c3a62e7a8b3cbb8d&pid=1-s2.0-S2666657X24000223-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142272394","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
Studies in fractal–fractional operators with examples 分形-分数算子研究与实例
Examples and Counterexamples Pub Date : 2024-12-01 Epub Date: 2024-06-29 DOI: 10.1016/j.exco.2024.100148
Rabha W. Ibrahim
{"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":"2024-12-01","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}
引用次数: 0
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