Examples and Counterexamples最新文献

{"title":"Example of a solution for Dorodnitzyn’s limit formula","authors":"C.V. Valencia-Negrete","doi":"10.1016/j.exco.2023.100114","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100114","url":null,"abstract":"<div><p>In the present paper, we show an example of a solution for Dorodnitzyn’s gaseous boundary layer limit formula. Oleinik’s <em>no back-flow</em> condition ensures the existence and uniqueness of solutions for the Prandtl equations in a rectangular domain <span><math><mrow><mi>R</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>. It also allowed us to find a limit formula for Dorodnitzyn’s stationary compressible boundary layer with constant total energy on a bounded convex domain in the plane <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Under the same assumption, we can give an approximate solution <span><math><mi>u</mi></math></span> for the limit formula if <span><math><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow><mo>&lt;</mo><mspace></mspace><mspace></mspace><mo>&lt;</mo><mspace></mspace><mspace></mspace><mo>&lt;</mo><mn>1</mn></mrow></math></span> such that: <span><span><span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>≅</mo><mi>δ</mi><mo>∗</mo><mi>c</mi><mo>∗</mo><mfenced><mrow><mi>z</mi><mo>+</mo><mfrac><mrow><mn>6</mn></mrow><mrow><mn>25</mn></mrow></mfrac><mi>⋅</mi><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>i</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></mfrac><mi>⋅</mi><mfrac><mrow><mn>4</mn><msup><mrow><mi>U</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac><msup><mrow><mi>z</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfenced><mo>+</mo><mi>o</mi><mrow><mo>(</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>that corresponds to an approximate horizontal velocity component when a small parameter <span><math><mi>ϵ</mi></math></span> given by the quotient of the maximum height of the domain divided by its length tends to zero. Here, <span><math><mrow><mi>c</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>, <span><math><mi>δ</mi></math></span> is the boundary layer’s height in Dorodnitzyn’s coordinates, <span><math><mi>U</mi></math></span> is the <em>free-stream</em> velocity at the upper boundary of the domain, and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is the absolute surface temperature.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100114"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203794","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}

{"title":"A condition for associativity of univariate resultants","authors":"D.A. Wolfram","doi":"10.1016/j.exco.2023.100113","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100113","url":null,"abstract":"<div><p>We provide a condition on the occurrences of variables in polynomials and show that it ensures associativity of univariate resultants in non-trivial cases. We give examples involving transformations and arithmetic with the zeros of polynomials. Associativity enables the composition of functions on the zeros of a polynomial by using resultants. The result is generalised to finite systems of polynomials.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100113"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50169005","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}

{"title":"Number of terms in the group determinant","authors":"Naoya Yamaguchi,&nbsp;Yuka Yamaguchi","doi":"10.1016/j.exco.2023.100112","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100112","url":null,"abstract":"<div><p>In this paper, we prove that when the number of terms in the group determinant of order odd prime <span><math><mi>p</mi></math></span> is divided by <span><math><mi>p</mi></math></span>, the remainder is 1. In addition, we give a table of the number of terms in <span><math><mi>k</mi></math></span>th power of the group determinant of the cyclic group of order <span><math><mi>n</mi></math></span> for <span><math><mrow><mi>n</mi><mo>≤</mo><mn>10</mn></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≤</mo><mn>6</mn></mrow></math></span>, and also give a table of one for every group of order at most 15. These tables raise some questions for us about the number of terms in the group determinants.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100112"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50169006","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}

{"title":"Some counterexamples for (strong) persistence property and (nearly) normally torsion-freeness","authors":"Mehrdad Nasernejad","doi":"10.1016/j.exco.2023.100111","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100111","url":null,"abstract":"<div><p>In this article, we present some rare counterexamples, which are related to (strong) persistence property and (nearly) normally torsion-freeness of monomial ideals. They may be useful for researchers in this field to construct the other counterexamples refuting some conjectures.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100111"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203797","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}

{"title":"Ulam–Hyers stability of some linear differential equations of second order","authors":"Idriss Ellahiani,&nbsp;Belaid Bouikhalene","doi":"10.1016/j.exco.2023.100110","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100110","url":null,"abstract":"<div><p>In this work we prove the Ulam–Hyers stability of the following equation <span><span><span><math><mrow><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msup><mrow><mi>ϕ</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>γ</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></math></span></span></span>where <span><math><mi>γ</mi></math></span> is a real number. The main purpose is to find a solution <span><math><mi>ϕ</mi></math></span> of (E) satisfying <span><math><mrow><mrow><mo>|</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>⩽</mo><mi>K</mi><mi>ɛ</mi></mrow></math></span>, where <span><math><mi>K</mi></math></span> is Ulam–Hyers-Stability constant and <span><math><mi>f</mi></math></span> is an exact solution of the associated inequality <span><span><span><math><mrow><mrow><mo>|</mo><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mi>γ</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>γ</mi><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>⩽</mo><mi>ɛ</mi><mo>,</mo></mrow></math></span></span></span>for any <span><math><mrow><mi>ɛ</mi><mo>&gt;</mo><mn>0</mn></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100110"},"PeriodicalIF":0.0,"publicationDate":"2023-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203796","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}

{"title":"A family of examples of harmonic maps into the sphere with one point singularity","authors":"Nobumitsu Nakauchi","doi":"10.1016/j.exco.2023.100107","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100107","url":null,"abstract":"<div><p>The radial map <span><math><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi></mrow><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow></mfrac></mrow></math></span> is a well-known example of a harmonic map into the spheres with a point singularity at <span><math><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></math></span>. In our previous paper (Misawa and Nakauchi, 2022) we give two examples of harmonic maps into the standard spheres of higher dimension with a singularity of a polynomial of <span><math><mrow><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow></mfrac><mo>,</mo><mspace></mspace><mo>⋯</mo><mspace></mspace><mfrac><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow><mrow><mo>‖</mo><mi>x</mi><mo>‖</mo></mrow></mfrac></mrow></math></span> of degree 2 and degree 3 respectively. In Fujioka (2020) uses our arguments to give an example of a harmonic map into the sphere with a singularity of a polynomial of degree 4. In this paper we give a family of examples of harmonic maps with a point singularity of a polynomial of higher degree.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100107"},"PeriodicalIF":0.0,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203801","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}

{"title":"Examples of identities and inequalities for the nonlinear term in the Navier–Stokes equation","authors":"Jorge Reyes","doi":"10.1016/j.exco.2023.100109","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100109","url":null,"abstract":"<div><p>In this paper, several examples of novel equations and inequalities for various forms of the non-linear term in the Navier–Stokes equations (NSE) are provided. The NSE are formulated from the conservation of linear momentum and mass conservation. However, they are also known to conserve energy, angular momentum, enstrophy in 2D, helicity in 3D, among other important physical quantities (Gresho and Sani, 1998) <span>[1]</span>. Depending on the desired quantity of interest, there are various representations of nonlinear term <span><math><mrow><mi>u</mi><mi>⋅</mi><mo>∇</mo><mi>u</mi></mrow></math></span> (e.g. convective, skew symmetric, rotational etc.) that can be implemented.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100109"},"PeriodicalIF":0.0,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203795","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}

{"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":"2023-04-06","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}

{"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":"2023-04-04","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}

{"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":"2023-04-04","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}