Examples and Counterexamples最新文献

{"title":"Non-existence of perturbed solutions under a second-order sufficient condition","authors":"Gerd Wachsmuth","doi":"10.1016/j.exco.2023.100122","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100122","url":null,"abstract":"<div><p>We present an optimization problem in infinite dimensions which satisfies the usual second-order sufficient condition but for which perturbed problems fail to possess solutions.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100122"},"PeriodicalIF":0.0,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49883264","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":"Counterexamples for Noise Models of Stochastic Gradients","authors":"Vivak Patel","doi":"10.1016/j.exco.2023.100123","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100123","url":null,"abstract":"<div><p>Stochastic Gradient Descent (SGD) is a widely used, foundational algorithm in data science and machine learning. As a result, analyses of SGD abound making use of a variety of assumptions, especially on the noise behavior of the stochastic gradients. While recent works have achieved a high-degree of generality on assumptions about the noise behavior of the stochastic gradients, it is unclear that such generality is necessary. In this work, we construct a simple example that shows that less general assumptions will be violated, while the most general assumptions will hold.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100123"},"PeriodicalIF":0.0,"publicationDate":"2023-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882666","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":"On the fractional Allee logistic equation in the Caputo sense","authors":"I. Area ,&nbsp;Juan J. Nieto","doi":"10.1016/j.exco.2023.100121","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100121","url":null,"abstract":"<div><p>In the framework of population models, logistic growth and fractional logistic growth has been analyzed. In some situations the so-called Allee effect gives more accurate approximation. In this work, fractional Allee differential equation in the Caputo sense is considered. The solution is obtained by considering formal power series. Numerical computations are presented to compare the truncating series with the classical Allee differential equation.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100121"},"PeriodicalIF":0.0,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882667","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":"Infinitesimal phase response functions can be misleading","authors":"Christoph Börgers","doi":"10.1016/j.exco.2023.100120","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100120","url":null,"abstract":"<div><p>Phase response functions are the central tool in the mathematical analysis of pulse-coupled oscillators. When an oscillator receives a brief input pulse, the phase response function specifies how its phase shifts as a function of the phase at which the input is received. When the pulse is weak, it is customary to linearize around zero pulse strength. The result is called the <em>infinitesimal</em> phase response function. These ideas have been used extensively in theoretical biology, and also in some areas of engineering. I give examples showing that the infinitesimal phase response function may predict that two oscillators, as they exchange pulses back and fourth, will converge to synchrony, yet this is false when the exact phase response function is used, for all positive interaction strengths. For short, the analogue of the Hartman–Grobman theorem that one might expect to hold at first sight is invalid. I give a condition under which the prediction derived using the infinitesimal phase response function does hold for the exact phase response function when interactions are sufficiently weak but of positive strength. However, I argue that this condition may often fail to hold.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100120"},"PeriodicalIF":0.0,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882573","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":"Diagonalization of the cross-product matrix","authors":"Oskar Maria Baksalary ,&nbsp;Götz Trenkler","doi":"10.1016/j.exco.2023.100118","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100118","url":null,"abstract":"<div><p>The paper considers diagonalization of the cross-product matrices, i.e., skew-symmetric matrices of order three. A procedure to determine a nonsingular matrix, which yields the diagonalization is indicated. Furthermore, a method to derive the inverse of a diagonalizing matrix is proposed by means of a formula for the Moore–Penrose inverse of any matrix, which is columnwise partitioned into two matrices having disjoint ranges. This rather nonstandard method to obtain the inverse of a nonsingular matrix is appealing, as it can be applied to any diagonalizing matrix, and not only of those originating from diagonalization of the cross-product matrices. The paper provides also comments and examples demonstrating applicability of the diagonalization procedure to calculate roots of a cross-product matrix.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100118"},"PeriodicalIF":0.0,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882668","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":"New example of strongly regular graph with parameters (81,30,9,12) and a simple group A5 as the automorphism group","authors":"Dean Crnković,&nbsp;Andrea Švob","doi":"10.1016/j.exco.2023.100119","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100119","url":null,"abstract":"<div><p>A new strongly regular graph with parameters <span><math><mrow><mo>(</mo><mn>81</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>9</mn><mo>,</mo><mn>12</mn><mo>)</mo></mrow></math></span> is found as a graph invariant under certain subgroup of the full automorphism group of the previously known strongly regular graph discovered in 1981 by J. H. van Lint and A. Schrijver.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100119"},"PeriodicalIF":0.0,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882574","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":"Solving recurrences for Legendre–Bernstein basis transformations","authors":"D.A. Wolfram","doi":"10.1016/j.exco.2023.100117","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100117","url":null,"abstract":"<div><p>The change of basis matrix <span><math><mi>M</mi></math></span> from shifted Legendre to Bernstein polynomials and <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> have applications in computer graphics. Algorithms use their properties to find the matrix elements efficiently. We give new functions for the elements of <span><math><mi>M</mi></math></span> as a summation, and a complete hypergeometric function. We find that Gosper’s algorithm does not produce closed-form expressions for the elements of either <span><math><mi>M</mi></math></span> or <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. Zeilberger’s algorithm produces four second-order recurrences for the elements of the matrices that enable them to be computed in <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> time, and for the derivation of closed-form functions by row and column for the elements. Two row recurrences are special cases of those found by Woźny (2013) who used a different method. We show that the recurrences for rows of <span><math><mi>M</mi></math></span> and columns of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> are equivalent. The recurrences for columns of <span><math><mi>M</mi></math></span> and rows of <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> generate functions that are the Lagrange interpolation polynomials of their elements. These polynomials are equal to hypergeometric functions, which are solutions of the recurrences.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"4 ","pages":"Article 100117"},"PeriodicalIF":0.0,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49882576","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 new doubly resolvable candelabra quadruple systems","authors":"Zhaoping Meng ,&nbsp;Qingling Gao","doi":"10.1016/j.exco.2023.100116","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100116","url":null,"abstract":"<div><p>Two resolutions of the same design are said to be orthogonal when each parallel class of one resolution has at most one block in common with each parallel class of the other resolution. If a candelabra quadruple system has two mutually orthogonal resolutions, the design is called doubly resolvable candelabra quadruple system and denoted by DRCQS. In this paper, we obtain a DRCQS<span><math><mrow><mo>(</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>5</mn></mrow></msup><mo>:</mo><mn>1</mn><mo>)</mo></mrow></math></span> by computer search.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100116"},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203805","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":"Sum structures in abelian groups","authors":"Robert Haas","doi":"10.1016/j.exco.2023.100101","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100101","url":null,"abstract":"<div><p>Any set <span><math><mi>S</mi></math></span> of elements from an abelian group produces a graph with colored edges <span><math><mi>G</mi></math></span>(S), with its points the elements of <span><math><mi>S</mi></math></span>, and the edge between points <span><math><mi>P</mi></math></span> and <span><math><mi>Q</mi></math></span> assigned for its “color” the sum <span><math><mrow><mi>P</mi><mo>+</mo><mi>Q</mi></mrow></math></span>. Since any pair of identically colored edges is equivalent to an equation <span><math><mrow><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><msup><mrow><mi>Q</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>, the geometric—combinatorial figure <span><math><mi>G</mi></math></span>(S) is thus equivalent to a system of linear equations. This article derives elementary properties of such “sum cographs”, including forced or forbidden configurations, and then catalogues the 54 possible sum cographs on up to 6 points. Larger sum cograph structures also exist: Points <span><math><mrow><mo>{</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow></math></span> in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> close up into a “Fibonacci cycle”–i.e. <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>k</mi></mrow></math></span>, <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math></span> for all integers <span><math><mrow><mi>i</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, and then <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>–provided that <span><math><mrow><mi>m</mi><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> is a Lucas prime, in which case actually <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msup><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow></math></span> for all <span><math><mrow><mi>i</mi><mo>≥</mo><mn>0</mn></mrow></math></span>.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100101"},"PeriodicalIF":0.0,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203806","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":"On the conjecture of Sombor energy of a graph","authors":"Harishchandra S. Ramane,&nbsp;Deepa V. Kitturmath","doi":"10.1016/j.exco.2023.100115","DOIUrl":"https://doi.org/10.1016/j.exco.2023.100115","url":null,"abstract":"<div><p>The Sombor matrix of a graph <span><math><mi>G</mi></math></span> with vertices <span><math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> is defined as <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>S</mi><mi>O</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>[</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msqrt><mrow><msubsup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>d</mi></mrow><mrow><mi>j</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msqrt></mrow></math></span> if <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is adjacent to <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> and <span><math><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span>, otherwise, where <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the degree of a vertex <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. The Sombor energy of a graph is defined as the sum of the absolute values of the eigenvalues of the Sombor matrix. N. Ghanbari (Ghanbari, 2022) conjectured that there is no graph with integer valued Sombor energy. In this paper we give some class of graphs for which this conjecture holds. Further we conjecture that there is no regular graph with adjacency energy equal to <span><math><mrow><mn>2</mn><mi>k</mi><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></math></span> where <span><math><mi>k</mi></math></span> is a positive integer.</p></div>","PeriodicalId":100517,"journal":{"name":"Examples and Counterexamples","volume":"3 ","pages":"Article 100115"},"PeriodicalIF":0.0,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50203710","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}