Aequationes Mathematicae最新文献

{"title":"Homomorphisms from Functional Equations: The Goldie Equation, II","authors":"N. H. Bingham,&nbsp;A. J. Ostaszewski","doi":"10.1007/s00010-024-01130-9","DOIUrl":"10.1007/s00010-024-01130-9","url":null,"abstract":"<div><p>This first of three sequels to <i>Homomorphisms from Functional equations: The Goldie equation</i> (Ostaszewski in Aequationes Math 90:427–448, 2016) by the second author—the second of the resulting quartet—starts from the Goldie functional equation arising in the general regular variation of our joint paper (Bingham et al. in J Math Anal Appl 483:123610, 2020). We extend the work there in two directions. First, we algebraicize the theory, by systematic use of certain groups—the <i>Popa groups</i> arising in earlier work by Popa, and their relatives the <i>Javor groups </i>. Secondly, we extend from the original context on the real line to multi-dimensional (or infinite-dimensional) settings.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"1 - 19"},"PeriodicalIF":0.9,"publicationDate":"2024-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01130-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A simple proof of the rationality of Takagi-like functions","authors":"Yangyang Chen,&nbsp;Nankun Hong,&nbsp;Hongyuan Yu","doi":"10.1007/s00010-024-01134-5","DOIUrl":"10.1007/s00010-024-01134-5","url":null,"abstract":"<div><p>Takagi function is a well-known continuous but nowhere differentiable function defined over real numbers. The Takagi function maps rational numbers to themselves. In this note, by applying Euler’s theorem, we give a simple proof of this property for Takagi-like functions, a slight generalization of the Takagi function.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"433 - 437"},"PeriodicalIF":0.9,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On strongly m-convex stochastic processes","authors":"Jaya Bisht,&nbsp;Rohan Mishra,&nbsp;Abdelouahed Hamdi","doi":"10.1007/s00010-024-01128-3","DOIUrl":"10.1007/s00010-024-01128-3","url":null,"abstract":"<div><p>In this paper, we introduce the concept of strongly <i>m</i>-convex stochastic processes and present some basic properties of these stochastic processes. We derive Hermite-Hadamard type inequalities for stochastic processes whose first derivatives in absolute values are strongly <i>m</i>-convex. The results presented in this paper are a generalization and extension of previously known results.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"655 - 667"},"PeriodicalIF":0.9,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01128-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maker-Breaker resolving game played on corona products of graphs","authors":"Tijo James,&nbsp;Sandi Klavžar,&nbsp;Dorota Kuziak,&nbsp;Savitha K. S,&nbsp;Ambat Vijayakumar","doi":"10.1007/s00010-024-01132-7","DOIUrl":"10.1007/s00010-024-01132-7","url":null,"abstract":"<div><p>The Maker-Breaker resolving game is a game played on a graph <i>G</i> by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of <i>G</i>. The goal of Resolver is to select all the vertices in a resolving set of <i>G</i>, while that of Spoiler is to prevent this from happening. The outcome <i>o</i>(<i>G</i>) of the game played is one of <span>(mathcal {R})</span>, <span>(mathcal {S})</span>, and <span>(mathcal {N})</span>, where <span>(o(G)=mathcal {R})</span> (resp. <span>(o(G)=mathcal {S})</span>), if Resolver (resp. Spoiler) has a winning strategy no matter who starts the game, and <span>(o(G)=mathcal {N})</span>, if the first player has a winning strategy. In this paper, the game is investigated on corona products <span>(Godot H)</span> of graphs <i>G</i> and <i>H</i>. It is proved that if <span>(o(H)in {mathcal {N}, mathcal {S}})</span>, then <span>(o(Godot H) = mathcal {S})</span>. No such result is possible under the assumption <span>(o(H) = mathcal {R})</span>. It is proved that <span>(o(Godot P_k) = mathcal {S})</span> if <span>(k=5)</span>, otherwise <span>(o(Godot P_k) = mathcal {R})</span>, and that <span>(o(Godot C_k) = mathcal {S})</span> if <span>(k=3)</span>, otherwise <span>(o(Godot C_k) = mathcal {R})</span>. Several results are also given on corona products in which the second factor is of diameter at most 2.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1221 - 1233"},"PeriodicalIF":0.9,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01132-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuous inverse ambiguous functions on Lie groups","authors":"David Schmitz,&nbsp;Sadman Rahman,&nbsp;Anthony Kindness","doi":"10.1007/s00010-024-01131-8","DOIUrl":"10.1007/s00010-024-01131-8","url":null,"abstract":"<div><p>In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an inverse ambiguous function on a group <i>G</i> to be a bijective function <span>(f: G rightarrow G)</span> satisfying the functional equation <span>(f^{-1}(x) = f(x^{-1}))</span> for all <span>(x in G)</span>. In this paper, we investigate the existence of continuous inverse ambiguous functions on various Lie groups. In particular, we look at tori, elliptic curves over various fields, vector spaces, additive matrix groups, and multiplicative matrix groups.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1357 - 1369"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Iteration of a class of multi-plateau mappings","authors":"Jiaju Huang,&nbsp;Liu Liu,&nbsp;Jialing Zhang","doi":"10.1007/s00010-024-01129-2","DOIUrl":"10.1007/s00010-024-01129-2","url":null,"abstract":"<div><p>Iteration is one of the most important topics in the field of functional equations, which is also concerned with dynamical systems and operator theory. In this paper we discuss a class of multi-plateau mappings with finitely many P-intervals whose range crosses the diagonals and classifies them into four classes. For each class the properties of all the orbits of the two endpoints of the P-intervals are investigated in a similar way as for symbolic dynamical systems, and finally the iteration of these multi-plateau mappings are given.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1065 - 1084"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Jensen’s inequality involving divided differences of convex functions","authors":"G. Aras-Gazić,&nbsp;Julije Jakšetić,&nbsp;Rozarija Mikić,&nbsp;J. Pečarić","doi":"10.1007/s00010-024-01127-4","DOIUrl":"10.1007/s00010-024-01127-4","url":null,"abstract":"<div><p>This research is motivated by Zwick’s work on the convexity of divided differences and one more recent result on this topis with the aim of extending the use of the Edmundson-Lah-Ribaric inequality. Applications to Wulbert’s result for 3-convex functions and extensions to functions of two variables are provided in the paper.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1125 - 1141"},"PeriodicalIF":0.9,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Clique trees with a given zero forcing number maximizing the (A_alpha )-spectral radius","authors":"Long Jin,&nbsp;Jianxi Li,&nbsp;Yuan Hou","doi":"10.1007/s00010-024-01125-6","DOIUrl":"10.1007/s00010-024-01125-6","url":null,"abstract":"<div><p>The <span>(A_alpha )</span>-spectral radius of a graph <i>G</i> is the largest eigenvalue of <span>(A_alpha (G):=alpha D(G)+(1-alpha ) A(G))</span> for any real number <span>(alpha in [0,1])</span>, where <i>A</i>(<i>G</i>) and <i>D</i>(<i>G</i>) are the adjacency matrix and the degree matrix of <i>G</i>, respectively. In this paper, we settle the problem of characterizing graphs which attain the maximum <span>(A_alpha )</span>-spectral radius over <span>({mathscr {B}}(n, k))</span>, the class of clique trees of order <i>n</i> with a zero forcing number <i>k</i>, where <span>(0 le alpha &lt;1)</span>, <span>(leftlfloor frac{n}{2}rightrfloor +1 le k le n-1)</span> and each block is a clique of size at least 3. Moreover, an estimation on the <span>(A_alpha )</span>-spectral radius of the extremal graph is also included. Our result covers a recent result of Das (2023).</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"335 - 350"},"PeriodicalIF":0.9,"publicationDate":"2024-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The number of independent sets in bipartite graphs and benzenoids","authors":"Michael Han,&nbsp;Sycamore Herlihy,&nbsp;Kirsti Kuenzel,&nbsp;Daniel Martin,&nbsp;Rachel Schmidt","doi":"10.1007/s00010-024-01124-7","DOIUrl":"10.1007/s00010-024-01124-7","url":null,"abstract":"<div><p>Given a graph <i>G</i>, we study the number of independent sets in <i>G</i>, denoted <i>i</i>(<i>G</i>). This parameter is known as both the Merrifield–Simmons index of a graph as well as the Fibonacci number of a graph. In this paper, we give general bounds for <i>i</i>(<i>G</i>) when <i>G</i> is bipartite and we give its exact value when <i>G</i> is a balanced caterpillar. We improve upon a known upper bound for <i>i</i>(<i>T</i>) when <i>T</i> is a tree, and study a conjecture that all but finitely many positive integers represent <i>i</i>(<i>T</i>) for some tree <i>T</i>. We also give a recursive formula for finding <i>i</i>(<i>G</i>) when <i>G</i> is a linear chain of hexagons and use this to study the number of independent sets in benzenoids. We also answer a conjecture relating <i>i</i>(<i>G</i>) when <i>G</i> is a linear chain of hexagons and the number of <span>(2times n)</span> matrices containing a 1 in the top left entry where all entries are integer values and adjacent entries differ by at most 1.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1175 - 1195"},"PeriodicalIF":0.9,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01124-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144074142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"System of equations and configurations in the Euclidean space","authors":"Annachiara Korchmaros","doi":"10.1007/s00010-024-01114-9","DOIUrl":"10.1007/s00010-024-01114-9","url":null,"abstract":"<div><p>In the 3-dimensional Euclidean space <span>({textbf{E}}^3)</span>, fix six pairwise distinct points </p><div><div><span>$$begin{aligned} begin{array}{ccc} A=(a_1,a_2,a_3), &amp; B=(b_1,b_2,b_3), &amp; C=(c_1,c_2,c_3), D=(d_1,d_2,d_3), &amp; E=(e_1,e_2,e_3), &amp; F=(f_1,f_2,f_3) end{array} end{aligned}$$</span></div></div><p>together with two further points <span>(X^*=(x_1^*,x_2^*,x_3^*))</span> and <span>(Y^*=(y_1^*,y_2^*,y_3^*))</span> in <span>({textbf{E}}^3)</span>. We show that System <span>((*))</span> consisting of the following six equations in the unknowns <span>(X=(x_1,x_2,x_3))</span> and <span>(Y=(y_1,y_2,y_3))</span></p><div><div><span>$$begin{aligned} frac{1}{Vert X-TVert ^2} +frac{1}{Vert Y-TVert ^2}=frac{1}{Vert X^*-TVert ^2} +frac{1}{Vert Y^*-TVert ^2}, quad Tin {A,B,C,D,E,F} end{aligned}$$</span></div><div>\u0000 (1)\u0000 </div></div><p>has only finitely many solutions provided that both of the following two conditions are satisfied: </p><ol>\u0000 <li>\u0000 <span>(i)</span>\u0000 \u0000 <p>No four of the fixed points <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i>, <i>E</i>, <i>F</i> are coplanar;</p>\u0000 \u0000 </li>\u0000 <li>\u0000 <span>(ii)</span>\u0000 \u0000 <p>No four of the six spheres of center <i>T</i> and radius <span>(1/sqrt{k_T})</span> with </p><div><div><span>$$begin{aligned} k_T=frac{1}{Vert X^*-TVert ^2} +frac{1}{Vert Y^*-TVert ^2} end{aligned}$$</span></div><div>\u0000 (2)\u0000 </div></div><p> share a common point in <span>({textbf{E}}^3)</span>.</p>\u0000 \u0000 </li>\u0000 </ol><p> Furthermore, we exhibit configurations <span>(ABCDEFX^*Y^*)</span>, showing that (i) is also necessary. This result is an improvement on [2, Theorem 1] where the finiteness of solutions of System <span>((*))</span> was only ensured for sufficiently generic choices of the points <span>(A,B,ldots ,F,X^*,Y^*.)</span> The extended System <span>((**))</span> associated to System <span>((*))</span> consists of seven equations (1) where <span>(Tin {A,B,C,D,E,E,F,G})</span> with a further point <span>(G=(g_1,g_2,g_3)in {textbf{E}}^3)</span>. We show that if (i) and (ii) hold for <span>(Tin {A,B,C,D,E,F})</span> and the associated extended System <span>((**))</span> has some solutions other than <span>((X^*,Y^*))</span> and <span>((Y^*,X^*))</span>, then <i>G</i> lies on a real affine surface only depending on <span>({A,B,ldots ,F})</span>. This result proves [2, Conjecture 1]. Motivation for studying the above problems comes from applications to genetics; see [2].</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1003 - 1023"},"PeriodicalIF":0.9,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01114-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144074134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}