Aequationes Mathematicae最新文献

{"title":"A class of functional equations for additive functions","authors":"Bruce Ebanks","doi":"10.1007/s00010-024-01105-w","DOIUrl":"https://doi.org/10.1007/s00010-024-01105-w","url":null,"abstract":"<p>The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form (<span>(*)</span>) <span>(sum _{j=1}^{k} x^{p_j}f_j(x^{q_j}) = 0)</span>, where the <span>(p_j,q_j)</span> are non-negative integers, the <span>(f_j:R rightarrow S)</span> are additive functions, <i>S</i> is a commutative ring, and <i>R</i> is a sub-ring of <i>S</i>. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form (<span>(*)</span>) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et al. (Results Math 73(2):27, 2018). It seems that this fact may have been overlooked, judging by some results about a particular case of (<span>(*)</span>) in recent publications. We also present a new method for the homogeneous case by combining the results above with [6], and we show how to solve non-homogeneous equations of the form (<span>(*)</span>).</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194456","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":"Correction to: Linked pairs of additive functions","authors":"Bruce Ebanks","doi":"10.1007/s00010-024-01108-7","DOIUrl":"10.1007/s00010-024-01108-7","url":null,"abstract":"","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1439 - 1440"},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410095","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 the solvability of the (SSIE) with operator ({D}_{x} {mathbf { * }}left( { s}_{ R}^{{textbf{0}}} right) _{{{{Sigma }} - {lambda I}}} {{ subset }}{ s}_{ R}^{{{0}}} ), involving the fine spectrum of an operator","authors":"Bruno de Malafosse,&nbsp;Eberhard Malkowsky,&nbsp;Vladinir Rakočević","doi":"10.1007/s00010-024-01111-y","DOIUrl":"10.1007/s00010-024-01111-y","url":null,"abstract":"<div><p>Given any sequence <span>(a=(a_{n})_{nge 1})</span> of positive real numbers and any set <i>E</i> of complex sequences, we write <span>(E_{a})</span> for the set of all sequences <span>(y=(y_{n})_{nge 1})</span> such that <span>(y/a=(y_{n}/a_{n})_{nge 1}in E)</span>; in particular, <span>(c_{a})</span> denotes the set of all sequences <i>y</i> such that <i>y</i>/<i>a</i> converges. In this paper, we use the sum operator <span>(Sigma )</span>, defined by <span>(Sigma _{n}y=sum _{k=1}^{n}y_{k})</span> for all sequences <i>y</i>, and we determine its spectrum over each of the sets <span>(s_{a}=(ell _{infty })_{a})</span> and <span>(s_{a}^{0}=(c_{0})_{a})</span>. Then we determine the point, residual and continuous spectra of the operator <span>(D_{1/R}Sigma D_{R})</span>, with <span>(R&gt;1)</span>, and we solve the special  <i>sequence spaces inclusion equations (SSIE),</i> (which are determined by an inclusion, for which each term is a <i>sum </i>or<i> a sum of products of sets of the form </i><span>((E_{a})_{mathcal {T}})</span><i> and </i> <span>(( E_{f(x)})_{mathcal {T}})</span> where <i>f</i> maps <span>(U^{+})</span> to itself, <i>E</i> is any linear space of sequences and <span>(mathcal {T})</span> is a triangle) <span>(D_{x}*(s_{R}^{0})_{Sigma -lambda I}subset s_{R}^{0})</span>, using the fine spectrum of this operator. The solvability of this (SSIE) consists in determining, for each <span>(lambda in mathbb {C})</span>, the set of all sequences <span>(xin omega )</span> that satisfy the next statement. For every <span>(yin omega )</span>, we have </p><div><div><span>$$begin{aligned} lim _{nrightarrow infty }frac{1}{R^{n}}left( sum _{k=1}^{n}y_{k}-lambda y_{n}right) =0Longrightarrow lim _{nrightarrow infty }x_{n}left( frac{y_{n}}{R^{n}}right) =0text {.} end{aligned}$$</span></div></div><p>Then, we solve this (SSIE) for <span>(R=1)</span>. Finally, we solve each (SSIE) <span>(D_{x}*( E_{R})_{Sigma -lambda I}subset s_{R})</span>, where <i>E</i> is successively equal to <span>(c_{0})</span>, <i>c</i>, and <span>(ell _{infty })</span>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"767 - 783"},"PeriodicalIF":0.9,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885057","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":"Some new minimax inequalities for centered convex bodies","authors":"Horst Martini, Zokhrab Mustafaev, Sakhavet M. Zarbaliev","doi":"10.1007/s00010-024-01109-6","DOIUrl":"https://doi.org/10.1007/s00010-024-01109-6","url":null,"abstract":"<p>The purpose of this manuscript is to derive some new minimax inequalities for centered convex bodies in <span>({mathbb {R}}^{d})</span>. As consequences, new characterizations of ellipsoids will be established as well. Some open problems related to the Busemann-Petty list of problems will also be discussed.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"74 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865325","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":"Another look at the Matkowski and Wesołowski problem yielding a new class of solutions","authors":"Janusz Morawiec, Thomas Zürcher","doi":"10.1007/s00010-024-01110-z","DOIUrl":"https://doi.org/10.1007/s00010-024-01110-z","url":null,"abstract":"<p>The following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions <span>(varphi :[0,1]rightarrow [0,1])</span>, distinct from the identity on [0, 1], such that <span>(varphi (0)=0)</span>, <span>(varphi (1)=1)</span> and <span>(varphi (x)=varphi (frac{x}{2})+varphi (frac{x+1}{2})-varphi (frac{1}{2}))</span> for every <span>(xin [0,1])</span>? By now, it is known that each of the de Rham functions <span>(R_p)</span>, where <span>(pin (0,1))</span>, is a solution of the MW-problem, and for any Borel probability measure <span>(mu )</span> concentrated on (0, 1) the formula <span>(phi _mu (x)=int _{(0,1)}R_p(x), dmu (p))</span> defines a solution <span>(phi _mu :[0,1]rightarrow [0,1])</span> of this problem as well. In this paper, we give a new family of solutions of the MW-problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW-problem that are not of the above integral form with any Borel probability measure <span>(mu )</span>.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865359","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 spectral irregularity of graphs","authors":"Lu Zheng,&nbsp;Bo Zhou","doi":"10.1007/s00010-024-01106-9","DOIUrl":"10.1007/s00010-024-01106-9","url":null,"abstract":"<div><p>The spectral radius <span>(rho (G))</span> of a graph <i>G</i> is the largest eigenvalue of the adjacency matrix of <i>G</i>. For a graph <i>G</i> with maximum degree <span>(Delta (G))</span>, it is known that <span>(rho (G)le Delta (G))</span> with equality when <i>G</i> is connected if and only if <i>G</i> is regular. So the quantity <span>(beta (G)=Delta (G)-rho (G))</span> is a spectral measure of irregularity of <i>G</i>. In this paper, we identify the trees of order <span>(nge 12)</span> with the first 15 largest <span>(beta )</span>-values, the unicyclic graphs of order <span>(nge 17)</span> with the first 16 largest <span>(beta )</span>-values, as well as the bicyclic graphs of order <span>(nge 30)</span> with the first 11 largest <span>(beta )</span>-values. We also determine the graphs with the largest <span>(beta )</span>-values among all connected graphs with given order and clique number.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1161 - 1176"},"PeriodicalIF":0.9,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745756","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":"Path factors in bipartite graphs from size or spectral radius","authors":"Yifang Hao,&nbsp;Shuchao Li","doi":"10.1007/s00010-024-01107-8","DOIUrl":"10.1007/s00010-024-01107-8","url":null,"abstract":"<div><p>Let <i>G</i> be a graph and let <span>(P_n)</span> be a path on <i>n</i> vertices. A spanning subgraph <i>H</i> of <i>G</i> is called a <span>({P_{3},P_{4},P_{5}})</span>-factor if every component of <i>H</i> is one of <span>(P_3,, P_4)</span> and <span>(P_5)</span>. In 1994, Wang (J Graph Theory 18(2):161–167, 1994) gave a sufficient and necessary condition to ensure that a bipartite graph contains a <span>({P_{3},P_{4},P_{5}})</span>-factor. In this paper, we use an equivalent form of Wang-type condition to establish two sufficient conditions to ensure that there exists a <span>({P_{3},P_{4},P_{5}})</span>-factor in a connected bipartite graph, in which one is based on the size, the other is based on the spectral radius of the bipartite graph.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1177 - 1210"},"PeriodicalIF":0.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745757","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":"Solutions of the functional difference Toda equation from centered Darboux transformation","authors":"Pierre Gaillard","doi":"10.1007/s00010-024-01097-7","DOIUrl":"https://doi.org/10.1007/s00010-024-01097-7","url":null,"abstract":"<p>By using a particular Darboux transformation which we can call centered Darboux transformation, we construct solutions of the functional difference Toda lattice equation in terms of Casorati determinants. We give a complete description of the method and the corresponding proofs. We construct some explicit solutions for the first orders.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"369 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572683","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":"Well-posedness for a class of pseudo-differential hyperbolic equations on the torus","authors":"Duván Cardona,&nbsp;Julio Delgado,&nbsp;Michael Ruzhansky","doi":"10.1007/s00010-024-01093-x","DOIUrl":"10.1007/s00010-024-01093-x","url":null,"abstract":"<div><p>In this paper we establish the well-posedness of the Cauchy problem for a class of pseudo-differential hyperbolic equations on the torus. The class considered here includes a space-like fractional order Laplacians. By applying the toroidal pseudo-differential calculus we establish regularity estimates, existence and uniqueness in the scale of the standard Sobolev spaces on the torus.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 4","pages":"1019 - 1038"},"PeriodicalIF":0.9,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01093-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572685","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":"Balanced Stirling numbers","authors":"Michael Maltenfort","doi":"10.1007/s00010-024-01087-9","DOIUrl":"10.1007/s00010-024-01087-9","url":null,"abstract":"<div><p>Hsu and Shiue (Adv Appl Math 20(3):366–384, 1998. https://doi.org/10.1006/aama.1998.0586) defined generalized Stirling numbers, which include as special cases a wide variety of combinatorial quantities. We prove that the two kinds of central factorial numbers are fundamentally different new special cases. Our approach also yields a previously unrecognized connection between the two kinds of central factorial numbers. In order to prove our main results, we introduce balanced Stirling numbers, which specialize the generalized Stirling numbers and can be further specialized into either kind of central factorial numbers.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"705 - 731"},"PeriodicalIF":0.9,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141572684","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}