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Alienation and stability of Jensen’s and other functional equations 詹森方程和其他函数方程的异化和稳定性
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-03-21 DOI: 10.1007/s00010-024-01046-4
Mohamed Tial, Driss Zeglami
{"title":"Alienation and stability of Jensen’s and other functional equations","authors":"Mohamed Tial,&nbsp;Driss Zeglami","doi":"10.1007/s00010-024-01046-4","DOIUrl":"10.1007/s00010-024-01046-4","url":null,"abstract":"<div><p>Let <i>S</i> be a semigroup and <span>(mathbb {K})</span> be the field of real or complex numbers. We deal with the stability and alienation of Cauchy’s multiplicative (resp. additive) and Jensen’s functional equations, starting from the inequalities </p><div><div><span>$$begin{aligned} left| f(xy)+f(xsigma y)+g(xy)-2f(x)-g(x)g(y)right|le &amp; {} varepsilon , ;x,yin S, left| f(xy)+f(xsigma y)+g(xy)-2f(x)-g(x)-g(y)right|le &amp; {} varepsilon , ;x,yin S, end{aligned}$$</span></div></div><p>where <span>(f,g:Srightarrow mathbb {K})</span> and <span>(sigma )</span> is an involutive automorphism on <i>S</i>. We also consider analogous problems for Jensen’s and the quadratic (resp. Drygas) functional equations with an involutive automorphism.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"275 - 286"},"PeriodicalIF":0.9,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203594","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}
引用次数: 0
A functional equation related to Wigner’s theorem 与维格纳定理有关的函数方程
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-03-07 DOI: 10.1007/s00010-024-01042-8
Xujian Huang, Liming Zhang, Shuming Wang
{"title":"A functional equation related to Wigner’s theorem","authors":"Xujian Huang,&nbsp;Liming Zhang,&nbsp;Shuming Wang","doi":"10.1007/s00010-024-01042-8","DOIUrl":"10.1007/s00010-024-01042-8","url":null,"abstract":"<div><p>An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation </p><div><div><span>$$begin{aligned} {Vert f(x)-beta f(y)Vert : beta in {mathbb {T}}_n}={Vert x-beta yVert : beta in {mathbb {T}}_n} quad (x,yin H) end{aligned}$$</span></div></div><p>where <span>(f: H rightarrow K)</span> is between two complex normed spaces and <span>({mathbb {T}}_n:={e^{ifrac{2kpi }{n}}: k=1, cdots ,n})</span> is the set of the <i>n</i>th roots of unity. With the aid of the celebrated Wigner’s unitary-antiunitary theorem, we show that if <span>(nge 3)</span> and <i>H</i> and <i>K</i> are complex inner product spaces, then <i>f</i> satisfies the above equation if and only if there exists a phase function <span>(sigma : Hrightarrow {mathbb {T}}_n)</span> such that <span>(sigma cdot f)</span> is a linear or anti-linear isometry. Moreover, if the solution <i>f</i> is continuous, then <i>f</i> is a linear or anti-linear isometry.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 3","pages":"885 - 894"},"PeriodicalIF":0.9,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055868","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}
引用次数: 0
Bisector fields and pencils of conics 圆锥的对分线域和铅笔
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-03-06 DOI: 10.1007/s00010-024-01033-9
{"title":"Bisector fields and pencils of conics","authors":"","doi":"10.1007/s00010-024-01033-9","DOIUrl":"https://doi.org/10.1007/s00010-024-01033-9","url":null,"abstract":"<h3>Abstract</h3> <p>We introduce the notion of a bisector field, which is a maximal collection of pairs of lines such that for each line in each pair, the midpoint of the points where the line crosses every pair is the same, regardless of choice of pair. We use this to study asymptotic properties of pencils of affine conics over fields and show that pairs of lines in the plane that occur as the asymptotes of hyperbolas from a pencil of affine conics belong to a bisector field. By including also pairs of parallel lines arising from degenerate parabolas in the pencil, we obtain a full characterization: Every bisector field arises from a pencil of affine conics, and vice versa, every nontrivial pencil of affine conics is asymptotically a bisector field. Our main results are valid over any field of characteristic other than 2 and hence hold in the classical Euclidean setting as well as in Galois geometries.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045917","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}
引用次数: 0
Symmetries in Dyck paths with air pockets 有气穴的戴克路径的对称性
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-03-06 DOI: 10.1007/s00010-024-01043-7
Jean-Luc Baril, Rigoberto Flórez, José L. Ramírez
{"title":"Symmetries in Dyck paths with air pockets","authors":"Jean-Luc Baril,&nbsp;Rigoberto Flórez,&nbsp;José L. Ramírez","doi":"10.1007/s00010-024-01043-7","DOIUrl":"10.1007/s00010-024-01043-7","url":null,"abstract":"<div><p>The main objective of this paper is to analyze symmetric and asymmetric peaks in <i>Dyck paths with air pockets</i> (DAPs). These paths are formed by combining each maximal run of down-steps in ordinary Dyck paths into a larger, single down-step. To achieve this, we present a trivariate generating function that counts the number of DAPs based on their length and the number of symmetric and asymmetric peaks they contain. We determine the total numbers of symmetric and asymmetric peaks across all DAPs, providing an asymptotic for the ratio of these two quantities. Recursive relations and closed formulas are provided for the number of DAPs of length <i>n</i>, as well as for the total number of symmetric peaks, weight of symmetric peaks, and height of symmetric peaks. Furthermore, a recursive relation is established for the overall number of DAPs, similar to that for classic Dyck paths. A DAP is said to be <i>non-decreasing</i> if the sequence of ordinates of all local minima forms a non-decreasing sequence. In the last section, we focus on the sets of non-decreasing DAPs and examine their symmetric and asymmetric peaks.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1235 - 1264"},"PeriodicalIF":0.9,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055755","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}
引用次数: 0
Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials 奇数强度球形设计达到法泽卡斯-列文斯丹边界的覆盖和普遍最小电位
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-03-06 DOI: 10.1007/s00010-024-01036-6
Sergiy Borodachov
{"title":"Odd strength spherical designs attaining the Fazekas–Levenshtein bound for covering and universal minima of potentials","authors":"Sergiy Borodachov","doi":"10.1007/s00010-024-01036-6","DOIUrl":"10.1007/s00010-024-01036-6","url":null,"abstract":"<div><p>We characterize the cases of existence of spherical designs of an odd strength attaining the Fazekas–Levenshtein bound for covering and prove some of their properties. We also find all universal minima of the potential of regular spherical configurations in two new cases: the demihypercube on <span>(S^d)</span>, <span>(dge 4)</span>, and the <span>(2_{41})</span> polytope on <span>(S^7)</span> (which is dual to the <span>(E_8)</span> lattice).\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 2","pages":"509 - 533"},"PeriodicalIF":0.9,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01036-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055579","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}
引用次数: 0
Quaternion-valued multiplicative functions on semigroups 半群上的四元值乘法函数
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-02-27 DOI: 10.1007/s00010-024-01040-w
Ayoub Ouhabi, Driss Zeglami, Mohamed Ayoubi
{"title":"Quaternion-valued multiplicative functions on semigroups","authors":"Ayoub Ouhabi,&nbsp;Driss Zeglami,&nbsp;Mohamed Ayoubi","doi":"10.1007/s00010-024-01040-w","DOIUrl":"10.1007/s00010-024-01040-w","url":null,"abstract":"<div><p>Our aim is to solve a system of functional equations closely related to trigonometric functional equations. This allows us to express quaternion-valued multiplicative functions in terms of complex-valued multiplicative functions. As an application of our results, we give the continuous quaternion-valued solutions of a functional equation on <span>(({mathbb {R}},cdot ))</span>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1565 - 1578"},"PeriodicalIF":0.9,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002362","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}
引用次数: 0
Homi-repair under iteration (II): oscillating discontinuities and pre-discontinuities 迭代下的同向修复（II）：振荡不连续和前不连续
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-02-25 DOI: 10.1007/s00010-024-01034-8
Xiaohua Liu, Liu Liu, Weinian Zhang
{"title":"Homi-repair under iteration (II): oscillating discontinuities and pre-discontinuities","authors":"Xiaohua Liu,&nbsp;Liu Liu,&nbsp;Weinian Zhang","doi":"10.1007/s00010-024-01034-8","DOIUrl":"10.1007/s00010-024-01034-8","url":null,"abstract":"<div><p>It is shown that removable and jumping discontinuities for functions having more than one but finitely many discontinuities have a second order <span>(C^0)</span> homi-repair. In this paper we study second order <span>(C^0)</span> homi-repair of oscillating discontinuities and pre-discontinuities for those functions and give necessary and sufficient conditions for <span>(C^0)</span> repair by second order iteration.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 2","pages":"381 - 397"},"PeriodicalIF":0.9,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139953229","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}
引用次数: 0
A note on convex solutions to an equation on open intervals 关于开区间方程凸解的说明
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-02-24 DOI: 10.1007/s00010-024-01038-4
Chaitanya Gopalakrishna
{"title":"A note on convex solutions to an equation on open intervals","authors":"Chaitanya Gopalakrishna","doi":"10.1007/s00010-024-01038-4","DOIUrl":"10.1007/s00010-024-01038-4","url":null,"abstract":"<div><p>The note is concerned with the functional equation </p><div><div><span>$$begin{aligned} lambda _1H_1(f(x))+lambda _2H_2(f^2(x))+cdots +lambda _nH_n(f^n(x))=F(x), end{aligned}$$</span></div></div><p>which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 4","pages":"1151 - 1159"},"PeriodicalIF":0.9,"publicationDate":"2024-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139953243","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}
引用次数: 0
Zonal labelings and Tait colorings from a new perspective 从新的角度看带状标记和泰特着色
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-02-21 DOI: 10.1007/s00010-024-01037-5
Andrew Bowling, Weiguo Xie
{"title":"Zonal labelings and Tait colorings from a new perspective","authors":"Andrew Bowling,&nbsp;Weiguo Xie","doi":"10.1007/s00010-024-01037-5","DOIUrl":"10.1007/s00010-024-01037-5","url":null,"abstract":"<div><p>Let <span>(G=(V(G), E(G), F(G)))</span> be a plane graph with vertex, edge, and region sets <i>V</i>(<i>G</i>), <i>E</i>(<i>G</i>),  and <i>F</i>(<i>G</i>) respectively. A zonal labeling of a plane graph <i>G</i> is a labeling <span>(ell : V(G)rightarrow {1,2}subset mathbb {Z}_3)</span> such that for every region <span>(Rin F(G))</span> with boundary <span>(B_R)</span>, <span>(sum _{vin V(B_R)}ell (v)=0)</span> in <span>(mathbb {Z}_3)</span>. It has been proven by Chartrand, Egan, and Zhang that a cubic map has a zonal labeling if and only if it has a 3-edge coloring, also known as a Tait coloring. A dual notion of cozonal labelings is defined, and an alternate proof of this theorem is given. New features of cozonal labelings and their utility are highlighted along the way. Potential extensions of results to related problems are presented.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1611 - 1625"},"PeriodicalIF":0.9,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139953231","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}
引用次数: 0
Homi-repair under iteration (I): removable and jumping cases 迭代（I）下的同向修复：可移动和跳跃情况
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-02-20 DOI: 10.1007/s00010-024-01035-7
Xiaohua Liu, Liu Liu, Weinian Zhang
{"title":"Homi-repair under iteration (I): removable and jumping cases","authors":"Xiaohua Liu,&nbsp;Liu Liu,&nbsp;Weinian Zhang","doi":"10.1007/s00010-024-01035-7","DOIUrl":"10.1007/s00010-024-01035-7","url":null,"abstract":"<div><p>It was found that a function with exactly one discontinuity may have a continuous iterate of second order, indicating that a discontinuity may be repaired to be a continuous one by its adjacent pair of functions of second order, called second order <img> sui-repair. If a function has more than one discontinuities, examples show that some discontinuities may be repaired to be continuous ones by the other’s adjacent pair of functions of second order, called second order <span>(C^{0})</span> homi-repair. In this paper we investigate second order <span>(C^{0})</span> homi-repairs of removable and jumping discontinuities for functions having more than one but finitely many discontinuities. We give necessary and sufficient conditions for removable and jumping discontinuities to be <span>(C^0)</span> repaired by the second order iteration.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 2","pages":"351 - 379"},"PeriodicalIF":0.9,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139917539","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}
引用次数: 0
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