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Alternative equations for generalized polynomials on plane curves 平面曲线上广义多项式的替代方程
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2025-05-13 DOI: 10.1007/s00010-025-01163-8
Bruce Ebanks
{"title":"Alternative equations for generalized polynomials on plane curves","authors":"Bruce Ebanks","doi":"10.1007/s00010-025-01163-8","DOIUrl":"10.1007/s00010-025-01163-8","url":null,"abstract":"<div><p>We study the Pexiderized alternative equation (PAE): <span>(f(x)g(y) = 0)</span> for all <span>((x,y) in S)</span>, where <span>(f,g:{mathbb {R}}rightarrow {mathbb {R}})</span> are generalized polynomials and <i>S</i> is a plane curve. This extends the study of the alternative equation (AE): <span>(f(x)f(y) = 0)</span> for <span>((x,y) in S)</span>, where <span>(f:{mathbb {R}}rightarrow {mathbb {R}})</span> is an additive function or other generalized polynomial. The main question about (AE) is whether <span>(f=0)</span> is the unique solution, and for (PAE) whether it implies that <span>(f=0)</span> or <span>(g=0)</span>. In the case of (AE) it is known that <span>(f=0)</span> is the unique additive solution when <i>S</i> is a circle centered at the origin, a curve with polynomial parametrization, or a certain form of hyperbola. Moreover some results are known for (AE) when <i>f</i> is assumed to be a generalized polynomial. Our findings generalize and extend those results to (PAE) and to other plane curves. As a consequence we also gain some new results about (AE).</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 4","pages":"1843 - 1853"},"PeriodicalIF":0.7,"publicationDate":"2025-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110555","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 study on Davison’s functional equation for set-valued functions 集值函数Davison泛函方程的研究
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2025-04-16 DOI: 10.1007/s00010-025-01157-6
Elham Mohammadi, Abbas Najati, Iz-iddine EL-Fassi
{"title":"A study on Davison’s functional equation for set-valued functions","authors":"Elham Mohammadi,&nbsp;Abbas Najati,&nbsp;Iz-iddine EL-Fassi","doi":"10.1007/s00010-025-01157-6","DOIUrl":"10.1007/s00010-025-01157-6","url":null,"abstract":"<div><p>Let <i>X</i> be a unital algebra with unit <i>e</i> and <i>Y</i> a real Hausdorff topological vector space. In this article, we obtain the general set-valued solution of the Davison functional equation </p><div><div><span>$$begin{aligned} F(xy+x)+F(y)=F(xy)+F(x+y) end{aligned}$$</span></div></div><p>for functions <i>F</i> defined on <i>X</i> with values in the set of nonempty compact and convex subsets of <i>Y</i>. Additionally, we investigate various extensions of the Davison set-valued functional equation.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 4","pages":"1799 - 1808"},"PeriodicalIF":0.7,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110558","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
Inequalities characterizing differential operators 表征微分算子的不等式
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2025-04-16 DOI: 10.1007/s00010-025-01158-5
Gerd Herzog, Peer Kunstmann
{"title":"Inequalities characterizing differential operators","authors":"Gerd Herzog,&nbsp;Peer Kunstmann","doi":"10.1007/s00010-025-01158-5","DOIUrl":"10.1007/s00010-025-01158-5","url":null,"abstract":"<div><p>We show that under certain inequality assumptions an arbitrary linear operator <span>(D:C^infty (mathbb {R}) rightarrow C(mathbb {R}))</span> is a differential operator, for example, if <i>D</i>[<i>f</i>] is nonnegative in local minima of <i>f</i>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 4","pages":"1773 - 1780"},"PeriodicalIF":0.7,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-025-01158-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110593","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
Cauchy product of iterative functional equation 迭代泛函方程的柯西积
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2025-04-15 DOI: 10.1007/s00010-025-01159-4
Akash Pradhan, Deepesh Kumar Patel, Hemant Kumar Nashine
{"title":"Cauchy product of iterative functional equation","authors":"Akash Pradhan,&nbsp;Deepesh Kumar Patel,&nbsp;Hemant Kumar Nashine","doi":"10.1007/s00010-025-01159-4","DOIUrl":"10.1007/s00010-025-01159-4","url":null,"abstract":"<div><p>This manuscript examines the existence and uniqueness of differentiable and continuous solutions of the iterative functional equation of the form </p><div><div><span>$$begin{aligned} sum limits _{i=0}^{n}lambda _{i}f^{i}(varkappa )f^{n-i}(varkappa )= F (varkappa ), quad varkappa in [a,b], end{aligned}$$</span></div></div><p>where <span>(lambda _{i})</span>’s are real constants and <span>( F )</span> is a given function. The novelty of this work lies in the generalization of the iterative root problem when <i>n</i> is even and all <span>(lambda _i)</span>’s are zero except for <span>(lambda _{n/2})</span>. This generalization offers the advantage of covering a wider class of functional equations. Numerical examples are presented to validate the existence results, and the stability of each solution is thoroughly analyzed.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 4","pages":"1585 - 1602"},"PeriodicalIF":0.7,"publicationDate":"2025-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110589","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
New recursive constructions of amoebas and their balancing number 阿米巴虫的新递归结构及其平衡数
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2025-04-10 DOI: 10.1007/s00010-025-01156-7
Laura Eslava, Adriana Hansberg, Tonatiuh Matos-Wiederhold, Denae Ventura
{"title":"New recursive constructions of amoebas and their balancing number","authors":"Laura Eslava,&nbsp;Adriana Hansberg,&nbsp;Tonatiuh Matos-Wiederhold,&nbsp;Denae Ventura","doi":"10.1007/s00010-025-01156-7","DOIUrl":"10.1007/s00010-025-01156-7","url":null,"abstract":"<div><p>Amoeba graphs are based on iterative <i>feasible edge-replacements</i>, where, at each step, an edge from the graph is removed and placed in an available spot so that the resulting graph is isomorphic to the original graph. Broadly speaking, amoebas are graphs that, by means of a chain of feasible edge-replacements, can be transformed into any other copy of itself on a given vertex set (depending on which they are defined as local or global amoebas). Global amoebas were born as examples of <i>balanceable</i> graphs, which appear with half of their edges in each color in any 2-edge coloring of a large enough complete graph with a sufficient amount of edges <i>k</i> in each color. The minimum value of <i>k</i> is called the <i>balancing number</i> of <i>G</i>. We provide a recursive construction to generate very diverse infinite families of local and global amoebas, which not only answers a question posed by Caro et al. but also yields an efficient algorithm that provides a chain of feasible edge-replacements that one can perform in order to move a local amoeba into an aimed copy in the same vertex set. All results are illustrated by three different families of local amoebas, including the Fibonacci-type trees. We express the balancing number of a global amoeba <i>G</i> in terms of the extremal number of a class of subgraphs of <i>G</i> and give a general lower bound. We provide linear lower and upper bounds for the balancing number of our three case studies.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1265 - 1299"},"PeriodicalIF":0.9,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073919","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
Analytical aspects of a q, r-analogue of poly-Stirling numbers of both kinds 两类聚斯特林数的q， r模拟的解析方面
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2025-02-22 DOI: 10.1007/s00010-024-01135-4
Takao Komatsu, Eli Bagno, David Garber
{"title":"Analytical aspects of a q, r-analogue of poly-Stirling numbers of both kinds","authors":"Takao Komatsu,&nbsp;Eli Bagno,&nbsp;David Garber","doi":"10.1007/s00010-024-01135-4","DOIUrl":"10.1007/s00010-024-01135-4","url":null,"abstract":"<div><p>The Stirling numbers of type <i>B</i> of the second kind count signed set partitions. In this paper, we provide new combinatorial and analytical identities regarding these numbers as well as Broder’s <i>r</i>-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion–exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized using results of Comtet and Lancaster. We generalize these theorems, which present equivalent conditions for a triangle of numbers to be a triangle of generalized Stirling numbers, to the case of the <i>q</i>, <i>r</i>-poly Stirling numbers, which are <i>q</i>-analogues of the restricted Stirling numbers defined by Broder and having a polynomial value appearing in their defining recursion. There are two ways to do this and these ways are related by a nice identity.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"287 - 320"},"PeriodicalIF":0.9,"publicationDate":"2025-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871400","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
On and around the balanced Cauchy equation 在平衡柯西方程上和周围
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2025-02-10 DOI: 10.1007/s00010-025-01155-8
Ekaterina Shulman
{"title":"On and around the balanced Cauchy equation","authors":"Ekaterina Shulman","doi":"10.1007/s00010-025-01155-8","DOIUrl":"10.1007/s00010-025-01155-8","url":null,"abstract":"<div><p>Let <i>G</i> be a unital semigroup, <span>((K, +))</span> an Abelian group. We extend to this case results of several authors on functions <span>(f: Grightarrow K)</span> satisfying the equations </p><div><div><span>$$begin{aligned} f(x_1x_2ldots x_n) = sum _{i=1}^n F_i(x_1, ldots ,widehat{x_i}, ldots ,x_n) end{aligned}$$</span></div></div><p>and </p><div><div><span>$$begin{aligned} sum _{k=0}^n(-1)^{k}sum _{|S|=n-k}f left( prod _Sx right) = 0. end{aligned}$$</span></div></div><p>We also study a more general class of equations: </p><div><div><span>$$begin{aligned} f(x_1x_2ldots x_n) = sum _{i=1}^n sum _{j=1}^{J_i}p_{ij}(x_i)(F_{ij}(x_1,ldots ,widehat{x_i}, ldots , x_n)), end{aligned}$$</span></div></div><p>where all <span>(p_{ij})</span> are polynomial maps from <i>G</i> to the group of all endomorphisms of <i>K</i>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 4","pages":"1763 - 1772"},"PeriodicalIF":0.7,"publicationDate":"2025-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110492","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
Orthogonality induced by norm derivatives: a new geometric constant and symmetry 范数导数诱导的正交性：一种新的几何常数和对称性
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2025-01-28 DOI: 10.1007/s00010-025-01154-9
Souvik Ghosh, Kallol Paul, Debmalya Sain
{"title":"Orthogonality induced by norm derivatives: a new geometric constant and symmetry","authors":"Souvik Ghosh,&nbsp;Kallol Paul,&nbsp;Debmalya Sain","doi":"10.1007/s00010-025-01154-9","DOIUrl":"10.1007/s00010-025-01154-9","url":null,"abstract":"<div><p>In this article we study the difference between orthogonality induced by norm derivatives (known as <span>(rho )</span>-orthogonality) and Birkhoff-James orthogonality in a normed linear space <span>(mathbb {X})</span> by introducing a new geometric constant, denoted by <span>(Gamma (mathbb {X}).)</span> We explore the relation between various geometric properties of the space and the constant <span>(Gamma (mathbb {X}).)</span> We also investigate the left symmetric and right symmetric elements of a normed linear space with respect to <span>(rho )</span>-orthogonality and obtain a characterization of the same. We characterize inner product spaces among normed linear spaces using the symmetricity of <span>(rho )</span>-orthogonality. Finally, we provide a complete description of both left symmetric and right symmetric elements with respect to <span>(rho )</span>-orthogonality for some particular Banach spaces.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"883 - 904"},"PeriodicalIF":0.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144073898","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
On composition and Right Distributive Law for formal power series of multiple variables 多变量形式幂级数的复合与右分配律
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2025-01-23 DOI: 10.1007/s00010-024-01152-3
Dariusz Bugajewski, Alessia Galimberti, Piotr Maćkowiak
{"title":"On composition and Right Distributive Law for formal power series of multiple variables","authors":"Dariusz Bugajewski,&nbsp;Alessia Galimberti,&nbsp;Piotr Maćkowiak","doi":"10.1007/s00010-024-01152-3","DOIUrl":"10.1007/s00010-024-01152-3","url":null,"abstract":"<div><p>In the first part of the paper we prove a necessary and sufficient condition for the existence of the composition of formal power series in the case when the outer series is a series of one variable while the inner one is a series of multiple variables. The aim of the second part is to remove ambiguities connected with the Right Distributive Law for formal power series of one variable as well as to provide analogues of that law in the multivariable case.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"21 - 35"},"PeriodicalIF":0.9,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143554185","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
Characterizations of weighted generalized inverses 加权广义逆的刻画
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2025-01-20 DOI: 10.1007/s00010-024-01151-4
Bibekananda Sitha, Ratikanta Behera, Jajati Keshari Sahoo, R. N. Mohapatra, Predrag Stanimirović, Alena Stupina
{"title":"Characterizations of weighted generalized inverses","authors":"Bibekananda Sitha,&nbsp;Ratikanta Behera,&nbsp;Jajati Keshari Sahoo,&nbsp;R. N. Mohapatra,&nbsp;Predrag Stanimirović,&nbsp;Alena Stupina","doi":"10.1007/s00010-024-01151-4","DOIUrl":"10.1007/s00010-024-01151-4","url":null,"abstract":"<div><p>The main objective of this study is to introduce unique representations and characterizations for the several classes of weighted generalized inverses of matrices. Proposed representations of the matrix-weighted core inverse will help us to discuss some results associated with the reverse order law for these inverses. Furthermore, this paper introduces an extension of the concepts of generalized bilateral inverse and their respective dual for complex rectangular matrices. Characteristics that lead to self-duality in weighted bilateral inverses are also examined. In addition, a W-weighted index-MP, W-weighted MP-index, and W-weighted MP-index-MP matrices for rectangular complex matrices are introduced.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 3","pages":"1301 - 1336"},"PeriodicalIF":0.9,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144074172","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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