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Alternative equations for generalized polynomials on hyperbolas 双曲线上广义多项式的替代方程
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-06-06 DOI: 10.1007/s00010-026-01302-9
Bruce Ebanks
{"title":"Alternative equations for generalized polynomials on hyperbolas","authors":"Bruce Ebanks","doi":"10.1007/s00010-026-01302-9","DOIUrl":"10.1007/s00010-026-01302-9","url":null,"abstract":"<div><p>In an earlier paper we studied the Pexiderized alternative equation <span>(f(x)g(y) = 0)</span> for <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. We showed that if <i>S</i> is an ellipse (in standard position) or is parametrized by polynomials then either <span>(f=0)</span> or <span>(g=0)</span>. For hyperbolas in standard position we were able to prove the corresponding result only for the case that <i>f</i> and <i>g</i> are generalized monomials. Here we prove some results on hyperbolas when <i>f</i> and <i>g</i> are certain generalized polynomials.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172898","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 Levi-Civita functional equation on non-abelian groups 非阿贝尔群上的Levi-Civita泛函方程
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-06-06 DOI: 10.1007/s00010-026-01303-8
Keltouma Belfakih, Elhoucien Elqorachi
{"title":"A Levi-Civita functional equation on non-abelian groups","authors":"Keltouma Belfakih,&nbsp;Elhoucien Elqorachi","doi":"10.1007/s00010-026-01303-8","DOIUrl":"10.1007/s00010-026-01303-8","url":null,"abstract":"<div><p>Let <i>G</i> be a topological group and <i>C</i>(<i>G</i>) the algebra of continuous, complex valued functions on <i>G</i>. We determine the solutions <span>(f,g,h,kin C(G))</span> of the functional equation </p><div><div><span>$$begin{aligned} f(xy)=f(x)h(y)+g(x)k(y),quad x,yin G, end{aligned}$$</span></div></div><p>which generalizes the sine and cosine addition formulas. This enables us to solve the Levi-Civita functional equation </p><div><div><span>$$begin{aligned} f(xy)=g_1(x)h_1(y)+g_2(x)h_2(y),,x,yin G, end{aligned}$$</span></div></div><p>where <span>(f,g_1,g_2,h_1,h_2in C(G))</span> are unknown. Our approach removes the usual centrality assumption, allowing us to characterize more general solutions.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172901","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
Functional equations characterizing totients in regular convolution rings 正则卷积环中特征子的泛函方程
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-06-06 DOI: 10.1007/s00010-026-01304-7
Pentti Haukkanen
{"title":"Functional equations characterizing totients in regular convolution rings","authors":"Pentti Haukkanen","doi":"10.1007/s00010-026-01304-7","DOIUrl":"10.1007/s00010-026-01304-7","url":null,"abstract":"<div><p>Let <i>A</i> be a regular convolution. An arithmetical function <i>f</i> is <i>A</i>-multiplicative if <span>(f(1)=1)</span> and <span>(f(mn) = f(m)f(n))</span> for all positive integers <i>m</i>, <i>n</i> with <span>(m, n in A(mn))</span>, and <i>f</i> is an <i>A</i>-totient if there exist <i>A</i>-multiplicative functions <span>(f_T)</span> and <span>(f_V)</span> such that <span>(f=f_Tmathbin {*_A}f_V^{-1})</span>. In this paper, we present various functional equations characterizing <i>A</i>-totients, derived from fundamental properties of Euler’s totient function. Our exposition also provides a historical perspective.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01304-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172983","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
A unified approach to multiplicative generators of pseudo-overlap functions with their applications on image edge detection 伪重叠函数乘法生成器的统一方法及其在图像边缘检测中的应用
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-06-04 DOI: 10.1007/s00010-026-01273-x
Juntao Wang, Yanhong She, Benjamín Bedregal, Qiang Huang
{"title":"A unified approach to multiplicative generators of pseudo-overlap functions with their applications on image edge detection","authors":"Juntao Wang,&nbsp;Yanhong She,&nbsp;Benjamín Bedregal,&nbsp;Qiang Huang","doi":"10.1007/s00010-026-01273-x","DOIUrl":"10.1007/s00010-026-01273-x","url":null,"abstract":"<div><p>The notion of pseudo-overlap functions is to generalize that of overlap functions by dropping the strong requirement of commutativity, and pseudo-overlap functions are to overlap functions just what pseudo t-norms are to t-norms. The main aim of this paper is to present some constructions of pseudo-overlap functions by means of their pseudo-overlap generator triples. More narrowly, we introduce the concept of pseudo-overlap generator triple for pseudo-overlap functions, and obtain some equivalent conditions under which two one-place real functions and a pseudo-overlap function to be a pseudo-overlap generator triple. Moreover, based on the pseudo-overlap generators, we prove that a pseudo-overlap function can be obtained from the distortion of a pseudo-overlap function and two pseudo-automorphisms. This method is indeed a generalization of the known method that construct the (general) overlap function by a pseudo-automorphism in the available literature. In addition, we prove that the continuous left and right-inverse of pseudo-automorphisms, if exist, are the same automorphisms, and improve the corresponding results for general pseudo-automorphisms in (Fuzzy Sets and Systems, 448:65-83,2022). Lastly, with the aid of pseudo-overlap generator triple of pseudo overlap functions, two kinds of fuzzy mathematical morphology operators were applied to image edge detection, and obtained a better results than existing algorithms.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148172197","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
Inscribed rhombi having diagonals collinear with specified points 有对角线与特定点共线的内切菱形
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-06-03 DOI: 10.1007/s00010-026-01307-4
Stephen E. Wright
{"title":"Inscribed rhombi having diagonals collinear with specified points","authors":"Stephen E. Wright","doi":"10.1007/s00010-026-01307-4","DOIUrl":"10.1007/s00010-026-01307-4","url":null,"abstract":"<div><p>We investigate the question of whether a simple closed curve in the plane must contain all four vertices of some rhombus having one diagonal collinear with a specified point. This complements previous research on whether there is a rhombus with a diagonal or side parallel to a given line. We obtain a new proof that a simple closed curve contains the vertices of uncountably many rhombi. We also explore conditions guaranteeing that all points in some region are collinear with diagonals of such rhombi.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01307-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148141661","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
Disjunctive total domination in maximal outerplanar graphs 极大外平面图的析取全支配
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-05-29 DOI: 10.1007/s00010-026-01301-w
Michael A. Henning, Paras Vinubhai Maniya, Dinabandhu Pradhan
{"title":"Disjunctive total domination in maximal outerplanar graphs","authors":"Michael A. Henning,&nbsp;Paras Vinubhai Maniya,&nbsp;Dinabandhu Pradhan","doi":"10.1007/s00010-026-01301-w","DOIUrl":"10.1007/s00010-026-01301-w","url":null,"abstract":"<div><p>A disjunctive total dominating set of a graph <i>G</i> is a set <span>(D subseteq V(G))</span> such that every vertex in <i>V</i>(<i>G</i>) has a neighbor in <i>D</i> or has at least two vertices in <i>D</i> at distance 2 from it. The disjunctive total domination number of <i>G</i>, denoted by <span>(gamma _t^d(G))</span>, is the minimum cardinality among all disjunctive total dominating sets of <i>G</i>. In this paper, we show that if <i>G</i> is a maximal outerplanar graph with <span>(nge 5)</span> vertices, then <span>(gamma _t^d(G)le lfloor frac{1}{3}(n+1)rfloor )</span>, and this bound is sharp.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148173617","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 necessary condition for cylindrical curves in terms of curvature and torsion 在曲率和扭转方面圆柱曲线的必要条件
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-05-19 DOI: 10.1007/s00010-026-01293-7
Rafael López
{"title":"A necessary condition for cylindrical curves in terms of curvature and torsion","authors":"Rafael López","doi":"10.1007/s00010-026-01293-7","DOIUrl":"10.1007/s00010-026-01293-7","url":null,"abstract":"<div><p>We establish necessary conditions for a regular curve to lie on a circular cylinder in terms of its curvature <span>(kappa )</span> and torsion <span>(tau )</span>. By identifying a fundamental function <span>(psi = sin ^2 alpha )</span>, representing the squared sine of the angle between the tangent vector and the axis of the cylinder, we reduce the geometric inclusion problem to a compatibility condition between an explicit eighth-degree polynomial equation and a differential equation for <span>(psi )</span>. This approach yields a single ODE involving only <span>(kappa )</span> and <span>(tau )</span> that governs the inclusion of the curve in the cylinder. The robustness of this framework is demonstrated through specific examples of cylindrical curves. Furthermore, we analyze the case of curves with constant curvature <span>(kappa _0)</span>, obtaining an explicit ODE for the torsion. Remarkably, we prove that if <span>(kappa _0 = 1/rho )</span>, this equation admits an explicit, exact solution for <span>(tau )</span>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01293-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148008829","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
On the Invariant Sphere Theorem 关于不变球定理
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-05-19 DOI: 10.1007/s00010-026-01298-2
Jaume Llibre, Xiang Zhang
{"title":"On the Invariant Sphere Theorem","authors":"Jaume Llibre,&nbsp;Xiang Zhang","doi":"10.1007/s00010-026-01298-2","DOIUrl":"10.1007/s00010-026-01298-2","url":null,"abstract":"<div><p>For a system in <span>(mathbb {R}^n)</span> consisting of an unstable star node and homogeneous nonlinearity, i.e. <span>(dot{x}=mu x+Q(x))</span> with <span>(mu &gt;0)</span> and <i>Q</i>(<i>x</i>) a homogeneous polynomial satisfying <span>(xQ(x)&lt;0)</span> for all <span>(xne 0)</span>, Field in 1989 proved the existence of a unique topological invariant sphere. Here restricted to <span>(n=2)</span>, we prove that the topological invariant circle is of star shape, and obtain the classification of all their phase portraits in the Poincaré disc when <i>Q</i> is of degree 5 and the invariant circle is the circle <span>({mathbb {S}}^1={(x,y,z) in mathbb {R}^3| x^2+y^2+z^2=1})</span>. Moreover, without the restriction <span>(xQ(x)&lt;0)</span> we characterize the relation between equilibria (both finite and infinite) and the invariant straight lines passing through the origin. These results extend and improve the previous results of other authors.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01298-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148008830","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
On continuous solutions of a generalized Goła̧b–Schinzel functional inequality 广义Goła - b-Schinzel泛函不等式的连续解
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-05-16 DOI: 10.1007/s00010-026-01299-1
Eliza Jabłońska, Zuzanna Żurek
{"title":"On continuous solutions of a generalized Goła̧b–Schinzel functional inequality","authors":"Eliza Jabłońska,&nbsp;Zuzanna Żurek","doi":"10.1007/s00010-026-01299-1","DOIUrl":"10.1007/s00010-026-01299-1","url":null,"abstract":"<div><p>In the paper we improve some results from [9] concerning the Goła̧b–Schinzel inequality <span>(f(x+f(x)y)le f(x)f(y))</span> for a continuous unknown function <span>(f:mathbb {R}rightarrow mathbb {R})</span>. We also show some properties of solutions of the generalized inequality <span>(f(x+g(x)y)le f(x)f(y))</span> for continuous unknown functions <span>(f,g:mathbb {R}rightarrow mathbb {R})</span> provided <span>(gge f)</span>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01299-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148008981","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
A geometric constant for quantifying the differences between approximate orthogonality types 一种几何常数，用于量化近似正交类型之间的差异
IF 0.7 3区 数学
Aequationes Mathematicae Pub Date : 2026-05-14 DOI: 10.1007/s00010-026-01295-5
Mahdi Dehghani
{"title":"A geometric constant for quantifying the differences between approximate orthogonality types","authors":"Mahdi Dehghani","doi":"10.1007/s00010-026-01295-5","DOIUrl":"10.1007/s00010-026-01295-5","url":null,"abstract":"<div><p>In this paper, we introduce and study a new geometric constant <span>(D_varepsilon (X))</span> (<span>(varepsilon in [0,1))</span>) to measure the difference between approximate isosceles orthogonality and approximate Birkhoff-James orthogonality in real normed linear spaces <span>((X,Vert cdot Vert ))</span>. A characterization of inner product spaces is provided in terms of the constant <span>(D_varepsilon (X))</span>. From the inequality concerning approximate isosceles orthogonality, we obtain both lower and upper bounds for <span>(D_varepsilon (X))</span>. More precisely, we show that </p><div><div><span>$$begin{aligned} varepsilon +2(sqrt{2(1-varepsilon )}-1)le D_varepsilon (X)le 1 end{aligned}$$</span></div></div><p>for all <span>(varepsilon in (0,2(sqrt{2}-1)))</span>. We further demonstrate that the derived lower bound for <span>(D_varepsilon (X))</span> is sharp.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147938644","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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