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Angular structure of Reuleaux cones 鲁莱欧锥的角度结构
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-25 DOI: 10.1007/s00010-024-01063-3
José Pedro Moreno, Alberto Seeger
{"title":"Angular structure of Reuleaux cones","authors":"José Pedro Moreno,&nbsp;Alberto Seeger","doi":"10.1007/s00010-024-01063-3","DOIUrl":"10.1007/s00010-024-01063-3","url":null,"abstract":"<div><p>In this note we exhibit some examples of proper cones that have the property of being of constant opening angle. In particular, we analyze the class of Reuleaux cones in <span>(mathbb {R}^n)</span> with <span>(nge 3)</span>. Such cones are constructed as intersection of <i>n</i> revolutions cones <span>(textrm{Rev}(g_1,psi ),ldots , textrm{Rev}(g_n,psi ))</span> whose incenters <span>(g_1,ldots , g_n)</span> are unit vectors forming a common angle. The half-aperture angle <span>(psi )</span> of each revolution cone corresponds to the common angle between the incenters. A major result of this work is that a Reuleaux cone in <span>(mathbb {R}^n)</span> is of constant opening angle if and only if <span>(n= 3)</span>. Reuleaux cones in dimension higher than 3 are not of constant opening angle, but such mathematical objects are still of interest. In the same way that a Reuleaux triangle is a “rounded” version of an equilateral triangle, a Reuleaux cone can be viewed as a rounded version of an equiangular simplicial cone and, therefore, it has a lot of symmetry in it.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"635 - 654"},"PeriodicalIF":0.9,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153159","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 the variable-order fractional derivatives with respect to another function 关于另一个函数的变阶分数导数
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-24 DOI: 10.1007/s00010-024-01082-0
Ricardo Almeida
{"title":"On the variable-order fractional derivatives with respect to another function","authors":"Ricardo Almeida","doi":"10.1007/s00010-024-01082-0","DOIUrl":"10.1007/s00010-024-01082-0","url":null,"abstract":"<div><p>In this paper, we present various concepts concerning generalized fractional calculus, wherein the fractional order of operators is not constant, and the integral kernel depends on a function. We observe that in the case of variable order, the concepts are distinct, and we present relations between them. Formulas for approximating fractional derivatives are provided, involving only integer-order derivatives. Finally, we conclude the work with some simulations to exemplify the method.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"805 - 822"},"PeriodicalIF":0.9,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01082-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141101425","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
Alienation of the quadratic, exponential and d’Alembert functional equations 二次函数方程、指数函数方程和达朗贝尔函数方程的异化
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-20 DOI: 10.1007/s00010-024-01084-y
Marcin Adam
{"title":"Alienation of the quadratic, exponential and d’Alembert functional equations","authors":"Marcin Adam","doi":"10.1007/s00010-024-01084-y","DOIUrl":"10.1007/s00010-024-01084-y","url":null,"abstract":"<div><p>Let <span>((S,+,0))</span> be a commutative monoid, <span>(sigma :Srightarrow S)</span> be an endomorphism with <span>(sigma ^2=id)</span> and let <i>K</i> be a field of characteristic different from 2. We study the solutions <span>(f,g,h:Srightarrow K)</span> of the Pexider type functional equation </p><div><div><span>$$begin{aligned} f(x+y)+f(x+sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) end{aligned}$$</span></div></div><p>resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces <i>f</i> and <i>g</i> to split back into the system of two equations </p><div><div><span>$$begin{aligned} left{ begin{array}{ll}f(x+y)+f(x+sigma y)=2f(x)+2f(y) g(x+y)=g(x)g(y)end{array}right. end{aligned}$$</span></div></div><p>for all <span>(x,yin S)</span> (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"411 - 432"},"PeriodicalIF":0.9,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01084-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141122168","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 note on homotopy extension KKM type maps 关于同调扩展 KKM 类型映射的说明
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-18 DOI: 10.1007/s00010-024-01081-1
Donal O’Regan
{"title":"A note on homotopy extension KKM type maps","authors":"Donal O’Regan","doi":"10.1007/s00010-024-01081-1","DOIUrl":"10.1007/s00010-024-01081-1","url":null,"abstract":"<div><p>In this paper we present a variety of continuation (homotopy) theorems for general classes of maps in the literature.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"555 - 576"},"PeriodicalIF":0.9,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060702","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
The Rhodes semilattice of a biased graph 偏置图的罗得半网格
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-15 DOI: 10.1007/s00010-024-01039-3
Michael J. Gottstein, Thomas Zaslavsky
{"title":"The Rhodes semilattice of a biased graph","authors":"Michael J. Gottstein,&nbsp;Thomas Zaslavsky","doi":"10.1007/s00010-024-01039-3","DOIUrl":"10.1007/s00010-024-01039-3","url":null,"abstract":"<div><p>We reinterpret the Rhodes semilattices <span>(R_n({mathfrak {G}}))</span> of a group <span>({mathfrak {G}})</span> in terms of gain graphs and generalize them to all gain graphs, both as sets of partition-potential pairs and as sets of subgraphs, and for the latter, further to biased graphs. Based on this we propose four different natural lattices in which the Rhodes semilattices and its generalizations are order ideals.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1677 - 1687"},"PeriodicalIF":0.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141060703","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
Shannon’s entropy and its bounds for some a priori known equiprobable states 香农熵及其对某些先验已知等价状态的约束
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-15 DOI: 10.1007/s00010-024-01068-y
Eleutherius Symeonidis, Flavia-Corina Mitroi-Symeonidis
{"title":"Shannon’s entropy and its bounds for some a priori known equiprobable states","authors":"Eleutherius Symeonidis,&nbsp;Flavia-Corina Mitroi-Symeonidis","doi":"10.1007/s00010-024-01068-y","DOIUrl":"10.1007/s00010-024-01068-y","url":null,"abstract":"<div><p>It is known that Shannon’s entropy is nonnegative and its maximum value is reached for equiprobable events. Adding or removing impossible events does not affect Shannon’s entropy. However, if we increase the number of events and consider not necessarily all of them equiprobable, but at least as many of them as the initial number of equiprobable events, how does Shannon’s entropy change? We study the lower bound of the interval where the probability value of the a priori assumed equiprobable states must belong when the entropy increases.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"237 - 242"},"PeriodicalIF":0.9,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01068-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140974128","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 an alternative additive-quadratic functional equation 关于另一种加二次函数方程
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-13 DOI: 10.1007/s00010-024-01074-0
Gian Luigi Forti, Bettina Wilkens
{"title":"On an alternative additive-quadratic functional equation","authors":"Gian Luigi Forti,&nbsp;Bettina Wilkens","doi":"10.1007/s00010-024-01074-0","DOIUrl":"10.1007/s00010-024-01074-0","url":null,"abstract":"<div><p>We consider a map <i>f</i> from one abelian group into another that satisfies either an additive or quadratic functional equation on any given pair of elements of its domain. Particular emphasis is placed on the possibility that <i>f</i> itself is neither additive nor quadratic and a complete description of all those cases is obtained.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"591 - 610"},"PeriodicalIF":0.9,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925574","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
Behavior of convex integrand at a d-apex of its Wulff shape and approximation of spherical bodies of constant width 凸积分在其 Wulff 形的 d-apex 处的行为和恒定宽度球体的近似值
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-11 DOI: 10.1007/s00010-024-01079-9
Huhe Han
{"title":"Behavior of convex integrand at a d-apex of its Wulff shape and approximation of spherical bodies of constant width","authors":"Huhe Han","doi":"10.1007/s00010-024-01079-9","DOIUrl":"10.1007/s00010-024-01079-9","url":null,"abstract":"<div><p>Let <span>(gamma : S^nrightarrow mathbb {R}_+)</span> be a convex integrand and <span>(mathcal {W}_gamma )</span> be the Wulff shape of <span>(gamma )</span>. A d-apex point naturally arises in a non-smooth Wulff shape, in particular, as a vertex of a convex polytope. In this paper, we study the behavior of the convex integrand at a d-apex point of its Wulff shape. We prove that <span>(gamma (P))</span> is locally maximum, and <span>(mathbb {R}_+ Pcap partial mathcal {W}_gamma )</span> is a d-apex point of <span>(mathcal {W}_gamma )</span> if and only if the graph of <span>(gamma )</span> around the d-apex point is a piece of a sphere with center <span>(frac{1}{2}gamma (P)P)</span> and radius <span>(frac{1}{2}gamma (P))</span>. As an application of the proof of this result, we prove that for any spherical convex body <i>C</i> of constant width <span>(tau &gt;pi /2)</span>, there exists a sequence <span>({C_i}_{i=1}^infty )</span> of convex bodies of constant width <span>(tau )</span>, whose boundaries consist only of arcs of circles of radius <span>(tau -frac{pi }{2})</span> and great circle arcs such that <span>(lim _{irightarrow infty }C_i=C)</span> with respect to the Hausdorff distance.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"397 - 410"},"PeriodicalIF":0.9,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925542","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 ideal C $$^*$$ -completions and amenability 关于理想 C $$^*$$ -补全和可亲和性的说明
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-10 DOI: 10.1007/s00010-024-01077-x
Tomasz Kochanek
{"title":"A note on ideal C $$^*$$ -completions and amenability","authors":"Tomasz Kochanek","doi":"10.1007/s00010-024-01077-x","DOIUrl":"https://doi.org/10.1007/s00010-024-01077-x","url":null,"abstract":"<p>For a discrete group <i>G</i>, we consider certain ideals <span>(mathcal {I}subset c_0(G))</span> of sequences with prescribed rate of convergence to zero. We show that the equality between the full group C<span>(^*)</span>-algebra of <i>G</i> and the C<span>(^*)</span>-completion <span>(textrm{C}^*_{mathcal {I}}(G))</span> in the sense of Brown and Guentner (Bull. London Math. Soc. 45:1181–1193, 2013) implies that <i>G</i> is amenable.\u0000</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"156 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942333","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 double Roman domination problem for several graph classes 关于几类图的双罗马支配问题
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-05-08 DOI: 10.1007/s00010-024-01071-3
Tatjana Zec, Dragan Matić, Marko Djukanović
{"title":"On double Roman domination problem for several graph classes","authors":"Tatjana Zec,&nbsp;Dragan Matić,&nbsp;Marko Djukanović","doi":"10.1007/s00010-024-01071-3","DOIUrl":"10.1007/s00010-024-01071-3","url":null,"abstract":"<div><p><i>A double Roman domination function</i> (DRDF) on a graph <span>(G=(V,E))</span> is a mapping <span>(f :Vrightarrow {0,1,2,3})</span> satisfying the conditions: (<i>i</i>) each vertex with 0 assigned is adjacent to a vertex with 3 assigned or at least two vertices with 2 assigned and (<i>ii</i>) each vertex with 1 assigned is adjacent to at least one vertex with 2 or 3 assigned. The weight of a DRDF <i>f</i> is defined as the sum <span>(sum _{vin V}f(v))</span>. The minimum weight of a DRDF on a graph <i>G</i> is called the <i>double Roman domination number</i> (DRDN) of <i>G</i>. This study establishes the values on DRDN for several graph classes. The exact values of DRDN are proved for Kneser graphs <span>(K_{n,k},nge k(k+2))</span>, Johnson graphs <span>(J_{n,2})</span>, for a few classes of convex polytopes, and the flower snarks. Moreover, tight lower and upper bounds on SRDN are proved for some convex polytopes. For the generalized Petersen graphs <span>(P_{n,3}, n not equiv 0,(mathrm {mod 4}))</span>, we make a further improvement on the best known upper bound from the literature.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 2","pages":"439 - 463"},"PeriodicalIF":0.9,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140925658","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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