Aequationes Mathematicae最新文献

{"title":"On some classes of multiplicative functions","authors":"Pentti Haukkanen","doi":"10.1007/s00010-024-01053-5","DOIUrl":"https://doi.org/10.1007/s00010-024-01053-5","url":null,"abstract":"<p>An arithmetical function <i>f</i> is multiplicative if <span>(f(1)=1)</span> and <span>(f(mn)=f(m)f(n))</span> whenever <i>m</i> and <i>n</i> are coprime. We study connections between certain subclasses of multiplicative functions, such as strongly multiplicative functions, over-multiplicative functions and totients. It appears, among others, that the over-multiplicative functions are exactly same as the totients and the strongly multiplicative functions are exactly same as the so-called level totients. All these functions satisfy nice arithmetical identities which are recursive in character.\u0000</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577947","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":"A note on the Radiant formula and its relations to the sliced Wasserstein distance","authors":"Gennaro Auricchio","doi":"10.1007/s00010-024-01049-1","DOIUrl":"10.1007/s00010-024-01049-1","url":null,"abstract":"<div><p>In this note, we show that the squared Wasserstein distance can be expressed as the average over the sphere of one dimensional Wasserstein distances. We name this new expression for the Wasserstein Distance <i>Radiant Formula</i>. Using this formula, we are able to highlight new connections between the Wasserstein distances and the Sliced Wasserstein distance, an alternative Wasserstein-like distance that is cheaper to compute.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01049-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578027","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":"Global centers of a family of cubic systems","authors":"Raul Felipe Appis,&nbsp;Jaume Llibre","doi":"10.1007/s00010-024-01051-7","DOIUrl":"10.1007/s00010-024-01051-7","url":null,"abstract":"<div><p>Consider the family of polynomial differential systems of degree 3, or simply cubic systems </p><div><div><span>$$ x' = y, quad y' = -x + a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x^3 + a_5 x^2 y + a_6 xy^2 + a_7 y^3, $$</span></div></div><p>in the plane <span>(mathbb {R}^2)</span>. An equilibrium point <span>((x_0,y_0))</span> of a planar differential system is a <i>center</i> if there is a neighborhood <span>(mathcal {U})</span> of <span>((x_0,y_0))</span> such that <span>(mathcal {U} backslash {(x_0,y_0)})</span> is filled with periodic orbits. When <span>(mathbb {R}^2setminus {(x_0,y_0)})</span> is filled with periodic orbits, then the center is a <i>global center</i>. For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797–2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01051-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577776","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":"On iterative roots of injective functions","authors":"Bojan Bašić,&nbsp;Stefan Hačko","doi":"10.1007/s00010-024-01047-3","DOIUrl":"10.1007/s00010-024-01047-3","url":null,"abstract":"<div><p>In 1951 Łojasiewicz found a necessary and sufficient condition for the existence of a <i>q</i>-iterative root of an arbitrary bijective function <i>g</i> for any <span>(qge 2)</span>. In this article we extend this result to the injective case. More precisely, a necessary and sufficient condition for the existence of an iterative root of an injective function is proved, and in the case of existence, the characterization and enumeration of all iterative roots are given. Furthermore, we devise a construction by which all iterative roots of an injective function can be constructed (provided that the considered function has at least one iterative root). As an illustration, we apply the developed theory to several results from the literature to obtain somewhat more elegant and shorter proofs of those results.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301117","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":"The minmin coalition number in graphs","authors":"Davood Bakhshesh, Michael A. Henning","doi":"10.1007/s00010-024-01045-5","DOIUrl":"https://doi.org/10.1007/s00010-024-01045-5","url":null,"abstract":"<p>A set <i>S</i> of vertices in a graph <i>G</i> is a dominating set if every vertex of <span>(V(G) setminus S)</span> is adjacent to a vertex in <i>S</i>. A coalition in <i>G</i> consists of two disjoint sets of vertices <i>X</i> and <i>Y</i> of <i>G</i>, neither of which is a dominating set but whose union <span>(X cup Y)</span> is a dominating set of <i>G</i>. Such sets <i>X</i> and <i>Y</i> form a coalition in <i>G</i>. A coalition partition, abbreviated <i>c</i>-partition, in <i>G</i> is a partition <span>({mathcal {X}} = {X_1,ldots ,X_k})</span> of the vertex set <i>V</i>(<i>G</i>) of <i>G</i> such that for all <span>(i in [k])</span>, each set <span>(X_i in {mathcal {X}})</span> satisfies one of the following two conditions: (1) <span>(X_i)</span> is a dominating set of <i>G</i> with a single vertex, or (2) <span>(X_i)</span> forms a coalition with some other set <span>(X_j in {mathcal {X}})</span>. Let <span>({{mathcal {A}}} = {A_1,ldots ,A_r})</span> and <span>({{mathcal {B}}}= {B_1,ldots , B_s})</span> be two partitions of <i>V</i>(<i>G</i>). Partition <span>({{mathcal {B}}})</span> is a refinement of partition <span>({{mathcal {A}}})</span> if every set <span>(B_i in {{mathcal {B}}} )</span> is either equal to, or a proper subset of, some set <span>(A_j in {{mathcal {A}}})</span>. Further if <span>({{mathcal {A}}} ne {{mathcal {B}}})</span>, then <span>({{mathcal {B}}})</span> is a proper refinement of <span>({{mathcal {A}}})</span>. Partition <span>({{mathcal {A}}})</span> is a minimal <i>c</i>-partition if it is not a proper refinement of another <i>c</i>-partition. Haynes et al. [AKCE Int. J. Graphs Combin. 17 (2020), no. 2, 653–659] defined the minmin coalition number <span>(c_{min }(G))</span> of <i>G</i> to equal the minimum order of a minimal <i>c</i>-partition of <i>G</i>. We show that <span>(2 le c_{min }(G) le n)</span>, and we characterize graphs <i>G</i> of order <i>n</i> satisfying <span>(c_{min }(G) = n)</span>. A polynomial-time algorithm is given to determine if <span>(c_{min }(G)=2)</span> for a given graph <i>G</i>. A necessary and sufficient condition for a graph <i>G</i> to satisfy <span>(c_{min }(G) ge 3)</span> is given, and a characterization of graphs <i>G</i> with minimum degree 2 and <span>(c_{min }(G)= 4)</span> is provided.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301216","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":"Alienation and stability of Jensen’s and other functional equations","authors":"Mohamed Tial, Driss Zeglami","doi":"10.1007/s00010-024-01046-4","DOIUrl":"https://doi.org/10.1007/s00010-024-01046-4","url":null,"abstract":"<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><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><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>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}