Aequationes Mathematicae最新文献

{"title":"About Sobolev spaces on fractals: fractal gradians and Laplacians","authors":"Alireza Khalili Golmankhaneh, Palle E. T. Jørgensen, Cristina Serpa, Kerri Welch","doi":"10.1007/s00010-024-01060-6","DOIUrl":"https://doi.org/10.1007/s00010-024-01060-6","url":null,"abstract":"<p>The paper covers the foundations of fractal calculus on fractal curves, defines different function classes, establishes vector spaces for <span>(F^{alpha })</span>-integrable functions, introduces local fractal integrable functions and fractal distribution functionals, defines the dual space of a fractal function space, proves completeness for <span>(F^{alpha })</span>-differentiable function spaces, defines Fractal Sobolev spaces, and introduces fractal gradian and fractal Laplace operators on fractal Hilbert spaces. It also presents criteria for the existence of unique solutions to fractal differential equations.\u0000</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"57 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608423","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":"Curves that allow the motion of a regular polygon","authors":"David Rochera","doi":"10.1007/s00010-024-01054-4","DOIUrl":"https://doi.org/10.1007/s00010-024-01054-4","url":null,"abstract":"<p>This paper characterizes curves where a regular polygon of either a variable side length or a constant side length is allowed to rotate during <i>k</i> full revolutions while having its vertices on the curve during the motion. A constructive method to generate these curves is given based on the curve described by the polygon centers (centroids) during the motion and some examples are shown. Moreover, if the regular polygon divides the curve perimeter into parts of equal length, it is proved that the curve is either a rotational symmetric curve in the case of a variable side length or a circle otherwise. Finally, in the case of a regular polygon of constant side length rotating along a curve, a simple relation between the algebraic areas of such a curve and the curve of polygon centers is revisited.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"90 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577789","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 characterization of the Euclidean ball via antipodal points","authors":"Xuguang Lu","doi":"10.1007/s00010-024-01055-3","DOIUrl":"10.1007/s00010-024-01055-3","url":null,"abstract":"<div><p>Arising from an equilibrium state of a Fermi–Dirac particle system at the lowest temperature, a new characterization of the Euclidean ball is proved: a compact set <span>(Ksubset {{{mathbb {R}}}^n})</span> (having at least two elements) is an <i>n</i>-dimensional Euclidean ball if and only if for every pair <span>(x, yin partial K)</span> and every <span>(sigma in {{{mathbb {S}}}^{n-1}})</span>, either <span>(frac{1}{2}(x+y)+frac{1}{2}|x-y|sigma in K)</span> or <span>(frac{1}{2}(x+y)-frac{1}{2}|x-y|sigma in K)</span>. As an application, a measure version of this characterization of the Euclidean ball is also proved and thus the previous result proved for <span>(n=3)</span> on the classification of equilibrium states of a Fermi–Dirac particle system holds also true for all <span>(nge 2)</span>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 3","pages":"637 - 660"},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01055-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577781","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":"The cosine addition and subtraction formulas on non-abelian groups","authors":"Omar Ajebbar,&nbsp;Elhoucien Elqorachi,&nbsp;Henrik Stetkær","doi":"10.1007/s00010-024-01052-6","DOIUrl":"10.1007/s00010-024-01052-6","url":null,"abstract":"<div><p>Let <i>G</i> be a topological group, and let <i>C</i>(<i>G</i>) denote the algebra of continuous, complex valued functions on <i>G</i>. We determine the solutions <span>(f,g,h in C(G))</span> of the Levi-Civita equation </p><div><div><span>$$begin{aligned} g(xy) = g(x)g(y) + f(x)h(y), x,y in G, end{aligned}$$</span></div></div><p>that extends the cosine addition law. As a corollary we obtain the solutions <span>(f,g in C(G))</span> of the cosine subtraction law <span>(g(xy^*) = g(x)g(y) + f(x)f(y))</span>, <span>(x,y in G)</span> where <span>(x mapsto x^*)</span> is a continuous involution of <i>G</i>. That <span>(x mapsto x^*)</span> is an involution, means that <span>((xy)^* = y^*x^*)</span> and <span>(x^{**} = x)</span> for all <span>(x,y in G)</span>.\u0000</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1657 - 1676"},"PeriodicalIF":0.9,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01052-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140577778","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 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":"86 1","pages":""},"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":"98 5","pages":"1317 - 1332"},"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":"98 5","pages":"1373 - 1389"},"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":"Means of Cauchy’s difference type","authors":"Janusz Matkowski","doi":"10.1007/s00010-024-01044-6","DOIUrl":"10.1007/s00010-024-01044-6","url":null,"abstract":"<div><p><i>k</i>-variable means which are the Cauchy differences of additive type generated by a real single variable function <i>f</i>, and denoted by <span>(C_{f,k})</span>, are examined. It is shown that <span>(C_{f,k})</span> is an increasing mean in <span>(left( 0,infty right) )</span> iff <i>f</i> is a convex solution of the (reflexivity) functional equation <span>(fleft( kxright) -kfleft( xright) =x)</span>, and a construction of a large class of such means is presented. The form of a unique homogeneous mean of the form <span>(C_{f,k})</span> is given. As corollaries, the suitable results for the Cauchy differences of exponential, logarithmic and multiplicative types are obtained. It is shown that there exists a unique continuous and differentiable at 0 function <i>f</i> such that <span>(Mleft( x,yright) :=fleft( x+yright) -fleft( xright) fleft( yright) )</span> is a bivariable premean in <span>(mathbb {R})</span>, and its analyticity is proved. Finding the explicit form of <i>f</i> is one of the proposed open questions.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"89 - 105"},"PeriodicalIF":0.9,"publicationDate":"2024-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140362502","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":"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":"98 3","pages":"697 - 726"},"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,&nbsp;Michael A. Henning","doi":"10.1007/s00010-024-01045-5","DOIUrl":"10.1007/s00010-024-01045-5","url":null,"abstract":"<div><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></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"99 1","pages":"223 - 236"},"PeriodicalIF":0.9,"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}