Aequationes Mathematicae最新文献

{"title":"On multiplicative functions which are additive on positive cubes","authors":"Poo-Sung Park","doi":"10.1007/s00010-024-01118-5","DOIUrl":"10.1007/s00010-024-01118-5","url":null,"abstract":"<div><p>Let <span>(k ge 3)</span>. If a multiplicative function <i>f</i> satisfies </p><div><div><span>$$begin{aligned} f(a_1^3 + a_2^3 + cdots + a_k^3) = f(a_1^3) + f(a_2^3) + cdots + f(a_k^3) end{aligned}$$</span></div></div><p>for all <span>(a_1, a_2, ldots , a_k in {mathbb {N}})</span>, then <i>f</i> is the identity function. The set of positive cubes is said to be a <i>k</i>-additive uniqueness set for multiplicative functions. But, the condition <span>(k=2)</span> can be satisfied by infinitely many multiplicative functions. In additon, if <span>(k ge 3)</span> and a multiplicative function <i>g</i> satisfies </p><div><div><span>$$begin{aligned} g(a_1^3 + a_2^3 + cdots + a_k^3) = g(a_1)^3 + g(a_2)^3 + cdots + g(a_k)^3 end{aligned}$$</span></div></div><p>for all <span>(a_1, a_2, ldots , a_k in {mathbb {N}})</span>, then <i>g</i> is the identity function. However, when <span>(k=2)</span>, there exist three different types of multiplicative functions.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1457 - 1474"},"PeriodicalIF":0.9,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194425","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":"Three inequalities that characterize the exponential function","authors":"David M. Bradley","doi":"10.1007/s00010-024-01115-8","DOIUrl":"https://doi.org/10.1007/s00010-024-01115-8","url":null,"abstract":"<p>Three functional inequalities are shown to uniquely characterize the exponential function. Each of the three inequalities is indispensable in the sense that no two of the three suffice.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"39 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194455","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":"Overpartitions in terms of 2-adic valuation","authors":"Mircea Merca","doi":"10.1007/s00010-024-01117-6","DOIUrl":"https://doi.org/10.1007/s00010-024-01117-6","url":null,"abstract":"<p>In this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of <i>m</i>-Modular Nim with two heaps are presented in this context.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"22 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194429","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":"Characterizing spanning trees via the size or the spectral radius of graphs","authors":"Jie Wu","doi":"10.1007/s00010-024-01112-x","DOIUrl":"10.1007/s00010-024-01112-x","url":null,"abstract":"<div><p>Let <i>G</i> be a connected graph and let <span>(kge 1)</span> be an integer. Let <i>T</i> be a spanning tree of <i>G</i>. The leaf degree of a vertex <span>(vin V(T))</span> is defined as the number of leaves adjacent to <i>v</i> in <i>T</i>. The leaf degree of <i>T</i> is the maximum leaf degree among all the vertices of <i>T</i>. Let |<i>E</i>(<i>G</i>)| and <span>(rho (G))</span> denote the size and the spectral radius of <i>G</i>, respectively. In this paper, we first create a lower bound on the size of <i>G</i> to ensure that <i>G</i> admits a spanning tree with leaf degree at most <i>k</i>. Then we establish a lower bound on the spectral radius of <i>G</i> to guarantee that <i>G</i> contains a spanning tree with leaf degree at most <i>k</i>. Finally, we create some extremal graphs to show all the bounds obtained in this paper are sharp.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 6","pages":"1441 - 1455"},"PeriodicalIF":0.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194454","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 class of functional equations for additive functions","authors":"Bruce Ebanks","doi":"10.1007/s00010-024-01105-w","DOIUrl":"https://doi.org/10.1007/s00010-024-01105-w","url":null,"abstract":"<p>The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form (<span>(*)</span>) <span>(sum _{j=1}^{k} x^{p_j}f_j(x^{q_j}) = 0)</span>, where the <span>(p_j,q_j)</span> are non-negative integers, the <span>(f_j:R rightarrow S)</span> are additive functions, <i>S</i> is a commutative ring, and <i>R</i> is a sub-ring of <i>S</i>. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form (<span>(*)</span>) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et al. (Results Math 73(2):27, 2018). It seems that this fact may have been overlooked, judging by some results about a particular case of (<span>(*)</span>) in recent publications. We also present a new method for the homogeneous case by combining the results above with [6], and we show how to solve non-homogeneous equations of the form (<span>(*)</span>).</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194456","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":"Correction to: Linked pairs of additive functions","authors":"Bruce Ebanks","doi":"10.1007/s00010-024-01108-7","DOIUrl":"10.1007/s00010-024-01108-7","url":null,"abstract":"","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1439 - 1440"},"PeriodicalIF":0.9,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142410095","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 the solvability of the (SSIE) with operator $${D}_{x} {mathbf { * }}left( { s}_{ R}^{{textbf{0}}} right) _{{{{Sigma }} - {lambda I}}} {{ subset }}{ s}_{ R}^{{{0}}} $$ , involving the fine spectrum of an operator","authors":"Bruno de Malafosse, Eberhard Malkowsky, Vladinir Rakočević","doi":"10.1007/s00010-024-01111-y","DOIUrl":"https://doi.org/10.1007/s00010-024-01111-y","url":null,"abstract":"<p>Given any sequence <span>(a=(a_{n})_{nge 1})</span> of positive real numbers and any set <i>E</i> of complex sequences, we write <span>(E_{a})</span> for the set of all sequences <span>(y=(y_{n})_{nge 1})</span> such that <span>(y/a=(y_{n}/a_{n})_{nge 1}in E)</span>; in particular, <span>(c_{a})</span> denotes the set of all sequences <i>y</i> such that <i>y</i>/<i>a</i> converges. In this paper, we use the sum operator <span>(Sigma )</span>, defined by <span>(Sigma _{n}y=sum _{k=1}^{n}y_{k})</span> for all sequences <i>y</i>, and we determine its spectrum over each of the sets <span>(s_{a}=(ell _{infty })_{a})</span> and <span>(s_{a}^{0}=(c_{0})_{a})</span>. Then we determine the point, residual and continuous spectra of the operator <span>(D_{1/R}Sigma D_{R})</span>, with <span>(R&gt;1)</span>, and we solve the special <i>sequence spaces inclusion equations (SSIE),</i> (which are determined by an inclusion, for which each term is a <i>sum </i>or<i> a sum of products of sets of the form </i><span>((E_{a})_{mathcal {T}})</span><i> and </i> <span>(( E_{f(x)})_{mathcal {T}})</span> where <i>f</i> maps <span>(U^{+})</span> to itself, <i>E</i> is any linear space of sequences and <span>(mathcal {T})</span> is a triangle) <span>(D_{x}*(s_{R}^{0})_{Sigma -lambda I}subset s_{R}^{0})</span>, using the fine spectrum of this operator. The solvability of this (SSIE) consists in determining, for each <span>(lambda in mathbb {C})</span>, the set of all sequences <span>(xin omega )</span> that satisfy the next statement. For every <span>(yin omega )</span>, we have </p><span>$$begin{aligned} lim _{nrightarrow infty }frac{1}{R^{n}}left( sum _{k=1}^{n-1}y_{k}-lambda y_{n}right) =0Longrightarrow lim _{nrightarrow infty }x_{n}left( frac{y_{n}}{R^{n}}right) =0text {.} end{aligned}$$</span><p>Then, we solve this (SSIE) for <span>(R=1)</span>. Finally, we solve each (SSIE) <span>(D_{x}*( E_{R})_{Sigma -lambda I}subset s_{R})</span>, where <i>E</i> is successively equal to <span>(c_{0})</span>, <i>c</i>, and <span>(ell _{infty })</span>.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"186 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141885057","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":"Some new minimax inequalities for centered convex bodies","authors":"Horst Martini, Zokhrab Mustafaev, Sakhavet M. Zarbaliev","doi":"10.1007/s00010-024-01109-6","DOIUrl":"https://doi.org/10.1007/s00010-024-01109-6","url":null,"abstract":"<p>The purpose of this manuscript is to derive some new minimax inequalities for centered convex bodies in <span>({mathbb {R}}^{d})</span>. As consequences, new characterizations of ellipsoids will be established as well. Some open problems related to the Busemann-Petty list of problems will also be discussed.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"74 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865325","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":"Another look at the Matkowski and Wesołowski problem yielding a new class of solutions","authors":"Janusz Morawiec, Thomas Zürcher","doi":"10.1007/s00010-024-01110-z","DOIUrl":"https://doi.org/10.1007/s00010-024-01110-z","url":null,"abstract":"<p>The following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions <span>(varphi :[0,1]rightarrow [0,1])</span>, distinct from the identity on [0, 1], such that <span>(varphi (0)=0)</span>, <span>(varphi (1)=1)</span> and <span>(varphi (x)=varphi (frac{x}{2})+varphi (frac{x+1}{2})-varphi (frac{1}{2}))</span> for every <span>(xin [0,1])</span>? By now, it is known that each of the de Rham functions <span>(R_p)</span>, where <span>(pin (0,1))</span>, is a solution of the MW-problem, and for any Borel probability measure <span>(mu )</span> concentrated on (0, 1) the formula <span>(phi _mu (x)=int _{(0,1)}R_p(x), dmu (p))</span> defines a solution <span>(phi _mu :[0,1]rightarrow [0,1])</span> of this problem as well. In this paper, we give a new family of solutions of the MW-problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW-problem that are not of the above integral form with any Borel probability measure <span>(mu )</span>.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"46 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141865359","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 spectral irregularity of graphs","authors":"Lu Zheng,&nbsp;Bo Zhou","doi":"10.1007/s00010-024-01106-9","DOIUrl":"10.1007/s00010-024-01106-9","url":null,"abstract":"<div><p>The spectral radius <span>(rho (G))</span> of a graph <i>G</i> is the largest eigenvalue of the adjacency matrix of <i>G</i>. For a graph <i>G</i> with maximum degree <span>(Delta (G))</span>, it is known that <span>(rho (G)le Delta (G))</span> with equality when <i>G</i> is connected if and only if <i>G</i> is regular. So the quantity <span>(beta (G)=Delta (G)-rho (G))</span> is a spectral measure of irregularity of <i>G</i>. In this paper, we identify the trees of order <span>(nge 12)</span> with the first 15 largest <span>(beta )</span>-values, the unicyclic graphs of order <span>(nge 17)</span> with the first 16 largest <span>(beta )</span>-values, as well as the bicyclic graphs of order <span>(nge 30)</span> with the first 11 largest <span>(beta )</span>-values. We also determine the graphs with the largest <span>(beta )</span>-values among all connected graphs with given order and clique number.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"98 5","pages":"1161 - 1176"},"PeriodicalIF":0.9,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745756","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}