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On the minimality of the Winterbottom shape 论温特巴顿形状的最小性
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-17 DOI: 10.1007/s00010-024-01122-9
Shokhrukh Yu. Kholmatov
{"title":"On the minimality of the Winterbottom shape","authors":"Shokhrukh Yu. Kholmatov","doi":"10.1007/s00010-024-01122-9","DOIUrl":"https://doi.org/10.1007/s00010-024-01122-9","url":null,"abstract":"<p>In this short note we prove that the Winterbottom shape (Winterbottom in Acta Metallurgica 15:303-310, 1967) is a volume-constraint minimizer of the corresponding anisotropic capillary functional.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266631","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
Two-sided delay-difference equations and evolution maps 双侧延迟差分方程和演化图
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-14 DOI: 10.1007/s00010-024-01121-w
Luís Barreira, Claudia Valls
{"title":"Two-sided delay-difference equations and evolution maps","authors":"Luís Barreira, Claudia Valls","doi":"10.1007/s00010-024-01121-w","DOIUrl":"https://doi.org/10.1007/s00010-024-01121-w","url":null,"abstract":"<p>We establish the equivalence of hyperbolicity and of two other properties for a two-sided linear delay-difference equation and its evolution map. These two properties are the admissibility with respect to various pairs of spaces, and the Ulam–Hyers stability of the equation, again with respect to various spaces. This gives characterizations of important properties of a linear dynamical system in terms of corresponding properties of the autonomous dynamical system determined by the associated evolution map.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142266632","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
Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime 广义立方分区和以质数为模数的过分区的算术性质
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-10 DOI: 10.1007/s00010-024-01116-7
Tewodros Amdeberhan, James A. Sellers, Ajit Singh
{"title":"Arithmetic properties for generalized cubic partitions and overpartitions modulo a prime","authors":"Tewodros Amdeberhan, James A. Sellers, Ajit Singh","doi":"10.1007/s00010-024-01116-7","DOIUrl":"https://doi.org/10.1007/s00010-024-01116-7","url":null,"abstract":"<p>A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194427","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
Min-phase-isometries on the unit sphere of $$mathcal {L}^infty (Gamma )$$ -type spaces $$mathcal {L}^infty (Gamma )$$ 型空间单位球上的最小相位等分线
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-10 DOI: 10.1007/s00010-024-01119-4
Dongni Tan, Lu Yuan, Peng Yang
{"title":"Min-phase-isometries on the unit sphere of $$mathcal {L}^infty (Gamma )$$ -type spaces","authors":"Dongni Tan, Lu Yuan, Peng Yang","doi":"10.1007/s00010-024-01119-4","DOIUrl":"https://doi.org/10.1007/s00010-024-01119-4","url":null,"abstract":"<p>We show that every surjective mapping <i>f</i> between the unit spheres of two real <span>(mathcal {L}^infty (Gamma ))</span>-type spaces satisfies </p><span>$$begin{aligned} min {Vert f(x)+f(y)Vert ,Vert f(x)-f(y)Vert }=min {Vert x+yVert ,Vert x-yVert }quad (x,yin S_X) end{aligned}$$</span><p>if and only if <i>f</i> is phase-equivalent to an isometry, i.e., there is a phase-function <span>(varepsilon )</span> from the unit sphere of the <span>(mathcal {L}^infty (Gamma ))</span>-type space onto <span>({-1,1})</span> such that <span>(varepsilon cdot f)</span> is a surjective isometry between the unit spheres of two real <span>(mathcal {L}^infty (Gamma ))</span>-type spaces, and furthermore, this isometry can be extended to a linear isometry on the whole space <span>(mathcal {L}^infty (Gamma ))</span>. We also give an example to show that these are not true if “min” is replaced by “max”.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194428","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
Multivariable generalizations of bivariate means via invariance 通过不变性对二元均值进行多变量概括
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-09 DOI: 10.1007/s00010-024-01113-w
Paweł Pasteczka
{"title":"Multivariable generalizations of bivariate means via invariance","authors":"Paweł Pasteczka","doi":"10.1007/s00010-024-01113-w","DOIUrl":"https://doi.org/10.1007/s00010-024-01113-w","url":null,"abstract":"<p>For a given <i>p</i>-variable mean <span>(M :I^p rightarrow I)</span> (<i>I</i> is a subinterval of <span>({mathbb {R}})</span>), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its <span>((p+1))</span>-variable <span>(beta )</span>-invariant extension as the unique solution <span>(K :I^{p+1} rightarrow I)</span> of the functional equation </p><span>$$begin{aligned}&amp;Kbig (M(x_2,dots ,x_{p+1}),M(x_1,x_3,dots ,x_{p+1}),dots ,M(x_1,dots ,x_p)big )&amp;quad =K(x_1,dots ,x_{p+1}), text { for all }x_1,dots ,x_{p+1} in I end{aligned}$$</span><p>in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194426","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 multiplicative functions which are additive on positive cubes 关于正立方体上相加的乘法函数
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-09 DOI: 10.1007/s00010-024-01118-5
Poo-Sung Park
{"title":"On multiplicative functions which are additive on positive cubes","authors":"Poo-Sung Park","doi":"10.1007/s00010-024-01118-5","DOIUrl":"https://doi.org/10.1007/s00010-024-01118-5","url":null,"abstract":"<p>Let <span>(k ge 3)</span>. If a multiplicative function <i>f</i> satisfies </p><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><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><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><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>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"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}
引用次数: 0
Three inequalities that characterize the exponential function 指数函数的三个不等式
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-06 DOI: 10.1007/s00010-024-01115-8
David M. Bradley
{"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":null,"pages":null},"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}
引用次数: 0
Overpartitions in terms of 2-adic valuation 从 2-adic 估值角度看过度分区
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-06 DOI: 10.1007/s00010-024-01117-6
Mircea Merca
{"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":null,"pages":null},"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}
引用次数: 0
Characterizing spanning trees via the size or the spectral radius of graphs 通过图的大小或谱半径确定生成树的特征
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-09-04 DOI: 10.1007/s00010-024-01112-x
Jie Wu
{"title":"Characterizing spanning trees via the size or the spectral radius of graphs","authors":"Jie Wu","doi":"10.1007/s00010-024-01112-x","DOIUrl":"https://doi.org/10.1007/s00010-024-01112-x","url":null,"abstract":"<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>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"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}
引用次数: 0
A class of functional equations for additive functions 一类加法函数方程
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-08-13 DOI: 10.1007/s00010-024-01105-w
Bruce Ebanks
{"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":null,"pages":null},"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}
引用次数: 0
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