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Semi-classical 2-dimensional Minkowski planes 半经典二维闵科夫斯基平面
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-23 DOI: 10.1007/s00010-024-01056-2
Günter F. Steinke
{"title":"Semi-classical 2-dimensional Minkowski planes","authors":"Günter F. Steinke","doi":"10.1007/s00010-024-01056-2","DOIUrl":"https://doi.org/10.1007/s00010-024-01056-2","url":null,"abstract":"","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140666367","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
Position of the centroid of a planar convex body 平面凸体的中心点位置
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-23 DOI: 10.1007/s00010-024-01058-0
Marek Lassak
{"title":"Position of the centroid of a planar convex body","authors":"Marek Lassak","doi":"10.1007/s00010-024-01058-0","DOIUrl":"10.1007/s00010-024-01058-0","url":null,"abstract":"<div><p>It is well known that any planar convex body <i>A</i> permits to inscribe an affine-regular hexagon <span>(H_A)</span>. We prove that the centroid of <i>A</i> belongs to the homothetic image of <span>(H_A)</span> with ratio <span>(frac{4}{21})</span> and the center in the center of <span>(H_A)</span>. This ratio cannot be decreased.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-024-01058-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140669872","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
Matrix inequalities between $$f(A)sigma f(B)$$ and $$Asigma B$$ $$f(A)sigma f(B)$$ 与 $$Asigma B$ 之间的矩阵不等式
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-23 DOI: 10.1007/s00010-024-01059-z
Manisha Devi, Jaspal Singh Aujla, Mohsen Kian, Mohammad Sal Moslehian
{"title":"Matrix inequalities between $$f(A)sigma f(B)$$ and $$Asigma B$$","authors":"Manisha Devi, Jaspal Singh Aujla, Mohsen Kian, Mohammad Sal Moslehian","doi":"10.1007/s00010-024-01059-z","DOIUrl":"https://doi.org/10.1007/s00010-024-01059-z","url":null,"abstract":"<p>Let <i>A</i> and <i>B</i> be <span>(ntimes n)</span> positive definite complex matrices, let <span>(sigma )</span> be a matrix mean, and let <span>(f: [0,infty )rightarrow [0,infty ))</span> be a differentiable convex function with <span>(f(0)=0)</span>. We prove that </p><span>$$begin{aligned} f^{prime }(0)(A sigma B)le frac{f(m)}{m}(Asigma B)le f(A)sigma f(B)le frac{f(M)}{M}(Asigma B)le f^{prime }(M)(Asigma B), end{aligned}$$</span><p>where <i>m</i> represents the smallest eigenvalues of <i>A</i> and <i>B</i> and <i>M</i> represents the largest eigenvalues of <i>A</i> and <i>B</i>. If <i>f</i> is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if <i>f</i>(<i>x</i>)/<i>x</i> is increasing, then </p><span>$$begin{aligned} |||f(A)+f(B)|||le frac{f(M)}{M} |||A+B|||le |||f(A+B)||| end{aligned}$$</span><p>holds for all <i>A</i> and <i>B</i> with <span>(Mle A+B)</span>. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski’s determinant inequality.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799775","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
Rotation number of 2-interval piecewise affine maps 两区间片断仿射映射的旋转数
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-22 DOI: 10.1007/s00010-024-01064-2
José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira
{"title":"Rotation number of 2-interval piecewise affine maps","authors":"José Pedro Gaivão, Michel Laurent, Arnaldo Nogueira","doi":"10.1007/s00010-024-01064-2","DOIUrl":"https://doi.org/10.1007/s00010-024-01064-2","url":null,"abstract":"<p>We study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps <span>(f_{varvec{p}})</span> are parametrized by a quintuple <span>(varvec{p})</span> of real numbers satisfying inequations. Viewing <span>(f_{varvec{p}})</span> as a circle map, we show that it has a rotation number <span>(rho (f_{varvec{p}}))</span> and we compute <span>(rho (f_{varvec{p}}))</span> as a function of <span>(varvec{p})</span> in terms of Hecke–Mahler series. As a corollary, we prove that <span>(rho (f_{varvec{p}}))</span> is a rational number when the components of <span>(varvec{p})</span> are algebraic numbers.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140799835","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
Set valued pexiderized quadratic functional equation 集合值化的二次函数方程
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-21 DOI: 10.1007/s00010-024-01067-z
Elham Mohammadi, Abbas Najati, Kazimierz Nikodem
{"title":"Set valued pexiderized quadratic functional equation","authors":"Elham Mohammadi, Abbas Najati, Kazimierz Nikodem","doi":"10.1007/s00010-024-01067-z","DOIUrl":"https://doi.org/10.1007/s00010-024-01067-z","url":null,"abstract":"<p>Consider a real vector space denoted as <i>X</i>, and let <i>cc</i>(<i>Y</i>) represent the collection of all convex and compact subsets of a real Hausdorff topological vector space <i>Y</i>. This paper investigates set-valued solutions of the Pexiderized quadratic functional equation </p><span>$$begin{aligned} f_1(x+y)+f_2(x-y)=f_3(x)+f_4(y), end{aligned}$$</span><p>for unknown functions <span>(f_1,f_2,f_3,f_4:Xrightarrow cc(Y))</span>. This functional equation incorporates many functional equations including the quadratic, Cauchy’s and Drygas’ equations. A characterization for set-valued solutions of this functional equation is presented in this paper.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623865","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
Generalizing the concept of bounded variation 推广有界变化的概念
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-20 DOI: 10.1007/s00010-024-01050-8
Angshuman R. Goswami
{"title":"Generalizing the concept of bounded variation","authors":"Angshuman R. Goswami","doi":"10.1007/s00010-024-01050-8","DOIUrl":"https://doi.org/10.1007/s00010-024-01050-8","url":null,"abstract":"<p>Let <span>([a,b]subseteq mathbb {R})</span> be a non-empty and non singleton closed interval and <span>(P={a=x_0&lt;cdots &lt;x_n=b})</span> is a partition of it. Then <span>(f:Irightarrow mathbb {R})</span> is said to be a function of <i>r</i>-bounded variation, if the expression <span>(sum nolimits ^{n}_{i=1}|f(x_i)-f(x_{i-1})|^{r})</span> is bounded for all possible partitions like <i>P</i>. One of the main results of the paper deals with the generalization of the classical Jordan decomposition theorem. We establish that for <span>(rin ]0,1])</span>, a function of <i>r</i>-bounded variation can be written as the difference of two monotone functions. While for <span>(r&gt;1)</span>, under minimal assumptions such a function can be treated as an approximately monotone function which can be closely approximated by a nondecreasing majorant. We also prove that for <span>(0&lt;r_1&lt;r_2)</span>, the function class of <span>(r_1)</span>-bounded variation is contained in the class of functions satisfying <span>(r_2)</span>-bounded variations. We go through approximately monotone functions and present a possible decomposition for <span>(f:I(subseteq mathbb {R}_+)rightarrow mathbb {R})</span> satisfying the functional inequality </p><span>$$f(x)le f(x)+(y-x)^{p}quad (x,yin I text{ with } x&lt;y text{ and } pin ]0,1[ ).$$</span><p>A generalized structural study has also been done in that specific section. On the other hand, for <span>(ell [a,b]ge d)</span>, a function satisfying the following monotonic condition under the given assumption will be termed as <i>d</i>-periodically increasing </p><span>$$f(x)le f(y)quad text{ for } text{ all }quad x,yin Iquad text{ with }quad y-xge d.$$</span><p>We establish that in a compact interval any function satisfying <i>d</i>-bounded variation can be decomposed as the difference of a monotone and a <i>d</i>-periodically increasing function. The core details related to past results, motivation, structure of each and every section are thoroughly discussed below.</p>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623861","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
About Sobolev spaces on fractals: fractal gradians and Laplacians 关于分形上的索波列夫空间：分形梯度和拉普拉奇
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-16 DOI: 10.1007/s00010-024-01060-6
Alireza Khalili Golmankhaneh, Palle E. T. Jørgensen, Cristina Serpa, Kerri Welch
{"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":null,"pages":null},"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}
引用次数: 0
Curves that allow the motion of a regular polygon 允许正多边形运动的曲线
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-15 DOI: 10.1007/s00010-024-01054-4
David Rochera
{"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":null,"pages":null},"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}
引用次数: 0
A characterization of the Euclidean ball via antipodal points 通过对跖点描述欧几里得球的特征
IF 0.9 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-11 DOI: 10.1007/s00010-024-01055-3
Xuguang Lu
{"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":null,"pages":null},"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}
引用次数: 0
The cosine addition and subtraction formulas on non-abelian groups 非阿贝尔群的余弦加减公式
IF 0.8 3区 数学
Aequationes Mathematicae Pub Date : 2024-04-10 DOI: 10.1007/s00010-024-01052-6
Omar Ajebbar, Elhoucien Elqorachi, Henrik Stetkær
{"title":"The cosine addition and subtraction formulas on non-abelian groups","authors":"Omar Ajebbar, Elhoucien Elqorachi, Henrik Stetkær","doi":"10.1007/s00010-024-01052-6","DOIUrl":"https://doi.org/10.1007/s00010-024-01052-6","url":null,"abstract":"<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><span>$$begin{aligned} g(xy) = g(x)g(y) + f(x)h(y), x,y in G, end{aligned}$$</span><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>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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