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The expanding universe of the geometric mean 不断扩大的几何平均宇宙
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-05-07 DOI: 10.1007/s44146-024-00133-x
Jimmie D. Lawson, Yongdo Lim
{"title":"The expanding universe of the geometric mean","authors":"Jimmie D. Lawson,&nbsp;Yongdo Lim","doi":"10.1007/s44146-024-00133-x","DOIUrl":"10.1007/s44146-024-00133-x","url":null,"abstract":"<div><p>In this paper the authors seek to trace in an accessible fashion the rapid recent development of the theory of the matrix geometric mean in the cone of positive definite matrices up through the closely related operator geometric mean in the positive cone of a unital <span>(C^*)</span>-algebra. The story begins with the two-variable matrix geometric mean, moves to the breakthrough developments in the multivariable matrix setting, the main focus of the paper, and then on to the extension to the positive cone of the <span>(C^*)</span>-algebra of operators on a Hilbert space, even to general unital <span>(C^*)</span>-algebras, and finally to the consideration of barycentric maps that grow out of the geometric mean on the space of integrable probability measures on the positive cone. Besides expected tools from linear algebra and operator theory, one observes a surprisingly substantial interplay with geometrical notions in metric spaces, particularly the notion of nonpositive curvature. Added features include a glance at the probabilistic theory of random variables with values in a metric space of nonpositive curvature, and the appearance of related means such as the inductive and power means. The authors also consider in a much briefer fashion the extension of the theory to the setting of Lie groups and briefer still to the positive symmetric cones of finite-dimensional Euclidean Jordan algebras.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 3-4","pages":"327 - 347"},"PeriodicalIF":0.5,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s44146-024-00133-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826332","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New multivariable mean from nonlinear matrix equation associated to the harmonic mean 从与调和平均值相关的非线性矩阵方程得出新的多变量平均值
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-29 DOI: 10.1007/s44146-024-00132-y
Vatsalkumar N. Mer, Sejong Kim
{"title":"New multivariable mean from nonlinear matrix equation associated to the harmonic mean","authors":"Vatsalkumar N. Mer,&nbsp;Sejong Kim","doi":"10.1007/s44146-024-00132-y","DOIUrl":"10.1007/s44146-024-00132-y","url":null,"abstract":"<div><p>Various multivariable means have been defined for positive definite matrices, such as the Cartan mean, Wasserstein mean, and Rényi power mean. These multivariable means have corresponding matrix equations. In this paper, we consider the following non-linear matrix equation: </p><div><div><span>$$begin{aligned} X = left[ sum _{i=1}^{n} w_{i} [ (1-t) X + t A_{i} ]^{-1} right] ^{-1}, end{aligned}$$</span></div></div><p>where <span>(t in (0,1])</span>. We prove that this equation has a unique solution and define a new mean, which we denote as <span>(G_{t}(omega ; mathbb {A}))</span>. We explore important properties of the mean <span>(G_{t}(omega ; mathbb {A}))</span> including the relationship with matrix power mean, and show that the mean <span>(G_{t}(omega ; mathbb {A}))</span> is monotone in the parameter <i>t</i>. Finally, we connect the mean <span>(G_{t}(omega ; mathbb {A}))</span> to a barycenter for the log-determinant divergence.\u0000</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 3-4","pages":"605 - 622"},"PeriodicalIF":0.5,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the spectrum of tridiagonal operators in the context of orthogonal polynomials 正交多项式下三对角算子的谱
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-28 DOI: 10.1007/s44146-023-00106-6
Rupert Lasser, Josef Obermaier
{"title":"On the spectrum of tridiagonal operators in the context of orthogonal polynomials","authors":"Rupert Lasser,&nbsp;Josef Obermaier","doi":"10.1007/s44146-023-00106-6","DOIUrl":"10.1007/s44146-023-00106-6","url":null,"abstract":"<div><p>The basis for our studies is a large class of orthogonal polynomial sequences <span>((P_n)_{nin {{mathbb {N}}}_0})</span>, which is normalized by <span>(P_n(x_0)=1)</span> for all <span>(nin {mathbb {N}}_0)</span> where the coefficients in the three-term recurrence relation are bounded. The goal is to check if <span>(x_0 in {mathbb {R}})</span> is in the support of the orthogonalization measure <span>(mu )</span>. For this purpose, we use, among other things, a result of G. H. Hardy concerning Cesàro operators on weighted <span>(l^2)</span>-spaces. These investigations generalize ideas from Lasser et al. (Arch Math 100:289–299, 2013).</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"91 1-2","pages":"95 - 108"},"PeriodicalIF":0.5,"publicationDate":"2024-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s44146-023-00106-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143938693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Comparison between power difference means and Heron means 幂差均值与赫伦均值的比较
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-24 DOI: 10.1007/s44146-024-00131-z
Hideki Kosaki
{"title":"Comparison between power difference means and Heron means","authors":"Hideki Kosaki","doi":"10.1007/s44146-024-00131-z","DOIUrl":"10.1007/s44146-024-00131-z","url":null,"abstract":"<div><p>Power difference means and Heron means are well-known numerical means with parameters. Their comparison in the positive definite sense is studied. More precisely, for a power difference mean <i>M</i> with each fixed parameter we try to determine the exact parameter range for Heron means majorizing <i>M</i> (in the positive definite sense). Since this order is known to determine validity of (unitarily invariant) norm inequalities between corresponding matrix power difference means and matrix Heron means, we obtain an abundance of very precise norm inequalities between these two matrix means.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 3-4","pages":"441 - 489"},"PeriodicalIF":0.5,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140662395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Spatial Numerical range of bounded operators on right quaternionic Banach spaces 右四元巴拿赫空间上有界算子的空间数值范围
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-15 DOI: 10.1007/s44146-024-00130-0
Somayya Moulaharabbi, Mohamed Barraa
{"title":"Spatial Numerical range of bounded operators on right quaternionic Banach spaces","authors":"Somayya Moulaharabbi,&nbsp;Mohamed Barraa","doi":"10.1007/s44146-024-00130-0","DOIUrl":"10.1007/s44146-024-00130-0","url":null,"abstract":"<div><p>In this paper, we establish and study some properties of the spatial numerical range of right linear bounded operators on a right quaternionic Banach space. To be more specific, we show that the spatial numerical range is circular and we give the relation between the spatial numerical range, the point S-spectrum and the approximate S-spectrum of an operator on a right quaternionic Banach space. We prove also that the S-spectrum of a quaternionic bounded operator is included in the closure of its spatial numerical range. To show this, we generalize the Bishop-Phelps theorem for quaternionic Banach spaces.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"91 1-2","pages":"109 - 119"},"PeriodicalIF":0.5,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140702138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fixed point and periodic point theorems 定点定理和周期点定理
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-09 DOI: 10.1007/s44146-024-00126-w
R. P. Pant, Vladimir Rakočević
{"title":"Fixed point and periodic point theorems","authors":"R. P. Pant,&nbsp;Vladimir Rakočević","doi":"10.1007/s44146-024-00126-w","DOIUrl":"10.1007/s44146-024-00126-w","url":null,"abstract":"<div><p>We introduce a weaker form of continuity which is a necessary and sufficient condition for the existence of fixed points. The obtained theorems exhibit interesting fixed point – eventual fixed point patterns. If we slightly weaken the conditions then the mappings admit periodic points besides fixed points and such mappings possess interesting combinations of fixed and periodic points. Our results are applicable to contractive type as well as non-expansive type mappings. Our theorems are independent of almost all the existing results for contractive type mappings. The last theorem of Sect. 2 is applicable to mappings having various geometric patterns as their domain and is perhaps the first result of its type that also opens up scope for the study of periodic points and periodic point structures. We also give an application of our theorem to obtain the solutions of a nonlinear Diophantine equation; and also show that various well-known fixed point theorems are not applicable in solving this equation.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 1-2","pages":"175 - 192"},"PeriodicalIF":0.5,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140723201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Almost unbounded L and M-weakly compact operators 几乎无界的 L 和 M 弱紧凑算子
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-08 DOI: 10.1007/s44146-024-00129-7
Somayeh Hazrati, Kazem Haghnejad Azar
{"title":"Almost unbounded L and M-weakly compact operators","authors":"Somayeh Hazrati,&nbsp;Kazem Haghnejad Azar","doi":"10.1007/s44146-024-00129-7","DOIUrl":"10.1007/s44146-024-00129-7","url":null,"abstract":"<div><p>In this paper, we introduce and investigate a new class of operators known as almost unbounded <i>L</i>-weakly compact (in shortly, <span>(_{au}L)</span>-weakly compact) and almost unbounded <i>M</i>-weakly compact (in shortly, <span>(_{au}M)</span>-weakly compact) operators. We explore the lattice properties related to this class and examine their relationships with other established operator classes, such as <i>L</i>-weakly compact operators and almost <i>L</i>-weakly compact operators. We demonstrate that every <i>L</i>-weakly compact operator is an <span>(_{au}L)</span>-weakly compact operator, but the reverse implication does not necessarily hold in all cases.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 1-2","pages":"251 - 267"},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140730844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Voting protocols on the star graph 星形图投票协议
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-08 DOI: 10.1007/s44146-024-00125-x
Kamilla Kátai-Urbán, András Pongrácz, Csaba Szabó
{"title":"Voting protocols on the star graph","authors":"Kamilla Kátai-Urbán,&nbsp;András Pongrácz,&nbsp;Csaba Szabó","doi":"10.1007/s44146-024-00125-x","DOIUrl":"10.1007/s44146-024-00125-x","url":null,"abstract":"<div><p>Let <span>(G=(V,E))</span> be a finite graph together with an initial assignment <span>(Vrightarrow {0,1})</span> that represents the opinion of each vertex. Then discordant push voting is a discrete, non-deterministic protocol that alters the opinion of one vertex at a time until a consensus is reached. More precisely, at each round a discordant vertex <i>u</i> (i.e., one that has a neighbor with a different opinion) is chosen uniformly at random, and then we choose a neighbor <i>v</i> with different vote uniformly at random, and force <i>v</i> to change its opinion to that of <i>u</i>. In case of the discordant pull protocol we simply choose a discordant vertex uniformly at random and change its opinion. In this paper, we give asymptotically sharp estimations for the worst expected runtime of the discordant push and pull protocols on the star graph.\u0000</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 1-2","pages":"41 - 51"},"PeriodicalIF":0.5,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s44146-024-00125-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140731535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Log-majorizations related to the spectral geometric and Rényi means 与光谱几何均值和雷尼均值有关的对数大化
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-06 DOI: 10.1007/s44146-024-00128-8
Raluca Dumitru, Jose A. Franco
{"title":"Log-majorizations related to the spectral geometric and Rényi means","authors":"Raluca Dumitru,&nbsp;Jose A. Franco","doi":"10.1007/s44146-024-00128-8","DOIUrl":"10.1007/s44146-024-00128-8","url":null,"abstract":"<div><p>In this article we study log-majorizations related to the spectral geometric and Rényi means. Our goal is to establish certain geometric properties for them with respect to the Thompson metric and Kim’s semi-metric on the cone of positive definite matrices. We also study geodesic in-betweenness type results for these two means and some Audenaert-type in-betweenness inequalities.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"90 3-4","pages":"551 - 563"},"PeriodicalIF":0.5,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140735204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Differentiation properties of class (L^{1}([0,1)^{2})) with respect to two different bases of rectangles 类 $$L^{1}([0,1)^{2})$$ 关于两个不同矩形基的微分性质
IF 0.5
ACTA SCIENTIARUM MATHEMATICARUM Pub Date : 2024-04-03 DOI: 10.1007/s44146-024-00127-9
Michihiro Hirayama, Davit Karagulyan
{"title":"Differentiation properties of class (L^{1}([0,1)^{2})) with respect to two different bases of rectangles","authors":"Michihiro Hirayama,&nbsp;Davit Karagulyan","doi":"10.1007/s44146-024-00127-9","DOIUrl":"10.1007/s44146-024-00127-9","url":null,"abstract":"<div><p>The Lebesgue differentiation theorem claims that the integral averages of <span>(fin L^{1}([0,1)^2))</span> with respect to the family of axis-parallel <i>squares</i> converge almost everywhere on <span>([0,1)^2)</span>. On the other hand, it is a well known result by Saks that there exist a function <span>(f in L^{1}([0,1)^2))</span> such that its integral averages with respect to the family of axis-parallel <i>rectangles</i> diverge everywhere on <span>([0,1)^2)</span>. In this paper, we address the following question: assume we have two different collections of rectangles; under which conditions does there exist a function <span>(f in L^{1}([0,1)^2))</span> so that its integral averages converge with respect to one collection and diverge with respect to another? More specifically, let <span>({varvec{C}}, {varvec{D}} subset (0,1])</span> and consider rectangles with side lengths respectively in <span>({varvec{C}})</span> and <span>({varvec{D}})</span>. We show that if the sets <span>({varvec{C}})</span> and <span>({varvec{D}})</span> occasionally become sufficiently “far” from each other, then such a function can be constructed. We also show that in the class of positive functions our condition is necessary for such a function to exist.</p></div>","PeriodicalId":46939,"journal":{"name":"ACTA SCIENTIARUM MATHEMATICARUM","volume":"91 1-2","pages":"121 - 152"},"PeriodicalIF":0.5,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140749144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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