For given continuous functions $gamma_{{}_{i}}: S^{n}to mathbb{R}_{+}$ ($i=1, 2$), the functions $gamma_{{}_{max}}$ and $gamma_{{}_{min}}$ can be defined naturally. In this paper, by applying the spherical method, we first show that the Wulff shape associated to $gamma_{{}_{max}}$ is the convex hull of the union of Wulff shapes associated to $gamma_{{}_1}$ and $gamma_{{}_2}$, if $gamma_{{}_1}$ and $gamma_{{}_2}$ are convex integrands. Next, we show that the Wulff shape associated to $gamma_{{}_{min}}$ is the intersection of Wulff shapes associated to $gamma_{{}_1}$ and $gamma_{{}_2}$. Moreover, relationships between their dual Wulff shapes are given.
{"title":"Maximum and minimum of support functions","authors":"Huhe HAN","doi":"10.14492/hokmj/2021-557","DOIUrl":"https://doi.org/10.14492/hokmj/2021-557","url":null,"abstract":"For given continuous functions $gamma_{{}_{i}}: S^{n}to mathbb{R}_{+}$ ($i=1, 2$), the functions $gamma_{{}_{max}}$ and $gamma_{{}_{min}}$ can be defined naturally. In this paper, by applying the spherical method, we first show that the Wulff shape associated to $gamma_{{}_{max}}$ is the convex hull of the union of Wulff shapes associated to $gamma_{{}_1}$ and $gamma_{{}_2}$, if $gamma_{{}_1}$ and $gamma_{{}_2}$ are convex integrands. Next, we show that the Wulff shape associated to $gamma_{{}_{min}}$ is the intersection of Wulff shapes associated to $gamma_{{}_1}$ and $gamma_{{}_2}$. Moreover, relationships between their dual Wulff shapes are given.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain upper and lower bounds for the $A$-numerical radius inequalities of operators and operator matrices which generalize and improve on the existing ones. We present new upper bounds for the $A$-numerical radius of the product of two operators. We also develop various inequalities for the $A$-numerical radius of $2 times 2 $ operator matrices.
{"title":"Improvements of $A$-numerical radius bounds","authors":"Raj Kumar NAYAK, Pintu BHUNIA, Kallol PAUL","doi":"10.14492/hokmj/2022-603","DOIUrl":"https://doi.org/10.14492/hokmj/2022-603","url":null,"abstract":"We obtain upper and lower bounds for the $A$-numerical radius inequalities of operators and operator matrices which generalize and improve on the existing ones. We present new upper bounds for the $A$-numerical radius of the product of two operators. We also develop various inequalities for the $A$-numerical radius of $2 times 2 $ operator matrices.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities is given. The bound means that concentration of electrons is prohibited due to the repulsion between the electrons. A relation between one-electron and two-electron densities resulting from the antisymmetry of the eigenfunctions plays a crucial role in the proof.
{"title":"Upper bounds of local electronic densities in molecules","authors":"Sohei ASHIDA","doi":"10.14492/hokmj/2021-577","DOIUrl":"https://doi.org/10.14492/hokmj/2021-577","url":null,"abstract":"The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities is given. The bound means that concentration of electrons is prohibited due to the repulsion between the electrons. A relation between one-electron and two-electron densities resulting from the antisymmetry of the eigenfunctions plays a crucial role in the proof.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The artin braid group actions on the set of spin structures on a surface","authors":"Gefei WANG","doi":"10.14492/hokmj/2021-570","DOIUrl":"https://doi.org/10.14492/hokmj/2021-570","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.
{"title":"An analogy of Jacobi's formula and its applications","authors":"Jun CHIBA, Keiji MATSUMOTO","doi":"10.14492/hokmj/2021-572","DOIUrl":"https://doi.org/10.14492/hokmj/2021-572","url":null,"abstract":"We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is motivated by the works of W.Y. Hsiang and H.B. Lawson [7], Pages $12$ and $13$. In this paper, we deal with the following double fibration: [ xymatrix@R-0.5cm @C-0.5cm{ & (G,g) ar[ld]_{pi_1} ar[rd]^{pi_2} & (G/H,h_1) && (Kbackslash G,h_2) } ] We will show that every $K$-invariant minimal or biharmonic hypersurface $M$ in $(G/H,h_1)$ induces an $H$-invariant minimal or biharmonic hypersurface $widetilde{M}$ in $(Kbackslash G,h_2)$ by means of $widetilde{M}:=pi_2(pi_1{}^{-1}(M))$ (cf. Theorems 3.2 and 4.1). We give a one to one correspondence between the class of all the $K$-invariant minimal or biharmonic hypersurfaces in $G/H$ and the one of all the $H$-invariant minimal or biharmonic hypersurfaces in $Kbackslash G$ (cf. Theorem 4.2).
{"title":"Harmonic maps and biharmonic maps for double fibrations of compact Lie groups","authors":"Hajime URAKAWA","doi":"10.14492/hokmj/2021-558","DOIUrl":"https://doi.org/10.14492/hokmj/2021-558","url":null,"abstract":"This work is motivated by the works of W.Y. Hsiang and H.B. Lawson [7], Pages $12$ and $13$. In this paper, we deal with the following double fibration: [ xymatrix@R-0.5cm @C-0.5cm{ & (G,g) ar[ld]_{pi_1} ar[rd]^{pi_2} & (G/H,h_1) && (Kbackslash G,h_2) } ] We will show that every $K$-invariant minimal or biharmonic hypersurface $M$ in $(G/H,h_1)$ induces an $H$-invariant minimal or biharmonic hypersurface $widetilde{M}$ in $(Kbackslash G,h_2)$ by means of $widetilde{M}:=pi_2(pi_1{}^{-1}(M))$ (cf. Theorems 3.2 and 4.1). We give a one to one correspondence between the class of all the $K$-invariant minimal or biharmonic hypersurfaces in $G/H$ and the one of all the $H$-invariant minimal or biharmonic hypersurfaces in $Kbackslash G$ (cf. Theorem 4.2).","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"152 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper investigates systems exhibiting two attractors, and we discuss the problem of steering the state from one attractor to the other attractor by our idea of associating with the stabilization problem an optimal control problem. We first formulate the steering problem and give partial answers for the problem in a two-dimensional case by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts. Furthermore, under some conditions, we verify that the phase space can be separated into some openly connected components depending on the asymptotic behavior of the orbit starting from initial points in their components. This classification of initial points suggests that the system enables robust stabilizable control. Moreover, we illustrate some numerical results for the control obtained by applying our focused system for the Bonhoeffer–Van der Pol model.
本文研究了具有两个吸引子的系统，并将镇定问题与最优控制问题联系起来，讨论了状态从一个吸引子转向另一个吸引子的问题。首先利用基于负折扣无限视界最优控制模型的常微分方程，给出了二维情况下的转向问题的部分解。此外，在某些条件下，我们验证了相位空间可以被分割成一些开放连接的分量，这取决于轨道从它们的分量中的初始点开始的渐近行为。这种初始点的分类表明，该系统能够实现鲁棒稳定控制。此外，我们还举例说明了将我们的聚焦系统应用于Bonhoeffer-Van der Pol模型所得到的一些数值结果。
{"title":"Accessibility and stabilization by infinite horizon optimal control with negative discounting","authors":"Fumihiko NAKAMURA","doi":"10.14492/hokmj/2021-551","DOIUrl":"https://doi.org/10.14492/hokmj/2021-551","url":null,"abstract":"The present paper investigates systems exhibiting two attractors, and we discuss the problem of steering the state from one attractor to the other attractor by our idea of associating with the stabilization problem an optimal control problem. We first formulate the steering problem and give partial answers for the problem in a two-dimensional case by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts. Furthermore, under some conditions, we verify that the phase space can be separated into some openly connected components depending on the asymptotic behavior of the orbit starting from initial points in their components. This classification of initial points suggests that the system enables robust stabilizable control. Moreover, we illustrate some numerical results for the control obtained by applying our focused system for the Bonhoeffer–Van der Pol model.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136204781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Certain Convexity Properties of Hurwitz-Lerch Zeta and Mittag-Leffler Functions","authors":"D. Bansal, R. K. Raina","doi":"10.14492/hokmj/2021-543","DOIUrl":"https://doi.org/10.14492/hokmj/2021-543","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48225472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On surjective homomorphisms from a configuration space group to a surface group","authors":"K. Sawada","doi":"10.14492/hokmj/2021-526","DOIUrl":"https://doi.org/10.14492/hokmj/2021-526","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42608374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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