We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.
{"title":"Extremal Area of Polygons, sliding along a circle","authors":"D. Siersma","doi":"10.14492/hokmj/2020-312","DOIUrl":"https://doi.org/10.14492/hokmj/2020-312","url":null,"abstract":"We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as `zigzag trains'.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46857910","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 consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $varepsilon in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski (1963) from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $varepsilon to +0$ of the solution itself depending on $varepsilon$ under rather high regularity assumptions on the initial data.
{"title":"Singular limit problem of abstract second order evolution equations","authors":"R. Ikehata, M. Sobajima","doi":"10.14492/hokmj/2021-504","DOIUrl":"https://doi.org/10.14492/hokmj/2021-504","url":null,"abstract":"We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $varepsilon in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski (1963) from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $varepsilon to +0$ of the solution itself depending on $varepsilon$ under rather high regularity assumptions on the initial data.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44002475","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}
Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models and are related with $beta$-Hermite and $beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$. Recently, several limit theorems for $ktoinfty$ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study the solutions of these ODEs which are are singular on the boundaries of their domains. In particular we prove that for a start in arbitrary boundary points, the ODEs always admit unique solutions in their domains for $t>0$.
{"title":"The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime","authors":"M. Voit, Jeannette H. C. Woerner","doi":"10.14492/hokmj/2020-307","DOIUrl":"https://doi.org/10.14492/hokmj/2020-307","url":null,"abstract":"Multivariate Bessel processes describe Calogero-Moser-Sutherland particle models and are related with $beta$-Hermite and $beta$-Laguerre ensembles. They depend on a root system and a multiplicity $k$. Recently, several limit theorems for $ktoinfty$ were derived where the limits depend on the solutions of associated ODEs in these freezing regimes. In this paper we study the solutions of these ODEs which are are singular on the boundaries of their domains. In particular we prove that for a start in arbitrary boundary points, the ODEs always admit unique solutions in their domains for $t>0$.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46615706","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}
Pub Date : 2019-10-01DOI: 10.14492/hokmj/1573722020
M. Luo, R. K. Raina
{"title":"The decompositional structure of certain fractional integral operators","authors":"M. Luo, R. K. Raina","doi":"10.14492/hokmj/1573722020","DOIUrl":"https://doi.org/10.14492/hokmj/1573722020","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48078381","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}
Pub Date : 2019-10-01DOI: 10.14492/hokmj/1573722018
T. Komatsu, V. Laohakosol, Pinthira Tangsupphathawat
In this paper, we introduce the truncated Euler-Carlitz numbers as analogues of hypergeometric Euler numbers. In a special case, Euler-Carlitz numbers are defined, which is an analogue of the classical Euler numbers. We give several interesting properties for these numbers. We also show some determinant expressions of Euler-Carlitz numbers.
{"title":"Truncated Euler-Carlitz numbers","authors":"T. Komatsu, V. Laohakosol, Pinthira Tangsupphathawat","doi":"10.14492/hokmj/1573722018","DOIUrl":"https://doi.org/10.14492/hokmj/1573722018","url":null,"abstract":"In this paper, we introduce the truncated Euler-Carlitz numbers as analogues of hypergeometric Euler numbers. In a special case, Euler-Carlitz numbers are defined, which is an analogue of the classical Euler numbers. We give several interesting properties for these numbers. We also show some determinant expressions of Euler-Carlitz numbers.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49209043","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}
Pub Date : 2019-10-01DOI: 10.14492/hokmj/1573722019
M. Kimura, S. Maeda, H. Tanabe
{"title":"Integral curves of the characteristic vector field of minimal ruled real hypersurfaces in non-flat complex space forms","authors":"M. Kimura, S. Maeda, H. Tanabe","doi":"10.14492/hokmj/1573722019","DOIUrl":"https://doi.org/10.14492/hokmj/1573722019","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46113508","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}
Pub Date : 2019-10-01DOI: 10.14492/hokmj/1573722013
J. Lapébie
{"title":"Degree formulas for the Euler characteristic of semialgebraic sets","authors":"J. Lapébie","doi":"10.14492/hokmj/1573722013","DOIUrl":"https://doi.org/10.14492/hokmj/1573722013","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42436122","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 matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jelic Milutinovic et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.
{"title":"Matching complexes of polygonal line tilings","authors":"Takahiro Matsushita","doi":"10.14492/hokmj/2019-213","DOIUrl":"https://doi.org/10.14492/hokmj/2019-213","url":null,"abstract":"The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jelic Milutinovic et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45125674","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}
Assuming initial data have small weighted $H^4times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and Hormander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the course of a priori estimates of solutions. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably low weight is small enough.
{"title":"Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight","authors":"K. Hidano, K. Yokoyama","doi":"10.14492/hokmj/2021-523","DOIUrl":"https://doi.org/10.14492/hokmj/2021-523","url":null,"abstract":"Assuming initial data have small weighted $H^4times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman. Compared with the work of Christodoulou, our result assumes smallness of data with respect to $H^4times H^3$ norm having a lower weight. Our proof uses the space-time $L^2$ estimate due to Alinhac for some special derivatives of solutions to variable-coefficient wave equations. It also uses the conformal energy estimate for inhomogeneous wave equation $Box u=F$. A new observation made in this paper is that, in comparison with the proofs of Klainerman and Hormander, we can limit the number of occurrences of the generators of hyperbolic rotations or dilations in the course of a priori estimates of solutions. This limitation allows us to obtain global solutions for radially symmetric data, when a certain norm with considerably low weight is small enough.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49074522","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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