The purpose of this paper is to study the famous Pappus configuration of $9$ lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the initial data of beginning configuration. We consider also the Pappus arrangements with some additional incidences and we establish algebraic conditions paralleling with these incidences.
{"title":"On the Pappus arrangement of lines, forth and back and to the point","authors":"Magdalena Lampa-Baczy'nska, D. W'ojcik","doi":"10.14492/hokmj/2021-521","DOIUrl":"https://doi.org/10.14492/hokmj/2021-521","url":null,"abstract":"The purpose of this paper is to study the famous Pappus configuration of $9$ lines and its dual arrangement. We show among others that by applying the Pappus Theorem to the dual arrangement we obtain the configuration corresponding to the initial data of beginning configuration. We consider also the Pappus arrangements with some additional incidences and we establish algebraic conditions paralleling with these incidences.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46777535","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 study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold. We show that for odd powers of the unit tangent sphere the mappings are fold maps. Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.
{"title":"Fold maps associated to geodesic random walks on non-positively curved manifolds","authors":"Pablo Lessa, L. Oliveira","doi":"10.14492/hokmj/2020-439","DOIUrl":"https://doi.org/10.14492/hokmj/2020-439","url":null,"abstract":"We study a family of mappings from the powers of the unit tangent sphere at a point to a complete Riemannian manifold with non-positive sectional curvature, whose behavior is related to the spherical mean operator and the geodesic random walks on the manifold. \u0000We show that for odd powers of the unit tangent sphere the mappings are fold maps. \u0000Some consequences on the regularity of the transition density of geodesic random walks, and on the eigenfunctions of the spherical mean operator are discussed and related to previous work.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42400431","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 : 2020-02-01DOI: 10.14492/hokmj/1591085014
Ha Huong Giang
{"title":"Uniqueness theorem of meromorphic mappings of a complete Kähler manifold into a projective space","authors":"Ha Huong Giang","doi":"10.14492/hokmj/1591085014","DOIUrl":"https://doi.org/10.14492/hokmj/1591085014","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42473630","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 : 2020-02-01DOI: 10.14492/hokmj/1591085013
S. Bannai, H. Tokunaga, Momoko Yamamoto
{"title":"Rational points of elliptic surfaces and the topology of cubic-line, cubic-conic-line arrangements","authors":"S. Bannai, H. Tokunaga, Momoko Yamamoto","doi":"10.14492/hokmj/1591085013","DOIUrl":"https://doi.org/10.14492/hokmj/1591085013","url":null,"abstract":"","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46149245","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 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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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