Shunsuke Kobayashi, T. Sakamoto, Yasuhide Uegata, S. Yazaki
An oscillatory hexagonal solution in a two component reaction-di¤usion system with a non-local term is studied. By applying the center manifold theory, we obtain a four-dimensional dynamical system that informs us about the bifurcation structure around the trivial solution. Our results suggest that the oscillatory hexagonal solution can bifurcate from a stationary hexagonal solution via the Hopf bifurcation. This provides a reasonable explanation for the existence of the oscillatory hexagon.
{"title":"A time-periodic oscillatory hexagonal solution in a 2-dimensional integro-differential reaction-diffusion system","authors":"Shunsuke Kobayashi, T. Sakamoto, Yasuhide Uegata, S. Yazaki","doi":"10.32917/hmj/1595901630","DOIUrl":"https://doi.org/10.32917/hmj/1595901630","url":null,"abstract":"An oscillatory hexagonal solution in a two component reaction-di¤usion system with a non-local term is studied. By applying the center manifold theory, we obtain a four-dimensional dynamical system that informs us about the bifurcation structure around the trivial solution. Our results suggest that the oscillatory hexagonal solution can bifurcate from a stationary hexagonal solution via the Hopf bifurcation. This provides a reasonable explanation for the existence of the oscillatory hexagon.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46173597","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}
A bstract . In this paper, we evaluate the asymptotic bias of C p type criterion for model selection in the GEE (generalized estimating equation) method when the sample and cluster sizes are large. We present the asymptotic properties of GEE estimator and the model selection criterion. Then, we present the order of the asymptotic bias of PMSEG (the prediction mean squared error in the GEE).
{"title":"Asymptotic bias of $C_p$ type criterion for model selection in the GEE when the sample size and the cluster sizes are large","authors":"Tomoharu Sato","doi":"10.32917/hmj/1595901629","DOIUrl":"https://doi.org/10.32917/hmj/1595901629","url":null,"abstract":"A bstract . In this paper, we evaluate the asymptotic bias of C p type criterion for model selection in the GEE (generalized estimating equation) method when the sample and cluster sizes are large. We present the asymptotic properties of GEE estimator and the model selection criterion. Then, we present the order of the asymptotic bias of PMSEG (the prediction mean squared error in the GEE).","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46432242","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}
A bstract . Let E ð 1 Þ be the first Johnson-Wilson spectrum at the prime two. In this paper, we calculate the homotopy groups of the E ð 1 Þ -localized sphere spectrum with a grading over the Picard group of the stable homotopy category of E ð 1 Þ -local spectra.
{"title":"The $E(1)$-local Picard graded homotopy groups of the sphere spectrum at the prime two","authors":"Ryo Kato","doi":"10.32917/hmj/1595901623","DOIUrl":"https://doi.org/10.32917/hmj/1595901623","url":null,"abstract":"A bstract . Let E ð 1 Þ be the first Johnson-Wilson spectrum at the prime two. In this paper, we calculate the homotopy groups of the E ð 1 Þ -localized sphere spectrum with a grading over the Picard group of the stable homotopy category of E ð 1 Þ -local spectra.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46174954","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}
A bstract . For a given angle y , consider the configuration space C n of equilateral n -gons in R 3 whose bond angles are equal to y except for two successive ones. We show that when n b 8 and y is su‰ciently close to the inner angle n (cid:1) 2 n p of the regular n -gon, C n is homeomorphic to the ð n (cid:1) 4 Þ -dimensional sphere S n (cid:1) 4 .
一个混蛋。For a赐予安格尔y,认为《configuration太空equilateral之C n n -gons在R 3是谁的邦德angles equal to y除了For two successive势力。我们表演那8当n b和y是苏‰ciently接近《内在安格尔n (cid): 1)普通之2 n p n -gon C, n是homeomorphicðn (cid》:1)4Þ-dimensional范围S n (cid: 1) 4。
{"title":"The configuration space of almost regular polygons","authors":"S. Goto, Kazushi Komatsu, Junya Yagi","doi":"10.32917/hmj/1595901626","DOIUrl":"https://doi.org/10.32917/hmj/1595901626","url":null,"abstract":"A bstract . For a given angle y , consider the configuration space C n of equilateral n -gons in R 3 whose bond angles are equal to y except for two successive ones. We show that when n b 8 and y is su‰ciently close to the inner angle n (cid:1) 2 n p of the regular n -gon, C n is homeomorphic to the ð n (cid:1) 4 Þ -dimensional sphere S n (cid:1) 4 .","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45875146","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}
A bstract . Let N g be the non-orientable surface of genus g , MCG ð N g Þ the mapping class group of N g , T ð N g Þ the index 2 subgroup generated by all Dehn twists of MCG ð N g Þ . We prove that for odd genus, (1) if g ¼ 4 k þ 3 ð k b 1 Þ , MCG ð N g Þ can be generated by three elements of finite order; (2) if g ¼ 4 k þ 1 ð k b 2 Þ , T ð N g Þ can be generated by three elements of finite order.
一个混蛋。让N g属non-orientable地表》成为g, MCGðN g绘图课集团》Þg, TðN gÞ《指数2 subgroup generated by所有Dehn扭曲了的MCGðN gÞ。我们证明那个奇怪的,(1)如果g¼属4 kþ3ðk b 1Þ,MCGðN gÞ可以成为generated byfinite秩序的三个文本;(2)如果g¼4 kþ1,2Þ,Tððk b N gÞ可以成为generated by三个finite命令的文本。
{"title":"The torsion generating set of the mapping class groups and the Dehn twist subgroups of non-orientable surfaces of odd genus","authors":"Xiaoming Du","doi":"10.32917/hmj/1595901627","DOIUrl":"https://doi.org/10.32917/hmj/1595901627","url":null,"abstract":"A bstract . Let N g be the non-orientable surface of genus g , MCG ð N g Þ the mapping class group of N g , T ð N g Þ the index 2 subgroup generated by all Dehn twists of MCG ð N g Þ . We prove that for odd genus, (1) if g ¼ 4 k þ 3 ð k b 1 Þ , MCG ð N g Þ can be generated by three elements of finite order; (2) if g ¼ 4 k þ 1 ð k b 2 Þ , T ð N g Þ can be generated by three elements of finite order.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47104146","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}
It is well-known that the second coefficient of the Alexander polynomial of any lens space knot in $S^3$ is $-1$. We show that the non-zero third coefficient condition of the Alexander polynomial of a lens space knot $K$ in $S^3$ confines the surgery to the one realized by the $(2,2g+1)$-torus knot, where $g$ is the genus of $K$. In particular, such a lens surgery polynomial coincides with $Delta_{T(2,2g+1)}(t)$.
{"title":"The third term in lens surgery\u0000 polynomials","authors":"M. Tange","doi":"10.32917/H2020050","DOIUrl":"https://doi.org/10.32917/H2020050","url":null,"abstract":"It is well-known that the second coefficient of the Alexander polynomial of any lens space knot in $S^3$ is $-1$. We show that the non-zero third coefficient condition of the Alexander polynomial of a lens space knot $K$ in $S^3$ confines the surgery to the one realized by the $(2,2g+1)$-torus knot, where $g$ is the genus of $K$. In particular, such a lens surgery polynomial coincides with $Delta_{T(2,2g+1)}(t)$.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44488644","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 the ring extensions R subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.
{"title":"A note on pairs of rings with the same prime ideals","authors":"Rahul Kumar, A. Gaur","doi":"10.32917/h2021037","DOIUrl":"https://doi.org/10.32917/h2021037","url":null,"abstract":"We study the ring extensions R subseteq T having the same set of prime ideals provided Nil(R) is a divided prime ideal. Some conditions are given under which no such T exist properly containing R. Using idealization theory, the examples are also discussed to strengthen the results.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48472060","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 spaces of formal power series of class M of Roumieu type and of Beurling type","authors":"Xiaoran Jin","doi":"10.32917/hmj/1583805652","DOIUrl":"https://doi.org/10.32917/hmj/1583805652","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47897261","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}
Recently, in real data analysis, we consider the data with correlation for many fields, for example medical science, economics and many other fields. Especially, the data what is measured repeatedly over times from same subjects, named longitudinal data, is widely used in those fields. In general, the data from same subject have correlation, on the other hand, the data from different subjects are independent.. Liang and Zeger (1986) introduce an extension of generalized linear model (Nelder and Wedderburn, 1972), named generalized estimating equation (GEE). GEE method is one of the methods to analyze the data with correlation. Defining features of the GEE method are that we can use working correlation matrix one can choose freely. We can get good estimation of parameters if working correlation matrix is correct or not. It is important that we don’t need a full specification of a joint distribution. In those reason, GEE method is widely used in many fields. ”Model selection” is also important problem, so we apply model selection to the GEE. In general, in model selection, we measure the goodness of fit by risk function, and choose the model with smallest risk function. Then, by using the asymptotically unbiased estimator of risk function, we consider the model selection criterion. For example, expected Kullback-Leibler information (Kullback and Leibler, 1951), and most famous Akaike’s information criterion (AIC) (Akaike, 1973, 1974) are used. The AIC is calculated by AIC = −2× (maximumloglikelihood)+2× (thenumberofparameters). Furthermore, the GIC what is expansion of the AIC proposed by Nishii (1984) and Rao (1988) is also applied for many fields. However, we can’t use the model selection criterion based likelihood as AIC or GIC because of we don’t specify joint distribution. Some model selection criteria like AIC and GIC in the GEE method have been already proposed. For example, Pan (2001) proposed the QIC based on the quasi-likelihood (defined by Wedderburn, 1974). Furthermore, the GCp proposed by Cantoni et al. (2005) is the generally extension of Mallow’s Cp (Mallows, 1973). The CIC proposed by Hin and Wang (2009) and Gosho et al. (2011) is criterion what select the correlation structure. Unfortunately, the above criteria are derived without consider the correlation structure so we regard to these criteria don’t reflect the correlation. From this background, in Inatsu and Imori (2013) proposed a new model selection criterion PMSEG (the prediction mean squared error in the GEE) using the risk function based on the prediction mean squared error (PMSE) normalized by the covariance matrix. Inatsu and Imori (2013) proposed this criterion when both correlation and scale parameters are known, but correlation and scale parameters are generally unknown so we consider this criterion when both correlation and scale parameters are unknown. In this paper, the main topic is to propose the model selection criterion considered correlation structure when both
{"title":"A $C_p$ type criterion for model selection in the GEE method when both scale and correlation parameters are unknown","authors":"Tomoharu Sato, Yu Inatsu","doi":"10.32917/hmj/1583805651","DOIUrl":"https://doi.org/10.32917/hmj/1583805651","url":null,"abstract":"Recently, in real data analysis, we consider the data with correlation for many fields, for example medical science, economics and many other fields. Especially, the data what is measured repeatedly over times from same subjects, named longitudinal data, is widely used in those fields. In general, the data from same subject have correlation, on the other hand, the data from different subjects are independent.. Liang and Zeger (1986) introduce an extension of generalized linear model (Nelder and Wedderburn, 1972), named generalized estimating equation (GEE). GEE method is one of the methods to analyze the data with correlation. Defining features of the GEE method are that we can use working correlation matrix one can choose freely. We can get good estimation of parameters if working correlation matrix is correct or not. It is important that we don’t need a full specification of a joint distribution. In those reason, GEE method is widely used in many fields. ”Model selection” is also important problem, so we apply model selection to the GEE. In general, in model selection, we measure the goodness of fit by risk function, and choose the model with smallest risk function. Then, by using the asymptotically unbiased estimator of risk function, we consider the model selection criterion. For example, expected Kullback-Leibler information (Kullback and Leibler, 1951), and most famous Akaike’s information criterion (AIC) (Akaike, 1973, 1974) are used. The AIC is calculated by AIC = −2× (maximumloglikelihood)+2× (thenumberofparameters). Furthermore, the GIC what is expansion of the AIC proposed by Nishii (1984) and Rao (1988) is also applied for many fields. However, we can’t use the model selection criterion based likelihood as AIC or GIC because of we don’t specify joint distribution. Some model selection criteria like AIC and GIC in the GEE method have been already proposed. For example, Pan (2001) proposed the QIC based on the quasi-likelihood (defined by Wedderburn, 1974). Furthermore, the GCp proposed by Cantoni et al. (2005) is the generally extension of Mallow’s Cp (Mallows, 1973). The CIC proposed by Hin and Wang (2009) and Gosho et al. (2011) is criterion what select the correlation structure. Unfortunately, the above criteria are derived without consider the correlation structure so we regard to these criteria don’t reflect the correlation. From this background, in Inatsu and Imori (2013) proposed a new model selection criterion PMSEG (the prediction mean squared error in the GEE) using the risk function based on the prediction mean squared error (PMSE) normalized by the covariance matrix. Inatsu and Imori (2013) proposed this criterion when both correlation and scale parameters are known, but correlation and scale parameters are generally unknown so we consider this criterion when both correlation and scale parameters are unknown. In this paper, the main topic is to propose the model selection criterion considered correlation structure when both","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2020-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45285736","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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