{"title":"投影空间中的平面等谱对和对合;2。广义射影等谱数据的分类","authors":"K. Thas","doi":"10.1088/0305-4470/39/42/004","DOIUrl":null,"url":null,"abstract":"In Am. Math. Monthly (73 1–23 (1966)), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. Giraud (2005 J. Phys. A: Math. Gen. 38 L477–83) observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions for n = 2, when the involutions fix the same number of points. In Thas (2006 J. Phys. A: Math. Gen. 39 L385–8) we showed that no other examples arise for any other dimension, still assuming that the involutions fix the same number of points. In this paper we study the problem for involutions not necessarily fixing the same number of points, and solve the problem completely.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2006-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Kac's question, planar isospectral pairs and involutions in projective space: II. Classification of generalized projective isospectral data\",\"authors\":\"K. Thas\",\"doi\":\"10.1088/0305-4470/39/42/004\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In Am. Math. Monthly (73 1–23 (1966)), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. Giraud (2005 J. Phys. A: Math. Gen. 38 L477–83) observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions for n = 2, when the involutions fix the same number of points. In Thas (2006 J. Phys. A: Math. Gen. 39 L385–8) we showed that no other examples arise for any other dimension, still assuming that the involutions fix the same number of points. In this paper we study the problem for involutions not necessarily fixing the same number of points, and solve the problem completely.\",\"PeriodicalId\":87442,\"journal\":{\"name\":\"Journal of physics A: Mathematical and general\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2006-10-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of physics A: Mathematical and general\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1088/0305-4470/39/42/004\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of physics A: Mathematical and general","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1088/0305-4470/39/42/004","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
摘要
在我。数学。每月(73 1-23(1966)),卡茨问了他著名的问题“人能听到鼓的形状吗?”,这最终在Gordon et al (1992 Invent)中得到了否定的回答。数学。110 - 22)通过构造平面等谱对。吉拉德(2005)。答:数学。gen 38 (L477-83)指出,大多数已知的例子都可以由某一方程的解生成,该方程涉及某一有限域上n维射影空间的一组对合。然后,他生成了n = 2时所有可能的解,当对合固定相同数量的点时。In Thas (2006) [j]。答:数学。gen 39 L385-8)我们证明了在任何其他维度上都不会出现其他例子,仍然假设这些对合固定了相同数量的点。本文研究了不一定固定相同点数的对合线问题,并彻底解决了这一问题。
Kac's question, planar isospectral pairs and involutions in projective space: II. Classification of generalized projective isospectral data
In Am. Math. Monthly (73 1–23 (1966)), Kac asked his famous question ‘Can one hear the shape of a drum?’, which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. Giraud (2005 J. Phys. A: Math. Gen. 38 L477–83) observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions for n = 2, when the involutions fix the same number of points. In Thas (2006 J. Phys. A: Math. Gen. 39 L385–8) we showed that no other examples arise for any other dimension, still assuming that the involutions fix the same number of points. In this paper we study the problem for involutions not necessarily fixing the same number of points, and solve the problem completely.