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TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES 非孤立实奇点的1参数变形拓扑
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-07-30 DOI: 10.1017/S0017089521000239
N. Dutertre, Juan Antonio Moya Pérez
{"title":"TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES","authors":"N. Dutertre, Juan Antonio Moya Pérez","doi":"10.1017/S0017089521000239","DOIUrl":"https://doi.org/10.1017/S0017089521000239","url":null,"abstract":"Abstract Let \u0000$f,{:},(mathbb R^n,0)to (mathbb R,0)$\u0000 be an analytic function germ with non-isolated singularities and let \u0000$F,{:}, (mathbb{R}^{1+n},0) to (mathbb{R},0)$\u0000 be a 1-parameter deformation of f. Let \u0000$ f_t ^{-1}(0) cap B_epsilon^n$\u0000 , \u0000$0 < vert t vert ll epsilon$\u0000 , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0017089521000239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47173321","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}
引用次数: 0
On homomorphisms into Weyl modules corresponding to partitions with two parts 关于与具有两个部分的分区对应的Weyl模的同态
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-07-25 DOI: 10.1017/S0017089522000246
M. Maliakas, D. Stergiopoulou
{"title":"On homomorphisms into Weyl modules corresponding to partitions with two parts","authors":"M. Maliakas, D. Stergiopoulou","doi":"10.1017/S0017089522000246","DOIUrl":"https://doi.org/10.1017/S0017089522000246","url":null,"abstract":"Abstract Let K be an infinite field of characteristic \u0000$p>0$\u0000 and let \u0000$lambda, mu$\u0000 be partitions, where \u0000$mu$\u0000 has two parts. We find sufficient arithmetic conditions on \u0000$p, lambda, mu$\u0000 for the existence of a nonzero homomorphism \u0000$Delta(lambda) to Delta (mu)$\u0000 of Weyl modules for the general linear group \u0000$GL_n(K)$\u0000 . Also, for each p we find sufficient conditions so that the corresponding homomorphism spaces have dimension at least 2.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44675613","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}
引用次数: 2
EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35 k = a·b = 21,33和35时卷积和的求值
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-07-16 DOI: 10.1017/S0017089521000203
K. Pushpa, K. R. Vasuki
{"title":"EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35","authors":"K. Pushpa, K. R. Vasuki","doi":"10.1017/S0017089521000203","DOIUrl":"https://doi.org/10.1017/S0017089521000203","url":null,"abstract":"\u0000 The article focuses on the evaluation of convolution sums \u0000 \u0000 \u0000 $${W_k}(n): = mathop sum nolimits_{_{m < {n over k}}} sigma (m)sigma (n - km)$$\u0000 \u0000 involving the sum of divisor function \u0000 \u0000 \u0000 $$sigma (n)$$\u0000 \u0000 for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form \u0000 \u0000 \u0000 $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$\u0000 \u0000 , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0017089521000203","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46454215","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}
引用次数: 0
Subrepresentations in the homology of finite covers of graphs 图的有限覆盖同调中的子表示
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-07-15 DOI: 10.1017/S0017089523000150
Xenia Flamm
{"title":"Subrepresentations in the homology of finite covers of graphs","authors":"Xenia Flamm","doi":"10.1017/S0017089523000150","DOIUrl":"https://doi.org/10.1017/S0017089523000150","url":null,"abstract":"Abstract Let \u0000$p ;:; Y to X$\u0000 be a finite, regular cover of finite graphs with associated deck group \u0000$G$\u0000 , and consider the first homology \u0000$H_1(Y;;{mathbb{C}})$\u0000 of the cover as a \u0000$G$\u0000 -representation. The main contribution of this article is to broaden the correspondence and dictionary between the representation theory of the deck group \u0000$G$\u0000 on the one hand and topological properties of homology classes in \u0000$H_1(Y;;{mathbb{C}})$\u0000 on the other hand. We do so by studying certain subrepresentations in the \u0000$G$\u0000 -representation \u0000$H_1(Y;;{mathbb{C}})$\u0000 . The homology class of a lift of a primitive element in \u0000$pi _1(X)$\u0000 spans an induced subrepresentation in \u0000$H_1(Y;;{mathbb{C}})$\u0000 , and we show that this property is never sufficient to characterize such homology classes if \u0000$G$\u0000 is Abelian. We study \u0000$H_1^{textrm{comm}}(Y;;{mathbb{C}}) leq H_1(Y;;{mathbb{C}})$\u0000 —the subrepresentation spanned by homology classes of lifts of commutators of primitive elements in \u0000$pi _1(X)$\u0000 . Concretely, we prove that the span of such a homology class is isomorphic to the quotient of two induced representations. Furthermore, we construct examples of finite covers with \u0000$H_1^{textrm{comm}}(Y;;{mathbb{C}}) neq ker!(p_*)$\u0000 .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41548595","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}
引用次数: 1
QUANTITATIVE ESTIMATE FOR THE MEASURE OF A SET OF REAL NUMBERS 对一组实数的度量的定量估计
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-07-06 DOI: 10.1017/S0017089521000197
N. Budarina
{"title":"QUANTITATIVE ESTIMATE FOR THE MEASURE OF A SET OF REAL NUMBERS","authors":"N. Budarina","doi":"10.1017/S0017089521000197","DOIUrl":"https://doi.org/10.1017/S0017089521000197","url":null,"abstract":"Abstarct An effective estimate for the measure of the set of real numbers for which the inequality |P(x)| {3 over 2}n + 1$ has a solution in integral polynomials P of degree n and of height H(P) at most $Q in {rm{mathbb N}}$ is obtained.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0017089521000197","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41770340","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}
引用次数: 0
Brave New World 美丽新世界
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-06-29 DOI: 10.4324/9781003174059-7
A. Gibb
{"title":"Brave New World","authors":"A. Gibb","doi":"10.4324/9781003174059-7","DOIUrl":"https://doi.org/10.4324/9781003174059-7","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87305203","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}
引用次数: 0
New Horizons 新视野号
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-06-29 DOI: 10.4324/9781003174059-4
A. Gibb
{"title":"New Horizons","authors":"A. Gibb","doi":"10.4324/9781003174059-4","DOIUrl":"https://doi.org/10.4324/9781003174059-4","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74695633","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}
引用次数: 0
Epilogue 后记
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-06-29 DOI: 10.4324/9781003174059-8
A. Gibb
{"title":"Epilogue","authors":"A. Gibb","doi":"10.4324/9781003174059-8","DOIUrl":"https://doi.org/10.4324/9781003174059-8","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73204104","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}
引用次数: 0
Iron in The Soul 灵魂中的铁
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-06-29 DOI: 10.4324/9781003174059-6
A. Gibb
{"title":"Iron in The Soul","authors":"A. Gibb","doi":"10.4324/9781003174059-6","DOIUrl":"https://doi.org/10.4324/9781003174059-6","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87048023","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}
引用次数: 0
Environment and Urban Roots 环境与城市根源
IF 0.5 4区 数学
Glasgow Mathematical Journal Pub Date : 2021-06-29 DOI: 10.4324/9781003174059-1
A. Gibb
{"title":"Environment and Urban Roots","authors":"A. Gibb","doi":"10.4324/9781003174059-1","DOIUrl":"https://doi.org/10.4324/9781003174059-1","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82723529","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}
引用次数: 0
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