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A Survey on Fixed Divisors 关于固定除数的综述
Confluentes Mathematici Pub Date : 2017-09-23 DOI: 10.5802/cml.54
Confluentes Mathematici, Devendra Prasad, K. Rajkumar, Satyanarayana Reddy, Devendra Prasad, K. Rajkumar
{"title":"A Survey on Fixed Divisors","authors":"Confluentes Mathematici, Devendra Prasad, K. Rajkumar, Satyanarayana Reddy, Devendra Prasad, K. Rajkumar","doi":"10.5802/cml.54","DOIUrl":"https://doi.org/10.5802/cml.54","url":null,"abstract":"In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of Z, progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n×n matrices over Z (or Dedekind domain) could lead to the generalization of fixed divisors in that setting. keywords Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81679415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Stability of stationary solutions of singular systems of balance laws 平衡律奇异系统平稳解的稳定性
Confluentes Mathematici Pub Date : 2017-09-14 DOI: 10.5802/cml.52
N. Seguin
{"title":"Stability of stationary solutions of singular systems of balance laws","authors":"N. Seguin","doi":"10.5802/cml.52","DOIUrl":"https://doi.org/10.5802/cml.52","url":null,"abstract":"The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77671202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory 形式格式的Drinfeld-Grinberg-Kazhdan定理与奇点理论
Confluentes Mathematici Pub Date : 2017-09-13 DOI: 10.5802/CML.35
David Bourqui, J. Sebag
{"title":"The Drinfeld-Grinberg-Kazhdan Theorem for formal schemes and singularity theory","authors":"David Bourqui, J. Sebag","doi":"10.5802/CML.35","DOIUrl":"https://doi.org/10.5802/CML.35","url":null,"abstract":"Let k be a field. In this article, we provide an extended version of the Drinfeld-Grinberg-Kazhdan Theorem in the context of formal geometry. We prove that, for every formal scheme V topologically of finite type over Spf(k[[T ]]), for every non-singular arc γ ∈ L∞(V )(k), there exists an affine noetherian adic formal k-scheme S and an isomorphism of formal k-schemes L∞(V )γ ∼= S ×k Spf(k[[(Ti)i∈N]]). We emphasize the fact that the proof is constructive and, when V is the completion of an affine algebraic k-variety, effectively implementable. Besides, we derive some properties of such an isomorphism in the direction of singularity theory.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"18 1","pages":"29-64"},"PeriodicalIF":0.0,"publicationDate":"2017-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89398834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
On Harder-Narasimhan filtrations and their compatibility with tensor products hard - narasimhan滤波及其与张量积的相容性
Confluentes Mathematici Pub Date : 2017-03-23 DOI: 10.5802/CML.49
C. Cornut
{"title":"On Harder-Narasimhan filtrations and their compatibility with tensor products","authors":"C. Cornut","doi":"10.5802/CML.49","DOIUrl":"https://doi.org/10.5802/CML.49","url":null,"abstract":"We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85449268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
The role of the Hilbert metric in a class of singular elliptic boundary value problem in convex domains 研究了一类凸域奇异椭圆边值问题中Hilbert度规的作用
Confluentes Mathematici Pub Date : 2017-02-01 DOI: 10.5802/CML.38
D. Serre
{"title":"The role of the Hilbert metric in a class of singular elliptic boundary value problem in convex domains","authors":"D. Serre","doi":"10.5802/CML.38","DOIUrl":"https://doi.org/10.5802/CML.38","url":null,"abstract":"In a recent paper [7], we were led to consider a distance over a bounded open convex domain. It turns out to be the so-called Thompson metric, which is equivalent to the Hilbert metric. It plays a key role in the analysis of existence and uniqueness of solutions to a class of elliptic boundary-value problems that are singular at the boundary.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"15 1","pages":"105-117"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76243621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sur l’inexistence d’ensembles minimaux pour le flot horocyclique@@@On the non-existence of minimal sets for the horocycle flow Sur l 'inexistence d 'ensembles minimaux pour le flot horcyclclique @@@关于不存在的最小集的horcyclclique流
Confluentes Mathematici Pub Date : 2017-01-01 DOI: 10.5802/CML.37
Masseye Gaye, Cheikh Lo
{"title":"Sur l’inexistence d’ensembles minimaux pour le flot horocyclique@@@On the non-existence of minimal sets for the horocycle flow","authors":"Masseye Gaye, Cheikh Lo","doi":"10.5802/CML.37","DOIUrl":"https://doi.org/10.5802/CML.37","url":null,"abstract":"","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"8 1","pages":"95-104"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81258154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces 曲面的编织和映射类群的嵌入和(虚)上同调维数
Confluentes Mathematici Pub Date : 2016-10-11 DOI: 10.5802/cml.45
D. Gonccalves, John Guaschi, Miguel A. Maldonado
{"title":"Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces","authors":"D. Gonccalves, John Guaschi, Miguel A. Maldonado","doi":"10.5802/cml.45","DOIUrl":"https://doi.org/10.5802/cml.45","url":null,"abstract":"In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and their (virtual) cohomological dimensions. We first generalise results of Birman and Chillingworth and of Gonc{c}alves and Guaschi to show that the mapping class group MCG(N ; k) of N relative to a k-point subset embeds in the mapping class group MCG(S; 2k) of S relative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all compact, connected aspherical surfaces without boundary. Finally, if the genus of N is greater than or equal to 2, we give upper bounds for the virtual cohomological dimension of MCG(N ; k).","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74036248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Almost all non-archimedean Kakeya sets have measure zero 几乎所有非阿基米德Kakeya集合的测度都为零
Confluentes Mathematici Pub Date : 2016-08-09 DOI: 10.5802/CML.44
X. Caruso
{"title":"Almost all non-archimedean Kakeya sets have measure zero","authors":"X. Caruso","doi":"10.5802/CML.44","DOIUrl":"https://doi.org/10.5802/CML.44","url":null,"abstract":"We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also discuss possible relations with the non-archimedean Kakeya conjecture.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"240 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74902817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator 波算符在弱色散性质下带导数损失的Strichartz估计
Confluentes Mathematici Pub Date : 2016-05-04 DOI: 10.5802/cml.56
V. Samoyeau
{"title":"Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator","authors":"V. Samoyeau","doi":"10.5802/cml.56","DOIUrl":"https://doi.org/10.5802/cml.56","url":null,"abstract":"This paper can be considered as a sequel of [BS14] by Bernicot and Samoyeau, where the authors have proposed a general way of deriving Strichartz estimates for the Schr{\"o}dinger equation from a dispersive property of the wave propagator. It goes through a reduction of H 1 -- BMO dispersive estimates for the Schr{\"o}dinger propagator to L 2 -- L 2 microlocalized (in space and in frequency) dispersion inequalities for the wave operator. This paper aims to contribute in enlightening our comprehension of how dispersion for waves imply dispersion for the Schr{\"o}dinger equation. More precisely, the hypothesis of our main theorem encodes dispersion for the wave equation in an uniform way, with respect to the light cone. In many situations the phenomena that arise near the boundary of the light cone are the more complicated ones. The method we present allows to forget those phenomena we do not understand very well yet. The second main step shows the Strichartz estimates with loss of derivatives we can obtain under those assumptions. The setting we work with is general enough to recover a large variety of frameworks (infinite metric spaces, Riemannian manifolds with rough metric, some groups, ...) where the lack of knowledge of the wave propagator is a restraint to our understanding of the dispersion phenomenon.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2016-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76169751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Basic Zariski Topology 基本的Zariski拓扑
Confluentes Mathematici Pub Date : 2016-02-03 DOI: 10.5802/CML.18
Davide Rinaldi, G. Sambin, P. Schuster
{"title":"The Basic Zariski Topology","authors":"Davide Rinaldi, G. Sambin, P. Schuster","doi":"10.5802/CML.18","DOIUrl":"https://doi.org/10.5802/CML.18","url":null,"abstract":"We present the Zariski spectrum as an inductively generated basic topology à la Martin-Löf and Sambin. Since we can thus get by without considering powers and radicals, this simplifies the presentation as a formal topology initiated by Sigstam. Our treatment includes closed subspaces and basic opens: that is, arbitrary quotients and singleton localisations. All the effective objects under consideration are introduced by means of inductive definitions. The notions of spatiality and reducibility are characterized for the class of Zariski formal topologies, and their nonconstructive content is pointed out: while spatiality implies classical logic, reducibility corresponds to a fragment of the Axiom of Choice in the form of Russell’s Multiplicative Axiom.","PeriodicalId":52130,"journal":{"name":"Confluentes Mathematici","volume":"14 1","pages":"55-81"},"PeriodicalIF":0.0,"publicationDate":"2016-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87923423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 7
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