Translations of Mathematical Monographs最新文献_第3页

Small oscillations, potential energy matrix and 𝐋-matrix, direct and inverse problems of the theory of small oscillations 小振荡，势能矩阵和𝐋-matrix，小振荡理论的正逆问题

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/08

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Solving the inverse problem of the theory of small oscillations numerically 用数值方法求解小振荡理论的反问题

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/20

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General solution for the inverse problem of spectral analysis for Hermitian matrices 厄米矩阵谱分析反问题的通解

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/10

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Properties of completely 𝑀-extendable sets 完全𝑀-extendable集合的属性

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/15

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Schmidt-Sonin orthogonalization process Schmidt-Sonin正交过程

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/04

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Direct problem of the oscillation theory of loaded strings 加载弦振荡理论的直接问题

Translations of Mathematical Monographs Pub Date : 2018-12-12 DOI: 10.1090/mmono/247/01

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Residue Map on a Brauer Group Brauer群上的残差映射

Translations of Mathematical Monographs Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/04

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Brauer Group of a Field 一个领域的布劳尔群

Translations of Mathematical Monographs Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/03

Jon Aycock

{"title":"Brauer Group of a Field","authors":"Jon Aycock","doi":"10.1090/mmono/246/03","DOIUrl":"https://doi.org/10.1090/mmono/246/03","url":null,"abstract":"In this paper we discuss the Brauer group of a field and its connections with cohomology groups. Definitions involving central simple algebras lead to a discussion of splitting fields, which are the important step in the connection of the Brauer group with cohomology groups. Finally, once the connection between the Brauer group and cohomology groups is established, specific examples of cocycles associated to central simple algebras are calculated 1. Central Simple Algebras and Splitting Fields Elements of the Brauer group are equivalence classes of central simple algebras. As such, it is important to have an understanding of these algebras to understand the Brauer group. This section aims to lay the foundation for the rest of our discussion; this foundation starts with the definition of a CSA. Definitions 1.1: Fix some field k. 1. An algebra over k is a ring A along with an embedding ψ : k ↪→ A, where 1 in k maps to 1 in A. This embedding induces a scalar product that allows A to have a vector space structure over k. The image ψ(k) is often denoted k · 1, or simply k. We require ψ(k) to commute with every element of the algebra, so that the “left scalar product” and the “right scalar product” will be the same. 2. The center of an algebra A is the set Z(A) = {z ∈ A : az = za ∀a ∈ A}. If this set is the subspace k · 1, A is said to be central. 3. A (two-sided) ideal of an algebra is a (two-sided) ideal of the algebra viewed as a ring. Note that (x ·1)i = x · i ∈ I for every x ∈ k, so it has the addition stipulation of being closed under scalar multiplication. If the only ideals of A are {0} and A, A is said to be simple. 4. An algebra is a central simple algebra if it is both central and simple. Central simple algebras are often referred to as CSAs, an abbreviation we will use often. In addition to these definitions, it is useful to know when an algebra is finite-dimensional. An algebra is finite-dimensional when it is finite-dimensional as a vector space over k. All algebras will be assumed to be finite-dimensional unless stated otherwise. Examples 1.2: Common examples of CSAs include: 1. Mn(k) is a CSA over k for all n > 0. The matrix ring is equipped with a scalar product operation that multiplies each entry by an element of k. 2. Central division algebras, including k itself, are CSAs over k as well. 3. The quaternion algebra ( a,b k ) , generated by i and j with i2 = a, j2 = b, ij = −ji is a central simple algebra over k. In fact, it is either a division algebra or it is isomorphic to M2(k). A discussion of these algebras and their properties is found in [Lam].","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126757603","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}

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Example of a Non-rational Stably Rational Variety 一个非理性的稳定理性变种的例子

Translations of Mathematical Monographs Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/09

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Non-rational Double Covers of ℙ³ 非理性的双罩

Translations of Mathematical Monographs Pub Date : 2018-09-10 DOI: 10.1090/mmono/246/07

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