Faithfully Quadratic Rings

by M. Dickmann
Faithfully Quadratic Rings
Author: M. Dickmann
Publisher: American Mathematical Soc.
Total Pages: 148
Release: 2015-10-27
Genre: Mathematics
ISBN: 1470414686
GET BOOK

In this monograph the authors extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where is not a sum of squares and is invertible. They accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of -isometry, where is a preorder of the given ring, , or . (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.

Relative Nonhomogeneous Koszul Duality

by Leonid Positselski
Relative Nonhomogeneous Koszul Duality
Author: Leonid Positselski
Publisher: Springer Nature
Total Pages: 303
Release: 2022-02-10
Genre: Mathematics
ISBN: 3030895408
GET BOOK

This research monograph develops the theory of relative nonhomogeneous Koszul duality. Koszul duality is a fundamental phenomenon in homological algebra and related areas of mathematics, such as algebraic topology, algebraic geometry, and representation theory. Koszul duality is a popular subject of contemporary research. This book, written by one of the world's leading experts in the area, includes the homogeneous and nonhomogeneous quadratic duality theory over a nonsemisimple, noncommutative base ring, the Poincare–Birkhoff–Witt theorem generalized to this context, and triangulated equivalences between suitable exotic derived categories of modules, curved DG comodules, and curved DG contramodules. The thematic example, meaning the classical duality between the ring of differential operators and the de Rham DG algebra of differential forms, involves some of the most important objects of study in the contemporary algebraic and differential geometry. For the first time in the history of Koszul duality the derived D-\Omega duality is included into a general framework. Examples highly relevant for algebraic and differential geometry are discussed in detail.

Proof of the 1-Factorization and Hamilton Decomposition Conjectures

by Béla Csaba
Proof of the 1-Factorization and Hamilton Decomposition Conjectures
Author: Béla Csaba
Publisher: American Mathematical Soc.
Total Pages: 176
Release: 2016-10-05
Genre: Mathematics
ISBN: 1470420252
GET BOOK

In this paper the authors prove the following results (via a unified approach) for all sufficiently large n: (i) [1-factorization conjecture] Suppose that n is even and D≥2⌈n/4⌉−1. Then every D-regular graph G on n vertices has a decomposition into perfect matchings. Equivalently, χ′(G)=D. (ii) [Hamilton decomposition conjecture] Suppose that D≥⌊n/2⌋. Then every D-regular graph G on n vertices has a decomposition into Hamilton cycles and at most one perfect matching. (iii) [Optimal packings of Hamilton cycles] Suppose that G is a graph on n vertices with minimum degree δ≥n/2. Then G contains at least regeven(n,δ)/2≥(n−2)/8 edge-disjoint Hamilton cycles. Here regeven(n,δ) denotes the degree of the largest even-regular spanning subgraph one can guarantee in a graph on n vertices with minimum degree δ. (i) was first explicitly stated by Chetwynd and Hilton. (ii) and the special case δ=⌈n/2⌉ of (iii) answer questions of Nash-Williams from 1970. All of the above bounds are best possible.

The $abc$-Problem for Gabor Systems

by Xin-Rong Dai
The $abc$-Problem for Gabor Systems
Author: Xin-Rong Dai
Publisher: American Mathematical Soc.
Total Pages: 116
Release: 2016-10-05
Genre: Mathematics
ISBN: 1470420155
GET BOOK

A longstanding problem in Gabor theory is to identify time-frequency shifting lattices aZ×bZ and ideal window functions χI on intervals I of length c such that {e−2πinbtχI(t−ma): (m,n)∈Z×Z} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above abc-problem for Gabor systems.

Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology

by Reiner Hermann:
Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology
Author: Reiner Hermann:
Publisher: American Mathematical Soc.
Total Pages: 158
Release: 2016-09-06
Genre: Mathematics
ISBN: 1470419955
GET BOOK

In this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links Ext-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces n-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties.

Rohlin Flows on von Neumann Algebras

by Toshihiko Masuda
Rohlin Flows on von Neumann Algebras
Author: Toshihiko Masuda
Publisher: American Mathematical Soc.
Total Pages: 128
Release: 2016-10-05
Genre: Mathematics
ISBN: 1470420163
GET BOOK

The authors will classify Rohlin flows on von Neumann algebras up to strong cocycle conjugacy. This result provides alternative approaches to some preceding results such as Kawahigashi's classification of flows on the injective type II1 factor, the classification of injective type III factors due to Connes, Krieger and Haagerup and the non-fullness of type III0 factors. Several concrete examples are also studied.

Descent Construction for GSpin Groups

by Joseph Hundley
Descent Construction for GSpin Groups
Author: Joseph Hundley
Publisher: American Mathematical Soc.
Total Pages: 138
Release: 2016-09-06
Genre: Mathematics
ISBN: 1470416670
GET BOOK

In this paper the authors provide an extension of the theory of descent of Ginzburg-Rallis-Soudry to the context of essentially self-dual representations, that is, representations which are isomorphic to the twist of their own contragredient by some Hecke character. The authors' theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin2n to GL2n.

Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Nonsmooth Parabolic Equations

by Genni Fragnelli
Carleman Estimates, Observability Inequalities and Null Controllability for Interior Degenerate Nonsmooth Parabolic Equations
Author: Genni Fragnelli
Publisher: American Mathematical Soc.
Total Pages: 96
Release: 2016-06-21
Genre: Mathematics
ISBN: 1470419548
GET BOOK

The authors consider a parabolic problem with degeneracy in the interior of the spatial domain, and they focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence.

The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup

by U. Meierfrankenfeld
The Local Structure Theorem for Finite Groups With a Large $p$-Subgroup
Author: U. Meierfrankenfeld
Publisher: American Mathematical Soc.
Total Pages: 356
Release: 2016-06-21
Genre: Mathematics
ISBN: 1470418770
GET BOOK

Let p be a prime, G a finite Kp-group S a Sylow p-subgroup of G and Q a large subgroup of G in S (i.e., CG(Q)≤Q and NG(U)≤NG(Q) for 1≠U≤CG(Q)). Let L be any subgroup of G with S≤L, Op(L)≠1 and Q⋬L. In this paper the authors determine the action of L on the largest elementary abelian normal p-reduced p-subgroup YL of L.