Basic facts about symmetric bilinear forms, and definition of the Witt ring

Wittrings ◽  
1982 ◽  
pp. 1-18
Author(s):  
Manfred Knebusch ◽  
Manfred Kolster
Keyword(s):  
Bilinear Forms ◽  
Witt Ring ◽  
Basic Facts ◽  
Definition Of

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

From Finding to Explaining

2018 ◽  
pp. 1535-1551
Author(s):  
Valentina Dragos
Keyword(s):  
Information Retrieval ◽  
Semantic Integration ◽  
Challenging Problem ◽  
Domain Experts ◽  
User Interests ◽  
Maritime Surveillance ◽  
Novel Approach ◽  
Basic Facts ◽  
Anomaly Analysis ◽  
Definition Of

Supporting anomaly analysis in the maritime field is a challenging problem because of the dynamic nature of the task: the definition of abnormal or suspicious behaviour is subject to change and depends on user interests. This paper provides a novel approach to support anomaly analysis in the maritime domain through the exploration of large collections of interpretation reports. Based on observables or more sophisticated patterns, the approach provides information retrieval strategies going from basic facts retrieval that guide short-term corrective actions to more complex networks of related concepts that help domain experts to understand or to explain abnormal vessel behaviours. Semantic integration is used to link various information sources, by using a commonly adopted standard. The paper seeks to explore different aspects of using information retrieval to support the analysis and interpretation of abnormal vessel behaviours for maritime surveillance.

Signatures and Semi Signatures of Abstract Witt Rings and Witt Rings of Semilocal Rings

1978 ◽  
Vol 30 (4) ◽  
pp. 872-895 ◽  
Author(s):  
Jerrold L. Kleinstein ◽  
Alex Rosenberg
Keyword(s):  
Residue Class ◽  
Point Of View ◽  
Prime Ideals ◽  
Bilinear Forms ◽  
Semilocal Ring ◽  
Witt Ring ◽  
Witt Rings ◽  
Full Knowledge ◽  
Carry Over

This paper originated in an attempt to carry over the results of [3] from the case of a field of characteristic different from two to that of semilocal rings. To carry this out, we reverse the point of view of [3] and do assume a full knowledge of the theory of Witt rings of classes of nondegenerate symmetric bilinear forms over semilocal rings as given, for example, in [10; 11]. It turns out that the rings WT of [3] are just the residue class rings of W(C), the Witt ring of a semilocal ring C, modulo certain intersections of prime ideals.

scholarly journals Symplectic bilinear forms on affine real algebraic surfaces

1989 ◽  
Vol 31 (2) ◽  
pp. 195-198
Author(s):  
W. Kucharz
Keyword(s):  
Commutative Ring ◽  
Algebraic Surfaces ◽  
Bilinear Forms ◽  
Witt Group ◽  
Real Algebraic Surfaces ◽  
Definition Of

Given a commutative ring A with identity, let W–1(A) denote the Witt group of skew-symmetric bilinear forms over A (cf. [1] or [7] for the definition of W–1 (A)).

scholarly journals Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rasoul Abazari
Keyword(s):  
Metric Space ◽  
Cauchy Sequence ◽  
Statistical Convergence ◽  
Metric Spaces ◽  
Asymptotic Density ◽  
Strong Topology ◽  
Generalized Metric Spaces ◽  
Generalized Metric ◽  
Basic Facts ◽  
Definition Of

AbstractIn this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of $\mathbb{N}^{l}$ N l is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

On the definition of singular bilinear forms and singular linear operators

1993 ◽  
Vol 45 (8) ◽  
pp. 1208-1214 ◽  
Author(s):  
W. Karwowski ◽  
V. Koshmanenko
Keyword(s):  
Linear Operators ◽  
Bilinear Forms ◽  
Definition Of

scholarly journals Lattice of ℤ-module

2016 ◽  
Vol 24 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Coding Theory ◽  
Positive Definite ◽  
Bilinear Forms ◽  
Reduction Algorithm ◽  
Real Numbers ◽  
Scalar Products ◽  
Definition Of

Summary In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].

Erratum to Profinite rigidity for twisted Alexander polynomials (J. reine angew. Math. 771 (2021), 171–192)

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jun Ueki
Keyword(s):  
Alexander Polynomials ◽  
Basic Facts ◽  
Definition Of

Abstract We clarify the definition of the divisorial hull and recollect some basic facts. Then we correct Lemma 4.2 and Theorem 11.2 (1)–(2) in the original article.

scholarly journals Generalized Topological Groupoids

2021 ◽  
pp. 117-131
Author(s):  
Mustafa Habil Gursoy
Keyword(s):  
Topological Space ◽  
Generalized Topology ◽  
Crossed Module ◽  
Topological Groupoid ◽  
Generalized Topological Space ◽  
Basic Facts ◽  
Definition Of ◽  
Topological Groupoids

Our aim in this paper is to give the notion of generalized topological groupoid which is a generalization of the topological groupoid by using the notion of generalized topology defined by Csasz ´ ar [6]. We in- ´ vestigate the basic facts in the groupoid theory in terms of generalized topological groupoids. We present the action of a generalized topological groupoid on a generalized topological space. We obtain some characterizations about this concept that is called the generalized topological action. Beside these, we give definition of a generalized topological crossed module by generalizing the concept of crossed module defined on topological groupoids. At the last part of the study, we show how a generalized topological crossed module can be obtained from a generalized topological groupoid and how a generalized topological groupoid can be obtained from a generalized topological crossed module.

From Finding to Explaining

2016 ◽  
Vol 7 (1) ◽  
pp. 40-55 ◽  
Author(s):  
Valentina Dragos
Keyword(s):  
Information Retrieval ◽  
Semantic Integration ◽  
Short Term ◽  
Domain Experts ◽  
User Interests ◽  
Maritime Surveillance ◽  
Novel Approach ◽  
Basic Facts ◽  
Anomaly Analysis ◽  
Definition Of

Supporting anomaly analysis in the maritime field is a challenging problem because of the dynamic nature of the task: the definition of abnormal or suspicious behaviour is subject to change and depends on user interests. This paper provides a novel approach to support anomaly analysis in the maritime domain through the exploration of large collections of interpretation reports. Based on observables or more sophisticated patterns, the approach provides information retrieval strategies going from basic facts retrieval that guide short-term corrective actions to more complex networks of related concepts that help domain experts to understand or to explain abnormal vessel behaviours. Semantic integration is used to link various information sources, by using a commonly adopted standard. The paper seeks to explore different aspects of using information retrieval to support the analysis and interpretation of abnormal vessel behaviours for maritime surveillance.

Sign in / Sign up

Export Citation Format

Share Document