Graded witt rings of elementary type

1985 ◽  
Vol 272 (2) ◽  
pp. 267-280 ◽  
Author(s):  
J�n Kr. Arason ◽  
Richard Elman ◽  
Bill Jacob
Keyword(s):  
Witt Rings ◽  
Elementary Type

Abstract Witt Rings When Certain Binary Forms Represent Exactly Four Elements

1993 ◽  
Vol 45 (6) ◽  
pp. 1184-1199 ◽  
Author(s):  
Craig M. Cordes
Keyword(s):  
Group Ring ◽  
Positive Characteristic ◽  
Finitely Generated ◽  
Binary Forms ◽  
Witt Ring ◽  
Witt Rings ◽  
Elementary Type ◽  
Characteristic 2

AbstractAn abstract Witt ring (R, G) of positive characteristic is known to be a group ring S[Δ] with ﹛1﹜ ≠ Δ ⊆ G if and only if it contains a form〈1,x〉, x ≠1, which represents only the two elements 1 and x. Carson and Marshall have characterized all Witt rings of characteristic 2 which contain binary forms representing exactly four elements. Such results which show R is isomorphic to a product of smaller rings are helpful in settling the conjecture that every finitely generated Witt ring is of elementary type. Here, some special situations are considered. In particular if char(R) = 8, |D〈l, 1〉| = 4, and R contains no rigid elements, then R is isomorphic to the Witt ring of the 2-adic numbers. If char(R) = 4, |D〈l,a〉| = 4 where a ∈ D〈1, 1〉, and R contains no rigid elements, then R is either a ring of order 8 or is the specified product of two Witt rings at least one of which is a group ring. In several cases R is realized by a field.

Gorenstein Witt Rings

1988 ◽  
Vol 40 (5) ◽  
pp. 1186-1202 ◽  
Author(s):  
Robert W. Fitzgerald
Keyword(s):  
Group Ring ◽  
Quotient Ring ◽  
Injective Resolution ◽  
Gorenstein Rings ◽  
Witt Ring ◽  
Witt Rings ◽  
Local Type ◽  
Elementary Type ◽  
Ring Extensions

Throughout R is a noetherian Witt ring. The basic example is the Witt ring WF of a field F of characteristic not 2 and finite. We study the structure of (noetherian) Witt rings which are also Gorenstein rings (i.e., have a finite injective resolution). The underlying motivation is the elementary type conjecture. The Gorenstein Witt rings of elementary type are group ring extensions of Witt rings of local type. We thus wish to compare the two classes of Witt rings: Gorenstein and group ring over local type. We show the two classes enjoy many of the same properties and are, in several cases, equal. However we cannot decide if the two classes are always equal.In the first section we consider formally real Witt rings R (equivalently, dim R = 1). Here the total quotient ring of R is R-injective if and only if R is reduced. Further, R is Gorenstein if and only if R is a group ring over Z. This result appears to be somewhat deep.

Gorenstein Witt Rings II

1997 ◽  
Vol 49 (3) ◽  
pp. 499-519
Author(s):  
Robert W. Fitzgerald
Keyword(s):  
Classification Problem ◽  
Witt Rings ◽  
Dimension Zero ◽  
Elementary Type

AbstractThe abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type.

Agent-Based Computational Model

2017 ◽  
Author(s):  
Joshua M. Epstein
Keyword(s):  
Computational Model ◽  
Bounded Rationality ◽  
Time Scales ◽  
Spatially Explicit ◽  
Agent Based ◽  
Affective Component ◽  
Elementary Type ◽  
Social Component ◽  
Different Time Scales

This part describes the agent-based and computational model for Agent_Zero and demonstrates its capacity for generative minimalism. It first explains the replicability of the model before offering an interpretation of the model by imagining a guerilla war like Vietnam, Afghanistan, or Iraq, where events transpire on a 2-D population of contiguous yellow patches. Each patch is occupied by a single stationary indigenous agent, which has two possible states: inactive and active. The discussion then turns to Agent_Zero's affective component and an elementary type of bounded rationality, as well as its social component, with particular emphasis on disposition, action, and pseudocode. Computational parables are then presented, including a parable relating to the slaughter of innocents through dispositional contagion. This part also shows how the model can capture three spatially explicit examples in which affect and probability change on different time scales.

scholarly journals Crystal growth and Witt rings

1991 ◽  
Vol 136 (1) ◽  
pp. 190-196 ◽  
Author(s):  
Mieczysław Kula
Keyword(s):  
Crystal Growth ◽  
Witt Rings

scholarly journals Combinatorial techniques and abstract Witt rings III

1991 ◽  
Vol 148 (1) ◽  
pp. 39-58 ◽  
Author(s):  
Robert Fitzgerald
Keyword(s):  
Witt Rings

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].

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