One-Relator Maximal Pro-p Galois Groups and the Koszulity Conjectures

Let p be a prime number and let ${\mathbb{K}}$ be a field containing a root of 1 of order p. If the absolute Galois group $G_{\mathbb{K}}$ satisfies $\dim\, H^1(G_{\mathbb{K}},\mathbb{F}_p)\lt\infty$ and $\dim\, H^{\,2}(G_{\mathbb{K}},\mathbb{F}_p)=1$, we show that L. Positselski’s and T. Weigel’s Koszulity conjectures are true for ${\mathbb{K}}$. Also, under the above hypothesis, we show that the $\mathbb{F}_p$-cohomology algebra of $G_{\mathbb{K}}$ is the quadratic dual of the graded algebra ${\rm gr}_\bullet\mathbb{F}_p[G_{\mathbb{K}}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{\mathbb{K}}]$, and these two algebras decompose as products of elementary quadratic algebras. Finally, we propose a refinement of the Koszulity conjectures, analogous to I. Efrat’s elementary type conjecture.

Classical History between Epic and Rhetoric

This chapter establishes the foundations on which the rest of the book is built since post-classical scholarship largely develops from classical models or is shaped by its reaction against those models. The chapter addresses a series of conceptual issues that have recurring relevance, including: differing conceptions of the nature of historical truth; the relationship between History and ethnography; the relationship between rhetoric and historianship; the relationship between philosophy, poetry, and History; and the relationship between ‘useful’ and ‘pleasurable’ Histories. In a more empirical vein the chapter discusses the relationship between Greek and Roman historianship and accounts for different tendencies in the development of historiography in each culture—tendencies like a greater or lesser interest in the outside world, and a greater or lesser interest in individuals as opposed to power structures or the study of society and culture. The question of the consciousness of qualitative historical change is also discussed in the case of a number of historians. In the 900 or so years of historianship covered in this chapter no rationale for History that is present at or near the outset was ruled out by the end, though of course many avenues of possibility were more fully explored. It is more than coincidence that the survey opens and closes with species of History as Identity, beginning with the most elementary type of that genre: genealogy.

Division probability for quaternion algebras over fields

Let [Formula: see text] be a field of [Formula: see text] with finitely many square classes. In this paper, we define a new rational valued invariant of [Formula: see text], and call it the division probability of [Formula: see text]. We compute it for all fields of elementary type. Further, we show that [Formula: see text], where [Formula: see text] is the number of Witt-equivalence classes of fields with [Formula: see text], and [Formula: see text] is the count of rational numbers that appear as division probabilities for fields [Formula: see text] of elementary type with [Formula: see text]. In the paper, we also determine [Formula: see text] for all [Formula: see text] and show that rational numbers of type [Formula: see text] always occur as division probability for a suitable field [Formula: see text].

Agent-Based Computational Model

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.

Gorenstein Witt Rings II

The 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.