The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.Comment: 15 page

Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties

In this paper we define a new presentation for the Dunkl-Opdam subalgebra of the rational Cherednik algebra. This shows that the Dunkl-Opdam subalgebra is a Drinfeld algebra. We use this fact to define Dirac cohomology for the DO subalgebra. We also formalise generalised graded Hecke algebras and define a Langlands classification to generalised graded Hecke algebras

Freedom of Thought, Offensive Fantasies and the Fundamental Human Right to Hold Deviant Ideas: Why the Seventh Circuit Got it Wrong in Doe v. City of Lafayette, Indiana

[Excerpt] “A precarious balance and considerable tension exists between two competing legal interests – the essential, First Amendment-grounded human right to freedom of thought, on the one hand, and the desire to prevent harm and injury that might occur if thought is converted to action, on the other. To understand this tension, it is useful to start by considering three different and disturbing factual scenarios. This article examines and critiques the majority opinion of the Seventh Circuit Court of Appeals in City of Lafayette. The majority held that the city’s ban of John Doe, a convicted sex offender, from its parks because he once fantasized about molesting children while watching them play in the park does not violate the First Amendment right to freedom of thought. The July 2004 en banc opinion reversed the result of an earlier vacated decision by a three-judge panel of the same court just thirteen months before. […] Part II of this article provides background on the right to freedom of thought, including a discussion of recent United States Supreme Court analysis on this right. Part III then describes, analyzes, and critiques the Seventh Circuit’s 2004 en banc opinion holding that the City of Lafayette did not violate John Doe’s right to freedom of thought. Importantly, Part III contends there are at least four separate reasons, each grounded in First Amendment jurisprudence tied to freedom of expression, why the majority erred in its conclusion. These reasons go far beyond those articulated in the dissenting opinion in City of Lafayette and thus suggest new and additional rationales for reversal. Next, Part IV demonstrates the dangerousness of the precedent set by the majority’s reasoning as it might apply to other scenarios, including the first two hypothetical fact patterns laid out at the beginning of this article. Finally, the article concludes in Part V that the United States Supreme Court should accept certiorari in this case and reverse the Seventh Circuit’s decision.

Young people's housing transitions in context

The transition to adulthood is frequently characterised as delayed or extended in the contemporary period, in the UK as elsewhere. Studies have addressed changing school-to-work transitions, for example, the extension of full-time education and expansion of higher education, as well as changing patterns in family formation and partnership, including the postponement of marriage and childbearing. Some of these changes have been associated with increasing rates of living alone or ‘solo- living’ and living in shared housing. However, the nature of young people’s housing transitions has received less attention and this paper provides a background to a study which specifically addresses the housing transitions of young people aged between 25 and 34 years old living in ‘non-family’ households, that is, living alone or sharing with others. The paper concludes with a short overview of the project and its main research focu

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

In this paper we will derive an explicit description of the genuine projective representations of the symmetric group $S_n$ using Dirac cohomology and the branching graph for the irreducible genuine projective representations of $S_n$. In 2015 Ciubotaru and He, using the extended Dirac index, showed that the characters of the projective representations of $S_n$ are related to the characters of elliptic graded modules. We derived the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties $\mathcal{B}_e$ of $\mathfrak{g}$ and were able to use Dirac cohomology to construct an explicit model for the projective representations. We also described Vogan's morphism for Hecke algebras in type A using spectrum data of the Jucys-Murphy elements

PAC Learning, VC Dimension, and the Arithmetic Hierarchy

We compute that the index set of PAC-learnable concept classes is $m$-complete $\Sigma^0_3$ within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable $\Pi^0_1$ classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension

'He came to her bed pretending courtship': sex, courtship and the making of marriage in Ulster, 1750-1844

This article is available open access.The history of sex and sexuality is underdeveloped in Irish historical studies, particularly for the period before the late-nineteenth century. While much has been written on rates of illegitimacy in Ireland, and its regional diversity, little research has been conducted on how ordinary women and men viewed sex and sexuality. Moreover, we still know little about the roles that sex played in the rituals of courtship and marriage. Drawing on a sample of Presbyterian church records, this article offers some new insights into these areas. It argues that sexual intercourse and other forms of sexual activity formed part of the normal courtship rituals for many young Presbyterian couples in Ulster. Courting couples participated in non-penetrative sexual practices, such as petting, groping and bundling. Furthermore, while sexual intercourse did not have a place in the formal route to marriage, many couples engaged in it regardless.Peer reviewedFinal Published versio