Operator pencils on the algebra of densities

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators, and the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO-pencil lifting to describe all regular proj-equivariant pencil liftings. In particular the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian. Within this paper the question of whether the pencil lifting factors through a full symbol map naturally arises.Comment: 23 pages, LaTeX file Small corrections are mad

Operator pencil passing through a given operator

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight \lo on a manifold $M$. One can consider a pencil of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator $\Delta$ such that any \Delta_\l is a linear differential operator acting on densities of weight \l. This pencil can be identified with a linear differential operator \hD acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of \diff(M)-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil

Joutsing at Windmills: Cervantes and the Quixotic Fight for Authorial Control

Achieving the appropriate balance between the right of first authors to control the later use of their work and freedom for follow-on authors to further develop from that text has long been challenging. Currently, under United States law in particular, fair use stands as a nebulous to buffer between the two creative camps, granting a significantly limited right to the second author to work from the first authors’ text. While that tension excites its own debate, a less considered aspect of this tension involves the degree to which the first author might be creatively and productively affected by the follow-on author, particularly in a context where absolutely no such mediating protection exists. If that lack of protection substantially improves and increases the original author’s output, by extension it puts the foundational reasoning for U.S. copyright’s limited monopoly in question. Don Miguel de Cervantes wrote Don Quixote in such a copyright-less landscapre. Cervantes’ bitter interplay with a follow-on author, Alonso Fernándex de Avellaneda—which substantially affected both the plot and general content of Cervantes’ original Don Quixote—provides a striking insight into what such an uncontrolled universe might produce creatively. Our focus here will first be to consider the interplay of these two authors through a close reading of their rival texts. Having provided the introduction to our topic in Part I, in Part II we will take a broader look more generally at the historical relationship of the author to his text. In Part III we will focus on Cervantes specifically, addressing his misattributions and non-attributions, which highlight the lack of authorial control for artists in his time. In Part IV we will consider Cervantes’ follow-on author, Avellaneda, and his work, often termed the “false Don Quixote.” We will conclude by considering what this bitter rivalry may suggest in terms of creative production, the core goal of the U.S. Copyright Clause

Perceived stress and emotional social support among women who are denied or receive abortions in the United States: a prospective cohort study.

BackgroundExamining women's stress and social support following denial and receipt of abortion furthers understanding of the effects of unwanted childbearing and abortion on women's well-being. This study investigated perceived stress and emotional social support over time among women who were denied wanted abortions and who received abortions, and compared outcomes between the groups.MethodsThe Turnaway Study is a prospective cohort study of women who sought abortions at 30 abortion facilities across the United States, and follows women via semiannual phone interviews for five years. Participants include 956 English or Spanish speaking women aged 15 and over who sought abortions between 2008 and 2010 and whose gestation in pregnancy fit one of three groups: women who presented up to three weeks beyond a facility's gestational age limit and were denied an abortion; women presenting within two weeks below the limit who received an abortion; and women who received a first trimester abortion. The outcomes were modified versions of the Perceived Stress Scale and the Multidimensional Scale of Perceived Social Support. Longitudinal mixed effects models were used to assess differences in outcomes between study groups over 30 months.ResultsWomen denied abortions initially had higher perceived stress than women receiving abortions near gestational age limits (1.0 unit difference on 0-16 scale, P = 0.003). Women receiving first-trimester abortions initially had lower perceived stress than women receiving abortions near gestational age limits (0.6 difference, P = 0.045). By six months, all groups' levels of perceived stress were similar, and levels remained similar through 30 months. Emotional social support scores did not differ among women receiving abortions near gestational limits versus women denied abortions or women having first trimester abortions initially or over time.ConclusionsSoon after being denied abortions, women experienced higher perceived stress than women who received abortions. The study found no longer-term differences in perceived stress or emotional social support between women who received versus were denied abortions