Several identities in statistical mechanics

In an earlier paper concerning a solvable Inodel in statistical mechanics, Miwa and Jimbo state a theta-function identity which they have checked to the 200th power, but of which they do not have a proof. The main objective of this note is to provide such a proof

On the sums of two cubes

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary operations on the set of solutions. We show that a particular subset of the set of solutions is ring-isomorphic to $\mathbb Z[e^{2 \pi i / 3}]$.Comment: Revised version, to appear in the International Journal of Number Theor

A double bounded key identity for Goellnitz's (big) partition theorem

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on Symbolic Computation

Homotopy Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it. On the other hand, those conditions are easy to check and provide a wide class of models some of which are listed in the paper.Comment: Corrected version of the published articl

Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

Exploring unwarranted clinical variation: The attitudes of midwives and obstetric medical staff regarding induction of labour and planned caesarean section

Background: Unexplained clinical variation is a major issue in planned birth i.e. induction of labour and planned caesarean section. Aim: To map attitudes and knowledge of maternity care professionals regarding indications for planned birth, and assess inter-professional (midwifery versus medical) and intra-professional variation. Methods: A custom-created survey of medical and midwifery staff at eight Sydney hospitals. Staff were asked to rate their level of agreement with 45 “evidence-based” statements regarding caesareans and inductions on a five-point Likert scale. Responses were grouped by profession, and comparisons made of inter- and intra-professional responses. Findings: Total 275 respondents, 78% midwifery and 21% medical. Considerable inter- and intra-professional variation was noted, with midwives generally less likely to consider any of the planned birth indications “valid” compared to medical staff. Indications for induction with most variation in midwifery responses included maternal characteristics (age≥40, obesity, ethnicity) and fetal macrosomia; and for medical personnel in-vitro fertilisation, maternal request, and routine induction at 39 weeks gestation. Indications for caesarean with most variation in midwifery responses included previous lower segment caesarean section, previous shoulder dystocia, and uncomplicated breech; and for medical personnel uncomplicated dichorionic twins. Indications with most inter-professional variation were induction at 41+ weeks versus 42+ weeks and cesarean for previous lower segment caesarean section. Discussion: Both inter- and intra-professional variation in what were considered valid indications reflected inconsistency in underlying evidence and/or guidelines. Conclusion: Greater focus on interdisciplinary education and consensus, as well as on shared decision-making with women, may be helpful in resolving these tensions

Small world effect in an epidemiological model

A model for the spread of an infection is analyzed for different population structures. The interactions within the population are described by small world networks, ranging from ordered lattices to random graphs. For the more ordered systems, there is a fluctuating endemic state of low infection. At a finite value of the disorder of the network, we find a transition to self-sustained oscillations in the size of the infected subpopulation

The homotopy theory of dg-categories and derived Morita theory

The main purpose of this work is the study of the homotopy theory of dg-categories up to quasi-equivalences. Our main result provides a natural description of the mapping spaces between two dg-categories $C$ and $D$ in terms of the nerve of a certain category of $(C,D)$-bimodules. We also prove that the homotopy category $Ho(dg-Cat)$ is cartesian closed (i.e. possesses internal Hom's relative to the tensor product). We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories $C$ and $D$ as the dg-category of $(C,D)$-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the \emph{classifying space of dg-categories} (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent complexes (resp. perfect) on their product.Comment: 50 pages. Few mistakes corrected, and some references added. Thm. 8.15 is new. Minor corrections. Final version, to appear in Inventione

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat