Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.Comment: 11 pages one important references is added, proof of lemma 2.1 (2) and many typos are correcte

A Generalization of Martin's Axiom

We define the $\aleph_{1.5}$ chain condition. The corresponding forcing axiom is a generalization of Martin's Axiom and implies certain uniform failures of club--guessing on $\omega_1$ that don't seem to have been considered in the literature before.Comment: 36 page

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

Cohomological descent theory for a morphism of stacks and for equivariant derived categories

In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.Comment: 28 page

Development and validation of a VISA tendinopathy questionnaire for greater trochanteric pain syndrome, the VISA-G

BACKGROUND Greater trochanteric pain syndrome (GTPS) is common, resulting in significant pain and disability. There is no condition specific outcome score to evaluate the degree of severity of disability associated with GTPS in patients with this condition. OBJECTIVE To develop a reliable and valid outcome measurement capable of evaluating the severity of disability associated with GTPS. METHODS A phenomenological framework using in-depth semi structured interviews of patients and medical experts, and focus groups of physiotherapists was used in the item generation. Item and format clarification was undertaken via piloting. Multivariate analysis provided the basis for item reduction. The resultant VISA-G was tested for reliability with the inter class co-efficient (ICC), internal consistency (Cronbach's Alpha), and construct validity (correlation co-efficient) on 52 naïve participants with GTPS and 31 asymptomatic participants. RESULTS The resultant outcome measurement tool is consistent in style with existing tendinopathy outcome measurement tools, namely the suite of VISA scores. The VISA-G was found to be have a test-retest reliability of ICC2,1 (95% CI) of 0.827 (0.638-0.923). Internal consistency was high with a Cronbach's Alpha of 0.809. Construct validity was demonstrated: the VISA-G measures different constructs than tools previously used in assessing GTPS, the Harris Hip Score and the Oswestry Disability Index (Spearman Rho:0.020 and 0.0205 respectively). The VISA-G did not demonstrate any floor or ceiling effect in symptomatic participants. CONCLUSION The VISA-G is a reliable and valid score for measuring the severity of disability associated GTPS.The study was funded through the Australian National University, Monash University and LaTrobe University. Prof Cook was supported by the Australian Centre for Research into Sports Injury and its Prevention, which is one of the International Research Centres for Prevention of Injury and Protection of Athlete Health supported by the International Olympic Committee (IOC). Prof Cook is a NHMRC practitioner fellow (ID 1058493)

The Universality of Einstein Equations

It is shown that for a wide class of analytic Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the so--called ``Palatini formalism'', i.e., treating the metric and the connection as independent variables, leads to ``universal'' equations. If the dimension $n$ of space--time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians $L=R^{n/2} \sqrt g$ and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2--dimensional space--time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi--Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.Comment: 15 pages, LaTeX, (Extended Version), TO-JLL-P1/9

Majority Dynamics and Aggregation of Information in Social Networks

Consider n individuals who, by popular vote, choose among q >= 2 alternatives, one of which is "better" than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. It follows from the law of large numbers that a plurality vote among the n individuals would result in the correct outcome, with probability approaching one exponentially quickly as n tends to infinity. Our interest in this paper is in a variant of the process above where, after forming their initial opinions, the voters update their decisions based on some interaction with their neighbors in a social network. Our main example is "majority dynamics", in which each voter adopts the most popular opinion among its friends. The interaction repeats for some number of rounds and is then followed by a population-wide plurality vote. The question we tackle is that of "efficient aggregation of information": in which cases is the better alternative chosen with probability approaching one as n tends to infinity? Conversely, for which sequences of growing graphs does aggregation fail, so that the wrong alternative gets chosen with probability bounded away from zero? We construct a family of examples in which interaction prevents efficient aggregation of information, and give a condition on the social network which ensures that aggregation occurs. For the case of majority dynamics we also investigate the question of unanimity in the limit. In particular, if the voters' social network is an expander graph, we show that if the initial population is sufficiently biased towards a particular alternative then that alternative will eventually become the unanimous preference of the entire population.Comment: 22 page

Azumaya Objects in Triangulated Bicategories

We introduce the notion of Azumaya object in general homotopy-theoretic settings. We give a self-contained account of Azumaya objects and Brauer groups in bicategorical contexts, generalizing the Brauer group of a commutative ring. We go on to describe triangulated bicategories and prove a characterization theorem for Azumaya objects therein. This theory applies to give a homotopical Brauer group for derived categories of rings and ring spectra. We show that the homotopical Brauer group of an Eilenberg-Mac Lane spectrum is isomorphic to the homotopical Brauer group of its underlying commutative ring. We also discuss tilting theory as an application of invertibility in triangulated bicategories.Comment: 23 pages; final version; to appear in Journal of Homotopy and Related Structure

Chern character, loop spaces and derived algebraic geometry.

International audienceIn this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere

Nano-sized SQUID-on-tip for scanning probe microscopy

We present a SQUID of novel design, which is fabricated on the tip of a pulled quartz tube in a simple 3-step evaporation process without need for any additional processing, patterning, or lithography. The resulting devices have SQUID loops with typical diameters in the range 75–300 nm. They operate in magnetic fields up to 0.6 T and have flux sensitivity of 1.8 μΦ0/Hz1/2 and magnetic field sensitivity of 10−7 T/Hz1/2, which corresponds to a spin sensitivity of 65 μB/Hz1/2 for aluminum SQUIDs. The shape of the tip and the small area of the SQUID loop, together with its high sensitivity, make our device an excellent tool for scanning SQUID microscopy: With the SQUID-on-tip glued to a tine of a quartz tuning fork, we have succeeded in obtaining magnetic images of a patterned niobium film and of vortices in a superconducting film in a magnetic field.Physic