Localization for nonabelian group actions

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction \xred = \mu^{-1}(0)/K. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X)$ of $X$ to the cohomology H^*(\xred). In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of \xred of any \eta_0 \in H^*(\xred) whose degree is the dimension of \xred, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ isComment: 42 pages, LaTex version no. 2.09, Introduction and Section 8 have been rewritten in revised versio

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e. by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.Comment: 22 page

Critical sets of the total variance of state detect all SLOCC entanglement classes

We present a general algorithm for finding all classes of pure multiparticle states equivalent under Stochastic Local Operations and Classsical Communication (SLOCC). We parametrize all SLOCC classes by the critical sets of the total variance function. Our method works for arbitrary systems of distinguishable and indistinguishable particles. We also discuss the Morse indices of critical points which have the interpretation of the number of independent non-local perturbations increasing the variance and hence entanglement of a state. We illustrate our method by two examples.Comment: 4 page

Stratifying quotient stacks and moduli stacks

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.Comment: 25 pages, submitted to the Proceedings of the Abel Symposium 201

Hodge polynomials of some moduli spaces of Coherent Systems

When $k<n$, we study the coherent systems that come from a BGN extension in which the quotient bundle is strictly semistable. In this case we describe a stratification of the moduli space of coherent systems. We also describe the strata as complements of determinantal varieties and we prove that these are irreducible and smooth. These descriptions allow us to compute the Hodge polynomials of this moduli space in some cases. In particular, we give explicit computations for the cases in which $(n,d,k)=(3,d,1)$ and $d$ is even, obtaining from them the usual Poincar\'e polynomials.Comment: Formerly entitled: "A stratification of some moduli spaces of coherent systems on algebraic curves and their Hodge--Poincar\'e polynomials". The paper has been substantially shorten. Theorem 8.20 has been revised and corrected. Final version accepted for publication in International Journal of Mathematics. arXiv admin note: text overlap with arXiv:math/0407523 by other author

Subconjunctival Injection of XG-102, a JNK Inhibitor Peptide, in Patients with Intraocular Inflammation: A Safety and Tolerability Study.

Abstract Purpose: We aimed to investigate the safety, tolerability, and systemic diffusion of a single escalating dose of XG-102 (a 31-D-amino-acid peptide inhibiting JNK pathway activation), administered subconjunctivally in the treatment of post-surgery or post-trauma intraocular inflammation. Methods: This is a dose-escalating, tolerance Phase Ib study. Twenty patients with post-surgery or post-traumatic intraocular inflammation were assigned to 1 of the 4 dose escalating (45, 90, 450, or 900 μg XG-102) groups of 5 patients each. Patients were evaluated at 24, 48 h, 8, and 28 days following the administration of XG-102, including laboratory tests, standard eye examinations, vital signs, and occurrence of adverse events. A single plasma quantification of XG-102 was performed 30 min after administration, according to previous pharmacokinetics studies performed on volunteers. Results: A total of 17 non-serious adverse events, considered unrelated to the study treatment, were reported for 10 patients. The adverse event incidence was not related to the drug dose. All patients experienced a decrease in intraocular inflammation as of 24 h post-administration and this decrease was sustained up to 28 days thereafter. No patient required local injection or systemic administration of corticoids following the administration of XG-102. XG-102 was undetectable in the first 3 dose groups. In the fourth-dose group (900 μg) the XG-102 plasma levels were above the limit of detection for 3 patients and above the limit of quantification for 1 patient. Conclusions: In this first clinical trial using XG-102, administered as a single subconjunctival injection as adjunct therapy, in patients with recent post-surgery or post-trauma intraocular inflammation is safe and well tolerated. Further studies are required to evaluate its efficacy

Quantization of Fayet-Iliopoulos Parameters in Supergravity

In this short note we discuss quantization of the Fayet-Iliopoulos parameter in supergravity theories. We argue that in supergravity, the Fayet-Iliopoulos parameter determines a lift of the group action to a line bundle, and such lifts are quantized. Just as D-terms in rigid N=1 supersymmetry are interpreted in terms of moment maps and symplectic reductions, we argue that in supergravity the quantization of the Fayet-Iliopoulos parameter has a natural understanding in terms of linearizations in geometric invariant theory (GIT) quotients, the algebro-geometric version of symplectic quotients.Comment: 21 pages, utarticle class; v2: typos and tex issue fixe

How do different lighting conditions affect the vision and quality of life of people with glaucoma? A systematic review

This article is a systematic review of evidence regarding the impact of different lighting conditions on the vision and quality of life (QoL) of people with primary open-angle glaucoma (POAG). A systematic literature search was carried out using CINAHL, MEDLINE, PsycARTICLES, PsycINFO, Embase, and Ovid Nursing Database for studies: published up to April 2019; including people diagnosed with POAG; and assessing visual function or QoL in response to changing lighting/luminance levels or glare. Two researchers independently screened studies for eligibility. Data were extracted from eligible studies regarding study design, participant characteristics, outcomes, and results. Quality of included studies was critically appraised. Of 8437 studies, 56 eligible studies were included. Studies investigated the effects of lighting on the following domains among people with POAG: QoL (18/56), psychophysical measures (16/56), functional vision (10/56), activities of daily living (10/56), and qualitative findings (2/56). POAG negatively affects low-luminance contrast sensitivity, glare symptoms, and dark adaptation time and extent. In vision-related QoL questionnaires, people with POAG report problems with lighting, glare, and dark adaptation more frequently than any other domain. These problems worsen with progressing visual field loss. Early-stage POAG patients experience significantly more difficulties in low-luminance or changing lighting conditions than age-matched controls (AMCs), challenging perceptions of early-stage POAG as asymptomatic. However, performance-based studies seldom show significant differences between POAG participants and AMCs on tasks simulating daily activities under non-optimal lighting conditions. Further research with larger samples is required to optimise ambient and task-oriented lighting that can support patients’ adaptation to POAG

A spectacular jet from the bright 244-440 Orion proplyd: the MUSE NFM view

In this work we present the highest spatial and spectral resolution integral field observations to date of the bipolar jet from the Orion proplyd 244-440 using MUSE NFM) observations on the VLT. We observed a previously unreported chain of six distinct knots in a roughly S-shaped pattern, and by comparing them with HST images we estimated proper motions in the redshifted knots of 9.5 mas yr$^{-1}$ with an inclination angle of $73^{\circ}$, though these quantities could not be measured for the blueshifted lobe. Analysis of the [FeII] and [NiII] lines suggests jet densities on the order of $\sim 10^5$ cm$^{-3}$. We propose that the observed S-shaped morphology originates from a jet launched by a smaller source with $M_\star < 0.2$ M$_{\odot}$ in orbital motion around a larger companion of $M_\star \simeq 0.5$ M$_{\odot}$ at a separation of 30-40 au. The measured luminosities of the knots using the [OI]$\lambda6300$ and [SII]$\lambda6731$ lines were used to estimate a lower limit to the mass-loss rate in the jet of $1.3 \times 10^{-11}$ M$_{\odot}$ yr$^{-1}$ and an upper limit of $10^{-9}$ M$_{\odot}$ yr$^{-1}$, which is typical for low-mass driving sources. While the brightness asymmetry between the redshifted and blueshifted lobes is consistent with external irradiation, further analysis of the [NiII] and [FeII] lines suggests that photoionization of the jet is not likely to be a dominant factor, and that the emission is dominated by collisional excitation. The dynamical age of the jet compared to the anticipated survival time of the proplyd demonstrates that photoevaporation of the proplyd occurred prior to jet launching, and that this is still an active source. These two points suggest that the envelope of the proplyd may shield the jet from the majority of external radiation, and that photoionization of the proplyd does not appear to impact the ability of a star to launch a jet.Comment: 30 pages, 20 figure

Are GPs under-investigating older patients presenting with symptoms of ovarian cancer? Observational study using General Practice Research Database

Background: Recent studies suggest that older patients in the United Kingdom are not benefiting as much from improvements in cancer treatments as their younger counterparts. We investigate whether this might be partly due to differential referral rates using ovarian cancer as an example. Methods: From the General Practice Research Database (GPRD), we identified all women aged 40–80 years on 1 June 2002 with a Read code for ovarian cancer between 1 June 2002 and 31 May 2007. Using these records, we compared the GPRD incidence of ovarian cancer with rates compiled from the UK cancer registries and investigated the relationship between age and coded investigations for suspected ovarian cancer. Results: The GPRD rates peaked earlier, at 70–74, and were lower than registry rates for nearly all ages particularly for patients over 59. The proportion investigated or referred by the GP decreased significantly with age and delays between first coded symptom and investigation showed a U-shaped distribution by age. Conclusions: GPs appear to be less likely to recognise and to refer patients presenting with ovarian cancer as they get older. If our findings extend to other cancers, lack of or delays in referral to secondary care may partly explain poor UK cancer mortality rates of older people