The exterior degree of a pair of finite groups

The exterior degree of a pair of finite groups $(G,N)$, which is a generalization of the exterior degree of finite groups, is the probability for two elements $(g,n)$ in $(G,N)$ such that $g\wedge n=1$. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups.Comment: To appear in Mediterr. J. Mat

Analytic models of plausible gravitational lens potentials

Gravitational lenses on galaxy scales are plausibly modelled as having ellipsoidal symmetry and a universal dark matter density profile, with a Sersic profile to describe the distribution of baryonic matter. Predicting all lensing effects requires knowledge of the total lens potential: in this work we give analytic forms for that of the above hybrid model. Emphasising that complex lens potentials can be constructed from simpler components in linear combination, we provide a recipe for attaining elliptical symmetry in either projected mass or lens potential. We also provide analytic formulae for the lens potentials of Sersic profiles for integer and half-integer index. We then present formulae describing the gravitational lensing effects due to smoothly-truncated universal density profiles in cold dark matter model. For our isolated haloes the density profile falls off as radius to the minus fifth or seventh power beyond the tidal radius, functional forms that allow all orders of lens potential derivatives to be calculated analytically, while ensuring a non-divergent total mass. We show how the observables predicted by this profile differ from that of the original infinite-mass NFW profile. Expressions for the gravitational flexion are highlighted. We show how decreasing the tidal radius allows stripped haloes to be modelled, providing a framework for a fuller investigation of dark matter substructure in galaxies and clusters. Finally we remark on the need for finite mass halo profiles when doing cosmological ray-tracing simulations, and the need for readily-calculable higher order derivatives of the lens potential when studying catastrophes in strong lenses.Comment: 24 pages, 10 figures, matches published versio

Constraints on cosmological models from strong gravitational lensing systems

Strong lensing has developed into an important astrophysical tool for probing both cosmology and galaxies (their structure, formation, and evolution). Using the gravitational lensing theory and cluster mass distribution model, we try to collect a relatively complete observational data concerning the Hubble constant independent ratio between two angular diameter distances $D_{ds}/D_s$ from various large systematic gravitational lens surveys and lensing by galaxy clusters combined with X-ray observations, and check the possibility to use it in the future as complementary to other cosmological probes. On one hand, strongly gravitationally lensed quasar-galaxy systems create such a new opportunity by combining stellar kinematics (central velocity dispersion measurements) with lensing geometry (Einstein radius determination from position of images). We apply such a method to a combined gravitational lens data set including 70 data points from Sloan Lens ACS (SLACS) and Lens Structure and Dynamics survey (LSD). On the other hand, a new sample of 10 lensing galaxy clusters with redshifts ranging from 0.1 to 0.6 carefully selected from strong gravitational lensing systems with both X-ray satellite observations and optical giant luminous arcs, is also used to constrain three dark energy models ($\Lambda$CDM, constant $w$ and CPL) under a flat universe assumption. For the full sample ($n=80$) and the restricted sample ($n=46$) including 36 two-image lenses and 10 strong lensing arcs, we obtain relatively good fitting values of basic cosmological parameters, which generally agree with the results already known in the literature. This results encourages further development of this method and its use on larger samples obtained in the future.Comment: 22 pages, 5 figures, 2 tables; accepted by JCA

Transmission of SARS and MERS coronaviruses and influenza virus in healthcare settings: the possible role of dry surface contamination

Viruses with pandemic potential including H1N1, H5N1, and H5N7 influenza viruses, and severe acute respiratory syndrome (SARS)/Middle East respiratory syndrome (MERS) coronaviruses (CoV) have emerged in recent years. SARS-CoV, MERS-CoV, and influenza virus can survive on surfaces for extended periods, sometimes up to months. Factors influencing the survival of these viruses on surfaces include: strain variation, titre, surface type, suspending medium, mode of deposition, temperature and relative humidity, and the method used to determine the viability of the virus. Environmental sampling has identified contamination in field-settings with SARS-CoV and influenza virus, although the frequent use of molecular detection methods may not necessarily represent the presence of viable virus. The importance of indirect contact transmission (involving contamination of inanimate surfaces) is uncertain compared with other transmission routes, principally direct contact transmission (independent of surface contamination), droplet, and airborne routes. However, influenza virus and SARS-CoV may be shed into the environment and be transferred from environmental surfaces to hands of patients and healthcare providers. Emerging data suggest that MERS-CoV also shares these properties. Once contaminated from the environment, hands can then initiate self-inoculation of mucous membranes of the nose, eyes or mouth. Mathematical and animal models, and intervention studies suggest that contact transmission is the most important route in some scenarios. Infection prevention and control implications include the need for hand hygiene and personal protective equipment to minimize self-contamination and to protect against inoculation of mucosal surfaces and the respiratory tract, and enhanced surface cleaning and disinfection in healthcare settings