On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

On the relation between standard and μ\mu-symmetries for PDEs

We give a geometrical interpretation of the notion of $\mu$-prolongations of vector fields and of the related concept of $\mu$-symmetry for partial differential equations (extending to PDEs the notion of $\lambda$-symmetry for ODEs). We give in particular a result concerning the relationship between $\mu$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $\mu$-symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.

C∞\mathcal{C}^{\infty}-structures in the integration of involutive distributions

For a system of ordinary differential equations (ODEs) or, more generally, an involutive distribution of vector fields, the problem of its integration is considered. Among the many approaches to this problem, solvable structures provide a systematic procedure of integration via Pfaffian equations that are integrable by quadratures. In this paper structures more general than solvable structures (named cinf-structures) are considered. The symmetry condition in the concept of solvable structure is weakened for cinf-structures by requiring their vector fields be just cinf-symmetries. For cinf-structures there is also an integration procedure, but the corresponding Pfaffian equations, although completely integrable, are not necessarily integrable by quadratures. The well-known result on the relationship between integrating factors and Lie point symmetries for first-order ODEs is generalized for cinf-structures and involutive distributions of arbitrary corank by introducing symmetrizing factors. The role of these symmetrizing factors on the integrability by quadratures of the Pfaffian equations associated with the \cinf-structure is also established. Some examples that show how these objects and results can be applied in practice are also presented

C∞\mathcal{C}^{\infty}-symmetries of distributions and integrability

An extension of the notion of solvable structure for involutive distributions of vector fields is introduced. The new structures are based on a generalization of the concept of symmetry of a distribution of vector fields, inspired in the extension of Lie point symmetries to $\mathcal{C}^{\infty}$-symmetries for ODEs developed in the recent years. These new objects, named $\mathcal{C}^{\infty}$-structures, play a fundamental role in the integrability of the distribution: the knowledge of a $\mathcal{C}^{\infty}$-structure for a corank $k$ involutive distribution permits to find its integral manifolds by solving $k$ successive completely integrable Pfaffian equations. These results have important consequences for the integrability of differential equations. In particular, we derive a new procedure to integrate an $m$th-order ordinary differential equation by splitting the problem into $m$ completely integrable Pfaffian equations. This step-by-step integration procedure is applied to integrate completely several equations that cannot be solved by standard procedures

Effects of Noise Reduction and Care Clustering on Quality of Sleep in Critical Care Patients

Introduction: Sleep deprivation has detrimental effects on critical patients’ health. A lack of sleep can affect multiple body systems. There are nursing interventions that can reduce sleep deprivation. However, there is inconclusive evidence on how to properly assess sleep deprivation and implement sleep promoting nursing interventions in clinical practice. Purpose: The purpose of this literature review is to examine the effects of noise reduction and nursing care clustering on improving the quality of patient sleep in the critical care setting. Methods: This literature review was conducted using 10 sources published within the last 5 years. Inclusion criteria consisted of articles about the effects of noise, quality of sleep, and implementation of nurse care clustering on various critical care populations. The 6 databases used for this research were UpToDate, CINAHL, PubMed, PsycInfo, Proquest, and CCForum. This research concentrated on examining articles containing nursing interventions for noise reduction and care clustering related to quality of sleep. Results: Noise has a negative effect on sleep by causing more arousals/awakenings, which greatly impacts the restorative function of the process. Noise is not the only sleep disturbing factor, but it has been shown to be significant. Some noise sources cannot be eliminated due to safety reasons, but interventions exist to help counteract the effects of noise. Nursing care interventions are as disruptive to sleep as noise. 13.9% of nursing interruptions could be safely omitted, and nurses should cluster care to promote sleep. Interventions to prevent sleep disruption can be practical in routine nursing, but nurses are less likely to implement them at night because prioritizing care clustering can require more time and effort. Discussion: Sleep deprivation causes major health concerns in critical care patients. Noise and nursing care interventions have been found to cause equal disruptions in sleep. Noise reduction and care clustering have been observed to reduce sleep deprivation. Further evidence is needed on how to effectively and practically implement these nursing interventions into daily nursing practice

Galaxy groups in the 2dF galaxy redshift survey: A Compactness Analysis of Groups

A comprehensive study on compactness has been carried out on the 2dF Galaxy Group Catalogue constructed by Merch\'an & Zandivarez. The compactness indexes defined in this work take into account different geometrical constraints in order to explore a wide range of possibilities. Our results show that there is no clear distinction between groups with high and low level of compactness when considering particular properties as the radial velocity dispersion, the relative fraction of galaxies per spectral type and luminosity functions of their galaxy members. Studying the trend of the fraction of galaxies per spectral type as a function of the dimensionless crossing time some signs of dynamical evolution are observed. From the comparison with previous works on compactness we realize that special care should be taken into account for some compactness criteria definitions in order to avoid possible biases in the identification.Comment: 11 pages, 14 figures, resubmitted to MNRAS after minor revisio

Redshifts in the Southern Abell Redshift Survey Clusters. I. The Data

The Southern Abell Redshift Survey contains 39 clusters of galaxies with redshifts in the range 0.0 < z < 0.31 and a median redshift depth of z = 0.0845. SARS covers the region 0 21h (while avoiding the LMC and SMC) with b > 40. Cluster locations were chosen from the Abell and Abell-Corwin-Olowin catalogs while galaxy positions were selected from the Automatic Plate Measuring Facility galaxy catalog with extinction-corrected magnitudes in the range 15 <= b_j < 19. SARS utilized the Las Campanas 2.5 m duPont telescope, observing either 65 or 128 objects concurrently over a 1.5 sq deg field. New redshifts for 3440 galaxies are reported in the fields of these 39 clusters of galaxies.Comment: 20 pages, 5 figures, accepted for publication in the Astronomical Journal, Table 2 can be downloaded in its entirety from http://trotsky.arc.nasa.gov/~mway/SARS1/sars1-table2.cs