Two-Dimensional Homogeneous Polynomial Vector Fields with Common Factors

AbstractInvariant characterizations are obtained for the existence of one or more common factors in two-dimensional homogeneous polynomial vector fields of arbitrary degree, r. The presence of a common factor is related to the existence of nonisolated critical points of the vector fields. The particular case where the highest common factor is of maximal degree, r, is studied further, from the invariant point of view. The linear (r = 1) and quadratic (r = 2) cases are then examined in the context of the general theory, and the results are contrasted with those which are obtained using more conventional approaches. The relevance of the investigation to certain inhomogeneous polynomial vector fields is briefly discussed. This brings to light an invariant characterization of the classical Jacobi differential equation

From 2-Dimensional Surfaces to Cosmological Solutions

We construct perfect fluid metrics corresponding to spacelike surfaces invariant under a 1-dimensional group of isometries in 3-dimensional Minkowski space. Under additional assumptions we obtain new cosmological solutions of Bianchi type II, VI_0 and VII_0. The solutions depend on an arbitrary function of time, which can be specified in order to satisfy an equation of state.Comment: 12 pages, no figures, LaTeX2e, to be published in Class. Quant. Gra

Local dynamics and gravitational collapse of a self-gravitating magnetized Fermi gas

We use the Bianchi-I spacetime to study the local dynamics of a magnetized self-gravitating Fermi gas. The set of Einstein-Maxwell field equations for this gas becomes a dynamical system in a 4-dimensional phase space. We consider a qualitative study and examine numeric solutions for the degenerate zero temperature case. All dynamic quantities exhibit similar qualitative behavior in the 3-dimensional sections of the phase space, with all trajectories reaching a stable attractor whenever the initial expansion scalar H_{0} is negative. If H_{0} is positive, and depending on initial conditions, the trajectories end up in a curvature singularity that could be isotropic(singular "point") or anisotropic (singular "line"). In particular, for a sufficiently large initial value of the magnetic field it is always possible to obtain an anisotropic type of singularity in which the "line" points in the same direction of the field.Comment: 6 pages, 3 figures (accepted in General Relativity and Gravitation

Mitigation of electroencephalographic and cardiovascular responses to castration in Bos indicus bulls following the administration of either lidocaine or meloxicam

Objective To investigate the mitigating effects of administration of local or systemic meloxicam on the electroencephalographic (EEG) and cardiovascular responses during surgical castration of Bos indicus bull calves. Study design Prospective, randomized, experimental study. Animals Thirty-six 6–8 month-old Bos indicus bull calves, with a mean ± standard deviation weight of 237 ± 19 kg. Methods Animals were randomly allocated to three groups of 12 (group L, 260 mg of 2% lidocaine subcutaneously and intratesticularly 5 minutes prior to castration; group M, 0.5 mg kg−1 of meloxicam subcutaneously 30 minutes prior to castration; group C, no pre-operative analgesia administered). Anaesthesia was induced and maintained with halothane (0.9–1.1%) in oxygen. Electroencephalogram, heart rate (HR) and mean blood pressure (MAP) were recorded for 300 seconds prior to (baseline, B) and from the start of surgery (first testicle incision, T1). HR and MAP were compared at 10 second intervals for 90 seconds from the start of T1. Median frequency (F50), spectral edge frequency (F95) and total power of the EEG (Ptot) were analysed using area under the curve comparing T1 to B. Results All EEG variables were significantly different between B and T1 (p ≤ 0.0001). No differences in F50 were found between groups during T1 (p = 0.6491). F95 and Ptot were significantly different between group L and groups C and M during T1 (p = 0.0005 and 0.0163, respectively). There were transient significant changes in HR and MAP in groups L and M compared to group C during the 20–50 second periods. Conclusions The EEG changes indicate nociceptive responses in all three groups during surgical castration, greater in group L compared to groups C and M. Both analgesics attenuated the peracute cardiovascular response. Lidocaine and meloxicam administered prior to castration attenuated these responses in Bos indicus bull calves. Clinical relevance These findings provide support for the pre-operative administration of lidocaine and potentially meloxicam for castration in Bos indicus bull calves

Time-Symmetrization and Isotropization of Stiff-Fluid Kantowski-Sachs Universes

It is shown that growing-entropy stiff-fluid Kantowski-Sachs universes become time-symmetric (if they start with time-asymmetric phase) and isotropize. Isotropization happens without any inflationary era during the evolution since there is no cosmological term here. It seems that this approach is an alternative to inflation since the universe gets bigger and bigger approaching 'flatness'.Comment: 9 pages, no figure

Exponential-Potential Scalar Field Universes I: The Bianchi I Models

We obtain a general exact solution of the Einstein field equations for the anisotropic Bianchi type I universes filled with an exponential-potential scalar field and study their dynamics. It is shown, in agreement with previous studies, that for a wide range of initial conditions the late-time behaviour of the models is that of a power-law inflating FRW universe. This property, does not hold, in contrast, when some degree of inhomogeneity is introduced, as discussed in our following paper II.Comment: 16 pages, Plain LaTeX, 1 Figure to be sent on request, to appear in Phys. Rev.

Shear free solutions in General Relativity Theory

The Goldberg-Sachs theorem is an exact result on shear-free null geodesics in a vacuum spacetime. It is compared and contrasted with an exact result for pressure-free matter: shear-free flows cannot both expand and rotate. In both cases, the shear-free condition restricts the way distant matter can influence the local gravitational field. This leads to intriguing discontinuities in the relation of the General Relativity solutions to Newtonian solutions in the timelike case, and of the full theory to the linearised theory in the null case. It is a pleasure to dedicate this paper to Josh Goldberg.Comment: 17 pages, no figures. For GRG special issue in honor of Josh Goldber

Duality properties of indicatrices of knots

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.Comment: 22 pages, 9 figure

Shear-free rotating inflation

We demonstrate the existence of shear-free cosmological models with rotation and expansion which support the inflationary scenarios. The corresponding metrics belong to the family of spatially homogeneous models with the geometry of the closed universe (Bianchi type IX). We show that the global vorticity does not prevent the inflation and even can accelerate it.Comment: Revtex, 12 pages; to appear in Phys. Rev.

Temperature and Polarization Patterns in Anisotropic Cosmologies

We study the coherent temperature and polarization patterns produced in homogeneous but anisotropic cosmological models. We show results for all Bianchi types with a Friedman-Robertson-Walker limit (i.e. Types I, V, VII$_{0}$, VII$_{h}$ and IX) to illustrate the range of possible behaviour. We discuss the role of spatial curvature, shear and rotation in the geodesic equations for each model and establish some basic results concerning the symmetries of the patterns produced. We also give examples of the time-evolution of these patterns in terms of the Stokes parameters $I$, $Q$ and $U$.Comment: 24 pages, 7 Figures, submitted to JCAP. Revised version: numerous references added, text rewritten, and errors corrected