Viscous Generalized Chaplygin Gas with Arbitrary α

In this paper we study generalized Chaplygin gas which has viscosity for the case of arbitrary α. We obtained modified time-dependent energy density due to bulk viscosity and generalized Chaplygin gas. We have obtained the energy density in terms of α, and also we plotted it with respect to time by different α.     

Geometries with the Second Poincar Symmetry

The second Poincar kinematical group serves as one of new ones in addition to the known possible kinematics.The geometries with the second Poincar'e symmetry is presented and their properties are analyzed.On the geometries,the new mechanics based on the principle of relativity with two universal constants(c,l) can be established.     

Local Poincaré Symmetry in Gauge Theory of Gravity

It is well known that the Poincaré gauge theories of gravity do not have the structure of a standard gauge theory. Nevertheless, we show that a general form of action for the gravitational gauge fields in the gauge theory does possess local Poincaré invariance.     

Mechanical systems with Poincaré invariance

Some years ago Ruijsenaars and Schneider initiated the study of mechanical systems exhibiting an action of the Poincaré algebra. The systems they discovered were far richer: their models were actually integrable and possessed a natural quantum version. We follow this early work finding and classifying mechanical systems with such an action. New solutions are found together with a new class of models exhibiting an action of the Galilean algebra. These are related to the functional identities underlying the various Hirzebruch genera. The quantum mechanics is also discussed.     

Time-Dependent Density of Viscous Modified Chaplygin Gas with Arbitrary α

In this paper we study modified Chaplygin gas which has viscosity for the case of arbitrary α instead of α=0.5. We obtain behavior of the energy density of modified Chaplygin gas with respect to the constant and time-dependent bulk viscosity. We find also, numerically, effect of α on the energy density.     

Poincaré vortices

Traditional interferometric methods for measuring the vortex structure of complex wave fields suffer from many intrinsic problems and seldom yield results of any accuracy. Using the unique properties of what I call Poincaré vortices, I develop a radically different method based on Stokes parameters that offers many practical advantages. The theory of this new method is discussed, and its unique capabilities are illustrated by reconstruction with high accuracy of the vortex structure of a simulated random field containing numerous vortices, including several closely spaced vortex pairs that would be difficult, if not impossible, to resolve by traditional means.     

Evolution of Variable Generalized Chaplygin Gas

We consider the variable Generalized Chaplygin gas （VGCG） proposal for unification of dark matter and dark energy with p = pdc and ρ= pdm ＋ ρdc. The equation of state of the VGCG is given by p = -A0a^-n/ρ^α, where a is the scale factor. Some cosmological quantities such as the fractional contributions of different components of the universe Ωi （i respectively denotes baryons, dark matter and dark energy） to the critical density, the deceleration parameter q are all obtained. The transition from deceleration to acceleration is described in this model. In addition, we find the behaviour of variable Generalized Chaplgin gas is similar to dust-like matter at early times and will be quiessence or phantom at late stage.     

Fields on the Poincaré Group: Arbitrary Spin Description and Relativistic Wave Equations

In this paper, starting from a pure group-theoretic point of view, we develop an approach to describing particles with different spins in the framework of a theory of scalar fields on the Poincaré group. Such fields can be considered as generating functions for conventional spin-tensor fields. The case of two, three, and four dimensions are elaborated in detail. Discrete transformations C, P, T are defined for the scalar fields as automorphisms of the Poincaré group. We classify the scalar functions, and obtain relativistic wave equations for particles with definite spin and mass. There exist two different types of scalar functions (which describe the same mass and spin), one related to a finite-dimensional nonunitary representation and the other to an infinite-dimensional unitary representation of the Lorentz subgroup. This allows us to derive both usual finite-component wave equations for spin-tensor fields and positive-energy, infinite-component wave equations.     

Pseudospin Symmetry in Relativistic Framework with Harmonic Oscillator Potential and Woods—Saxon Potential

Based on the Dirac equation,we discuss the exact pseudospin symmetry and some examples are presented.For harmonic oscillator potential there exist bound states under the condition of the exact pseudospin symmetry.Vs Vv=0,and even usual intruder orbits will have degenerated pseudospin parners,Apart from the harmonic oscillator potential,a Woods-Saxon potential is introduced to eliminate the redundant degeneracies due to the speciality of harmonic oscillator potential except the pseudospin degeneracy.     

Curvilinear Poincaré vector beams

We develop a method for completely shaping optical vector beams with controllable amplitude, phase, and polarization gradients along three-dimensional freestyle trajectories. We design theoretically and demonstrate experimentally curvilinear Poincaré vector beams that exhibit high intensity gradients and accurate state of polarization prescribed along the beam trajectory.     

Poincaré Invariant Three-Body Scattering

Relativistic Faddeev equations for three-body scattering are solved at arbitrary energies in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated withing the framework of Poincaré invariant quantum mechanics. Based on a Malfliet–Tjon interaction, observables for elastic and breakup scattering are calculated and compared to non-relativistic ones.     

On the Poincaré and Gauge Symmetry of a Model where Vector and Axial Vector Interaction get Mixed up with Different Weight

A (1+1) dimensional model where vector and axial vector interaction get mixed up with different weight is considered with a generalized masslike term for gauge field. Through Poincaré algebra it has been made confirm that only a Lorentz covariant masslike term leads to a physically sensible theory as long as the number of constraints in the phase space is two. With that admissible masslike term, phase space structure of this model with proper identification of physical canonical pair has been determined using Diracs’ scheme of quantization of constrained system. The bosonized version of the model remains gauge non-invariant to start with. Therefore, with the inclusion of appropriate Wess-Zumino term it is made gauge symmetric. An alternative quantization has been carried out over this gauge symmetric version to determine the phase space structure in this situation. To establish that the Wess-Zumino fields allocates themselves in the un-physical sector of the theory an attempts has been made to get back the usual theory from the gauge symmetric theory of the extended phase-space without hampering any physical principle. It has been found that the role of gauge fixing is crucial for this transmutation.     

Full vectorial feature of second-harmonic generation with full Poincaré beams

We demonstrate the full vectorial feature of second-harmonic generation(SHG), i.e., from infrared full Poincarébeams to visible full Poincaré beams, based on two cascading type I phase-matching beta barium borate crystals of orthogonal optical axes. We visualize the structured features of the vectorial SHG wave by using Stokes polarimetry and show the interesting doubling effect of the polarization topological index, i.e., a low-order full Poincaré beam is converted to a high-order one. However, the polarization singularities of both C points and L lines are found to keep invariant during the SHG process. Our scheme could offer a deeper understanding on the interaction of vectorial light fields with media and can be generalized to other nonlinear optical effects.     

Flattop focusing with full Poincaré beams under low numerical aperture illumination

In this Letter, we propose and experimentally demonstrate the generation of second-order full Poincaré beams and its applications in two-dimensional flattop beam shaping with spatially variant polarization under low NA focusing condition. High-quality flattop profiles with steep edge roll-off can be obtained with this technique. The experiment results also demonstrate that flattop profile can be maintained for different input beam sizes by conveniently rotating a half-wave plate.     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.