It was shown by Oberlack and Rosteck [Discr. Cont. Dyn. Sys. S, 3, 451 2010] that the infinite set of multipoint correlation (MPC) equations of turbulence admits a considerable extended set of Lie point symmetries compared to the Galilean group, which is implied by the original set of equations of fluid mechanics. Specifically, a new scaling group and an infinite set of translational groups of all multipoint correlation tensors have been discovered. These new statistical groups have important consequences for our understanding of turbulent scaling laws as they are essential ingredients of, e.g., the logarithmic law of the wall and other scaling laws, which in turn are exact solutions of the MPC equations. In this paper we first show that the infinite set of translational groups of all multipoint correlation tensors corresponds to an infinite dimensional set of translations under which the Lundgren-Monin-Novikov (LMN) hierarchy of equations for the probability density functions (PDF) are left invariant. Second, we derive a symmetry for the LMN hierarchy which is analogous to the scaling group of the MPC equations. Most importantly, we show that this symmetry is a measure of the intermittency of the velocity signal and the transformed functions represent PDFs of an intermittent (i.e., turbulent or nonturbulent) flow. Interesting enough, the positivity of the PDF puts a constraint on the group parameters of both shape and intermittency symmetry, leading to two conclusions. First, the latter symmetries may no longer be Lie group as under certain conditions group properties are violated, but still they are symmetries of the LMN equations. Second, as the latter two symmetries in its MPC versions are ingredients of many scaling laws such as the log law, the above constraints implicitly put weak conditions on the scaling parameter such as von Karman constant κ as they are functions of the group parameters. Finally, let us note that these kind of statistical symmetries are of much more general type, i.e., not limited to MPC or PDF equations emerging from Navier-Stokes, but instead they are admitted by other nonlinear partial differential equations like, for example, the Burgers equation when in conservative form and if the nonlinearity is quadratic.
We give an explicit form of the symplectic groupoid G(S 2 , π) that integrates the semiclassical standard Podles sphere (S 2 , π). We show that Sheu's groupoid G S , whose convolution C * -algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of G(S 2 , π). By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.
We study a reduction procedure for describing the symplectic groupoid of a Poisson homogeneous space obtained by quotient of a coisotropic subgroup. We perform it as a reduction of the Lu-Weinstein symplectic groupoid integrating Poisson Lie groups, that is suitable even for the non complete case.Comment: 22 pages, minor change
Under the assumption of total absorption and dominance of the imaginary part of the scattering amplitude, we present a sum rule for any hadronic elastic differential cross-section dσ dt : the dimensionless quantity 1 2 (dt) 1 π dσ dt → 1, at asymptotic energies. Experimental data from ISR and Tevatron confirm a trend towards its saturation and some estimates are presented for LHC. Its universality and further consequences for the nature of absorption in QCD based models for elastic and total cross-sections are explored.Ab initio calculations of hadronic elastic amplitudes and total cross-sections (in QCD) are presently difficult due to our meager understanding of "soft" physics, that is, the non-perturbative and confinement region of QCD. Hence, the need to invoke general principles such as analyticity and unitarity to obtain bounds and restrictions on these amplitudes [1][2][3]. Analyticity and unitarity are expected to hold for finite-ranged hadron dynamics, only massive hadrons being the bound states of quarks and glue. In the following, we find that, under rather mild assumptions, a universal behavior for all hadrons is likely to emerge at asymptotic energies. Consider the amplitude for an elastic process, be the square of the CM energy, t = (p a − p c ) 2 , be the momentum transfer and let us normalize the amplitude so that the differential and total cross-sections are given byTo enforce (direct or s-channel) unitarity and incorporate the knowledge that most of the hadronic scatterings at high energies are peaked in the forward direction, an eikonal formalism is convenient. The elastic amplitude may be expanded in the impact parameter (b-space) in the usual fashionin terms of the "partial b-wave" amplitudesFwhere the inelasticity factor η(s, b) lies between 0 and 1, and δ R (s, b) is the real part of the phase shift. The dimensionless "b-wave cross-sections" are given byandEqs.(3) show explicitly the maximum permissible rise for the different cross-sections due to unitarity. For complete absorption of "low" partial waves at asymptotic energies (which translates into η(s, b) → 0 for b → 0 and s → ∞), one obtains the geometric limit (including the contribution from shadow scattering):1
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