Algebraic models for the hierarchy structure of evolution equations at small x

We explore several models of QCD evolution equations simplified by considering only the rapidity dependence of dipole scattering amplitudes, while provisionally neglecting their dependence on transverse coordinates. Our main focus is on the equations that include the processes of pomeron splittings. We examine the algebraic structures of the governing equation hierarchies, as well as the asymptotic behavior of their solutions in the large-rapidity limit.Comment: 12 pages, 5 figures; minor changes in the revised versio

Identification of Boundary Conditions Using Natural Frequencies

The present investigation concerns a disc of varying thickness of whose flexural stiffness $D$ varies with the radius $r$ according to the law $D=D_0 r^m$, where $D_0$ and $m$ are constants. The problem of finding boundary conditions for fastening this disc, which are inaccessible to direct observation, from the natural frequencies of its axisymmetric flexural oscillations is considered. The problem in question belongs to the class of inverse problems and is a completely natural problem of identification of boundary conditions. The search for the unknown conditions for fastening the disc is equivalent to finding the span of the vectors of unknown conditions coefficients. It is shown that this inverse problem is well posed. Two theorems on the uniqueness and a theorem on stability of the solution of this problem are proved, and a method for establishing the unknown conditions for fastening the disc to the walls is indicated. An approximate formula for determining the unknown conditions is obtained using first three natural frequencies. The method of approximate calculation of unknown boundary conditions is explained with the help of three examples of different cases for the fastening the disc (rigid clamping, free support, elastic fixing). Keywords: Boundary conditions, a disc of varying thickness,inverse problem, Plucker condition.Comment: 19 page

Tunneling transition to the Pomeron regime

We point out that, in some models of small-x hard processes, the transition to the Pomeron regime occurs through a sudden tunneling effect, rather than a slow diffusion process. We explain the basis for such a feature and we illustrate it for the BFKL equation with running coupling by gluon rapidity versus scale correlation plots.Comment: 17 pages, 5 figures, mpeg animations available from http://www.lpthe.jussieu.fr/~salam/tunneling/ . v2 includes additional reference

Deep inelastic scattering and "elastic" diffraction

We examine the total cross section of virtual photons on protons, $\sigma_{\gamma^* p}(W^2,Q^2)$, at low $x \cong Q^2/W^2 \ll 1$ and its connection with ``elastic'' diffractive production $\gamma^*_{T,L}p \to X^{J=1}_{T,L} p$ in the two-gluon exchange dynamics for the virtual forward Compton scattering amplitude. Solely based on the generic structure of two-gluon exchange, we establish that the cross section is described by the (imaginary part of the) amplitude for forward scattering of $q \bar q$ vector states, $(q \bar q)^{J=1}_{T,L} p \to (q \bar q)^ {J=1}_{T,L} p$. The generalized vector dominance/color dipole picture (GVD/CDP) is accordingly established to only rest on the two-gluon-exchange generic structure. This is explicitly seen by the sum rules that allow one to directly relate the total cross section to the cross section for elastic diffractive forward production, $\gamma^*_{T,L} p\to (q \bar q)^{J=1}_{T,L} p$, of vector states.Comment: 24 pages, latex file with three eps figures. BI-TP 2002/2

Quantum interference in nanofractals and its optical manifestation

We consider quantum interferences of ballistic electrons propagating inside fractal structures with nanometric size of their arms. We use a scaling argument to calculate the density of states of free electrons confined in a simple model fractal. We show how the fractal dimension governs the density of states and optical properties of fractal structures in the RF-IR region. We discuss the effect of disorder on the density of states along with the possibility of experimental observation.Comment: 19 pages, 6 figure

Traveling wave fronts and the transition to saturation

We propose a general method to study the solutions to nonlinear QCD evolution equations, based on a deep analogy with the physics of traveling waves. In particular, we show that the transition to the saturation regime of high energy QCD is identical to the formation of the front of a traveling wave. Within this physical picture, we provide the expressions for the saturation scale and the gluon density profile as a function of the total rapidity and the transverse momentum. The application to the Balitsky-Kovchegov equation for both fixed and running coupling constants confirms the effectiveness of this method.Comment: 9 pages, 3 figures, references adde

Patchiness and Demographic Noise in Three Ecological Examples

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy

Topological Field Theories and Geometry of Batalin-Vilkovisky Algebras

The algebraic and geometric structures of deformations are analyzed concerning topological field theories of Schwarz type by means of the Batalin-Vilkovisky formalism. Deformations of the Chern-Simons-BF theory in three dimensions induces the Courant algebroid structure on the target space as a sigma model. Deformations of BF theories in $n$ dimensions are also analyzed. Two dimensional deformed BF theory induces the Poisson structure and three dimensional deformed BF theory induces the Courant algebroid structure on the target space as a sigma model. The deformations of BF theories in $n$ dimensions induce the structures of Batalin-Vilkovisky algebras on the target space.Comment: 25 page

Signal and System Approximation from General Measurements

In this paper we analyze the behavior of system approximation processes for stable linear time-invariant (LTI) systems and signals in the Paley-Wiener space PW_\pi^1. We consider approximation processes, where the input signal is not directly used to generate the system output, but instead a sequence of numbers is used that is generated from the input signal by measurement functionals. We consider classical sampling which corresponds to a pointwise evaluation of the signal, as well as several more general measurement functionals. We show that a stable system approximation is not possible for pointwise sampling, because there exist signals and systems such that the approximation process diverges. This remains true even with oversampling. However, if more general measurement functionals are considered, a stable approximation is possible if oversampling is used. Further, we show that without oversampling we have divergence for a large class of practically relevant measurement procedures.Comment: This paper will be published as part of the book "New Perspectives on Approximation and Sampling Theory - Festschrift in honor of Paul Butzer's 85th birthday" in the Applied and Numerical Harmonic Analysis Series, Birkhauser (Springer-Verlag). Parts of this work have been presented at the IEEE International Conference on Acoustics, Speech, and Signal Processing 2014 (ICASSP 2014

Wilson line correlator in the MV model: relating the glasma to deep inelastic scattering

In the color glass condensate framework the saturation scale measured in deep inelastic scattering of high energy hadrons and nuclei can be determined from the correlator of Wilson lines in the hadron wavefunction. These same Wilson lines give the initial condition of the classical field computation of the initial gluon multiplicity and energy density in a heavy ion collision. In this paper the Wilson line correlator in both adjoint and fundamental representations is computed using exactly the same numerical procedure that has been used to calculate gluon production in a heavy ion collision. In particular the discretization of the longitudinal coordinate has a large numerical effect on the relation between the color charge density parameter g^2 mu and the saturation scale Qs. Our result for this relation is Qs = 0.6 g^2 mu, which results in the classical Yang-Mills value for the "gluon liberation coefficient" c = 1.1.Comment: 8 pages, 10 figures, RevTEX4, V2: typo corrections, V3: small clarifications, to be published in EPJ