High Density QCD

The dynamics of high partonic density QCD is presented considering, in the double logarithm approximation, the parton recombination mechanism built in the AGL formalism, developed including unitarity corrections for the nucleon as well for nucleus. It is shown that these corrections are under theoretical control. The resulting non linear evolution equation is solved in the asymptotic regime, and a comprehensive phenomenology concerning Deep Inelastic Scattering like $F_2$, $F_L$, $F_2^c$. $\partial F_2/ \partial \ln Q^2$, $\partial F^A_2/ \partial \ln Q^2$, etc, is presented. The connection of our formalism with the DGLAP and BFKL dynamics, and with other perturbative (K) and non-perturbative (MV-JKLW) approaches is analised in detail. The phenomena of saturation due to shadowing corrections and the relevance of this effect in ion physics and heavy quark production is emphasized. The implications to e-RHIC, HERA-A, and LHC physics and some open questions are mentioned.Comment: 41 pages, 13 figures. Plenary Talk presented at XXI ENFPC, Sao Lourenco, Brasil, October, 24th (2000

Representing Symmetric Rank Two Updates

Various quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function). In this paper we examine several ways to express a symmetric rank 2 matrix [delta] as the sum of rank 1 matrices. We show that it is easy to compute rank 1 matrices [delta1] and [delta2] such that [delta] = [delta1] + [delta2] and [the norm of delta1]+ [the norm of delta2] is minimized, where ||.|| is any inner product norm. Such a representation recommends itself for use in those computer programs that maintain A explicitly, since it should reduce cancellation errors and/or improve efficiency over other representations. In the common case where [delta] is indefinite, a choice of the form [delta1] = [delta2 to the power of T] = [xy to the power of T] appears best. This case occurs for rank 2 quasi- Newton updates [delta] exactly when [delta] may be obtained by symmetrizing some rank 1 update; such popular updates as the DFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category.

On Modifying Singular Values to Solve Possible Singular Systems of Non-Linear Equations

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations f(x) = 0 whose Jacobian matrix J(x) exists but maybe singular. The main idea is to modify small singular values of J(x) in such away that the modified Jacobian matrix J^(x) has a continuous pseudoinverse J^+(x)and that a solution x* of f(x) = 0 may be found by determining an asymptote of the solution to the initial value problem x(0) = x[sub0}, x’(t) = -J^+(x)f(x). We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and J(x*) is non-singular, then x is unique), algorithms using(x) may enjoy a larger region of convergence than those that require(an approximation to) J[to the -1 power[(x).

Some Convergence Properties of Broyden's Method

In 1965 Broyden introduced a family of algorithms called(rank-one) quasi—New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every second iteration, then the solution is found in at most 2n steps. Specializing to the particular family member known as Broyden’s (good) method, we use this result to show that Broyden's method enjoys local 2n-step Q-quadratic convergence on nonlinear problems.

Investigating diffractive W production in hadron-hadron collisions at high energies

We compute the hard diffractive hadroproduction of bosons W at high energies using Regge factorization and taking into account multiple Pomeron exchange corrections. The ratio of diffractive to non-diffractive W production agrees with the current Tevatron data and a prediction for the LHC is presented.Comment: 4 pages, 1 table. To appear in the proceedings of 10th International Workshop on Hadron Physics (X Hadron Physics), Florianopolis, Brazil, 26-31 Mar 200