Pre-Cluster Dynamics In Multifragmentation

The initial production and dynamical expansion of hot spherical nuclei are examined as the first stage both in the projectile-multifragmentation and in central collision processes. The initial temperatures, which are necessary for entering the adiabatic spinodal region, as well as the minimum temperatures and densities, which are reached in the expansion, significantly differ for hard and soft equations of state. Additional initial compression, occurring in central collisions leads most likely to a qualitatively different multifragmentation mechanism. Recent experimental data are discussed in relation to the results of the proposed model.Comment: 6 pages LaTeX with 3 EPSF figures; Proceedings of the 1st Catania Relativistic Ion Studies: Critical Phenomena and Collective Observables, Acicastello, May 27-31, 1996, to be published by World Scientific Publ. C

Bilinearity rank of the cone of positive polynomials and related cones

For a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials

Bilinearity rank of the cone of positive polynomials and related cones

For a proper cone K ⊂ Rn and its dual cone K the complementary slackness condition xT s = 0 defines an n-dimensional manifold C(K) in the space { (x, s) | x ∈ K, s ∈ K^* }. When K is a symmetric cone, this manifold can be described by a set of n bilinear equalities. When K is a symmetric cone, this fact translates to a set of n linearly independent bilinear identities (optimality conditions) satisfied by every (x, s) ∈ C(K). This proves to be very useful when optimizing over such cones, therefore it is natural to look for similar optimality conditions for non-symmetric cones. In this paper we define the bilinearity rank of a cone, which is the number of linearly independent bilinear identities valid for the cone, and describe a linear algebraic technique to bound this quantity. We examine several well-known cones, in particular the cone of positive polynomials P2n+1 and its dual, the closure of the moment cone M2n+1, and compute their bilinearity ranks. We show that there are exactly four linearly independent bilinear identities which hold for all (x,s) ∈ C(P2n+1), regardless of the dimension of the cones. For nonnegative polynomials over an interval or half-line there are only two linearly independent bilinear identities. These results are extended to trigonometric and exponential polynomials

Random Matrix Filtering in Portfolio Optimization

We study empirical covariance matrices in finance. Due to the limited amount of available input information, these objects incorporate a huge amount of noise, so their naive use in optimization procedures, such as portfolio selection, may be misleading. In this paper we investigate a recently introduced filtering procedure, and demonstrate the applicability of this method in a controlled, simulation environment.Comment: 9 pages with 3 EPS figure

Chiral Disorder and Diffusion of Light Quarks in the QCD Vacuum

We give a pedagogical introduction to the concept that light quarks diffuse in the QCD vacuum following the spontaneous breaking of chiral symmetry. By analogy with disordered electrons in metals, we show that the diffusion constant for light quarks in QCD is D=2F_{\pi}^2/|\la\bar{q}q\to| which is about 0.22 fm. We comment on the correspondence between the diffusive phase and the chiral phase as described by chiral perturbation theory, as well as the cross-over to the ergodic phase as described by random matrix theory. The cross-over is identified with the Thouless energy $E_c=D/\sqrt{V_4}$ which is the inverse diffusion time in an Euclidean four-volume $V_4$.Comment: 9 pages in APPB sty (included). Invited talk by MAN at the Workshop on the Structure of Mesons, Baryons and Nuclei, Cracow, May 26-30, 199