Uniform convergence of Vapnik--Chervonenkis classes under ergodic sampling

We show that if $\mathcal{X}$ is a complete separable metric space and $\mathcal{C}$ is a countable family of Borel subsets of $\mathcal{X}$ with finite VC dimension, then, for every stationary ergodic process with values in $\mathcal{X}$, the relative frequencies of sets $C\in\mathcal{C}$ converge uniformly to their limiting probabilities. Beyond ergodicity, no assumptions are imposed on the sampling process, and no regularity conditions are imposed on the elements of $\mathcal{C}$. The result extends existing work of Vapnik and Chervonenkis, among others, who have studied uniform convergence for i.i.d. and strongly mixing processes. Our method of proof is new and direct: it does not rely on symmetrization techniques, probability inequalities or mixing conditions. The uniform convergence of relative frequencies for VC-major and VC-graph classes of functions under ergodic sampling is established as a corollary of the basic result for sets.Comment: Published in at http://dx.doi.org/10.1214/09-AOP511 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Beam-Energy and System-Size Dependence of Dynamical Net Charge Fluctuations

We present measurements of net charge fluctuations in $Au + Au$ collisions at $\sqrt{s_{NN}} =$ 19.6, 62.4, 130, and 200 GeV, $Cu + Cu$ collisions at $\sqrt{s_{NN}} =$ 62.4, 200 GeV, and $p + p$ collisions at $\sqrt{s} =$ 200 GeV using the net charge dynamical fluctuations measure $\nu_{+-,dyn}$. The dynamical fluctuations are non-zero at all energies and exhibit a rather modest dependence on beam energy. We find that at a given energy and collision system, net charge dynamical fluctuations violate $1/N_{ch}$ scaling, but display approximate $1/N_{part}$ scaling. We observe strong dependence of dynamical fluctuations on the azimuthal angular range and pseudorapidity widths.Comment: 4 pages, 4 figures, presented at the 19th International Conference on Ultra-Relativistic Nucleus-Nucleus Collisions, "Quark Matter 2008", Jaipur, India, February 4-10, 200

Interpretation of AIS Images of Cuprite, Nevada Using Constraints of Spectral Mixtures

A technique is outlined that tests the hypothesis Airborne Imaging Spectrometer (AIS) image spectra are produced by mixtures of surface materials. This technique allows separation of AIS images into concentration images of spectral endmembers (e.g., surface materials causing spectral variation). Using a spectral reference library it was possible to uniquely identify these spectral endmembers with respect to the reference library and to calibrate the AIS images

Systematic study of the jet fragmentation function for inclusive jet-production in p+p collisions at sqrt{s}=200 GeV in STAR

Jet fragmentation functions measured in e^+e^- and p+\bar{p} experiments are well-described on an inclusive hadron level by QCD-based calculations. Fragmentation is expected to be modified by the presence of a strongly interacting medium, but full theoretical description of this modification must still be developed. It has recently been suggested that particle-identified fragmentation functions may provide additional insight into the processes underlying jet quenching. To assess the applicability of QCD-based fragmentation calculations to RHIC data, and to provide a baseline with which to compare fragmentation function measurements in heavy ion collisions, we present the first measurements of charged hadron and particle-identified fragmentation functions of jets reconstructed via a midpoint-cone algorithm from p+p collisions at 200 GeV in STAR. We study the dependence on jet cone-size and jet-energy, and compare the results to PYTHIA simulations based on the Modified Leading Log Approximation (MLLA).Comment: 6 pages, 5 figures, proceedings of Hard Probes 2008 conferenc

Hyperspherical Harmonics, Separation of Variables and the Bethe Ansatz

The relation between solutions to Helmholtz's equation on the sphere $S^{n-1}$ and the [{\gr sl}(2)]^n Gaudin spin chain is clarified. The joint eigenfuctions of the Laplacian and a complete set of commuting second order operators suggested by the $R$--matrix approach to integrable systems, based on the loop algebra \wt{sl}(2)_R, are found in terms of homogeneous polynomials in the ambient space. The relation of this method of determining a basis of harmonic functions on $S^{n-1}$ to the Bethe ansatz approach to integrable systems is explained.Comment: 14 pgs, Plain Tex, preprint CRM--2174 (May, 1994

Pseudorapidity dependence of parton energy loss in relativistic heavy ion collisions

We analyze the recent data from the BRAHMS Collaboration on the pseudorapidity dependence of nuclear modification factors in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV by using the full three dimensional hydrodynamic simulations for the density effects on parton energy loss. We first compute the transverse spectra at $\eta=0$ and 2.2, and next take a ratio $R_{\eta}=R_{AA}(\eta=2.2)/R_{AA}(\eta=0)$, where $R_{AA}$ is a nuclear modification factor. It is shown that hydrodynamic components account for $R_{\eta}\simeq 1$ at low $p_\mathrm{T}$ and that quenched pQCD components lead $R_{\eta} < 1$ at high $p_\mathrm{T}$ which are consistent with the data. Strong suppression at $\eta=2.2$ is compatible with the parton energy loss in the final state.Comment: 5 pages, 4 figures; one figure adde