Zero biasing and growth processes

The tools of zero biasing are adapted to yield a general result suitable for analyzing the behavior of certain growth processes. The main theorem is applied to prove central limit theorems, with explicit error terms in the L^1 metric, for certain statistics of the Jack measure on partitions and for the number of balls drawn in a Polya-Eggenberger urn process.Comment: 21 pages. Error in one term of the bound of the main theorem has been corrected, resulting in some changes to the bound for urn proces

On the fairness of the main galaxy sample of SDSS

Flux-limited and volume-limited galaxy samples are constructed from SDSS data releases DR4, DR6 and DR7 for statistical analysis. The two-point correlation functions $\xi(s)$, monopole of three-point correlation functions $\zeta_0$, projected two-point correlation function $w_p$ and pairwise velocity dispersion $\sigma_{12}$ are measured to test if galaxy samples are fair for these statistics. We find that with increment of sky coverage of SDSS, $\xi(s)$ of flux-limited sample is extremely robust and insensitive to local structures at low redshift. But for volume-limited samples fainter than $L^*$ at large scales s>\sim 10\hmpc, deviation of $\xi(s)$ and $\zeta_0$ of DR7 to those of DR4 and DR6 increases with larger absolute magnitude. In the weakly nonlinear regime, there is no agreement between $\zeta_0$ of different data releases in all luminosity bins. Furthermore, $w_p$ of volume-limited samples of DR7 in luminosity bins fainter than $-M_{r,0.1}=[18.5,19.5]$ are significantly larger, and $\sigma_{12}$ of the two faintest volume-limited samples of DR7 display very different scale dependence than results of DR4 and DR6. Our findings call for cautions in understanding clustering analysis results of SDSS faint galaxy samples, and higher order statistics of SDSS volume-limited samples in the weakly nonlinear regime. The first zero-crossing points of $\xi(s)$ of volume-limited samples are also investigated and discussed.Comment: 16 pages, 12 figures, accepte

Finite-temperature critical point of a glass transition

We generalize the simplest kinetically constrained model of a glass-forming liquid by softening kinetic constraints, allowing them to be violated with a small finite rate. We demonstrate that this model supports a first-order dynamical (space-time) phase transition, similar to those observed with hard constraints. In addition, we find that the first-order phase boundary in this softened model ends in a finite-temperature dynamical critical point, which we expect to be present in natural systems. We discuss links between this critical point and quantum phase transitions, showing that dynamical phase transitions in $d$ dimensions map to quantum transitions in the same dimension, and hence to classical thermodynamic phase transitions in $d+1$ dimensions. We make these links explicit through exact mappings between master operators, transfer matrices, and Hamiltonians for quantum spin chains.Comment: 10 pages, 5 figure

Identification of gene pathways implicated in Alzheimer's disease using longitudinal imaging phenotypes with sparse regression

We present a new method for the detection of gene pathways associated with a multivariate quantitative trait, and use it to identify causal pathways associated with an imaging endophenotype characteristic of longitudinal structural change in the brains of patients with Alzheimer's disease (AD). Our method, known as pathways sparse reduced-rank regression (PsRRR), uses group lasso penalised regression to jointly model the effects of genome-wide single nucleotide polymorphisms (SNPs), grouped into functional pathways using prior knowledge of gene-gene interactions. Pathways are ranked in order of importance using a resampling strategy that exploits finite sample variability. Our application study uses whole genome scans and MR images from 464 subjects in the Alzheimer's Disease Neuroimaging Initiative (ADNI) database. 66,182 SNPs are mapped to 185 gene pathways from the KEGG pathways database. Voxel-wise imaging signatures characteristic of AD are obtained by analysing 3D patterns of structural change at 6, 12 and 24 months relative to baseline. High-ranking, AD endophenotype-associated pathways in our study include those describing chemokine, Jak-stat and insulin signalling pathways, and tight junction interactions. All of these have been previously implicated in AD biology. In a secondary analysis, we investigate SNPs and genes that may be driving pathway selection, and identify a number of previously validated AD genes including CR1, APOE and TOMM40