Universality of a family of Random Matrix Ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of $U(N)$ invariant Random Matrix Ensembles which is characterized by a parameter $\lambda$ describing logarithmic soft-confinement potentials $V(H) \sim [\ln H]^{(1+\lambda)} \:(\lambda>0$). We showed that we can study eigenvalue correlations of these "$\lambda$-ensembles" based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function $\exp[- (\ln x)^{1+\lambda}]$. In this work, we expand our previous work and show that: i) the eigenvalue density is given by a power-law of the form $\rho(x) \propto [\ln x]^{\lambda-1}/x$ and ii) the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called "ghost-correlation peak", is controlled by the parameter $\lambda$; decreasing $\lambda$ increases the anomaly. We also identify the two-level kernel of the $\lambda$-ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for $\lambda=1$. Finally, we discuss the universality of the $\lambda$-ensembles, which includes Wigner-Dyson universality ($\lambda \to \infty$ limit), the uncorrelated Poisson-like behavior ($\lambda \to 0$ limit), and a critical behavior for all the intermediate $\lambda$ ($0<\lambda<\infty$) in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the $N$ dependence of the two-level kernel of the fat-tail random matrices.Comment: 10 pages, 13 figure

Rotationally invariant family of L\'evy like random matrix ensembles

We introduce a family of rotationally invariant random matrix ensembles characterized by a parameter $\lambda$. While $\lambda=1$ corresponds to well-known critical ensembles, we show that $\lambda \ne 1$ describes "L\'evy like" ensembles, characterized by power law eigenvalue densities. For $\lambda > 1$ the density is bounded, as in Gaussian ensembles, but $\lambda <1$ describes ensembles characterized by densities with long tails. In particular, the model allows us to evaluate, in terms of a novel family of orthogonal polynomials, the eigenvalue correlations for L\'evy like ensembles. These correlations differ qualitatively from those in either the Gaussian or the critical ensembles.Comment: 9 pages, 5 figure

Integrative functional genomic analysis of human brain development and neuropsychiatric risks

INTRODUCTION The brain is responsible for cognition, behavior, and much of what makes us uniquely human. The development of the brain is a highly complex process, and this process is reliant on precise regulation of molecular and cellular events grounded in the spatiotemporal regulation of the transcriptome. Disruption of this regulation can lead to neuropsychiatric disorders. RATIONALE The regulatory, epigenomic, and transcriptomic features of the human brain have not been comprehensively compiled across time, regions, or cell types. Understanding the etiology of neuropsychiatric disorders requires knowledge not just of endpoint differences between healthy and diseased brains but also of the developmental and cellular contexts in which these differences arise. Moreover, an emerging body of research indicates that many aspects of the development and physiology of the human brain are not well recapitulated in model organisms, and therefore it is necessary that neuropsychiatric disorders be understood in the broader context of the developing and adult human brain. RESULTS Here we describe the generation and analysis of a variety of genomic data modalities at the tissue and single-cell levels, including transcriptome, DNA methylation, and histone modifications across multiple brain regions ranging in age from embryonic development through adulthood. We observed a widespread transcriptomic transition beginning during late fetal development and consisting of sharply decreased regional differences. This reduction coincided with increases in the transcriptional signatures of mature neurons and the expression of genes associated with dendrite development, synapse development, and neuronal activity, all of which were temporally synchronous across neocortical areas, as well as myelination and oligodendrocytes, which were asynchronous. Moreover, genes including MEF2C, SATB2, and TCF4, with genetic associations to multiple brain-related traits and disorders, converged in a small number of modules exhibiting spatial or spatiotemporal specificity. CONCLUSION We generated and applied our dataset to document transcriptomic and epigenetic changes across human development and then related those changes to major neuropsychiatric disorders. These data allowed us to identify genes, cell types, gene coexpression modules, and spatiotemporal loci where disease risk might converge, demonstrating the utility of the dataset and providing new insights into human development and disease

Generational status, social capital and family engagement in immigrant families

This study highlighted the importance of social capital in understanding the disparity in family engagement across immigrant generations. Using the national representative data, the ELS:2002, confirmatory factor analysis (CFA) and structural equation modeling (SEM) was used to examine the relationships among generational status, social capital, and home- and school-based family engagement. The results suggested that social capital played an important role in immigrant home- and schoolbased family engagement. The findings of specific pathways through social capital in and outside the family to home- and school-based family engagement might make a tangible contribution to understanding of family engagement and immigrant generations. Further, the present research suggested that immigrant families were not only constrained from participating in their children's education, but also had their own strengths for family engagement such as positive expectations for and extensive communications with their children.Includes bibliographical references

Network Analysis of Genome-Wide Selective Constraint Reveals a Gene Network Active in Early Fetal Brain Intolerant of Mutation

Using robust, integrated analysis of multiple genomic datasets, we show that genes depleted for non-synonymous de novo mutations form a subnetwork of 72 members under strong selective constraint. We further show this subnetwork is preferentially expressed in the early development of the human hippocampus and is enriched for genes mutated in neurological Mendelian disorders. We thus conclude that carefully orchestrated developmental processes are under strong constraint in early brain development, and perturbations caused by mutation have adverse outcomes subject to strong purifying selection. Our findings demonstrate that selective forces can act on groups of genes involved in the same process, supporting the notion that purifying selection can act coordinately on multiple genes. Our approach provides a statistically robust, interpretable way to identify the tissues and developmental times where groups of disease genes are active

The 72-member constrained gene subnetwork is enriched for canonical pathways reflecting neuronal and immune functionality and basic aspects of cell cycle control.

<p>We tested pathways from two sources (the Reactome database and KEGG, the Kyoto Encyclopedia of Genes and Genomes), assessing how many genes are in each pathway (All), how many map onto the 9729 inteconnected genes in our analysis (Mapped), and how many are present in the constrained subnetwork (Subnetwork). We assess significance using both the GSEA approach of a Kolmogorov-Smirnov (KS) test and a simple hypergeometric (HG) test of expected overlaps.</p