A note on isoparametric polynomials

We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-M\"unzner polynomial.Comment: 6 page

Association between SNPs and gene expression in multiple regions of the human brain

Identifying the genetic cis associations between DNA variants (single-nucleotide polymorphisms (SNPs)) and gene expression in brain tissue may be a promising approach to find functionally relevant pathways that contribute to the etiology of psychiatric disorders. In this study, we examined the association between genetic variations and gene expression in prefrontal cortex, hippocampus, temporal cortex, thalamus and cerebellum in subjects with psychiatric disorders and in normal controls. We identified cis associations between 648 transcripts and 6725 SNPs in the various brain regions. Several SNPs showed brain regional-specific associations. The expression level of only one gene, PDE4DIP, was associated with a SNP, rs12124527, in all the brain regions tested here. From our data, we generated a list of brain cis expression quantitative trait loci (eQTL) genes that we compared with a list of schizophrenia candidate genes downloaded from the Schizophrenia Forum (SZgene) database (http://www.szgene.org/). Of the SZgene candidate genes, we found that the expression levels of four genes, HTR2A, PLXNA2, SRR and TCF4, were significantly associated with cis SNPs in at least one brain region tested. One gene, SRR, was also involved in a coexpression module that we found to be associated with disease status. In addition, a substantial number of cis eQTL genes were also involved in the module, suggesting eQTL analysis of brain tissue may identify more reliable susceptibility genes for schizophrenia than case–control genetic association analyses. In an attempt to facilitate the identification of genetic variations that may underlie the etiology of major psychiatric disorders, we have integrated the brain eQTL results into a public and online database, Stanley Neuropathology Consortium Integrative Database (SNCID; http://sncid.stanleyresearch.org)

Spectrum Estimates and Applications to Geometry

In 1867, E. Beltrami (Ann Mat Pura Appl 1(2):329\u2013366, 1867, [12]) introduced a second order elliptic operator on Riemannian manifolds, extending the Laplace operator on Rn, called the Laplace\u2013Beltrami operator. The Laplace\u2013Beltrami operator became one of the most important operators in Mathematics and Physics, playing a fundamental role in differential geometry, geometric analysis, partial differential equations, probability, potential theory, stochastic process, just to mention a few. It is in important in various differential equations that describe physical phenomena such as the diffusion equation for the heat and fluid flow, wave propagation, Laplace equation and minimal surfaces