Recently MacCluer and Shapiro [6] have characterized the compact composition operators in Hardy and weighted Bergman spaces of the disc, and MacCluer [5] has made an extensive study of these opertors in the unit ball of Cⁿ. Angular derivatives and Carleson measures have played an essential role in these studies. In this article we study composition operators in poly discs and characterize those operators which are bounded or compact in Hardy and weighted Bergman spaces. In addition to Carleson measure theorems resembling those of [5], [6], we give necessary and sufficient conditions satisfied by the maps inducing bounded or compact composition operators. We conclude by considering the role of angular derivatives on the compactness question explicitly.

It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variational theory. It allows us, in particular, to obtain efficient conditions via the system data supporting the sufficiency in Brockett's theorem and ensuring local exponential stabilization by means of continuous stationary feedback laws. Furthermore, we derive new necessary conditions for local exponential and asymptotic stabilization of continuous-time control systems by using both continuous and continuously differentiable stationary feedback laws and establish also some counterparts of the obtained sufficient conditions for local asymptotic stabilization by continuous stationary feedback laws in the case of nonlinear discrete-time control systems.

Background: Proliferation and expansion of security risks necessitates new measures to ensure authenticity and validation of GMOs. Watermarking and other cryptographic methods are available which conceal and recover the original signature, but in the process reveal the authentication information. In many scenarios watermarking and standard cryptographic methods are necessary but not sufficient and new, more advanced, cryptographic protocols are necessary. Results: Herein, we present a new crypto protocol, that is applicable in broader settings, and embeds the authentication string indistinguishably from a random element in the signature space and the string is verified or denied without disclosing the actual signature. Results show that in a nucleotide string of 1000, the algorithm gives a correlation of 0.98 or higher between the distribution of the codon and that of E. coli, making the signature virtually invisible. Conclusions: This algorithm may be used to securely authenticate and validate GMOs without disclosing the actual signature. While this protocol uses watermarking, its novelty is in use of more complex cryptographic techniques based on zero knowledge proofs to encode information.

RNA sequencing (RNA-seq) is the conventional genome-scale approach used to capture the expression levels of all detectable genes in a biological sample. This is now regularly used for population-based studies designed to identify genetic determinants of various diseases. Naturally, the accuracy of these tests should be verified and improved if possible. In this study, we aimed to detect and correct for expression level-dependent errors which are not corrected by conventional normalization techniques. We examined several RNA-seq datasets from the Cancer Genome Atlas (TCGA), Stand Up 2 Cancer (SU2C), and GTEx databases with various types of preprocessing. By applying local averaging, we found expression-level dependent biases that differ from sample to sample in all datasets studied. Using simulations, we show that these biases corrupt gene-gene correlation estimations and t tests between subpopulations. To mitigate these biases, we introduce two different nonlinear transforms based on statistical considerations that correct these observed biases. We demonstrate that these transforms effectively remove the observed per-sample biases, reduce sample-to-sample variance, and improve the characteristics of gene-gene correlation distributions. Using a novel simulation methodology that creates controlled differences between subpopulations, we show that these transforms reduce variability and increase sensitivity of two population tests. The improvements in sensitivity and specificity were of the order of 3-5% in most instances after the data was corrected for bias. Altogether, these results improve our capacity to understand gene-gene relationships, and may lead to novel ways to utilize the information derived from clinical tests.