scholarly journals Subadditive ergodic theorems for random sets in infinite dimensions

2000 ◽  
Vol 50 (4) ◽  
pp. 409-416 ◽  
Author(s):  
Jennie C. Hansen ◽  
Paul Hulse
1994 ◽  
Vol 14 (1) ◽  
pp. 141-149
Author(s):  
Yenkun Huang

AbstractWe generalize a result of Bourgain and a result of Huang. We also give a positive solution to A. Bellow's question: the a.e. convergence of the averages for σn = 1/n. On the other hand, we establish a sufficient and necessary condition for random sets in Z+ with asymptotic density zero which almost surely satisfy a mean ergodic theorem.


Author(s):  
Paweł Wójcik

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.


Author(s):  
Aniket Bhattacharya ◽  
Vineet Jha ◽  
Khushboo Singhal ◽  
Mahar Fatima ◽  
Dayanidhi Singh ◽  
...  

Abstract Alu repeats contribute to phylogenetic novelties in conserved regulatory networks in primates. Our study highlights how exonized Alus could nucleate large-scale mRNA-miRNA interactions. Using a functional genomics approach, we characterize a transcript isoform of an orphan gene, CYP20A1 (CYP20A1_Alu-LT) that has exonization of 23 Alus in its 3’UTR. CYP20A1_Alu-LT, confirmed by 3’RACE, is an outlier in length (9 kb 3’UTR) and widely expressed. Using publically available datasets, we demonstrate its expression in higher primates and presence in single nucleus RNA-seq of 15928 human cortical neurons. miRanda predicts ∼4700 miRNA recognition elements (MREs) for ∼1000 miRNAs, primarily originated within these 3’UTR-Alus. CYP20A1_Alu-LT could be a potential multi-miRNA sponge as it harbors ≥10 MREs for 140 miRNAs and has cytosolic localization. We further tested whether expression of CYP20A1_Alu-LT correlates with mRNAs harboring similar MRE targets. RNA-seq with conjoint miRNA-seq analysis was done in primary human neurons where we observed CYP20A1_Alu-LT to be downregulated during heat shock response and upregulated in HIV1-Tat treatment. 380 genes were positively correlated with its expression (significantly downregulated in heat shock and upregulated in Tat) and they harbored MREs for nine expressed miRNAs which were also enriched in CYP20A1_Alu-LT. MREs were significantly enriched in these 380 genes compared to random sets of differentially expressed genes (p = 8.134e-12). Gene ontology suggested involvement of these genes in neuronal development and hemostasis pathways thus proposing a novel component of Alu-miRNA mediated transcriptional modulation that could govern specific physiological outcomes in higher primates.


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