scholarly journals Egoroff's theorems for random sets on non-additive measure spaces

2021 ◽  
Vol 6 (5) ◽  
pp. 4803-4810
Author(s):  
Tao Chen ◽  
◽  
Hui Zhang ◽  
Jun Li ◽  
Keyword(s):  
Random Sets ◽  
Additive Measure ◽  
Measure Spaces

EGOROFF'S THEOREM ON MONOTONE NON-ADDITIVE MEASURE SPACES

2004 ◽  
Vol 12 (01) ◽  
pp. 61-68 ◽  
Author(s):  
JUN LI ◽  
MASAMI YASUDA
Keyword(s):  
Measure Space ◽  
Measure Theory ◽  
Sufficient Condition ◽  
Additive Measure ◽  
Necessary And Sufficient Condition ◽  
Converse Problem ◽  
Almost Everywhere ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Function Sequence

In this paper, the well-known Egoroff's theorem in classical measure theory is established on monotone non-additive measure spaces. Taylor's theorem, which concerns almost everywhere convergence of measurable function sequence in classical measure theory, is also generalized. The converse problem of the theorems are discussed, and a necessary and sufficient condition for the Egoroff's theorem is obtained on semicontinuous fuzzy measure space with S-compactness.

ON TAU-SMOOTH MEASURE SPACES WITHOUT THICK LINDELÖF SUBSETS

1991 ◽  
Vol 17 (1) ◽  
pp. 379 ◽  
Author(s):  
Aldaz
Keyword(s):  
Smooth Measure ◽  
Measure Spaces

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

Depth and outliers for samples of sets and random sets distributions

2021 ◽  
Author(s):  
Ignacio Cascos ◽  
Qiyu Li ◽  
Ilya Molchanov
Keyword(s):  
Random Sets

scholarly journals Some Weighted Estimates on Gaussian Measure Spaces

2021 ◽  
Author(s):  
Boning Di ◽  
Qianjun He ◽  
Dunyan Yan
Keyword(s):  
Gaussian Measure ◽  
Weighted Estimates ◽  
Measure Spaces

scholarly journals Scott functions, their representations on domains, and applications to random sets

2021 ◽  
Vol 498 (2) ◽  
pp. 124976
Author(s):  
Motoya Machida ◽  
Alexander Y. Shibakov
Keyword(s):  
Random Sets

scholarly journals Multiple Alu exonization in 3’UTR of a primate specific isoform of CYP20A1 creates a potential miRNA sponge

2020 ◽  
Author(s):  
Aniket Bhattacharya ◽  
Vineet Jha ◽  
Khushboo Singhal ◽  
Mahar Fatima ◽  
Dayanidhi Singh ◽  
...  
Keyword(s):  
Heat Shock ◽  
Cortical Neurons ◽  
Regulatory Networks ◽  
Large Scale ◽  
Neuronal Development ◽  
Random Sets ◽  
Rna Seq ◽  
Orphan Gene ◽  
Mirna Sponge ◽  
Human Neurons

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.

Non-singular -actions: an ergodic theorem over rectangles with application to the critical dimensions

2020 ◽  
pp. 1-18
Author(s):  
ANTHONY H. DOOLEY ◽  
KIERAN JARRETT
Keyword(s):  
Ergodic Theorem ◽  
Natural Action ◽  
Critical Dimensions ◽  
Measure Spaces ◽  
Metric Isomorphism

Abstract We adapt techniques developed by Hochman to prove a non-singular ergodic theorem for $\mathbb {Z}^d$ -actions where the sums are over rectangles with side lengths increasing at arbitrary rates, and in particular are not necessarily balls of a norm. This result is applied to show that the critical dimensions with respect to sequences of such rectangles are invariants of metric isomorphism. These invariants are calculated for the natural action of $\mathbb {Z}^d$ on a product of d measure spaces.

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