RintC: fast and accuracy-aware decomposition of distributions of RNA secondary structures with extended logsumexp

The analysis of secondary structures is essential to understanding the function of RNAs. Because RNA molecules thermally fluctuate, it is necessary to analyze the probability distribution of secondary structures. Existing methods, however, are not applicable to long RNAs owing to their high computational complexity. Additionally, previous research has suffered from two numerical difficulties: overflow and significant numerical error. In this research, we reduced the computational complexity in calculating the landscape of the probability distribution of secondary structures by introducing a maximum-span constraint. In addition, we resolved numerical computation problems through two techniques: extended logsumexp and accuracy-guaranteed numerical computation. We analyzed the stability of the secondary structures of 16S ribosomal RNAs at various temperatures without overflow. The results obtained are consistent with in vivo assay results reported in previous research. Furthermore, we quantitatively assessed numerical stability using our method. These results demonstrate that the proposed method is applicable to long RNAs. Source code is available on https://github.com/eukaryo/rintc.

Weak Hardy spaces and paraproducts

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] The purpose of this dissertation is to provide a new square function characterization of weak Hardy spaces in the full range of exponents possible and use this characterization in applications on endpoint estimates for multilinear paraproducts. Additionally, we prove several maximal characterizations of weak Hardy spaces and obtain several properties of these spaces. Our main result is a Littlewood-Paley square function characterization of the Hardy spaces. Our proof is based on a Calderon-Zygmund type decomposition of distributions in Hardy spaces and on interpolation. Our results allow us to obtain endpoint estimates for several operators in terms of square function characterizations of weak L^p norms. As an application of this technique, we prove endpoint boundedness for mutlilinear paraproducts.

The product of independent random variables with slowly varying truncated moments

The Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.