Asymptotic enumeration of RNA secondary structure

Based on the new representation of RNA secondary structures, we obtain the basic relations about secondary structures with a prescribed size m for hairpin loops and minimum stack length l. Furthermore, we make an asymptotic analysis on RNA secondary structures with certain additional constrains.     

一类RNA二级结构的计数

多核苷酸的二级结构可视为一类顶点标号平面图，通常通过枚举每类RNA二级结构图的各种子图来计算其递推公式。本文作者给出了限制端环长度的RNA二级结构的递推公式，并运用隐式估计法计算它的渐近值。     

Symmetric circular matchings and RNA folding

RNA secondary structures can be computed as optimal solutions of certain circular matching problems. An accurate treatment of this energy minimization problem has to account for the small — but non-negligible — entropic destabilization of secondary structures with non-trivial automorphisms. Such intrinsic symmetries are typically excluded from algorithmic approaches; however, because the effects are small, they play a role only for RNAs with symmetries at sequence level, and they appear only in particular settings that are less frequently used in practical application, such as circular folding or the co-folding of two or more identical RNAs. Here, we show that the RNA folding problem with symmetry terms can still be solved with polynomial-time algorithms. Empirically, the fraction of symmetric ground state structures decreases with chain length, so that the error introduced by neglecting the symmetry terms affects fewer and fewer predictions. We then explore the combinatorics of symmetric secondary structures in detail. Surprisingly, the singularities of the generating function coincide between symmetric and non-symmetric structures. Furthermore, generating functions and explicit asymptotic results for both the circular and the co-folding version are derived.     

Loop homology of bi-secondary structures II

Journal of Algebraic Combinatorics - In this paper, we analyze the homology of the simplicial complex induced by a given pair of RNA secondary structures, $$R=(S,T)$$ . Such a pair induces a...     

Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations

We consider stationary solutions with internal transition layers (contrast structures) for a singularly perturbed elliptic equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an efficient algorithm for constructing an asymptotic approximation to the localization curve of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions.     

具有小周期孔隙复合材料弹性结构的双尺度有限元分析

对于具有小周期孔隙复合材料弹性结构,在双尺度渐近分析理论结果的基础上提出了双尺度有限元计算格式,并给出了严格的误差估计.     

Progress on asymptotic behavior of the Yang-Mills-Higgs flow

In this note we will introduce our recent work on the existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles, and the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity.     

Hypothesis tests for high-dimensional covariance structures

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

     

Life span and secondary critical exponent for degenerate and singular parabolic equations

In this paper, we investigate the large time behavior to the Cauchy problem of degenerate and singular parabolic equations. Firstly, we establish the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Secondly, we give the large time behavior of the global solution. Finally, the precise estimate of life span for the blow-up solution is obtained.     

A bijection between ordered trees and 2-Motzkin paths and its many consequences

A new bijection between ordered trees and 2-Motzkin paths is presented, together with its numerous consequences regarding ordered trees as well as other combinatorial structures such as Dyck paths, bushes, {0,1,2}-trees, Schröder paths, RNA secondary structures, noncrossing partitions, Fine paths, and Davenport-Schinzel sequences.RésuméUne nouvelle bijection entre arbres ordonnés et chemins de Motzkin bicolorés est présentée, avec ses nombreuses conséquences en ce qui concerne les arbres ordonnés ainsi que d'autres structures combinatoires telles que chemins de Dyck, buissons, arbres de type {0,1,2}, chemins de Schröder, structures secondaires de type RNA, partitions non croisées, chemins de Fine, et enfin suites de Davenport-Schinzel.