Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4281

Central and Local Limit Theorems for RNA Structures

Emma Y. Jin·Christian M. Reidys
View original source ↗Read source document ↗Export metadata ↓

Abstract

A k-noncrossing RNA pseudoknot structure is a graph over $\{1,...,n\}$ without 1-arcs, i.e. arcs of the form (i,i+1) and in which there exists no k-set of mutually intersecting arcs. In particular, RNA secondary structures are 2-noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3-noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of \cite{Reidys:07rna1} and \cite{Reidys:07rna2}, where the generating function of k-noncrossing RNA pseudoknot structures and the asymptotics for its coefficients have been derived. The results of this paper explain the findings on the numbers of arcs of RNA secondary structures obtained by molecular folding algorithms and predict the distributions for k-noncrossing RNA folding algorithms which are currently being developed.

Explore related subjects

math.COq-bio.QM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emma Y. Jin, Christian M. Reidys. 2007-08-01. Central and Local Limit Theorems for RNA Structures. https://arxiv.org/abs/0707.4281

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Rainbow Saturation Number of Cycles

An edge-colored graph $(G,\mathcal C)$ is $F$-rainbow saturated if it contains no rainbow copy of $F$, but adding any nonedge in any prescribed color creates one. The rainbow saturation number $\operatorname{rsat}(n,F)$ is the minimum number of edges in such a graph of order $n$. In this paper we study $\operatorname{rsat}(n,C_r)$. We determine the value for $C_4$ exactly, and establish bounds for every fixed $C_r$ with $r\geq5$.

math.CO↗
arXiv2026

Moreira's Theorem for Image Partition Regular Matrices

The famous Hindman conjecture says that for any finite coloring of natural numbers, there exists a monochromatic copy of the form $\{x,x+y,xy\}.$ In a celebrated article, Moreira gave a partial answer to this conjecture by showing that every finite coloring of the natural numbers contains a monochromatic configuration of the form $\{x, x+y, xy\}$. In this article we prove matrix versions (both finite and infinite) of Moreira's theorem. A matrix $A$ is said to be an image partition regular matrix if for any finite coloring of naturals, there exists a monochromatic image of $A,$ i.e. there exists a vector $\vec X$ such that all the entries of $A\vec X$ are monochromatic. From a recent paper of Bowen, one can derive the finite matrix version of the Moreira theorem: if $A$ and $B$ are two finite image partition regular matrices of the same order, then under any finite coloring of $\mathbb{N}$, there exist vectors $\vec{X}$ and $\vec{Y}$ such that all entries in the union of $A\vec{X}, A\vec{X} + B\vec{Y}, A\vec{X} \cdot B\vec{Y}$ are monochromatic, where $A\vec{X} \cdot B\vec{Y}$ denote the vector each of its entries are pointwise multiplication of the coordinates of $A\vec{X} \text{ and } B\vec{Y}$. In this article, we give a short combinatorial proof of this result, and then we extend it to infinite image partition regular matrices.

math.CO↗
arXiv2026

Explicit enumeration and large-valence asymptotics of even-valent maps

Let $\mathscr{N}_g(2ν,j)$ denote the number of connected labelled $2ν$-valent maps of genus $g$ with $j$ vertices. Explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ have been available only in the planar and toroidal cases. Using a structural formula of Ercolani et al (2023), we translate the problem of determining an explicit bivariate formula for $\mathscr{N}_g(2ν,j)$, $g \geq 2$, to finding finitely many counts with a fixed number of vertices. For $g=2,3$ and $4$ we determine these counts using the associated orthogonal polynomials, yielding explicit bivariate formulae for $\mathscr{N}_g(2ν,j)$ in these genera. Furthermore, the same method applies for every $g\ge5$ at the cost of additional computation. From these formulae we obtain the leading-order asymptotics of $\mathscr{N}_g(2ν,j)$ as $ν\to\infty$ for $g=2,3,4$, and we conjecture the structure of these formulae in general genus. In addition, we establish an analogous reduction to finitely many counts, derive explicit formulae and large-valence asymptotics, and formulate corresponding conjectures for two-legged even-valent maps.

math.CO↗