RNA structure prediction using positive and negative evolutionary information

Knowing the structure of conserved structural RNAs is important to elucidate their function and mechanism of action. However, predicting a conserved RNA structure remains unreliable, even when using a combination of thermodynamic stability and evolutionary covariation information. Here we present a method to predict a conserved RNA structure that combines the following three features. First, it uses significant covariation due to RNA structure and removes spurious covariation due to phylogeny. Second, it uses negative evolutionary information: basepairs that have variation but no significant covariation are prevented from occurring. Lastly, it uses a battery of probabilistic folding algorithms that incorporate all positive covariation into one structure. The method, named CaCoFold (Cascade variation/covariation Constrained Folding algorithm), predicts a nested structure guided by a maximal subset of positive basepairs, and recursively incorporates all remaining positive basepairs into alternative helices. The alternative helices can be compatible with the nested structure such as pseudoknots, or overlapping such as competing structures, base triplets, or other 3D non-antiparallel interactions. We present evidence that CaCoFold predictions are consistent with structures modeled from crystallography.

How to Obtain Maximal and Minimal Subranges of Two-Dimensional Vector Measures

Let (X, ℱ) be a measurable space with a nonatomic vector measure µ =(µ1, µ2). Denote by R(Y) the subrange R(Y)= {µ(Z): Z ∈ ℱ, Z ⊆ Y }. For a given p ∈ µ(ℱ) consider a family of measurable subsets ℱp = {Z ∈ ℱ : µ(Z)= p}. Dai and Feinberg proved the existence of a maximal subset Z* ∈ Fp having the maximal subrange R(Z*) and also a minimal subset M* ∈ ℱp with the minimal subrange R(M*). We present a method of obtaining the maximal and the minimal subsets. Hence, we get simple proofs of the results of Dai and Feinberg.

The subset of R3 not realizing metrics on the curve complex

In [F. Zhang, R. Qiu and Y. Zou, The subset of [Formula: see text] realizing metrics on the curve complex, Topology Appl. 193 (2015) 259–269], they defined a subset [Formula: see text] of [Formula: see text] and a metric [Formula: see text] through each point of [Formula: see text] for the curve complex [Formula: see text]. For further understanding the curve complex, we concern the whole set [Formula: see text]. With adaption to the flow theory on torus, we prove that for any point of [Formula: see text], the [Formula: see text] is not a metric on [Formula: see text] or [Formula: see text]. This means that the [Formula: see text] is the maximal subset of [Formula: see text] realizing metrics on the curve complex.

ON MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS IN FINITE -GROUPS

A subset $X$ of a finite group $G$ is a set of pairwise noncommuting elements if $xy\neq yx$ for all $x\neq y\in X$. If $|X|\geq |Y|$ for any other subset $Y$ of pairwise noncommuting elements, then $X$ is called a maximal subset of pairwise noncommuting elements and the size of such a set is denoted by ${\it\omega}(G)$. In a recent article by Azad et al. [‘Maximal subsets of pairwise noncommuting elements of some finite $p$-groups’, Bull. Iran. Math. Soc.39(1) (2013), 187–192], the value of ${\it\omega}(G)$ is computed for certain $p$-groups $G$. In the present paper, our aim is to generalise these results and find ${\it\omega}(G)$ for some more $p$-groups of interest.

ON NONCOMMUTING SETS AND CENTRALISERS IN INFINITE GROUPS

A subset$X$of a group$G$is a set of pairwise noncommuting elements if$ab\neq ba$for any two distinct elements$a$and$b$in$X$. If$|X|\geq |Y|$for any other set of pairwise noncommuting elements$Y$in$G$, then$X$is called a maximal subset of pairwise noncommuting elements and the cardinality of such a subset (if it exists) is denoted by${\it\omega}(G)$. In this paper, among other things, we prove that, for each positive integer$n$, there are only finitely many groups$G$, up to isoclinism, with${\it\omega}(G)=n$, and we obtain similar results for groups with exactly$n$centralisers.

MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS OF SOME p-GROUPS OF MAXIMAL CLASS

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy≠yx for any two distinct elements x and y in X. If |X|≥|Y | for any other set of pairwise noncommuting elements Y in G, then X is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements for some p-groups of maximal class. Specifically, we determine this cardinality for all 2 -groups and 3 -groups of maximal class.

MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS OF THREE-DIMENSIONAL GENERAL LINEAR GROUPS

Let G be a group. A subset N of G is a set of pairwise noncommuting elements if xy⁄=yx for any two distinct elements x and y in N. If ∣N∣≥∣M∣ for any other set of pairwise noncommuting elements M in G, then N is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements in a three-dimensional general linear group. Moreover, we show how to modify a given maximal subset of pairwise noncommuting elements into another maximal subset of pairwise noncommuting elements that contains a given ‘generating element’ from each maximal torus.

ON VARIOUS NOTIONS OF PARALLELISM IN P SYSTEMS

We consider the following definition (different from the standard definition in the literature) of "maximal parallelism" in the application of evolution rules in a P system G: Let R = {r1, …rk} be the set of (distinct) rules in the system. G operates in maximally parallel mode if at each step of the computation, a maximal subset of R is applied, and at most one instance of any rule is used at every step (thus at most k rules are applicable at any step). We refer to this system as a maximally parallel system. We look at the computing power of P systems under three semantics of parallelism. For a positive integer n ≤ k, define: n-Max-Parallel: At each step, nondeterministically select a maximal subset of at most n rules in R to apply (this implies that no larger subset is applicable). ≤ n-Parallel: At each step, nondeterministically select any subset of at most n rules in R to apply. n-Parallel: At each step, nondeterministically select any subset of exactly n rules in R to apply. In all three cases, if any rule in the subset selected is not applicable, then the whole subset is not applicable. When n = 1, the three semantics reduce to the Sequential mode. We focus on two popular models of P systems: multi-membrane catalytic systems and communicating P systems. We show that for these systems, n-Max-Parallel mode is strictly more powerful than any of the following three modes: Sequential, ≤ n-Parallel, or n-Parallel. For example, it follows from the result in [9] that a maximally parallel communicating P system is universal for n = 2. However, under the three limited modes of parallelism, the system is equivalent to a vector addition system, which is known to only define a recursive set. These generalize and refine the results for the case of 1-membrane systems recently reported in [3]. Some of the present results are rather surprising. For example, we show that a Sequential 1-membrane communicating P system can only generate a semilinear set, whereas with k membranes, it is equivalent to a vector addition system for any k ≥ 2 (thus the hierarchy collapses at 2 membranes - a rare collapsing result for nonuniversal P systems). We also give another proof (using vector addition systems) of the known result [8] that a 1-membrane catalytic system with only 3 catalysts and (non-prioritized) catalytic rules operating under 3-Max-Parallel mode can simulate any 2-counter machine M. Unlike in [8], our catalytic system needs only a fixed number of noncatalysts, independent of M. A simple cooperative system (SCO) is a P system where the only rules allowed are of the form a → v or of the form aa → v, where a is a symbol and v is a (possibly null) string of symbols not containing a. We show that a 9-Max-Parallel 1-membrane SCO is universal.