Conjugacy invariant pseudo-norms, representability and RNA secondary structures

To a reasonable approximation, a secondary structures of RNA is determined by Watson-Crick pairing without pseudo-knots in such a way as to minimise the number of unpaired bases: We show that this minimal number is determined by the maximal conjugacy-invariant pseudo-norm on the free group on two generators subject to bounds on the generators. This allows us to construct lower bounds on the minimal number of unpaired bases by constructing conjugacy invariant pseudo-norms. We show that one such construction, based on isometric actions on metric spaces, gives a sharp lower bound. A major goal here is to formulate a purely mathematical question, based on considering orthogonal representations, which we believe is of some interest independent of its biological roots.

Minimal Triangulations of Manifolds

Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following:(i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices.In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3,we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.

An invariant subspace-based approach to the random eigenvalue problem of systems with clustered spectrum

The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non-unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal-based reduced-order model of the system. Copyright (C) 2012 John Wiley & Sons, Ltd.

Asymptotics of the invariant measure in mean field models with jumps

We consider the asymptotics of the invariant measure for the process of spatial distribution of N coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of transition rates on the spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. Our model is also applicable in the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution converges weakly to this equilibrium. Using a control-theoretic approach, we examine the question of a large deviation from this equilibrium.

Poincare invariant quantum field theories with twisted internal symmetries

Following up the work of 1] on deformed algebras, we present a class of Poincare invariant quantum field theories with particles having deformed internal symmetries. The twisted quantum fields discussed in this work satisfy commutation relations different from the usual bosonic/fermionic commutation relations. Such twisted fields by construction are nonlocal in nature. Despite this nonlocality we show that it is possible to construct interaction Hamiltonians which satisfy cluster decomposition principle and are Lorentz invariant. We further illustrate these ideas by considering global SU(N) symmetries. Specifically we show that twisted internal symmetries can provide a natural-framework for the discussion of the marginal deformations (beta-deformations) of the N = 4 SUSY theories.

Role of invariant water molecules and water-mediated ionic interactions in D-xylose isomerase from Streptomyces rubiginosus

The enzyme, D-xylose isomerase (D-xylose keto-isomerase; EC 5.3.1.5) is a soluble enzyme that catalyzes the conversion of the aldo-sugar D-xylose to the keto-sugar D-xylulose. A total of 27 subunits of D-xylose isomerase from Streptomyces rubiginosus were analyzed in order to identify the invariant water molecules and their water-mediated ionic interactions. A total of 70 water molecules were found to be invariant. The structural and/or functional roles of these water molecules have been discussed. These invariant water molecules and their ionic interactions may be involved in maintaining the structural stability of the enzyme D-xylose isomerase. Fifty-eight of the 70 invariant water molecules (83%) have at least one interaction with the main chain polar atom.

On Wilking's criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators , which are nonnegative in a suitable sense, to every invariant subset . In this article we show that if is an invariant subset of such that is closed and denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in also admits a metric with curvature operator in (b) The normalized Ricci flow on any compact Riemannian manifold with curvature operator in converges to a metric of constant positive sectional curvature. We also point out that if is an arbitrary subset, then is contained in the cone of curvature operators with nonnegative isotropic curvature.

An infinite family of tight triangulations of manifolds

We give explicit construction of vertex-transitive tight triangulations of d-manifolds for d >= 2. More explicitly, for each d >= 2, we construct two (d(2) + 5d + 5)-vertex neighborly triangulated d-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated manifolds currently known is the series of non-simply connected triangulated d-manifolds with 2d + 3 vertices constructed by Kuhnel. The manifolds we construct are strongly minimal. For d >= 3, they are also tight neighborly as defined by Lutz, Sulanke and Swartz. Like Kuhnel complexes, our manifolds are orientable in even dimensions and non-orientable in odd dimensions. (c) 2013 Elsevier Inc. All rights reserved.

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce k-stellated spheres and consider the class W-k(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W-k*; (d), consisting of the (k + 1)-neighbourly members of W-k(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W-k(d) for d >= 2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W-k*(d) for d >= 2k + 2. As another application, we prove that, when d not equal 2k + 1, all members of W-k*(d) are tight. We also characterize the tight members of W-k*(2k + 1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W-1(d) provide the equality case. This generalizes a result (the d = 4 case) due to Walkup and Kuhnel. As a consequence, it is shown that every tight member of W-1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal. (C) 2013 Elsevier Ltd. All rights reserved.

Minimal crystallizations of 3-manifolds

We have introduced the weight of a group which has a presentation with number of relations is at most the number of generators. We have shown that the number of facets of any contracted pseudotriangulation of a connected closed 3-manifold M is at least the weight of the fundamental group of M. This lower bound is sharp for the 3-manifolds RP3, L(3, 1), L(5, 2), S-1 x S-1 x S-1, S-2 x S-1, S-2 (x) under bar S-1 and S-3/Q(8), where Q(8) is the quaternion group. Moreover, there is a unique such facet minimal pseudotriangulation in each of these seven cases. We have also constructed contracted pseudotriangulations of L(kq - 1, q) with 4(q + k - 1) facets for q >= 3, k >= 2 and L(kq + 1, q) with 4(q + k) facets for q >= 4, k >= 1. By a recent result of Swartz, our pseudotriangulations of L(kg + 1, q) are facet minimal when kg + 1 are even. In 1979, Gagliardi found presentations of the fundamental group of a manifold M in terms of a contracted pseudotriangulation of M. Our construction is the converse of this, namely, given a presentation of the fundamental group of a 3-manifold M, we construct a contracted pseudotriangulation of M. So, our construction of a contracted pseudotriangulation of a 3-manifold M is based on a presentation of the fundamental group of M and it is computer-free.

Proper holomorphic mappings between hyperbolic product manifolds

We prove a result on the structure of finite proper holomorphic mappings between complex manifolds that are products of hyperbolic Riemann surfaces. While an important special case of our result follows from the ideas developed by Remmert and Stein, the proof of the full result relies on the interplay of the latter ideas and a finiteness theorem for Riemann surfaces.

Non-existence of tight neighborly triangulated manifolds with beta(1)=2

All triangulated d-manifolds satisfy the inequality ((f0-d-1)(2)) >= ((d+2)(2))beta(1) for d >= 3. A triangulated d-manifold is called tight neighborly if it attains equality in this bound. For each d >= 3, a (2d + 3)-vertex tight neighborly triangulation of the Sd-1-bundle over S-1 with beta(1) = 1 was constructed by Kuhnel in 1986. In this paper, it is shown that there does not exist a tight neighborly triangulated manifold with beta(1) = 2. In other words, there is no tight neighborly triangulation of (Sd-1 x S-1)(#2) or (Sd-1 (sic) S-1)(#2) for d >= 3. A short proof of the uniqueness of K hnel's complexes for d >= 4 under the assumption beta(1) not equal 0 is also presented.

Matrix-Assisted Refolding, Purification and Activity Assessment Using a `Form Invariant' Assay for Matrix Metalloproteinase 2 (MMP2)

Matrix metalloproteinases expression is used as biomarker for various cancers and associated malignancies. Since these proteinases can cleave many intracellular proteins, overexpression tends to be toxic; hence, a challenge to purify them. To overcome these limitations, we designed a protocol where full length pro-MMP2 enzyme was overexpressed in E. coli as inclusion bodies and purified using 6xHis affinity chromatography under denaturing conditions. In one step, the enzyme was purified and refolded directly on the affinity matrix under redox conditions to obtain a bioactive protein. The pro-MMP2 protein was characterized by mass spectrometry, CD spectroscopy, zymography and activity analysis using a simple in-house developed `form invariant' assay, which reports the total MMP2 activity independent of its various forms. The methodology yielded higher yields of bioactive protein compared to other strategies reported till date, and we anticipate that using the protocol, other toxic proteins can also be overexpressed and purified from E. coli and subsequently refolded into active form using a one step renaturation protocol.