Surfaces of Bounded Mean Curvature In Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F subset of M, together with bounds on the geometry of M, give an upper bound on the diameter of F. Our proof is modelled on Gromov's compactness theorem for J-holomorphic curves.

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.

Path-integral formulation for Levy flights: Evaluation of the propagator for free, linear, and harmonic potentials in the over- and underdamped limits

Levy flights can be described using a Fokker-Planck equation, which involves a fractional derivative operator in the position coordinate. Such an operator has its natural expression in the Fourier domain. Starting with this, we show that the solution of the equation can be written as a Hamiltonian path integral. Though this has been realized in the literature, the method has not found applications as the path integral appears difficult to evaluate. We show that a method in which one integrates over the position coordinates first, after which integration is performed over the momentum coordinates, can be used to evaluate several path integrals that are of interest. Using this, we evaluate the propagators for (a) free particle, (b) particle subjected to a linear potential, and (c) harmonic potential. In all the three cases, we have obtained results for both overdamped and underdamped cases. DOI: 10.1103/PhysRevE.86.061105

SOLVING VOLUME INTEGRAL EQUATION USING ScaLAPACK

In this article, we investigate the performance of a volume integral equation code on BlueGene/L system. Volume integral equation (VIE) is solved for homogeneous and inhomogeneous dielectric objects for radar cross section (RCS) calculation in a highly parallel environment. Pulse basis functions and point matching technique is used to convert the volume integral equation into a set of simultaneous linear equations and is solved using parallel numerical library ScaLAPACK on IBM's distributed-memory supercomputer BlueGene/L by different number of processors to compare the speed-up and test the scalability of the code.

Site-specific N-Linked Glycosylation of Receptor Guanylyl Cyclase C Regulates Ligand Binding, Ligand-mediated Activation and Interaction with Vesicular Integral Membrane Protein 36, VIP36

Guanylyl cyclase C (GC-C) is a multidomain, membrane-associated receptor guanylyl cyclase. GC-C is primarily expressed in the gastrointestinal tract, where it mediates fluid-ion homeostasis, intestinal inflammation, and cell proliferation in a cGMP-dependent manner, following activation by its ligands guanylin, uroguanylin, or the heat-stable enterotoxin peptide (ST). GC-C is also expressed in neurons, where it plays a role in satiation and attention deficiency/hyperactive behavior. GC-C is glycosylated in the extracellular domain, and differentially glycosylated forms that are resident in the endoplasmic reticulum (130 kDa) and the plasma membrane (145 kDa) bind the ST peptide with equal affinity. When glycosylation of human GC-C was prevented, either by pharmacological intervention or by mutation of all of the 10 predicted glycosylation sites, ST binding and surface localization was abolished. Systematic mutagenesis of each of the 10 sites of glycosylation in GC-C, either singly or in combination, identified two sites that were critical for ligand binding and two that regulated ST-mediated activation. We also show that GC-C is the first identified receptor client of the lectin chaperone vesicular integral membrane protein, VIP36. Interaction with VIP36 is dependent on glycosylation at the same sites that allow GC-C to fold and bind ligand. Because glycosylation of proteins is altered in many diseases and in a tissue-dependent manner, the activity and/or glycan-mediated interactions of GC-C may have a crucial role to play in its functions in different cell types.

Some questions on integral geometry on noncompact symmetric spaces of higher rank

Let be a noncompact symmetric space of higher rank. We consider two types of averages of functions: one, over level sets of the heat kernel on and the other, over geodesic spheres. We prove injectivity results for functions in which extend the results in Pati and Sitaram (Sankya Ser A 62:419-424, 2000).

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.

Reaction dynamics under confinement: an exact path integral treatment of a two-stage model of stochastic gene expression

Gene expression in living systems is inherently stochastic, and tends to produce varying numbers of proteins over repeated cycles of transcription and translation. In this paper, an expression is derived for the steady-state protein number distribution starting from a two-stage kinetic model of the gene expression process involving p proteins and r mRNAs. The derivation is based on an exact path integral evaluation of the joint distribution, P(p, r, t), of p and r at time t, which can be expressed in terms of the coupled Langevin equations for p and r that represent the two-stage model in continuum form. The steady-state distribution of p alone, P(p), is obtained from P(p, r, t) (a bivariate Gaussian) by integrating out the r degrees of freedom and taking the limit t -> infinity. P(p) is found to be proportional to the product of a Gaussian and a complementary error function. It provides a generally satisfactory fit to simulation data on the same two-stage process when the translational efficiency (a measure of intrinsic noise levels in the system) is relatively low; it is less successful as a model of the data when the translational efficiency (and noise levels) are high.

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.