Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere.

Investigation of the effect of nonlocal scale on ultrasonic wave dispersion characteristics of a monolayer graphene

In this article, an ultrasonic wave propagation in graphene sheet is studied using nonlocal elasticity theory incorporating small scale effects. The graphene sheet is modeled as an isotropic plate of one-atom thick. For this model, the nonlocal governing differential equations of motion are derived from the minimization of the total potential energy of the entire system. An ultrasonic type of wave propagation model is also derived for the graphene sheet. The nonlocal scale parameter introduces certain band gap region in in-plane and flexural wave modes where no wave propagation occurs. This is manifested in the wavenumber plots as the region where the wavenumber tends to infinite or wave speed tends to zero. The frequency at which this phenomenon occurs is called the escape frequency. The explicit expressions for cutoff frequencies and escape frequencies are derived. The escape frequencies are mainly introduced because of the nonlocal elasticity. Obviously these frequencies are function of nonlocal scaling parameter. It has also been obtained that these frequencies are independent of y-directional wavenumber. It means that for any type of nanostructure, the escape frequencies are purely a function of nonlocal scaling parameter only. It is also independent of the geometry of the structure. It has been found that the cutoff frequencies are function of nonlocal scaling parameter (e(0)a) and the y-directional wavenumber (k(y)). For a given nanostructure, nonlocal small scale coefficient can be obtained by matching the results from molecular dynamics (MD) simulations and the nonlocal elasticity calculations. At that value of the nonlocal scale coefficient, the waves will propagate in the nanostructure at that cut-off frequency. In the present paper, different values of e(o)a are used. One can get the exact e(0)a for a given graphene sheet by matching the MD simulation results of graphene with the results presented in this paper. (C) 2010 Elsevier B.V. All rights reserved.

Constructing Reeb Graphs using Cylinder Maps

The Reeb graph of a scalar function represents the evolution of the topology of its level sets. In this video, we describe a near-optimal output-sensitive algorithm for computing the Reeb graph of scalar functions defined over manifolds. Key to the simplicity and efficiency of the algorithm is an alternate definition of the Reeb graph that considers equivalence classes of level sets instead of individual level sets. The algorithm works in two steps. The first step locates all critical points of the function in the domain. Arcs in the Reeb graph are computed in the second step using a simple search procedure that works on a small subset of the domain that corresponds to a pair of critical points. The algorithm is also able to handle non-manifold domains.

Spectral assignments based on the 13C NMR spectra of mixed liquid crystals of opposite diamagnetic anisotropies

The proton-decoupled 13C NMR spectra of mixtures of liquid crystals with opposite diamagnetic anisotropies have been studied in the natural abundance of 13C. A new method to assign the spectral lines to specific carbons in the liquid crystalline phase has been developed. For this purpose, the assignments of lines in the isotropic media are required, and they were obtained from two-dimensional hetero-COSY experiments. From the spectra in the �critical� mixtures where both the orientations of the liquid crystal directors, with the alignments along and perpendicular to the direction of the magnetic field, �coexist,� the 13C chemical-shift anisotropies have been determined, assuming uniaxial symmetry.

An accurate numerical integration scheme for finite rotations using rotation vector parametrization

A numerical integration procedure for rotational motion using a rotation vector parametrization is explored from an engineering perspective by using rudimentary vector analysis. The incremental rotation vector, angular velocity and acceleration correspond to different tangent spaces of the rotation manifold at different times and have a non-vectorial character. We rewrite the equation of motion in terms of vectors lying in the same tangent space, facilitating vector space operations consistent with the underlying geometric structure. While any integration algorithm (that works within a vector space setting) may be used, we presently employ a family of explicit Runge-Kutta algorithms to solve this equation. While this work is primarily motivated out of a need for highly accurate numerical solutions of dissipative rotational systems of engineering interest, we also compare the numerical performance of the present scheme with some of the invariant preserving schemes, namely ALGO-C1, STW, LIEMIDEA] and SUBCYC-M. Numerical results show better local accuracy via the present approach vis-a-vis the preserving algorithms. It is also noted that the preserving algorithms do not simultaneously preserve all constants of motion. We incorporate adaptive time-stepping within the present scheme and this in turn enables still higher accuracy and a `near preservation' of constants of motion over significantly longer intervals. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

An Experimental and Numerical Study of Calliphora Wing Structure

Experiments are performed to determine the mass and stiffness variations along the wing of the blowfly Calliphora. The results are obtained for a pairs of wings of 10 male flies and fresh wings are used. The wing is divided into nine locations along the span and seven locations along the chord based on venation patterns. The length and mass of the sections is measured and the mass per unit length is calculated. The bending stiffness measurements are taken at three locations, basal (near root), medial and distal (near tip) of the fly wing. Torsional stiffness measurements are also made and the elastic axis of the wing is approximately located. The experimental data is then used for structural modeling of the wing as a stepped cantilever beam with nine spanwise sections of varying mass per unit lengths, flexural rigidity (EI) and torsional rigidity (GJ) values. Inertial values of nine sections are found to approximately vary according to an exponentially decreasing law over the nine sections from root to tip and it is used to calculate an approximate value of Young's modulus of the wing biomaterial. Shear modulus is obtained assuming the wing biomaterial to be isotropic. Natural frequencies, both in bending and torsion, are obtained by solving the homogeneous part of the respective governing differential equations using the finite element method. The results provide a complete analysis of Calliphora wing structure and also provide guidelines for the biomimetic structural design of insect-scale flapping wings.

Bearing Capacity of Clays whose Cohesion Increases Linearly with Depth

Experimental studies (Bishop 1966) show that for saturated normally consolidated clay the cohesion varies linearly with depth. The ultimate bearing capacity in such soil is generally obtained by limit equilibrium and limit analysis methods. Sokolovski's (1960) approach to the method of characteristics is used to find the ultimate bearing capacity of saturated clay whose cohesion is homogeneous and isotropic. In this technical note, the method of characteristics is further developed to determine ultimate bearing capacity of clay whose cohesion varies linearly with depth.

On the relation between rotation increments in different tangent spaces

In computational mechanics, finite rotations are often represented by rotation vectors. Rotation vector increments corresponding to different tangent: spaces are generally related by a linear operator, known as the tangential transformation T. In this note, we derive the higher order terms that are usually left out in linear relation. The exact nonlinear relation is also presented. Errors via the linearized T are numerically estimated. While the concept of T arises out of the nonlinear characteristics of the rotation manifold, it has been derived via tensor analysis in the context of computational mechanics (Cardona and Geradin, 1988). We investigate the operator T from a Lie group perspective, which provides a better insight and a 1-1 correspondence between approaches based on tensor analysis and the standard matrix Lie group theory. (C) 2010 Elsevier Ltd. All rights reserved.

Infinite dimensional slow modulations in a well known delayed model for cutting tool vibrations

We apply the method of multiple scales (MMS) to a well-known model of regenerative cutting vibrations in the large delay regime. By ``large'' we mean the delay is much larger than the timescale of typical cutting tool oscillations. The MMS up to second order, recently developed for such systems, is applied here to study tool dynamics in the large delay regime. The second order analysis is found to be much more accurate than the first order analysis. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy in that plotted solutions of moderate amplitudes are visually near-indistinguishable. The advantages of the present analysis are that infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space; lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS; the strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly; and though certain parameters are treated as small (or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation.

Optical and magnetic properties of the exact PPP states of biphenyl

The low-lying singlets and triplets of biphenyl are obtained exactly within the PPP model using the diagrammatic valence bond method. The energy gaps within the singlet manifold as well as the lowest singlet-triplet gap are found to be in good agreement with experimental results. The two weak absorptions between 4·1 and 4·2 eV reported experimentally are attributed to the two states lying below the optical gap that become weakly allowed on breaking electron-hole and inversion symmetries. The observed blue shift of the spectral lines, attributed to a change in dihedral angle, on going from crystalline to solution to vapour phase is also well reproduced within the PPP model. The bond orders show that the ground singlet state is benzenoidal while the dipole excited state as well as the lowest triplet state are quinonoidal and planar. Comparison with the experimental spin densities and the fine structure constants D and E in the triplet state point to slightly weaker correlations than assumed by the PPP model. The introduction of a 1-8 bond to mimic poly(paraphenylene)s gives an optical gap that is in good agreement with experiment.

Stress and strength analysis of pin joints in laminated anisotropic plates

Mechanical joints in composites can be tailored to achieve improved performance and better life by appropriately selecting the laminate parameters. In order to gain the best advantage of this possibility of tailoring the laminate, it is necessary to understand the influence of laminate parameters on the behaviour of joints in composites. Most of the earlier studies in this direction were based on simplified assumptions regarding load transfer at the pin-plate interface and such studies were only carried out on orthotropic and quasi-isotropic laminates. In the present study, a more rigorous analysis is carried out to study pin joints in laminates with anisotropic properties. Two types of laminates with (0/ + ?4/90)s and (0/ ± ?2/90)s layups made out of graphite epoxy T300/5208 material system are considered. The analysis mainly concentrates on clearance fit in which the pin is of smaller diameter compared to the hole. The main aspect of the analysis of pin joints is the changing contact between the pin and the plate with increasing load levels. The analysis is carried out by an iterative finite element technique and a computationally efficient routine is developed for this purpose. Numerical studies indicate that the location and magnitude of the peak stresses along the hole boundary are functions of fibre angle and the overall anisotropic properties. It is also shown that the conventional assumption of cosine distribution for the contact pressure between pin and the plate in the analysis lead to underestimation of bearing failure load and overestimation of shear and tensile failure loads in typical (0/905)s cross-ply laminates.

NMR of bicyclic diazines oriented in a lyotropic mesophase

Restricted Access.. Proton NMR spectra of bicyclic diazines such as phthalazine and quinoxaline have been studied in lyotropic liquid crystalline solvents. Values of the indirect spin-spin couplings which could not be derived from studies in the isotropic medium have been obtained. Geometrical information has been obtained in both the cases. The results are indicative of significant solvcntsolute interactions in phthalazine but not in quinoxaline.

Visualization of enantiomers and determination of homo- and hetero-nuclear residual dipolar and scalar couplings: The natural abundant C-13 edited J/D-resolved NMR techniques

The simple two dimensional C-13-satellite J/D-resolved experiments have been proposed for the visualization of enantiomers, extraction of homo- and hetero-nuclear residual dipolar couplings and also H-1 chemical shift differences between the enantiomers in the anisotropic medium. The significant advantages of the techniques are in the determination of scalar couplings of bigger organic molecules. The scalar couplings specific to a second abundant spin such as F-19 can be selectively extracted from the severely overlapped spectrum. The methodologies are demonstrated on a chiral molecule aligned in the chiral liquid crystal medium and two different organic molecules in the isotropic solutions. (C) 2010 Elsevier B.V. All rights reserved.

An improved iterative finite element solution for pin joints

Accurate, reliable and economical methods of determining stress distributions are important for fastener joints. In the past the contact stress problems in these mechanically fastened joints using interference or push or clearance fit pins were solved using both inverse and iterative techniques. Inverse techniques were found to be most efficient, but at times inadequate in the presence of asymmetries. Iterative techniques based on the finite element method of analysis have wider applications, but they have the major drawbacks of being expensive and time-consuming. In this paper an improved finite element technique for iteration is presented to overcome these drawbacks. The improved iterative technique employs a frontal solver for elimination of variables not requiring iteration, by creation of a dummy element. This automatically results in a large reduction in computer time and in the size of the problem to be handled during iteration. Numerical results are compared with those available in the literature. The method is used to study an eccentrically located pin in a quasi-isotropic laminated plate under uniform tension.

Dynamic Structure Factor of Sponge Phases

We predict the dynamic light scattering intensity S(q,t) for the L3 phase (anomalous isotropic phase) of dilute surfactant solutions. Our results are based on a Landau-Ginzburg approach, which was previously used to explain the observed static structure factor S(q, 0). In the extreme limit of small q, we find a monoexponential decay with marginal or irrelevant hydrodynamic interactions. In most other regimes the decay of S(q,t) is strongly nonexponential; in one case, it is purely algebraic at long times.