Variational principles in evolution

For a one-locus selection model, Svirezhev introduced an integral variational principle by defining a Lagrangian which remained stationary on the trajectory followed by the population undergoing selection. It is shown here (i) that this principle can be extended to multiple loci in some simple cases and (ii) that the Lagrangian is defined by a straightforward generalization of the one-locus case, but (iii) that in two-locus or more general models there is no straightforward extension of this principle if linkage and epistasis are present. The population trajectories can be constructed as trajectories of steepest ascent in a Riemannian metric space. A general method is formulated to find the metric tensor and the surface-in the metric space on which the trajectories, which characterize the variations in the gene structure of the population, lie. The local optimality principle holds good in such a space. In the special case when all possible linkage disequilibria are zero, the phase point of the n-locus genetic system moves on the surface of the product space of n higher dimensional unit spheres in a certain Riemannian metric space of gene frequencies so that the rate of change of mean fitness is maximum along the trajectory. In the two-locus case the corresponding surface is a hyper-torus.

Structure-potential relationship and nature of disorder in liquids and glassy solids

A method based on the minimal-spanning tree is extended to a collection of points in three dimensions. Two parameters, the average edge length and its standard deviation characterize the disorder. The structural phase diagram for a monatomic system of particles and the characteristic values for the uniform random distribution of points have been obtained. The method is applied to hard spheres and Lennard-Jones systems. These systems occupy distinct regions in the structural phase diagram. The structure of the Lennard-Jones system approaches that of the defective close-packed arrangements at low temperatures whereas in the liquid regime, it deviates from the close-packed configuration.

On Walkup's class K (d) and a minimal triangulation of (S-3 (sic) S-1)(#3)

For d >= 2, Walkup's class K (d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d - 1)-spheres. Kalai showed that for d >= 4, all connected members of K (d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic chi satisfies f(1) >= 5f(0) - 15/2 chi, with equality only for X is an element of K(4). Kuhnel observed that this implies f(0)(f(0) - 11) >= -15 chi, with equality only for 2-neighborly members of K(4). Kuhnel also asked if there is a triangulated 4-manifold with f(0) = 15, chi = -4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S-3 (sic) S-1. Because of Kuhnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d + 3)-vertex sphere products Sd-1 X S-1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result. (C) 2011 Elsevier B.V. All rights reserved.

Velocity distribution function for a dilute granular material in shear flow

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle-wall collisions is large compared to particle-particle collisions. An asymptotic analysis is used in the small parameter epsilon, which is naL in two dimensions and na(2)L in three dimensions, where; n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle-wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution e(t) and e(n) in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (u(x), u(y)) = (+/-V, O) after repeated collisions with the wall, where u(x) and u(y) are the velocities tangential and normal to the wall, V = (1 - e(t))V-w/(1 + e(t)), and V-w and -V-w, are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions :in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to epsilon(1/2) in the limit epsilon --> 0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to epsilon(1/2) in the limit epsilon --> 0. The moments of the velocity distribution function are evaluated, and it is found that [u(x)(2)] --> V-2, [u(y)(2)] similar to V-2 epsilon and -[u(x)u(y)] similar to V-2 epsilon log(epsilon(-1)) in the limit epsilon --> 0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.

Boundary conditions at a rigid wall for rough granular gases

We derive boundary conditions at a rigid wall for a granular material comprising rough, inelastic particles. Our analysis is confined to the rapid flow, or granular gas, regime in which grains interact by impulsive collisions. We use the Chapman-Enskog expansion in the kinetic theory of dense gases, extended for inelastic and rough particles, to determine the relevant fluxes to the wall. As in previous studies, we assume that the particles are spheres, and that the wall is corrugated by hemispheres rigidly attached to it. Collisions between the particles and the wall hemispheres are characterized by coefficients of restitution and roughness. We derive boundary conditions for the two limiting cases of nearly smooth and nearly perfectly rough spheres, as a hydrodynamic description of granular gases comprising rough spheres is appropriate only in these limits. The results are illustrated by applying the equations of motion and boundary conditions to the problem of plane Couette flow.

Helical structures: The geometry of protein helices and nanotubes

In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit. A plane hexagonal lattice wrapped round a cylinder provides a useful starting point for describing the helical conformations of protein molecules, for investigating the geometrical properties of carbon nanotubes, and for certain types of dense packings of equal spheres.

Rayleigh-Benard convection during solidification of an eutectic solution cooled from the top

The interaction between laminar Rayleigh-Benard convection and directional solidification is studied for the case of an eutectic solution kept in a rectangular cavity cooled from the top. Experiments and numerical simulations are carried out using an NH4Cl-H2O solution as the model fluid. The flow is visualized using a sheet of laser light scattered by neutrally buoyant, hollow-glass spheres seeded in the fluid. The numerical modeling is performed using a pressure-based finite-volume method according to the SIMPLER algorithm. The present configuration enables us to visualize flow vortices in the presence of a continuously evolving solid/liquid interface. Clear visualization of the Rayleigh-Benard convective cells and their interaction with the solidification front are obtained. It is observed that the convective cells are characterized by zones of up-flow and down-flow, resulting in the development of a nonplanar interface. Because of the continuous advancement of the solid/liquid interface, the effective liquid height of the cavity keeps decreasing. Once the height of the fluid layer falls below a critical value, the convective cells become weaker and eventually die out, leading to the growth of a planar solidification front. Results of flow visualization and temperature measurement are compared with those from the numerical simulation, and a good agreement is found.

Studies on compressive failure features in syntactic foam material

Syntactic foam made by mechanical mixing of glass hollow spheres in epoxy resin matrix is characterized for compressive properties in the present study. Volume fraction of hollow spheres in the syntactic foam under investigation is kept at 67.8%. Effect of specimen aspect ratio on failure behavior and stress-strain curve of the material is highlighted. Considerable differences are noted in the macroscopic fracture features of the specimen and the stress-strain curve with the variation in specimen aspect ratio, although compressive yield strength values were within a narrow range. Post compression test scanning electron microscopic observations coupled with the macroscopic observations taken during the test helped in explaining the deviation in specimen behavior and in gathering support for the proposed arguments.

Particle dynamics in a turbulent particle–gas suspension at high Stokes number. Part 1. Velocity and acceleration distributions

The effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas–solid suspension is analysed in the low-Reynolds-number and high Stokes number limits, where the particle relaxation time is long compared with the correlation time for the fluid velocity fluctuations, and the drag force on the particles due to the fluid can be expressed by the modified Stokes law. The direct numerical simulation procedure is used for solving the Navier–Stokes equations for the fluid, the particles are modelled as hard spheres which undergo elastic collisions and a one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. The particle mean and root-mean-square (RMS) fluctuating velocities, as well as the probability distribution function for the particle velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette (where the velocity profile is linear and the RMS velocities are nearly constant), are examined. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the spanwise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the particle velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. It is also found that the acceleration distribution on the particles is in very good agreement with the distribution that is calculated from the velocity fluctuations in the fluid, using the Stokes drag law, indicating that there is very little correlation between the fluid velocity fluctuations and the particle velocity fluctuations in the presence of one-way coupling. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise.

Collision Cones for Quadric Surfaces

The problem of collision prediction in dynamic environments appears in several diverse fields, which include robotics, air vehicles, underwater vehicles, and computer animation. In this paper, collision prediction of objects that move in 3-D environments is considered. Most work on collision prediction assumes objects to be modeled as spheres. However, there are many instances of object shapes where an ellipsoidal or a hyperboloid-like bounding box would be more appropriate. In this paper, a collision cone approach is used to determine collision between objects whose shapes can be modeled by general quadric surfaces. Exact collision conditions for such quadric surfaces are obtained in the form of analytical expressions in the relative velocity space. For objects of arbitrary shapes, exact representations of planar sections of the 3-D collision cone are obtained.

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.

The γ-brass structure and the Boerdijk–Coxeter helix

The γ-brass structure was for a long time regarded as a modified bcc structure. It is more accurately described in terms of a 26-atom cluster consisting of four interpenetrating icosahedral clusters. An alternative description in terms of a 38-atom cluster is also illuminating. We discuss the γ-brass structure in terms of the packing of spheres and the packing of ‘almost regular’ tetrahedra and demonstrate a close relationship to the helical sphere packings investigated by Boerdijk, who considered the configuration of touching spheres centred at the vertices of a Coxeter helix, and extended it by adding an extra layer of spheres. Adding a further layer of spheres gives a rod-like structure in which every sphere of the original helix is surrounded by twelve others, configured as a somewhat distorted icosahedron. Thus each tetrahedron of the initial structure is then shared by four icosahedra. This 26-sphere cluster is a slightly distorted form of the 26-atom γ-brass cluster.

Generalization of the collision cone approach for motion safety in 3-D environments

Avoidance of collision between moving objects in a 3-D environment is fundamental to the problem of planning safe trajectories in dynamic environments. This problem appears in several diverse fields including robotics, air vehicles, underwater vehicles and computer animation. Most of the existing literature on collision prediction assumes objects to be modelled as spheres. While the conservative spherical bounding box is valid in many cases, in many other cases, where objects operate in close proximity, a less conservative approach, that allows objects to be modelled using analytic surfaces that closely mimic the shape of the object, is more desirable. In this paper, a collision cone approach (previously developed only for objects moving on a plane) is used to determine collision between objects, moving in 3-D space, whose shapes can be modelled by general quadric surfaces. Exact collision conditions for such quadric surfaces are obtained and used to derive dynamic inversion based avoidance strategies.

The Nature of Bond Critical Points in Dinuclear Copper(I) Complexes

Closed-shell contacts between two copper(I) ions are expected to be repulsive. However, such contacts are quite frequent and are well documented. Crystallographic characterization of such contacts in unsupported and bridged multinuclear copper(I) complexes has repeatedly invited debates on the existence of cuprophilicity. Recent developments in the application of Baders theory of atoms-in-molecules (AIM) to systems in which weak hydrogen bonds are involved suggests that the copper(I)copper(I) contacts would benefit from a similar analysis. Thus the nature of electron-density distributions in copper(I) dimers that are unsupported, and those that are bridged, have been examined. A comparison of complexes that are dimers of symmetrical monomers and those that are dimers of two copper(I) monomers with different coordination spheres has also been made. AIM analysis shows that a bond critical point (BCP) between two Cu atoms is present in most cases. The nature of the BCP in terms of the electron density, ?, and its Laplacian is quite similar to the nature of critical points observed in hydrogen bonds in the same systems. The ? is inversely correlated to Cu?Cu distance. It is higher in asymmetrical systems than what is observed in corresponding symmetrical systems. By examining the ratio of the local electron potential-energy density (Vc) to the kinetic energy density (Gc), |Vc|/Gc at the critical point suggests that these interactions are not perfectly ionic but have some shared nature. Thus an analysis of critical points by using AIM theory points to the presence of an attractive metallophilic interaction similar to other well-documented weak interactions like hydrogen bonding.

Geosphere laminations in free groups

We construct for free groups, which are codimension one analogues of geodesic laminations on surfaces. Other analogues that have been constructed by several authors are dimension-one instead of codimension-one. Our main result is that the space of such laminations is compact. This in turn is based on the result that crossing, in the sense of Scott-Swarup, is an open condition. Our construction is based on Hatcher's normal form for spheres in the model manifold.