Spanning trees short or small

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of decomposable graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to provide a polynomial-time solution for finding $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.Comment: 27 page

Significance of thermal fluctuations and hydrodynamic interactions in receptor-ligand mediated adhesive dynamics of a spherical particle in wall bound shear flow

The dynamics of adhesion of a spherical micro-particle to a ligand-coated wall, in shear flow, is studied using a Langevin equation that accounts for thermal fluctuations, hydrodynamic interactions and adhesive interactions. Contrary to the conventional assumption that thermal fluctuations play a negligible role at high P$\acute{e}$clet numbers, we find that for particles with low surface densities of receptors, rotational diffusion caused by fluctuations about the flow and gradient directions aids in bond formation, leading to significantly greater adhesion on average, compared to simulations where thermal fluctuations are completely ignored. The role of wall hydrodynamic interactions on the steady state motion of a particle, when the particle is close to the wall, has also been explored. At high P$\acute{e}$clet numbers, the shear induced force that arises due to the stresslet part of the Stokes dipole, plays a dominant role, reducing the particle velocity significantly, and affecting the states of motion of the particle. The coupling between the translational and rotational degrees of freedom of the particle, brought about by the presence of hydrodynamic interactions, is found to have no influence on the binding dynamics. On the other hand, the drag coefficient, which depends on the distance of the particle from the wall, plays a crucial role at low rates of bond formation. A significant difference in the effect of both the shear force and the position dependent drag force, on the states of motion of the particle, is observed when the P$\acute{e}$let number is small.Comment: The manuscript has been accepted as an article in Physical Review E Journa

Wet and dry internal friction can be measured with the Jarzynski equality

The existence of two types of internal friction wet and dry is revisited, and a simple protocol is proposed for distinguishing between the two types and extracting the appropriate internal friction coefficient. The scheme requires repeatedly stretching a polymer molecule, and measuring the average work dissipated in the process by applying the Jarzynski equality. The internal friction coefficient is then estimated from the average dissipated work in the extrapolated limit of zero solvent viscosity. The validity of the protocol is established through analytical calculations on a one-dimensional free-draining Hookean spring-dashpot model for a polymer, and Brownian dynamics simulations of: (a) a single-mode nonlinear spring-dashpot model for a polymer, and (b) a finitely extensible bead-spring chain with cohesive intra-chain interactions, both of which incorporate fluctuating hydrodynamic interactions. Well-established single-molecule manipulation techniques, such as optical tweezer-based pulling, can be used to implement the suggested protocol experimentally.Comment: 27 pages, 17 figure

An exposition on Friedmann Cosmology with Negative Energy Densities

How would negative energy density affect a classic Friedmann cosmology? Although never measured and possibly unphysical, certain realizations of quantum field theories leaves the door open for such a possibility. In this paper we analyze the evolution of a universe comprising varying amounts of negative energy forms. Negative energy components have negative normalized energy densities, $\Omega < 0$. They include negative phantom energy with an equation of state parameter $w<-1$, negative cosmological constant: $w=-1$, negative domain walls: $w=-2/3$, negative cosmic strings: $w=-1/3$, negative mass: $w=0$, negative radiation: $w=1/3$ and negative ultralight: $w > 1/3$. Assuming that such energy forms generate pressure like perfect fluids, the attractive or repulsive nature of negative energy components are reviewed. The Friedmann equation is satisfied only when negative energy forms are coupled to a greater magnitude of positive energy forms or positive curvature. We show that the solutions exhibit cyclic evolution with bounces and turnovers.The future and fate of such universes in terms of curvature, temperature, acceleration, and energy density are reviewed. The end states are dubbed Big Crunch, Big Void, or Big Rip and further qualified as "Warped", "Curved", or "Flat", "Hot" versus "Cold", "Accelerating" versus "Decelerating" versus "Coasting". A universe that ends by contracting to zero energy density is termed "Big Poof." Which contracting universes "bounce" in expansion and which expanding universes "turnover" into contraction are also reviewed.Comment: Abridged version with minor correction

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

Analysis of radial segregation of granular mixtures in a rotating drum

This paper considers the segregation of a granular mixture in a rotating drum. Extending a recent kinematic model for grain transport on sandpile surfaces to the case of rotating drums, an analysis is presented for radial segregation in the rolling regime, where a thin layer is avalanching down while the rest of the material follows rigid body rotation. We argue that segregation is driven not just by differences in the angle of repose of the species, as has been assumed in earlier investigations, but also by differences in the size and surface properties of the grains. The cases of grains differing only in size (slightly or widely) and only in surface properties are considered, and the predictions are in qualitative agreement with observations. The model yields results inconsistent with the assumptions for more general cases, and we speculate on how this may be corrected.Comment: 12 pages inclusive of 10 PostScript (*.eps) figures, uses svjour, psfrag and graphicx. Submitted for publication to Euro. Phys. J.

Universality of the collapse transition of sticky polymers

The universality of the swelling of the radius of gyration of a homopolymer relative to its value in the $\theta$ state, independent of polymer-solvent chemistry, in the crossover regime between $\theta$ and athermal solvent conditions, is well known. Here we study, by Brownian dynamics, a polymer model where a subset of monomers is labelled as "stickers". The mutual interaction of the stickers is more attractive than those of the other ("backbone") monomers, and has the additional important characteristic of "functionality" $\varphi$, i.e., the maximum number of stickers that can locally bind to a given sticker. A saturated bond formed in this manner remains bound until it breaks due to thermal fluctuations, a requirement which can be viewed as an additional Boolean degree of freedom that describes the bonding. This, in turn, makes the question of the order of the collapse transition a non-trivial one. Nevertheless, for the parameters that we have studied (in particular, $\varphi=1$), we find a standard second-order $\theta$ collapse, using a renormalised solvent quality parameter that takes into account the increased average attraction due to the presence of stickers. We examine the swelling of the radius of gyration of such a sticky polymer relative to its value in the altered $\theta$ state, using a novel potential to model the various excluded volume interactions that occur between the monomers on the chain. We find that the swelling of such sticky polymers is identical to the universal swelling of homopolymers in the thermal crossover regime. Additionally, for our model, the Kuhn segment length under $\theta$ conditions is found to be the same for chains with and without stickers.Comment: 13 pages, 10 figures, supplementary material (see ancillary directory), to appear in Soft Matte

Thresholded Covering Algorithms for Robust and Max-Min Optimization

The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case cost (summed over both days) is minimized? Feige et al. and Khandekar et al. considered the k-robust model where the possible outcomes tomorrow are given by all demand-subsets of size k, and gave algorithms for the set cover problem, and the Steiner tree and facility location problems in this model, respectively. In this paper, we give the following simple and intuitive template for k-robust problems: "having built some anticipatory solution, if there exists a single demand whose augmentation cost is larger than some threshold, augment the anticipatory solution to cover this demand as well, and repeat". In this paper we show that this template gives us improved approximation algorithms for k-robust Steiner tree and set cover, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios (except for multicut) are almost best possible. As a by-product of our techniques, we also get algorithms for max-min problems of the form: "given a covering problem instance, which k of the elements are costliest to cover?".Comment: 24 page

Gaussian approximation for finitely extensible bead-spring chains with hydrodynamic interaction

The Gaussian Approximation, proposed originally by Ottinger [J. Chem. Phys., 90 (1) : 463-473, 1989] to account for the influence of fluctuations in hydrodynamic interactions in Rouse chains, is adapted here to derive a new mean-field approximation for the FENE spring force. This "FENE-PG" force law approximately accounts for spring-force fluctuations, which are neglected in the widely used FENE-P approximation. The Gaussian Approximation for hydrodynamic interactions is combined with the FENE-P and FENE-PG spring force approximations to obtain approximate models for finitely-extensible bead-spring chains with hydrodynamic interactions. The closed set of ODE's governing the evolution of the second-moments of the configurational probability distribution in the approximate models are used to generate predictions of rheological properties in steady and unsteady shear and uniaxial extensional flows, which are found to be in good agreement with the exact results obtained with Brownian dynamics simulations. In particular, predictions of coil-stretch hysteresis are in quantitative agreement with simulations' results. Additional simplifying diagonalization-of-normal-modes assumptions are found to lead to considerable savings in computation time, without significant loss in accuracy.Comment: 26 pages, 17 figures, 2 tables, 75 numbered equations, 1 appendix with 10 numbered equations Submitted to J. Chem. Phys. on 6 February 200