Isoperimetric Problem and Meta-fibonacci Sequences

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$ , such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as b e (i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and b v (i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of b e (i, G) (b v (i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation b c (i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T 2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T 2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that b c (i, T 2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.

Time and Energy Complexity of Distributed Computation of a Class of Functions in Wireless Sensor Networks

We consider a scenario in which a wireless sensor network is formed by randomly deploying n sensors to measure some spatial function over a field, with the objective of computing a function of the measurements and communicating it to an operator station. We restrict ourselves to the class of type-threshold functions (as defined in the work of Giridhar and Kumar, 2005), of which max, min, and indicator functions are important examples: our discussions are couched in terms of the max function. We view the problem as one of message-passing distributed computation over a geometric random graph. The network is assumed to be synchronous, and the sensors synchronously measure values and then collaborate to compute and deliver the function computed with these values to the operator station. Computation algorithms differ in (1) the communication topology assumed and (2) the messages that the nodes need to exchange in order to carry out the computation. The focus of our paper is to establish (in probability) scaling laws for the time and energy complexity of the distributed function computation over random wireless networks, under the assumption of centralized contention-free scheduling of packet transmissions. First, without any constraint on the computation algorithm, we establish scaling laws for the computation time and energy expenditure for one-time maximum computation. We show that for an optimal algorithm, the computation time and energy expenditure scale, respectively, as Theta(radicn/log n) and Theta(n) asymptotically as the number of sensors n rarr infin. Second, we analyze the performance of three specific computation algorithms that may be used in specific practical situations, namely, the tree algorithm, multihop transmission, and the Ripple algorithm (a type of gossip algorithm), and obtain scaling laws for the computation time and energy expenditure as n rarr infin. In particular, we show that the computation time for these algorithms scales as Theta(radicn/lo- g n), Theta(n), and Theta(radicn log n), respectively, whereas the energy expended scales as , Theta(n), Theta(radicn/log n), and Theta(radicn log n), respectively. Finally, simulation results are provided to show that our analysis indeed captures the correct scaling. The simulations also yield estimates of the constant multipliers in the scaling laws. Our analyses throughout assume a centralized optimal scheduler, and hence, our results can be viewed as providing bounds for the performance with practical distributed schedulers.

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one-dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008.

Optimal multicommodity flow through the complete graph with random edge capacities

We consider a multicommodity flow problem on a complete graph whose edges have random, independent, and identically distributed capacities. We show that, as the number of nodes tends to infinity, the maximumutility, given by the average of a concave function of each commodity How, has an almost-sure limit. Furthermore, the asymptotically optimal flow uses only direct and two-hop paths, and can be obtained in a distributed manner.

Enzymatic Degradation of Poly(soybean oil-g-methyl methacrylate)

This study discusses grafting of methyl methacrylate units from thepolymeric soybean oil peroxide to produce poly(soybean oil-graft-methyl methacrylate) (PSO-g-PMMA). The degradation of this copolymer in solution was evaluated in the presence of different lipases, viz Candida rugosa (CR), Lipolase 100T (LP), Novozym 435 (N435) and Porcine pancreas (PP), at different temperatures The copolymer degraded by specific chain end scission and the mass fraction of the specific product evolved was determined The degradation was modeled using continuous distribution kinetics to determine the rate coefficients ofmenzymatic chain end scission and deactivation of the enzyme The enzymes, CR. LP and N435 exhibited maximum activity for the degradation of PSO-g-PMMA at 60 degrees C, while PP was most active at 50 degrees C. The thermal degradability of the copolymer, assessed by thermo-gravimetry, indicated that the activation energy of degradation of the copolymer was 154 kJ mol(-1), which was lesser than that of the PMMA homopolymer.

G.W. Leibniz and Scientific Societies

The famous philosopher Leibniz (1646-1716) was also active in the (cultural) politics of his time. His interest in forming scientific societies never waned and his efforts led to the founding of the Berlin Academy of Sciences. He also played a part in the founding of the Dresden Academy of Science and the St. Petersburg Academy of Science. Though Leibniz's models for the scientific society were the Royal Society and the Royal Science Academy of France, his pansophistic vision of scientific cooperation sometimes took on utopian dimensions. In this paper, I will present Leibniz's ideas of scientific cooperation as a kind of religious activity and discuss his various schemes for the founding of such scientific societies.

Temperature dependence of g values in tetragonal ionic Cu++ salts

A temperature dependence has been observed in the spin-Hamiltonian parameters of the Cu++ ion in a tetragonal crystal field and the variation has been interpreted in terms of vibronic effects.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

A new class of separators and planarity of chordal graphs

We introduce a new class of clique separators, called base sets, for chordal graphs. Base sets of a chordal graph closely reflect its structure. We show that the notion of base sets leads to structural characterizations of planar k-trees and planar chordal graphs. Using these characterizations, we develop linear time algorithms for recognizing planar k-trees and planar chordal graphs. These algorithms are extensions of the Lexicographic_Breadth_First_Search algorithm for recognizing chordal graphs and are much simpler than the general planarity checking algorithm. Further, we use the notion of base sets to prove the equivalence of hamiltonian 2-trees and maximal outerplanar graphs.

Effect of coordinated ions on structure and flexibility of parallel G-quadruplexes: A molecular dynamics study

ingle tract guanine residues can associate to form stable parallel quadruplex structures in the presence of certain cations. Nanosecond scale molecular dynamics simulations have been performed on fully solvated fibre model of parallel d(G(7)) quadruplex structures with Na+ or K+ ions coordinated in the cavity formed by the O6 atoms of the guanine bases. The AMBER 4.1 force field and Particle Mesh Ewald technique for electrostatic interactions have been used in all simulations. There quadruplex structures are stable during the simulation, with the middle four base tetrads showing root mean square deviation values between 0.5 to 0.8 Angstrom from the initial structure as well the high resolution crystal structure. Even in the absence of any coordinated ion in the initial structure, the G-quadruplex structure remains intact throughout the simulation. During the 1.1 ns MD simulation, one Nai counter ion from the solvent as well as several water molecules enter the central cavity to occupy the empty coordination sites within the parallel quadruplex and help stabilize the structure. Hydrogen bonding pattern depends on the nature of the coordinated ion, with the G-tetrad undergoing local structural variation to accommodate cations of different sizes. in the absence of any coordinated ion. due to strong mutual repulsion, O6 atoms within G-tetrad are forced farther apart from each other, which leads to a considerably different hydrogen bonding scheme within the G-tetrads and very favourable interaction energy between the guanine bases constituting a G-tetrad. However, a coordinated ion between G-tetrads provides extra stacking energy for the G-tetrads and makes the quadruplex structure more rigid. Na+ ions, within the quadruplex cavity, are more mobile than coordinated K+ ions. A number of hydrogen bonded water molecules are observed within the grooves of all quadruplex structures.

Symmetrical Bisbenzimidazoles with Benzenediyl Spacer: The Role of the Shape of the Ligand on the Stabilization and Structural Alterations in Telomeric G-Quadruplex DNA and Telomerase Inhibition

The extremities of chromosomes end in a G-rich single-stranded overhang that has been implicated in the onset of the replicate senescence. The repeated sequence forming a G-overhang is able to adopt a four-stranded DNA structure called G-quadruplex, which is a poor substrate for the enzyme telomerase. Small molecule based ligands that selectively stabilize the telomeric G-quadruplex DNA, induce telomere shortening eventually leading to cell death. Herein, we have investigated the G-quadruplex DNA interaction with two isomeric bisbenzimidazole-based compounds that differ in terms of shape (V-shaped angular vs linear).While the linear isomer induced some stabilization of the intramolecular G-quadruplex structure generated in the presence of Na+ the other, having V-shaped central planar core, caused a dramatic structural alteration of the latter, above a threshold concentration. This transition was evident from the pronounced changes observed in the circular dichroism spectra and from the get mobility shift assa involving the G-quadruples DNA. Notably, this angular isomer could also induce the G-quadruplex formation in the absence of any added cation. The ligand-quadruples complexes were investigated by computational molecular modeling, providing further information on structure-activity relationships. Finally, TRAP (telomerase repeat amplification protocol) experiments demonstrated that the angular isomer is selective toward the inhibition of telomerase activity.

Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs

Let G - (V, E) be a weighted undirected graph having nonnegative edge weights. An estimate (delta) over cap (u, v) of the actual distance d( u, v) between u, v is an element of V is said to be of stretch t if and only if delta(u, v) <= (delta) over cap (u, v) <= t . delta(u, v). Computing all-pairs small stretch distances efficiently ( both in terms of time and space) is a well-studied problem in graph algorithms. We present a simple, novel, and generic scheme for all-pairs approximate shortest paths. Using this scheme and some new ideas and tools, we design faster algorithms for all-pairs t-stretch distances for a whole range of stretch t, and we also answer an open question posed by Thorup and Zwick in their seminal paper [J. ACM, 52 (2005), pp. 1-24].