Cubicity and Bandwidth

A unit cube in (or a k-cube in short) is defined as the Cartesian product R (1) x R (2) x ... x R (k) where R (i) (for 1 a parts per thousand currency sign i a parts per thousand currency sign k) is a closed interval of the form a (i) , a (i) + 1] on the real line. A k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that two vertices in G are adjacent if and only if their corresponding k-cubes have a non-empty intersection. The cubicity of G is the minimum k such that G has a k-cube representation. From a geometric embedding point of view, a k-cube representation of G = (V, E) yields an embedding such that for any two vertices u and v, ||f(u) - f(v)||(a) a parts per thousand currency sign 1 if and only if . We first present a randomized algorithm that constructs the cube representation of any graph on n vertices with maximum degree Delta in O(Delta ln n) dimensions. This algorithm is then derandomized to obtain a polynomial time deterministic algorithm that also produces the cube representation of the input graph in the same number of dimensions. The bandwidth ordering of the graph is studied next and it is shown that our algorithm can be improved to produce a cube representation of the input graph G in O(Delta ln b) dimensions, where b is the bandwidth of G, given a bandwidth ordering of G. Note that b a parts per thousand currency sign n and b is much smaller than n for many well-known graph classes. Another upper bound of b + 1 on the cubicity of any graph with bandwidth b is also shown. Together, these results imply that for any graph G with maximum degree Delta and bandwidth b, the cubicity is O(min{b, Delta ln b}). The upper bound of b + 1 is used to derive upper bounds for the cubicity of circular-arc graphs, cocomparability graphs and AT-free graphs in terms of the maximum degree Delta.

On Using Nanoporous Alumina Films as Tribological Coating

Anodization of aluminum alloys is a common surface treatment procedure employed for the protection against corrosion. A thin amorphous layer of alumina is formed on the surface of alloy, which seals the alloy surface from the surrounding. This alumina layer being harder than the base aluminum alloy can be useful as a tribological coating. But since this alumina layer is randomly formed with disordered voids and pores, predicting the mechanical properties is difficult. Specific anodizing conditions can be used to form highly ordered anodic nanoporous alumina films 1] on the aluminum alloy surface. These nanoporous alumina layer can be effectively used as a tribological coating, because of the highly ordered controllable geometry and the empty pores which can be used as reservoirs for lubricant.

Calculation of Chemical Potentials and Occupancies in Clathrate Hydrates through Monte Carlo Molecular Simulations

The flexibility of the water lattice in clathrate hydrates and guest-guest interactions has been shown in previous studies to significantly affect the values of the thermodynamic properties, such as chemical potentials and free energies. Here we describe methods for computing occupancies, chemical potentials, and free energies that account for the flexibility of water lattice and guest-guest interactions in the hydrate phase. The methods are validated for a wide variety of guest molecules, such as methane, ethane, carbon dioxide, and tetrahydrodfuran by comparing the predicted occupancy values of guest molecules with those obtained from isothermal isobaric semigrand Monte Carlo simulations. The proposed methods extend the van der Waals and Platteuw theory for clathrate hydrates, and the Langmuir constant is calculated based on the structure of the empty hydrate lattice. These methods in combination with development of advanced molecular models for water and guest molecules should lead to a more thermodynamically consistent theory for clathrate hydrates.

Finding hidden females in a crowd: Mate recognition in fig wasps

Multi-species mating aggregations are crowded environments within which mate recognition must occur. Mating aggregations of fig wasps can consist of thousands of individuals of many species that attain sexual maturity simultaneously and mate in the same microenvironment, i.e, in syntopy, within the close confines of an enclosed globular inflorescence called a syconium - a system that has many signalling constraints such as darkness and crowding. All wasps develop within individual galled flowers. Since mating mostly occurs when females are still confined within their galls,, male wasps have the additional burden of detecting conspecific females that are ``hidden'' behind barriers consisting of gall walls. In Ficus racemosa, we investigated signals used by pollinating fig wasp males to differentiate conspecific females from females of other syntopic fig wasp species. Male Ceratosolen fusciceps could detect conspecific females using cues from galls containing females, empty galls, as well as cues from gall volatiles and gall surface hydrocarbons. In many figs, syconia are pollinated by single foundress wasps, leading to high levels of wasp inbreeding due to sibmating. In F. racemosa, as most syconia contain many foundresses, we expected male pollinators to prefer non-sib females to female siblings to reduce inbreeding. We used galls containing females from non-natal figs as a proxy for non-sibs and those from natal figs as a proxy for sibling females. We found that males preferred galls of female pollinators from natal figs. However, males were undecided when given a choice between galls containing non-pollinator females from natal syconia and pollinator females from non-natal syconia, suggesting olfactory imprinting by the natal syconial environment. (C) 2013 Elsevier Masson SAS. All rights reserved.

Two-point correlation function of an exclusion process with hole-dependent rates

We consider an exclusion process on a ring in which a particle hops to an empty neighboring site with a rate that depends on the number of vacancies n in front of it. In the steady state, using the well-known mapping of this model to the zero-range process, we write down an exact formula for the partition function and the particle-particle correlation function in the canonical ensemble. In the thermodynamic limit, we find a simple analytical expression for the generating function of the correlation function. This result is applied to the hop rate u(n) = 1 + (b/n) for which a phase transition between high-density laminar phase and low-density jammed phase occurs for b > 2. For these rates, we find that at the critical density, the correlation function decays algebraically with a continuously varying exponent b - 2. We also calculate the two-point correlation function above the critical density and find that the correlation length diverges with a critical exponent nu = 1/(b - 2) for b < 3 and 1 for b > 3. These results are compared with those obtained using an exact series expansion for finite systems.

Remarkable enhancement in hydrogen storage on free-standing Ti3B and BC3 supported Ti-3 clusters

Hydrogen storage capacity of Tin-1B (n = 3-7) clusters is studied and compared with that of the pristine Ti-n (n = 3-7), using density functional theory (DFT) based calculations. Among these clusters, Ti3B shows the most significant enhancement in the storage capacity by adsorbing 12 H-2, out of which three are dissociated and the other nine are stored as dihydrogen via Kubas-interaction. The best storage in Ti3B is owed to a large charge transfer from Ti to B along with the largest distance of Ti empty d-states above the Fermi level, which is a distinct feature of this particular cluster. Furthermore, the effect of substrates on the storage capacity of Ti3B was assessed by calculating the number of adsorbed H-2 on Ti-3 cluster anchored onto B atoms in the B-doped graphene, BC3, and BN substrates. Similar to free-standing Ti3B, Ti-3 anchored onto boron atom in BC3, stores nine di-hydrogen via Kubas interaction, at the same time eliminating the total number of non-useful dissociated hydrogen. Gibbs energy of adsorption as a function of H-2 partial pressure, indicated that at 250 K and 300 K the di-hydrogens on Ti-3@BC3 adsorb and desorb at ambient pressures. Importantly, Ti-3@BC3 avoids the clustering, hence meeting the criteria for efficient and reversible hydrogen storage media. Copyright (C) 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.

Improving the Rigor and Consistency of the Thermodynamic Theory for Clathrate Hydrates through Incorporation of Movement of Water Molecules of Hydrate Lattice

Current applications of statistical thermodynamic theories for clathrate hydrates do not incorporate the translational and rotational movement of water molecules of the hydrate lattice,in a rigorous manner. Previous studies have shown that the movement of water molecules has a significant effect on the properties of clathrate hydrates. In this Article, a method is presented to incorporate the effect of water movement with as much rigor as possible. This method is then used to calculate the Langmuir constant of the guest species in a clathrate hydrate. Unlike previous studies on modeling of clathrate hydrate thermodynamics, the method presented in this paper does not regress either the intermolecular potentials or the properties of the empty hydrate from clathrate phase equilibria data. Also the properties of empty hydrate used in the theory do not depend on the nature and composition of the guest molecules. The predicted phase equilibria from the resulting theory are shown to be highly accurate and thermodynamically consistent by comparing them with the phase equilibria computed directly from molecular simulations.

Two player game variant of the Erdos-Szekeres problem

The classical Erdos-Szekeres theorem states that a convex k-gon exists in every sufficiently large point set. This problem has been well studied and finding tight asymptotic bounds is considered a challenging open problem. Several variants of the Erdos-Szekeres problem have been posed and studied in the last two decades. The well studied variants include the empty convex k-gon problem, convex k-gon with specified number of interior points and the chromatic variant. In this paper, we introduce the following two player game variant of the Erdos-Szekeres problem: Consider a two player game where each player playing in alternate turns, place points in the plane. The objective of the game is to avoid the formation of the convex k-gon among the placed points. The game ends when a convex k-gon is formed and the player who placed the last point loses the game. In our paper we show a winning strategy for the player who plays second in the convex 5-gon game and the empty convex 5-gon game by considering convex layer configurations at each step. We prove that the game always ends in the 9th step by showing that the game reaches a specific set of configurations.

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces M-X (r, c(1), c(2)) parametrizing slope semistable vector bundles of rank r and fixed Chern classes c(1), c(2) on a ruled surface whose base is a rational nodal curve. We showthat under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space M-X (r, c(1), c(2)) is rational as a variety defined over R.