A note on the slow rotation of a disc in a bounded fluid with a surfactant surface layer

A method involving eigenfunction expansion and collocation is employed to solve the axisymmetric problem of a slowly and steadily rotating circular disc in a fluid of finite extent whose surface is covered with a surfactant film. The present method (originally described by Wang (Acta Mech. 94, 97, 1992)) is observed to produce results of practical importance associated with the problem more quickly and more easily than the one used earlier by Shail and Gooden (Int. J. Multiphase Flow 7, 245, 1992). (C) 1994 Academic Press, Inc.

Average lattices and aperiodic structures

Statistically averaged lattices provide a common basis to understand the diffraction properties of structures displaying deviations from regular crystal structures. An average lattice is defined and examples are given in one and two dimensions along with their diffraction patterns. The absence of periodicity in reciprocal space corresponding to aperiodic structures is shown to arise out of different projected spacings that are irrationally related, when the grid points are projected along the chosen coordinate axes. It is shown that the projected length scales are important factors which determine the existence or absence of observable periodicity in the diffraction pattern more than the sequence of arrangement.

Surfaces of Bounded Mean Curvature In Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F subset of M, together with bounds on the geometry of M, give an upper bound on the diameter of F. Our proof is modelled on Gromov's compactness theorem for J-holomorphic curves.

Bounded Unpopularity Matchings

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching M is popular if there is no matching M' such that more people prefer M' to M than the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied by Abraham et al. (SIAM J. Comput. 37(4):1030-1045, 2007). If there is no popular matching, a reasonable substitute is a matching whose unpopularity is bounded. We consider two measures of unpopularity-unpopularity factor denoted by u(M) and unpopularity margin denoted by g(M). McCutchen recently showed that computing a matching M with the minimum value of u(M) or g(M) is NP-hard, and that if G does not admit a popular matching, then we have u(M) >= 2 for all matchings M in G. Here we show that a matching M that achieves u(M) = 2 can be computed in O(m root n) time (where m is the number of edges in G and n is the number of nodes) provided a certain graph H admits a matching that matches all people. We also describe a sequence of graphs: H = H(2), H(3), ... , H(k) such that if H(k) admits a matching that matches all people, then we can compute in O(km root n) time a matching M such that u(M) <= k - 1 and g(M) <= n(1 - 2/k). Simulation results suggest that our algorithm finds a matching with low unpopularity in random instances.

Repulsive fermions in optical lattices: Phase separation versus coexistence of antiferromagnetism and d-wave superfluidity

We investigate a system of fermions on a two-dimensional optical square lattice in the strongly repulsive coupling regime. In this case, the interactions can be controlled by laser intensity as well as by Feshbach resonance. We compare the energetics of states with resonating valence bond d-wave superfluidity, antiferromagnetic long-range order, and a homogeneous state with coexistence of superfluidity and antiferromagnetism. Using a variational formalism, we show that the energy density of a hole e(hole)(x) has a minimum at doping x = x(c) that signals phase separation between the antiferromagnetic and d-wave paired superfluid phases. The energy of the phase-separated ground state is, however, found to be very close to that of a homogeneous state with coexisting antiferromagnetic and superfluid orders. We explore the dependence of the energy on the interaction strength and on the three-site hopping terms and compare with the nearest-neighbor hopping t-J model.

Density matrix renormalization group algorithm for Bethe lattices of spin-1/2 or spin-1 sites with Heisenberg antiferromagnetic exchange

A density matrix renormalization group (DMRG) algorithm is presented for the Bethe lattice with connectivity Z = 3 and antiferromagnetic exchange between nearest-neighbor spins s = 1/2 or 1 sites in successive generations g. The algorithm is accurate for s = 1 sites. The ground states are magnetic with spin S(g) = 2(g)s, staggered magnetization that persists for large g > 20, and short-range spin correlation functions that decrease exponentially. A finite energy gap to S > S(g) leads to a magnetization plateau in the extended lattice. Closely similar DMRG results for s = 1/2 and 1 are interpreted in terms of an analytical three-site model.

Product Dimension of Forests and Bounded Treewidth Graphs

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has product dimension at most 1.441 log n + 3. This improves the best known upper bound of 3 log n for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a well-known result on the existence of orthogonal Latin squares to show that every graph on n vertices with treewidth at most t has product dimension at most (t + 2) (log n + 1). We also show that every k-degenerate graph on n vertices has product dimension at most inverted right perpendicular5.545 k log ninverted left perpendicular + 1. This improves the upper bound of 32 k log n for the same by Eaton and Rodl.

Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications

A series expansion for Heckman-Opdam hypergeometric functions phi(lambda) is obtained for all lambda is an element of alpha(C)*. As a consequence, estimates for phi(lambda) away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The L-P-theory for the hypergeometric Fourier transform is developed for 0 < p < 2. In particular, an inversion formula is proved when 1 <= p < 2. (C) 2013 Elsevier Inc. All rights reserved.

Frustration in Optical Lattices of Interacting Bosons

The realization of optical lattices of cold atoms has opened up the possibility of engineering interacting lattice systems of bosons and fermions, stimulating a frenzy of research over the last decade. More recently, experimental techniques have been developed to apply synthetic gauge fields to these optical lattices. As a result, it has become possible to study quantum Hall physics and the effects of frustration in lattices of cold atoms. In this article we describe the combined effect of frustration and interactions on the superfluidity of bosons. By focussing on a frustrated ladder of interacting bosons, we show that the effect of frustration is for ``chiral'' order to develop, which manifests itself as an alternating pattern of circulating supercurrents. Remarkably, this order persists even when superfluidity is lost and the system enters a Mott phase giving rise to a novel chiral Mott insulator. We describe the combined physics of frustration and interactions by studying a fully frustrated one dimensional model of interacting bosons. The model is studied using mean-field theory, a direct quantum simulation and a higher dimensional classical theory in order to offer a full description of the different quantum phases contained in it and transitions between the different phases. In addition, we provide physical descriptions of the chiral Mott insulator as a vortex-anitvortex super solid and indirect excitonic condensate in addition to obtaining a variational wavefunction for it. We also briefly describe the chiral Mott states arising in other microscopic models.

PROPER HOLOMORPHIC MAPS BETWEEN BOUNDED SYMMETRIC DOMAINS REVISITED

We prove that a proper holomorphic map between two nonplanar bounded symmetric domains of the same dimension, one of them being irreducible, is a biholomorphism. Our methods allow us to give a single, all-encompassing argument that unifies the various special cases in which this result is known. We discuss an application of these methods to domains having noncompact automorphism groups that are not assumed to act transitively.

SEPARATION DIMENSION OF BOUNDED DEGREE GRAPHS

The separation dimension of a graph G is the smallest natural number k for which the vertices of G can be embedded in R-k such that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family F of total orders of the vertices of G such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on n vertices is Theta(log n). In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree d is at most 2(9) (log*d)d. We also demonstrate that the above bound is nearly tight by showing that, for every d, almost all d-regular graphs have separation dimension at least d/2]

Homogeneous Hermitian holomorphic vector bundles and the Cowen-Douglas class over bounded symmetric domains

It is known that all the vector bundles of the title can be obtained by holomorphic induction from representations of a certain parabolic group on finite-dimensional inner product spaces. The representations, and the induced bundles, have composition series with irreducible factors. We write down an equivariant constant coefficient differential operator that intertwines the bundle with the direct sum of its irreducible factors. As an application, we show that in the case of the closed unit ball in C-n all homogeneous n-tuples of Cowen-Douglas operators are similar to direct sums of certain basic n-tuples. (c) 2015 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Kinetic and Size Control of Polystyrene and Polyacrylic Octadecyl Ester Lattices Via Polymerization in O/W Microemulsions

The kinetic studies of the acrylic octadecyl ester and styrene polymerization in microemulsion systems, (1) cetyl pyridine bromide (CPDB)/t-butanol/styrene/water; (2) CPDB/t-butanol/toluene + acrylic octadecyl ester (1:1, w/v)/ water; (3) cetyl pyridine bromide/styrene/formamide, were made by using dynamic laser light scattering techniques (DLS). The mechanisms of nucleation of latex particles were discussed. The most possible nucleation location of the styrene and acrylic octadecyl ester microlatex particles in aqueous microemulsion system is in aqueous phase via homogeneous nucleation. Meanwhile, parts of microlatex particles are possibly produced via swollen micelles (microemulsions) and monomer droplets nucleation. On the other hand, the most possible nucleation location of the styrene microlatex particles in nonaqueous microemulsion system is inside monomer droplets. The relationship between the amount of monomer and the size of microlatex was also investigated. It has been found that the size of microlatex particles could be controlled by changing the amount of monomer. (C) 2002 Elsevier Science B.V. All rights reserved.

Optimal bounded control of first-passage failure of quasi-integrable Hamiltonian systems with wide-band random excitation

The optimal bounded control of quasi-integrable Hamiltonian systems with wide-band random excitation for minimizing their first-passage failure is investigated. First, a stochastic averaging method for multi-degrees-of-freedom (MDOF) strongly nonlinear quasi-integrable Hamiltonian systems with wide-band stationary random excitations using generalized harmonic functions is proposed. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximizinig reliability and maximizing mean first-passage time are formulated based on the averaged It$\ddot{\rm o}$ equations by applying the dynamical programming principle. The optimal control law is derived from the dynamical programming equations and control constraints. The relationship between the dynamical programming equations and the backward Kolmogorov equation for the conditional reliability function and the Pontryagin equation for the conditional mean first-passage time of optimally controlled system is discussed. Finally, the conditional reliability function, the conditional probability density and mean of first-passage time of an optimally controlled system are obtained by solving the backward Kolmogorov equation and Pontryagin equation. The application of the proposed procedure and effectiveness of control strategy are illustrated with an example.