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**ABSTRACT:**We construct Q-operators for the open spin-1/2 XXX Heisenberg spin chain with diagonal boundary matrices. The Q-operators are defined as traces over an infinite-dimensional auxiliary space involving novel types of reflection operators derived from the boundary Yang-Baxter equation. We argue that the Q-operators defined in this way are polynomials in the spectral parameter and show that they commute with transfer matrix. Finally, we prove that the Q-operators satisfy Baxter's TQ-equation and derive the explicit form of their eigenvalues in terms of the Bethe roots.Nuclear Physics B 09/2015; DOI:10.1016/j.nuclphysb.2015.10.010 - [Show abstract] [Hide abstract]

**ABSTRACT:**We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.Advances in Mathematics 01/2015; 268:306-338. DOI:10.1016/j.aim.2014.09.023 - [Show abstract] [Hide abstract]

**ABSTRACT:**Motivated by single molecule experiments on biopolymers we explore equilibrium morphologies and force-extension behavior of copolymers with hydrophobic segments using Langevin dynamics simulations. We find that the interplay between different length scales, namely, the persistence length $\ell_{p}$, and the disorder correlation length $p$, in addition to the fraction of hydrophobic patches $f$ play a major role in altering the equilibrium morphologies and mechanical response. In particular, we show a plethora of equilibrium morphologies for this system, \textit{e.g.} core-shell, looped (with hybridised hydrophilic-hydrophobic sections), and extended coils as a function of these parameters. A competition of bending energy and hybridisation energies between two types of beads determines the equilibrium morphology. Further, mechanical properties of such polymer architectures are crucially dependent on their native conformations, and in turn on the disorder realisation along the chain backbone. Thus, for flexible chains, a globule to extended coil transition is effected via a tensile force for all disorder realisations. However, the exact nature of the force-extension curves are different for the different disorder realisations. In contrast, we find that force-extension behavior of semi-flexible chains with different equilibrium configurations \textit{e.g.} core-shell, looped, \textit{etc.} reveal a cascade of force-induced conformational transitions.Macromolecular Theory and Simulations 05/2014; 23(4). DOI:10.1002/mats.201300154 - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ is the non-negative integers and $S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of $X_n$, and that, roughly speaking, $\eta_n$ is close to being Markov when $X_n$ is large. This departure from much of the literature, which assumes that $\eta_n$ is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for $X_n$ given $\eta_n$. We give a recurrence classification in terms of increment moment parameters for $X_n$ and the stationary distribution for the large-$X$ limit of $\eta_n$. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between $X_n$ (rescaled) and $\eta_n$. Our results can be seen as generalizations of Lamperti's results for non-homogeneous random walks on $\mathbb{Z}_+$ (the case where $S$ is a singleton). Motivation arises from modulated queues or processes with hidden variables where $\eta_n$ tracks an internal state of the system.Stochastic Processes and their Applications 02/2014; 124(10). DOI:10.1016/j.spa.2014.05.005 - [Show abstract] [Hide abstract]

**ABSTRACT:**Convection induced by the selective absorption of radiation is investigated, where the internal heat source is concentration dependent. Regions of very large subcritical instabilities, i.e. where agreement between the linear instability thresholds and nonlinear stability thresholds is poor, are studied by solving for the full three-dimensional system. The results indicate that linear theory is very accurate in predicting the onset of convective motion, and thus, regions of stability.Applied Mathematics and Computation 01/2014; 227:92–101. DOI:10.1016/j.amc.2013.11.007 - [Show abstract] [Hide abstract]

**ABSTRACT:**The purpose of this paper is to study the effect of a heat source on the solution to the equations for an incompressible heat conducting viscous fluid. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using three dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the steady state increases significantly as the Rayleigh number tends to the linear threshold.International Journal of Engineering Science 01/2014; 74:91–102. DOI:10.1016/j.ijengsci.2013.08.011 - [Show abstract] [Hide abstract]

**ABSTRACT:**The challenge of understanding complex systems often gives rise to a mul-tiplicity of models. It is natural to consider whether the outputs of these models can be combined to produce a system prediction that is more infor-mative than the output of any one of the models taken in isolation. And, in particular, to consider the relationship between the spread of model out-puts and system uncertainty. We describe a statistical framework for such a combination, based on the exchangeability of the models, and their co-exchangeability with the system. We demonstrate the simplest implemen-tation of our framework in the context of climate prediction. Throughout we work entirely in means and variances, to avoid the necessity of specifying higher-order quantities for which we often lack well-founded judgements.Journal of the American Statistical Association 09/2013; 108(503). DOI:10.1080/01621459.2013.802963 - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider parity-odd transport in 2+1 dimensional charged fluids restricting attention to the class of non-dissipative fluids. We show that there is a two parameter family of such non-dissipative fluids which can be derived from an effective action, in contradistinction with a four parameter family that can be derived from an entropy current analysis. The effective action approach allows us to extract the adiabatic transport data, in particular the Hall viscosity and Hall conductivity amongst others, in terms of the thermodynamic functions that enter as 'coupling constants'. Curiously, we find that Hall viscosity is forced to vanish, whilst the Hall conductivity is generically a non-vanishing function of thermodynamic data determined in terms of the hydrodynamic couplings.Journal of High Energy Physics 05/2013; 2013(10). DOI:10.1007/JHEP10(2013)074 - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the random walk in a random environment on Z+={0,1,2,…}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)β, for β∈(1,∞), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution.Stochastic Processes and their Applications 04/2013; DOI:10.1016/j.spa.2007.04.011 - [Show abstract] [Hide abstract]

**ABSTRACT:**The theory of classically integrable nonlinear wave equations, and the Bethe Ansatz systems describing massive quantum field theories defined on an infinite cylinder, are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper we shall describe this link for the case of the classical and quantum versions of the (Tzitz\'eica-)Bullough-Dodd model.Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences 03/2013; 371(1989):20120052. DOI:10.1098/rsta.2012.0052

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