Bounds on Lifting Continuous-State Markov Chains to Speed Up Mixing

It is often possible to speed up the mixing of a Markov chain \(\{ X_{t} \}_{t \in \mathbb {N}}\) on a state space \(\Omega \) by lifting, that is, running a more efficient Markov chain \(\{ \widehat{X}_{t} \}_{t \in \mathbb {N}}\) on a larger state space \(\hat{\Omega } \supset \Omega \) that projects to \(\{ X_{t} \}_{t \in \mathbb {N}}\) in a certain sense. Chen et al. (Proceedings of the 31st annual ACM symposium on theory of computing. ACM, 1999) prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure of \(\{X_t\}_{t \in \mathbb {N}}\). Unfortunately, this extra factor makes the bound in Chen et al. (1999) very loose for Markov chains on large state spaces and useless for Markov chains on continuous state spaces. In this paper, we develop an extension of the evolving set method that allows us to refine this extra factor and find bounds for Markov chains on continuous state spaces that are analogous to the bounds in Chen et al. (1999). These bounds also allow us to improve on the bounds in Chen et al. (1999) for some chains on finite state spaces.     

Vanishing pseudo–Schur complements,reverse order laws,absorption laws and inheritance properties

The problem of vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implying that the other (generalized) Schur complement (corresponding to the trailing principal subblock) is zero, is revisited. Absorption laws for two important classes of generalized inverses are considered next. Inheritance properties of the generalized Schur complements in relation to the absorption laws are derived. Inheritance by the generalized principal pivot transform is also studied.     

Convex duality and the Skorokhod Problem. I

The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝ ⁿ . Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified. We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem. These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems. Received: 17 April 1998 / Revised version: 8 January 1999     

Locating the minimum: Approach to equilibrium in a disordered, symmetric zero range process

We consider the dynamics of the disordered, one-dimensional, symmetric zero range process in which a particle from an occupied site k hops to its nearest neighbor with a quenched rate w(k). These rates are chosen randomly from the probability distribution f(w) ∼ (w − c) ⁿ , where c is the lower cutoff. For n>0, this model is known to exhibit a phase transition in the steady state from a low density phase with a finite number of particles at each site to a high density aggregate phase in which the site with the lowest hopping rate supports an infinite number of particles. In the latter case, it is interesting to ask how the system locates the site with globally minimum rate. We use an argument based on the local equilibrium, supported by Monte Carlo simulations, to describe the approach to the steady state. We find that at large enough time, regions with a smooth density profile are described by a diffusion equation with site-dependent rates, while the isolated points where the mass distribution is singular act as the boundaries of these regions. Our argument implies that the relaxation time scales with the system size L as L ^z with z = 2 + 1/(n + 1) for n>1 and suggests a different behavior for n<1.     

A Spectrophotometric Determination of Ascorbic Acid

A new, simple and sensitive method for the spectrophotometric indirect determination of ascorbic acid in fruits, beverages, and pharmaceuticals is described. In this method, the ascorbic acid reduces Cu²⁺ to Cu⁺ and reacts with 2,9‐dimethyl‐1,10‐phenanothroline (neucoproine) to form Cu (neucoproine)⁺ complex, and it was extracted with N‐phenylbenzimidoylthiourea (PBITU) in chloroform. The apparent value of molar absorptivity of the complex in terms of ascorbic acid is (3.52) × 10⁴ L mole^?1 cm^?1 at λ_max, 460. The detection limit of ascorbic acid is 40 μg L^?1 and the method obeys Beer's law over the concentration range of 0.1–4.0 μg mL^?1. The proposed method was successfully applied for the determination of ascorbic acid in various samples. The validity of the present method was checked by the flow injection analysis (FIA) method.     

DABCO-catalyzed synthesis of 3-substituted-3-hydroxyindolin-2-ones in aqueous media

An efficient and greener protocol for easy access to 3-susbstituted-3-hydroxy-2-oxindoles by the reaction with various substituted isatins and acetophenones is described. This protocol is widely applicable for a variety of isatins and acetophenones using water as a reaction media and 1,4-diazabicyclo[2.2.2]octane (DABCO) as catalyst with shorter reaction time and good to excellent yield of products.     

Multi-stimuli programmable FRET based RGB absorbing antennae towards ratiometric temperature,pH and multiple metal ion sensing

A red-green-blue (RGB) multichromophoric antenna 1 consisting of energy donors naphthalimides and perylenediimides and a central aza-BODIPY energy acceptor along with two subchromophoric red-blue (RB 6) and green-blue (GB 12) antennae was designed that showed efficient cascade Förster resonance energy transfer (FRET). RGB antenna 1 showed pronounced temperature-dependent emission behaviour where emission intensities in green and red channels could be tuned in opposite directions by temperature giving rise to unique ratiometric sensing with a temperature sensitivity of 0.4% °C. RGB antenna 1 showed reversible absorption modulation selectively in the blue region (RGB ↔ RG) upon acid/base addition giving rise to pH sensing behaviour. Furthermore, RGB antenna 1 was utilized to selectively sense metal ions such as Co²⁺ and Fe³⁺ through a FRET turn-off mechanism induced by a redox process at the aza-BODIPY site that resulted in the selective spectral modulation of the red band (i.e., RGB → GB). Model antenna RB 6 showed white light emission with chromaticity coordinates (0.32, 0.33) on acid addition. Antennae 1, 6 and 12 also exhibited solution state electrochromic switching characterized by distinct colour changes upon changing the potential. Finally, antennae 1, 6 and 12 served as reversible fluorescent inks in PMMA/antenna blends whereby the emission colours could be switched or tuned using different stimuli such as acid vapour, temperature and metal ions.

RGB antennae consisting of naphthalimides, perylenediimides and aza-BODIPY with efficient FRET show unique ratiometric temperature sensing, metal sensing (FRET-off) and pH sensing through various stimuli sensitive band tuning.     

Convex duality and the Skorokhod Problem. II

In this paper we consider Skorokhod Problems on polyhedral domains with a constant and possibly oblique constraint direction specified on each face of the domain, and with a corresponding cone of constraint directions at the intersection of faces. In part one of this paper we used convex duality to develop new methods for the construction of solutions to such Skorokhod Problems, and for proving Lipschitz continuity of the associated Skorokhod Maps. The main alternative approach to Skorokhod Problems of this type is the reflection mapping technique introduced by Harrison and Reiman [8]. In this part of the paper we apply the theory developed in part one to show that the reflection mapping technique of [8] is restricted to a slight generalization of the class of problems originally considered in [8]. We further illustrate the power of the duality approach by applying it to two other classes of Skorokhod Problems – those with normal directions of constraint, and a new class that arises from a model of processor sharing in communication networks. In particular, we prove existence of solutions to and Lipschitz continuity of the Skorokhod Maps associated with each of these Skorokhod Problems. Received: 17 April 1998 / Revised: 8 January 1999     

Preparation of Fe/Pt Films with Perpendicular Magnetic Anisotropy

We have investigated the microstructures and magnetic properties of L1₀ ordered equi-atomic FePt thin films prepared by ion beam sputtering and subsequent annealing. It is observed from X-ray reflectivity and X-ray diffraction measurements that the mixing at Fe/Pt interfaces starts to occur with annealing and leading to the FePt alloy phase formation. The rapid increase in the coercivity values above 275°C, obtained from vibrating sample magnetometer (VSM) measurements, confirms the formation of the ordered L1₀ FCT FePt phase.