Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

Padé approach to potential transients: Part 1. Electron transfer without and with coupling to first-order homogeneous reactions at planar electrodes

Potential transients are obtained by using “Padé approximants” (an accurate approximation procedure valid globally — not just perturbatively) for all amplitudes of concentration polarization and current densities. This is done for several mechanistic schemes under constant current conditions. We invert the non-linear current-potential relationship in the form (using the Lagrange or the Ramanujan method) of power series appropriate to the two extremes, namely near reversible and near irreversible. Transforming both into the Pad́e expressions, we construct the potential-time profile by retaining whichever is the more accurate of the two. The effectiveness of this method is demonstrated through illustrations which include couplings of homogeneous chemical reactions to the electron-transfer step.

Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence with polymer additives

We carry out a direct numerical simulation (DNS) study that reveals the effects of polymers on statistically steady, forced, homogeneous, and isotropic fluid turbulence. We find clear manifestations of dissipation-reduction phenomena: on the addition of polymers to the turbulent fluid, we obtain a reduction in the energy dissipation rate; a significant modification of the fluid-energy spectrum, especially in the deep-dissipation range; and signatures of the suppression of small-scale structures, including a decrease in small-scale vorticity filaments. We also compare our results with recent experiments and earlier DNS studies of decaying fluid turbulence with polymer additives.

Instability and dewetting of ultrathin solid viscoelastic films on homogeneous and heterogeneous substrates

Instability and dewetting engendered by the van der Waals force in soft thin (<100 nm) linear viscoelastic solid (e. g., elastomeric gel) films on uniform and patterned surfaces are explored. Linear stability analysis shows that, although the elasticity of the film controls the onset of instability and the corresponding critical wavelength, the dominant length-scale remains invariant with the elastic modulus of the film. The unstable modes are found to be long-wave, for which a nonlinear long-wave analysis and simulations are performed to uncover the dynamics and morphology of dewetting. The stored elastic energy slows down the temporal growth of instability significantly. The simulations also show that a thermodynamically stable film with zero-frequency elasticity can be made unstable in the presence of physico-chemical defects on the substrate and can follow an entirely different pathway with far fewer holes as compared to the viscous films. Further, the elastic restoring force can retard the growth of a depression adjacent to the hole-rim and thus suppress the formation of satellite holes bordering the primary holes. These findings are in contrast to the dewetting of viscoelastic liquid films where nonzero frequency elasticity accelerates the film rupture and promotes the secondary instabilities. Thus, the zero-frequency elasticity can play a major role in imposing a better-defined long-range order to the dewetted structures by arresting the secondary instabilities. (C) 2011 American Institute of Physics. doi: 10.1063/1.3554748]

A classification of homogeneous operators in the Cowen-Douglas class

An explicit construction of all the homogeneous holomorphic Hermitian vector bundles over the unit disc D is given. It is shown that every such vector bundle is a direct sum of irreducible ones. Among these irreducible homogeneous holomorphic Hermitian vector bundles over D, the ones corresponding to operators in the Cowen-Douglas class B-n(D) are identified. The classification of homogeneous operators in B-n(D) is completed using an explicit realization of these operators. We also show how the homogeneous operators in B-n(D) split into similarity classes. (C) 2011 Elsevier Inc. All rights reserved.

Systematics of the magnetic-Prandtl-number dependence of homogeneous, isotropic magnetohydrodynamic turbulence

We present the results of our detailed pseudospectral direct numerical simulation (DNS) studies, with up to 1024(3) collocation points, of incompressible, magnetohydrodynamic (MHD) turbulence in three dimensions, without a mean magnetic field. Our study concentrates on the dependence of various statistical properties of both decaying and statistically steady MHD turbulence on the magnetic Prandtl number Pr-M over a large range, namely 0.01 <= Pr-M <= 10. We obtain data for a wide variety of statistical measures, such as probability distribution functions (PDFs) of the moduli of the vorticity and current density, the energy dissipation rates, and velocity and magnetic-field increments, energy and other spectra, velocity and magnetic-field structure functions, which we use to characterize intermittency, isosurfaces of quantities, such as the moduli of the vorticity and current density, and joint PDFs, such as those of fluid and magnetic dissipation rates. Our systematic study uncovers interesting results that have not been noted hitherto. In particular, we find a crossover from a larger intermittency in the magnetic field than in the velocity field, at large Pr-M, to a smaller intermittency in the magnetic field than in the velocity field, at low Pr-M. Furthermore, a comparison of our results for decaying MHD turbulence and its forced, statistically steady analogue suggests that we have strong universality in the sense that, for a fixed value of Pr-M, multiscaling exponent ratios agree, at least within our error bars, for both decaying and statistically steady homogeneous, isotropic MHD turbulence.

Interaction analysis of strip footing resting on a non-homogeneous elastic medium

In this paper, a plane stress solution for the interaction analysis of strip footing resting on (i) a non-homogeneous elastic half-plane and (ii) a non-homogeneous elastic layer resting on a rigid stratum has been presented. The analysis has been done using a combined analytical and FEM method in which the discretization of the half-plane is not required and thereby minimizes the computational efforts considerably. The contact pressure distribution and the settlement profile for the selected cases of varying modulus half-plane, which has more relevance to foundation engineering, have been given. Experimental verification through a photoelastic method of stress analysis has been carried out for the case of footing on Gibson elastic half-plane, and the contact pressure distribution thus obtained has been compared with the theoretical results. Copyright (C) 1996 Elsevier Science Ltd

Covalent Grafting of Polydimethylsiloxane over Surface-Modified Alumina Nanoparticles

The synthesis of ``smart structured'' conducting polymers and the fabrication of devices using them are important areas of research. However, conducting polymeric materials that are used in devices are susceptible to degradation due to oxygen and moisture. Thus, protection of such devices to ensure long-term stability is always desirable. Polymer nanocomposites are promising materials for the encapsulation of such devices. Therefore, it is important to develop suitable polymer nanocomposites as encapsulation materials to protect such devices. This work presents a technique based on grafting between surface-decorated gamma-alumina nanoparticles and polymer to make nanocomposites that can be used for the encapsulation of devices. Alumina was functionalized with allyltrimethoxysilane and used to conjugate polymer molecules (hydride-terminated polydimethylsiloxane) through a platinum-catalyzed hydrosilylation reaction. Fourier transform infrared spectroscopy, X-ray-photoelectron spectroscopy, and Raman spectroscopy were used to characterize the surface chemistry of the nanoparticles after surface modification. The grafting density of alkene groups on the surface of the modified nanoparticles was calculated using CHN and thermogravimetric analyses. The thermal stability of the composites was also evaluated using thermogravimetric analysis. The nanoindentation technique was used to analyze the mechanical characteristics of the composites. The densities of the composites were evaluated using a density gradient column, and the morphology of the composites was evaluated by scanning electron microscopy. All of our studies reveal that the composites have good thermal stability and mechanical flexibility and, thus, can potentially be used for the encapsulation of organic photovoltaic devices.

Oxidation of aqueous sulphur dioxide catalysed by poly-4-vinylpyridine-Cu(II) complex .2. Combined homogeneous and heterogeneous phase oxidation

Aqueous phase oxidation of sulphur dioxide at low concentrations catalysed by a PVP-Cu complex in the solid phase and dissolved Cu(II) in the liquid phase is studied in a rotating catalyst basket reactor (RCBR). The equilibrium adsorption of Cu(II) and S(VI) on PVP particles is found to be of the Langmuir-type. The diffusional effects of S(IV) species in PVP-Cu resin are found to be insignificant whereas that of product S(VI) are found to be significant. The intraparticle diffusivity of S(VI) is obtained from independent tracer experiments. In the oxidation reaction HSO3- is the reactive species. Both the S(IV) species in the solution, namely SO2(aq) and HSO3- get adsorbed onto the active PVP-Cu sites of the catalyst, but only HSO3- undergoes oxidation. A kinetic mechanism is proposed based on this feature which shows that SO2(aq) has a deactivating effect on the catalyst. A rate model is developed for the three-phase reaction system incorporating these factors along with the effect of concentration of H2SO4 on the solubility of SO2 in the dilute aqueous solutions of Cu(II). Transient oxidation experiments are conducted at different conditions of concentration of SO2 and O-2 in the gas phase and catalyst concentration, and the rate parameters are estimated from the data. The observed and calculated profiles are in very good agreement. This confirms the deactivating effect of nonreactive SO2(aq) on the heterogeneous catalysis.

Some recent advances in the theory of homogeneous isotropic turbulence

We review some advances in the theory of homogeneous, isotropic turbulence. Our emphasis is on the new insights that have been gained from recent numerical studies of the three-dimensional Navier Stokes equation and simpler shell models for turbulence. In particular, we examine the status of multiscaling corrections to Kolmogorov scaling, extended self similarity, generalized extended self similarity, and non-Gaussian probability distributions for velocity differences and related quantities. We recount our recent proposal of a wave-vector-space version of generalized extended self similarity and show how it allows us to explore an intriguing and apparently universal crossover from inertial- to dissipation-range asymptotics.

Axially homogeneous, zero mean flow buoyancy-driven turbulence in a vertical pipe

We report an experimental study of a new type of turbulent flow that is driven purely by buoyancy. The flow is due to an unstable density difference, created using brine and water, across the ends of a long (length/diameter=9) vertical pipe. The Schmidt number Sc is 670, and the Rayleigh number (Ra) based on the density gradient and diameter is about 108. Under these conditions the convection is turbulent, and the time-averaged velocity at any point is ‘zero’. The Reynolds number based on the Taylor microscale, Reλ, is about 65. The pipe is long enough for there to be an axially homogeneous region, with a linear density gradient, about 6–7 diameters long in the midlength of the pipe. In the absence of a mean flow and, therefore, mean shear, turbulence is sustained just by buoyancy. The flow can be thus considered to be an axially homogeneous turbulent natural convection driven by a constant (unstable) density gradient. We characterize the flow using flow visualization and particle image velocimetry (PIV). Measurements show that the mean velocities and the Reynolds shear stresses are zero across the cross-section; the root mean squared (r.m.s.) of the vertical velocity is larger than those of the lateral velocities (by about one and half times at the pipe axis). We identify some features of the turbulent flow using velocity correlation maps and the probability density functions of velocities and velocity differences. The flow away from the wall, affected mainly by buoyancy, consists of vertically moving fluid masses continually colliding and interacting, while the flow near the wall appears similar to that in wall-bound shear-free turbulence. The turbulence is anisotropic, with the anisotropy increasing to large values as the wall is approached. A mixing length model with the diameter of the pipe as the length scale predicts well the scalings for velocity fluctuations and the flux. This model implies that the Nusselt number would scale as Ra1/2Sc1/2, and the Reynolds number would scale as Ra1/2Sc−1/2. The velocity and the flux measurements appear to be consistent with the Ra1/2 scaling, although it must be pointed out that the Rayleigh number range was less than 10. The Schmidt number was not varied to check the Sc scaling. The fluxes and the Reynolds numbers obtained in the present configuration are much higher compared to what would be obtained in Rayleigh–Bénard (R–B) convection for similar density differences.

Grafting of m-isopropenyl-alpha, alpha-dimethylbenzyl-isocyanate (m-TMI) onto isotactic polypropylene: Synthesis and characterization

A novel vinyl monomer with an isocyanate functional group, m-isopropenyl-alpha,alpha-dimethylbenzyl-isocyanate (m-TMI), was grafted onto isotactic polypropylene (i-PP) using dicumyl peroxide (DCP) as the initiator. This would open up the possibility of using the grafted polymer with the reactive isocyanate group as compatibilizer for blending carbohydrates such as cellulose with. polypropylene. The grafting was carried out in a Brabender Plasticoder at 180degreesC. The effects of monomer and initiator concentrations on the yield of grafting were investigated by performing statistical analysis. While the grafting yield increased with the concentration of DCP at any given concentration of m-TMI, the variation of the grafting yield with m-TMI concentration, for a given concentration of DCP, went through a maximum, the optimum yield of 7.8% (w/w) being obtained at 10 wt.% concentration of both DCP and m-TMI. The grafting reaction is. accompanied by considerable chain scission of I-PP, resulting in a decrease in the molecular weight of the grafted polymer. While the molecular weight drops sharply even at a low concentration of DCP, there occurs no further significant change in the molecular weight even at much higher concentrations of the initiator.