Raman investigation of hydrostatic and nonhydrostatic compressions of OH‐ and F‐apophyllites up to 8 GPa

Layer silicates F‐ and OH‐apophyllites, KCa₄Si₈O₂₀(F, OH)·8H₂O, have been investigated by Raman spectroscopy at hydrostatic and nonhydrostatic pressures up to ~8 GPa in diamond anvil cells using a 4 : 1 methanol–ethanol mix as pressure‐transmitting medium. Our experiments show that at hydrostatic compression, apophyllites retain their crystalline states (i.e. no amorphization) up to 5 GPa. The wavenumbers of most bands exhibit linear dependences on pressure, except for a few ones, e.g. at 162 and 3565 cm^–1 in OH‐form (160.5 and 3558 cm^–1 in F‐form) that show nonlinear dependences. Nonhydrostatic compression with additional uniaxial loading induces amorphization of apophyllites. The majority of the bands in OH‐apophyllite decreases markedly in intensity and shows considerable broadening under nonhydrostatic compression up to 3–6 GPa. In addition, the wavenumbers of several bands at nonhydrostatic compression exhibit considerable nonlinear dependences on pressure with strong hysteresis. These bands are mainly associated with vibrations of the interlayer ions and molecules and also of stretching and bending silicate sheets, hence being highly sensitive to the interlayer distance. Finally, we have calculated the lattice dynamics of F‐apophyllite and interpreted the majority of bands, and these data are used to explain the complex baric behavior of the bands. Copyright © 2011 John Wiley & Sons, Ltd.     

Logical omniscience as infeasibility

Logical theories for representing knowledge are often plagued by the so-called Logical Omniscience Problem. The problem stems from the clash between the desire to model rational agents, which should be capable of simple logical inferences, and the fact that any logical inference, however complex, almost inevitably consists of inference steps that are simple enough. This contradiction points to the fruitlessness of trying to solve the Logical Omniscience Problem qualitatively if the rationality of agents is to be maintained. We provide a quantitative solution to the problem compatible with the two important facets of the reasoning agent: rationality and resource boundedness. More precisely, we provide a test for the logical omniscience problem in a given formal theory of knowledge. The quantitative measures we use are inspired by the complexity theory. We illustrate our framework with a number of examples ranging from the traditional implicit representation of knowledge in modal logic to the language of justification logic, which is capable of spelling out the internal inference process. We use these examples to divide representations of knowledge into logically omniscient and not logically omniscient, thus trying to determine how much information about the reasoning process needs to be present in a theory to avoid logical omniscience.     

On some extremal problems of approximation theory of functions on the real axis I

A number of extremal problems of approximation theory of functions have been solved on the real line $ \mathbb{R} $ . Exact constants in the Jackson-type inequalities in the space L ₂( $ \mathbb{R} $ ) are calculated. The exact values of average ν-widths are obtained for the classes of functions from L ₂( $ \mathbb{R} $ ) defined by averaged moduli of continuity of the k-th order, as well as for the classes of functions defined by K-functionals. The quite complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and for the whole real axis is carried out in the chronological order.     

Nonmonotone travelling waves in a single species reaction-diffusion equation with delay

We prove the existence of a continuous family of positive and generally nonmonotone travelling fronts for delayed reaction-diffusion equations , when g∈C²(R₊,R₊) has exactly two fixed points: x₁=0 and x₂=K>0. Recently, nonmonotonic waves were observed in numerical simulations by various authors. Here, for a wide range of parameters, we explain why such waves appear naturally as the delay h increases. For the case of g with negative Schwarzian, our conditions are rather optimal; we observe that the well known Mackey-Glass-type equations with diffusion fall within this subclass of (∗). As an example, we consider the diffusive Nicholson's blowflies equation.     

Link Homologies and the Refined Topological Vertex

We establish a direct map between refined topological vertex and sl(N) homological invariants of the of Hopf link, which include Khovanov-Rozansky homology as a special case. This relation provides an exact answer for homological invariants of the Hopf link, whose components are colored by arbitrary representations of sl(N). At present, the mathematical formulation of such homological invariants is available only for the fundamental representation (the Khovanov-Rozansky theory) and the relation with the refined topological vertex should be useful for categorizing quantum group invariants associated with other representations (R ₁, R ₂). Our result is a first direct verification of a series of conjectures which identifies link homologies with the Hilbert space of BPS states in the presence of branes, where the physical interpretation of gradings is in terms of charges of the branes ending on Lagrangian branes.     

KAM theory and the 3D Euler equation

We prove that the dynamical system defined by the hydrodynamical Euler equation on any closed Riemannian 3-manifold M   is not mixing in the C^k$C^k$ topology (k>4$k>4$ and non-integer) for any prescribed value of helicity and sufficiently large values of energy. This can be regarded as a 3D version of Nadirashvili's and Shnirelman's theorems showing the existence of wandering solutions for the 2D Euler equation. Moreover, we obtain an obstruction for the mixing under the Euler flow of C^k$C^k$-neighborhoods of divergence-free vectorfields on M  . On the way we construct a family of functionals on the space of divergence-free C¹$C^1$ vectorfields on the manifold, which are integrals of motion of the 3D Euler equation. Given a vectorfield these functionals measure the part of the manifold foliated by ergodic invariant tori of fixed isotopy types. We use the KAM theory to establish some continuity properties of these functionals in the C^k$C^k$-topology. This allows one to get a lower bound for the C^k$C^k$-distance between a divergence-free vectorfield (in particular, a steady solution) and a trajectory of the Euler flow.     

Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.      

ATR-FTIR imaging for the analysis of organic materials in paint cross sections: case studies on paint samples from the National Gallery, London

The potential of attenuated total reflection Fourier transform infrared (ATR-FTIR) imaging for the characterisation of the chemical components of paint cross sections from old master paintings was investigated. Three cross sections were chosen to cover a variety of the analytical problems encountered in samples from paintings. The binding medium and degradation products in a green paint sample from a fifteenth-century Florentine painting were imaged, as well as a thin layer within a cross-section from a fifteenth-century German painting, and multiple thin surface coatings on a painting of the 1760s by Peter Romney. The application of chemometric methods for further analysis of the large data set generated for each sample was also explored. The study demonstrated the advantages of ATR-FTIR imaging, which allowed images to be obtained with high spatial resolution (ca. 3-4 microm) without the need to microtome the sample. The gain in sensitivity in detecting trace materials and the information derived from the location of these compounds in the sample was especially valuable, improving interpretation of the FTIR analysis and extending knowledge of the sample composition beyond that obtainable with other analytical techniques.     

High-throughput study of poly(ethylene glycol)/ibuprofen formulations under controlled environment using FTIR imaging

Simultaneous analysis of many samples under identical conditions improves the effectiveness of research and accelerates product design. A novel spectroscopic imaging approach using a multichannel detector has been developed for parallel analysis of pharmaceutical formulations under controlled environments. Samples of formulations of ibuprofen in poly(ethylene glycol) have been prepared with ibuprofen concentrations ranging from 0 to 100% using a microdroplet deposition approach. The concentration of ibuprofen in PEG at which dimerization of ibuprofen molecules can be avoided has been determined via simultaneous measurement of all samples using in situ FTIR spectroscopic imaging. FTIR spectra from all samples have been analyzed to assess the molecular state of the drug and the degree of polymer swelling as a function of drug concentration. The effect of elevated temperature on the stability of all formulations was also studied. This high-throughput approach identified the concentration range for stable formulations and provided evidence that hydrogen bonding between ibuprofen and the polymer is responsible for enhanced stability at higher temperatures. This high-throughput imaging approach, based on a miniature sampling system, significantly reduces the experimental time by allowing many (potentially a few thousand) experiments to be run in parallel and increases the accuracy by minimizing variations between experiments.