Stochastic convergence of weighted sums in normed linear spaces

Let X be a (real) separable Banach space, let {V_k} be a sequence of random elements in X, and let {a_nk} be a double array of real numbers such that lim_n→∞ a_nk = 0 for all k and Σ^∞_k=1 |a_nk| ≤ 1 for all n. Define S_n = Σⁿ_k=1 a_nk(V_k − EV_k). The convergence of {S_n} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {V_k}. Convergence with probability one for {S_n} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {a_nk} but without distribution conditions for the random elements {V_k}. Finally, examples of arrays {a_nk}, spaces, and applications of these results are considered.     

On the Stereoselectivity of Fluorine and Acetylhypofluorite Additions to Glycals: The Synthesis of 2-Deoxy-2-Fldorohexoses

Electrophilic syn additions of fluorine and acetylhypofluorite across double bonds in 3,4,6-tri-0-acetyl-D-glucal (1a) and D-glucal (1b) followed by acid hydrolysis gave mixtures of 2-deoxy-2-fluoro-D-glucose (8) and 2-deoxy-2-fluoro-D-mannose (9). These addition reactions were conducted in various solvents with a view to investigating the reaction mechanism based on the product distribution analysis by ¹⁹F NMR. Tight ion pair intermediates (4 and 5) have been invoked to explain the stereospecific characteristics of the addition of fluorine or acetylhypofluorite to glycals. The relative stabilities of these intermediates control the product distributions and are governed by a) the anomeric effect (axial vs equatorial preference of C(1) electronegative substituents in pyranose rings), b) dipole-dipole interactions of the lone pairs of electrons on the ring oxygen and the electronegative substituents on C(2), and c) the gauche relationship that exists between the C(2) fluorine and polar groups in the molecule. The overall contribution of these three factors largely depends upon the polarity of the solvent. A rationale for the ¹⁹F NMR chemical shifts as well as the anomeric distributions of the α and β anomers of 2-FDG (8) and 2-FDM (9) has been proposed.     

Nonuniform Crank‐Nicolson scheme for solving the stochastic Kawarada equation via arbitrary grids

This article studies a nonuniform finite difference method for solving the degenerate Kawarada quenching‐combustion equation with a vibrant stochastic source. Arbitrary grids are introduced in both space and time via adaptive principals to accommodate the uncertainty and singularities involved. It is shown that, under proper constraints on mesh step sizes, the positivity, monotonicity of the solution, and numerical stability of the scheme developed are well preserved. Numerical experiments are given to illustrate our conclusions. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1305–1328, 2017     

Fractality of light's darkness

Natural light fields are threaded by lines of darkness. For monochromatic light, the phenomenon is familiar in laser speckle, i.e., the black points that appear in the scattered light. These black points are optical vortices that extend as lines throughout the volume of the field. We establish by numerical simulations, supported by experiments, that these vortex lines have the fractal properties of a Brownian random walk. Approximately 73% of the lines percolate through the optical beam, the remainder forming closed loops. Our statistical results are similar to those of vortices in random discrete lattice models of cosmic strings, implying that the statistics of singularities in random optical fields exhibit universal behavior.     

Hydrogenation of

The course of the hydrogenation of [5]- and [6]metacyclophane (1b and 1c) and their thermochemistry is described. Both compounds are hydrogenated rapidly (within 10 s) to furnish the bridgehead olefins 13b and 12c. The accompanying hydrogenation enthalpies are -220 and -141 kJmol(-1), respectively. Strain energies (SE) and olefinic strains (OS) of a number of bridgehead olefins have been evaluated by DFT calculations; it was concluded that 13b belongs to the class of hyperstable olefins which correlates nicely with its reluctance to undergo hydrogenation. By combining experimental hydrogenation enthalpies and DFT calculations, SE of 187 and 121 kJmol(-1) were derived for 1b and 1c.     

Efficient generation of Bessel beam arrays by means of an SLM

We use a Spatial Light Modulator (SLM) to produce arrays of Bessel beams by using multiple axicon phase-masks on the SLM. This approach utilises the whole of the SLM, rather than just a thin annular region (which is the case if the SLM is in the far-field of the generated Bessel beams). Using the whole SLM rather than just an annular region means that the required intensity on the SLM is an order of magnitude lower for a given power in the Bessel beams. Spreading the power over the whole SLM is important for high-power applications such as laser micromachining. We allow the axicons to overlap and interfere in the hologram, so the axial length of the Bessel beam core is maintained as we add more beams to the array.