Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc and a complex-valued function , is the uniform algebra on generated by z and F equal to ? When F∈C¹(D), this question is complicated by the presence of points in the surface that have complex tangents. Such points are called CR singularities. Let p∈S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.     

A modified Brent’s method for finding zeros of functions

Brent’s method, also known as zeroin, has been the most popular method for finding zeros of functions since it was developed in 1972. This method usually converges very quickly to a zero; for the occasional difficult functions encountered in practice, it typically takes $O(n)$ iterations to converge, where $n$ is the number of steps required for the bisection method to find the zero to approximately the same accuracy. While it has long been known that in theory Brent’s method could require as many as $O(n^2)$ iterations to find a zero, such behavior had never been observed in practice. In this paper, we first show that Brent’s method can indeed take $O(n^2)$ iterations to converge, by explicitly constructing such worst case functions. In particular, for double precision accuracy, Brent’s method takes $2{,}914$ iterations to find the zero of our function, compared to the $77$ iterations required by bisection. Secondly, we present a modification of Brent’s method that places a stricter complexity bound of $O(n)$ on the search for a zero. In our extensive testing, this modification appears to behave very similarly to Brent’s method for all the common functions, yet it remains at worst five times slower than the bisection method for all difficult functions, in sharp contrast to Brent’s method.     

A Finite Deformation Microsphere Model for Magneto-Visco-Elastic Response in Magnetorheological Elastomers

In this work we present a novel approach to the modeling of magnetorheological elastomers (MREs) for finite deformations. Keeping in mind the composite nature at the microscale, we employ the microsphere model as an effective tool to capture the constitutive response of the material. The microsphere model has been successfully applied to the modelling of rubber-like materials. Here, we extend this approach by taking into account the effect of the magnetic dipole-dipole interactions on the orientation of the polymer chains. Thus, the presented microsphere model is directly motivated by considering the underlying phenomena at the microscale level. Finally the material model is embedded in a finite element framework and the results of a boundary value problems is presented. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

From a Binary to a Quaternary Cocrystal: An Unusual Supramolecular Synthon

Formation of a stoichiometric quaternary cocrystal consisting of resorcinol ( RES ), tetramethylpyrazine ( TMP ), phenazine ( PHE ) and pyrene ( PYR ) is described. A closed tetrameric resorcinol‐heterocycle synthon, unusual in that it has two different linker bases rather than just one, is observed in this four‐component solid. The tetrameric synthon is formed by two RES molecules and the two pyridine bases TMP and PHE . The stoichiometric quaternary cocrystal grows in an epitaxial fashion on the surfaces of a RES.PHE binary cocrystal which is initially obtained from the mother liquor. By indexing the common crystal faces of the binary and quaternary cocrystals, and noting that no ternary solid is obtained, a plausible mechanism has been proposed for the formation of this rare supramolecular architecture.     

Evolution of absorption energy per unit thickness of damaged sandstone

Journal of Thermal Analysis and Calorimetry - This paper experimentally studies the evolution of absorption energy characteristics per unit thickness of Jhiri sandstone at variable temperatures...     

Rotational dynamics of propylene inside Na-Y zeolite cages

We report here the quasielastic neutron scattering (QENS) studies on the dynamics of propylene inside Na-Y zeolite using triple axis spectrometer (TAS) at Dhruva reactor, Trombay. Molecular dynamics (MD) simulations performed on the system had shown that the rotational motion involves energy larger than that involved in the translational motion. Therefore, rotational motion was not observed in our earlier QENS studies on propylene adsorbed Na-Y zeolite using a higher resolution spectrometer at Dhruva. Analysis of the TAS spectra revealed that the quasielastic broadening observed in propylene-loaded zeolite spectra is due to the rotational motion of the propylene molecules. This is consistent with our simulation result. Further, the rotational motion is found to be isotropic. The rotational diffusion coefficient has been obtained.      

Lauricella hypergeometric series $$F_A^{(n)}$$ F A ( n ) over finite fields

The Ramanujan Journal - In this paper, we develop a finite field analogue for one of the Lauricella series, $$F^{(n)}_A$$ . Extending results of Greene, a finite field analogue for the multinomial...     

Controlling Proton and Electron Transfer Rates to Enhance the Activity of an Oxygen Reduction Electrocatalyst

An electrochemical approach is developed that allows for the control of both proton and electron transfer rates in the O₂ reduction reaction (ORR). A dinuclear Cu ORR catalyst was prepared that can be covalently attached to thiol‐based self‐assembled monolayers (SAMs) on Au electrodes using azide–alkyne click chemistry. Using this architecture, the electron transfer rate to the catalyst is modulated by changing the length of the SAM, and the proton transfer rate to the catalyst is controlled with an appended lipid membrane modified with proton carriers. By tuning the relative rates of proton and electron transfer, the current density of the lipid‐covered catalyst is enhanced without altering its core molecular structure. This electrochemical platform will help identify optimal thermodynamic and kinetic parameters for ORR catalysts and catalysts of other reactions that involve the transfer of both protons and electrons.