Multigrid methods for the computation of singular solutions and stress intensity factors II: Crack singularities

We consider the Poisson equation −Δu=f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Ω with cracks. Multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed. The convergence rate for the stress intensity factors is whenfεL ²(Ω) and whenfεH ¹(Ω). The convergence rate in the energy norm is in the first case and in the second case. The costs of these multigrid methods are proportional to the number of elements in the triangulation. The general case wherefεH ^m (Ω) is also discussed. The work of the first author was partially supported by NSF under grant DMS-96-00133     

On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal differentiable functions and such that the Zalcwasser rank and the Kechris-Woodin rank of are but the Denjoy rank of is 2 and the Denjoy rank and the Kechris-Woodin rank of are but the Zalcwasser rank of is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.

     

Natural and edible biopolymer poly-gamma-glutamic acid: synthesis, production, and applications

Poly-gamma-glutamic acid (gamma-PGA) is a very promising biodegradable polymer that is produced by Bacillus subtilis. Gamma-PGA is water-soluble, anionic, biodegradable, and edible. This paper reviews the production of a strain of gamma-PGA and recent developments with respect to applications in terms of Ca absorption, moisturizing properties, gamma-PGA conjugation, super absorbent polymer, and so on. Our recent research shows that gamma-PGA can be used as an immune-stimulating and anti-tumor agent, especially at high molecular weight.     

Pyrene-appended calix[4]crowned logic gates involving normal and reverse PET: NOR, XNOR and INHIBIT

As a novel sensing system, N-(1-pyrenylmethyl) amide-appended calix[4]crown-5 (2) and crown-6 (3) have been newly synthesized. Judging from the fluorescence changes upon the addition of cations, 3 having crown-6 ring showed the Pb²⁺ ion selectivity over other cations tested regarding fluorescence quenching. Upon the Pb²⁺ ion coordination to two amide oxygen atoms with aid of crown ring, a reverse-photo-induced electron transfer (PET) occurs in such a way that electrons transfer from the pyrene groups to the electron deficient amide oxygen atoms to give a quenched fluorescence. By the addition of either HClO₄ or triethylamine in the solution of 3, the fluorescence intensity decreased because of the reverse-PET from pyrene groups to protonated amide oxygen atoms and because of normal PET from the nitrogen anion formed by triethylamine to pyrene groups, respectively. For 3, NOR logic gate in which the strong fluorescence signal appears at 395 nm (output: 1) is operated only when neither of triethylamine nor Pb(ClO₄)₂ (inputs A and B) is added (A=B: 0). XNOR gate is also operated only when both of two inputs are added (triethylamine and HClO₄, A=B: 1) or when neither of two inputs is added (A=B: 0). Then, for 3, new INHIBIT gate system was also designed using such combinational inputs as HClO₄, Pb(ClO₄)₂ and triethylamine.     

Acid–Base Chemistry at the Ice Surface: Reverse Correlation Between Intrinsic Basicity and Proton‐Transfer Efficiency to Ammonia and Methyl Amines

Proton transfer from the hydronium ion to NH₃, CH₃NH₂, and (CH₃)₂NH is examined at the surface of ice films at 60 K. The reactants and products are quantitatively monitored by the techniques of Cs⁺ reactive‐ion scattering and low‐energy sputtering. The proton‐transfer reactions at the ice surface proceed only to a limited extent. The proton‐transfer efficiency exhibits the order NH₃>(CH₃)NH₂=(CH₃)₂NH, which opposes the basicity order of the amines in the gas phase or aqueous solution. Thermochemical analysis suggests that the energetics of the proton‐transfer reaction is greatly altered at the ice surface from that in liquid water due to limited hydration. Water molecules constrained at the ice surface amplify the methyl substitution effect on the hydration efficiency of the amines and reverse the order of their proton‐accepting abilities.     

Photoaddition reactions of 1,4-diphenylbutadiyne with olefins

1,4-Diphenylbutadiyne(DPB) and several olefins such as 2,3-dimethyl-2-butene, 1,4-cyclohexadiene and dimethyl fumarate were irradiated with 300 nm UV light to obtain interesting cross-cycloaddition products.