Fractal interpolation on the Sierpinski Gasket

We prove for the Sierpinski Gasket (SG) an analogue of the fractal interpolation theorem of Barnsley. Let V₀={p₁,p₂,p₃} be the set of vertices of SG and the three contractions of the plane, of which the SG is the attractor. Fix a number n and consider the iterations uw=uw₁uw₂?uwn for any sequence w=(w₁,w₂,…,wn)∈n{1,2,3}. The union of the images of V₀ under these iterations is the set of nth stage vertices Vn of SG. Let F:Vn→R be any function. Given any numbers αw(w∈n{1,2,3}) with 0<|αw|<1, there exists a unique continuous extension of F, such that

f(uw(x))=αwf(x)+hw(x)     

On Non-M-Cosingular Completely ⊕-Supplemented Modules

In this paper, it is shown that any non--cosingular -supplemented module is if and only if has the summand intersection property. Let be any module such that has a coclosure in . Then we prove that is (completely) -supplemented if and only if for some submodule of such that and both are (completely) -supplemented.     

Novel and enantioselective syntheses of (R)- and (S)-3-hydroxy-4-(2,4,5-trifluorophenyl)butanoic acid: a synthon for sitagliptin and its derivatives

A new synthetic strategy for (R)- and (S)-3-hydroxy-4-(2,4,5-trifluorophenyl)butanoic acid, a building block in the preparation of sitagliptin and its derivatives, was developed. Pd(OAc)₂ catalyzed coupling of 2,4,5-trifluoro-1-iodobenzene with allyl alcohol gave 3-(2,4,5-trifluorophenyl)propanal in a yield of 95%. l-Proline catalyzed reaction of the 3-phenylpropanal (in only 1.2 molar equiv) with nitrosobenzene followed by reduction with NaBH₄ and Pd/C catalyzed hydrogenation gave (R)-3-(2,4,5-trifluorophenyl)propane-1,2-diol with >99% ee and 65% yield. Selective tosylation of primary hydroxyl group of the 1,2-propandiol unit followed by cyanide displacement afforded (R)-3-hydroxy-4-(2,4,5-trifluorophenyl)butanenitrile (80%). The nitrile was converted to the title β-hydroxy acid under basic hydrolysis in a yield of 90%. Thus, (R)-3-hydroxy-4-(2,4,5-trifluorophenyl)butanoic acid was prepared enantioselectively from the starting material in four steps and 45% overall yield. The reaction sequence was repeated with d-proline as the catalyst to give (S)-3-hydroxy-4-(2,4,5-trifluorophenyl)butanoic acid in 45% overall yield and >99% enantiomeric excess.     

Surface spin disorder and spin-glass-like behaviour in manganese-substituted cobalt ferrite nanoparticles

Due to the increasing use of silver nanoparticles (AgNPs) in consumer products, it is essential to understand how variables, such as light exposure, may change the physical and chemical characteristics of AgNP suspensions. To this end, the effect of 300?nm ultraviolet (UV) light on (20, 40, 60 and 80)?nm citrate-capped AgNP suspensions has been investigated. As a consequence of irradiation, the initial yellow hue of the AgNP suspensions is transformed towards a near colorless solution due to the loss of the surface plasmon resonance (SPR) absorbance. The decrease in SPR absorbance followed a first-order decay process for all particle sizes with a rate constant that increased linearly with the AgNP specific surface area and non-linearly with light intensity. The rate of loss of the SPR absorbance decreased with increasing citrate concentration, suggesting a surface-mediated transformation. Absorbance, atomic force microscopy, and dynamic light scattering results all indicated that AgNP photolysis was accompanied by a diameter decrease and occasional aggregation. Furthermore, in situ transmission electron microscopy imaging using a specialized liquid cell also showed a decrease in the particle size and the formation of a core?Cshell structure in UV-exposed AgNPs. X-ray photoelectron spectroscopy analysis suggested that this shell consisted of oxidized silver. The SPR in UV-exposed AgNP suspensions could be regenerated by addition of a strong reducing agent (NaBH₄), supporting the idea that oxidized silver is present after photolysis. Evidence for UV-enhanced dissolution and the production of silver ions was obtained with the Donnan membrane technique. This study reveals that the physico-chemical properties of aqueous AgNP suspensions will change significantly upon exposure to UV light, with implications for environmental health and safety risk assessments.