Remark on factorials that are products of factorials

In a paper published in 1993, Erdos proved that if n! = a! b!, where 1 < a a parts per thousand currency sign b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + epsilon)/ log 2) log log n and generalize it to the equation a (1)!a (2)! ... a (k) ! = n!. In a recent paper, F. Luca proved that n - b = 1 for large enough n provided that the ABC-hypothesis holds.

Development of radiosynthesis methods for 18F-labelled radiopharmaceuticals : General considerations and production of a dopamine transporter radioligand

Positron emission tomography (PET) is a molecular imaging technique that utilises radiopharmaceuticals (radiotracers) labelled with a positron-emitting radionuclide, such as fluorine-18 (18F). Development of a new radiotracer requires an appropriate radiosynthesis method: the most common of which with 18F is nucleophilic substitution with [18F]fluoride ion. The success of the labelling reaction is dependent on various factors such as the reactivity of [18F]fluoride, the structure of the target compound in addition to the chosen solvent. The overall radiosynthesis procedure must be optimised in terms of radiochemical yield and quality of the final product. Therefore, both quantitative and qualitative radioanalytical methods are essential in developing radiosynthesis methods. Furthermore, biological properties of the tracer candidate need to be evaluated by various pre-clinical studies in animal models. In this work, the feasibility of various nucleophilic 18F-fluorination strategies were studied and a labelling method for a novel radiotracer, N-3-[18F]fluoropropyl-2beta-carbomethoxy-3beta-4-fluorophenyl)nortropane ([18F]beta-CFT-FP), was optimised. The effect of solvent was studied by labelling a series of model compounds, 4-(R1-methyl)benzyl R2-benzoates. 18F-Fluorination reactions were carried out both in polar aprotic and protic solvents (tertiary alcohols). Assessment of the 18F-fluorinated products was studied by mass spectrometry (MS) in addition to conventional radiochromatographic methods, using radiosynthesis of 4-[18F]fluoro-N-[2-[1-(2-methoxyphenyl)-1-piperazinyl]ethyl-N-2-pyridinyl-benzamide (p-[18F]MPPF) as a model reaction. Labelling of [18F]beta-CFT-FP was studied using two 18F-fluoroalkylation reagents, [18F]fluoropropyl bromide and [18F]fluoropropyl tosylate, as well as by direct 18F-fluorination of sulfonate ester precursor. Subsequently, the suitability of [18F]beta-CFT-FP for imaging dopamine transporter (DAT) was evaluated by determining its biodistribution in rats. The results showed that protic solvents can be useful co-solvents in aliphatic 18F-fluorinations, especially in the labelling of sulfonate esters. Aromatic 18F-fluorination was not promoted in tert-alcohols. Sensitivity of the ion trap MS was sufficient for the qualitative analysis of the 18F-labelled products; p-[18F]MPPF was identified from the isolated product fraction with a mass-to-charge (m/z) ratio of 435 (i.e. protonated molecule [M+H]+). [18F]beta-CFT-FP was produced most efficiently via [18F]fluoropropyl tosylate, leading to sufficient radiochemical yield and specific radioactivity for PET studies. The ex vivo studies in rats showed fast kinetics as well as the specific uptake of [18F]beta-CFT-FP to the DAT rich brain regions. Thus, it was concluded that [18F]beta-CFT-FP has potential as a radiotracer for imaging DAT by PET.

Three different co-ordination geometries in the pentacopper(II) unit of [Cu5(OH)2(H2O)(O2CMe)6(Him)4][ClO4]2(Him = imidazole)

The reaction of imidazole (Him) with [Cu2(µ-O2CMe)4(H2O)2] in water–NaClO4 led to the formation of a polynuclear copper(II) complex, [Cu5(OH)2(H2O)(O2CMe)6(Him)4][ClO4]21, in which the pentanuclear units, showing four, five and six co-ordination geometries for the copper(II) centres and Cu Cu distances of 3.043(1), 3.178(1) and 3.578(1)Å, were linked by aqua bridges to give an intra-chain inter-unit Cu Cu separation of 4.507(1)Å.

Chemical toughening of epoxies. I. Structural modification of epoxy resins by reaction with hydroxy-terminated poly (butadiene-co-acrylonitrile)

In an attempt to toughen the epoxy resin matrix for fiber-reinforced composite applications, a chemical modification procedure of a commercially available bisphenol-A-based epoxy resin using reactive liquid rubber HTBN [hydroxy-terminated poly(butadiene-co-acrylonitrile)] and TDI (tolylene diisocyanate) is described. The progress of the reaction and the structural changes during modification process are studied using IR spectroscopy, viscosity data, and chemical analysis (epoxy value determination). The studies support the proposition that TDI acts as a coupling agent between the epoxy and HTBN, forming a urethane linkage with the former and an oxazolidone ring with the latter. The chemical reactions that possibly take place during the modification are discussed.

An unusual axial co-ordination of phenolate oxygen to copper(II): crystal structure of chloro(2-[bis(2-pyridylmethyl)aminomethyl]-4-nitrophenolato)copper(II)

A tripod ligand possessing two pyridine moieties and a phenolato group as pendants forms a mononuclear complex with an axial copper(II)–phenolate co-ordination in a square-pyramidal environment.

A porphyrin bearing tetrahydrothiophene pickets; co-ordination behaviour and electron transfer

A novel ‘picket-fence’ porphyrin, 5,10,15,20-tetrakis[o-(tetrahydro-2-thenoylamino)phenyl]porphyrin (H2L) with ligating tetrahydrothiophene rings disposed perpendicular to the porphyrin plane has been synthesised. Its zinc(II) derivative, [ZnL], binds two silver(I) ions co-operatively with a dissociation constant of 4.8 × 10–8 dm3 mol–1. Time-resolved fluorescence lifetime measurements reveal the presence of intramolecular photoexcited electron transfer in this donor–acceptor system.

Electron energy loss spectroscopic study of Cr(CO)6, Mo(CO)6 and W(CO)6 in the vapour phase

Electron energy loss spectra (EELS) of Cr, Mo and W hexacarbonyls in the vapour phase are reported. Most of the bands observed are similar to those in optical spectra, but the two high energy transitions in the 9·8–11·2 eV region are reported here for the first time. Based on the orbital energies from the ultraviolet photoelectron spectra and the electronic transition energies from EELS and earlier optical studies, the molecular energy level schemes of these molecules are constructed.

Chemical toughening of epoxies. II. Mechanical, thermal, and microscopic studies of epoxies toughened with hydroxyl-terminated poly(butadiene-co-acrylonitrile)

Diglycidyl ether–bisphenol-A-based epoxies toughened with various levels (0–12%) of chemically reacted liquid rubber, hydroxyl-terminated poly(butadiene-co-acrylonitrile) (HTBN) were studied for some of the mechanical and thermal properties. Although the ultimate tensile strength showed a continuous decrease with increasing rubber content, the toughness as measured by the area under the stress-vs.-strain curve and flexural strength reach a maximum around an optimum rubber concentration of 3% before decreasing. Tensile modulus was found to increase for concentrations below 6%. The glass transition temperature Tg as measured by DTA showed no variation for the toughened formulations. The TGA showed no variations in the pattern of decomposition. The weight losses for the toughened epoxies at elevated temperatures compare well with that of the neat epoxy. Scanning electron microscopy revealed the presence of a dual phase morphology with the spherical rubber particles precipitating out in the cured resin with diameter varying between 0.33 and 6.3 μm. In contrast, a physically blended rubber–epoxy showed much less effect towards toughening with the precipitated rubber particles of much bigger diameter (0.6–21.3 μm).

Chemical toughening of epoxies. I. Structural modification of epoxy resins by reaction with hydroxy-terminated poly(butadiene-co-acrylonitrile)

In an attempt to toughen the epoxy resin matrix for fiber-reinforced composite applications, a chemical modification procedure of a commercially available bisphenol-A-based epoxy resin using reactive liquid rubber HTBN [hydroxy-terminated poly(butadiene-co-acrylonitrile)] and TDI (tolylene diisocyanate) is described. The progress of the reaction and the structural changes during modification process are studied using IR spectroscopy, viscosity data, and chemical analysis (epoxy value determination). The studies support the proposition that TDI acts as a coupling agent between the epoxy and HTBN, forming a urethane linkage with the former and an oxazolidone ring with the latter. The chemical reactions that possibly take place during the modification are discussed.

Analysis of heat and mass transfer during bulk hydraircooling of spherical food products

Hydraircooling is a technique used for precooling food products. In this technique chilled water is sprayed over the food products while cold unsaturated air is blown over them. Hydraircooling combines the advantages of both air- and hydrocooling. The present study is concerned with the analysis of bulk hydraircooling as it occurs in a package filled with several layers of spherical food products with chilled water sprayed from the top and cold unsaturated air blown from the bottom. A mathematical model is developed to describe the hydrodynamics and simultaneous heat and mass transfer occurring inside the package. The non-dimensional governing equations are solved using the finite difference numerical methods. The results are presented in the form of time-temperature charts. A correlation is obtained to calculate the process time in terms of the process parameters.

Combustion Synthesis and Properties of Fine Particle Spinel Manganites

Fine particle spinel manganites have been prepared by thermal decomposition of the precursors N2H5M1/3Mn2/3(N2H3COO)3 · H2O (M = Co and Ni) and M1/3 Mn2/3(N2H3COO)2 · 2H2O (M = Mg and Zn), as well as by the combustion of redox mixtures containing M(II) nitrate (M = Mg, Co, Ni, Cu, and Zn), Mn(II) nitrate, and maleic hydrazide (MH) in the required molar ratio. Both the precursor and redox mixtures undergo self-propagating, gas-producing, exothermic reactions once ignited at 250-375°C to yield corresponding manganites in less than 5 min. Formation of single phase products was confirmed by X-ray powder diffraction patterns. The manganites are of submicrometer size and have surface area in the range 20-76 m2/g.