The Bartle-Dunford-Schwartz theorem and projection operators in certain banach lattices

The projection map used successfully by Kakutani in his classical characterization of abstract L-spaces is used to establish a variant for certain Banach lattices of a theorem of Bartle, Dunford and Schwartz which dealth with absolute continuity in spaces of measures. Entrata in Redazione il 27 settembre 1999.     

Soap films spanning rectangular prisms

In this paper, we consider soap films spanning rectangular prisms with regular n-gon bases. As the number of edges n varies, we show that there are significant changes in the qualitative properties of the spanning soap films as well as a change in the number of spanning soap films whose existence we can prove: We can find two nontrivial soap films for n = 3, 4, 5 but only one for n ≥ 6. We also prove some results concerning the interval of aspect ratios through which the soap films exist: The interval is finite if n = 3, 4, 5 and infinite if n ≥ 6. Furthermore, for n > 6, we have that the spanning soap film converges to a soap film spanning the vertical lines through the vertices of a regular n-gon as the aspect ratio goes to infinity. We can also make sense of the case n = ∞. Here, we discover some interesting singly and triply periodic soap films spanning singly and doubly periodic sets of vertical lines or spanning singly periodic sets of vertical line segments connected by pairs of parallel, horizontal lines. Finally, for n = 3, 4, 5, 6, we can derive parameterizations for the spanning soap films, and these parameterizations are explicit up to knowing the aspect ratio.      

Aggregation and interfacial properties of an isomerically pure alkylaryl sulfonate

Sodium dodecylorthoxylene sulfonate (SDOXS) was studied as an isomerically pure compound in micellar and microemulsion solutions. The particular isomer of SDOXS used in this work is one in which the sulfonate group is located ortho to the alkyl chain on the xylene ring. Aggregation of SDOXS in deuterated water was followed by specific conductivity and ²³Na NMR measurements at several temperatures. The aggregate size distribution at low concentrations agrees with the picture expected for ideal cooperative micellization but the data show deviations from this behavior at higher concentrations. This suggests either an increase in the average aggregation number with increasing concentration or an enhanced counterion binding at higher concentrations. ¹³C NMR spectra indicate that the benzene-ring motion slows down and the head-group solvation changes at high SDOXS concentrations or with the addition of electrolyte. A similar result was found for a microemulsion solution of SDOXS, n-octane, and NaCl brine. We discuss our results in view of the positional substitution pattern of the SDOXS aromatic ring.     

Good candidates for least area soap films

Soap films are presented as potential global area minimizers subject to a topological constraint. Experimentally, this constraint is the shape of the soapy water in a soap film experiment. In this context, soap films which are probable area minimizers for rectangular n-prisms are described. By allowing area minimizers which arise as deformations of higher genus surfaces, we are able to discover previously unknown soap films spanning rectangular n-prisms with large aspect ratios and n ≥ 5. For n = 3, 4, 5, we show that the central film contracts to a point as the aspect ratio of the prism increases. We also prove that the area of the central hexagon for a soap film spanning a tall 6-prism approaches zero like (height)^?4 as the height approaches infinity, provided we fix the length of the hexagon base. Finally, we prove that, if the aspect ratio is large enough, the soap film produced experimentally spanning a 4-prism has films which look planar but in reality are non-planar.     

Oligomerization of ethylene to branched alkenes using neutral phosphinosulfonamide nickel(II) complexes

A series of neutral phosphinosulfonamide complexes of nickel(II) were synthesized that catalyzed the oligomerization of ethylene to branched oligomers with average degrees of polymerization between 10 and 35. Branching numbers varied from 17 to 80 branches per 1000 carbons, depending on the catalyst structure and reaction conditions. The catalysts were active in a variety of solvents, including toluene, CH₂Cl₂, tetrahydrofuran, ethyl acetate, and methanol, but showed decreasing activity at temperatures higher than 40 °C. Electron‐rich phosphinosulfonamides produced the highest catalyst activities in a series of structure–reactivity studies. The mechanism of oligomer formation was investigated with ¹H NMR spectroscopy, which indicated that branching arose from the isomerization of the nickel alkyl species during propagation rather than the reincorporation of α‐olefin products. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 4627–4640, 2000     

The preparation and reactions of C(α),N-dilithiocarboalkoxyhydrazones

C(α),N-Carboalkoxyhydrazones were metalated with an excess of lithium diisopropylamide (LDA), and the resulting dianions were condensed with methyl iodide, an unsubstituted ketone, a benzoate ester, salicylates, a benzoylacetate, or ethyl chloroformate.     

Synthesis, characterization, and X-ray crystal structures of W(VI) alkyl complexes with chelating diamide and imido co-ligands

The complex W(NPh)Cl₂[o-(NSiMe₃)₂C₆H₄] 3 was synthesized from PhN = WCl₄ · OEt₂ and N,N′-(Li₂[o-(NSiMe₃)₂C₆H₄] and reacts with Lewis bases to form the adducts W(NPh)Cl₂[o-(NSiMe₃)₂C₆H₄](L) (L = PMe₃, THF, 3-picoline, ^tBuNC, MeCN) 4a–e. Crystals of 4a are triclinic, space group P1¯, with a = 9.562(1), b = 10.277(1), c = 14.920(2) Å, = 82.15(1), β = 80.18(1), γ = 80.41(1)°, and Z = 2. The structure was solved by the heavy atom method and refined to R = 0.0408 for 4224 observed (I > 2σ(I)) reflections. The dialkyl complexes W(NPh)R₂[o-(NSiMe₃)₂C₆H₄] (R = Me, Et, CH₂Ph, CH₂CMe₃, CH₂CMe₂Ph) 5–9 are formed through subsequent reactions of 3 with the corresponding Grignard reagent. Crystals of complex 5 are monoclinic, space group P2(1)/n, with a = 10.3545(2), b = 17.9669(1), c = 13.3168(1) Å, β = 103.826(1)°, and Z = 4. The structure of complex 5 was solved by direct methods in SHELXTL5 and refined to R = 0.0247 for 4572 observed reflections. Compound 5 has a square pyramidal geometry in which the imido ligand occupies the apical position and reacts with PMe₃ to form the adduct W(NPh)Me₂[o-(NSiMe₃)₂C₆H₄](PMe₃) 5a. Crystals of complex 5a are monoclinic, space group C2/m, with a = 13.5336(1), b = 14.4291(1), c = 15.3785(1) Å, β = 110.365(1)°, and Z = 4. The structure of compound 5a was solved by direct methods in shelxtl5 and refined to R = 0.0272 for 3057 observed reflections. Crystals of the bis-neopentyl complex 8 are monoclinic, space group P2(1)/n, with a = 10.6992(4), b = 18.3144(7), c = 16.0726(6) Å, β = 92.042(1)°, and Z = 4. The structure of 8 was solved by direct methods in shelxtl5 and refined to R = 0.0261 for 5881 observed reflections. Complex 8 has a trigonal bipyramidal geometry with both neopentyl groups and one amido nitrogen in the equatorial plane.