Separately harmonic and subharmonic functions

Let u(x, y) be defined in B ₁×B ₂ where B ₁ ^m and B ₂ ⁿ , and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C ² for each y then u is subharmonic in B ₁×B ₂ in both variables jointly.     

Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W ^1,2 (D) and W ^1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on being indistinguishable (in distribution) from RBM on . This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on is indistinguishable from the RBM on , or equivalently, W ^1,2(D\E)=W^1,2(D). An example of such kind is: D=² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483.     

Influence of organic modifier and temperature on the enantiomeric separation of γ-lactams by HPLC

Summary A high-performance liquid chromatographic method is reported for the resolution of the enantiomers of a series of fused 31h2550038n4923/xxlarge947.gif" alt="gamma" align="MIDDLE" BORDER="0">-lactams (2,7-diaza-3-oxo[3.3.0]octan-6-ones) with probable anti-HIV and anticancer activity. Resolution was achieved on a Chiralcel^® OD column, cellulose tris-(3,5-dimethylphenyl carbamate) adsorbed on macroporous silica gel; mixtures ofn-hexane and isopropyl alcohol in different proportions were used as the mobile phase. The analysis was studied at different temperatures.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Aryl-bis(triorganylphosphine)nickel phenoxides

Compounds of general formulatrans-ArNi(PR₃)₂OAr' (R = Et, cyclohexyl; Ar = 2-MeC₆H₄, 2-FC₆H₄; Ar' = 4-FC₆H₄, 4-NO₂C₆H₄) were synthesized by the reaction of Ar'OK with cationic nickel complexes generated by treatment of ArNi(PR₃)₂Cl with TlBF₄. Syntheses of 4-fluorophenoxide complexes, ArNi(PR₃)₂OC₆H₄F-4, additionally give some quantities oftrans-[ArNi(PR₃)₂OC₆H₄F-4][HOC₆H₄F-4] adducts. Exchange reactions MeC₆H₄Ni(PEt₃)₂OC₆H₄F-4 + XC₆H₄OH 2-MeC₆H₄Ni(PEt₃)₂OC₆H₄X + 4-FC₆H₄OH were studied in THF. The equilibrium is shifted to the right as the acidity of ArOH increases. A linear relationship between lgK _eq and pK _a of XC₆H₄OH in DMSO was found. A conclusion concerning the strong polarization of the Ni-O bond was made on the basis of an analysis of the chemical shifts of fluorine atoms in 2-MeC₆H₄Ni(PEt₃)₂OC₆H₄F-4.Translated fromIvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 2266–2271, November, 1995.     

Correlations of13C chemical shifts and geometry modifications

The effect of geometry modifications of¹³C chemical shifts has been investigated in a small subset of molecules using both LO-INDO and Gaussian 70 (4–31) calculations. The Gaussian calculations, while known to give poor absolute shifts, compare well to the reparameterized semi-empirical INDO determinations in calculated shift changes. In virtually all cases the signs of the shift changes were found to be opposite to that of the changes in the calculated electronic energy.     

Synthesis and characterization of mixed-morphology CePO4 nanoparticles

Cerium phosphate nanoparticles with diameters of 10-180 nm were synthesized by a variety of solution techniques. X-ray diffraction (XRD) determined the crystalline phase(s) present in each sample. Population, shift, and spin-lattice relaxation ³¹P solid-state nuclear magnetic resonance (NMR) measurements accounted for all the ³¹P nuclei expected in each sample, and were able to distinguish between phosphorous nuclei in different environments and phases. Transmission electron microscopy (TEM) characterized the morphology and crystallinity of the powder samples as well as of the sintered compacts of the powders. In conjunction with TEM, energy-dispersive spectroscopy (EDS) provided a measure of the composition of the bulk intergranular regions within each CePO₄ sample. The presence of an amorphous, phosphate-rich intergranular phase was found in those samples prepared by dissolution of ceria in H₃PO₄ under various conditions.     

DFT Study on the （S）-Proline-catalyzed Direct Aldol Reaction between Acetone and 4-Nitrobenzaldehyde

1 INTRODUCTION Aldol reaction, one of the most important C–C bond-formation reactions, is widely utilized in orga- nic synthesis[1]. Extensive efforts have been donated to the development of catalytic stereo-selective va- riants of this reaction[2]. However, the direct aldol reaction between unmodified carbonyles would be the most attractive synthetic alternative. Proline-catalyzed intermolecular direct aldol reac- tion between acetone and 4-nitrobenzaldehyde was first reported by List…