Natural bound in Kwiecinski's criterion for flatness

Kwiecinski has proved a geometric criterion for flatness: A morphism of germs of analytic spaces is not flat if and only if its fibre power has a vertical component, for some . We show how to bound using Hironaka's local flattener: If is not flat, then has a vertical component, where is the minimal number of generators of the ideal in of the flattener of .

     

On sums of Darboux and nowhere constant continuous functions

We show that the property

(P)

for every Darboux function there exists a continuous nowhere constant function such that is Darboux

follows from the following two propositions:

(A)

for every subset of of cardinality there exists a uniformly continuous function such that ,

(B)

for an arbitrary function whose image contains a non-trivial interval there exists an of cardinality such that the restriction of to is uniformly continuous,

which hold in the iterated perfect set model.

     

Continuous monitoring of thermooxidative degradation products of polystyrene by membrane extraction with sorbent interface and gas chromatography

A novel method for the continuous monitoring of thermooxidative degradation products of polystyrene by membrane extraction with sorbent interface (MESI) and gas chromatography (GC) is developed. The results are compared with solid-phase microextraction-GC, which can extract gases, vapors, and aerosols. The volatile and semivolatile degradation products are identified by mass spectrometry. The membrane used in the MESI-GC analysis shows a high permeation for volatile aromatic hydrocarbons; a low permeation for corresponding volatile aldehydes; and no permeation for less volatile alcohols, acids, and degradation products with a high molecular weight, thus reducing significantly the number of compounds detected from MESI-GC. Sensitivity of the method depends on the time of trapping, which is limited by the breakthrough of the trap. By heating the trap at fixed intervals of time, consecutive gas chromatograms are obtained.     

High-efficiency coupling structure for a single-line-defect photonic-crystal waveguide

We present the design and fabrication of a planar structure for coupling light from a multimode feed waveguide into a single-line-defect photonic-crystal waveguide. Finite-difference time-domain calculations predict a coupling efficiency of greater than 90%, and preliminary experimental results indicate successful coupling through a single-line-defect photonic-crystal waveguide. Device design, fabrication, and characterization are presented.     

Novel and elegant modes of SbF3 coordination in the complexes with azacrown ethers

Synthesis and single-crystal X-ray structures are recorded for three adducts of SbF₃ with different azacrown ethers. [SbF₃·H₂O·L₁], 1, (L₁ = 12,13-benz-1,10-di(azamethyl)-4,7-dioxacyclotetradecane-1,14-dione), triclinic, , a = 11.234(2), b = 11.691(2), c = 8.869(2) Å, = 94.66(3), = 113.12(3), = 67.82(3)°, Z = 2. [SbF₃Cl·H₂O·(HL₂)], 2, (L₂ = monoaza-18-crown-6), orthorhombic, P2₁2₁2₁, a = 8.763(2), b = 13.003(3), a = 16.836(3) Å, Z = 4. [(SbF₃)₂Cl₂·(H₂L₃)], 3, (L₃ = 1,10-diaza-18-crown-6), triclinic, , a = 8.284(2), b = 9.016(2), c = 9.134(2) Å, = 82.92(3), = 65.24(3), = 63.38(3)°, Z = 1. All three structures include SbF₃ neutral molecules in the pyramidal mode and the antimony second coordination sphere is completed up to six in different fashions. In 1 the dimeric (SbF₃)₂ adducts are made up due to the involvement of the symmetry-related fluorine atom in coordination. The distorted octahedron is then completed by water molecule and carbonyl oxygen of L₁. The neutral molecules are joined by coordination and hydrogen bonds in the infinite ribbons. 2 and 3 are both comprised of neutral and charged species also bounded via coordination and hydrogen bonds. L₂ and L₃ in the complexes adopt the form of single and double-charged cations, respectively. The inorganic backbone unites the neutral SbF₃ molecules and chloride anions in the alternative mode. The sixth coordination site in the antimony polyhedron is completed by the water molecule in 2 and the ethereal oxygen atom in 3. Alignment of the inorganic entities within the structures of 2 and 3 is strikingly similar. Hydrogen bonding patterns are discussed.     

On the number of solutions of an algebraic equation on the curve , and a consequence for o-minimal structures

We prove that every polynomial of degree has at most zeros on the curve . As a consequence we deduce that the existence of a uniform bound for the number of zeros of polynomials of a fixed degree on an analytic curve does not imply that this curve belongs to an o-minimal structure.

     

Anisochronism of large geminal groups in the 1H and 13C nuclear magnetic resonance spectra

The role of steric factors in magnetic non-equivalence anisochronism of complex geminal groups and of their fragments in compounds of the general formula RCH(Ph)CH(COOR′)₂ has been investigated. CMR anisochronism is more sensitive to conformational and other steric changes than is PMR.