Simulation of femtosecond pulse propagation in air

We present numerical simulations of the propagation of intense spatio-temporal fields propagating in air. Characteristic parameters of the propagation like peak intensities, plasma densities and diameters of the plasma channels are obtained. The evolution of the spectral content of the pulses is investigated and, in agreement with recent experiments, white-light supercontinuum generation is observed. We show in which areas of the pulse the supercontinuum is generated and conclude that self-phase modulation is responsible for the spectral broadening. Furthermore we investigate azimuthal instabilities of intense rotationally symmetric pulsed beams propagating in air. Although the spatial-temporal evolution of the field is strongly influenced by the onset of plasma generation, the instabilities are basically caused by the Kerr effect. We conclude that calculations assuming rotational symmetry become unrealistic due to the fast growth of azimuthal instabilities in the focal region.     

[{LiCH(CH2CH2OMen)SO2Ph}4] (OMen = Menthyl): Synthesis and First Structure of a Chiral Solvent Free Lithiated Sulfone with a Direct Li–C Bond

R*OCH₂CH₂CH₂SO₂Ph (R*OH = MenOH, (–)‐menthol, ( 3a ); BorOH, (1S)‐(–)‐borneol, ( 3b )) were found to react with n‐BuLi in n‐pentane/n‐hexane and toluene/n‐hexane under deprotonation yielding LiCH(CH₂CH₂OR*)SO₂Ph (R* = Men, ( 4a ); Bor, ( 4b )) which reacted with n‐Bu₃SnCl forming the requisite tri(n‐butyl)tin compounds n‐Bu₃SnCH(CH₂CH₂OR*)SO₂Ph (R* = Men, ( 5a ); Bor, ( 5b )) as diastereomeric mixtures. The identities of 5a and 5b were unambiguously proved by ¹H, ¹³C and ¹¹⁹Sn NMR spectroscopic measurements. Solutions of 4a afforded crystals of [{LiCH(CH₂CH₂OMen)SO₂Ph}₄] ( 4a′ ) for which the structure was determined by single‐crystal X‐ray crystallography. Complex 4a′ crystallized in a tetrameric structure without any additional solvent molecules. There were found direct Li–C bonds (Li1–C1/Li2–C20 2.231(9)/2.236(9) Å). The tetrahedral donor set of Li is completed by three oxygen atoms. One oxygen atom comes from the OMen substituent via intramolecular coordination and two oxygen atoms come from SO₂ groups of neighboured LiCH(CH₂CH₂OMen)SO₂Ph moieties. Thus, a heterocubane structure with a Li₄S₄ core is built up.     

A note on the Undirected Rural Postman Problem polytope

This note refers to the article by G. Ghiani and G. Laporte ``A branch-and-cut algorithm for the Undirected Rural Postman Problem', Math. Program. 87 (2000). We show that some conditions for the facet-defining property of the basic non-trivial inequalities are not sufficient and that the Rural Postman Problem polytope is more complex even when focusing on canonical inequalities only.     

Contamination caused by ion-exchange resin!? Consequences for ultra-trace analysis

Well known sources of contamination are the laboratory environment, vessels, reagents and handling of the sample. Attention is drawn to the importance of cationic impurities included in the ion-exchangers used for trace-matrix separations (TMS) at the sub-ng/g-level. The usual rinsing and conditioning of the resins does not remove all adhering metals and thus leads to contaminated eluates in analytical work. High and badly reproducible blanks are observed, in particular for Na, Mg, Al, K, Cr, Fe, Zn and Ba. They are introduced during the polymerization process and are not ionically bound but obviously included in the polymer matrix. Any time when another “nest” of inclusion is opened in the resin column by mechanical stress, the next irreproducible event of contamination occurs.     

The complexity of computing the tutte polynomial on transversal matroids

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.