Chord techniques and Newton's method for discrete bifurcation problems

Summary The numerical treatment of discrete bifurcation problems (2) with chord methods or Newton's method is a question of constructing appropriate initial approximations to prevent the sequence from converging to the trivial solution. This problem is being discussed under conditions which are satisfied for quite a few examples arising in applications (see Sect. 3).     

Ein elementarer Beweis des Zariski-Lipman-Problems für graduierte analytische Algebren

Ohne Zusammenfassung     

Elektrochemische Bildung von Δ1,2-Norbornen

Electrochemical Formation of Δ^1,2-Norbornene The electrochemical reduction of 1,2-dihalogen norbornanes in tetrahydrofuran/furan leads to a mixture of two isomeric cycloadducts 6 and 7. The ratio of these adducts corresponds to those which have been found in reductive bisdehalogenation of 1 and 2 by butyllithium.     

Zur elektronenstruktur vicinaler triketone: Bicyclo[3.1.0]hexan-2.3.4-trione

Bicyclo[3.1.0]hexane-2.3.4-trione and its 1.5-dimethyl derivative have been synthesized and investigated by means of photoelectron spectroscopy.     

Ultralow-jitter, 1550-nm mode-locked semiconductor laser synchronized to a visible optical frequency standard

Using high-bandwidth feedback, we have synchronized the pulse train from a mode-locked semiconductor laser to an external optical atomic clock signal and achieved what is to our knowledge the lowest timing jitter to date (22 fs, integrated from 1 Hz to 100 MHz) for such devices. The performance is limited by the intrinsic noise of the phase detector used for timing-jitter measurement. We expect such a highly stable device to play an important role in fiber-network-based precise time/frequency distribution.     

The four-electron diamagnetic ring current of porphycene

Porphycene, an isomer that can replace porphin in chemical and biochemical contexts, is predicted by ab initio calculation to exhibit a global diatropic pi ring current with bifurcation across the four pyrrole units of the macrocycle. Analysis of the orbital contributions to the current density in porphycene reveals that the global current, with its bifurcation feature, is attributable to the four electrons of the near-degenerate HOMO levels, the same set of active electrons that feature in the well-known four-orbital model of the electronic spectra of porphyrins. Integration of the current density gives 1H, 13C and 15N NMR shieldings that are compatible with the observed low-field shifts of peripheral and bridge protons and high-field shift of the internal NH protons, assignment of the 13C NMR spectrum and the single average 15N chemical shift resulting from rapid NH tautomerism. Geometries were calculated with the DFT B3LYP functional, the current density maps were calculated with the ipsocentric coupled-Hartree-Fock CTOCD-DZ method, and the shieldings with the CTOCD-PZ2 variant, all in the same 6-31G** basis.     

Carbon Monoxide Triplet Rydberg Series in the f Complex Region

Using a multistep state-selective excitation scheme, fluorescence-dip spectra of carbon monoxide have been recorded in the energy region between 106 100 and 112 200 cm(-1). Prominent groups of bands were tentatively identified as transitions into the 6f em leader10f (v=0) and 9psigma em leader11psigma (v=0) triplet Rydberg series. For the f complex states, molecular constants have been calculated. The triplet f levels show an unusually large rotational constant and are located at higher energies than their singlet counterparts. Copyright 2001 Academic Press.     

High dimensional integration

Advances in Computational Mathematics -     

Probing the exohedral magnetic properties of C20 derivatives by through space NMR shieldings (TSNMRS)

The through space NMR shieldings (TSNMRS) of dodecahedrane C₂₀H₂₀, of the isomeric hydrocarbons C₂₀H₁₂, of the ions C₂₀H₁₂²⁺ and C₂₀H₁₂^2?, of the fluxional fullerene C₂₀ and of its dication C₂₀²⁺ have been ab initio calculated employing the NICS concept on basis of MP2/6-31G1 geometries and visualized as iso-chemical-shielding/deshielding surfaces (ICSSs). TSNMRS values were employed to study the exohedral magnetic properties of the compounds studied. Hereby, the curved π-conjugation in the compounds studied could be quantified.