Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, wedescribe the complexified twisted equivariant K-theory of aspace with a compact Lie group action in terms of fixed-pointdata. We apply this to the case of a compact group acting onitself by conjugation and relate the result to the Verlindealgebra and to the Kac numerator at q=1. Verlinde's formulais also discussed in this context. Received February 28, 2007.     

Probing the structures of gas-phase rhodium cluster cations by far-infrared spectroscopy

The geometric structures of small cationic rhodium clusters Rh(n)(+) (n = 6-12) are investigated by comparison of experimental far-infrared multiple photon dissociation spectra with spectra calculated using density functional theory. The clusters are found to favor structures based on octahedral and tetrahedral motifs for most of the sizes considered, in contrast to previous theoretical predictions that rhodium clusters should favor cubic motifs. Our findings highlight the need for further development of theoretical and computational methods to treat these high-spin transition metal clusters.     

Synthesis of 1,3-diarylsubstituted indazoles utilizing a Suzuki cross-coupling/deprotection/N-arylation sequence

Herein we report a general synthesis of 1,3-diarylsubstituted indazoles utilizing a two-step Suzuki cross-coupling/deprotection/N-arylation sequence. This procedure proceeds in excellent overall yield starting from the 3-iodo-N-Boc indazole derivative allowing for rapid access to these compounds.     

Avoiding the classical resolution during the synthesis of MeO-BIPHEP and 3,3′-disubstituted derivatives

The Ullmann coupling of 1 (R = H) gives a 2:1 mixture of diastereomers 2 (R = H) in 81% yield that are easily separated by silica gel chromatography. This procedure avoids the generally cumbersome and sometimes difficult resolution step with DBTA. Similar Ullmann couplings and separation of the corresponding diastereomers are employed with other derivatives of 1 (R = OtBu, iPr, Ph, and mesityl) ultimately affording a new series of 3,3′-disubsituted-MeO-BIPHEP derivatives. The use of these new derivatives in palladium-catalyzed asymmetric Heck reaction, Pd-catalyzed polyene cyclizations and rhodium-catalyzed hydrogenations is also reported.     

Nilpotence and finite H-spaces

Zabrodsky asked when is the iterated commutator mapX ⁿ →X for a connected associative H-spaceX a null map. In this paper we reduce this question to a cohomological question and answer it in several cases. Supported by an NSF Presidential Young Investigator award and the Sloan Foundation.     

Infrared-induced reactivity of N2O on small gas-phase rhodium clusters

Far- and mid-infrared multiple photon dissociation spectroscopy has been employed to study both the structure and surface reactivity of isolated cationic rhodium clusters with surface-adsorbed nitrous oxide, Rh(n)N(2)O(+) (n = 4-8). Comparison of experimental spectra recorded using the argon atom tagging method with those calculated using density functional theory (DFT) reveals that the nitrous oxide is molecularly bound on the rhodium cluster via the terminal N-atom. Binding is thought to occur exclusively on atop sites with the rhodium clusters adopting close-packed structures. In related, but conceptually different experiments, infrared pumping of the vibrational modes corresponding with the normal modes of the adsorbed N(2)O has been observed to result in the decomposition of the N(2)O moiety and the production of oxide clusters. This cluster surface chemistry is observed for all cluster sizes studied except for n = 5. Plausible N(2)O decomposition mechanisms are given based on DFT calculations using exchange-correlation functionals. Similar experiments pumping the Rh-O stretch in Rh(n)ON(2)O(+) complexes, on which the same chemistry is observed, confirm the thermal nature of this reaction.     

High resolution laser spectroscopy of the titanium monohalides, TiCl and TiBr

The electronic spectra of the titanium monohalides, TiCl and TiBr, have been investigated in the blue-violet region using a laser ablation molecular beam spectrometer. Five subbands assigned as ⁴Γ_5/2-X⁴Φ_3/2 (0, 0), ⁴Γ_5/2-X⁴Φ_3/2 (1, 1), ⁴Γ_5/2-X⁴Φ_5/2 (0, 0), ⁴Γ_7/2-X⁴Φ_5/2 (0, 0) and ⁴Γ_7/2-X⁴Φ_7/2 (0, 0) were observed and recorded at both low and high-resolution for titanium monochloride. A Hund’s case (a) rotational analysis has been carried out for the ⁴⁸Ti³⁷Cl and ⁴⁸Ti³⁵Cl isotopic species, and polynomial analyses for these, as well as the ⁴⁶Ti³⁵Cl, ⁴⁷Ti³⁵Cl, ⁴⁹Ti³⁵Cl, and ⁵⁰Ti³⁵Cl isotopologues have been completed. The same spectral region yielded several molecular transitions for titanium monobromide, 10 of which were recorded at high resolution. Six of these have been attributed to a ⁴Γ-X⁴Φ electronic transition at 23 484 cm⁻¹, while the remaining four have been assigned to a second ⁴Γ-X⁴Φ electronic transition at 23 613 cm⁻¹. A Hund’s case (a) global analysis has been carried out for the ⁴⁸Ti⁷⁹Br and ⁴⁸Ti⁸¹Br isotopologues.     

Note on a family of economical sixth order finite difference methods for non-linear 2-point boundary value problems

In this note the sixth order method of Chawla [1] is generalized to give a parameterized family of methods.