A Hecke Algebra Quotient and Some Combinatorial Applications

Let (W, S) be a Coxeter group associated to a Coxeter graph which has no multiple bonds. Let H be the corresponding Hecke Algebra. We define a certain quotient \-H of H and show that it has a basis parametrized by a certain subset W _cof the Coxeter group W. Specifically, W _cconsists of those elements of W all of whose reduced expressions avoid substrings of the form sts where s and t are noncommuting generators in S. We determine which Coxeter groups have finite W _cand compute the cardinality of W _cwhen W is a Weyl group. Finally, we give a combinatorial application (which is related to the number of reduced expressions for w W _cof an exponential formula of Lusztig which utilizes a specialization of a subalgebra of \-H.     

Normal subgroups of

We classify the normal subgroups of of index less than 960; they are all congruence subgroups.

     

Compact flat manifolds with holonomy group

In this paper we construct a family of compact flat manifolds, for all dimensions , with holonomy group isomorphic to and first Betti number zero.

     

Derivative of the exponential mapping for infinite dimensional lie groups

It is proved that for infinite dimensional Lie groups in the sense of the differential calculus of Frölicher and Kriegl the derivative of the exponential mappings is given by the formula d(exp)(X)Y=d_exp(X)(e) ₀ ¹ Ad_exp(–tX) Y dt, where stands for the left translation ande is the neutral element.This work was supported by the Alexander von Humboldt-Stiftung.     

Synthesis of β-tosylethylhydrazine and its use in preparation of N-protected pyrazoles and 5-aminopyrazoles

β-Tosylethylhydrazine (6) can be prepared efficiently in one step from commercially available p-tolyl vinyl sulfone (7) and hydrazine hydrate. This hydrazine reacts with both 1,3-diketones and conjugated ynones in glacial acetic acid to provide a variety of N-tosylethyl-protected (TSE) pyrazoles in good yields. The TSE group can be removed from the pyrazoles using potassium t-butoxide in THF at −30 °C-rt. In addition, hydrazine 6 condenses with β-ketonitriles and β-aminoacrylonitriles to afford 5-aminopyrazoles, which can be deprotected by brief treatment with NaOEt in EtOH/DMSO at 45 °C.     

V. Thioanalogues of sparteine lactams. (+)-2-Thiono-17-oxosparteine and (+)-2,17-dithionosparteine

The X-ray and spectroscopic results clearly indicate that the (+)-2-thiono-17-oxosparteine (1) and (+)-2,17-ditihionosparteine (2) are conformationally rigid. In order to analyze deviations of lactam/thiolactam groups from planarity induced by ring constraints, the Dunitz-Winkler approach has been used. The lactam and thiolactam groups are close to planarity, only the lactam group in one of the two independent molecules of 1 is markedly non-planar. The bond angles in the thiolactam and lactam groups are highly diverse. Rings A and C adopt a distorted sofa conformation in both compounds. The distortions in the molecules of 2 as compared with those in the related monothiolactams correspond to the unusual chemical shifts of H5(eq), H5(ax) and H11, as well as to the extremely low J_5ax-6 and extremely large J_5eq-6 coupling constants. Also chemical shifts show a similar regularity being extremely high and low for C2 and C17, respectively.     

Theory of electron tunneling through a bridge molecule with two electronic levels at low temperatures

A theory of electron tunneling through a single-center bridge redox group that has two electronic levels participating in the electron transfer process is presented. The temperature is presumed to be low enough to ignore activation redox conversions of the bridge group. Salient features of this system are due both to the presence of two electroactive states of the bridge group and to relaxation processes along the reaction coordinate. The processes in question make the tunneling current time-dependent at fixed potentials and can bring about hysteresis in current-voltage curves when cycling the bias potential. Effects of inelastic tunneling with excitation of vibrations of a local quantum degree of freedom are considered.     

Raman spectroscopy of tris-(hydroxymethyl)aminomethane as a model system for the studies of α-chymotrypsin activation by crown ether in organic solvents

Raman spectroscopy is employed to study tris-(hydroxymethyl)aminomethane and its complexes with 18-crown-6. The results obtained are used to interpret the known effect of α-chymotrypsin activation by crown ether in organic solvents. Raman spectra of the samples lyophilized from aqueous solutions at various pH values are measured in solid state, acetonitrile and cyclohexane.     

Photocleavage studies of fluorescent amino acid conjugates bearing different types of linkages

The synthesis and photochemistry of 3-oxo-3H-benzopyran derivatives linked through ester, anhydride, urethane and carbonate bonds to representative l-amino acids, at the amino and carboxylic acid groups at the main chain or the hydroxyl group at the side chain, were carried out. The stability to photolysis of the resulting conjugates was studied at different wavelengths of irradiation (254, 300 and 350 nm), the anhydride and ester linkages being the most sensitive in the studied conditions.     

A polystyrene-supported triflating reagent for the synthesis of aryl triflates

An insoluble polystyrene-supported triflating reagent has been prepared by suspension co-polymerization of N-(4-vinylphenyl)trifluoromethanesulphonimide, styrene and the JandaJel^® cross-linker. This reagent, in the presence of triethylamine, allows for the efficient synthesis of aryl triflates from a wide range of phenols in a process that permits the desired product to be isolated from the reaction mixture in essentially pure form via several filtration and concentration operations. Adding to the utility of this reagent is its ability to be easily recovered, regenerated and reused. Both soluble and insoluble bifunctional polymers containing trialkylamine moieties in addition to triflimide groups were also prepared and examined as triflating reagents. Unfortunately these reagents afforded only modest yields of the desired products in representative reactions.