Facets for the cut cone I

We study facets of the cut coneC _n , i.e., the cone of dimension 1/2n(n – 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a lifting procedure for constructing facets ofC _n+1 from given facets of the lower dimensional coneC _n . After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented.     

Cotype of operators fromC(K)

Summary Forq>2, an operator fromC(K) toX is of cotypeq if and only if it factors through the Lorentz space . Forq=2, ifX is a rearrangement invariant space on [0, 1], the injectionC([0, 1])X is of cotype 2 if and only if it factors through the Lorentz space ; but there is a cotype 2 operator C(K) that does not factor through . If a Banach latticeX satisfies the Orlicz property, any bounded lattice operatorT:C(K)X is of cotype 2. We however construct a Banach lattice with the Orlicz property, but that fails to be of cotype 2.Oblatum 4-VII-1990 & 18-IV-1991Work partially supported by an NSF grant     

On vector-valued random Fourier series

Consider a (complex) Banach spaceX, such thatX C_O, and vectors(X _i ) _i ofX. Consider an independent standard normal sequence(g _i ) _i . Then if anX-valued random Fourier series _{|k| n} e ^ikt g _k x _k satisfies

Capture by stabilized continuum: Classical and quantum aspects

We review here the results of our investigations concerning chaotic atomic scattering in the presence of a laser field. Particular emphasis is put on the existence of classical stable resonance structures, induced by the intense laser field, which are embedded in the field-free continuum. We show that phase space structures in the vicinity of a resonance island play an important role in the chaotic scattering behavior and form the basis for a mechanism to enhance the lifetimes of the collisional partners. Quantum calculations, based on a wave packet propagation method, show that quantum solutions are strongly influenced by the classical phase space structures. More specifically, a wave packet is found to spread differently in the regular and chaotic regions; in the latter case it spreads exponentially with time until saturation occurs, defining the saturation time. We also investigate the dependence of the spreading rates in both the regular and chaotic regimes. Calculations with an ensemble of classical trajectories are also presented to further illustrate the smoothing effects of varying.     

Galactose oxidase models: solution chemistry, and phenoxyl radical generation mediated by the copper status

Galactose oxidase (GO) is an enzyme that catalyzes two-electron oxidations. Its active site contains a copper atom coordinated to a tyrosyl radical, the biogenesis of which requires copper and dioxygen. We have recently studied the properties of electrochemically generated mononuclear Cu(II)-phenoxyl radical systems as model compounds of GO. We present here the solution chemistry of these ligands under various copper and dioxygen statuses: N(3)O ligands first chelate Cu(II), leading, in the presence of base, to [Cu(II)(ligand)(CH(3)CN)](+) complexes (ortho-tert-butylated ligands) or [(Cu(II))(2)(ligand)(2)](2+) complexes (ortho-methoxylated ligands). Excess copper(II) then oxidizes the complex to the corresponding mononuclear Cu(II)-phenoxyl radical species. N(2)O(2) tripodal ligands, in the presence of copper(II), afford directly a copper(II)-phenoxyl radical species. Addition of more than two molar equivalents of copper(II) affords a Cu(II)-bis(phenoxyl) diradical species. The donor set of the ligand directs the reaction towards comproportionation for ligands possessing an N(3)O donor set, while disproportionation is observed for ligands possessing an N(2)O(2) donor set. These results are discussed in the light of recent results concerning the self-processing of GO. A path involving copper(II) disproportionation is proposed for oxidation of the cross-linked tyrosinate of GO, supporting the fact that both copper(I) and copper(II) activate the enzyme.     

Photochemistry of 3-Substituted 1-Iminopyridinium Ylides. Regiospecific versus non-regiospecific photoisomerization patterns [1]

3-Substituted 1-iminopyridinium ylides 1 undergo photo-induced ring enlargement to 1H-1,2-diazepines. With strongly electron-withdrawing substituents the ring expansion process is regiospecific and leads exclusively to 4-substituted 1 H-1, 2-diazepines. Weak electron-donating substituents, like a methyl group and halogen atoms, do not have any directing effect since both 4- and 6-substituted 1 H -1,2-diazepines are obtained. With strong electron-donating substituents no diazepines are formed; instead one observes photo-induced isomerization to the 2-aminopyridine derivatives, the process being non-regiospecific. Regiospecific photo-induced ring expansion processes are explained in terms of a simple HMO model.     

Molecular tectonics. Selective exchange of cations in porous anionic hydrogen-bonded networks built from derivatives of tetraphenylborate

Tetraphenylmethanes with multiple hydrogen-bonding sites are known to associate to form robust porous supramolecular networks. Analogous anionic networks can be built from the corresponding tetraphenylborates. Crystallization of the tetraphenylphosphonium salt of tetraphenylborate 2 produces an anionic network in which 74% of the volume is available for including cations and neutral guests. Other salts of anion 2 with diverse cations crystallize consistently to form the same network, whereas a neutral analogue of anion 2, tetraphenylmethane 1, produces an uncharged network that is far less open. Cations can be exchanged in single crystals of salts of tetraphenylborate 2 with retention of crystallinity and with selectivities similar to those observed in typical zeolites. Together, these observations provide new strategies for making ordered molecular materials by design, and they reveal that constructing such materials from charged subunits offers special advantages.     

Gear effect—10 : Conformational aspects of the positive or negative buttressing effects of methyl groups: polymethylpyridines

The effect of the shape of a methyl group on reactivity, which cannot be accounted for by considering a methyl group as a spherical substituent with the appropriate van der Waals radius, was considered in kinetics of alkylalion of substituted pyridines and barriers to rotation and ground state conformations of an isopropyl group attached to a planar framework. The perturbation of a methyl group by an o-methyl group is accounted for by a unique conformational explanation which involves the polyhedral shape of the methyl group.     

Direct evidence for highly organized networks of circular surface micelles of surfactant at the air-water interface

We report the first direct evidence for the formation of circular surface micelles (hemimicelles) on the surface of water. These highly monodisperse 30 nm hemimicelles, made from a semifluorinated alkane deposited as a Langmuir monolayer, form organized hexagonal arrays as determined by small-angle X-ray diffraction conducted directly on the water surface at grazing incidence.