The gauge theorem for a class of additive functionals of zero energy

Summary In earlier works, the gauge theorem was proved for additive functionals of Brownian motion of the form ₀ ^t q(B _s )ds, whereq is a function in the Kato class. Subsequently, the theorem was extended to additive functionals with Revuz measures in the Kato class. We prove that the gauge theorem holds for a large class of additive functionals of zero energy which are, in general, of unbounded variation. These additive functionals may not be semi-martingales, but correspond to a collection of distributions that belong to the Kato class in a suitable sense. Our gauge theorem generalizes the earlier versions of the gauge theorem.Research supported in part by NSA grant MDA-92-H-30324     

On sparse spanners of weighted graphs

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v V, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.The work of G. Das and D. Joseph was supported by NSF PYI Grant DCR-8402375. The work of D. Dobkin was supported by NSF Grant CCR-8700917. The work of J. Soares was supported by CNPq proc 203039/87.4 (Brazil) and NSF Grant CCR-9014562. This research was accomplished while G. Das was a student at the University of Wisconsin-Madison. A preliminary version was presented at the Second Scandinavian Workshop on Algorithm Theory, Bergen, Norway, 1990, under the title Generating Sparse Spanners for Weighted Graphs, and proceedings appear in the series Lecture Notes in Computer Science, Springer-Verlag. The preliminary version also appears as Princeton University Technical Report CS-TR-261-90, and as University of Wisconsin-Madison Computer Sciences Technical Report 882.     

Synthesis and reactivity of a series of [14]azaannulenes

Using a metal template reaction, a series of [14]azaannulenes formed from a β-diketone and an aromatic diamine in the presence of nickel(II) have been synthesized. Attempted cyclizations with bulky β-diketones and certain substituted diamines showed that the cyclization is sensitive to steric effects. At least one of the nickel complexes readily reacts with a series of nucleophilic reagents to give methine-substituted products.     

Synthesis of ansamycins: An approach to the naphthoquinone portion of the rifamycins and streptovaricins

Naphthoquinone $\text{1}$, a fully functionalized model for intermediates in ansamycin synthesis, was prepared by the intramolecular Claisen condensation of benzofuran $\text{9}$ followed by oxidation of the novel tricyclic compound $\text{10}$.     

A new tri-orthogonal strategy for peptide cyclization

A solid phase tri-orthogonal protection/cleavage strategy that uses acidic, basic, and neutral conditions is described. Strategically protected alpha-azido-gamma-9-fluorenylmethyl-L-glutamate (1) and alpha-azido-epsilon-N-Fmoc-L-lysine (2) were incorporated into growing peptides on Wang resin using a novel azide protection strategy. These residues, separated by 1-3 monomers, were deprotected at the side chains and cyclized via lactam formation. The N-terminus was further functionalized to extend the chain. This method represents a straightforward protocol for peptide cyclization on solid support.     

“Statistical” symmetry with applications to phase transitions

Hermann proposed that mesomorphic media should be classified by assigning certain statistical symmetry groups to each possible partially ordered array. Two translational groups introduced were called superordinate and subordinate. We find that the average density in such a partially ordered medium has the superordinate symmetry ₁, while the pair correlation function has the subordinate symmetry ₂. A complete listing is made of all compatible combinations of ₁ and ₂ in two and three dimensions. This leads to more possible symmetries than Hermann obtained, e.g., also to nonstoichiometric crystals. The order parameter space for the systems is found to be the quotient space ₁/₂. In most cases it is identical to the order parameter space of low-dimensionalXY spin systems. The Landau free energy is expanded as functional of the two-particle correlation functionK; the translation group is found to be ₁×₂. A Landau mean-field theory can then be carried out by expanding the system free energy into a series of invariants of the active irreducible representations ofK and mapping the free energy onto that for anXY planar spin system. We predict novel critical behavior for transitions between mesomorphic phases and go nogo selection rules for continuous transitions. We give the structure factors for X-ray scattering so changes in all such phase transitions are observable. The statistical symmetry groups, which describe point and translational symmetries of the mesophases, are classified. Proposals are made to include quasi-long-range or topological order in the classification scheme.This work supported in part by National Science Foundation (Division of International Programs), the PSC-BHE—Faculty Research Award CUNY and Deutsche Forschungsgemeinschaft.     

Matchings and radon transforms in lattices. I. Consistent lattices

An element in a lattice is join-irreducible if x=ab implies x=a or x=b. A meet-irreducible is a join-irreducible in the order dual. A lattice is consistent if for every element x and every join-irreducible j, the element xj is a join-irreducible in the upper interval [x, î]. We prove that in a finite consistent lattice, the incidence matrix of meet-irreducibles versus join-irreducibles has rank the number of join-irreducibles. Since modular lattices and their order duals are consistent, this settles a conjecture of Rival on matchings in modular lattices.     

Optical electron transfer processes. The dependence of intervalence line shape and transition energy on chromophore concentration

Optical electron transfer in the mixed-valence cation of biferrocenylacetylene (BF⁺) has been examined in CD₂Cl₂ solvent. The intervalence absorption line shape is relatively narrow at both low and high chromophore concentrations, but broader at intermediate concentrations. The transition energy for metal-to-metal charge transfer increases from ≈4440 cm⁻¹ at infinite dilution to 5995 cm⁻¹ for 3.8 mM BF⁺. Related effects exist due to added electrolyte. Neither the electrolyte nor chromophore concentration effects are expected from a simple reading of electron transfer theories. Nevertheless, both phenomena can be understood and within the context of theory upon careful consideration of the effects of ion-pairing (and tripling) equilibria upon electron-transfer energetics.     

Total synthesis of (+)-aspidospermidine: a new strategy for the enantiospecific synthesis of aspidosperma alkaloids

A new strategy was developed for the enantiospecific synthesis of aspidosperma alkaloids. The key steps involve a novel ketene-lactonization reaction of a chiral vinyl sulfoxide to efficiently set up the quaternary carbon center, and a tandem Michael addition-alkylation reaction sequence to form the polycyclic core structure. This new strategy was employed in the total synthesis of natural product (+)-aspidospermidine.