Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik–Zamolodchikov (KZ) equations. In the case of a principal series module, we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions.For the spin representation of the affine Hecke algebra of type C, the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-\({\frac{1}{2}}\) XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter’s 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik–Zamolodchikov–Bernard (KZB) equations.     

Amine Catalysis with Substrates Bearing N-Heterocyclic Moieties Enabled by Control over the Enamine Pyramidalization Direction

Stereoselective organocatalytic C−C bond formations that tolerate N-heterocycles are valuable since these moieties are common motifs in numerous chiral bioactive compounds. Such transformations are, however, challenging since N-heterocyclic moieties can interfere with the catalytic reaction. Here, we present a peptide that catalyzes conjugate addition reactions between aldehydes and nitroolefins bearing a broad range of different N-heterocyclic moieties with basic and/or H-bonding sites in excellent yields and stereoselectivities. Tuning of the pyramidalization direction of the enamine intermediate enabled the high stereoselectivity.     

Necessary conditions for the compensation approach for a random walk in the quarter-plane

Queueing Systems - We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of...     

Equiangular tight frames with centroidal symmetry

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others.     

Wetting phenomena at the CO2/water/glass interface

A novel high-pressure apparatus and technique were developed to measure CO2/water/solid contact angles (theta) in situ for pressures up to 204 bar. For two glass substrates with different hydrophilicities, theta increased significantly with CO2 pressure. As the pressure was increased, an increase in the cohesive energy density of CO2 caused the substrate/CO2 and water/CO2 interfacial tensions (gamma) to decrease, whereas the water/substrate gamma value increased. theta for the more hydrophobic substrate was predicted accurately from the experimental water/CO2 gamma value and an interfacial model that included only long-range forces. However, for the more hydrophilic substrate, short-range specific interactions due to capping of the silanol groups by physisorbed CO2 resulted in an unusually large increase in the water/substrate gamma value, which led to a much larger increase in theta than predicted by the model. A novel type of theta hysteresis was discovered in which larger theta values were observed during depressurization than during pressurization, even down to ambient pressure. Effective receding angles were observed upon pressurization, and effective advancing angles were observed upon depressurization on the basis of movement of the three-phase contact line. The greater degree of hysteresis for the more hydrophilic silica can be attributed in part to the capping of silanol groups with CO2. The large effects of CO2 on the various interfacial energies play a key role in the enhanced ability of CO2, relative to many organic solvents, to dry silica surfaces as reported previously on the basis of FTIR spectroscopy (Tripp, C. P.; Combes, J. R. Langmuir 1998, 14, 7348-7352).     

Synthesis and characterisation of BODIPY radical anions

The redox processes associated with BODIPY analogues are studied by electrochemical and spectroscopic methods revealing a characteristic profile for the persistent BODIPY radical and quenching of fluorescence upon reduction.