Notes on highest weight modules of the elliptic algebra Aq,p(sl^2){\cal A}_{q,p}\left(\widehat{sl}_2\right)

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.Comment: Nonstandard macro package eliminate

Fermionic solution of the Andrews-Baxter-Forrester model II: proof of Melzer's polynomial identities

We compute the one-dimensional configuration sums of the ABF model using the fermionic technique introduced in part I of this paper. Combined with the results of Andrews, Baxter and Forrester, we find proof of polynomial identities for finitizations of the Virasoro characters $\chi_{b,a}^{(r-1,r)}(q)$ as conjectured by Melzer. In the thermodynamic limit these identities reproduce Rogers--Ramanujan type identities for the unitary minimal Virasoro characters, conjectured by the Stony Brook group. We also present a list of additional Virasoro character identities which follow from our proof of Melzer's identities and application of Bailey's lemma.Comment: 28 pages, Latex, 7 Postscript figure

Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4\nu)

We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.Comment: 41 pages, harvmac, no figures; new identities, proofs and comments added; misprints eliminate

Energy Transduction of Isothermal Ratchets: Generic Aspects and Specific Examples Close to and Far from Equilibrium

We study the energetics of isothermal ratchets which are driven by a chemical reaction between two states and operate in contact with a single heat bath of constant temperature. We discuss generic aspects of energy transduction such as Onsager relations in the linear response regime as well as the efficiency and dissipation close to and far from equilibrium. In the linear response regime where the system operates reversibly the efficiency is in general nonzero. Studying the properties for specific examples of energy landscapes and transitions, we observe in the linear response regime that the efficiency can have a maximum as a function of temperature. Far from equilibrium in the fully irreversible regime, we find a maximum of the efficiency with values larger than in the linear regime for an optimal choice of the chemical driving force. We show that corresponding efficiencies can be of the order of 50%. A simple analytic argument allows us to estimate the efficiency in this irreversible regime for small external forces.Comment: 16 pages, 10 figure

Level-0 structure of level-1 Uq(sl^2)U_q(\widehat{sl}_2)-modules and Macdonald polynomials

The level-$1$ integrable highest weight modules of $U_q(\widehat{sl}_2)$ admit a level-$0$ action of the same algebra. This action is defined using the affine Hecke algebra and the basis of the level-$1$ module generated by components of vertex operators. Each level-$1$ module is a direct sum of finite-dimensional irreducible level-$0$ modules, whose highest weight vector is expressed in terms of Macdonald polynomials. This decomposition leads to the fermionic character formula for the level-$1$ modules.Comment: 22 pages, LaTeX 2.09 (and amsfonts preferably). (Minor corrections

Membranes by the Numbers

Many of the most important processes in cells take place on and across membranes. With the rise of an impressive array of powerful quantitative methods for characterizing these membranes, it is an opportune time to reflect on the structure and function of membranes from the point of view of biological numeracy. To that end, in this article, I review the quantitative parameters that characterize the mechanical, electrical and transport properties of membranes and carry out a number of corresponding order of magnitude estimates that help us understand the values of those parameters.Comment: 27 pages, 12 figure

Understanding how excess lead iodide precursor improves halide perovskite solar cell performance

The presence of excess lead iodide in halide perovskites has been key for surpassing 20% photon-to-power conversion efficiency. To achieve even higher power conversion efficiencies, it is important to understand the role of remnant lead iodide in these perovskites. To that end, we explored the mechanism facilitating this effect by identifying the impact of excess lead iodide within the perovskite film on charge diffusion length, using electron-beam-induced current measurements, and on film formation properties, from grazing-incidence wide-angle X-ray scattering and high-resolution transmission electron microscopy. Based on our results, we propose that excess lead iodide in the perovskite precursors can reduce the halide vacancy concentration and lead to formation of azimuthal angle-oriented cubic alpha-perovskite crystals in-between 0 degrees and 90 degrees. We further identify a higher perovskite carrier concentration inside the nanostructured titanium dioxide layer than in the capping layer. These effects are consistent with enhanced lead iodide-rich perovskite solar cell performance and illustrate the role of lead iodide

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction