On the work of Basil Gordon

A review is given of several aspects of the work of Basil Gordon. These include: Rogers-Ramanujan identities, plane partitions, the method of weighted words, modular forms and partition congruences, and the asymptotics of partitions and related q-series.     

Hasse invariants for the Clausen elliptic curves

Gauss’s hypergeometric function gives periods of elliptic curves in Legendre normal form. Certain truncations of this hypergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we prove that certain truncations of and in $\mathbb {F}_{p}[x]$ are related to the characteristic p Hasse invariants.     

Design of polylactide-grafted copolymeric stabilizer for dispersion polymerization of D,L-lactide

Poly(D,L-lactide) (PDLLA) microspheres with narrow diameter distribution were prepared by dispersion polymerization of D,L-lactide in xylene/heptane (1:2, v/v) using poly(dodecyl methacrylate)-g-poly(D,L-lactide) (PDMA-g-PDLLA) as a dispersion stabilizer. The particle diameters of PDLLA microspheres were controlled from 200 nm to 5 μm by altering the concentration and the graft chain number of PDMA-g-PDLLA. The effect of the copolymer composition on the particle diameter was investigated to clarify an important factor of the copolymer structure for the control of the particle diameter. As a result, it was necessary for anchor block in diblock copolymer as a dispersion stabilizer to have low solubility in the solution rather than the compatibility with particles. Moreover, we confirmed by dynamic light scattering measurement that PDMA-g-PDLLA formed micelles in the solution. In conclusion, it was clarified that PDLLA microspheres with a wide range of particle diameter were prepared due to the different kinetic stability of micelles.     

Systematic Investigation of Molecular Arrangements and Solid‐State Fluorescence Properties on Salts of Anthracene‐2,6‐disulfonic Acid with Aliphatic Primary Amines

Organic salts of anthracene‐2,6‐disulfonic acid (ADS) with a wide variety of primary amines have been fabricated, and their arrangements of anthracene molecules and solid‐state fluorescence properties investigated. Single‐crystal X‐ray studies reveal that the salts show seven types of crystal forms and corresponding molecular arrangements of anthracene moieties depending on the amine, while anthracene shows only one form and arrangement in the solid state. Depending on the molecular arrangements, the ADS salts exhibit various solid‐state fluorescence properties: spectral shift (30 nm) and suppression and enhancement of the fluorescence intensity. Especially the ADS salt with n‐heptylamine (nHepA), which shows discrete anthracene moieties in the crystal, exhibits the highest quantum yield (Φ_F=46.1±0.2 %) in the series of ADS salts, which exceeds that of anthracene crystal (Φ_F=42.9±0.2 %). From these systematic investigations on the arrangements and the solid‐state properties, the following factors are essential for high fluorescence quantum yield in the solid state: prevention of contact between π planes of anthracene moieties and immobilization of anthracene rings. In addition, such organic salts have potential as a system for modulating the molecular arrangements of fluorophores and the concomitant solid‐state properties. Thus, systematic investigation of this system constructs a library of arrangements and properties, and the library leads to remarkable strategies for the development of organic solid materials.     

Some characters of Kac and Wakimoto and nonholomorphic modular functions

Answering a question of Kac, we relate the character formulas for certain sℓ(m, 1)^ modules to automorphic forms. We show that these q-series are the “holomorphic parts” of nonholomorphic modular functions. The authors thank the NSF, and the second author thanks the Manasse family, and the Hilldale Foundation for their support.     

Valley polarization in Si(100) at zero magnetic field

The valley splitting, which lifts the degeneracy of the lowest two valley states in a SiO(2)/Si(100)/SiO(2) quantum well, is examined through transport measurements. We demonstrate that the valley splitting can be observed directly as a step in the conductance defining a boundary between valley-unpolarized and -polarized regions. This persists to well above liquid helium temperature and shows no dependence on magnetic field, indicating that single-particle valley splitting and valley polarization exist in (100) silicon even at zero magnetic field.     

Electronic inhomogeneity and breakdown of the universal thermal conductivity of cuprate superconductors

We report systematic, high-precision measurements of the low-T (down to 70 mK) thermal conductivity kappa of YBa2Cu3O(y), La(2-x)Sr(x)CuO4, and Bi2Sr2CaCu2O(8+delta). Careful examinations of the Zn- and hole-doping dependences of the residual thermal conductivity kappa0/T, as well as the in-plane anisotropy of kappa0/T in Bi2Sr2CaCu2O(8+delta), indicate a breakdown of the universal thermal conductivity, a notable theoretical prediction for d-wave superconductors. Our results point to an important role of electronic inhomogeneities, which are not considered in the standard perturbation theory for thermal conductivity, in the underdoped to optimally doped regime.     

Scanning Josephson tunneling microscopy of single-crystal Bi_{2}Sr_{2}CaCu_{2}O_{8+delta} with a conventional superconducting tip

We have performed both Josephson and quasiparticle tunneling in vacuum tunnel junctions formed between a conventional superconducting scanning tunneling microscope tip and overdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+delta} single crystals. A Josephson current is observed with a peak centered at a small finite voltage due to the thermal-fluctuation-dominated superconducting phase dynamics. Josephson measurements at different surface locations yield local values for the Josephson I_{C}R_{N} product. Corresponding energy gap measurements were also performed and a surprising inverse correlation was observed between the local I_{C}R_{N} product and the local energy gap.     

Dynamic instability of flying head slider and stabilizing design for near-contact magnetic recording

The dynamic instability of a flying head slider is described, and a design guideline is presented for a slider that can stably fly at a height of less than 5 nm. The current use of shrouded Rayleigh step-bearing slider has been a good way to increase air-bearing stiffness and damping while reducing meniscus adhesion and friction forces, but does not have enough ability to achieve a small flying height less than 5 nm. On the basis of meniscus interfacial force theory, I proposed a spherical pad slider that is inherently stable in the near-contact regime and that has high durability against intermittent contact, which is necessary for future high-density magnetic recording. This design concept is being partly realized by thermal flying height control technique.