Rank differences for overpartitions

In 1954, Atkin and Swinnerton-Dyer proved Dyson's conjectures on the rank of a partition by establishing formulas for the generating functions for rank differences in arithmetic progressions. In this paper, we prove formulas for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series.Comment: 17 pages, final version, accepted for publication in the Quarterly Journal of Mathematic

Fluctuation dynamics of a single magnetic chain

"Tunable" fluids such as magnetorheological "MR" and electrorheological "ER" fluids are comprised of paramagnetic or dielectric particles suspended in a low-viscosity liquid. Upon the application of a magnetic or electric field, these fluids display a dramatic, reversible, and rapid increase of the viscosity. This change in viscosity can, in fact, be tuned by varying the applied field, hence the name "tunable fluids". This effect is due to longitudinal aggregation of the particles into chains in the direction of the applied field and the subsequent lateral aggregation into larger semisolid domains. A recent theoretical model by Halsey and Toor "HT" explains chain aggregation in dipolar fluids by a fluctuation-mediated long-range interaction between chains and predicts that this interaction will be equally efficient at all applied fields. This paper describes video-microscopy observations of long, isolated magnetic chains that test HT theory. The measurements show that, in contrast to the HT theory, chain aggregation occurs more efficiently at higher magnetic field strength (H0) and that this efficiency scales as H0½. Our experiments also yield the steady-state and time-dependent fluctuation spectra C(x,x')≡ [h(x)-h(x')]²>½ and C(x,x',t,t')≡ ½ for the instantaneous deviation h(x,t) from an axis parallel to the field direction to a point x on the chain. Results show that the steady-state fluctuation growth is similar to a biased random walk with respect to the interspacing ͉ |x-x'| along the chain, C(x,x')≈|x-x'| α, with a roughness exponent α =0.53±0.02. This result is partially confirmed by Monte Carlo simulations. Time-dependent results also show that chain relaxation is slowed down with respect to classical Brownian diffusion due to the magnetic chain connectivity, C(x,x',t,t')≈|t-t'|β, with a growth exponent β=0.35±0.05<½. All data can be collapsed onto a single curve according to C(x,x',t,t')≈|x-x'| α ψ (|t-t'| / |x-x'| z ), with a dynamic exponent z= α /β≅ 1.42

A connectionist account of analogical development

We present a connectionist model that provides a mechanistic account of the development of simple relational analogy completion. Drawing analogies arises as a bi-product of pattern completion in a network that learns input/output pairings representing relational information. Analogy is achieved by an initial example of a relation priming the network such that the subsequent presentation of an input produces the correct analogical response. The results show that the model successfully solves simple A:B::C:D analogies and that its developmental trajectory closely parallels that of children. Finally, the model makes two strong empirical predictions